
Citation 
 Permanent Link:
 http://digital.auraria.edu/AA00000137/00001
Material Information
 Title:
 Assessment for students' conceptual readiness and multiplicative reasoning
 Creator:
 Legnard, Ericka J. ( author )
 Place of Publication:
 Denver, Colo.
 Publisher:
 University of Colorado Denver
 Publication Date:
 2013
 Language:
 English
 Physical Description:
 1 electronic resource. : ;
Thesis/Dissertation Information
 Degree:
 Master's ( Master of Science in Education)
 Degree Grantor:
 University of Colorado Denver
 Degree Divisions:
 School of Education and Human Development, CU Denver
 Degree Disciplines:
 Mathematics Education
 Committee Chair:
 Tzur, Ron
 Committee Members:
 Johnson, Heather
Subjects
 Subjects / Keywords:
 Mathematical readiness ( lcsh )
Mathematics  Study and teaching ( lcsh ) Mathematical readiness ( fast ) Mathematics  Study and teaching ( fast )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Abstract:
 This research thesis addressed the problem of how teachers can assess students' readiness to begin learning and progress through multiplicative reasoning. Addressing this problem is important, particularly because this type of reasoning serves as a conceptual foundation for higherlevel mathematical topics, such as fractions, ratios, proportions, and algebra. When students and teachers struggle with these higherlevel topics, assessing conceptual prerequisites plays a key role in identifying and eliminating the root causes of difficulty. This constructivist premise, of the need to assess students' current knowledge as a basis for implementing instruction adaptive to the students, is considered essential for effective teaching and underlies this thesis study. Accordingly, a first aim of this study was to determine the reasoning students need, both additive and multiplicative, to learn the aforementioned higherlevel concepts. A second aim was to create an assessment instrument for teachers and researchers that would provide essential information about students' preparedness to engage in further, meaningful study of mathematics. The researcher used studies on students' thinking and learning to create assessment items that would either bring to light their available reasoning or reveal its absence. Additionally, a theory that describes the stages of student understanding in developing multiplicative reasoning was used to better place concepts in a framework that depicts current knowledge, and what is needed before instruction can fruitfully move ahead. The assessment items were presented in a format and in situations that should be accessible to elementary and middle grades students and teachers. An instrument consisting of these items was then developed to serve assessors of students' multiplicative reasoning. This instrument includes differentiating items to fit with various grade levels and to determine the extent to which students can solve these tasks independently or with assistive prompting.
 Thesis:
 Thesis (M.S.)University of Colorado Denver. Mathematics education
 Bibliography:
 Includes bibliographic references.
 General Note:
 School of Education and Human Development
 Statement of Responsibility:
 by Ericka J. Legnard.
Record Information
 Source Institution:
 University of Colorado Denver
 Holding Location:
 Auraria Library
 Rights Management:
 All applicable rights reserved by the source institution and holding location.
 Resource Identifier:
 879677624 ( OCLC )
ocn879677624

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ASSESSMENT FOR STUDENTS5 CONCEPTUAL READINESS AND
MULTIPLICATIVE REASONING
by
ERICKA J. LEGNARD
B.S., Colorado State University, 1992
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science in Education in Mathematics Education
Mathematics Education
2013
This thesis for the Master of Science in Education in Math Education degree by
Ericka J. Legnard
has been approved for the
Mathematics Education Department
by
Ron TzurAdvisor
Heather Johnson
7/2/2013
LegnardEricka J. (M.S.Ed.Mathematics Education)
Assessment for Students Conceptual Readiness and Multiplicative Reasoning
Thesis directed by Professor Ron Tzur.
ABSTRACT
This research thesis addressed the problem of how teachers can assess students
readiness to begin learning and progress through multiplicative reasoning. Addressing
this problem is important, particularly because this type of reasoning serves as a
conceptual foundation for higherlevel mathematical topics, such as fractions, ratios,
proportionsand algebra. When students and teachers struggle with these higherlevel
topics, assessing conceptual prerequisites plays a key role in identifying and eliminating
the root causes of difficulty. This constructivist premiseof the need to assess students'
current knowledge as a basis for implementing instruction adaptive to the studentsis
considered essential for effective teaching and underlies this thesis study. Accordingly, a
first aim of this study was to determine the reasoning students need, both additive and
multiplicative, to learn the aforementioned higherlevel concepts. A second aim was to
create an assessment instalment for teachers and researchers that would provide essential
information about students' preparedness to engage in further, meaningful study of
mathematics. The researcher used studies on students' thinking and learning to create
assessment items that would either bring to light their available reasoning or reveal its
absence. Additionally, a theory that describes the stages of student understanding in
developing multiplicative reasoning was used to better place concepts in a framework
that depicts current knowledgeand what is needed before instruction can fruitfully move
ahead. The assessment items were presented in a format and in situations that should be
li
accessible to elementary and middle grades students and teachers. An instalment
consisting of these items was then developed to serve assessors of students' multiplicative
reasoning. This instalment includes differentiating items to fit with various grade levels
and to determine the extent to which students can solve these tasks independently or with
assistive prompting.
The form and content of this abstract are approved. I recommend its publication.
Approved: Ron Tzur
TABLE OF CONTENTS
CHAPTER
I. INTRODUCTION..........................................................1
II. REVIEW OF LITERATURE..................................................5
Constructivist Theory of Learning.....................................5
Construction of Schemes...............................................6
Fractions as Problematic for Students and Teachers....................9
Practices for Developing Multiplicative Understanding of Fractions...10
Multiplicative Reasoning.............................................16
Schemes for Reasoning Multiplicatively...............................19
Activity Effect Relationship.........................................20
Anticipatory and Participatory Stages................................23
FineGrained Assessment..............................................26
Previous Research on Assessment of Multiplicative Reasoning..........27
III. METHODS................................................................30
Development of Tasks.................................................31
Composite Unit Iteration Task........................................32
Additive Tasks.......................................................33
Multiplicative Tasks.................................................37
Task Refinement......................................................38
Construction of Prompts..............................................39
Selection of Administration Method...................................43
Development of Administration Guide..................................46
IV
Scoring Considerations.............................................46
IV. RESULTS...............................................................48
Tasks..............................................................48
Administration Protocol............................................53
Administration Guide...............................................62
Scoring Guide......................................................74
Reliability and Validity...........................................82
V. DISCUSSION............................................................84
Contributions to Research and Practice.............................84
Limitations........................................................87
Implications for Further Research..................................89
Concluding Remarks.................................................90
REFERENCES................................................................92
APPENDIX..................................................................97
A. Feedback Form for Assessment....................................97
v
CHAPTER I
INTRODUCTION
This research thesis addressed the problem of how students5 conceptual readiness
for and progression through reasoning multiplicatively can be assessed. The research
problem grew out of an interest in the issue of student conceptual preparedness for
learning fractions. The teaching and learning of concepts such as fraction and ratio seem
quite ineffective in many American school settings (Fan and Zhou, 2006). To learn
fractionsmany researchers believe that students need to have previously constructed
ways of reasoning multiplicatively (Confrey and Smith, 1995). It may be that some
students are not prepared with the necessary foundation, namely multiplicative reasoning,
for instruction in more advanced conceptssuch as fractionsat the time that the
instruction is delivered. Tzurs (2004) work indicated that one of the crucial steps a
teacher can take to best serve his/her students is to assess the underlying reasoning
necessary for students to build a concept before attempting to teach that concept.
Assessment of students prior knowledge is considered important before instruction in
any case, therefore, if we consider multiplicative reasoning as an important basis for
fraction, assessing students5 conceptual preparedness for multiplicative reasoning seems
critical for enabling their progress through the study of fractions.
Weak foundational skills can hinder a student for years to come (Confrey and
Smith, 1995). In particular, a lack of sufficient prerequisite knowledge may cause
students to learn the mles and procedures for concepts, such as fraction operations,
without being able to take in the fundamentals of fraction concepts. This may allow the
student to be somewhat successful in the short term, but as time goes on there can be
frustration in trying to understand the concepts that come later. The need for a rich
thorough learning of fractions is clear when we consider the wealth of future
mathematical topics that students will encounter that require an understanding of
multiplicative reasoning and fractions (Thompson and Saldanha, 2003). Ratio and
proportion follow soon after fraction instructionbut the concepts of similarityslope
exponential and logarithmic functions, just to name a few, also draw on multiplicative
schemes. Confrey and Smith (1995) traced such schemes extending through calculus and
beyond, and noted that they should be based on multiplicative concepts instead of
additive ones. Teachers often lament that introducing an algebra problem including
fractions seems to dissuade many students, even if using algebra with whole numbers
seems solid for those students This indicates that there may be a problem in the way
that students learn the foundational knowledge for fractions, and that there is an urgent
need to correct this problem.
Students who are able to efficiently and independently solve problems involving
multiplicative situations may be more able to create conceptions that support the learning
of future concepts, and those who do not can find these future topics increasingly difficult
to understand (Confrey and Smith, 1995). It seems an important issue that students are
prepared with a solid conception of multiplicative reasoning, leading to a solid
conception of fraction, and provided a clear path to future studies such as algebra.
In the context of a regular classroom experience, it is difficult for educators to
correctly determine the reasoning used as students solve problems. There are factors,
discussed in this section, which make this especially tme when the assessment is taking
place mainly with pencil and paper tests, or when it is done informally through brief
2
conversations with individuals or groups of students. This is not to say that these means
of formative assessment are not valid. Rather, that when teachers need to know that
students are prepared to undertake what seems to be such important new learning, there is
a need to get an authenticdetailed picture of students readiness.
There are many factors that make the assessment of reasoning difficult in the
classroom. These include the timeconsuming need for careful selection of the problems
to be used, unintentional prompting within the problem or by the teacher or other
students, and the ways that students can demonstrate what seems to be evidence that they
understand, when in fact they may not. The ability to remove as many of these factors as
possible and to assess a student in a way that is likely to give a clear picture of each
students capabilities is important.
As a middle school teacher, I have come to recognize that teachers need to know
the importance of this multiplicative foundation, how to detect it, and ways to support
students at their current level.A userfriendly assessment of multiplicative reasoning,
which gives specific information about what students do and do not understand, would be
a tool to help teachers pinpoint where their students may need more experience before
fraction instruction begins. It is proposed in this thesis that having these assessment
results is important for teachers and students. This is not a simple matter, however,
because students who appear to have multiplicative reasoning may not have it. A reliable
assessment that helps clarify the depth of students5 understanding could be valuable to
ensure their conceptual preparedness.
The instrument developed as part of this thesis studythe Assessment for
Multiplicative Reasoning (AMR), was designed to allow assessors (e.g., teachers) to
3
determine not only the students level of multiplicative reasoningbut also whether a
student is able to reason independently or with support. The results of the AMR allow a
teacher to determine not only studentspreparedness for fractions but also whether what
is needed is to lay the groundwork for a multiplicative scheme, building on the previous
scheme, or help to progress to the next stage within a new scheme. This is something
that as a teacher I will certainly incorporate into my practice. The knowledge and tools
that I have gained in this study, both the awareness of multiplicative reasoning as an
essential part of math education, as well as a way to assess this knowledge, will serve my
students well in the future. I feel better prepared to address the issues that too often arise
when I begin the teaching of fractions. I intend to use this assessment to determine the
readiness of students according to the schemes that they can access, as well as their level
of understanding within the schemeto better target instruction for my students.
A discussion of the meaning of scheme in this context will follow in the Review
of Literature section in Chapter II The next section depicts the conceptual framework
that guided this researchbecause some of the constructs from this framework are needed
to better delineate the problem addressed in this study.
4
CHAPTER II
REVIEW OF LITERATURE
In this chapter, I situate the research problem depicted in the previous chapter
within the larger body of research literature. I begin by summarizing the constructivist
theory of learning that underlies all other ideas in this work. Next, I discuss the
prevailing theories of best practice in fraction instruction. A discussion of the facets of
multiplicative reasoning and why it is important for fraction concepts follows, to link the
best practices to what we know about multiplicative understanding. Then, I will detail
the learning theory that I suggest using to support students as these practices are
employed, as well as an explanation of the participatory (promptdependent) and
anticipatory (promptless) stages of understanding that students would arrive at using this
theory. Finally, I will explore methods of assessment that can be used to determine a
students knowledgeas well as the previous work on assessments specific to
multiplicative reasoning.
Constructivist Theory of Learning
This thesis used a constructivist theory of learning as a guiding framework
because the focus of the assessment is on cognitive aspects of students5 learning, namely,
reasoning multiplicatively. A constructivist theory asserts that students must construct
their own knowledge, and that it cannot be transferred to them through some other means
(Dewey, 1938; Piaget, 1985; von Glasersfeld, 1995). With this idea in place, it seems
that to guide students in the learning of new concepts, such as fraction, ratio, and the like,
teachers need to be aware of activities that help students make their own meaning and
understandings, and allow students to do the work necessary to create this new
knowledge. In constructivismthe students thinking is a key goal for any lessonand
effective lessons and teachers will help the student to do the thinking required to learn.
Piaget (1985)the originator of constructivismargued that anything that is to be
understood should be rooted in practical experience. Von Glasersfeld (1995) interpreted
this belief put forth by PiagetRadical Constructivismas knowledge being built up by
the cognizing subjectp. 51). If teachers are not able to relate multiplicationor
fractions, or ratios, to something that students already understand, and help them see how
to operate on fractional quantities using something tangible or figural that is well
understood, then it is unlikely that students will be able to create meaningful
understandings of fractions, let alone apply any of their understandings to other
situations. A further development of these constmctivist principles follows in the next
section about the development of schemes, a key building block of logicomathematical
thinking in the constmctivist theory.
Construction of Schemes
One of the foundations in the constmctivist theory of learning is the concept of
scheme (Piaget, 1985). A scheme is thought of as a 'miniature framework5 in the mind,
which allows thoughts, experiences, and knowledge to be organized and connected in a
way that makes sense to the learner. Von Glasersfeld (1995) described the action on
schemes as a threepart system, comprised of a perceived situation, an activity, and a
result. The perceived situation is the learners reality in the momentwhich may be
something that they notice on his or her own, or their view of a situation created by a
teacher to inspire learning. The activity is something that is done to an object in that
situation, which may involve physical manipulation, such as pouring of liquid in a
6
geometric model, or a mental activity completed on an available scheme, such as
considering the effect of adding two negative integers when the addition of two positive
integers is understood. The result is what happens after the activity is completed, or what
the learner notices about these effects, and how they relate to the object or scheme. This
threepart system describes the way that assimilation and accommodation can take place
in Piagefs constructivist theory.
Schemes begin as simple things, such as the scheme for what a table might be,
and then move on to be much more complex, such as a scheme for things that are related
proportionally. They can involve the concrete or the abstract, but the fundamental idea
regarding schemes is that they are necessary for new learning to take place. New
information, such as the relative size of fractional quantities, must be taken in to the
consciousness of the student in one of two ways. The processes of assimilation and
accommodation, described below, take place to allow students to learn via transforming
their existing schemes into novel ones (Piaget, 1985).
The student assimilates new information into an existing scheme, meaning that it
relates so naturally for the student that they incorporate it into what they already know.
For instance, if students already know the concept of unit fractions such as 1/5,1/8 and
1/6, and the unit fraction is well understood, when they encounter a new unit fraction,
such as 1/7, this should be a simple assimilation. The new idea of 1/7 can be
incorporated right into the present scheme because it fits nicely with the rules that the
student has constructed to understand unit fractions.
If, however, the student does not have a scheme for unit fractions, they will need
to accommodate, or make sufficient changes in an existing scheme, such as how a whole
7
number is composed by iterating the unit of 1,so that they can develop an understanding
of the unit fraction 1/7 (Piaget, 1985). Accommodation is a much more complex and
laborious taskwhat we usually call learningand requires the student to alter the rules
which currently govern their scheme to allow the understanding of the new information,
or to create a new scheme that will house the new information. This accommodation, or
linking of the known to the unknown, is what is referred to in the future sections
describing the ways that multiplicative reasoning can be used as the scheme which will
be accommodated for a student to understand fractions.
It was further conjectured by Piaget (1985) that students relate new information to
what they already know, even if what they know is incomplete. If not properly guided,
this connection may be tenuous and without any mathematical meaning, causing the new
learning to be unstable or misdirected. Teachers see this phenomenon when students
express misconceptions or convoluted reasoning in arriving at a solution that seems to the
teacher to be inexplicable. Presumably, the student made some connections in past work,
which functioned for them at the time in some way, but perhaps only coincidentally or
tangentially. This may happen if the underlying scheme is not properly developed, such
as may be the case with multiplicative reasoning and future concepts.
In the context of this thesis, the development of a scheme as an accommodation of
previous schemes is central, because it builds the case for educators to be mindful of the
schemes that student have available, through some sort of assessment, and how we expect
them to elaborate those schemes to include new knowledge. This understanding helps to
guide the later sections, to provide a foundation for the need that drove the creation of
this assessmentand explore the theories with which it was constructed. Giving students
8
a foundation in multiplicative reasoning, and creating experiences for them that will show
fractional quantities as multiplicative relations may be a solid path towards real
understanding of concepts such as fractions, ratios, rates and proportions. I suggest in the
following sections that one of the problems that teachers face in teaching fractions is that
the multiplicative schemes needed to proceed are not fully in place, and cause students to
have great difficulty in learning. Evidence for that difficulty is presented next.
Fractions as Problematic for Students and Teachers
As students are attempting to assimilate and accommodate their schemes
throughout their mathematics education, some seem to come more easily than others.
The conception of fraction is often difficult, as the studies in this section will show.
The familiar International Mathematics Studies, including SIMS in 1976 and
TIMSS beginning in 1995, as well as the International Assessments of Educational
Progress (IAEP1 and 2), showed US students lacking in mathematics in general, and
fractions, ratios and multiplicative reasoning, a prominent part of all of these
assessments, were no exception (Fan and Zhou, 2006). The National Mathematics
Advisory Panel stated, "The most important foundational skill not presently developed
appears to be proficiency with fractions 2008, p.18). It seems that this is an area
where American teachers and students need assistance and continued focus to create
improvement.
Fractions are a topic often dreaded by students and their parents, and cause
teachers from middle school through high school difficulty and frustration when students
who should have learned the concepts long ago struggle with using them to solve
problems. Students have been seen in studies to avoid operations such as multiplying by
9
fractions, even when this would be the most efficient strategy for solving a problem, and
they have had experience with these methods before (Fischbein et al., 1985). This is
presumably because they felt uncomfortable with the meaning of such an operation and
often preferred to cling to more familiar, additive methods. This may be due, in part, to
the evidence that multiplicative quantities are not intuitive in the way that additive ones
are, and that students will generally not develop these understandings without careful and
explicit instruction (Dehaene1997; Fischbein et al.1985; Steffe2002). It has been seen
in multiple studies that in order to come to a clear understanding, students should be
assisted in their learning process, usually in school by their teachers and through social
interactions with others, in order to develop the schemes necessary (Dewey, 1938).
While there is no single universal theory for what should be done to bolster the
fraction proficiency of US students, there are many theories that intend to guide educators
in this endeavor. Many researchers have examined the teaching and learning of fractions
as they attempted to unravel its complexities. Some of the main theories are presented
next in order to understand which among them seems to be the most promising, and what
prerequisite knowledge is required in each case.
Practices for Developing Multiplicative Understanding of Fractions
If fraction education is problematic, it makes sense to explore the prevailing
research about how the teaching and learning of fractions might be approached and
understood. The theories presented here are highlighted to show the rationale for the type
of reasoning that I tried to assess using the AMR. There are many options for
introducing fractional concepts to students, including, but not limited to, partofawhole,
fractions as division operations, whole number bias, and the reorganization hypothesis
10
(Siegler et al.2011; Stafylidou and Vosniadou, 2004; Steffe, 2002; Thompson and
Saldanha, 2003;). As a teacher, I found exploring these theories indispensable for
understanding the different ways of thinking about how to teach fractions, and to
determine which theory seemed most likely to produce effective learning. The sections
that follow explore what types of schemes students need to develop during the study of
fractions, and then to consider the underlying schemes that would need to support that
study. From this understanding, the assessment developed to focus on these underlying
schemes.
PartofaWhole Model
A prevalent method used to help students think about fractions in American
schools is the partofawhole model. In this model, the student sees a whole divided into
a given number of equalsize parts, and consider the fraction as one part, or several parts,
of that greater whole. This partofawhole conception breaks down, however, when
students are confronted by a quantity that does not fit this model, such as 8/5, because 8
cannot be a part of five (Thompson and Saldanha, 2003). When students are asked to
operate on these often mysterious quantities, they may be forced to rely on taught or
assumed mles, which may have little grounding in mathematical theory. This process is
understandably confusing, and may contribute to student thinking that math is a set of
arbitrary rules to be followedregardless of intuition or prior knowledge (Erlwanger
1973). The tme understanding of mathematical concepts is a major purpose of valuable
mathematics education (NCTM, 2000), and when students are not able to make sufficient
meaning, not only does their understanding break down, but their enthusiasm for the
discipline as a whole may break down as well. It is considered crucial in this thesis that
11
students understand fractions as multiplicative quantities instead of as partofawhole.
For example, they need to think of a unit fraction, say 1/6, as a unit such that the whole is
6 times as much.
Fractions as a Division Operation
A related understanding of fractions as multiplicative constructsin place of or in
addition to the part of whole model, is to see them as the effects of a division operation.
In such a model,5/8 means five divided by eight, which is a valid way to consider this
notation in certain circumstances, and which will also yield the decimal form of this
quantity. However, such a notion can be problematic when one encounters, for example,
5/ (8/3), because the operation to be completed is not clear (Thompson and Saldanha,
2003). This way of introducing fractions may also be dangerous to student understanding
because it does not provide a way to think about the value of the quantity presented, nor a
way to compare and order fractions without completing the operation and comparing the
decimal. This lack of understanding of the magnitude of fractional quantities can create
difficulty for estimation and using intuitive means to determine whether the solution to a
problem involving a fraction operation is reasonable or not. On the other hand, 5/8 could
be thought of as a unit that is 5 times as much of another (1/8), where the whole is 8
times greater than 1/8, as explained for unit fractions. Such a conception (e.g., 5 units of
1/8) supports understanding of fractions in a way that does not seem possible with the
division operation model alone.
Linking Additive and Multiplicative Situations
Another theory, proposed by the Rational Number Project (Behr, et al., 1983;
Cramer et al., 1997), is that the comparison of additive and multiplicative situations be
12
made explicit, so that students see examples of each, side by side. The theory supposes
that the students are then better able to use this knowledge to understand that fractions are
multiplicative quantities and can be treated as such. This avoids the common lack of
understanding by students, for example, that in making an equivalent fraction you may
just as easily add or subtract the same quantity to or from the numerator as multiply or
divide by the same quantity. This type of misconception may show that the student does
not see the fraction as a quantity involving a multiplicative relation.
One theory related to this stance is whole number bias, or whole number
interference (Stafylidou and Vosniadou, 2004). In this approach it is argued that when
students apply what they already understand about whole numbers to understand
fractions, confusion occurs, and that we should not try and connect the two for students.
For example, students may mistakenly believe that division by a fraction smaller than 1
results in a quantity smaller than the dividend, similarly to the result of division by a
whole number. While drawing upon prior knowledge is the only way for students to
create new knowledge in the constructivist viewit does present problems if students
make associations that are not helpful in understanding the concept. Stafylidou and
Vosniadou (2004) and Streefland (1991) stated that the conceptions students have about
natural numbers create problems for students when they encounter fractions, because they
generally draw incorrect conclusions, or make connections between ideas that are not
mathematically sound. For example, students sometimes reason that as the numbers in a
fraction get largerthe value of the fraction become larger as well (e.g.9/10 vs. 90/100
respectively), regardless of the invariant, multiplicative relation between numerator and
denominator. The study conducted by Stafylidou and Vosniadou (2004) asked students
13
to compare fractions and give a rationale for their conclusions. The responses were
categorized according to their conceptions. The conception called Relation between
Numerator/Denominator was the only one in which students showed a comprehensive
understanding and strong conception for all types of fraction relationships, including
improper fractions. Interestingly, even though this theory contrasted with the one that
follows, the importance of multiplicative reasoning prevailed in both cases.
The ReorRanization Hypothesis.
An alternative theory to the whole number bias, and that supports the
development of fraction from whole number multiplication, is called the reorganization
hypothesis (Steffe, 2002). It states that the number system that children learn in early
grades, should be expanded, or reorganized, to include all rational numbers instead of
separating whole number and fractional quantities for students (Siegler et al.2011;Steffe,
2002). This theory finds value in the struggle to integrate fractions and whole numbers as
members of a continuous number system, and to create a comprehensive mental number
line through which students not only gain a strong understanding of the values of rational
numbersbut also a richer understanding of whole numbers. This marriage of whole
number and fraction is also supported by recent neurological research which indicates
that the areas of the brain in which fraction magnitude decisions are made occurs in the
same area of the brain as whole number processing, and that absolute magnitude and
relative magnitude are processed with the same analog code in the brain (Jacob and
Nieder, 2009, p. 4656).
Steffes reorganization model (2002) asks students to equipartition to find the unit
fraction, and then iterate it to create the desired fraction. As Tzur (2004) emphasized, the
14
key in such an approach is building on the child5 s activity part of a scheme (here,
iteration of unitwhich underlies the creation of whole numbers) instead of on the childs
abstracted construct of number per se. Another aspect of Steffes theory is to use
splitting, which asks students to coordinate both partitioning and iterating as inverse
operations. This is contextindependent and applicable to many other situations,
including improper fractions, and underscores the multiplicative relationship by focusing
on the relation between the unit rate and the number of units.
Ni and Zhou (2005) also advocate a merging of teaching whole number and
fraction concepts concurrently, to emphasize and link the known whole number concepts
with the newer fraction concepts. This can allow students to accommodate their whole
number schemes effectively to understand rational numbers.
The basis for all of these strategies is multiplicative in nature, and calls for
students to have a strong background in the ways quantities can be related
multiplicatively in order to make sense of the tasks presented. It is considered for the
purpose of this thesisand the associated instrumentthat reorganization is preferable as a
method for guiding students in their understanding of fraction. This thinking, guided the
development of the assessment, discerning between schemes available to students that
can be reorganized further toward intended mathematical concepts.
It is important to note the conclusions of researchers regarding the most effective
ways for students to learn fractions, because many students, as well as adults, assume that
multiplication is simply repeated addition. Such a conception does not seem to support
fraction understanding as effectively as a more robust conception of multiplicative
relationships. The next section focuses on ways of reasoning multiplicatively that were
15
seen to be paramount in the theory of reorganization, and how such reasoning may
support students in their future learning.
Multiplicative Reasoning
In the previous section, I have provided a rationale for multiplicative reasoning as
the conceptual basis for strong fraction understanding. In this section, I provide more
details about what that reasoning entails. The understanding of multiplicative reasoning
for whole numbers that is used in this paper was summarized by Steffe (1992) in the
following statement; For a situation to be established as multiplicativeit is always
necessary at least to coordinate two composite units in such a way that one of the
composite units is distributed over the elements of the other. To make sense of this
foundational assertion, one must understand the idea of number as an abstract,
symbolized, composite unit.
A composite unit is an abstract constructor schemethat a person uses to
symbolize potential (but not actual) results of counting, such as a numeral or number
word, which means for him or her, a collection of units of ones. When a student says
4and is using it to refer to a unit of four ones considered as a groupthe student
understands composite unit. In earlier stages of development, a student may recite the
word four in rote countingor know that it goes with the numeral 4, but she must
develop the idea that the number four is a group of four ones which can be assembled and
reassembled (Olive, 2001). For example, understanding that 5x7 is not simply 5
repeated 7 times, but five units, each composed of seven units of 1,or seven units of five
units of 1,and why they yield the same number of Is in total, is necessary for students to
understand multiplication.
16
A repeated addition model misses the coordinated units idea of multiplicative
reasoning. Several of the core ideas that help students understand multiplication in a
flexible, complete manner, such as the importance of units as a point of reference, are not
supported by repeated addition. In a repeated addition model for multiplication, we can
consider, for example, three pies made of six apples each, which total eighteen apples (or,
in repetition, 6 apples + 6 apples + 6 apples =18 apples). In a multiplicative model, we
coordinate the one composite unit of six apples per pie (unit rate) with three pies as the
number of composite units, which when distributed as explained above produce eighteen
apples as the total of Is, coordinating two different units to yield a third unit). In the
multiplicative situation, one must keep track of whether the number refers to apples,
apples per pie or pies themselves, and distribute the six apples across each of three pies to
find the result. When thinking of this situation as repeated addition, this coordination can
be completely missed by students.
The apparent weakness of repeated addition is confirmed by Ni and Zhou (2005),
who stated that not only does repeated addition prove a weak and limited view of
multiplication, but that this reasoning is specifically harmful to students in learning
fractions. One issue is that seeing 5x7 as five groups of seven is conceptual, which is
essential for building schemes and understandings, while seeing 5x7 as seven added five
times is simply a command to act. When seen as a command, an expression such as 5
times x, or 5x, becomes problematic for students, as they are not able to carry out the
specified computation (Thompson and Saldanha, 2003).
Unfortunately, multiplication as the coordination of units explained above is not
something that teachers can take for granted as understood by their students. In a study
17
by Clark and Kamii (1996), only 49% of fifth graders had strong multiplicative
reasoningat a time when formal instruction in multiplication is likely finished in the
classroom. Furthermore, Siemon, Breed, and Virgona (2005) found that far too many
students at the secondary level were using simplistic additive models when multiplication
would have been more efficient. It was considered likely by the authors of the study that
part of the reason for their lacking proficiency was that they understood multiplication as
repeated addition. Steffe (1994) said that when students use an additive model to
multiply, it is an indication that they have yet to dissociate the idea of multiplication from
more primitive counting schemes.
The difficulties pointed to by the aforementioned studies show that there could be
benefit derived from an instalment to determine where students seem to be on the
continuum from additive to multiplicative reasoning before multiplication is considered
solid enough to begin further instruction in fractions. In this sensemultiplicative
reasoning is a conceptual prerequisite for middle grades mathematics (Thompson &
Saldnha, 2003; Tzur et al., 2013). Its absence may create significant problems for
students who are not yet equipped to construct these understandings. Most importantly
an assessment instalment is needed because teachers may not realize that this gap in
reasoning multiplicatively is present. It is often assumed that students are ready for
further instruction if they are proficient in executing calculations of multiplication and
division, and can give some reasoning that supports their work. The notion that these
students may not indeed have a full multiplicative understanding, complete with the
ability to track and anticipate the coordination of units involved in problem situations,
was paramount in the development of the AMR.
18
Another reason to develop an assessment instrument of multiplicative reasoning is
that many teachers may not be aware of the schemes and stages involved in multiplicative
understanding. This is especially tme at the secondary level, in which these concepts are
not usually taught. Jacob and Willis (2003) commented on the need for teachers to be
aware of and able to distinguish between additive and multiplicative reasoning in order to
better serve their students. Seeing students skip countwhich may reflect essentially a
facility with repeated addition, can give the impression that students understand the
concept of counting groups of objects. However, if this is simply a learned method, and
not rooted in an understanding, it may be only an indication of a willing student. The
next section will focus on the specific schemes that students develop when constructing
meaning for multiplicative situations.
Schemes for Reasoning Multiplicatively
The multiplicative reasoning described above does not develop all at once, but on
a continuum of schemes. The framework for student understandings discussed in this
work, came from the six schemes of multiplicative reasoning proposed by Tzur et al.,
(2012). These schemes begin with multiplicative Double Counting (mDC), during
which students are able to keep track of two quantities when one is distributed over
another. This first of six schemes, mDC, is the highest level assessed in the AMR
instrumentbecause it focuses on the shift in thinking from additive reasoning to
multiplicative reasoning for the student. For example, being able to find the total number
of flowers in four vasesif there are three flowers in eachwould require a student to be
able to count how many composite units of three flowers are present, while at the same
time counting how many units of 1 single flower are accumulating respectively. They
19
would then need to knowingly stop when the count of the composite units of threes
reaches four, and understand that the number obtained is total flowers, not vases nor
flowers per vase.
This distribution is an advancement in thinking from the additive, in which a
student would either need to count each floweror add 3 + 3 + 3 + 3 to find the result of
12, but misses the idea of the units being coordinated. The meanings of the answer and
the two numbers 3 and 4 are critical, as well as which is being distributed across the
other. Failure to understand this can become problematic in later studies if this concept is
not solidified early on. The six schemes, of which mDC is first, have laid a foundation for
educators to not only determine where students may be in relation to multiplicative
reasoning, but to ensure that understandings necessary for meaningful work on fractions
are in place.
Activity Effect Relationship
One elaboration of the constructivist learning theory was articulated by Tzur and
Simon (2004) in their depiction of the mechanism of learning, namely, reflection on
activityeffect relationship (AER). This depiction described the way that a student would
engage in an activity, which may be visible to an observer or occur mentally, to achieve a
goal or desired outcome that the learner has in mind. This activity is performed on an
object, which again may be physical or simply a prior conception (i.e., number as an
abstract, composite unit). The effect is what follows from the activity performed on the
object. It stands to reason that if there is not a viable object, the activity cannot proceed
fruitfullyand that the teacher must be aware of the mental objects available to the
student. When the object is acted upon an effect will be created, but the effect and
20
learning may not be optimal, and possibly even harmful, if it creates misunderstandings
on the part of the student due to insufficient prior conceptions.
To illustrate the working of this theory, consider a student who has a
multiplicative scheme such as mDC, as may be measured by the AMR, attempting to
solve a fraction problemand how that student may be instructed using the model of
AER. A student with solid multiplicative reasoning could independently solve a problem
such as this one: Maria wants to have juice boxes at her birthday party for all of her
guests. The juice boxes come in packs of 4, and the store has 6 packages on the shelf.
How many juice boxes will Maria have if she buys all 6 packages of four?
A student who has multiplicative reasoning could solve this problem by iterating
the composite unit of four six times to efficiently find that there are 24 juice boxes to be
purchased. When it has been established that the student is able to operate in this way,
the multiplicative scheme being used can be extended to allow the student to think about
fractions, such as considering what fraction is just one pack of all 24 items). This is
exactly the type of reasoning that the AMR can detect to determine whether a student is
ready to advance to fraction. A student may then be presented with a problem such as
this one: Maria is cleaning up from her party, and finds that some of the juice boxes she
served her friend are not finished. She finds five juice boxes that are each H full. If she
puts them all together, how many boxes could she fill completely?
Using a continuation of the previous problem can help students see the connection
between the two, and begin to activate the scheme, but they will likely need further
direction to fully utilize the current scheme for this problem. The student may be
prompted to draw the boxes, and to describe what they see, redrawing the juice boxes
21
when combined. The student could then be prompted to use the same idea that was used
in the multiplication problem to describe the total. By reflecting on the commonalities of
six units of four juice boxes, and five units of !/4 juice boxes, a student can come to see
that the total would be 5*1/4 or 5/4 of a juice box. This can mean for the student that it
takes four of these units to make a whole, and that five are being considered, so I will be
able to fill up one juice box with four units and will then have one left over, the same as 1
H juice boxes. If the student is comfortable with the multiplication to find the number of
juice boxes in the previous problem as 6 4 or 24 juice boxes, this same conception can
be used to understand fraction operations.
This transition is not likely to occur quickly or easily, but can occur with
continued focus on the conceptions that students have, and continued emphasis by the
teacher on the common patterns with iteration of whole numbers and fractions. By
asking the student to reflect upon their work for each type of problem and reporting on
their findings, a solid foundation for the understanding of fractions can occur. The key in
this process is for the student to reflect upon, and make sense of what they see as it
relates to what they already know. It is important that they do this reflection while being
directed toward reactivating the preexisting schemes that will best support this new
learning, using multiplication to support an understanding of fraction. This
understanding may begin with students needing support and prompting from the teacher,
but with assistance, students will become independent even when the situation or context
is different. The reflection on AER model helps us to understand these two stages, called
anticipatory and participatory (see below), as well as the need for finegrained
assessment, which is described in a later section as a way to discern between these stages.
22
Anticipatory and Participatory Stages
That students may be at different stages when constructing a new scheme was
considered in the development of the assessment in this thesis, and these two stages are
explained in the section that follows. Participatory and anticipatory stages were
considered important because in order for students to be able to function effectively in
their multiplicative schemes, they would need to be at the later stage described here.
Also included in this discussion are the ways to determine the stage in which a student
seems to be functioning, a critical part of the AMR.
Tzur & Simon (2004) postulated that students may have constructed a new
scheme at a stage, called participatory, which requires prompting in order for them to
access, and use, the newly forming scheme. Later, the students may progress to a more
advanced stage, called anticipatory, in which they can independently use the new scheme.
This distinction implies that students may be able to solve a problem given
manipulatives, following a lesson on that concept, or with prompting, but they would not
be able to solve independently, or when the memory of the activity has faded. Tzur &
Lambert (2011)related this cognitive stage to Vygotskys notion of Zone of Proximal
Development (ZPD)suggesting that the ZPD is a reflection of students cognitive need
for prompts.
A student who is at the participatory stage may be able to solve a problem about
the number of flowers in four vases with three flowers eachduring or immediately
following a lesson in which these types of problems were posed, and worked through
with guidance from the teacher (e.g.How about using our fingers to keep track of how
many vases and how many flowers we have in all?^). After some time, however, the
23
student may seem not to be able to solve problems of this type at all. This, sometimes
called the nextday phenomenonis a common stumbling block for teachers and
studentscausing frustration for both parties. Possiblythe issue here is that the student
might have been only at the participatory stage, meaning that they must be prompted, at
least mentally, to be able to reason about the problem. The student seems to understand,
and may be able to solve independently after prompting, but the next day in class the
student may not remember how they should proceed on a similar problem. For example
if the teacher asks questions such as How many flowers were there in one vase? Now
how many would be in two vases? to lead the students thinkingit may quickly result in
a students ability to resume their previous way of reasoning and solving the problem.
The student continues to need prompting, however, and will likely not be able to solve
these tasks as time goes on and the activity becomes a distant memory.
At a higher level of the participatory stage, a student may prompt oneself upon
realizing that he made an error, sometimes called the 'oops5 experience. For example,
students may begin the vase problem by saying there are seven flowers (e.g.adding
4+3), and then correct themselves as they realize that there are three flowers in each of
the four vases and answer correctly. This is still not considered anticipatory, because the
student needs to selfprompt. However, this is a higher level of the participatory stage,
because the prompt has now come from within the students mental system (Tzur &
Lambert, 2011). One difficulty in discerning between the levels is that unless the teacher
is very involved with the student during the solution process, the selfprompting may not
be evident. The student will correct and then move on, leading the teacher to assume
independence in using the new scheme, and thus anticipatory status.
24
Unlike at the participatory stage, a student at the anticipatory stage does not need
to consider the activity, or to be prompted. Their understanding is more solid because it
occurs in the students mind as a wellformed schemeindependent of working through
the activity in the familiar context. In the example above, the relationship between
flowers and vases is clear to the studentand they dont hesitate to proceedanticipating
the entire process from the start. They anticipate the need to coordinate counting of one
composite unit (4 vases) with another (3 flowers/vase) to calculate the total (e.g.,1 vase
is 3 flowers2is6, 3is9, 4is12). This anticipatory stage can become a conceptual
foundation for meaningful memorization of the fact 4x3 =12.
It may be that in the classroom some students do not reach the anticipatory stage
of a conceptsuch as multiplicative double countingbefore the instruction moves
forward. For students not at the anticipatory stage a problem arises, because though they
have been exposed to a concept, and made progress toward understanding at the
participatory stage, they are not prepared to independently use that understanding as a
building block for new learning before the topic of study changes. This situation is also
frustrating for teacherswho feel as if the student should have learned the material
already, and showed signs of understanding, but cannot access this concept or scheme
when they attempt to move forward. In the teaching of concepts such as fraction and
ratio, a participatory understanding of multiplicative relationships is not sufficient to
create understanding about the meaning of fractions, so the detailed assessment of this
reasoning is crucial. The next section discusses the type of detailed assessment that can
shed light on these distinctions, and was used in the development of the AMR.
25
FineGrained Assessment
The assessment technique used in the AMR is one developed by Tzur (2007) to
better assess the precise level of student understanding, by distinguishing between the
anticipatory and participatory stages explained in the previous section. In finegrained
assessment, it is important that the items begin without prompting, and progress to less
difficulty and more support, in the form of prompts and questioning, to determine
whether the student is able to reason independently. When students are presented with
problems, it is without any prompting or introduction by the assessor. If the student is
unable to solve the problem, prompts are then given in a way that gives the student clues
about how to act, while being cautious to give as little away as possible. The intent is to
find the highest level at which the student can function. If the student is given an easier
problem firstor is led toward that higher understanding by instructionsthe reasoning
seen may be as a result of the unintentional prompting the student receives, instead of the
student being able to anticipate the entire process on her or his own.
For example, if one desires to know if a student can find a missing second
addend, the problem should be presented without any prelude. The student may be
askedI had three candiesand my friend gave me some more. Now I have seven
candies. How many did my friend give me? If the student is able to provide reasoning
and solve this independently, they can be considered at the anticipatory stage. If they are
not able to solve this problem, the assessor could prompt, beginning with calling attention
to the numbers in the problem and what they mean. The assessor might ask: How many
candies did I have to start with? and then ask the student if they can solve. A follow up
could be: How many candies do I have at the end? and then: Can you use the number
26
I have now and the number I started with to find how many my friend gave me? If the
student can use these prompts to solve, they are at the participatory stage for this
conception. They have some knowledge of how this problem can be solved, but need to
be prompted to arrive at a solution. If the student is not able to solve even after
prompting, it is likely that they have not begun to build the concept for themselves at all.
Additionally, it is important that student assessment not follow an activity in
which they were engaged in solving similar problems. This is to avoid the participatory
student appearing to be anticipatory. Immediately following a lesson, the student can
simply recall the activity, and use it to solve the problem. It may be impossible to tell
whether the student can reason this way independently or only following the lesson.
The importance of this distinction between participatory and anticipatory students
is that even though the participatory students may appear to understand the concept in the
context of the current class period, or with subtle prompting, these participatory students
will likely not be able to draw upon this reasoning to develop later concepts, and could be
left behind. This shows the importance of an assessment that can distinguish between
these stages, so that planning for these students can be optimal and directed specifically
at their area of need. The next section shows previous work in assessing multiplicative
reasoning, which was used to guide the development of the AMR, and to show the
differences between these attempts and mine.
Previous Research on Assessment of Multiplicative Reasoning
There are some existing instalments that have been developed for the assessment
of multiplicative reasoning necessary for future concepts. Here I will present these
instalments, with what I consider to be their strengths, and then explain how the AMR
27
attempts to go one step further to achieve a finegrained assessment of multiplicative
reasoning.
One instrumentdeveloped by Siemon and Breed (2010)was a pencil and paper
test to address many of the same concerns expressed in this paper. The assessment
considers many of the same conceptions, including distributing one composite unit across
another. That instalment has excellent potential to determine whether students are
reasoning multiplicatively, and includes rich tasks for them to explore, moving beyond
what is assessed in the AMR to include proportional reasoning. The intent of the Siemon
and Breeds (2010) assessment is similar to the AMR in that it tries to capture student
thinking and elicit strategies used to determine the sophistication with which the student
is approaching the problem, and the resources that they employ to solve them. Scoring of
this assessment was rubric basedand completed after students had finished the problems.
Their instrumenthoweverdid not include a distinction between participatory and
anticipatory stages in a childs use of the mDC schemeand did not provide the same
feedback as a clinical interview situation could.
Another assessment, developed by Bright, Joyner and Wallis (2003), has similar
features to the Siemon and Breen instalment. This assessment included a multiple choice
paper and pencil test, but mentions interviews to follow up and question students about
their answers to delve deeper into their thinking. It was noted in their work that it is
possible that questions which are easier to answer, such as multiple choice questions,
may induce more sophisticated reasoning, while open ended questions may cause
students to revert to more primary methods which they feel most comfortable using.
Their point may be captured when using the participatory/anticipatory stage distinction,
28
for which specific, finer considerations and ordering of the tasks is needed (Tzur, 2007).
This consideration guided the development of the AMR, to allow assessors to
differentiate between anticipatory and participatory stages of mDC. The AMR attempts
to be more specific than the previous instalments in assessing the smaller increments in
reasoning. In the Methods section, the development of the AMR is detailed, using this
stage distinction as an important part of the assessment.
29
CHAPTER III
METHODS
This chapter chronicles the development the AMR, including the selection of
tasks, the administration method, the administration guide, and scoring. The Assessment
for Multiplicative Reasoning (AMR) was developed to give teachers and researchers a
way to assess the level of a students multiplicative reasoningor preparedness to engage
in that reasoning. The intent is that instruction can then be focused at the appropriate
level to give students the experiences that they need to develop the next level of
reasoning required to support multiplication, or to indicate that the student is ready with a
strong basis to experience other math concepts for which multiplicative reasoning is a
prerequisite, such as fractions, ratio and proportion, and slope.
The AMR was developed with assistance from experts, and then tested briefly on
a small sample of fourth grade students. The full final version of the assessment to date
with all of the components mentioned here is provided in Chapter IV, but examples are
given in this chapter to illustrate the development process. This chapter explains the
process by which the assessment grew from tasks to target specific schemes of
multiplicative reasoning, possible distractors that were taken into consideration, and the
design of prompts to bring to light the possible participatory and anticipatory stages at
each conception. Later sections describe the development of an administration guide for
assessors and considerations for scoring the results. Refinement based on feedback from
experts and from the initial testing are discussed in each section as appropriate, as well as
their implications for the improvement of the AMR.
30
Development of Tasks
Tasks were created based upon theoretical, literaturebased analysis and
on my interpretation of the performance that students should exhibit if they have
completely developed the scheme in question (Zazkis and Hazzan, 1998). Through
conversations with Dr. Ron Tzur, I decided to include prior schemes in the assessment:
Counting On, Missing Addend, and Composite Unit Iteration (shown in Chapter IV), to
screen students and determine whether the additive schemes, rooted in number as
composite unit and necessary for multiplicative concepts, were accessible to the student.
One multiplicative scheme was included as well, namely Multiplicative Double Counting
(mDC), to assess the progression of students from additive to multiplicative reasoning.
The tasks were differentiated into grade bands, in an attempt to allow for
flexibility in administration by providing multiple entry points for students and assessors
to use, being responsive to previous knowledge about student understanding. For
example, a student at the fifth grade level may be tested at the 34 grade band, if the
teacher or assessor knows that the student will struggle with the numbers involved at their
grade level. It is also advised, and stated in the Administration Guide in Chapter IV, that
in general the assessor should begin with the grade level task, but move down to the
previous level if the student is unable to answer. The intent here is to determine whether
the struggle is due to the numbers involvedor to the conceptual scheme or stage being
assessed. The following section provides a description of the tasks used to assess these
schemes, and information about the choices made in their development.
31
Composite Unit Iteration Task
Composite unit was explained earlier, and is important for both the additive and
multiplicative schemes. The Composite Unit Iteration task asks students to use these
units and iterate them to find a new quantity, without counting all of the items. This
requires students to coordinate two quantities, the unit to be iterated and the number of
iterations that take place. This task is administered first because it involves composite
unit, which is important to both additive and multiplicative thinking, but also because it
represents a midpoint in the continuum of the assessment as a whole. This approach was
used although there was not enough data from student responses to create a proper
Guttman Scale (Trochim & Donnelly2008) for these tasks. Such a scale requires one to
use data to create a linear ordering of tasks. However, an attempt was made to arrange the
tasks in a developmental order that would most likely occur for the student conceptually.
Thus, if a student can properly solve and reason about the first task the assessor can move
to higher levels, and if not, the assessor would go back to prior conceptions. For
instance, it is unlikely that a student who can perform Multiplicative Double Counting
would be unable to solve a Counting On problem.
The original Composite Unit Iteration Task I had created, seen in Figure 3.1, was
problematic for students. It was intended to give students a visual model and see if they
could use the groupings of four beads to iterate and find the total, but students were
generally not able to use this information to iterate correctly. It was suggested by Dr.
Tzur that I eliminate the task, and it no longer appears in the final version. Instead, a task
involving iteration of the smallest composite unit (i.e., 2) was created, with a context
32
using pairs of socks that seemed familiar to any student. This task is further described in
Chapter IV.
Figure 3.1. Previous Version of Composite Unit Iteration Task
Additive Tasks
The additive portion of the assessment, including the schemes of Counting On
and Missing Addend, is not intended to be an exhaustive assessment of additive
reasoning, but to give a rough idea of where a student might be functioning in regard to
additive schemes in general, and in terms of her use of composite units in particular.
These schemes are explained next.
Counting On refers to the process by which a student can begin with a quantity,
such as 7, and then add to that quantity, perhaps 4 more, and find the total. The student
does this not by starting to count from 1,but by beginning with the known number 7 and
counting from there. This counting on strategy is sometimes called double counting, to
emphasize that the student is purposely keeping track of Is in the second addend and
knowing to stop after they have counted four more (Tzur & Lambert, 2011). The student
might say something like this: I have 7 alreadyand four more; so eight is 1nine is 2
ten is 3, and eleven is 4. I have eleven all together. The Counting On tasks in this
assessment were constructed so that the secondadded quantity is smaller. This was done
to make representing these problems with figural, pictorial or manipulative strategies
more accessible. Assessment of students abilities to reverse the quantities for counting
33
on when the larger quantity is given first was not a part of this assessment due to the
limited nature of this portion.
The next two schemes are assessed using Missing Addend problems, where a
student is given one of the quantities to be added and the total in a problem situation, and
is asked to find the other added quantity. In the first type of missing addend problems,
the student is given the first addend and the total, and is asked to find the second addend.
In the second type of missing addend problems, the student is given the second addend
and the total, and is asked to find the first addend. These problems are considered
different for students, as the Missing First Addend requires more sophisticated thinking.
The Missing First Addend Tasks are considered more difficult, because of the way that
the composite units are nested within the whole (Fuson et al., 1988). If the second
addend is missing, the student can know where to start her counting, as they do in
counting on, and proceed by counting up from the first addend unit the total is reached.
If the first addend is missing, students do not have a way to start an activity of counting
upward using their previous way of operating. They thus need to either reverse the order
of the addends (in effect, creating a missing second addend situation), or count down
from the total until they reach the second addend. Both of these operations are more
advanced and less likely to be available to many students due to insufficient abstraction
of the total as a composite unit containing of both addends as subcomposite units.
Additional considerations for the additive tasks are discussed next, in an attempt to clear
away distractions that might cause the assessment to give invalid results.
It is understood that some tasks may be much too simple for many students in the
upper grade bands, but to be thorough, they are included. It is also understood that the
34
concept of multiplicative reasoning assessed later in the instalment may not be present in
students in the earlier grades. Yet, the task is designed in such a way that if younger
students are able to reason multiplicatively, the numbers and situations should not be a
barrier.
The levels in each task were created to target values and situations that were
developmentally appropriate for students who are achieving at the gradelevels indicated.
The example seen in Figure 3.1,the Missing First Addend task, is differentiated to four
levels to accommodate grades K through 8. I chose the bouncy ball situation in the same
way that all situations were chosen, with an attempt to provide familiarity and context to
the problems. The numbers for each level (gradeband) were selected (a) in consultation
with Dr. Tzur, (b) using expert feedback from 15 teachers, and (c) guided by the work of
Sherin and Fuson (2005). The latter, empirical study focused on ways in which chosen
numbers and problem types affect an assessment. An effort was made not to repeat and
addition or multiplication facts, so that regardless of which grade level tasks were given,
or in which order, there would not be any repeats. Also I did not include consecutive
facts, such as 3 x 5 and 4 x 5, so that there would not be reliance on the previous
problems to answer the later problems.
For example, in the Missing First Addend Task (Figure 3.2), the lower grade
levels are given the addend 3 and a sum of 8, so that the total does not pass ten and
involve place value. These are numbers that younger students are likely to be familiar
with, yet avoid the facts that they might know, such as 5 + 5 =10. For the oldest grade
band (78), the AMR uses the numbers 25 and 38 to discourage use of known facts and
encourage reasoning. It is unlikely that the student has the math fact 25 + 13 = 38 in their
35
memory, as they may have 4 + 5 = 9, and therefore will make it more likely that they will
use additive reasoning to find the solution (e.g.25+1035, and 3 more is 38a strategy
indicating the students operation on 13 by decomposing it strategically into 10 and 3).
Each task was designed with these considerations in mind, in an attempt to hone in on the
reasoning available without interference from memorized facts or inaccessible numbers
and situations.
Missing first addend task
Level1,K2
10. Anna has some bouncy balls. Her mom gives her three (3) more bouncy
balls. She then has eight (8) bouncy balls. How many bouncy balls did Anna
have in the beginning?
Level 2, grades 34
11. Anna has some bouncy balls. Her mom gives her thirteen (13) more bouncy
balls. She then has nineteen (19) bouncy balls. How many bouncy balls did
Anna have in the beginning?
Level 3, grades 56
12. Anna has some bouncy balls. Her mom gives her sixteen (16) more bouncy
balls. She then has twentyfour (24) bouncy balls. How many bouncy balls
did Anna have in the beginning?
Level 4, grades 78
13. Anna has some bouncy balls. Her mom gives her twentyfive (25) more
bouncy balls. She then has thirtyeight (38) bouncy balls. How many bouncy
balls did Anna have in the beginning?
Figure 3.1.Missing First Addend Task
There are some prevailing strategies commonly taught in schools to handle these
types of addition problems. The strategies seen in early trials of the problems were using
known doubles, such as 4 and 4 make 8, and building on them, so that 4+5 could be seen
as the double of 4 plus one. Another strategy is to 'break5 or decompose the numbers to
make ten, so that 8+5 could be seen as 8+(2+3) = (8+2) + 3 = 10 + 3 = 13. Clearly, many
of these strategies that do not involve counting on allow students to use them in any
situation, and could not be avoided completely. A resourceful student can use these
36
strategies on any problem. In fact, this adaptability shows a high level of proficiency with
additive situations, particularly the operation on and with composite units, and thus
indicates that the student is ready to proceed.
These problems require students to be able to either use a modification of the
countingon procedure, where they know to stop at the given total, and must find how
many more they need to count from the given addend. Alternatively, they can use
counting down, or beginning at the total and counting backwards until they reach the
known addend. The Missing Addend problems were created with the missing addend as
a quantity that is less than ten. As mentioned in the section on Counting On, this was
done because students are being asked to show their reasoning using figural or
manipulative means, so it was necessary that the missing portion be kept to a number that
would be reasonable for students to represent in this way. Only two of these tasks were
included, both considered Joining Problems by Carpenter et al.(1981). The reason is
twofoldthe AMR was not intended to be a comprehensive assessment of additive
reasoningand the study by Carpenter indicated that this deviation in problem structure
did not affect the solution methods of students. It is, intended that both tasks be
administered as a reliability measure. It is possible that a student could miss the first
problems, the easier missing second addend problem, for various reasons, but correctly
solve the more difficult second problem with a missing first addend.
Multiplicative Tasks
Multiplicative Double Counting, as described in the section on Multiplicative
Schemes in Chapter II, represents the first stage in the transition from additive to
37
multiplicative thinking. This is the first and only multiplicative stage assessed by the
AMR in this prototype.
Due to the fact that Multiplicative Reasoning was the primary goal of the
development of this assessment, there are more tasks to lend reliability to this section.
The three tasks were all created with the unit rate, how many items per group, being less
than the number of composite units given in the problem. For example, in the first MDC
task for level1,there are three cars per box and four boxes. The tasks were constructed
in this way to make it more conducive to the creation of a figural, pictorial or
manipulative solution. At the time of the defense of this thesis, it was suggested by Dr.
Heather Johnson, that the multiplicative tasks be adjusted to be consistent as to the
relative size of the unit compared to the unit rate. This additional expert feedback for
task adjustment was made and is reflected in the final instrument.
Task Refinement
Initial refinement of these tasks was done through feedback from 15 teachers of
K12 students. One suggestion they gave was to include picture cards for younger
students. A picture was included on the Composite Unit Iteration task, a frequent
suggestion, to help clarify the meaning of a pair of socks mentioned in the problem. I
decided not to include picture cards for the remainder of the problems, because it may
provide too much prompting via visible units. This would fail to show whether a student
has the reasoning or simply relies on the picture, counting Is while visually and silently
scanning the picture. I believed that part of the scheme necessary for students would be
the ability to create the pictorial or figural representation, and that providing a picture
might detract from their ability to show this crucial step in operating on units. It would
38
be appropriate for a kindergartener to be unable to solve even the additive reasoning
problems, so I did not feel that providing this scaffolding was necessary.
The remainder of the teachers5 suggestions involved the numbers chosen for the
tasks. There were some repeats in the number families chosen in the initial draft, such as
4+5=9 being used three times on different problems. This was corrected in the later
version to include more addition facts, carefully avoiding the doubles and other patterns
described earlier. It was also suggested that a problem on the initial version involving
earrings be changed to a more genderneutral item such as mittens. This change was
made to have the problem ask about socks, after much consideration as to an item that
students might have multiple pairs of at home.
After receiving the expert feedback from Dr. Tzur and the K12 teachers, the
instrument was tested with fourth graders. As a result of that initial testingthe grade
bands were narrowed to accommodate students in a more differentiated way. The first
draft of this assessment included three levels, K2, 35 and 68. During the initial testing
of the AMR it was found that some fourth grade students struggled with the values given
and that the grade bands may have been too large. This prompted the further
differentiation of levels in elementary grades to include only two grade levels instead of
one. The current structure includes levels for Kl23, 45 and 68.
Construction of Prompts
In this section, I will explain the way that prompts were developed for each of the
aforementioned tasks. These prompts are meant to assist a student who may not be at the
anticipatory stage in solving the problem with support, and thus gaining data about the
participatory or preparticipatory stage in which the student operates for the scheme being
39
tested. This part of the assessment, in which prompts are provided gradually, is designed
to give the assessor an idea of where the student is on the participatory continuum, and
what specific support they may need. This is important because a student who has no
prior knowledge of the reasoning at this stage will need to be approached differently than
one who is nearly anticipatory, and a nearly anticipatory student will likewise need
different learning experiences than one who is just beginning to be participatory. This
type of stage distinction was the goal of scripting the prompts to be given while the tasks
are being completed, instead of simply presenting the problem and prompts to the student
as typically occurs in a pencil and paper test.
As explained in the FineGrained Assessment section of Chapter II, the first step
is to administer the problem without prelude or prompts. This is intended to allow
students who have achieved an anticipatory stage for the scheme to demonstrate it. Using
the example of the Missing First Addend task above, the student would be given the
problem as is and asked to solve while explaining their thinking. If independently doing
so, it allows the assessor to infer construction of the scheme at the anticipatory stage.
Whether or not the problem is solved successfully, the student may then be
prompted to use figural, pictorial or manipulative means to prove their solution in an
attempt to understand the reasoning used by the student. This may be omitted sufficient
evidence was given that the student understands the concept, but it provides an
opportunity to delve deeper into what the student is thinking about the task. The use of
these means can also be a way for the student to selfprompt, or catch their own mistakes.
For a student who has used known facts, drawing or using manipulatives allow the
40
assessor to see if these facts are supported by reasoning, or simply reflect rote
memorization.
If the problem cannot be successfully solved independently, the assessor will then
begin to prompt the student in an attempt to determine whether the student has
constructed a participatory stage for the scheme as indicated by solving the problem with
support. The prompts are scripted purposely to provide the student information that
should help to orient their thinking to the activity involved in the scheme and thus to a
correct solution if they have begun forming that underlying conception. To the extent
possible, assessors are expected to avoid deviating significantly from these prompts, in
part because a teacher or researcher who works regularly with students may
unintentionally give additional information that would lead the student to score more
highly than is accurate.
An example of the prompting created for the Missing First Addend Task (Figure
3.2), which begins after the student has attempted to solve independently, is seen in
Figure 3.3. Some prompts were included to elicit specific reasoning from the student, if
the correct answer is given but sufficient reasoning is not.
41
For both correct responses without reasoning and incorrect responses:
o Ask students to explain their thinking on this task. It is important to discourage
counting all as a solution. Prompt the student to use counting on by covering the
first composite unit, or asking them to make a drawing that does not include all of
the ones in the unit. If the student is using marks or manipulatives, it may be
necessary to cover the first composite unit and then see if the student can then
proceed to count on.
o If at any time during the administration of this task you believe that the number
values are impeding the student, move down to a lower grade band and test again,
o If at any time the student answers correctly using counting on or counting back, go
to the next problem.
If response is correct, but the students use an unknown strategycannot articulate
one, use counting all or known math facts, prompt student using these questions:
1. How many did she have before her mom gave her more?
2. How many does she have in the end? Can you use the number in that she had
before as a starting place to see how many more she got from her mom? How?
3. Can you use the number she had in all and count backwards to find out how
many she got from her mom? How?
If the student solves correctly after prompting, move to the next problem
Figure 3.3. Prompts for Correct Responses
A second level of prompts was developed for students who do not answer
correctly, seen in Figure 3.4, to determine if they have a high or low participatory level of
understanding, or perhaps no understanding of this stage at all.
If response is incorrect:
Ask student to explain their thinking on this task. Produce manipulatives, and
ask the student to solve with the manipulatives.
If they does not solve correctly with manipulatives, prompt students using the
following questions:
1. How many did she have before her mom gave her more?
2. How many does she have in the end? Can you use the number in that she had
before as a starting place to see how many more she got from her mom? How?
3. Can you use the number she had in all and count backwards to find out how
many she got from her mom? How?
If the student solves correctly after prompting, move to the next problem
If the student cannot solve this problem after prompting, end testing.
Figure 3.4. Prompts for Incorrect Responses
The prompts here were designed to carefully lead a student through the
activityeffect thinking process that can help them arrive at a solution. The intent
42
of this stepwise approach is, again, to discern the specific level of the student
within the participatory stage. In the next section the choice of Administration
Method is discussed, with a rationale for the interview format.
Selection of Administration Method
To distinguish between the anticipatory and participatory stages in finegrained
assessment, conducting a facetoface interview is needed to provide the assessor with the
most specific feedback about what the student knows (Tzur, 2007). The format of the
assessment is a clinical interview, a technique often used to delve in to the thinking of
subjects, using what Zazkis and Hazzan (1998) called performance questions, followed
by Why questions. The interviews are to be conducted with one to three students at a
time, and are intended to give the students a task, observe their responses, and if needed,
prompt with questions that will clarify whether they are showing evidence of the scheme
or not (Ginsberg1981).
A clinical interview situation (Ginsberg, 1981) is best suited to this type of
assessment, because it allows the assessor to control and adapt the prompts and
information given to the student. The interview format provides the assessor with the
unique opportunity to ask students questions as they are solving, listen to what is being
verbalized, analyze and clarify figural representations created by the student in real time,
and discern any movements or quiet vocalizations that the student may produce.
While a paper and pencil task is efficient from a time standpoint, it does not
provide the rich data that can be gleaned from an interview setting. Furthermore, in a
pencil and paper test, all of the information is present for the student to use, and it is
extremely difficult to discern whether the student was able to solve a problem
43
independently, whether they used information given in previous or subsequent problems,
what kind of reasoning was used, or whether they could have solved with a little more
support. Asking students to explain their answers in writing during a paper and pencil
test is another optionbut it often provides inaccurate results (e.g.when the students
first language is not English). Students can often explain verbally what they would not
write on a test, and may know more than they show on paper.
The tasks were ordered so that students would be unlikely to be able to complete a
task if they were unable to complete the previous task, a Guttmanlike scale (Trochim &
Donnelly2008). This scaling is discussed further in Chapter IV. For that reason, it is
unnecessary for a student to complete the entire assessment. Rather, the assessor can
simply discontinue testing when the children being assessed reach a level they are
evidently unable to complete, even with prompts. As explained previously, the tasks
should begin at the Composite Unit Iteration task (pairs of socks), and then either move
forward to the multiplicative tasks if students reason independently, or begin the admtive
reasoning tasks if they do not. One reason for using the Composite Iteration Task as a
starting point is that composite unit is a conception that is necessary throughout the tasks.
The ability to consider a group of Is as its own unit is something that a student would
need to be able to do to reason multiplicatively or adaitively, but the iteration of these
composite units is a more advanced concept than simply recognizing composite units.
This allows a 'middle ground5 for students to begin at a level that is most appropriate for
them, without subjecting more advanced students to problems that are too easy for them,
and possibly losing their interest, or intimidating students who are unable to reason
multiplicatively. This can also save time in the administration procedure for both
44
teachers and students. Dr. Johnson suggested, with Dr. Alan Davis5 support, this change
in task order at the time of thesis defense, and it was implemented in the Administration
Guide.
Students from time to time may struggle with the tasks not because of the
concept, but because of the numbers involved. To accommodate these students and
assess their reasoning, it is possible in the course of administering the test to allow the
student to respond to a task that is at the grade level below the one being assessed. For
instanceif a fifth grade student is struggling to complete the Multiplicative Double
Counting task, they could be administered the level 2 task, intended for second and third
graders. If the student could then answer the problem, it could be surmised that they
began forming the necessary multiplicative scheme, but may need some work to
generalize it for larger quantities. This flexibility was included so that the assessor can be
responsive to the needs of their students.
If or when a student is unable to complete the task with prompting and/or use of
manipulatives, the assessor terminates the testing. Such inability indicates that the
reasoning of the student has been exhausted and that they need not be tested any further.
It is possible that a student who multiplies easily, and has some explanations as to what is
occurring may not score as highly as expected. This result might indicate that while the
student has a strong procedural knowledge or fact memorization, some of the prerequisite
schemes may not be solid enough to support their future learning. As discussed in
Chapter IIstudents will use available schemes to construct knowledge (Piaget1985)
but if not carefully monitored, may use schemes such as additive reasoning or partofa
whole, which might not support them as well as multiplicative reasoning could. Next I
45
will explain how the Administration Guide was created to walk assessors through the
instrument as it is being given.
Development of Administration Guide
The administration guide is shown in Chapter IV, but this section will serve as a
brief description of the thinking behind its development. The guide was created to be
userfriendly, complete with scripted prompts and a description of what a person using
the assessment should look for in the childs solution to each problem. The task is listed
with each grade band and all prompts, in an attempt to provide for the assessor a clear
picture of what will happen during the assessment, and the options they have during
administration. It was also created with an eye toward some technological component
being involved in the future, and considering some decisions the technology would need
to make based on the students response. The teacher or assessor can use the instalment
as is to pinpoint a students understandings and create a finegrained assessment (Tzur
2007). Such assessment should be helpful to determine the next steps that need to occur
for the student, and help the assessor to complete the scoring guide described in the next
section.
Scoring Considerations
As with the administration guide, the Scoring Guide is seen in Chapter IV, but in
this section I will describe the thought process behind its development. The scoring of
this assessment needed to be both detailed and simple to complete so that teachers and
researchers can obtain the needed information while minimizing the time taken for the
assessment. Initially, I had envisioned the assessor scoring in real time, as the
administration proceeded. In testing of the instrumenthoweverit was noted by Mr.
46
Evan McClintock during this first administration that it might be simpler to have an
observer score as the assessor interacts with the children. This was a very useful
suggestion, and it was added as a possibility for the scoring process. It was also noted
during the testing process that if students are videotaped, as they were in our first trial,
scoring could happen later. I do not believe that this later scoring would be ideal,
however, as the scoring process may drive further questioning of the student in real time
that may not be possible after the fact. That being said, it may be used if necessary, and
if care is taken to thoroughly prompt the student.
The next chapter shows the final result of the work and refinement processes
described in this Methods Chapter. This instrument is still in development; but it is
presented in its complete and final form at the time of submitting this thesis to the
Graduate School of the University of Colorado Denver.
47
CHAPTER IV
RESULTS
This chapter presents the final version of the AMR, including tasks,
administration guide, student problem strips, and the scoring guide. Also included is a
discussion of the initial attempts made at discerning the reliability and validity of this
instrument.
Tasks
The tasks, seen in Figure 4.1, are listed in the order of increasing conceptual
difficulty, but not necessarily in the order that they would be administered to a student in
an ordinary testing situation, because this order will vary depending on the result of each
problem. These tasks are also included in the Administration uideand in the Student
Problem Strips.
48
Composite Unit Iteration Task
Blanca has 8 pairs of socks. If she counts each sock, how many will there be?
A Pair of Socks
Counting on task
Level1K2
2. Lisa had five (5) pencils. She finds four (4) more pencils. How many pencils
does Lisa have?
Level 2, grades 34
3. Lisa had seventeen (17) pencils. She finds four (4) more pencils. How many
pencils does Lisa have?
Level 3, grades 56
4. Lisa had twentyfour (24) pencils. She finds seven (7) more pencils. How many
pencils does Lisa have?
Level 3, grades 68
5. Lisa had fiftyseven (57) pencils. She finds nine (9) more pencils. How many
pencils does Lisa have?
Figure 4.1. Tasks
49
Missing second addend task
Level1 K2
6. Joseph has six (6) pieces of gum. His friend gives him some more pieces of
gum. Now Joseph has nine (9) pieces of gum. How many pieces of gum did
his friend give him?
Level 2, grades 34
7. Joseph has (8) pieces of gum. His friend gives him some more pieces of gum.
Now Joseph has fifteen (15) pieces of gum. How many pieces of gum did his
friend give him?
8. Level 3, grades 56
Joseph has fifteen (15) pieces of gum. His friend gives him some more pieces
of gum. Now Joseph has twentythree (23) pieces of gum. How many pieces
of gum did his friend give him?
9. Level 4, grades 68
Joseph has twentyfive (25) pieces of gum. His friend gives him some more
pieces of gum. Now Joseph has thirtysix (36) pieces of gum. How many
pieces of gum did his friend give him?
Missing first addend task
Level1K2
10. Anna has some bouncy balls. Her mom gives her three (3) more bouncy
balls. She then has eight (8) bouncy balls. How many bouncy balls did Anna
have in the beginning?
Level 2, grades 34
11. Anna has some bouncy balls. Her mom gives her thirteen (13) more bouncy
balls. She then has nineteen (19) bouncy balls. How many bouncy balls did
Anna have in the beginning?
Level 3, grades 56
12. Anna has some bouncy balls. Her mom gives her sixteen (16) more bouncy
balls. She then has twentyfour (24) bouncy balls. How many bouncy balls
did Anna have in the beginning?
Level 4, grades 78
13. Anna has some bouncy balls. Her mom gives her twentyfive (25) more
bouncy balls. She then has thirtyeight (38) bouncy balls. How many bouncy
balls did Anna have in the beginning?
Figure 4.1. Tasks contd
50
mDC Task 1
Level1,K2
14. Tony has his toy cars in small boxes in his room. He has four (4)
boxes, and each box has three (3) cars in it. How many cars does Tony
have in all?
Level 2, grades 34
15. Tony has his toy cars in small boxes in his room. He has six (6) boxes,
and each box has three (3) cars in it. How many cars does Tony have
in all?
Level 3, grades 56
16. Tony has his toy cars in small boxes in his room. He has six (6) boxes,
and each box has five (5) cars in it. How many cars does Tony have in
all?
Level 4, grades 78
17. Tony has his toy cars in small boxes in his room. He has seven (7)
boxes, and each box has six (6) cars in it. How many cars does Tony
have in all?
mDC Task 2
Level1,K2
18. Ella has five (5) baskets for her friends. She wants to put three (3)
pieces of candy in each basket. How many pieces of candy will Ella
need?
Level 2, 34
19. Ella has five (5) baskets for her friends. She wants to put four (4)
pieces of candy in each basket. How many pieces of candy will Ella
need?
Level 3, 56
20. Ella has seven (7) baskets for her friends. She wants to put four (4)
pieces of candy in each basket. How many pieces of candy will Ella
need?
Level 4, 78
21. Ella has eight (8) baskets for her friends. She wants to put seven (7)
pieces of candy in each basket. How many pieces of candy will Ella
need?
Figure 4.1. Tasks contd
51
mDC Task 3
Level1K2
22. Aaron wants to make three (3) birdhouses, and he needs two (2) nails
for each birdhouse. How many nails will he need in all?
Level 2, 34
23. Aaron wants to make six (6) birdhouses, and he needs four (4) nails for
each birdhouse. How many nails will he need in all?
Level 3156
24. Aaron wants to make eight (8) birdhouses, and he needs four (4) nails
for each birdhouse. How many nails will he need in all?
Level 4, 78
25. Aaron wants to make nine (9) birdhouses, and he needs eight (8) nails
for each birdhouse. How many nails will he need in all?
Figure 4.1. Tasks contd
52
Assessment Protocol
Students are tested using the following protocol. The student(s) should be
provided with pencil and paper, or some electronic alternative that allows them to write
or draw only. A card containing the problem is handed to the student. These are
formatted and printed with the problem number for ease of reference. Instructions for the
cutting out and assembly of these problems strips are in Figure 4.2.
Student Problem Strips
To use these strips, print out on cardstock and cut along the lines that border the tasks.
Stack each group of problems (found on the same page) with the highest gradeband in
the back.
This should result in a stack of strips that exposes the problem at the top of the strip.
These can be left loose or stapled on the right or left into small booklets.
The Composite Unit Iteration Task on this page is not part of any of the booklets.
1.Composite Unit Iteration Task
Blanca has 8 pairs of socks. If she counts each sock, how many will there be?
A Pair of Socks
Figure 4.2. Student Problem Strips
53
2. Counting on task Level1,grades K2
Lisa had five (5) pencils. She finds four (4) more pencils. How many
pencils does Lisa have?
3. Counting on task Level 2, grades 34
Lisa had seventeen (17) pencils. She finds four (4) more pencils. How many
pencils does Lisa have?
4. Counting on task Level 3, grades 56
Lisa had twentyfour (24) pencils. She finds seven (7) more pencils. How
many pencils does Lisa have?
5. Counting on task Level 4, grades 68
Lisa had fiftyseven (57) pencils. She finds nine (9) more pencils. How many
pencils does Lisa have?
Figure 4.2. Student Problem Strips contd
54
6. Missing Second Addend Task Level1,grades K2
Joseph has six (6) pieces of gum. His friend gives him some more pieces of gum.
Now Joseph has nine (9) pieces of gum. How many pieces of gum did his friend
give him?
7. Missing Second Addend Task Level 2, grades 34
Joseph has (8) pieces of gum. His friend gives him some more pieces of gum.
Now Joseph has fifteen (15) pieces of gum. How many pieces of gum did his
friend give him?
8. Missing Second Addend Task Level 3, grades 56
Joseph has fifteen (15) pieces of gum. His friend gives him some more pieces of
gum. Now Joseph has twentythree (23) pieces of gum. How many pieces of
gum did his friend give him?
9. Missing Second Addend Task Level 4, grades 68
Joseph has twentyfive (25) pieces of gum. His friend gives him some more
pieces of gum. Now Joseph has thirtysix (36) pieces of gum. How many pieces
of gum did his friend give him?
Figure 4.2. Student Problem Strips cont'd
55
10. Missing First Addend Task Level1,grades K2
Anna has some bouncy balls. Her mom gives her three (3) more bouncy balls. She
then has eight (8) bouncy balls. How many bouncy balls did Anna have in the
beginning?
11.Missing First Addend Task Level 2, grades 34
Anna has some bouncy balls. Her mom gives her thirteen (13) more bouncy
balls. She then has nineteen (19) bouncy balls. How many bouncy balls did Anna
have in the beginning?
12. Missing First Addend Task Level 3, grades 56
Anna has some bouncy balls. Her mom gives her sixteen (16) more bouncy
balls. She then has twentyfour (24) bouncy balls. How many bouncy balls did
Anna have in the beginning?
13. Missing First Addend Task Level 4, grades 68
Anna has some bouncy balls. Her mom gives her twentyfive (25) more bouncy
balls. She then has thirtyeight (38) bouncy balls. How many bouncy balls did
Anna have in the beginning?
Figure 4.2. Student Problem Strips contd
56
14. Multiplicative Double Counting Task 1 Level1,grades K2
Tony has his toy cars in small boxes in his room. He has four (4) boxes,
and each box has three (3) cars in it. How many cars does Tony have in
all?
15. Multiplicative Double Counting Task 1 Level 2, grades 34
Tony has his toy cars in small boxes in his room. He has six (6) boxes,
and each box has three (3) cars in it. How many cars does Tony have in
all?
16. Multiplicative Double Counting Task 1 Level 3, grades 56
Tony has his toy cars in small boxes in his room. He has six (6) boxes,
and each box has five (5) cars in it. How many cars does Tony have in
all?
17. Multiplicative Double Counting Task 1 Level 4, grades 68
Tony has his toy cars in small boxes in his room. He has seven (7) boxes,
and each box has six (6) cars in it. How many cars does Tony have in all?
Figure 4.2. Student Problem Strips contd
57
18. Multiplicative Double Counting Task 2 Level1,grades K2
Ella has five (5) baskets for her friends. She wants to put three (3) pieces
of candy in each basket. How many pieces of candy will Ella need?
19. Multiplicative Double Counting Task 2 Level 2, grades 34
Ella has five (5) baskets for her friends. She wants to put four (4) pieces of candy
in each basket. How many pieces of candy will Ella need?
20. Multiplicative Double Counting Task 2 Level 3, grades 56
Ella has seven (7) baskets for her friends. She wants to put four (4) pieces of
candy in each basket. How many pieces of candy will Ella need?
21.Multiplicative Double Counting Task 2 Level 4, grades 68
Ella has eight (8) baskets for her friends. She wants to put seven (7)
pieces of candy in each basket. How many pieces of candy will Ella
need?
Figure 4.2. Student Problem Strips contd
58
22. Multiplicative Double Counting Task 3 Level1,grades K2
Aaron wants to make three (3) birdhouses, and he needs two (2) nails for
each birdhouse. How many nails will he need in all?
23. Multiplicative Double Counting Task 3 Level 2, grades 34
Aaron wants to make six (6) birdhouses, and he needs four (4) nails for each
birdhouse. How many nails will he need in all?
24. Multiplicative Double Counting Task 3 Level 3, grades 56
Aaron wants to make eight (8) birdhouses, and he needs four (4) nails for each
birdhouse. How many nails will he need in all?
25. Multiplicative Double Counting Task 3 Level 4, grades 68
Aaron wants to make nine (9) birdhouses, and he needs eight (8) nails for
each birdhouse. How many nails will he need in all?
Figure 4.2. Problem Strips for Students contd
59
Once the card is given to the student, the assessor reads the problem aloud.
Clarification about what the problem is asking is permissible, as long as it does not point
students toward a solution. The student is asked to solve the problem by any means they
wishthinking aloud as they do so. If students do not share their thinking during the
solution process, the assessor should ask the student after an answer is given what they
were thinking as they solved, encouraging making marks or drawings on their paper, or
using manipulatives to show their thinking.
Students should be encouraged to use writing, drawings or marks first before
using manipulatives. This is because using manipulatives, such as Unifix cubes, may
encourage students to count all instead of using a composite unit (Carpenter et al.1981).
When students are representing the composite units, it is preferable if they can do so
without drawing in or building each of the Is needed to construct that unit. Treating the
unit as a whole, in and of itself, is important in the understanding of composite unit
(Tzur, et al., 2013). For example, if the child began drawing 6 circles with 3 dots in each,
the assessor should encourage him or her to first try solving the problem by just drawing
the first 12 circles and then just draw the circles without dots. Such an intervention is
done because students may count the Is even if their reasoning supports activity on the
composite units without the Is, simply because the Is are available to count. These
instructions are seen in the Administration Guide presented later in this chapter. The
manipulatives used could vary, but most commonly used are Unifix cubes and paper and
pencil for drawing pictures by the student. Students can solve problems with
manipulatives that they would not be able to solve without them, and so this intervention
is seen as similar to a prompt (Steffe, 1970).
60
The assessor should do her or his best to avoid indicating to the student whether
their answer or their thinking is correct or incorrect. The burden of proof is on the
student, to be able to support their answer with reasoning, including figural or
manipulative means. This approach is needed to uphold the finegrained assessment
technique (Tzur, 2007), which proceeds from asking the student to solve with the
minimum amount of support necessary. In order to determine whether the student is at an
anticipatory or participatory stage, it is cmcial to allow the student to solve as
independently as possible, anticipating as much as they are able before prompts,
including those about correctness of solution, are given.
By following the guide, the assessor progresses through the questions using the
prompts, either moving down a level or using questioning to determine whether a student
may be able to solve a similar problem with different numbers. This will help teachers
determine whether what is lacking is the ability to reason about these tasks, or the
familiarity with the quantities presented. At the end of each task the assessor can find
directions about the next step in the process. Generally, if the problem was solved
correctly, the student moves to the next problem, if the student could not solve the
problem with prompting, testing is terminated. This is due to the attempt at a Guttman
like organization (Trochim & Donnelly2008). This instrument is certainly not qualified
as using the Guttman scale, which would be based on data from previous administrations,
and not require that the student complete each item to give a reliable result. Details about
this process are seen in the Administration Guide in the next section.
61
Administration Guide
This section provides the complete version to date of the Administration Guide,
intended for use by teachers or other assessorsand showing the instructionstasks
prompts and things to look for at each stage. This document would guide the
administration of the assessment for students at any level. Using the guide, seen in
Figure 4.3, should direct teachers and other assessors in how to administer the
assessment, and help in the completion of the Scoring Guide seen in the following section
62
Administration Guide
Assessment for Multiplicative Reasoning
General Directions
Please read each task exactly as written. Students may also view the task and read it
on their own from the problem strip as you read it aloud.
Administration begins with the Composite Unit Iteration task, and from there will
move to either Counting On or Multiplicative Double Counting, depending on student
response. The progression is shown in the flow chart for Administration Protocol, and
is indicated in this guide.
The problem numbers are listed on the far left for easy reference.
Directions and numbers in task may be repeated indefinitely, and students may write
down important information or numerals as needed. Clarification about what is meant
by the problem is permitted, but students should do all solving. If possible, stick to
the given prompts.
Students may use scratch paper, and all marks made by students should be kept and
labeled with the problem number.
When administering these tasks, ask the student to explain their thinking, and record
everything the student does on the scoring sheet.
You may use questions such as:
What were you thinking as you solved this problem?
What strategies did you use?
Did you picture something in your head to help you solve this problem?
Begin with the level appropriate for the childs grade level. If they are not successful
at this task, move down one level and test again.
Please explain to the student that you will be recording their responses and their work
so that we can learn more about how they solve problems.
Do not indicate to the student whether they are correct or incorrect as they work.
Figure 4.3. Administration Guide
63
Administration Sequence
/ \
Composite Unit
Iteration Task
(Problem 1)
\_____________________/
Incorrect Correct
Solution Solution
r ^ r a
Additive Tasks Multiplicative Tasks
(Problems 213) (Problems 1425)
r a
r Multiplicative Double
Counting On Counting
(Problems 25) (Problems 1425)
V
Missing Addend Tasks
(Problems 613)
If students solve the addend tasks correctly, begin the Multiplicative Tasks. It may be helpful to begin at a lower grade band if the students cannot reason with the numbers provided.
Figure 4.3. Administration Guide contd
64
Composite Unit Iteration Task
Please solve this problem, and explain as much as you can about what you're thinking
as you do so.
(Clarify the meaning of pair.
1. Blanca has 8 pairs of socks. If she counts each sock, how many will there be?
A Pair of Socks
If the student counts one by one, and not by composite units of two, ask if they
can use the way the socks are grouped into pairs.
If the students cannot solve after this prompt, begin with the Counting On Task
(problem 2).
If the student is able to iterate the socks to find that there are lb in all, skip to the
Multiplicative Double Counting Tasks (problem 14).
Counting on task
Please solve this problem, and explain as much as you can about what you're thinking
as you do so.
Level1K2
2. Lisa had five (5) pencils. She finds four (4) more pencils. How many pencils
does Lisa have?
Level 2, grades 34
3. Lisa had seventeen (17) pencils. She finds four (4) more pencils. How many
pencils does Lisa have?
Level 3, grades 56
4. Lisa had twentyfour (24) pencils. She finds seven (7) more pencils. How many
pencils does Lisa have?
Level 3, grades 68
5. Lisa had fiftyseven (57) pencils. She finds nine (9) more pencils. How many
pencils does Lisa have?
o If the student has answered correctly, and shown evidence of counting on, go to the
next problem.
Figure 4.3. Administration uide contd.
65
For both correct responses without reasoning and incorrect responses:
o Ask students to explain their thinking on this task. It is important to discourage
counting all as a solution. Prompt the student to use counting on by covering the
first composite unit, or asking them to make a drawing that does not include all of
the ones in the unit. If the student is using marks or manipulatives, it may be
necessary to cover the first composite unit and then see if the student can then
proceed to count on.
o If at any time during the administration of this task you believe that the number
values are impeding the progress of the students progressmove down to a lower
grade band and test again.
o If at any time the student answers correctly using counting on, go to the next
problem.
If response is correct, but the students use an unknown strategy, cannot articulate
one, use counting all or known math facts, prompt student using these questions:
1. How many did she have in the beginning?
2. How many did she find?
3. Can you use the number she had in the beginning as a starting place to see how
many in all? How?
If the student uses counting on after prompting, move to the next problem.
If response is incorrect:
Ask students to explain their thinking on this task. Produce manipulatives, and
ask the student to solve with the manipulatives.
If solved correctly with manipulatives, go to the next problem.
If you believe that the number values are impeding the progress of the student,
move down to a lower grade band and test again.
If they do not solve correctly, prompt students using the following questions:
How many did she have in the beginning?
How many did she find?
Can you use the number she had in the beginning as a starting place to see how
many in all? How?
If the student uses counting on after prompting, move to the next problem
If the student cannot solve this problem after prompting, end testing.
66
Additive Tasks
Missing Addend Tasks
Administer both of these tasks, regardless of the results, to assess the
additive reasoning of the student.
Missing second addend task
Please solve this problem, and explain as much as you can about what you're thinking
as you do so.
Level1 K2
6. Joseph has six (6) pieces of gum. His friend gives him some more pieces of
gum. Now Joseph has nine (9) pieces of gum. How many pieces of gum did
his friend give him?
Level 2, grades 34
7. Joseph has (8) pieces of gum. His friend gives him some more pieces of gum.
Now Joseph has fifteen (15) pieces of gum. How many pieces of gum did his
friend give him?
8. Level 3, grades 56
Joseph has fifteen (15) pieces of gum. His friend gives him some more pieces
of gum. Now Joseph has twentythree (23) pieces of gum. How many pieces
of gum did his friend give him?
9. Level 4, grades 68
Joseph has twentyfive (25) pieces of gum. His friend gives him some more
pieces of gum. Now Joseph has thirtysix (36) pieces of gum. How many
pieces of gum did his friend give him?
If the student uses counting on or counting back, move to the next problem.
For both correct responses without reasoning and incorrect responses:
o Ask students to explain their thinking on this task. It is important to discourage
counting all as a solution. Prompt the student to use counting on by covering the
first composite unit, or asking them to make a drawing that does not include all of
the ones in the unit. If the student is using marks or manipulatives, it may be
necessary to cover the first composite unit and then see if the student can then
proceed to count on.
o If at any time during the administration of this task you believe that the number
values are impeding the student, move down to a lower grade band and test again,
o If at any time the student answers correctly using counting on, go to the next
problem.
If response is correct, but the students use an unknown strategy, cannot articulate
one, use counting all or known math facts, prompt student using these questions:
1. How many did he have in the beginning? How many does he have in the end?
Can you use the number in that he had in the beginning as a starting place see
how many more he got from his friend? How?
2. Can you use the number he had in all and count backwards to get to find out
how many he got from his friend? How?
If the student solves correctly after prompting, move to the next problem.
If the student cannot solve this problem after prompting, end testing.
If response is incorrect:
Ask student to explain their thinking on this task. Produce manipulatives, and
ask the student to solve with the manipulatives.
1.If solved correctly with manipulatives, go to the next problem.
2. If you believe that the number values are impeding the student, move down
to a lower grade band and test again.
3. If they do not solve correctly, prompt students using the following questions:
4. How many did he have in the beginning? How many does he have in the
end? Can you use the number in that he had in the beginning as a starting
place see how many more he got from his friend? How?
5. Can you use the number he had in all and count backwards to get to find out
how many he got from his friend? How?
If the student solves correctly after prompting, move to the next problem
If the student cannot solve this problem after prompting, end testing.
Missing first addend task
Please solve this problem, and explain as much as you can about what you're thinking
as you do so.
Level1,K2
10. Anna has some bouncy balls. Her mom gives her three (3) more bouncy
balls. She then has eight (8) bouncy balls. How many bouncy balls did Anna
have in the beginning?
Level 2, grades 34
11. Anna has some bouncy balls. Her mom gives her thirteen (13) more bouncy
balls. She then has nineteen (19) bouncy balls. How many bouncy balls did
Anna have in the beginning?
Level 3, grades 56
12. Anna has some bouncy balls. Her mom gives her sixteen (16) more bouncy
balls. She then has twentyfour (24) bouncy balls. How many bouncy balls
did Anna have in the beginning?
Level 4, grades 78
13. Anna has some bouncy balls. Her mom gives her twentyfive (25) more
bouncy balls. She then has thirtyeight (38) bouncy balls. How many bouncy
balls did Anna have in the beginning?
If the student uses counting on or counting back, end assessment.
For both correct responses without reasoning and incorrect responses:
o Ask students to explain their thinking on this task. It is important to discourage
counting all as a solution. Prompt the student to use counting on by covering the
first composite unit, or asking them to make a drawing that does not include all of
the ones in the unit. If the student is using marks or manipulatives, it may be
necessary to cover the first composite unit and then see if the student can then
proceed to count on.
o If at any time during the administration of this task you believe that the number
values are impeding the student, move down to a lower grade band and test again,
o If at any time the student answers correctly using counting on or counting back, go
to the next problem.
If response is correct, but the students use an unknown strategy, cannot articulate
one, use counting all or known math facts, prompt student using these questions:
1. How many did she have before her mom gave her more?
2. How many does she have in the end? Can you use the number in that she had
before as a starting place to see how many more she got from her mom? How?
3. Can you use the number she had in all and count backwards to find out how
many she got from her mom? How?
If the student solves correctly after prompting, move to the next problem
If response is incorrect:
Ask student to explain their thinking on this task. Produce manipulatives, and
ask the student to solve with the manipulatives.
If they does not solve correctly with manipulatives, prompt students using the
following questions:
1. How many did she have before her mom gave her more?
2. How many does she have in the end? Can you use the number in that she had
before as a starting place to see how many more she got from her mom? How?
3. Can you use the number she had in all and count backwards to find out how
many she got from her mom? How?
If the student solves correctly after prompting, move to the next problem
If the student cannot solve this problem after prompting, end testing.
69
Multiplicative Tasks
Multiplicative Double Counting Tasks
There are three tasks in this section. A successful solution to two out of three is
considered evidence that the student has access to the Multiplicative Double Counting
Scheme, and testing should be concluded when the student has successfully completed
two.
mDC Task 1
Level1K2
14. Tony has his toy cars in small boxes in his room. He has four (4)
boxes, and each box has three (3) cars in it. How many cars does Tony
have in all?
Level 2, grades 34
15. Tony has his toy cars in small boxes in his room. He has six (6) boxes,
and each box has three (3) cars in it. How many cars does Tony have
in all?
Level 3, grades 56
16. Tony has his toy cars in small boxes in his room. He has six (6) boxes,
and each box has five (5) cars in it. How many cars does Tony have in
all?
Level 4, grades 78
17. Tony has his toy cars in small boxes in his room. He has seven (7)
boxes, and each box has six (6) cars in it. How many cars does Tony
have in all?
If response is correct:
Ask student to explain their thinking on this task. Student may use
manipulatives, fingers, or pencil and paper to prove their solution.
If the student gives an answer without units, ask what the number tells. Is it number
of cars? Number of boxes?
If correct, and using mDC strategy to find the answer, proceed to mDC task 2.
If they use counting all, or known math facts, prompt student using these
questions:
1. How many cars in one box?
2. Repeat original question
3. How many cars would be in two boxes?
4. Repeat original question
Figure 4.3. Administration Guide contd
70
If response is incorrect:
Ask student to explain their thinking on this task. Student may use
manipulatives, fingers, or pencil and paper to prove their solution.
As soon as the student is able to give the answer correctly, move to mDC task 2.
Prompt using these questions, encouraging the student to use scratch paper, fingers
or manipulatives to help him/her keep track.
1. How many cars in one box?
2. Repeat original question
3. How many cars would be in two boxes?
4. Repeat original question
Regardless of whether the student answers correctly or incorrectly, go on to mDC
Task 2.
mDC Task 2
Level1K2
18. Ella has five (5) baskets for her friends. She wants to put three (3)
pieces of candy in each basket. How many pieces of candy will Ella
need?
Level 2, 34
19. Ella has five (5) baskets for her friends. She wants to put four (4)
pieces of candy in each basket. How many pieces of candy will Ella
need?
Level 31 56
20. Ella has seven (7) baskets for her friends. She wants to put four (4)
pieces of candy in each basket. How many pieces of candy will Ella
need?
Level 4, 78
21. Ella has eight (8) baskets for her friends. She wants to put seven (7)
pieces of candy in each basket. How many pieces of candy will Ella
need?
If response is correct:
Ask student to explain their thinking on this task. Student may use
manipulatives, fingers, or pencil and paper to prove their solution.
If the student gives an answer without units, ask what the number tells. Is it
number of baskets? Number of candy pieces?
If correct, and using mDC strategy to find the answer, conclude testing.
If they use counting all, or known math facts, prompt student using these
questions:
1. How many candy pieces in one bag?
2. Repeat original question
3. How many candy pieces would be in two bags?
4. Repeat original question
Figure 4.3. Administration Guide contd
71
If response is incorrect:
Ask student to explain their thinking on this task. Student may use
manipulatives, fingers, or pencil and paper to prove their solution.
As soon as the student is able to give the answer correctly, move to mDC task 3.
Prompt using these questions, encouraging the student to use scratch paper, fingers
or manipulatives to help him/her keep track.
1. How many candy pieces in one bag?
2. Repeat original question
3. How many candy pieces would be in two bags?
4. Repeat original question
If both mDC tasks were answered correctly, conclude testing
If one mDC task is answered correctly, administer mDC Task 3.
If both mDC tasks were answered incorrectly, conclude testing.
mDC Task 3
Level1K2
22. Aaron wants to make three (3) birdhouses, and he needs two (2) nails
for each birdhouse. How many nails will he need in all?
Level 2, 34
23. Aaron wants to make six (6) birdhouses, and he needs four (4) nails for
each birdhouse. How many nails will he need in all?
Level 31 56
24. Aaron wants to make eight (8) birdhouses, and he needs four (4) nails
for each birdhouse. How many nails will he need in all?
Level 4, 78
25. Aaron wants to make nine (9) birdhouses, and he needs eight (8) nails
for each birdhouse. How many nails will he need in all?
If response is correct:
Ask student to explain their thinking on this task. Student may use
manipulatives, fingers, or pencil and paper to prove their solution.
If the student gives an answer without units, ask what the number tells. Is it
number of nails? Number of birdhouses?
If correct, and using mDC strategy to find the answer, conclude testing.
If the student uses counting all or known math facts, prompt student using these
questions:
1. How many nails does he need for one birdhouse?
2. Repeat original question
3. How many nails does he need for two birdhouses?
4. Repeat original question
Figure 4.3. Administration Guide contd
72
If response is incorrect:
Ask student to explain their thinking on this task. Student may use
manipulatives, fingers, or pencil and paper to prove their solution.
As soon as the student is able to give the answer correctly, conclude testing.
Prompt using these questions, encouraging the student to use scratch paper, fingers
or manipulatives to help him/her keep track.
1. How many nails does he need for one birdhouse?
2. Repeat original question
3. How many nails does he need for two birdhouses?
4. Repeat original question.
End of Test
Figure 4.3. Administration Guide contd
Scoring Guide
Scoring of students5 reasoning based on their solutions, including both answer and
explanation, for the tasks can optimally occur during the interview process, either by the
assessor or another observer. The assessment is meant to be adapted to what is seen in
student responses (e.g., counting Is or composite units, visible or figural, etc.), so that
care is taken to administer appropriate tasks to appropriate students. While the time and
care needed to do this type of assessment is significant, the benefits of having realtime
scoring and the ability to probe and question a student as they are working gives a better
understanding of the thinking and level of reasoning being employed by the student
(Ginsberg, 1981; Steffe, 2002). As students complete tasks, the assessor will record the
response on the scoring guide seen in Figure 4.4, including but not limited to the use of
figural items, manipulatives, drawings, or computational procedures. The extra space
provided in the middle of each page is intended for assessors note taking.
73
Scoring Guide
Assessment for Multiplicative Reasoning
Grade Level of Student
Composite Unit Iteration Task (CU)
Indicate any procedures that students may use, including but not limited
to the following:
Counts on fingers
Draws or uses marks
Other behaviors (please describe)
Student Response Stage Indicated by this Result
Correct Response to grade level problem without prompts. Anticipatory at grade level for this scheme
Correct Response to grade level problem with prompts. Participatory at grade level for this scheme
Correct Response to below grade level problem without prompts. Anticipatory, but below grade level for this scheme
Correct Response to below grade level problem with prompts. Participatory and below grade level for this scheme
Incorrect Response with prompts. PreParticipatory for this scheme
Figure 4.4. Scoring Guide
74
Countingon task
Indicate any procedures that students may use, including but not
limited to the following:
Counts on fingers
Draws or uses marks
Writes algorithm (Orient the student to avoid using an algorithm
before other methods of computing mentally have been exhausted.)
Other behaviors (please describe)
Student Response Stage Indicated by this Result
Correct Response to grade level problem without prompts. Anticipatory at grade level for this scheme
Correct Response to grade level problem with prompts. Participatory at grade level for this scheme
Correct Response to below grade level problem without prompts. Anticipatory, but below grade level for this scheme
Correct Response to below grade level problem with prompts. Participatory and below grade level for this scheme
Incorrect Response with prompts. PreParticipatory for this scheme
Figure 4.4. Scoring Guidecontd.
Missing Second Addend (MSA)
Indicate any procedures that students may use, including but not limited
to the following:
Counts on fingers
Draws or uses marks
Writes algorithm (Orient the student to avoid using an algorithm
before other methods of computing mentally have been exhausted.)
Other behaviors (please describe)
Student Response Stage Indicated by this Result
Correct Response to grade level problem without prompts. Anticipatory at grade level for this scheme
Correct Response to grade level problem with prompts. Participatory at grade level for this scheme
Correct Response to below grade level problem without prompts. Anticipatory, but below grade level for this scheme
Correct Response to below grade level problem with prompts. Participatory and below grade level for this scheme
Incorrect Response with prompts. PreParticipatory for this scheme
Figure 4.4 Scoring Guide contd
Missing First Addend
Indicate any procedures that students may use, including but not limited
to the following:
Counts on fingers
Draws or uses marks
Writes algorithm (Orient the student to avoid using an algorithm
before other methods of computing mentally have been exhausted.)
Other behaviors (please describe)
Student Response Stage Indicated by this Result
Correct Response to grade level problem without prompts. Anticipatory at grade level for this scheme
Correct Response to grade level problem with prompts. Participatory at grade level for this scheme
Correct Response to below grade level problem without prompts. Anticipatory, but below grade level for this scheme
Correct Response to below grade level problem with prompts. Participatory and below grade level for this scheme
Incorrect Response with prompts. PreParticipatory for this scheme
Figure 4.4. Scoring Guide, confd
Multiplicative Double Counting 1(mDCl)
Indicate any procedures that students may use, including but not limited
to the following:
Counts on fingers
Draws or uses marks
Writes algorithm (Orient the student to avoid using an algorithm
before other methods of computing mentally have been exhausted.)
Other behaviors (please describe)
Student Response Stage Indicated by this Result
Correct Response to grade level problem without prompts. Anticipatory at grade level for this scheme
Correct Response to grade level problem with prompts. Participatory at grade level for this scheme
Correct Response to below grade level problem without prompts. Anticipatory, but below grade level for this scheme
Correct Response to below grade level problem with prompts. Participatory and below grade level for this scheme
Incorrect Response with prompts. PreParticipatory for this scheme
Figure 4.4. Scoring Guide, confd.
78
Multiplicative Double Counting 2 (mDC2)
Indicate any procedures that students may use, including but not limited
to the following:
Counts on fingers
Draws or uses marks
Writes algorithm (Orient the student to avoid using an algorithm
before other methods of computing mentally have been exhausted.)
Other behaviors (please describe)
Student Response Stage Indicated by this Result
Correct Response to grade level problem without prompts. Anticipatory at grade level for this scheme
Correct Response to grade level problem with prompts. Participatory at grade level for this scheme
Correct Response to below grade level problem without prompts. Anticipatory, but below grade level for this scheme
Correct Response to below grade level problem with prompts. Participatory and below grade level for this scheme
Incorrect Response with prompts. PreParticipatory for this scheme
Figure 4.4. Scoring Guide, confd.
79
Multiplicative Double Counting 3 (mDC3)
Indicate any procedures that students may use, including but not limited
to the following:
Counts on fingers
Draws or uses marks
Writes algorithm (Orient the student to avoid using an algorithm
before other methods of computing mentally have been exhausted.)
Other behaviors (please describe)
Student Response Stage Indicated by this Result
Correct Response to grade level problem without prompts. Anticipatory at grade level for this scheme
Correct Response to grade level problem with prompts. Participatory at grade level for this scheme
Correct Response to below grade level problem without prompts. Anticipatory, but below grade level for this scheme
Correct Response to below grade level problem with prompts. Participatory and below grade level for this scheme
Incorrect Response with prompts. PreParticipatory for this scheme
Figure 4.4. Scoring Guide.contd.
80
Using the Scoring guide should allow assessors to determine at which stage,
participatory or anticipatory, students seem to operate in each scheme tested. It may also
help to guide prompts during the interview process, as it becomes clear the evidence
needed to prove competence at each level.
Reliability and Validity
The AMR is workinprogress in its preliminary stages. It has not yet been fully
tested for reliability or validity; however, attempts made to increase these two measures
of the instalment are discussed here. The validity measures taken in the development of
the AMR were based upon content validity according to expert opinion, both classroom
teachers and researchers. The items were developed under the advisement of Dr. Ron
Tzur of the University of Colorado, Denver. His guidance focused on ensuring that the
tasks were created to reflect the theoretical stances described in Chapter 2. Using
information about the numbers that would be most appropriate at each grade band, and
tying the tasks closely to the six Schemes of Multiplicative Reasoning discussed earlier,
the items are believed to be good indicators of the reasoning at each level. As explained
in Chapter III, these items were refined both by feedback from teachers and through
informal testing with fourthgrade students. I was able to observe students as they
participated in the assessment, and also received feedback from the reports of Dr. Tzur
and Mr. McClintock on later administrations of the tasks I was unable to witness.
Additionally, I solicited critique by way of a Feedback Form from classroom teachers, as
discussed in context in Chapter IV. The Form used to collect this data is found in the
Appendix to this thesis.
81
Another reliability measure is the twofold problem delivery of verbal directions
and printed student problem strips shown earlier in this section. These seemingly
redundant steps are to remove as many barriers to the students5 interpretation of the
problem as possible. Students who have difficulty interpreting verbal questions have
access to the printed problem, while those who have difficulty with reading have the
verbal instructions read by the assessor. The ability to refer to the card and clarify
numbers or any other aspect of the problem, as well as review any information, was
deliberately included to ensure that incorrect solutions were not due to misunderstandings
or difficulty remembering the information given in the tasks.
The choice of numbers for each problem type and each grade level was
purposeful, as supported by the work of Sherin and Fuson (2005). The strategy used by
students in that study varied depending on the numbers and context used. It showed that
the choice of each number used in a task must be made mindfully to elicit the strategy for
which it is designed. If the numbers and situations for these problems were not chosen
carefully, students could use some learned strategies, or known facts, that do not
necessarily reflect their understanding of composite unit, such as how to count on. Sherin
and Fuson also note that the only route to this clear, complex, flexible knowledge is the
passage through earlier stages, through which they develop strategies rooted in
understanding. Clearly, an understanding of the problem is an important facet, and was
treated as such using these methods, to provide a reliable assessment.
Because the focus of this thesis was on multiplicative reasoning, there are three
tasks to add validity to this portion of the assessment. If a student is able to correctly and
independently solve two tasks in a row, testing is terminated, and the student is
82
considered to have access to the scheme for Multiplicative Double Counting. If,
however, the student solves only one out of the first two tasks independently and
correctly, the third task is administered to determine whether the student can be
considered to have this scheme, or at which stage. A score of two out of three correct
would indicate the students readiness to move to the next level of multiplicative
reasoning.
These steps were taken to further increase reliability within the given time and
resources available, and could certainly help to improve the AMR further in future
endeavors. The following section includes a discussion of the implications and benefits
of the development of the AMR, as well as a further discussion of the limitations
indicated here.
83
CHAPTER V
DISCUSSION
Contributions to Research and Practice
This section will focus on the ways that the work seen in Chapter IV can benefit
the field of mathematics education in terms of research and classroom teaching. The
ability to test for and determine not only the multiplicative schemes available for students
but also the anticipatory or participatory stages within each scheme could be a great
benefit to teachers and researchers alike. In both the classroom and the field of research,
there is a need to find out what students understand about multiplicative reasoning, and
whether they have constructed a scheme at the participatory or anticipatory stage.
A classroom teacher could use this instalment to determine what students need to
be able to move ahead, into multiplicative reasoning, or whether they are solid in their
initial stage of multiplicative reasoning, and are ready to move on to more advanced
topics. For example, a student could be presented with the Missing Second Addend task
for the 34 gradeband. It reads: Joseph brings eight (8) pieces of gum to school. His
friend gives him some more pieces of gum. When Joseph counts them all, he finds that
he has fifteen (15) pieces of gum. How many pieces of gum did his friend give him?
The students response allows a teacher to distinguish whether a students is at the
anticipatory or participatory stage of using composite units within a given, encompassing
composite unit. If at the participatory, (i.e., can only solve the problem with prompting),
teaching would then focus on moving the student toward an anticipatory stage, using her
current knowledge and making prompting explicit so that she will begin to selfprompt.
However, if the student is at the anticipatory stage (can solve independent of any
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prompting while using strong reasoning), teaching can move forward to missing second
addend tasks, by using the missing first addend concepts as a bridge to these missing
second addend concepts. It may also indicate moving ahead to teaching mDC while
keeping in mind to focus on the students way of reasoning with Is vs. with composite
units.
In addition, the assessment should be flexible enough to include a wide range of
learners. An advanced first grader could be assessed for compaction of curriculum or
advancement to a more challenging math group based on the AMR results. A struggling
seventh grader might be able to receive needed intervention based on results that show he
or she is not able to perform Multiplicative Double Counting. A fifth grade student who
does have the availability of the appropriate scheme, but does not compute fluently with
whole numbers, could test at a lower grade band and show this reasoning in spite of their
computational difficulties.
I see this awareness as essential for teachers, because the connection between
multiplicative reasoning and topics such as fractions is not always a focus of instruction
in those topics, though it might be a way to improve it. Fractions are multiplicative
quantities because they require a reference to the unit. When fractions are an extension
of multiplication, the idea that there are multiple ratios to consider in multiplication
(Confrey and Smith, 1995) can help create a bridge from the whole number to the
fractional concepts. Some of the ratios indicated in 5 x 7 are each composite unit of five
to seven ones, as the relation of the seven ones in one group and the five groups to the
thirtyfive ones in total. This understanding of multiplication as multiple ratios builds the
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foundation for students to understand fractional quantities in a way that repeated addition
does not.
When the meaning of five groups of seven is conceptualized by the student, the
understanding that 2/3 means that there are some units, of which the whole is three times
as muchand that we are considering two of these unitsas 2 1/3 or 2 groups of 1/3
can equip a student to compare relative values of these fractions. This thinking also
allows a student to understand why in a problem such as 2/3 + 4/5, we must use a
common measure for operating on both thirds and fifths. When students recognize thirds
and fifths as different units, and see this as a way to build upon their knowledge of
multiplication to iterate those units, the search for a common measure ('denominator5) is
no longer some mysterious invention of teachers of mathematics, but a necessary
operation of multiplicative unit coordination needed to find common units. Lamon
(1996) saw a similar conception being built in students who use equal sharing tasks to
develop their fraction concepts. When students began to see certain quantities as
equivalent, they were using multiplicative rather than additive concepts to understand the
size of the portion. Being able to see and assess this reasoning could go a long way for
classroom teachers seeking to improve the teaching and learning of fractions and/or
multiplicative reasoning in their classrooms.
In research the AMR may be used for a similar purpose, or to find correlations
between multiplicative reasoning and other types of mathematical conceptions. For
example, it could be investigated whether a student who has a strong knowledge of area
and perimeter was more equipped with multiplicative reasoning than one who does not.
The AMR could be used to quickly determine the level of a student and then use it to
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correlate with other factors. There are some limitations to this instrumenthoweverand
they are discussed next.
Limitations
While the AMR provides benefits for teachers and researchers, it is important to
be aware of its limitations. It is understood that the time involved in completing an
assessment of this type is limiting for most classroom situations. One way to decrease
the amount of time needed for testing is to administer, as a written test, the portion that
the teacher believes the student has already constructed as a solid conception. In this way
the AMR5s tasks function as a preliminary screener. For example, if a third grade teacher
believes students are proficient in missing second addend problems, the counting on and
missing second addend section could be given to students simply as a written test. The
students who show clearly that they are anticipatory for these tasks can begin with the
missing first addend for the interview. Those who did not show that they are anticipatory
could then be tested further, by interview, to determine if they may be participatory for
that stage or do not yet have the conception at all. As explained in Chapter II, the
distinction between anticipatory and participatory stages requires the task to be first given
without prompts, to determine whether the students can anticipate the entire process, and
if not, with prompts to assess for a participatory level of understanding.
The AMR is a prototype, and only addresses the first scheme in the multiplicative
reasoning continuum delineated by Tzur et al.(2013)namelyMultiplicative Double
Counting. This was done for simplicity, and to allow testing and feedback to determine
the foci of later instalments. The remaining stages of multiplicative reasoning could be
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created in similar ways, and would show where more advanced students fall on the
continuum of multiplicative reasoning.
Instead of progressing from mDC to more advanced multiplicative schemes, the
AMR instalment includes schemes that are prerequisite to multiplicative reasoning, such
as counting on. These were included to determine the extent to which students have
constructed composite units needed for their journey from additive to multiplicative
reasoning, or are still developing the understandings on which additive reasoning should
be built (Steffe, 1994). In a study conducted with 5th grade students designated by their
school system as learning disabled, who did not reason multiplicatively (Tzur, 2010), it
was found that one of the major impediments to their progress was the lack of a concept
of number as an abstract, symbolic, composite unit.
The AMR is meant to begin with these prerequisite skills to save teachers time
when doing assessments. Clearly the additive portion of the test is not comprehensive
enough to account for all aspects of additive reasoning and all of the different types of
problem structures and what they may reveal about student thinking. Ratherthe additive
portion was designed to give a glimpse into the schemes available to students, and
whether they have the conceptual prerequisites needed to engage in and learn to reason
multiplicatively. With this in mind, the assessment only covers counting on, missing
second addend problems with the missing portion being smaller, and a missing first
addend with the missing potion being smaller. It is understood that this spans a limited
range in the additive abilities of students, but does so while providing essential
information about the students use and coordination of units. An assessor who finds that
students are in this general stage of reasoning and would like to further pinpoint their
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development, would need to consult a more comprehensive additive reasoning
assessment.
It is also important to note that this instrument has never been tested in its
entirety. Significant changes were made following the initial testing, up to and including
the defense phase. The following section suggests possibilities for future research, some
of which address the aforementioned limitations.
Implications for Future Research
A continuation of the assessment to include the remaining items in the framework
for multiplicative reasoning, outlined by Tzur et al.(2013), would be helpful to teachers
who need to be able to assess all schemes, and stages within them, up to the point in
which they may be ready to learn fractions. In particular, items could be created for the
remaining schemes in this framework: Same Unit Coordination (SUC), Unit
Differentiation and Selection (UDS), Mixed Unit Coordination (MUC), Quotitive
Division (QD), and Partitive Division (PD), so the assessment is expanded for the entire
scope of these schemes. When a student has progressed through all of these schemes,
they may be considered proficient in multiplicative reasoning.
Additionally, the prototype of the AMR was written such that a computerbased
assessment could be built to either complete the preliminary screening questions, used to
decide where a student might need to begin testing, or as a tool to help the assessor. A
software program that could record responses of students, coded by the assessor, and
indicate the next task and level that should be administered, could be a valuable tool. For
example, if the student answers a question correctly with strong reasoning, the
technological component could advance the student to the next appropriate question. It
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seems unlikely that the assessment could be productively administered without the
human interview component, due to the importance of the interaction between the
assessor and student, and the careful attention to student thinking gleaned from bodily
gestures and whispers, which may not be detected, let alone understood, by existing
technology.
Additionally, teachers may want to know how to meet student needs once their
reasoning, both scheme and stage, has been assessed. This is an excellent opportunity for
future work, and would greatly assist teachers in using the data obtained to foster the
understandings that they now know students need to prepare them for future work.
Concluding Remarks
This project was born out of my own frustrationas a middle school teacherwith
the low level of knowledge that my students often have about fractional quantities. Too
often I saw that though students gained procedural knowledge with repetition and
practice, their conceptual understanding remained weak, despite activities that I designed
to try to build these conceptions. I truly believe that for teachers and other educators to
first and foremost become aware of the gaps in reasoning that cause students in middle
school, high school and even higher education to not only dread, but show a limited
understanding of the meaning of fraction, is key. A teachers possession of both this
awareness and the knowledge about schemes used for reasoning multiplicatively, and of
anticipatory and participatory stages within each scheme, could go a long way towards
improving the teaching and learning of fraction and ratio concepts. Further, I am glad as
an educator to have an instalment that allows me to begin assessing the underlying
multiplicative reasoning that my students may or may not have. I foresee more
90
productive lessons when students are equipped with the necessary conceptual
prerequisites with which to construct fraction knowledgeas opposed to asking them to
operate on conceptions that they are not able to access due to incomplete multiplicative
schemes.
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Full Text 
PAGE 1
! ASSESSMENT FOR STUDENTS' CONCEPTUAL READINESS AND MULTIPLICATIVE REASONING by ERICKA J. LEGNARD B.S. Colorado State University, 1992 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science in Education in Mathematics Education Mathematics Education 2013
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! "" This thesis for the Master of Science in Education in Math Education degree by Ericka J. Legnard has been approved for the Ma t hematics Education Department by Ron Tzur, Advisor Heather Johnson 7/2/2013
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! "" Legnard, Ericka J. (M.S.Ed., Mathematics Education) Assessment f or Students' Conceptual Readiness a nd Multiplicative Reasoning Thesis directed by Professor Ron Tzur. ABSTRACT This research thesis addressed the problem of how teachers can assess students' readiness to begin learning and progress through multiplicative reasoning. Addressing this problem is important, particularly because this type of reasoning serves as a conceptual foundation for higher level mathematical topics, such as fractions, ratios, proportions, and algebra. When students and teachers struggle with these higher level topics, assessing conceptual prerequisites play s a key role in identifying and eli minating the root causes of difficulty. This constructivist premise, of the need to assess students' current knowledge as a basis for implementing instruction adaptive to the students, is considered essential for effecti ve teaching and underlies this thesis study. Accordingly, a first aim of this study was to determine the reasoning students need, both additive and multiplicative, to learn the aforementioned highe r level concepts. A second aim was to create an assessmen t instrument for teachers and researchers that would provide essential information about students' preparedness to engage in further, meaningful study of mathematics. The researcher used studies on students' thinking and learning to create assessment item s that would either bring to light their available reasoning or reveal its absence. Additionally, a theory that describes the stages of student understanding in developing multiplicative reasoning was used to better place concepts in a framework that depi cts current knowledge, and what is needed before instruction can fruitfully move ahead. The assessment items were presented in a format and in situations that should be
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! """ accessible to elementary and middle grades students and teachers. An instrument cons isting of these items was then developed to serve assessors of students' multiplicative reasoning. This instrument includes differentiating items to fit with various grade levels and to determine the extent to which students can solve these tasks independ ently or with assistive prompting. The form and content of this abstract are approved. I recommend its publication. Approved: Ron Tzur
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! "# TABLE OF CONTENTS CHAPTER I. INTRODUCTION ..1 II. REVIE W OF LITERATURE 5 Constructivist Theory of Learning ....5 Construction of Schemes. ...6 Fractions as Problematic for Students and Teachers.9 Practices for Developing Multiplicative Understanding of Fraction s ... ..10 Multiplicative Reasoning.1 6 Schemes for Reasoning Multiplicatively ...1 9 Activity Effect Relationsh ip......... 20 Anticipatory and Particip atory Stages..2 3 Fine Grained A ssessment 2 6 Previous Research on Assessment of M ultiplicative Reasoning.27 III. METHO DS 30 Development of Task s......... 3 1 Composite Unit Iteration Task.3 2 Additive Tasks. 3 3 Multiplicative Tasks.3 7 Task Refinement..3 8 Construction of Prompts .....3 9 Selection of Administra tion Method....4 3 Development of Administration Guide ...4 6
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! # Scoring Considerations ........4 6 IV. RESULTS ...4 8 Tasks 4 8 Administration Protocol ...5 3 Administration Guide... 62 Scoring Gui de...7 4 Reliability and Validity 82 V. DISCUSSION.. ... 8 4 Contributions to Research and Practice ....8 4 Limitations ........8 7 Implications for F urthe r Research ..... 89 Concluding R e marks.. 90 REFERENCES ..9 2 APPENDIX 9 7 A. Feedback Form for Assessment...9 7
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! 1 CHAPTER I INTRODUCTION This research thesis addressed the problem of how students' conceptual readiness for and progression through reasoning multiplicatively can be assessed. The research problem grew out of an interest in the issue of student conceptual preparedness for learning fractions. The teaching and learning of concepts such as fraction and ratio seem quite ineffective in man y American school settings (Fan and Zhou, 2006). To learn fractions, many researchers believe that students need to have previously constructed ways of reasoning multiplicatively (Confrey and Smith, 1995). It may be that some students are not prepared wit h the necessary foundation, namely multiplicative reasoning, for instruction in more advanced concepts, such as fractions, at the time that the instruction is delivered. Tzur 's (2004) work indicated that one of the crucial steps a teacher can take to best serve his/her students is to assess the underlying reasoning necessary for students to build a concept before attempting to teach that concept. Assessment of students' prior knowledge is considered important before instruction in any case, therefore, if we consider multiplicative reasoning as an important basis for fraction, assessing students' conceptual preparedness for multiplicative reasoning seems critical for enabling their progress through the study of fractions Weak foundational skills can hinder a student for years to come (Confrey and Smith, 1995 ) In particular a lack of sufficient prerequisite knowledge may cause students to learn the rules and procedures for concepts, such as fraction operations without being able to take in the fundamenta ls of fraction concepts. This may allow the student to be somewhat successful in the short term, but as time goes on there can be
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! 2 frustration in trying to understand the concepts that come later. The need for a rich, thorough learning of fractions is cle ar when we consider the wealth of future math ematical topics that s tudents will encounter that require an understanding of multiplicative reasoning and fractions (Thompson and Saldanha, 2003). Ratio and proportion follow soon after fraction instruction, but the concepts of similarity, slope, exponential and logarithmic functions, just to name a few, also draw on multiplicative schemes. Confrey and Smith (1995) trace d such scheme s e xtending through c alculus and beyond, and note d that they should be based on multiplicative concepts instead of additive ones. Teachers often lament that introducing an algebra problem including fractions seems to dissuade many students, even if using algebra with whole numbers seems solid for those s tudents This indicates that there may be a problem in the way that students learn the foundational knowledge for fractions, and that there is an urgent need to correct this problem. Students who are able to efficiently and independently solve problems involving multiplicative situations may be more able to create conceptions that support the learning of future concepts, and those who do not can find these future topics increasingly difficult to understand (Confrey and Smith, 1995). It seems an importa nt issue that students are prepared with a solid conception of multiplicative reasoning, leading to a solid conception of fraction, and provided a clear path to future studies such as algebra In the context of a regular classroom experience, it is diffic ult for educators to correctly determine the reasoning used as student s solve problems There are factors, discussed in this section, which make this especially true when the assessment is taking place mainly with pencil and paper tests or when it is done informally through brief
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! 3 conversations with individuals or groups of students. This is not to say that these means of formative assessment are not valid Rather that when teachers need to know that student s are prepared to undertake what seems to be such important new learning, there is a need to get an authentic detailed picture of student s' readiness. There are many factors that make the assessment of reasoning difficult in the classroom These includ e the time consuming need for careful selection of the problems to be used, unintentional prompting within the problem or by the teacher or other students, and the ways that students can demonstrate what seems to be evidence that they understand, when in fact they may not. The abilit y to remove as many of these factors as possible and to assess a student in a way that is likely to give a clear picture of each student's capabilities is important. As a middle school teacher, I have come to recognize that t eachers need to know the import ance of this multiplicative foundation, how to detect it, and ways to support students at their current level. A user friendly assessment of multiplicative reasoning, which gives specific information about what students do and do not understand, would be a tool to help teachers pinpoint where their students may need more experience before fraction instruction begins It is proposed in this thesis that having these assessment results is important for teachers and students. This is not a simple matter, howe ver, because students who appear to have multiplicative reasoning may not have it. A reliable assessment that help s clarify the depth of student s' understanding could be valuable to ensure their conceptual preparedness. The instrument developed as part of this thesis study, the Assessment for Multiplicative Reasoning (AMR) was designed to allow assessors (e.g., teachers) to
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! 4 determine not only the student's level of multiplicative reasoning, but also whether a student is able to reason independently or with support. The results of the AMR allow a teacher to determine not only students' preparedness for fractions but also whether what is needed is to lay the groundwork for a multiplicative scheme, building on the previous s cheme, or help to progress to the next stage within a new scheme This is something that as a teacher I will certainly incorporate into my practice. The knowledge and tools that I have gained in this study, both the awareness of multiplicative reasoning as an essential part of math education, as well as a way to assess this knowledge, will serve my students well in the future. I feel better prepared to address the issues that too often arise when I begin the teaching of fractions. I intend to use this a ssessment to determine the readiness of students according to the schemes that they can access as well as their level of understanding within the scheme, to better target instruction for my students A discussion of the meaning of scheme in this context will follow in the Review of Literature section in Chapter II The next section depicts the conceptual framework that guided this research, because some of the constructs from this framework are needed to better delineate the problem addressed in this s tudy.
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! 5 CHAPTER II REVIEW OF LITERATURE In this chapter, I situate the research problem depicted in the previous chapter within the larger body of research literature. I begin by summarizing the constructivist theory of learning that underlies all other ideas in this work. Next, I discuss the prevailing theories of best practice in fraction instruction. A discussion of the facets of multiplicative reasoning and why it is important for fraction concepts follows, to link the best practices to what we know about multiplicative understanding. Then I will detail the learning theory that I suggest using to support students as these practices are employed, as well as an explanation of the participatory (prompt dependent) and anticipatory (prompt less) stages of understanding that students would arrive at using this theory. Finally, I will explore methods of assessment that can be used to determine a student's knowledge, as well as the previous work on assessments specific to multiplicative reasoning. Construc tivist Theory of Learning This thesis used a constructivist theory of learning as a guiding framework, because the focus of the assessment is on cognitive aspects of students' learning, namely, reasoning multiplicatively. A constructivist theory asserts t hat students must construct their own knowledge, and that it cannot be transferred to them through some other means (Dewey, 1938 ; Piaget, 1985; von Glasersfeld, 1995 ). With this idea in place, it seems that to guide students in the learning of new concept s, such as fraction, ratio, and the like, teachers need to be aware of activities that help students make their own meaning and understandings, and allow students to do the work necessary to create this new
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! 6 knowledge. In constructivism, the student's thin king is a key goal for any lesson and effective lessons and teachers will help the student to do the thinking required to learn. Piaget (1985) the originator of constructivism, argued that anything that is to be understood should be rooted in practical e xperience. Von Glasersfeld (1 99 5) interpreted this belief put forth by Piaget, Radical Con s tructivism, as "knowledge being built up by the cognizing subject" (p. 51) If teachers are not able to relate multiplication, or fractions or ratios to something that students already understand, and help them see how to operate on fractional quantities using something tangible or figural that is well understood, then it is unlikely that students will be able to create meaningful understandings of fra ctions, let alone apply any of their understandings to other situations. A further development of these constructivist principles follows in the next section about the development of schem es a key building block of logico mathematical thinking in the con structivist theory. Construction of Schemes One of the foundations in the constructivist theory of learning is the concept of scheme (Piaget, 1985) A scheme is thought of as a miniature framework in the mind, which allows thoughts, experiences and kno wledge to be organized and connected in a way that makes sense to the learner. Von Glasersfeld (1995) described the action on schemes as a three part system, comprised of a perceived situation, an activity, and a result. The perceived situation is the lea rner's reality in the moment, which may be something that they notice on his or her own, or their view of a situation created by a teacher to inspire learning. The activity is something that is done to an object in that situation, which may involve physic al manipulation, such as pouring of liquid in a
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! 7 geometric model, or a mental activity completed on an available scheme, such as considering the effect of adding two negative integers when the addition of two positive integers is understood. The result is what happens after the activity is completed, or what the learner notices about these effects, and how they relate to the object or scheme. This three part system describes the way that assimilation and accommodation can take place in Piaget's constructiv ist theory. S chemes begin as simple things, such as the scheme for what a table might be, and then move on to be much more complex, such as a scheme for things that are related proportionally. They can i nvolve the concrete or the abstract, but the fundamental idea regarding schemes is that they are necessary for new learning to take place. New information, such as the relative size of fractional quantities, must be taken in to the consciousness of the student in one of two ways. The processes of assimilation and acco mmodation, described below, take place to allow students to learn via transforming their existing schemes into novel ones (Piaget, 1985). The student assimilate s new information into an existing scheme, meanin g that it relates so naturally for the student that they incorporate it into what they already know. For instance, if students already know the concept of unit fractions such as 1/5, 1/8 and 1/6, and the unit fraction is well understood, when they encount er a new unit fraction, such as 1/7, this should be a simple assimilation. The new idea of 1/7 can be incorporated right into the present scheme because it fits nicely with the "rules" that the student has constructed to understand unit fractions. If, h owever, the student does not have a scheme for unit fractions, they will need to accommodate, or make sufficient changes in an existing scheme such as how a whole
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! 8 number is composed by iterating the unit of 1, so that they can develop an understanding of the unit fraction 1/7 (Piaget, 1985) Accommodation is a much more complex and laborious task, what we usually call learning, and requires the student to alter the "rules" which currently govern their scheme to allow the understanding of the new informati on, or to create a new scheme that will house the new information. This accommodation, or linking of the known to t he unknown, is what is referred to in the future sections describing the ways that multiplicative reasoning can be used as the scheme which will be accommodated for a student to understand fractions. It was further conjectured by Piaget (1985) that student s relate new information to what they already know, even if what they know is incomplete. If not properly guided, this connection may be tenuous and without any mathematical meaning, causing the new learning to be unstable or misdirected. Teachers see this phenomenon when students express misconceptions or convoluted reasoning in arriving at a solution that seems to the teacher to be inexp licable. Presumably, the student made some connections in past work, which functioned for them at the time in some way, but perhaps only coincidentally or tangentially. This may happen if the underlying scheme is not properly developed, such as may be th e case with multiplicative reasoning and future concepts. In the context of this thesis the development of a scheme as an accommodation of previous schemes is central, because it builds the case for educators to be mindful of the schemes that student have available through some sort of assessment, and how we expect them to elaborate those schemes to include new knowledge. This understanding helps to guide the later se ctions, to provide a foundation for the need that drove the creation of this assessment, and explore the theories with which it was constructed. Giving students
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! 9 a foundation in multiplicative reasoning, and creating experiences for them that will show fra ctional quantities as multiplicative relations may be a solid path towards real understanding of concepts such as fractions, ratios, rates and proportions. I suggest in the following sections that one of the problems that teachers face in teaching fractio ns is that the multiplicative schem es needed to proceed are not fully in place, and cause students to have great difficulty in learning. Evidence for that difficulty is presented next. Fractions as Problematic for Students and Teachers As students are at tempting to assimilate and accommodate their schemes throughout their mathematics education, some seem to come more easily than others. The conception of fraction is often difficult, as the studies in this section will show. The familiar International Mat hematics Studies, including SIMS in 1976 and TIMSS beginning in 1995, as well as the International Assessments of Educational Progress (IAEP1 and 2) showed US students lacking in mathematics in general, and fractions, ratios and multiplicative reasoning, a prominent part of all of these assessments, were no exception (Fan and Zhou, 2006). The National Mathematics Advisory Panel stated, "The most important foundational skill not presently developed appears to be proficiency with fractions (2008, p.18) It seems that this is an area where American teachers and students need assistance and continued focus to create improvement. Fractions are a topic often dreaded by students and their parents, and cause teachers from middle school through high school difficulty and frustration when students who should have learned the concepts long ago struggle with using them to solve problems. Students have been seen in studies to avoid operations such as multiplying by
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! 10 fractions, even when this would be the most efficient strategy for solving a problem, and they have had experience with these methods before (Fischbein et al. 1985). This is presumably because they fel t uncomfortable with the meaning of such an operation and often prefer red to cling to more familiar, additive methods. This may be due in part to the evidence that multiplicative quantities are not intuitive in the way that additive ones are, and that students will generally not develop these understandings without ca reful and explicit instruction ( Dehaene, 1997; Fischbein et al. 1985 ; Steffe, 2002). It has been seen in multiple studies that in order to come to a clear understanding, students should be assisted in their learning process, usually in school by their teachers and through social interactions with others, in order to develop the schemes necessary ( Dewey, 1938) While there is no single universal theory for what should be done to bolster the fraction proficiency of US students, there are many theories that intend to guide educators in this endeavor. Many researchers have examined the teaching and learning of fractions as they attempted to unravel its complexities. Some of the main theories are presented next in order to understand which among them seems to be the most promising, and what prerequisite knowledge is required in each case. Practices for Developing Multiplicative Understanding of Fraction s If fraction education is problematic, it makes sense to explore the prevailing research about how the teaching and learning of fractions might be approached and understood. The theories presented here are highlighted to show the rationale for t he type of reasoning that I tried to assess using the AMR. There are many options for introducing fractional concepts to students, including, but not limited to, part of a whole fractions as division operations, whole number bias and the reorganization hypothesis
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! 11 ( Siegler et al. 2011; Stafylidou and Vosniadou 2004 ; Steffe, 2002 ; Thompson and Saldanha, 2003 ; ) As a teacher, I found exploring these theories indispensable for understand ing the different ways of thinking about how to teach fractions and to determine which theory seemed most likely to produce effective learning. The sections that follow explore what types of schemes students need to develop during the study of fraction s and then to consider the underlying schemes that would need to suppo rt that study From this understanding, the assessment developed to focus on these underlying schemes. Part of a Whole Model A prevalent method used to help students think about fractions in American schools is the part of a whole model. In this model, the student sees a whole divided into a given number of equal size parts, and consider the fraction as one part or several parts, of that greater whole. Th is part of a whole conception breaks down, however, when students are confronted by a quantity that does not fit this model, such as 8/5, because 8 cannot be a part of five (Thompson and Saldanha, 2003). When students are asked to operate on these often mysterious quantities, they may be forced to rely on taught or assumed rules, which may have little grounding in mathematical theory This process is understandably confusing, and may contribute to student thinking that math is a set of arbitrary rules to be followed, regardless of intuition or prior knowledge (Erlwanger, 19 73) The true understanding of mathematical concepts is a major purpose of valuable mathematics education (NCTM, 2000) and when students are not able to make sufficient meaning, not only does their understanding break down, but their enthusiasm for the d iscipline as a whole may break down as well. It is considered crucial in this thesis that
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! 12 students understand fractions as multiplicative quantities instead of as part of a whole For example, they need to think of a unit fraction, say 1/6, as a unit suc h that the whole is 6 times as much Fractions as a Division Operation A related understanding of fractions as multiplicative constructs in place of or in addition to the part of whole model, is to see them as the effects of a division operation. In such a model 5/8 means five divided by eight, which is a valid way to consider this notation in certain circumstances, and which will also yield the decimal form of this quantity However, such a notion can be problematic when one encounters for example, 5/ (8/3), because the operation to be completed is not clear (Thompson and Saldanha, 2003). This way of introducing fractions may also be dangerous to student understanding because it does not provide a way to think about the value of the quantity presented, nor a way to compare and order fractions without completing the operation and comparing the decimal. This lack of understanding of the magnitude of fractional quantities can create difficulty for estimation and using intuitive means to determine whether the solution to a problem involving a fraction operation is reasonable or not. On the other hand, 5/8 could be thought of as a unit that is 5 times as much of another (1/8), where the whole is 8 times greater than 1/8, as explained for unit fractions. Such a conception (e.g., 5 units of 1/8) supports understanding of fractions in a way that does not seem possible with the division operation model alone Linking Additive and Multiplicative Situations Another theory, proposed by the Rational Number Project ( Behr, et al., 1983; Cramer et al. 199 7 ), is that the comparison of additive a nd multiplicative situations be
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! 13 made explicit so that students see examples of each side by side. The theory supposes that the students are then better able to use this knowledge to understand that fractions are multiplicative quantities and can be trea ted as such. This avoids the common lack of understanding by students, for example, that in making an equivalent fraction you may just as easily add or subtract the same quantity to or from the numerator as multiply or divide by the same quantity. This t ype of misconception may show that the student does not see the fraction as a quantity involving a multiplicative relation One theory related to this stance is whole number bias, or whole number interference ( Stafylidou and Vosniadou 2004) In this approach it is argued that when students apply what they already understand about whole numbers to understand fractions, confusion occurs, and that we should not try and connect the two for students. For example, students may mistakenly believe that divis ion by a fraction smaller than 1 results in a quantity smaller than the dividend, similarly to the result of division by a whole number. While drawing upon prior knowledge is the only way for students to create new knowledge in the constructivist view, it does present problems if students make associations that are not helpful in understanding the concept. Stafylidou and Vosniadou (2004) and Streefland (1991) state d that the conceptions students have about natural numbers create problems for students when they encounter fractions, because they generally draw incorrect conclusions, or make connections between ideas that are not mathematically sound. For example, students sometimes reason that as the numbers in a fraction get larger, the value of the fracti on become larger as well (e.g., 9/10 vs. 90/100 respectively ) regardless of the invariant, multiplicative relation between numerator and denominator. The study conducted by Stafylidou and Vosniadou (2004) asked students
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! 14 to compare fractions and give a rationale for their conclusions. The responses were categorized according to their conceptions. The conception called Relation between Numerator/Denominator was the only one in which students showed a compr ehensive understanding and s trong conception for all types of fraction relationships, including improper fractions. Interestingly, even though this theory contrast ed with the one that follows, the importance of multiplicative reasoning prevail ed in both c ases. The Reorganization Hypothesis A n alternative theory to the whole number bias, and that supports the development of fraction from whole number multiplication, is called the reorganization hypothesis (Steffe, 2002) It states that the number system that children learn in early grades, should be expanded or reorganized to include all rational numbers instead of separating whole number and fractional quantities for students (Siegler et al. 2011 ; Steffe, 2002). This theory finds value in the struggle to integrate fractions and whole numbers as members of a continuous number system, and to create a comprehensive mental number line through which students not only gain a strong understanding of the values of rational numbers, but also a richer understand ing of whole numbers. This marriage of whole number and fraction is also supported by recent neurological research which indicates that the areas of the brain in which fraction magnitude decisions are made occurs in the same area of the brain as whole n umber processing, and that absolute magnitude and relative magnitude are processed with "the same analog code" in the brain (Jacob and Nieder, 2009, p. 4656). Steffe's reorganization model (2002) asks students to equipartition to find the unit fraction, a nd then iterate it to create the desired fraction. As Tzur (2004) emphasized, the
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! 15 key in such an approach is building on the child 's activity part of a scheme (here, iteration of unit, which underlies the creation of whole numbers) instead of on the child 's abstracted construct of number per se. Another aspect of Steffe's theory is to use splitting, which asks students to coordinate both partitioning and iterating as inverse operations. This is context independent and applicable to many other situations, including improper fractions and underscores the multiplicative relationship by focusing on the relation between the unit rate and the number of units. Ni and Zhou (2005) also advocate a merging of teaching whole number and fraction concepts concurrently to emphasize and link the known whole number concepts with the newer fraction concepts. This can allow students to accommodate their whole number schemes effectively to understand rational numbers. The basis for all of these strategies is multiplicati ve in nature, and calls for students to have a strong background in the ways quantities can be related multiplicatively in order to make sense of the tasks presented. It is considered for the purpose of this thesis, and the associated instrument that reo rganization is preferable as a method for guiding students in their understanding of fraction This thinking, guided the development of the assessmen t, discerning between schemes available to students that can be reorganized further toward intended mathem atical concepts It is important to note the conclusions of researchers regarding the most effective ways for students to learn fractions because many students, as well as adults, assume that multiplication is simply repeated addition Such a conception does not see m to support fraction understanding as effectively as a more robust conce p tion of multiplicative relationships The next section focuses on ways of reasoning multiplicative ly that were
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! 16 seen to be paramount in the theor y of reorganization and how such reasoning may support students in their future learning. Multiplicative Reasoning I n the previous section, I have provided a rationale for multiplicative reasoning as the conceptual basis for strong fraction understanding In this section I provide more detail s about what that reasoning en tails. The understanding of multiplicative reasoning for whole numbers that is used in this paper was summarized by Steffe (1992) in the following statement; "For a situation to be established as multiplic ative, it is always necessary at least to coordinate two composite units in such a way that one of the composite units is distributed over the elements of the other." To make sense of this foundational assertion, one must understand the idea of number as a n abstract, symbolized, composite unit A composite unit is a n abstract construct, or scheme, that a person uses to symbolize potential (but not actual) results of counting such as a numeral or number word, which means for him or her, a collection of u nits of one s When a student says "4", and is using it to refer to a unit of four ones considered as a group, the student understands composite unit. In earlier stages of development, a student may recite the word "four" in rote counting, or know that it goes with t he numeral 4, but she must develop the idea that the number four is a group of four ones which can be assembled and re assembled (Olive, 2001). For example, understanding that 5x7 is not simply 5 repeated 7 times, but five unit s, each composed of seven un its of 1, or seven units of five units of 1, and why they yield the same number of 1s in total, is necessary for students to understand multiplication.
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! 17 A rep eated addition model misses the coordinated units idea of multiplicative reasoning S everal of t he core ideas that help students understand multiplication in a flexible, complete manner, such as the importance of units as a point of reference are not supported by repeated addition In a repeated addition model for multiplication, we can consider f or example, three pies made of six apples each which total eighteen apples (or, in repetition, 6 apples + 6 apples + 6 apples =18 apples) In a multiplicative model, we coordinate the one composite unit of six apples per pie (unit rate) with three pies a s the number of composite units which when distributed as explained above produce eighteen apples as the total of 1s coordinating two different units to yield a third unit) In the multiplicative situation, one must keep track of whether the number refers to apples, apples per pie or pies themselves, and distribute the six apples across each of three pies to find the result When thinking of this situation as repeated addi tion, this coordination can be completely missed by students The apparent weakness of repeated addition is confirmed by Ni and Zhou (2005) who state d that not only does repeated addition prove a weak and limited view of multiplication, but that this rea soning is specifically harmful to students in learning fractions. One issue is that seeing 5x7 as five groups of seven is conceptual, which is essential for building schemes and understandings, while seeing 5x7 as seven added five times is simply a comman d to act. When seen as a command an expression such as 5 times x or 5x, becomes problematic for students, as they are not able to carry out the specified computation (Thompson and Saldanha, 2003). Unfortunately, multiplication as the coordination of un its explained above is not something that teachers can take for granted as understood by their students. In a study
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! 18 by Clark and Kamii (1996), only 49% of fifth graders had strong multiplicative reasoning, at a time when formal instruction in multiplicat ion is likely finished in the classroom. Further more, Siemon, Breed and Virgona (2005) found that far too many students at the secondary level were using simplistic additive models when multiplication would have been more efficient. It was considered lik ely by the authors of the study that part of the reason for their lacking proficiency was that they understood multiplication as repeated addition. Steffe (1994) said that when students use an additive model to multiply, it is an indication that the y have yet to dissociate the idea of multiplication from more primitive counting schemes. The difficulties pointed to by the aforementioned studies show that there could be benefit derived from an instrument to determine where students seem to be on t he contin uum from additive to multiplicative reasoning before multiplication is considered solid enough to begin further instruction in fractions. In this sense, multiplicative reasoning is a conceptual prerequisite for middle grades mathematics (Thompson & Saldnh a, 2003; Tzur et al., 2013) Its absence may create significant problems for students who are not yet equipped to construct these understandings. Most importantly, an assessment instrument is needed because teachers may not realize that this gap in reaso ning multiplicatively is present. It is often assumed that students are ready for further instruction if they are proficient in executing calculations of multipl ication and division and can give some reasoning that supports their work. The notion that t hese students may not indeed have a full multiplicative understanding, complete with the ability to track and anticipate the coordination of units involved in problem situations was paramount in the development of the AMR
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! 19 Another reason to develop an assessment instrument of multiplicative reasoning is that m any teachers may not be aware of the schemes and stages involved in multiplicative understanding This is especially true at the secondary level in which these concepts are not usually taught. Ja cob and Willis (2003) commented on the need for teachers to be aware of and able to distinguish between additive and multiplicative reasoning in order to better serve their students. Seeing students "skip count which may reflect essentially a facility w ith repeated addition, can give the impression that students understand the concept of counting groups of objects However if this is simply a learned method, and not rooted in an understanding, it may be only an indication of a willing student. The next section will focus on the specific schemes that stud ents develop when constructing meaning for multiplicative situations. Schemes for Reasoning Multiplicative ly The multiplicative reasoning described above does not develop all at once, but on a continuum of schemes. The framework for student understandings discussed in this work, came from the six schemes of multiplicative reasoning proposed by Tzur et al., (2012). These schemes begin with multiplicative Double Counting (mDC), during which students are able to keep track of two quantities when one is distributed over another. Th is first of six scheme s mDC, is the highest level assessed in the AMR instrument because it focuses on the shift in thin king from additive reasoning to multiplicative reasoning for the student. For example, being able to find the total number of flowers in four vases, if there are three flowers in each would require a student to be able to count how many composite units of three flowers are present, while at the same time counting how many units of 1 single flower are accumulating respectively They
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! 20 would then need to knowingly stop when the count of the composite units of t hree s reaches four, and understand that the number obtained is total flowers, not vases nor flowers per vase. This distribution is an advancement in thinking from the additive in which a student would either need to count each flower, or add 3 + 3 + 3 + 3 to find the result of 12, but mi sses the idea of the units being coordinated The meaning s of the answer and the two numbers 3 and 4 are critical, as well as which is being distributed across the other. Failure to understand this can become problematic in later studies if this concept is not solidified early on. The six schemes of which mDC is first have laid a foundation for educators to not only determine where students may be in relation to multiplicative reasoning, but to ensure that understandings necessary for meaningful work on fractions are in place. Activity Effect Relationship One elaboration of the constructivist learning theory was articulated by Tzur and Simon (2004) in their depiction of the mechanism of learning, namely, reflection on ac tivity effect relationship ( AER). This depiction describe d the way that a student would engage in an activity, which may be visible to an observer or occur mentally, to achieve a goal or desired outcome that the learner has in mind. This activity is performed on an object, which ag ain may be physical or simply a prior conception (i.e., number as an abstract, composite unit). The effect is what follows from the activity performed on the object. It stands to reason that if there is not a viable object the activity cannot proceed fr uitfully, and that the teacher must be aware of the mental objects available to the student. When the object is acted upon an effect will be created, but the effect and
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! 21 learning may not be optimal, and possibly even harmful if it creates misunderstanding s on the part of the student due to insufficient prior conceptions. To illustrate the working of this theory, consider a student who has a multiplicative scheme such as mDC as may be measured by the AMR, attempt ing to solve a fraction problem, and how t hat student may be in structed using the model of AER. A student with solid multiplicative reasoning could independently solve a problem such as this one: Maria wants to have juice boxes at her birthday party for all of her guests. The juice boxes come in packs of 4, and the store has 6 packages on the shelf. How many juice boxes will Mar ia have if s he buys all 6 packages of four? A student who has multiplicative reasoning could solve this problem by iterating the composite unit of four six times to eff iciently find that there are 24 juice boxes to be purchased. When it has been established that the student is able to operate in this way, the multiplicative scheme being used can be extended to allow the student to think about fractions such as consider ing what fraction is just one pack of all 24 items) This is exactly the type of reasoning that the AMR can detect to determine whether a student is ready to advance to fraction. A student may then be presented with a problem such as this one: Maria is cleaning up from her party, and finds that some of the juice boxes she served her friend are not finished. She finds five juice boxes that are each full. If she puts them all together, how many b oxes could she fill completely? Using a continuation of t he previous problem can help students see the connection between the two, and begin to activate the scheme, but they will likely need further direction to fully utilize the current scheme for this problem. The student may be prompted to draw the boxes, an d to describe what they see, redrawing the juice boxes
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! 22 when combined. The student could then be prompted to use the same idea that was used in the multiplication problem to describe the total. By reflecting on the commonalities of six units of four juice boxes, and five units of juice boxes, a student can come to see that the total would b e 5*1/4 or 5/4 of a juice box. This can mean for the student that it takes four of these units to make a whole, and that five are being considered, so I will be abl e to fill up one juice box with four units and will then have one left over, the same as 1 juice boxes. If the student is comfortable with the multiplication to find the number of juice boxes in the previous problem as 6 4 or 24 juice boxes this same conception can be used to understand fraction operation s This transition is not likely to occur quickly or easily but can occur with continued focus on the conceptions that students have, and continued emphasis by the teacher on the common patterns with iteration of whole numbers and fractions By asking the student to reflect upon their work for each type of problem and reporting on their findings, a solid foundation for the understanding of fractions can occur. The key in this process is for the student to reflect upon, and make sense of what they see as it relates to what they already know. It is important that they do this reflection while being directed toward reactivating the pre existing schemes that will best su pport this new learning, usin g multiplication to support an understanding of fraction. This understanding may begin with students needing support and prompting from the teacher, but with assistance students will become independent even when the situation or context is different. Th e ref l ection on AER model helps us to understand the se two stages, called anticipatory and participatory (see below) as well as the need for fine grained assessment which is described in a later section as a way to discern between these stages.
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! 23 Anticipatory and Participatory Stages That students may be at different stages when constructing a new scheme was considered in the development of the assessment in this thesis and these two stages are explained in the section that follows. Par ticipatory and an ticipatory stages were considered important because in order for students to be able to function effectively in their multiplicative schemes, they would need to be at the later stage described here. Also included in this discussion are the ways to determine the stage in which a student seems to be functioning, a critical part of the AMR. Tzur & Simon (2004) postulated that students may have constru cted a new scheme at a stage, called participatory, which requires prompting in order for them to acces s, and use, the newly forming scheme. Later, the students may progress to a more advanced stage, called anticipatory, in which they can independently use the new scheme. This distinction implies that students may be able to solve a problem given manipulat ives, following a lesson on that concept, or with prompting, but they would not be able to solve independently, or when the memory of the activity has faded. Tzur & Lambert (2011) related this cognitive stage to Vygotsky's notion of Zone of Proximal Devel opment (ZPD), suggesting that the ZPD is a reflection of students' cognitive need for prompts. A student who is at the participatory stage may be able to solve a problem about the number of flowers in four vases with three flowers each, during or immediately following a lesson in which these types of problems were posed and worked through with guidance from the teacher (e.g., "How about using our fingers to keep track of how many vases and how many flowers we have in all? ) After some time, however, the
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! 24 student may seem not to be able to solve problems of this type at all. This, sometimes called the "next day phenomenon", is a common stumbling block for teachers and students, causing frustration for both parties. Possibly, t he issue here is that the student might have been only at the participatory stage, meaning that they must be prompted, at least mentally, to be able to reason about the problem. The student seems to understand, and may be able to solve independently after prompting, but the next day in class the student may not remember how they should proceed on a similar problem. For example if the teacher asks questions such as "How many flowers were there in one vase?" "Now how many would be in two vases?" to lead the student's thin king, it may quickly result in a studen t's ability to resume their previous way of reasoning and solving the problem T he student continues to need prompting, however, and will likely not be able to solve these tasks as time goes on and the activity becom es a distant memory. At a higher level of the participatory stage, a student may prom pt oneself upon realizing that he made an error sometimes called the oops' experience For example, students may begin the vase problem by saying there are seven flow ers (e.g., adding 4+3), and then correct themselves as they realize that there are three flowers in each of the four vases and answer correctly. This is still not considered anticipatory, because the student needs to self prompt. However, this is a higher level of the participatory stage, because the prompt has now come from within the students' mental system (Tzur & Lambert, 2011). One difficulty in discerning between the levels is that unless the teacher is very involved with th e student during the solution process, the self prompting may not be evident. The student will correct and then move on, leading the teacher to assume independence in using the new scheme, and thus anticipatory status.
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! 25 Unlike at the participatory st age a student at the anticipatory stage does not need to consider the activity, or to be prompted. Their understanding is more solid because it occurs in the student's mind as a well formed scheme, independent of working through the activity in the familiar c ontext. In the example above, the relationship between flowers and vases is clear to the student, and they don't hesitate to proceed, anticipating the entire process from the start. They anticipate the need to coordinate counting of one composite unit (4 vases) with another (3 flower s/vase) to calculate the total (e.g., 1 vase is 3 flowers, 2 is 6, 3 is 9, 4 is 12). This anticipatory stage can become a conceptual foundation for meani ngful memorization of the fact 4 x 3 =12 It may be that in the classroo m some students do not reach the anticipatory stage of a concept, such as multiplicative double counting before the instruction moves forward. For students not at the anticipatory stage a problem arises, because though they have been exposed to a concept and made progress toward understanding at the participatory stage they are not prepared to independently use that understanding as a building block for new learning before the topic of study changes This situation is also frustrating for teachers, who feel as if the student should have learned the material already, and showed signs of understanding, but cannot access this concept or scheme when they attempt to move forward. In the teaching of concepts such as fraction and ratio, a participatory unders tanding of multiplicative relationships is not sufficient to create understanding about the meaning of fractions, so the detailed assessment of this reasoning is crucial. The next section discusses the type of detailed assessment that can shed light on th ese distinctions, and was used in the development of the AMR.
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! 26 Fine Grained Assessment The assessment technique used in the AMR is one developed by Tzur (2007) to better assess the precise level of student understanding, by distinguishing between the antic ipatory and participatory stages explained in the previous section. In fine grained assessment, it is important that the items begin with out prompting and progress to less difficulty and more support, in the form of prompts and questioning, to determine whether the student is able to reason independently When students are presented with problems, it is without any prompting or introduction by the ass essor. If the student is unable to solve the problem, prompts are then given in a way that gives the student clues about how to act, while being cautious to give as little away as possible. The intent is to find the highest level at which the student can function. If the student is given an easier problem first, or is led toward that higher understanding by instructions, the reasoning seen may be as a result of the unintentional prompting the student receives, instead of the student being able to anticip ate the entire process on her or his own. For example, if one desires to know if a student can find a missing second addend, the problem should be presented without any prelude. The student may be asked "I had three candies, and my friend gave me some more. Now I have seven candies. How many did my friend give me?" If the student is able to provide reasoning and solve this independently, they can be considered at the anticipatory stage. If they are not able to solve this problem, the assessor could prompt, beginning with calling attention to the numbers in the problem and what they mean. The assessor might ask: "How many candies did I have to start with?" and then ask the student if they can solve. A follow up could be: "How many candies do I have at the end?" and then: "Can you use the number
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! 27 I have now and the number I started with to find how many my friend gave me?" If the student can use these prompts to solve, they are at the participatory stage for this conception. They have some knowledge of how this problem can be solved, but need to be prompted to arrive at a solution. If the student is not able to solve even after prompting, it is likely that they have not begun to build the concept for themselves at all. Additionally, it is important that student assessment not follow an activity in which they were engaged in solving similar problems. This is to avoid the participatory student appearing to be anticipatory. Immediately following a lesson, the student can simply recall the activity, an d use i t to solve the problem. It may be impossible to tell whether the student can reason this way independently or only following the lesson. The importance of this distinction between participatory and anticipatory students is that even though the part icipatory students may appear to understand the concept in the context of the current class period, or with subtle prompting, these participatory students will likely not be able to draw upon this reasoning to develop later concepts and could be left behi nd. This shows the importance of an assessment that can distinguish between these stages, so that planning for these students can be optimal and directed specifically at their area of need. The next section shows previous work in assessing multiplicative reasoning, which was used to guide the development of the AMR, and to show the differences between these attempts and mine. Previous Research on Assessment of Multiplicative Reasoning There are some existing instruments that have been developed for the as sessment of multiplicative reasoning necessary for future concepts. Here I will present these instruments, with what I consider to be their strengths, and then explain how the AMR
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! 28 attempts to go one step further to achieve a fine grained assessment of mul tiplicative reasoning. One instrument developed by Siemon and Breed (2010), was a pencil and paper test to address many of the same concerns expressed in this paper. The assessment considers many of the same conceptions, including distributing one composite unit across another. That instrum ent has excellent potential to determine whether students are reasoning multiplicatively, and includes rich tasks for them to explore, moving beyond what is assessed in the AMR to include proportional reasoning. The intent of the Siemon and Breed's (2010) assessment is similar to the AMR in that it tries to capture student thinking and elicit strategies used to determine the sophistication with which the student is approaching the problem, and the resources that they employ to solve them. Scoring of this assessment was rubric based, and completed after students had finished the problems. Their instrument, however, did not include a distinction between participatory and anticipatory stages in a child's use of the mDC scheme, and did not provide the same fe edback as a clinical interview situation could Another assessment, developed by Bright, Joyner and Wallis (2003), has similar features to the Siemon and Breen instrument. This assessment included a multiple choice paper and pencil test, but mentions int erviews to follow up and question students about their answers to delve deeper into their thinking. It was noted in their work that it is possible that questions which are easier to answer, such as multiple choice questions, may induce more sophisticated reasoning, while open ended questions may cause students to revert to more primary methods which they feel most comfortable using. Their point may be captured when using the participatory/anticipatory stage distinction,
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! 29 for which specific, finer considera tions and ordering of the tasks is needed (Tzur, 2007). This consideration guided the development of the AMR, to allow assessors to differentiate between anticipatory and participatory stages of mDC. The AMR attempts to be more specific than the previous instruments in assessing the smaller increments in reasoning. In the Methods section, the development of the AMR is detailed, using this stage distinction as an important part of the assessment.
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! 30 CHAPTER III METHODS This chapter chronicles the develop ment the AMR, including the selection of tasks, the administration method, the administration guide and scoring. The Assessment for Multiplicative Reasoning (AMR) was developed to give teachers and researchers a way to assess the level of a studen t's multiplicative reasoning, or preparedness to engage in that reasoning. The intent is that instruction can then be focused at the appropriate level to give students the experiences that they need to develop the next level of reasoning required to suppo rt multiplication or to indicate that the student is ready with a strong basis to experience other math concepts for which multiplicative reasoning is a prerequisite, such as fractions, ratio and proportion and slope. The AMR was developed with assistan ce from experts, and then tested briefly on a small sample of fourth grade students. The full final version of the assessment to date with all of the components mention ed here is provided in Chapter IV but examples are given in this chapter to illustrate the development process. Th i s chapter explains the process by which the assessment grew from tasks to target specific schemes of multiplicative reasoning, possible distractors that were taken into consideration and the design of prompts to bring to light the possible participatory and anticipatory stages at each conception Later sections describe the development of an administration guide for assessors and considerations for scoring the results. Refinement ba sed on feedback from experts and from the initial testing are discussed in each section as appropriate, as well as their implications for the improvement of the AMR.
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! 31 Development of Tasks Tasks were created based upon theoretical literature based analysi s and on my interpretation of the performance that student s should exhibit if they have completely developed the scheme in question (Zazkis and Hazzan, 1998). Through conversations with Dr. Ron Tzur, I decided to include prior schemes in the assessment : C ounting On, Missing Addend, and Composite Unit Iteration (shown in Chapter IV ) t o screen students and determine whether the additive schemes rooted in number as composite unit and necessary for multiplicat ive concepts were accessible to the student One multiplicative scheme was included as well, namely Multiplicative Double Counting (mDC) to assess the progression of students from additive to multiplicative reasoning The tasks were differentiated into grade band s in an attempt to allow for flexi bility in administration by providing multiple entry points for students and assessors to use, being responsive to previous knowledge about student understanding. For example, a student at the fifth grade level may be tested at the 3 4 grade band, if the teacher or assessor knows that the student will struggle with the numbers involved at their grade level. It is also advised, and stated in the A dministration Guide in Chapter IV that in general the assessor should begin with the grade level task, but move down to the previous level if the student is unable to answer. The intent here is to determine whether the struggle is due to the numbers involved, or to the concept ual scheme or stage being as sessed. The following section provides a description of the tasks used to assess these schemes, and information about the choices made in their development.
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! 32 Composite Unit Iteration Task Composite unit was explained earlier, and is important for both th e additive and multiplicative schemes. The Composite Unit Iteration task asks students to use these units and iterate them to find a new quantity, without counting all of the items. This requires studen ts to coordinate two quantities the unit to be iter ated and the number of iterations that take place. This task is administered first because it involves composite unit, which is important to both additive and multiplicative thinking, but also because it represents a midpoint in the continuum of the asses sment as a whole. Th is approach was used although th ere w as not enough data from student responses to create a proper Guttman Scale ( Trochim & Donnelly, 2008) for these tasks Such a scale requires one to use data to create a linear ordering of tasks However an attempt was made to arrange the tasks in a developmental order that would most likely occur for the student conceptually. Thus, if a student can properly solve and reason about the first task the assessor can move to higher levels and if not the assessor would go back to prior conceptions. For instance, it is unlikely that a student who can perform Multiplicative Double Counting would be unable to solve a Counting On problem. T he original Composit e Unit It eration Task I ha d created seen in Figure 3.1 was problematic for students. It was intended to give students a visual model and see if they could use the groupings of four beads to iterate and find the total but students were generally not able to use this information to iterate corre ctly It was suggested by Dr. Tzur that I eliminate the task, and it no longer appears in the final version. Instead, a task involving iteration of the smallest composite unit (i.e., 2) was created, with a context
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! 33 using pairs of socks that seemed familia r to any student T his task is further described in Chapter IV How many beads will you need to make this necklace as shown? Figure 3. 1 Previous Version of Composite Unit Iteration Task Additive Tasks The additive portion o f the assessment including the schemes of Counting On and Missing Addend, is not intended to be an exhaustive assessment of additive reasoning, but to give a rough idea of where a student might be functioning in regard to additive schemes in general, and in terms of her u se of composite units in particular The se schemes are explained next. Counting On refers to the process by which a student can begin with a quantity, such as 7 and then add to that quantity, perhaps 4 more, and find the total Th e student does th is not by starting to count from 1, but by beginning with the known number 7 and counting from there This counting on strategy is sometimes called double counting, to emphasize that the student is purposely keeping track of 1s in the second addend and knowin g to stop after they have counted four more (Tzur & Lambert, 2011) The student might say something like this: "I have 7 already, and four more ; so eight is 1, nine is 2, ten is 3, and eleven is 4. I have eleven all together." The Counting On tasks in this assessment were constructed so that the second added quantity is smaller. This was done to make representing these problems with figural, pictorial or manipulative strategies more accessible. Assessment of students' abilities to reverse the quantit ies for counting
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! 34 on when the larger quantity is given first was not a part of this assessment due to the limited nature of this portion. The next two schemes are assessed using Missing Addend problems, where a student is given one of the quantities to be added and the total in a problem situation, and is asked to find the other added quantity In the first type of missing addend problems the student is given the first addend and the total, and is asked to find the second adden d. In the second type of missing addend problems the student is given the second addend and the total, and is asked to find the first addend. These problems are considered different for students, as the Missing First Addend requires more sophisticated t hinking The Missing First Addend Tasks are considered more difficult, because of the way that the composite units are nested within the whole (Fuson et al. 1988). If the second addend is missing, the student can know where to start her counting as the y do in counting on and proceed by count ing up from the first addend unit the total is reached I f the first addend is missing student s do not have a way to start an activity of counting upward using their previous way of operating. They thus need to either reverse the order of the addends (in effect, creating a missing second addend situation), or count down from the total until they reach the second addend. Both of these operations are more advanced and less likely to be available to many st udents due to insufficient abstraction of the total as a composite unit containing of both addends as sub composite units. Additional considerations for the additive tasks are discussed next, in an attempt to clear away distractions that might cause the a ssessment to give invalid results. It is understood that some tasks may be much too simple for many students in the upper grade bands, but to be thorough, they are included. It is also understood that the
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! 35 concept of multiplicative reasoning assessed later in the instrument may not be present in students in the earlier grades Yet the task is designed in such a way that if younger students are able to reason multiplicatively the numbers and situations should not be a barrier. The levels in each task were created to target values and situations that were developmentally appropriate for students who are achieving at the grade level s in dicated. The example seen in Figure 3.1, the Missing First Addend task is differentiated to four levels to accomm odate grades K through 8. I chose the bouncy ball situation in the same way that all situations were chosen, with an attempt to provide familiarity and context to the problems. The numbers for each level (grade band) were selected (a) in consultation wi th Dr. Tzur, (b) using expert feedback from 15 teachers, and (c) guided by the work of Sherin and Fuson (2005) The latter, empirical study focused on ways in which chosen numbers and problem types affect an assessment. An effort was made not to repeat and addition or multiplication facts, so that regardless of which grade level tasks were given, or in which order, there would not be any repeats. Also I did not include consecutive facts, such as 3 x 5 and 4 x 5, so that there would not be reliance on the previous problems to answer the later problems. F or example, in the Missing First Addend Task (Figure 3.2) the lo wer grade levels are given the addend 3 and a sum of 8 so that the total does not pass ten and involve pla ce value. These are numbers that younger students are likely to be familiar with, yet avoid the facts that they might know, such as 5 + 5 = 10. For t he oldest grade band (7 8), the AMR uses the numbers 25 and 38 to discourage use of known facts and encou rage reasoning. It is unlikely that the student has the math fact 25 + 13 = 38 in th eir
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! 36 memory, as they may have 4 + 5 = 9, and therefore will make it more likely that they will use additive reasoning to find the solution (e.g., 25+10 35, and 3 more is 38 a strategy indicating the student's operation on 13 by decomposing it strategically into 10 and 3) Each task was designed with these considerations in mind, in an attempt to hone in on the reasoning available without interference from memorized facts or inaccessible numbers and situations. Figure 3.1 .Missing First Addend Task There are some prevailing strategies commonly taught in schools to handle these types of addition problems. The strategies seen in early trials of the problems were using known doubles, such as 4 and 4 make 8, and building on them, so that 4+5 could be seen as the double of 4 plus one. Another strategy is to break or decompose the numbers to make ten, so that 8+5 could be seen as 8+(2+3) = (8+2) + 3 = 10 + 3 = 13. Clearly, many of these strategies that do not involve counting on allow students to use them in any situation and could not be avoided completely. A re sourceful student can use these Missing first addend task Level 1, K 2 10. Anna has some bouncy balls. Her mom gives her three ( 3 ) more bouncy balls. She then has eight ( 8 ) bouncy balls. How many bouncy balls did Anna have in the beginning? Level 2, grades 3 4 11. Anna has some bouncy balls Her mom gives her thirteen (13 ) more bouncy balls. She then has nineteen (19) bouncy balls. How many bouncy balls did Anna have in the beginning? Level 3, grades 5 6 12. Anna has some bouncy balls Her mom gives her sixteen (16 ) more bouncy balls. She then has twenty four (24 ) bouncy balls. How many bouncy balls did Anna have in the beginning? Level 4, grades 7 8 13. Anna has some bouncy balls. Her mom gives her twenty five (25) more bouncy balls. She then has thirty eight (38) bouncy balls. How many bouncy balls did Anna have in the beginning?
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! 37 strategies on any problem In fact this adaptability shows a high le vel of proficiency with additive situations particularly the operation on and with composite units, and thus indicates that the student is ready to procee d. These problems require students to be able to either use a modification of the counting on procedure, where they know to stop at the given total and must find how many more they need to count from the given addend. Alternatively, they can use counting down, or beginning at the total and counting backwards until the y reach the known addend. The Missing Addend problems were created with the missing addend as a quantity that is less than ten As mentioned in the section on Counting On, t his was done bec ause students are being asked to show their reasoning using figural or manipulative means, so it was necessary that the missing portion be kept to a number that would be reasonable for students to repre sent in this way Only two of these tasks were includ ed, both considered Joining Problems by Carpenter et al. (1981) The reason is twofold the AMR was not intended to be a comprehensive assessment of additive reasoning, and the study by Carpenter indicated that this deviation in problem structure did not affect the solution methods of students It is intended that both tasks be administered as a reliability measure It is possible that a student could miss the first problems, the easier missing second addend problem for various reasons, but correctly s olve the more difficult second problem with a missing first adden d Multiplicative Tasks Multiplicative Double Counting, as described in the section on Mul tiplicative Schemes in Chapter II represents the first stage in the transition from additive to
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! 38 multiplicative thinking. This is the first and only multiplicative stage assessed by the AMR in this prototype. Due to the fact that Multiplicative Reasoning was the primary goal of the development of this assessment, there are more tasks to lend reliabil ity to this section. The three tasks were all created with the unit rate how many items per grou p, being less than the number of composite units given in the problem For example, in the first MDC task for level 1, there are three cars per box and four boxes. The tasks were constructed in this way to make it more conducive to the creation of a figural, pictorial or manipulative solution At the time of the defense of this thesis, it was suggested by Dr. Heather Johnson, that the multiplicative tasks be adjusted to be consistent as to the relative size of the unit compared to the unit rate. This additional expert feedback for task adjustment was made and is reflected in the final instrument. Task Refinement Initial refinement of these tasks was done through feedback from 15 teachers of K 12 students. One suggestion they gave was to include picture cards for younger students. A picture was included on the Composite Unit Iteration task, a frequent suggestion, to help clarify the meaning of a pair of socks mentioned in the problem. I decided not to include picture cards for the remainder of the problems, because it may provide too much prompting via visible units This would fail to show whether a student has the reasoning or simply relies on the picture counting 1s while visually and silently scanning the picture I believed that part of the scheme necessary for students would be the ability to create the pictorial or figural representation and that providi ng a picture might detract from their ability to show this crucial step in operating on units It would
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! 39 be appropriate for a kindergartener to be unable to solve even the additive reasoning problems, so I did not feel that providing this scaffolding was necessary. The remainder of the teachers' suggestions involved the numbers chosen for the tasks. There were some repeats in the number families chosen in the initial draft, such as 4+5=9 being used three times on different problems. This was corrected in the later version to include more addition facts, carefully avoiding the doubles and other patterns described earlier. It was also suggested that a problem on the initial version involving earrings be changed to a more gender neutral item such as mittens. This change was made to have the problem ask about socks after much consideration as to an item that students might have multiple pairs of at home. After receiving the expert feedback from Dr. Tzur and the K 12 teachers, the instru ment was tested with fourth graders As a result of th at initial testing, the grade bands were nar rowed to accommodate students in a more differentiated way The first draft of this assessment included thr ee levels, K 2, 3 5 and 6 8. During the initial testing of the AMR it was found that some fourth grade students struggled with the values given, and that the grad e bands may have been too large. This prompted the further differentiation of levels in elementary grades to include only two grade levels in stead of one. The current structure includes levels for K 1, 2 3, 4 5 and 6 8. Construction of Prompts In this section, I will explain the way that prompts were developed for each of the aforementioned tasks These prompts are meant to assist a student who may not be at the anticipatory stage in solv ing the problem with support and thus gaining data about the participatory or pre participatory stage in which the student operates for the scheme being
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! 40 tested This part of the assessme nt in which prompts are provided gradually, is designed to give the assessor an idea of where the student is on the participatory continuum, and what s pecific support they may need. This is important because a student who has no prior knowledge of the re asoning at this stage will need to be approached differently than one who is nearly anticipatory, and a nearly anticipatory student will likewise need different learning experiences than one who is just beginning to be participatory. This type of stage distinction was the goal of scripting the prompts to be given while the tasks are being completed instead of simply presen ting the problem and prompts to the student as typically occurs i n a pencil and paper test. As explained in the Fine Grained Assessment section of Chapter II the first step is to administer the problem without prelude or prompts. This is intended to allow students who have achieved an anticipatory stage f or the scheme to demonstrate it Using the example of the Missing First Addend task above, the student would be given the problem as is and asked to solve while explaining their thinking. If independently doing so it allows the assessor to infer construction of the scheme at the anticipatory stage. Whether or not the problem is solved successfully, the student may then be prompted to use figural, pictorial or manipulative means to prove their solution in an attempt to understand the reasoning used by the student This may be omitted sufficient evidence was given that the student understands the concept, but it provides an opportunity to delve deeper into what the student is thinking about the task The use of these means can also be a way for the student to sel f prompt, or catch their own mistakes. For a studen t who has used known facts, drawing or using manipulatives allow the
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! 41 assessor to see if these facts are supported by reasoning, or simply reflect rote memor ization If the problem cannot be successfully s olved independently, the assessor will then begin to prompt the student in an attempt to determine whether the student has constructed a participatory stage for the scheme as indicated by solving th e problem with support. The prompts are scripted purposel y to provide the student information that should help to orient their thinking to the activity involved in the scheme and thus to a correct solution if they have beg u n forming that underlying conception. To the extent possible, a ssessors are expected to avoid deviat ing significantly from these prompts, in part because a teacher or researcher who works regularly with students may unintentionally give additional information that would lead the student to score more highly than is accurate. An example of the prompting created for the Missin g First Addend Task (Figure 3.2) which begin s after the student has attempted to solve independently is seen in Figure 3.3 Some prompts were included to elicit specific reasoning from the student, if the correct answer is given but sufficient reasoning is not.
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! 42 Figure 3.3 Prompts for Correct Responses A second level of prompts was developed for students who do not answer correctly, seen in Figure 3.4, to determine if they have a high or low participatory level of understanding, or perhaps no understanding of this stage at all. Figure 3.4 Prompts for In correct Responses The prompts here were designed to carefully lead a student through the activity effect thinking process that can help them arrive at a solution. The intent If response is incorrect: Ask student to explain their thinking on this task. Produce manipulatives, and ask the student to solve with the manipulatives. If they does not solve correctly with manipulatives, prompt students usin g the following questions: 1. How many did she have before her mom gave her more? 2. How many does she have in the end? Can you use the number in that she had before as a starting place to see how many more she got from her mom? How? 3. Can you use the number she had in all and count backwards to find out how many she got from her mom? How? If the student solves correctly after prompting, move to the next problem If the student cannot solve this problem after prompting, end testing. For both correct responses without reasoning and incorrect responses: o Ask students to explain their thinking on this task. It is important to discourage counting all as a solution. Prompt the student to use counting on by covering the first composite unit, or asking them to make a drawing that does not include all of the o nes in the unit. If the student is using marks or manipulatives, it may be necessary to cover the first composite unit and then see if the student can then proceed to count on. o If at any time during the administration of this task you believe that the nu mber values are impeding the student, move down to a lower grade band and test again. o If at any time the student answers correctly using counting on or counting back, go to the next problem. If response is correct, but the students use an unknown strategy, cannot articulate one, use counting all or known math facts, prompt student using these questions: 1. How many did she have before her mom gave her more? 2. How many does she have in the end? Can you use the number in that she had before as a starting place to see how many more she got from her mom? How? 3. Can you use the number she had in all and count backwards to find out how many she got from her mom? How? If the student solves correctly after prompting, move to the next problem
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! 43 of this stepwise approach is, again, to discern the specific level of the student within the participatory stage. In the next section the choice of Administration Method is discussed, with a rationale for the interview format. Selection of Administration Method To distinguish between the anticipatory and participatory stages in fine grained assessment, conducting a face to face interview is needed to provide the assessor with the most specific feedback about what the student knows (Tzur, 2007) Th e format of the as sessment is a clinical interview, a technique often used to delve in to the thinking of subjects, using what Zazkis and Hazzan (1998) call ed performance questions followed by "Why" questions. The interviews are to be conducted with one to three students at a time, and are intended to give the students a task, observe their responses, and if needed, prompt with questions that will clarify whether they are showing evidence of the scheme or not (Ginsberg, 1981). A clinical interview situation (Ginsberg, 19 81) is best suited to this type of assessment, because it allows the assessor to control and adapt the prompts and i nformation given to the student The interview format provides the assessor with the unique opportunity to ask students quest ions as they a re solving, listen to what is being verbalized, analyze and clarify figural representations created by the student in real time, and discern any movements or quiet vocalizations that the student may produce. While a paper and pencil task is efficient fro m a time standpoint, it does not provide the rich data that can be gleaned from an interview setting. Furthermore, i n a pencil and paper test, all of the information is present for the student to use, and it is extremely difficult to discern whether the st udent was able to s olve a problem
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! 44 independently, whether they used information given in previous or subsequent problems, what kind of reasoning was used, or whether they could have solved with a little more support. Asking students to explain their answers in writing during a paper and pencil test is another option, but it often provide s in accurate results (e.g., when the student's first language is not English) Students can often explain verbally what they would n ot write on a test, and may know more than they show on paper. The tasks were ordered so that students would be unlikely to be able to complete a task if they were unable to complete the previous task a Guttman like scale ( Trochim & Donnelly, 2008 ) This scaling is discussed further in Chapter IV. For that reason, it is unnecessary for a student to complete the entire assessment Rather the assessor can simply discontinue testing when the children being assessed reach a level they are evidently unab le to complete even with prompts As explained previously, t he tasks should begin at the Composite Unit Iteration task (pairs of socks) and then either move forward to the multiplicative tasks if student s reason independently or begin t he additive reas oning tasks if they do not. One reason for using the Composite Iteration Task as a starting point is that composite unit is a conception that is necessary throughout the tasks. The ability to consider a group of 1s as its own unit is something that a student would need to be able to do to reason multiplicatively or additively but the iteration of these composite units is a more advanced concept than simply recognizing composite unit s This allows a middle ground for students to beg in at a level that is most appropriate for them, without subjecting more advanced students to problems that are too easy for them and possibly losing their interest or intimidating students who are unable to reason multiplicatively. This can also save t ime in the administration procedure for both
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! 45 teachers and students. Dr. Johnson suggested with Dr. Alan Davis' support, this change in task order at the time of thesis defense and it was implemented in the Administration Guide Students from time to time may struggle with the tasks not because of the concept, but because of the numbers involved. To accommodate these students and assess their reasoning it is possible in the course of administering the test to allow the student to respond to a task th at is at the grade level below the one being assessed. For instance, if a fifth grade student is struggling to complete the Multiplicative Double Counting task, they could be administered the level 2 task, intended for second and third graders. If the st udent could then answer the problem, it could be surmised that they began forming the necessary multiplicative scheme, but may need some work to generalize it for larger quantities This flexibility was included so that the assessor can be responsive to t he needs of their students. If or when a student is unable to complete the task with prompting and/ or use of manipulatives, the assessor terminates the testing. Such inability indicates that the reasoning of the student has been exhausted and that they n eed not be tested any further. It is possible that a student who multiplies easily, and has some explanations as to what is occurring may not score as highly as expected. T his result might indicate that while the student has a strong procedural knowledge or fact memorization some of the prerequisite schemes may not be solid enough to support their future learning. As discussed in Chapter II, students will use available schemes to construct knowledge (Piaget, 1985) but if not carefully monitored, may use schemes such as additive reasoning or part of a whole which might not support them as well as mul t ip licati ve reasoning could. Next I
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! 46 will explain how the Administration G uide was created to walk assessors through the instrument as it is being given. Development of Administration Guide The administration gui de is shown in Chapter IV but this section will serve as a brief description of the thinking behind its development. The guide was created to be user friendly, complete with scripted prompts and a description of what a person using the assessment should look for in the child's solution to each problem. The task is listed with each grade band and all prompts, in an attempt to provide for the assess or a clear picture of what will happen during the assessment, and the options they have during administration. It was also created with an eye toward some technological component being involved in the future and considering some decisions the technology would need to make based on the student's response. The teacher or assessor can use the instrument as is to pinpoint a student's understandings and create a fine grained assessment (Tzur, 2007) Such assessment should be helpful to determine the next steps that need to occu r for the student and help the assessor to complete the scoring guide described in the next section. Scoring Considerations As with the administration guide, the Scoring Guide is seen in Cha pter IV but in this section I will describe the thought process behind its development. The scoring of this assessment needed to be both detailed and simple to complete so that teachers and researchers can obtain the needed information while minimizing t he time taken for the assessment Initially, I had envisioned the assessor scoring in real time, as the administration proceed ed In testing of the instrument, however, it was noted by Mr.
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! 47 Evan McClintock during this first administration that it might be simpler to have an observer score as the assessor interacts with the children. This was a very useful suggestion, and it was added as a possibility for the scoring process. It was also noted during the testing process that if students are videotaped, as they were in our first trial, scoring could happen later. I do not believe that this later scoring would be ideal, ho wever, as the scoring process may drive further questioning of th e student in real time that may not be possible after the fact. That be ing said it may be used if necessary, and if care is taken to thoroughly prompt the student. The next chapter shows the final result of the work and refinement processes described in this Methods Chapter This instrument is still in development ; but it is presented in its complete and final form at the time of submitting this thesis to the Graduate School of the University of Colorado Denver
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! 48 CHAPTER IV RESULTS T his chapter presents the final version of the AMR, including tasks, administration guide, student problem strips and the scoring guide Also included is a discussion of the initial attempts made at discerning the reliability and validity of this instrument. Tasks T he t asks seen in Figure 4.1, are listed in the order of increasing conceptu al difficulty, but not necessarily in the order that they would be administered to a student in an ordinary testing situation because this order will vary depending on the result of each problem These tasks are a l so included in the Administration Guide, and in the Student Problem Strips
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! 49 Figure 4.1. Tasks Composite Unit Iteration Task 1. Blanca has 8 pairs of socks. If she counts each sock, how many will there be? Counting on task Level 1, K 2 2. Lisa had five ( 5 ) pencils. She finds four ( 4 ) more pencils. How many pencils does Lisa have ? Level 2, grades 3 4 3. Lisa had seventeen (17) pencils. She finds four (4) more pencils. How many pencils does Lisa have ? Level 3, grades 5 6 4. Lisa h ad twenty four (24) pencils. She finds seven (7) more pencils. How many pencils does Lisa have ? Level 3, grades 6 8 5. Lisa had fifty seven (57) pencils. She finds nine (9) more pencils. How many pencils does Lisa have ?
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! 50 Figure 4.1. Tasks cont'd Missing second addend task Level 1 K 2 6. Joseph has six ( 6 ) pieces of gum. His friend gives h im some more pieces of gum. Now Joseph has nine ( 9 ) pieces of gum. How many pieces of gum did his friend give him? Level 2, grades 3 4 7. Joseph has (8) pieces of gum. His friend gives h im some more pieces of gum. Now Joseph has fifteen (15) pieces of gum. How many pieces of gum did his friend give him? 8. Level 3, grades 5 6 Joseph has fifteen (15) pieces of gum. His friend gives h im some more pi eces of gum. Now Joseph has twenty three (23) pieces of gum. How many pieces of gum did his friend give him? 9. Level 4, grades 6 8 Joseph has twenty five (2 5) pieces of gum. His friend gives h im some more pieces of gum. Now Joseph has thirty six (36 ) pieces of gum. How many pieces of gum did his friend give him? Missing first addend task Level 1, K 2 10. Anna has some bouncy balls. Her mom gives her three ( 3 ) more bouncy balls. She then has eight ( 8 ) bouncy balls. How many bouncy balls did Ann a have in the beginning? Level 2, grades 3 4 11. Anna has some bouncy balls Her mom gives her thirteen (13 ) more bouncy balls. She then has nineteen (19) bouncy balls. How many bouncy balls did Anna have in the beginning? Level 3, grades 5 6 12. Anna has some bouncy balls Her mom gives her sixteen (16 ) more bouncy balls. She then has twenty four (24 ) bouncy balls. How many bouncy balls did Anna have in the beginning? Level 4, grades 7 8 13. Anna has some bouncy balls. Her mom gives her twenty five (25) more bouncy balls. She then has thirty eight (38) bouncy balls. How many bouncy balls did Anna have in the beginning?
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! 51 Figure 4.1. Tasks cont'd mDC Task 1 Level 1, K 2 14 Tony has his toy cars in small boxes in his room. He has four (4) boxes, and each box has three (3) cars in it. How many cars does Tony have in all? Level 2, grades 3 4 15. Tony has his toy cars in small boxes in his room. He has six (6 ) boxes, and each box has three (3) cars in it. How many cars does Tony have in all ? Level 3, grades 5 6 16. Tony has his toy cars in small boxes in his room. He has six (6 ) boxes, and each box has five (5) cars in it. How many cars does Tony have in all? Level 4, grades 7 8 17. Tony has his toy cars in small bo xes in his room. He has seven (7 ) boxes, and each box has six (6 ) cars in it. How many cars does Tony have in all? mDC Task 2 Level 1, K 2 18. Ella has five (5) baskets for her friends. She wants to put three (3) pieces of candy in each basket. How many pieces of candy will Ella need? Level 2, 3 4 19. Ella has five (5 ) baskets for her fr iends. She wants to put four (4 ) pieces of candy in each basket. How many pieces of candy will Ella need? Level 3, 5 6 20. Ella has seven (7 ) baskets for her friends. She wants to put four (4 ) pieces of candy in each basket. How many pieces of candy will Ella need? Level 4, 7 8 21. Ella has eight (8 ) baskets for her fri ends. She wants to put seven (7 ) pieces of candy in each basket. How many pieces of candy will Ella need?
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! 52 Figure 4.1. Tasks cont'd mDC Task 3 Level 1, K 2 22. Aaron wants to make three (3 ) birdhouses and he needs two (2 ) nails for each birdhouse. H ow many nails will he need in all ? Level 2, 3 4 23. Aaron wants to make six (6 ) birdhouses and he needs four (4 ) nails for each birdhouse. H ow many nails will he need in all ? Level 3, 5 6 24 Aaron wants to make eight (8 ) birdhouses and he needs four (4 ) nails for each birdhouse. H ow many nails will he need in all ? Level 4, 7 8 25. Aaron wants to make nine (9 ) birdhouses and he needs eight (8 ) nails for each birdhouse. H ow many nails will he need in all ?
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! 53 Assessment Protocol Students are tested using the following protocol. The student(s) should be provided with pencil and paper, or some electronic alternative that allows them to write or draw only. A card containing the problem is handed to the student. These are formatted and printed with the problem number for ease of reference. Instructions for the cutting out and assembly of these problems strips are in Figure 4.2. Student Problem Strips To use these strips, print out on cardstock and cut along the lines that border the tasks. Stack each group of problems (found on the same page) with the highest grade band in the back. This should result in a stack of strips that exposes the problem at the top of the strip. These can be left loose or stapled on the right or left into small booklets. The Composite Unit Iteration Task on this page is not part of any of the booklets. 1. Composite Unit Iteration Task Blanca has 8 pairs of socks. If she counts each sock, how many will there be? Figure 4.2 Student Problem Strips
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! 54 Figure 4.2. Student Problem Strips cont'd 2 Counting on task Level 1, grades K 2 Lisa had five ( 5 ) pencils. She finds four ( 4 ) more pencils. How many pencils does Lisa have? 3. Counting on task Level 2, grades 3 4 Lisa had seventeen (17) pencils. She finds four (4) more pencils. How many pencils does Lisa have? 4. Counting on task Level 3, grades 5 6 Lisa had twenty four (24) pencils. She finds seven (7) more pencils. How many pencils does Lisa have? 5. Counting on task Level 4, grades 6 8 Lisa had fifty seven (57) pencils. She finds nine (9) more pencils. How many pencils does Lisa have?
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! 55 Figure 4 .2 Student Problem Strips $%&'()! 6 Missing Second Addend Task Level 1, grades K 2 Joseph has six ( 6 ) pieces of gum. His friend gives h im some more pieces of gum. Now Joseph has nine ( 9 ) pieces of gum. How many pieces of gum did his friend give him? 7 Missing Second Addend Task Level 2, grades 3 4 Joseph has (8) pieces of gum. His friend gives h im some more pieces of gum. Now Joseph has fifteen (15) pieces of gum. How many pieces of gum did his friend give him? 8. Missing Second Addend Task Level 3, grades 5 6 Joseph has fifteen (15) pieces of gum. His friend gives h im some more pieces of gum. Now Joseph has twenty three (23) pieces of gum. How many pieces of gum did his friend give him? 9 Missing Second Addend Task Level 4, grades 6 8 Joseph has twenty five (2 5) pieces of gum. His friend gives h im some more pieces of gum. Now Joseph has thirty six (36 ) pieces of gum. How many pieces of gum did his friend give him?
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! 56 Figure 4.2. Student Problem Strips cont'd 10 Missing First Addend Task Level 1, grades K 2 Anna has some bouncy balls. Her mom gives her three ( 3 ) more bouncy balls. She then has eight ( 8 ) bouncy balls. How many bouncy balls did Anna have in the beginning? 11 Missing First Addend Task Level 2, grades 3 4 A nna has some bouncy balls Her mom gives her thirteen (13 ) more bouncy balls. She then has nineteen (19) bouncy balls. How many bouncy balls did Anna have in the beginning? 12 Missing First Addend Task Level 3, grades 5 6 Anna has some bouncy balls Her mom gives her sixteen (16 ) more bouncy balls. She then has twenty four (24 ) bouncy balls. How many bouncy balls did Anna have in the beginning? 13. Missing First Addend Task Level 4, grades 6 8 Anna has some bouncy balls. Her mom gives her twenty five (25) more bouncy balls. She then has thirty eight (38) bouncy balls. How many bouncy balls did Anna have in the beginning?
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! 57 Figure 4.2. Student Problem Strips cont'd 14 Multiplicative Double Counting Task 1 Level 1, grades K 2 Tony has his toy cars in small boxes in his room. He has four (4) boxes, and each box has three (3) cars in it. How many cars does Tony have in all? 15 Multiplicative Double Counting Task 1 Level 2, grades 3 4 Tony has his toy cars in small boxes in his room. He has six (6 ) boxes, and each box has three (3) cars in it. How many cars does Tony have in all? 16 Multiplic ative Double Counting Task 1 Level 3, grades 5 6 Tony has his toy cars in small boxes in his room. He has six (6 ) boxes, and each box has five (5) cars in it. How many cars does Tony have in all? 17. Multiplicative Double Counting Task 1 Level 4, grades 6 8 Tony has his toy cars in small bo xes in his room. He has seven (7 ) boxes, and each box has six (6 ) cars in it. How many cars does Tony have in all?
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! 58 Figure 4.2. Student Problem Strips cont'd 18 Multiplicative Double Counting Task 2 Level 1, grades K 2 Ella has five (5) baskets for her friends. She wants to put three (3) pieces of candy in each basket. How many pieces of candy will Ella need? 19 Multiplicative Double Counting Task 2 Level 2, grades 3 4 Ella has five (5 ) baskets for her fr iends. She wants to put four (4 ) pieces of candy in each basket. How many pieces of candy will Ella need? 20 Multiplicative Double Counting Task 2 Level 3, grades 5 6 Ella has seven (7 ) baskets for her friends. She wants to put four (4 ) pieces of candy in each basket. How many pieces of candy will Ella need? 21. Multiplicative Double Counting Task 2 Level 4, grades 6 8 Ella has eight (8 ) baskets for her fri ends. She wants to put seven (7 ) pieces of candy in each basket. How many pieces of candy will Ella need?
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! 59 Figure 4.2 Problem Strips for Students cont'd 22 Multiplicative Double Counting Task 3 Level 1, grades K 2 Aaron wants to make three (3 ) birdhouses and he needs two (2 ) nails for each birdhouse. H ow many nails will he need in all ? 23 Multiplicative Double Counting Task 3 Level 2, grades 3 4 Aaron wants to make six (6 ) birdhouses and he needs four (4 ) nails for each birdhouse. H ow many nails will he need in all ? 24 Multiplicative Double Counting Task 3 Level 3, grades 5 6 Aaron wants to make eight (8 ) birdhouses and he needs four (4 ) nails for each birdhouse. H ow many nails will he need in all ? 25. Multiplicative Double Counting Task 3 Level 4, grades 6 8 Aaron wants to make nine (9 ) birdhouses and he needs eight (8 ) nails for each birdhouse. H ow many nails will he need in all ?
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! 60 Once the card is given to the student the assessor reads the problem aloud. Clarification about what the problem is asking is permissible, as long as it does not point students toward a solution. The student is asked to solve the problem by any means they wish, think ing aloud as they do so. If students do not share their thinking during the solution process the assessor should ask the student after an answer is given what they were thinking as they solved, encouraging making marks or dr awings on their paper, or using manipulatives to show their thinking. Students should be encouraged to use writing, drawings or marks first before using manipulatives. This is because using manipulatives, such as U nifix cubes, may encourage students to count all instead of using a composite unit ( Carpenter et al., 1981). When students are representing the composite units, it is preferable if they can do so without drawing in or building each of the 1s needed to construct that unit. T reating the unit as a whole in and of itself is important in the understanding of composite unit (Tzur, et al., 2013). For example, if the child began drawing 6 circles with 3 dots in each, the assessor should encourage him or her to first try solving the problem by just d rawing the first 1 2 circles and then just draw the circles without dots. Such an intervention is done because s tudents may count the 1s even if the ir reasoning support s activit y on the composite units without the 1s sim ply because the 1s are available t o count The se instructions are seen in the Administration Guide presented later in this chapter. The manipulatives used could vary but most commonly used are U nifix cubes and paper and pencil for drawing pictures by the student Students can solve problems with manipulatives that they would not be able to solve without them, and so this intervention is seen as similar to a prompt (Steffe, 1970).
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! 61 The assessor should do her or his best to avoid indicat ing to the student whether their answer or thei r thinking is correct or incorrect. The "burden of proof" is on the student, to be able to support their answer with reasoning, including figural or manipulative means. This approach is needed to uphold the fine grained assessment technique (Tzur, 2007) which proceeds from asking the student to solve with the minimum amount of support necessary. In order to determine whether the student is at an anticipatory or participatory stage, it is crucial to allow the student to solve as independently as possible anticipating as much as they are able before prompts including those about correctness of soluti on, are given By following the guide, the assessor progresses through the questions using the prompts, either moving down a level or using questioning to determine whether a student may be able to solve a similar problem with different numbers. This will help teachers determine whether what is lacking is the ability to reason about these tasks, or the familiarity with the quantities presented. At the end of each task the assessor can find directions about the next step in the process. Generally, if the problem was solved correctly, the student moves to the next problem, if the student could not solve the problem with prompting, testing is terminated. Thi s is due to the attempt at a Guttman like organization ( Trochim & Donnelly, 2008 ). This instrument is certainly not qualified as using the Guttman scale, which would be based on data from previous administrations, and not require that the student complete each item to give a reliable result. Details about this process are seen in the Administration Guide in the next section.
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! 62 Administration Guide T his section provides the complete version to date of the Administration Guide, intended for use by teachers or other assessors, and showing the instructions, tasks, prompts and things to look for at each stage. This document would guide the administration of the assessment for students at any level Using the guide seen in Figure 4.3, should direct teachers and other assessors in how to administer the assessment, and help in the completion of the Scoring Guide seen in the following section
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! 63 Figure 4.3 Administration Guide Administration Guide Assessment for Multiplicative Reasoning Gene ral Directions Please read each task exactly as written. Students may also view the task and read it on their own from the problem strip as you read it aloud. Administration begins with the Composite Unit Iteration task, and from there will move to eith er Counting On or Multiplicative Double Counting, depending on student response. The progression is shown in the flow chart for Administration Protocol, and is indicated in this guide. The problem numbers are listed on the far left for easy reference. Directions and numbers in task may be repeated indefinitely, and students may write down important information or numerals as needed. Clarification about what is meant by the problem is permitted, but students should do all solving. If possible, stick t o the given prompts. Students may use scratch paper, and all marks made by students should be kept a nd labeled with the problem number When administering these tasks, ask the student to explain their thinking, and record everything the student does on t he scoring sheet. You may use questions such as: "What were you thinking as you solved this problem?" "What strategies did you use?" "Did you picture something in your head to help you solve this problem?" Begin with the level appropriate for the child's grade level. If they are not successful at this task, move down one level and test again. Please explain to the student that you will be recording their responses and their work so that we can learn more about how they solve problems. Do not indicate to the student whether they are correct or incorrect as they work.
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! 64 Administration Sequence Composite *&"'! +',.'"%&!/.01 23%45,6!78 Incorrect Solution Correct Solution Additive Tasks (Problems 2 13) Multiplicative Tasks (Problems 14 25) Counting On (Problems 2 5) Missing Addend Tasks (Problems 6 13) Multiplicative Double Counting (Problems 14 25) Figure 4.3. Administration Guide cont'd If students solve the addend tasks correctly, begin the Multiplicative Tasks. It may be helpful to begin at a lower grade band if the students cannot reason with the numbers provided.
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! 65 Figure 4.3. Administration Guide cont'd. Composite Unit Iteration Task Please solve this problem, and explain as much as you can about what you're thinking as you do so. (Clarify the meaning of "pair".) 1. Blanca has 8 pairs of socks. If she counts each sock, how many will there be? If the student counts one by one, and not by composite units of two, ask if they can use the way the socks are grouped into pairs. If the students cannot solve after this prompt, begin with the Counting On Task (problem 2). If the student is able to iterate the socks to find that there are 16 in all, skip to the Multiplicative Double Counting Tasks (problem 14). Counting on task Please solve this problem, and explain as much as you can about what you're thinking as you do so. Level 1, K 2 2. Lisa had five ( 5 ) pencils. She finds four ( 4 ) more pencils. How many pencils does Lisa have? Level 2, grades 3 4 3. Lisa had seventeen (17) pencils. She finds four (4) more pencils. How many pencils does Lisa have? Level 3, grades 5 6 4. Lisa had twenty four (24) pencils. She finds seven (7) more pencils. How many pencils does Lisa have? Level 3, grades 6 8 5. Lisa had fifty seven (57) pencils. She finds nine (9) more pencils. How many pencils does Lisa have? o If the stude nt has answered correctly, and shown evidence of counting on, go to the next problem.
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! 66 For both correct responses without reasoning and incorrect responses: o Ask students to explain their thinking on this task. It is important to discourage counting all as a solution. Prompt the student to use counting on by covering the first composite unit, or asking them to make a drawing that does not include all of the o nes in the unit. If the student is using marks or manipulatives, it may be necessary to cover the first composite unit and then see if the student can then proceed to count on. o If at any time during the administration of this task you believe that the nu mber values are impeding the progress of the student's progress move down to a lower grade band and test again. o If at any time the student answers correctly using counting on, go to the next problem. If response is correct, but the students use an unknown strategy, cannot articulate one, use counting all or known math facts, prompt student using these questions: 1. How many did she have in the beginning? 2. How many did she find? 3. Can you use the number she had in the beginning as a starting place to s ee how many in all? How? If the student uses counting on after prompting, move to the next problem. If response is incorrect: Ask students to explain their thinking on this task. Produce manipulatives, and ask the student to solve with the manipulatives If solved correctly with manipulatives, go to the next problem. If you believe that the number values are impeding the progress of the student move down to a lower grade band and test again. If they do not solve correctly, prompt students using the foll owing questions: How many did she have in the beginning? How many did she find? Can you use the number she had in the beginning as a starting place to see how many in all? How? If the student uses counting on after prompting, move to the next problem If the student cannot solve this problem after prompting, end testing.
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! 67 Additive Tasks Missing Addend Tasks Administer both of these tasks, regardless of the results, to assess the additive reasoning of the student. Missing second addend task Please solve this problem, and explain as much as you can about what you're thinking as you do so. Level 1 K 2 6. Joseph has six ( 6 ) pieces of gum. His friend gives him some more pieces of gum. Now Joseph has nine ( 9 ) pieces of gum. How many pieces of gum did his friend give him? Level 2, grades 3 4 7. Joseph has (8) pieces of gum. His friend gives him some more pieces of gum. Now Joseph has fifteen (15) pieces of gum. How many pieces of gum did his friend give him? 8. Level 3, grades 5 6 J oseph has fifteen (15) pieces of gum. His friend gives him some more pieces of gum. Now Joseph has twenty three (23) pieces of gum. How many pieces of gum did his friend give him? 9. Level 4, grades 6 8 Joseph has twenty five (25) pieces of gum. His fr iend gives him some more pieces of gum. Now Joseph has thirty six (36) pieces of gum. How many pieces of gum did his friend give him? If the student uses counting on or counting back, move to the next problem For both correct responses without reasoni ng and incorrect responses: o Ask students to explain their thinking on this task. It is important to discourage counting all as a solution. Prompt the student to use counting on by covering the first composite unit, or asking them to make a drawing that d oes not include all of the ones in the unit. If the student is using marks or manipulatives, it may be necessary to cover the first composite unit and then see if the student can then proceed to count on. o If at any time during the administration of this task you believe that the number values are impeding the student, move down to a lower grade band and test again. o If at any time the student answers correctly using counting on, go to the next problem.
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! 68 If response is correct, but the students use an unknown strategy, cannot articulate one, use counting all or known math facts, prompt student using these questions: 1. How many did he have in the beginning? How many does he have in the end? Can you use the number in that he had in the beginning as a sta rting place see how many more he got from his friend? How? 2. Can you use the number he had in all and count backwards to get to find out how many he got from his friend? How? If the student solves correctly after prompting, move to the next problem. If the student cannot solve this problem after prompting, end testing. If response is incorrect: Ask student to explain their thinking on this task. Produce manipulatives, and ask the student to solve with the manipulatives. 1. If solved correctly with manipula tives, go to the next problem. 2. If you believe that the number values are impeding the student, move down to a lower grade band and test again. 3. If they do not solve correctly, prompt students using the following questions: 4. How many did he have in the beginn ing? How many does he have in the end? Can you use the number in that he had in the beginning as a starting place see how many more he got from his friend? How? 5. Can you use the number he had in all and count backwards to get to find out how many he got from his friend? How? If the student solves correctly after prompting, move to the next problem If the student cannot solve this problem after prompting, end testing. Missing first addend task Please solve this problem, and explain as much as you can about what you're thinking as you do so. Level 1, K 2 10. Anna has some bouncy balls. Her mom gives her three ( 3 ) more bouncy balls. She then has eight ( 8 ) bouncy balls. How many bouncy balls did Anna have in the beginning? Level 2, grades 3 4 11. Anna has some bouncy balls. Her mom gives her thirteen (13) more bouncy balls. She then has nineteen (19) bouncy balls. How many bouncy balls did Anna have in the beginning? Level 3, grades 5 6 12. Anna has some bouncy balls. Her mom gives her sixteen (16) m ore bouncy balls. She then has twenty four (24) bouncy balls. How many bouncy balls did Anna have in the beginning? Level 4, grades 7 8 13. Anna has some bouncy balls. Her mom gives her twenty five (25) more bouncy balls. She then has thirty eight (38) bouncy balls. How many bouncy balls did Anna have in the beginning?
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! 69 If the student uses counting on or counting back, end assessment. For both correct responses without reasoning and incorrect responses: o Ask students to explain their thinking on this task. It is important to discourage counting all as a solution. Prompt the student to use counting on by covering the first composite unit, or asking them to make a drawing that does not include all of the o nes in the unit. If the student is using marks or manipulatives, it may be necessary to cover the first composite unit and then see if the student can then proceed to count on. o If at any time during the administration of this task you believe that the nu mber values are impeding the student, move down to a lower grade band and test again. o If at any time the student answers correctly using counting on or counting back, go to the next problem. If response is correct, but the students use an unknown strategy cannot articulate one, use counting all or known math facts, prompt student using these questions: 1. How many did she have before her mom gave her more? 2. How many does she have in the end? Can you use the number in that she had before as a starting place t o see how many more she got from her mom? How? 3. Can you use the number she had in all and count backwards to find out how many she got from her mom? How? If the student solves correctly after prompting, move to the next problem If response is incorrect: Ask student to explain their thinking on this task. Produce manipulatives, and ask the student to solve with the manipulatives. If they does not solve correctly with manipulatives, prompt students using the following questions: 1. How many did she have befor e her mom gave her more? 2. How many does she have in the end? Can you use the number in that she had before as a starting place to see how many more she got from her mom? How? 3. Can you use the number she had in all and count backwards to find out how many s he got from her mom? How? If the student solves correctly after prompting, move to the next problem If the student cannot solve this problem after prompting, end testing.
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! 70 Figure 4.3. Administration Guide cont'd Multiplicative Tasks Multiplicative Double Counting Tasks There are three tasks in this section. A successful solution to two out of three is considered evidence that the student has access to the Multiplicative Double Counting Scheme, and testing should be concluded when the student has successfully completed two. mDC Task 1 Level 1, K 2 14 Tony has his toy cars in small boxes in his room. He has four (4) boxes, and each box has three (3) cars in it. How many cars does Tony have in all? Level 2, grades 3 4 15. Tony has his toy cars in small boxes in his room. He has six (6) boxes, and each box has three (3) cars in it. How many cars does Tony have in all? Level 3, grades 5 6 16. Tony has his toy cars in small boxes in his room. He has six (6) boxes, and each box has f ive (5) cars in it. How many cars does Tony have in all? Level 4, grades 7 8 17. Tony has his toy cars in small boxes in his room. He has seven (7) boxes, and each box has six (6) cars in it. How many cars does Tony have in all? If response is correct : Ask student to explain their thinking on this task. Student may use manipulatives, fingers, or pencil and paper to prove their solution. If the student gives an answer without units, ask what the number tells. Is it number of cars? Number of boxes? If correct, and using mDC strategy to find the answer, proceed to mDC task 2. If they use counting all, or known math facts, prompt student using these questions: 1. How many cars in one box? 2. Repeat original question 3. How many cars would be in two boxes? 4. Repeat original question
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! 71 Figure 4.3. Administration Guide cont'd If response is incorrect: Ask student to explain their thinking on this task. Student may use manipulatives, fingers, or pencil and paper to prove their solution. As soon as the student is able to give the answer correctly, move to mDC task 2. Prompt using these questions, encouraging the student to use scratch paper, fingers or manipulatives to help him/her keep track. 1. How many cars in one box? 2. Repeat original question 3. How ma ny cars would be in two boxes? 4. Repeat original question Regardless of whether the student answers correctly or incorrectly, go on to mDC Task 2. mDC Task 2 Level 1, K 2 18. Ella has five (5) baskets for her friends. She wants to put three (3) pieces of candy in each basket. How many pieces of candy will Ella need? Level 2, 3 4 19. Ella has five (5) baskets for her friends. She wants to put four (4) pieces of candy in each basket. How many pieces of candy will Ella need? Level 3, 5 6 20. Ella has seve n (7) baskets for her friends. She wants to put four (4) pieces of candy in each basket. How many pieces of candy will Ella need? Level 4, 7 8 21. Ella has eight (8) baskets for her friends. She wants to put seven (7) pieces of candy in each basket. Ho w many pieces of candy will Ella need? If response is correct: Ask student to explain their thinking on this task. Student may use manipulatives, fingers, or pencil and paper to prove their solution. If the student gives an answer without units, ask what the number tells. Is it number of baskets? Number of candy pieces? If correct, and using mDC strategy to find the answer, conclude testing. If they use counting all, or known math facts prompt student using these questions: 1. How many candy pieces in one bag? 2. Repeat original question 3. How many candy pieces would be in two bags? 4. Repeat original question
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! 72 Figure 4.3. Administration Guide cont'd If response is incorrect: Ask student to explain their thinking on this task. Student may use manipulatives, fingers, or pencil and paper to prove their solution. As soon as the student is able to give the answer correctly, move to mDC task 3. Prompt using these questions, encouraging the student to use scratch paper, fingers or manipulatives to he lp him/her keep track. 1. How many candy pieces in one bag? 2. Repeat original question 3. How many candy pieces would be in two bags? 4. Repeat original question If both mDC tasks were answered correctly, conclude testing If one mDC task is answered correctly, a dminister mDC Task 3. If both mDC tasks were answered incorrectly, conclude testing. mDC Task 3 Level 1, K 2 22. Aaron wants to make three (3) birdhouses, and he needs two (2) nails for each birdhouse. How many nails will he need in all? Level 2, 3 4 23. Aaron wants to make six (6) birdhouses, and he needs four (4) nails for each birdhouse. How many nails will he need in all? Level 3, 5 6 24 Aaron wants to make eight (8) birdhouses, and he needs four (4) nails for each birdhouse. How many nails wil l he need in all? Level 4, 7 8 25. Aaron wants to make nine (9) birdhouses, and he needs eight (8) nails for each birdhouse. How many nails will he need in all? If response is correct: Ask student to explain their thinking on this task. Student may use manipulatives, fingers, or pencil and paper to prove their solution. If the student gives an answer without units, ask what the number tells. Is it number of nails? Number of birdhouses? If correct, and using mDC strategy to find the answer, conclude te sting. If the student uses counting all or known math facts, prompt student using these questions: 1. How many nails does he need for one birdhouse? 2. Repeat original question 3. How many nails does he need for two birdhouses? 4. Repeat original question
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! 73 Figure 4.3. Administration Guide cont'd Scoring Guide Scoring of students' reasoning based on their solutions including both answer and explanation for the tasks can optimally occur during the interview process, either by the assessor or another observer. The assessment is meant to be adapted to what is seen in student responses (e.g., counting 1s or composite units, visible or figural, etc.) so that care is taken to administe r appropriate tasks to appropriate students. While the time and care needed to do this type of assessment is significant, the benefits of having real time scoring and the ability to probe and question a student as they are working gives a better understan ding of the thinking and level of reasoning being employed by the student (Ginsberg, 1981 ; Steffe, 2002 ) As students complete tasks, the assessor will record the respons e on the scoring guide seen in Figure 4.4 including but not limited to the use of fi gural items, manipulatives, drawings or computational procedures. The extra space provided in the middle of each page is intended for assessor's note taking. If response is incorrect: Ask student to explain their thinking on this task. Student may use manipulatives, fingers, or pencil and paper to prove their solution. As soon as the student is able to give the answer correctly, conclude testing. Prompt using these questions, encouraging the student to use scratch paper, fingers or manipulatives to help him/her keep track. 1. How many nails does he need for one birdhouse? 2. Repeat original question 3. How many nails does he need for two birdhouses? 4. Repeat original question. End of Test
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! 74 Figure 4.4 Scoring Guide Scoring Guide Assessment for Multiplicative Reasoning Grade Level of Student ______________ Composite Unit Iteration Task (CU) Indicate any procedures that students may use, including but not limited to the following: Counts on fingers Draws or uses marks Other behaviors (please describe) S tudent Response Stage Indicated by this Result Correct Response to grade level problem without prompts. Anticipatory at grade level for this scheme Correct Response to grade level problem with prompts. Participatory at grade level for this scheme Correct Response to below grade level problem without prompts. Anticipatory, but below grade level for this scheme Correct Response to below grade level problem with prompts. Participatory and below grade level for this scheme Incorrect Response with prompts. Pre Participatory for this scheme
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! 75 Figure 4.4 Scoring Guide cont'd. Counting on task Indicate any procedures that students may use, including but not limited to the following: Counts on fingers Draws or uses marks Writes algorithm ( Orient the student to avoid us ing an algorithm before other methods of computing mentally have been exhausted.) Other behaviors (please describe) S tudent Response Stage Indicated by this Result Correct Response to grade level problem without prompts. Anticipatory at grade level for this scheme Correct Response to grade level problem with prompts. Participatory at grade level for this scheme Correct Response to below grade level problem without prompts. Anticipatory, but below grad e level for this scheme Correct Response to below grade level problem with prompts. Participatory and below grade level for this scheme Incorrect Response with prompts. Pre Participatory for this scheme
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! 76 Figure 4.4 Scoring Guide cont'd Missing Second Addend (MSA) Indicate any procedures that students may use, including but not limited to the following: Counts on fingers Draws or uses marks Writes algorithm ( Orient the student to avoid us ing an algorithm before other methods of computing mentally have been exhausted.) Other behaviors (please describe) S tudent Response Stage Indicated by this Result Correct Response to grade level problem without prompts. Anticipatory at grade level for this scheme Correct Response to grade level problem with prompts. Participatory at grade level for this scheme Correct Response to below grade level problem without prompts. Anticipatory, but below grade level for this scheme Correct Response to below grade level problem with prompts. Participatory and below grade level for this scheme Incorrect Response with prompts. Pre Participatory for this scheme
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! 77 Figure 4.4 Scoring Guide cont' d Missing First Addend Indicate any procedures that students may use, including but not limited to the following: Counts on fingers Draws or uses marks Writes algorithm ( Orient the student to avoid us ing an algorithm before other methods of computing mentally have b een exhausted.) Other behaviors (please describe) S tudent Response Stage Indicated by this Result Correct Response to grade level problem without prompts. Anticipatory at grade level for this scheme Correct Response to grade level problem with prompts. Participatory at grade level for this scheme Correct Response to below grade level problem without prompts. Anticipatory, but below grade level for this scheme Correct Response to below grade level problem with prompts. Participatory and below grade level for this scheme Incorrect Response with prompts. Pre Participatory for this scheme
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! 78 Figure 4.4 Scoring Guide cont'd Multiplicative Double Counting 1 (mDC1) Indicate any procedures that students may use, including but not limited to the following: Counts on fingers Draws or uses marks Writes algorithm ( Orient the student to avoid us ing an algorithm before other methods of computing mentally have been exhausted.) Other behaviors (please describe) Student Response Stage Indicated by this Result Correct Response to grade level problem without prompts. Anticipatory at grade level for this scheme Correct Response to grade level problem with prompts. Participatory at grade level for this scheme Correct Response to below grade level problem without prompts. Anticipatory, but below grade level for this scheme Correct Response to below grade level problem with prompts. Participatory and below grade level for this scheme Incorrect Response with prompts. Pre Participatory for this scheme
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! 79 Figure 4.4 Scoring Guide cont'd. Multiplicative Double Counting 2 (mDC2) Indicate any procedures that students may use, including but not limited to the following: Counts on fingers Draws or uses marks Writes algorithm (Orient the student to avoid us ing an algorithm before other methods of computing mentally have been exhausted.) Other behaviors (please describe) Student Response Stage Indicated by this Result Correct Response to grade level problem without prompts. Anticipatory at grade level for this scheme Correct Response to grade level problem with prompts. Participatory at grade level for this scheme Correct Response to below grade level problem without prompts. Anticipatory, but below grade level for this scheme Correct Response to below grade level problem with prompts. Participatory and below grade level for this scheme Incorrect Response with prompts. Pre Participatory for this scheme
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! 80 Figure 4.4 Scoring Guide cont'd. Multiplicative Double Counting 3 (mDC3) Indicate any procedures that students may use, including but not limited to the following: Counts on fingers Draws or uses marks Writes algorithm (Orient the student to avoid us ing an algorithm before other methods of computing mentally have been exhausted.) Other behaviors (please describe) Student Response Stage Indicated by this Result Correct Response to grade level problem without prompts. Anticipatory at grade level for this scheme Correct Response to grade level problem with prompts. Participatory at grade level for this scheme Correct Response to below grade level problem without prompts. Anticipatory, but below grade level for this scheme Correct Response to below grade level problem with prompts. Participatory and below grade level for this scheme Incorrect Response with prompts. Pre Participatory for this scheme
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! 81 Using the Scoring guide should allow assessors to determine at which stage, participatory or anticipatory students seem to operate in each scheme tested It may also help to guide prompts during the interview process, as it becomes clear the evidence needed to prove competence at each level. Reliability and Validity The A MR is work in progress in its preliminary stages It has not yet been fully tested for reliab ility or valid ity ; h owever, attempts made to increase these two measures of the instrument are discussed here. The validity measures taken in the development of the AMR were based upon content validity according to expert opinion, both classroom teachers and researchers. The items were developed under the advisement of Dr. Ron Tzur of the University of Colorado, D enver. His guidance focused on ensuring that the tasks were created to reflect the theoretical stances described in Chapter 2. Using information about the numbers that would be most appropriate at each grade band, and tying the tasks closely to the six S chemes of Multiplicative Reasoning discussed earlier, the items are believed to be good indicators of the reasoning at each level. As explained in Chapter III t hese items were refined both by feedback from teachers and through informal testing with fourth grade students. I was able to observe students as they participated in the assessment and also received feedback from the reports of Dr. T zur and Mr. McClintock on later administrations of the tasks I was unable to witness Additionally, I solicite d critique by way of a F eedback F orm from classroom teachers, as d iscussed in context in Chapter IV The F orm used to collect this data is found in the Appendix to this thesis
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! 82 Another reliability measure is th e two fold problem delivery of verbal directi ons and printed student problem strips shown earlier in this section. These seemingly redundant steps are to remove as many barriers to the students' interpretation of the problem as possible. Students who have difficulty interpreting verbal questions have access to the printed problem, while those who have difficulty with reading have the verbal instructions read by the assessor The ability to refer to the card and clarify numbers or any other aspect of the problem, as well as review any information, was deliberately included to ensure that incorrect solutions were not due to misunderstandings or difficulty remembering the information given in the tasks The choice of numbers for each problem type and each grade level was purposeful as supported b y the wor k of Sherin and Fuson (2005). The strategy used by students in that study varied depending on the numbers and context used It showed that the choice of each number used in a task must be made mindfully to elicit the strategy for which it is desi gned If the numbers and situations for these problems were not chosen carefully, students could use some learned strategies, or known facts, that do not necessarily reflect their understanding of composite unit such as how to count on. Sherin and Fuson also note that the only route to this clear, complex, flexible knowledge is the passage through earlier stages, through which they develop strategies rooted in understanding. Clearly, an understanding of the problem is an important facet, and was treated as such using these methods, to provide a reliable assessment. Because the focus of this thesis was on multiplicative reasoning, there are three tasks to add validity to this portion of the assessment. If a student is able to correctly and independently solve two tasks in a row, testing is terminated, and the student is
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! 83 considered to have access to the scheme for Multiplicative Double Counting. If however, the student solves only one out of the first two tasks independently and correctly, the third task is administered to determine whether the student can be considered to have this schem e, or at which stage A score of two out of three correct would indicate the student s read iness to move to the next level of multiplicative reasoning. These steps were taken to further increase reliab ility within the given time and resources available, and could certainly help to improve the AMR further in future endeavors. The following section includes a discussion of the implications and benefits of the development of the AMR, as well as a further discussion of the limitations indicated here.
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! 84 C HAPTER V DISCUSSION Contributions to Research and Practice This section will focus on the ways that the work seen in Chapter IV can benefit the field of mathematics education in terms of research and classroom teaching. The ability to test for and determine not only the multiplicative schemes available for students but also the anticipatory or participatory stages within each sch eme could be a great benefit to teachers and researchers alike. In both the classroom and the field of research, there is a need to find out what students understand about multiplicative reasoning, and whether they have constructed a scheme at the partici patory or anticipatory stage A classroom teacher could use this instrument to determine what students need to be able to move ahead, into multiplicative reasoning, or whether they are solid in their initial stage of multiplicative reasoning and are rea dy to move on to more advanced topic s For example, a student could be presented with the Missing Second Addend task for the 3 4 grade band It reads: "Joseph brings eight (8) pieces of gum to school. His friend gives him some more pieces of gum. When Joseph counts them all, he finds that he has fifteen (15) pieces of gum. How many pieces of gum did his friend give him?" The student's response allows a teacher to distinguish whether a students is at the anticipatory or participatory stage of using composite units within a given, encompassing composite unit If at the participatory, (i.e., can only solve the problem with prompting ) teaching would then focus on moving the student toward an anticipatory stage using her current knowled ge and making prompting explicit so that she will begin to self prompt. However, if the student is at the anticipatory stage ( can solve independent of any
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! 85 prompting while using strong reasoning ) teach ing can move forward to missing second addend tasks, by using the missing first addend concep ts as a bridge to these missing second addend concepts. It may also indicate moving ahead to teaching mDC while keeping in mind to focus on the student's way of reasoning with 1s vs. with composite units. In addition, the assessment should be flexible enough to include a wide range of learners. An advanced first grader could be assessed for compaction of curriculum or advancement to a more challenging math group based on the AMR results. A struggling seventh grader m ight be able to receive needed intervention based on results that show he or she is not able to perform Multiplicative Double Counting. A fifth grade stud ent who does have the availability of the appropriate scheme, but does not compute fluently with whole numbers, could test at a lower grade band and show this reasoning in spite of their computational difficulties. I see t his awareness a s essential for teachers because the connection between multiplicative reasoning and topics such as fractions is n ot always a focus of instruction in those topics though it might be a way to improve it. Fractions are multiplicative quantities because they require a reference to the unit. When fractions are an extension of multiplication, the idea that there are mul tiple ratios to consider in multiplication (Confrey and Smith, 1995) can help creat e a bridge from the whole number to the fractional concepts Some of the ratios indicated in 5 x 7 are each composite unit of five to seven ones, as the relation of the sev en ones in one group and the five groups to the thirty five ones in total. This understanding of multiplication as multiple ratios builds the
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! 86 foundation for students to understand fractional quantities in a way that repeated addition does not. When the meaning of five groups of seven is conceptualized by the student, the understanding that 2/3 means that there are some units of which the whole is three times as much and that we are considering two o f these units as 2 1/3 or 2 groups of 1/ 3, can equip a student to compare relative values of these fractions This thinking also allows a student to understand why in a problem such as 2/3 + 4/5, we must use a common measure for operating on both thirds and fi fth s When students recognize thir ds and fifths as different units, and see this as a way to build upon their knowledge of multiplication to iterate those units, the search for a common measure ( denominator ') is no longer some mysterious invention of teachers of mathematics, but a necessary operation of multiplicative unit coordination need ed to find common units. Lamon ( 1996) saw a similar conception being built in students who use equal sharing tasks to develop their fraction concepts. When students began to see certain quantities as equivalent, they were using multiplicative rather than additive concepts to understand the size of the portion. Being able to see and assess this reasoning could go a long way for classroom teachers seeki ng to improve the teaching and learning of fractions and/or multiplicative reasoning in their classrooms. In research th e AMR may be used for a similar purpose, or to find correlations between multiplicative reasoning and other types of mathematical conceptions. For example, it could be investigated whether a student who has a strong knowledge of area and perimeter was more equ ipped with multiplicative reasoning than one who does not. The AMR could be used to quickly determine the level of a student and then use it to
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! 87 correlate with other factors. There are some limitations to this instrument, however, and they are discussed n ext. Limitations While the AMR provides benefits for teachers and researchers, it is important to be aware of its limitations. It is understood that the time involved in completing an assessment of this type is limiting for most classroom situations. One way to decrease the amount of time needed for testing is to administer, as a written test, the portion that the teacher believes the student has already constructed as a solid conception. In this way the AMR's tasks function as a preliminary screener. For example, if a third grade teacher believes students are proficient in missing second addend problems, the counting on and missing second addend section could be given to students simply as a written test. The students who show clearly that they are an ticipatory for these tasks can begin with the missing first addend for the interview T hose who did not show that they are anticipatory could then be tested further by interview to determine if they may be participatory for that stage or do not yet have the conception at all. As explained in Chapter II the distinction between anticipatory and participatory stages requires the task to be first given without prompts, to d etermine whether the students can anticipate the entire process and if not with pr ompts to assess for a participatory level of understanding. The AMR is a prototype, and only addresses the first scheme in the multiplicative re asoning continuum delineated by Tzur et al. (2013 ), namely, Multiplicative Double Counting. This was done for simplicity, and to allow testing and feedback to determine the foci of later instruments. The remaining stages of multiplicative reasoning could be
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! 88 created in similar way s and would show where more adva nced students fall on the continuum of multiplicative reasoning. Instead of progressing from mDC to more advanced multiplicative schemes, the AMR instrument includes schemes that are prerequisite to multiplicative reasoning, such as counting on. These wer e included to determine the extent to which students have constructed composite units needed for their journey from additive to multiplicative reasoning, or are still developing the understandings on which additive reasoning should be built (Steffe, 1994). In a study conducted with 5 th grade students designated by their school system as learning disabled who did not reason multiplicatively (Tz ur, 2010) it was found that one of the major impediments to their progress was the lack of a concept of number as an abstract, symbolic, composite unit. The AMR is meant to begin with these prerequisite skills to save teachers time when doing assessment s. Clearly the additive portion of the test is not comprehensive enough to account for all aspects of additive reasoning and all of the different types of problem structures and what they may reveal about student thinking. Rather, the additive portion wa s designed to give a glimpse into the schemes available to students, and whether they have the conceptual prerequisites needed to engage in and learn to reason multiplicative ly With this in mind, the assessment only covers counting on, missing second add end problems with the missing portion being smaller, and a missing first addend with the missing potion being smaller. It is understood that this spans a limited range in the additive abilities of students, but does so while providing essential informatio n about the students' use and coordination of units An assessor who finds that students are in this general stage of reasoning and would like to further pinpoint their
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! 89 development, would need to consult a more comprehensive additive reasoning assessment. It is also important to note that this instrument has never been tested in its entirety. Significant changes were made following the initial testing, up to and including the defense phase. The following section suggests possibilities for future resear ch, some of which address the aforementioned limitations. Implications for Future Research A continuation of the assessment to include the remaining items in the framework for multiplicative r easoning, outlined by Tzur et al. (2013 ), would be helpful to t eachers who need to be able to assess all schemes, and stages within them, up to the point in which they may be ready to learn fractions. In particular items could be created for the remaining schemes in thi s frame wor k: S ame Unit Coordination (SUC), Unit Differentiation and Selection ( UDS), Mixed Unit Coordinati on (MUC), Quotitive Division (QD), and Partitive Division (PD), so the assessment is expanded for the entire scope of these schemes. When a student has progressed through all of these s chemes, they may be considered proficient in multiplicative reasoning. Additionally, the prototype of the AMR was written such that a computer based assessment could be built to either complete the preliminary screening questions, used to decide where a st udent might need to begin testing, or as a tool to help the assessor. A software program that could record responses of students, coded by the assessor, and indicate the next task and level that should be administered, could be a valuable tool. For examp le, if the student answers a question correctly with strong reasoning, the technological component could advance the student to the n ext appropriate question. It
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! 90 seems unlikely that the assessment could be productively administered without the human inter view component, due to the importance of the interaction between the assessor and student, and the careful attention to student thinking gleaned from bodily gestures and whispers, which may not be detected, let alone understood, by existing technology. Ad ditionally, teachers may want to know how to meet student needs once their reasonin g, both scheme and stage h as been assessed. This is an excellent opportunity for future work, and would greatly assist teachers in using the data obtained to foster the understandings that they now know students need to prepare them for future work. Concluding Remarks This proje ct was born out of my own frustration, as a middle school teacher, with the low level of knowledge that my students often have about fractional quantities. Too often I saw that though students gained procedural knowledge with repetition and practice, their conceptual understanding remained weak, despite activities that I designed to try to build these conceptions. I truly believe that for teachers and other educators to first and foremost become aware of the gaps in reasoning that cause students in middle school, high school and even higher education to not only dread, but show a limited understanding of the m eaning of fraction, is key. A teacher's possession of both this awareness and the knowledge about schemes used for reasoning multiplicatively, and of anticipatory and participatory stages within each scheme, could go a long way towards improving the teaching and learning of fraction and ratio concepts. Further, I am glad as an educator to have an instrument that allows me to begin assessing the underl ying multiplicative reasoning that my students may or may not have. I foresee more
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! 91 productive lessons when students are equipped with the necessary conceptual prerequisites with which to construct fraction knowledge, as opposed to asking them to operate o n conceptions that they are not able to access due to incomplete multiplicative schemes.
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! 92 REFERENCES Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts. Acquisition of mathematics concepts and processes (pp. 91 126 ) New York: Academic Press. Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. A. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15 (5), 323 341. Bonato, M., Fabbri, S., Umil ta, C., & Zorzi, M. (2007). The mental representation of numerical fractions: Real or integer? Journal of Experimental Psychology Human Perception and Performance 33 (6), 1410 1419. Bright, G. W., Joyner, J. M., & Wallis, C. (2003). Assessing proportional thinking. Mathematics Teaching in the Middle School 9 (3), 166 172. Bryant, B. R., & Maddox, T. (1996). Using alternative assessment techniques to plan and evaluate mathematics instruction." Ariel 129, 6 1. Carpenter, T. P., Hiebert, J., & Moser, J. M. (1981). Problem structure and first grade children's initial solution processes for simple addition and subtraction problems. Journal for Research in Mathematics Education 12 (1), 27 39. Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in grades 1 5. Journal for Research in Mathematics Education, 27 (1), 41 51. Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential f unctions. Journal for Research in Mathematics Education, 26 (1), 66 86. Cramer, K., Behr, M., Post, T., & Lesh, R. (1997a). "Rational Number Project: Fraction Lessons for the Middle Grades: Level 1." Dubuque, IA: Kendall/Hunt Publishing. Cramer, K., Behr, M., Post, T., & Lesh, R. (1997b). "Rational Number Project: Fraction Lessons for the Middle Grades: Level 2." Dubuque, IA: Kendall/Hunt Publishing. De Corte, E., & Verschaffel, L. (1987). The effect of semantic structure on first graders strategies for solving addition and subtraction word problems. Journal for Research in Mathematics Education 18 (5), 363 381. Dehaene, S. (1997). The number sense: How the mind creates mathematics NY: Oxford University. Dewey, J. (1938). Experience and education New York: Collie Erlwanger, S. H. (1973). Benny's Conception of Rules and Answers in IPI Mathematics. Journal of Children's Mathematical Behavior, 1 (2), 7 26. Fan, L., & Zhou, L. (2006). How Have Chinese Students Performed in Mathematics? A Perspective from Large Scale International Mathematics Comparisons. In L. Fan, N. Y. Wong, J. Cai and S. Li (Eds.), How Chinese Learn Mathematics: Perspectives from Insiders (pp. 3 25).
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! 94 Norton, A., & Hackenberg, A. J. (2010). Continuing research on students' fraction schemes. In L.P. Steffe and J. Olive (Eds.), Children's Fractional Knowledge (pp. 341 352). New York, NT: Springer US. Olive, J. (2001). Children's number sequences: An explanation of Steffe's constructs and an extrapolation to rational numbers of arithmetic. The Mathematics Educator 11 (1), 4 9. Piaget, J. (1985). The equilibration of cognitive structures: The central pro blem of intellectual development (T. Brown & K.J. Thampy, Trans.). Chicago: The University of Chicago. Pirie, S. E. B., & Kieren, T. E. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studie s in Mathematics, 26 (2 3), 165 190. Raykov, T ., Marcoulides, G A. (2011). Introduction to Psychometric Theory Taylor & Francis. Retrieved 11 June 2013, from < http://lib.myilibrary.com?ID=303716 > Sherin, B., & Fuson, K. (2005). Multiplication strategies and the appropriation of computational resources. Journal for research in mathematics education 36(4), 347 395. Siegler, R. S., Fazio, L. K., Bailey, D. H., & Zhou, X. (2012). Fractions: the new fr ontier for theories of numerical development. Trends in cognitive sciences 17 ( 1), 13 19. Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive psychology 62 (4), 273 296. Siemon D., & Breed, M. (2010, May). Assessing multiplicative thinking using rich tasks. In A nnual C onference of the Australian Association for Research in Education. (Vol. 13). Downloaded from http://publications.aare.edu.au/06pap/sie06375.pdf Siemon, D., Breed, M., & Virgona, J. (2005). From additive to multiplicative thinking the big challenge of the middle years. In Proceedings of the 42nd Conference of the Mathematical Association of Victoria Siemon, D., Izard, J., Breed, M., & Virgona, J. (2006). The derivation of a learning assessment framework for multiplicative thinking. Sousa, D. A. (2008). How the brain learns mathematics Thousand Oaks, CA: Corwin (Sage). Stafylidou, S., & Vosniadou, S. (2004). The development of students' understandi ng of the numerical value of fractions. Learning and Instruction 14 (5), 503 518. Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4 (3), 259 309. Steffe, L. P. (2002). A new hypothesis concerning children's fractional knowledge. Journal of Mathematical Behavior, 20 1 41. Steffe, L. P., & Cobb, P. (1998). Multiplicative and divisional schemes. Focus on Learning Problems in Mathematics, 20 (1), 45 62. Steffe, L. P. (1970). Differential performance of first grade children when solving arithmetic addition problems. Journal for Research in Mathematics Education 1 (3), 144 161.
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! 95 Steffe, Leslie P. (2002). The constructivist teaching experiment: Illustration s and implications. In Radical constructivism in mathematics education (pp. 177 194). Netherlands: Springer. Streefland, L. (1991). Fractions in Realistic Mathematics Education: A Paradigm of Developmental Research Dordrecht, The Netherlands: Kluwer. Streefland, L. (1993). Fractions: A realistic approach. In Carpenter, Thomas P., Fennema, Elizabeth, and Romberg, Thomas A. (Eds.), Rational numbers: An integration of research 289 325. New York, NY :Routledge. Thompson, P. W., & Saldanha, L. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A research companion to Principles and standards for school mathematics (pp. 95 113). Reston, VA: Torbeyns, J., GhesquiÂre, P., & Verschaffel L. (2009). Efficiency and flexibility of indirect addition in the domain of multi digit subtraction. Learning and Instruction 19 (1), 1 12. Trochim, W. M., & Donnelly, J. P. (2008). Research methods knowledge base Mason, OH: Atomic Dog/Cengage Learning. Tzur, R. (2004). Teacher and students' joint production of a reversible fraction conception. Journal of Mathematical Behavior, 23 93 114. Tzur, R. (2007). Fine grain assessment of students' mathematical understanding: Participatory and anticipatory stages in learning a new mathematical conception. Educational Studies in Mathematics, 66 (3), 273 291. Tzur, R., Johnson, H. L., McClintock, E., Kenney, R. H., Xin, Y. P., Si, L., .... Jin, X. (2013). Distinguishing schemes and tasks in children's development of multiplicative reasoning. PNA, 7(3), 85 101. Tzur, R., Johnson, H., McClintock, E., Xin, Y. P., Si, L., Kenney, R., et al. (2013 ). Children's development of multiplicative reasoning: A schemes and tasks framework. In T. Y. Tso (Ed.), Proceedings of the 36th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp 155 162). Taipei, Taiwan: National Taiwan Normal University. Tzur, R., & Lambert, M. A. (2011). Intermediate participatory stages as Zone of Proximal Development correlate in constructing counting on: A plausible conceptual source for children's transito ry regress' to counting all. Journal for Research in Mathematics Education, 42(5), 418 450. Tzur, R., & Simon, M. A. (2004). Distinguishing two stages of mathematics conceptual learning. International Journal of Science and Mathematics Education, 2 287 3 04. Tzur, R., Xin, Y. P., Si, L., Kenney, R., & Guebert, A. (2010). Students with learning disability in math are left behind in multiplicative reasoning? Number as abstract composite unit is a likely "culprit" Paper presented at the American Educational Research Association Retrieved from http://www.eric.ed.gov/
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! 96 Tzur, R., Xin, Y. P., Si, L., Woodward, J., & Jin, X. (2009). Promoting transition from participatory to anticipatory stage: Chad's case of multiplicative mixed unit coordination (MMUC). In M. Tzekaki, M. Kaldrimidou & H. Sakon idis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 249 256). Thessaloniki, Greece: PME. Tzur, R (Personal Communication, November 15, 2012) von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning Washington, D.C.: Falmer. Wilkins, J. L., & Norton, A. (2011). The splitting loope. Journal for Research in Mathematics Educatio n 42 (4), 386 416. Zazkis, R., & Hazzan, O. (1998). Interviewing in mathematics education research: Choosing the questions. The Journal of Mathematical Behavior 17 (4), 429 439. Zhou, Z., Peverly, S. T., & Lin, J. (2005). Understanding early mathematical c ompetencies in American and Chinese children. School Psychology International 26 (4), 413 427.
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! 97 APPENDIX A Assessment for Multiplicative Reasoning Feedback Dear Expert/Teacher: This assessment will be taken by students to determine their level of multiplicative reasoning. Please provide me with your feedback regarding the tasks. Validity Will this task allow me to tell if a student can reason in this way? Grade level appropriateness Are the problem settings and numbers used appropriate for the grade level indicated? Wording (Comments) Please suggest changes that will make the problem clearer for students. Keep/change/omit : Indicate if item can/should be kept in the assessment as i s (1), maintained with some changes (2 add changes in the Comments column), or omitted (3 provide reason). Task Validity? Y/N Grade Level Appropriate? Y/N Comments, Wording Suggestions Keep/Change/Omit Counting On Missing Second Addend Missing First Addend Composite Unit 1 Composite Unit 2 mDC 1 mDC 2 mDC 3

