REPRESENTATIONS IN THE LEARNING OF INFERENTIAL STATISTICS IN
AN ADVANCED PLACEMENT STATISTICS CLASS
by
Steven C. Coddington
B.Sc., Colorado School of Mines, 1975
M.Sc., Colorado School of Mines, 1984
A dissertation submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Educational Leadership and Innovation
2004
iAi
This thesis for the Doctor of Philosophy
degree by
Steven Charles Coddington
has been approved
by
iam U Goodwin
Apr A g
Date
James Loats
Coddington, Steven Charles (Ph.D., Educational Leadership and Innovation)
Representations in the Learning of Inferential Statistics in an Advanced
Placement Statistics Class
Dissertation directed by Professor William Jurascheck
ABSTRACT
Interviews were conducted with six students in an Advanced Placement
Statistics class. The interviews were in the context of working problems in
inferential statistics. Analysis of the interviews showed that a representational
framework is useful in describing learning and in detecting problems associated
with learning. Representation use in five categories was detected. These categories
are imagistic, formal, syntactic, executive control, and affective. This dissertation
presents evidence of both growth and regression in the quality of representations
held by students and evidence of connections and conflicts between representations.
Connections were instances in which problem solving was supported by
representations from two systems. Conflicts were instances in which students were
able to simultaneously hold two conflicting ideas in two categories of
representation. Recommendations for better teaching based on the idea of
representation are presented.
This abstract accurately represents the content of the candidate's thesis. I
recommend its publication.
Signed
in
CONTENTS
Figures.............................................................viii
Tables..............................................................ix
CHAPTER
1. INTRODUCTION.......................................................1
Purpose of the Study................................................1
The Significance of the Problem.....................................2
Conceptual Framework and Operational Definitions....................4
Representations..................................................4
Growth of Representations........................................6
Research Questions.................................................16
Overview of the Methodology........................................16
Structure of the Dissertation......................................17
2. REVIEW OF THE LITERATURE..........................................18
Representations in Mathematics Learning Theory.....................18
External Representations........................................19
Internal Representations........................................20
The Connection Between External and Internal Representations....23
Development of Representations.................................... 24
Representation in the Context of Learning Theories.................25
Pedagogical Considerations Regarding Representations...............31
Other Related Concepts in Cognitive Science........................35
Literature on TaskBased Interviews................................37
Interviews for the Purpose of Inferring Internal Representations...39
Summary............................................................40
3. METHODOLOGY.......................................................41
Approach and Rationale.............................................41
The Role of the Researcher.........................................42
Data Collection Methods............................................42
Prompts.........................................................43
Subjects........................................................52
Site Selection..................................................53
Data Analysis......................................................53
xv
Clinical Interviewing.................................................53
Reliability...........................................................57
Threats to Reliability................................................57
Validity..............................................................59
Internal Threats to Validity.......................................60
External Threats to Validity.......................................62
4. RESULTS OF INTERVIEWS................................................64
Representation 1 Imagistic..........................................65
Subject 1 LM.....................................................71
Subject 2 LW.....................................................82
Subject 3 SH.....................................................94
Subject 4 CR....................................................102
Subject 5 SP....................................................109
Subject 6RM......................................................115
Representation 2 Formal Representations............................118
Subject 1 LM....................................................120
Subject 2LW......................................................125
Subject 3SH......................................................135
Subject 4 CR.................................................. 137
Subject 5 SP....................................................144
Subject 6 RM....................................................146
Representation 3 Syntactic.........................................148
Subject 1 LM....................................................149
Subject 2LW......................................................150
Subject3SH.......................................................153
Subject 4 CR....................................................154
Subject 5 SP....................................................155
Subject 6RM......................................................155
Representation 4 Executive Control.................................156
Subject 1 LM....................................................157
Subject 2LW......................................................161
Subject 3 SH....................................................165
Subject 4 CR....................................................167
Subject 5 SP....................................................167
Subject 6RM......................................................170
Representation 5 Affective Representation..........................171
Subject 1 LM....................................................172
Subject 2LW......................................................176
Subject3SH.......................................................178
Subject 4CR......................................................180
Subject 5SP....................................................181
Subject 6RM....................................................182
Conclusion........................................................183
5. CONCLUSIONS.......................................................184
The Three Research Questions......................................184
Recommendations...................................................186
Ambiguities In the Model........................................186
Transfer of Knowledge to Other Domains..........................187
Connection of Three Scales on a Sampling Distribution Graph.....188
Inability to Name the Midpoint of a Confidence Interval.........190
Identification of Means With t Distribution.....................191
Use of Difference Scale When Comparing Two Samples..............192
Importance of Large Values for p, t, and z......................192
p and Complement of p for Reversed Alternate Hypothesis.........193
Assigning Importance to Commonly Used Numbers...................194
Importance of Interviews in Assessment of Understanding.........195
Summary of Conclusions and Recommendations......................195
APPENDIX
A. Student Written Work.............................................197
Subject 1 LM....................................................197
Subject 2 LW.....................................................205
Subject 3 SH.....................................................212
Subject 4 CR....................................................219
Subject5SP.......................................................229
Subject 6 RM....................................................236
B. Transcripts......................................................241
Subject 1 LM....................................................241
Prompt 1: Residuals and Regression..............................241
Prompt 2: Central Limit Theorem and Sampling Distributions......244
Prompt 3: Inference With One Proportion.........................246
Prompt 4: Inference With One Mean...............................248
Prompt 5: Inference With Two Means..............................253
Subject 2 LW....................................................256
Prompt 1: Residuals and Regression..............................256
.Prompt 2: Central Limit Theorem and Sampling Distributions.....259
Prompt 2: Central Limit Theorem and Sampling Distributions......260
Prompt 3: Inference with One Proportion.........................266
vi
Prompt 4: Inference with One Mean..................................270
Prompt 5: Inference with Two Means.................................275
Subject 3SH.........................................................281
Prompt 1: Residuals and Regression.................................281
Prompt 2: Central Limit Theorem and Sampling Distributions.........284
Prompt 3: Inference with One Proportion............................288
Prompt 4: Inference with One Mean..................................291
Prompt 5: Inference with Two Means.................................293
Subject 4 CR.......................................................296
Prompt 1: Residuals and Regression.................................296
Prompt 2: Central Limit Theorem and Sampling Distributions.........300
Prompt 3: Inference with One Proportion............................304
Prompt 4: Inference with One Mean..................................308
Prompt 5: Inference with Two Means.................................312
Subject 5 SP.......................................................316
Prompt 1: Residuals and Regression.................................316
Prompt 2: Central Limit Theorem and Sampling Distributions.........319
Prompt 3: Inference with One Proportion............................323
Prompt 4: Inference with One Mean..................................327
Prompt 5: Inference with Two Means.................................330
Subject 6 RM.......................................................333
Prompt 1: Residuals and Regression.................................333
Prompt 3: Inference with One Proportion............................337
Prompt 4: Inference with One Mean..................................339
Prompt 5: Inference with Two Means.................................341
REFERENCES...............................................................343
Vll
FIGURES
Figure
3.1 Prompt I...........................................................45
vm
TABLES
Table
2.1 Descriptions of Growth Corresponding to the Initial Stage..........9
2.2 Descriptions of Growth Corresponding to the Development Stage.....11
2.3 Descriptions of Growth Corresponding to the Completion Stage......13
2.4 Growth Framework for Current Project..............................15
IX
CHAPTER 1
INTRODUCTION
This is an observational study of the internal representations formed by
students in an Advanced Placement (AP) Statistics class. Representations are the
cognitive objects used to hold concepts in memory. They provide an organizing
framework for studying the development of concepts in statistics. Through the use
of interviews and observation of problem solving during oneonone interviews, the
researcher inferred the existence and development of representations as posited in
several studies (Goldin, 1987; Goldin & Shteingold, 2001; Janvier, 1987; Kaput,
1987).
This chapter includes a description of the problem being studied, the
significance of the problem, the conceptual framework, research questions,
operational definitions, an overview of the methodology, and a description of the
structure of this dissertation.
Purpose of the Study
This study was undertaken to examine the use of representations by students
in an introductory statistics class. Representations provide an organizing framework
for the study of knowledge acquisition. The research has the goal of describing
1
learning using the idea of representation and making recommendations for teaching
based on the findings of the study.
The Significance of the Problem
Beginning in the 1980's, and continuing to the present, documents such as A
Nation at Risk (National Commission of Excellence in Education, 1983), the
SCANS report (United States Department of Labor: The Secretary's Commission on
Achieving Necessaiy Skills, 1992) and the Curriculum and Evaluation Standards
(National Council of Teachers of Mathematics, 1989) identified a need to improve
mathematics instruction at the school level in the United States. Among the calls
for reform was the need to include statistical methods in the K12 curriculum. In
Colorado, the state standards call for students to have experience in descriptive and
inferential statistics (Colorado Department of Education, 1991) including the design
and interpretation of experiments, combinatorics and probability, and the use of
informal inferential statistics.
The American Statistical Association (1994) has proposed a challenging
syllabus of elements of statistics that students graduating from high school should
have studied. These elements include planning a survey or experiment (and
therefore, understanding issues of sampling and design of experiments), parameter
estimation including confidence intervals, hypothesis testing, and regression. While
the American Statistical Association recommendations recognize that these topics
2
should not be taught in the same way that they are taught in introductory statistics
classes in college, the syllabus is quite similar to that type of course and is very
challenging when compared with current secondary curricula.
Statistics is notoriously difficult for most students to grasp (Wilensky, 1997).
The reform of teaching at the college level has received some attention within the
milieu of the general mathematical reform efforts. The Mathematical Association of
America (Gordon & Gordon, 1992) has called for the reform of syllabi and teaching
methods in introductory college courses. Advanced Placement courses are
considered equivalent to college level courses.
Students of statistics often fail to understand the underlying principles of the
discipline. It is a commonplace for students to be able to apply algorithms and
obtain answers without understanding what they are doing. Wilinsky (1997)
reported that students experience "epistemological anxiety" when they apply
methods without understanding. He identified reasons why statistics is a difficult
subject for students. Among these are the difficulties in conceiving of a data set as a
unit, difficulties with the philosophical basis of probability, and difficulty in
selecting procedures and applying the multiple steps of procedures. The idea of
"representation" can be used by teachers to help students be more successful at
learning and at problem solving.
3
Conceptual Framework and Operational Definitions
Representations
Representations are a relatively new concept in cognitive science and in
mathematics education (Janvier, 1987; Kosslyn, 1980; Palmer, 1977).
Representations, as used in mathematics education, have been defined (Goldin &
Janvier, 1998) to include the following four categories: external structured situations
in the physical environment, linguistic embodiments, formal mathematical
statements, and internal mental states.
Internal representations are the focus of the present study, but external
representations will also be described when they are used to report, explain, or
support internal representations. According to Goldin and Kaput (1996):
There are five types of internal representations. These are
a) verbal/syntactic, b) imagistic, c) formal systems, d) a system of
planning, monitoring, and executive control, and e) affective. It is
claimed that these are fundamental in that they exist in all learners,
and with the possible exception of formal systems, are present in all
cognitive activity ( p.417).
Verbal/syntactic are representations stored as sentences or words. They
include word to word correspondence, including synonyms, and other dictionary
type information. There may be statements, such as "a squared plus b squared
equals c squared." Such statements may or may not be related to information about
right triangles and the ability to do computations. Rather, they may be rote
4
memorizations that are not connected to imagistic or formal representations.
Imagistic systems include visual/spatial, auditory/rhythmic, and
tactile/kinesthetic systems. Imagistic systems may be necessary for the meaningful
interpretation of verbal representations. Visual/spatial representations include
quasipictorial information such as maps and geometric figures. Internal
auditory/rhythmic representation is observed when children learning to count or
recite multiplication facts do so in rhythm. Tactile/kinesthetic representation refers
to imagined physical actions by or on the person.
Formal notational systems are usually seen as external representations, for
example, equations. The competencies used to write, manipulate, and interpret
these equations are internal representations. These are observed when a learner
talks about the rules for operating within a formal system.
A system of planning, monitoring, and executive control includes the
heuristic processes used to monitor and control the problem solving process. These
can be inferred to exist when a student starts applying particular methods, pauses to
check intermediate results, changes course in the middle of the solution process, or
checks a final answer against some standard, such as reasonableness.
A system of affective representation refers to the affective aspects of
problem solving and of student's feelings towards mathematics or education in
general. These representations are dynamic and rapidly changing. They include
5
curiosity, surprise, frustration, anxiety, fear, encouragement, pleasure, and
satisfaction.
The existence of these types of representation is inferred through interview
and observation of problem solving. The interview method is necessary because,
although there are technological methods of observing some mental activity such as
functional magnetic resonance imaging, there is no method allowing for the direct
observation of the types of internal mental structures with which this study is
concerned.
Growth of Representations
Representations are built up in stages. There are several theories of concept
attainment in the mathematics education literature. Five of these (Cifarelli, 1988;
Dubinsky, 1991; Goldin, 1988; Pirie & Kieren, 1994; Sfard, 1991) have been used
to synthesize the descriptions of levels of growth in the present study. Of these,
only Goldins is written to explicitly describe the growth of representations. The
others refer to the development of mathematical ideas in general, without referring
to representations. Since they deal with cognitive development, they are applicable
to representations. Five of these frameworks for development are summarized in
Tables 1,2, and 3. The most finely divided of these frameworks is the one proposed
by Pirie and Kieren (1994). This framework has nine levels of development. The
needs of this study are met by a much simpler scheme. The definitions that I will
6
use are a synthesis of these five frameworks. I will be vising three stages to
categorize the attainment level of the subjects of this study: initial, development,
and completion.
The initial stage is characterized by struggling to make new combinations, or
construct new internal representations from external stimuli. This is a creative
stage. The learner may form an adequate representation but it is seen in terms of its
parts, not as a whole. It is possible to have representations that are incorrect, or that
are in logical conflict with other representations. Connections with related
representations may not exist. It is possible for a subject to hold conflicting ideas in
two different representational systems. If connections do exist, they are supportive
and not relational. That is, an older more established representation may be needed
to support the new initial stage representation. An algorithm is seen as a series of
steps that are not solidly connected. The steps are connected due to learned,
external rules. They are not connected by logical necessity, and the algorithm is
seen only as a series of steps, and not as a separate object. The learner is using
previous knowledge as the building blocks of the new knowledge. The learner
cannot imagine working through the process to arrive at a conclusion without
actually doing so, and will therefore be unable to make a reasonable estimate of the
outcome of a calculation. The learner will also have difficulty in recognizing
unreasonable answers to problems. The growth descriptions corresponding to the
initial stage of the five authors listed above are given in Table 1. If the authors use
7
more than three stages, all of the stages corresponding to the initial stage are
included.
8
Table 2.1 Descriptions of Growth Corresponding to the Initial Stage
Goldin, 1988 An inventive or semiotic stage: New characters are created or learned and are used to symbolize aspects of a prior representational system or systems. The prior system may be more primitive or less robust than the one created. An example could be the development of the idea of the normal distribution from experience with the binomial. In this phase the symbols and algorithms for using them are separate.
Sfard, 1991 Interiorization is the stage in which the learner performs operations on lower level mathematical objects. While in this stage the learner must perform an operation in order to think about it. As an example, when learning to perform the long division algorithm, a person in the interiorization stage must consciously think about the steps of the algorithm. When the learner has completed this stage, the process becomes automatic. When the learner divides without thinking about the operations the process has been interiorized.
Ciffareli, 1988 The first level of reflective abstraction as defined by Cifarelli is recognition. Cifarelli described die recognition level as the ability to recognize characteristics of a previously solved problem in a new situation and to believe that one can do again what one did before. Solvers operating at this level would not be able to anticipate sources of difficulty and would be surprised by complications that might occur as they attempted their solution. A student operating at this level would not be able to mentally run through a solution method in order to confirm or reject its usefulness.
Piaget and Beth as summarized in Dubinsky, 1991 Empirical abstraction has to do with experiences that seem entirely external. One can have experience of objects at this level, only by doing something such as looking, touching, or hearing. Different persons under different conditions might perceive differences in property values.
Pirie and Kieren, 1994 Primitive knowing is the starting point at which the teacher believes the learner can operate without initial instruction. Image making is the process of making distinctions about previous knowledge and using it in new ways.
9
The second stage is the development stage. It is characterized by transition
from the initial to the complete stage. The concept is not yet a fully formed object.
The process is beginning to be seen as a subroutine but this transition is not
complete. The process is seen as made from elements of previous learning, and is
still connected to those elements. The process has not yet become an autonomous
object. Students will not be able to think of inputs and outputs of the concept
without thinking of the process itself. The steps of the process cannot be ignored
until the completion stage. Connections with other representations are forming, but
are not complete. It may be possible to arrive at correct answer to problems, but not
be able to explain the answer or to recognize unreasonable answers nor to explain
the connections with other representations. The descriptions of the growth stages
corresponding to the development stage from five other authors are summarized in
Table 2.
10
Table 2.2
Descriptions of Growth Corresponding to the Development Stage
Goldin, 1988 In the period of structural development the development or construction is driven principally by structural features of die earlier system. This process makes use of the symbolization that was established in the first stage. In this way configurations are built up from characters, and a syntax for the new system is constructed. Rules for describing the new system are learned.
Sfard, 1991 Condensation is the stage during which a complex process in reduced into a form that is easier to use and think about It is an idea that is similar to the idea of subroutines in computer programming. The term "squeezing" to describe the process of reducing long algorithms to more manageable units. While in this stage the idea is tied to a procedure. The learner is becoming able to generalize and make comparisons, and is increasingly able to move between various representations of the concept In statistics, the learner has attained condensation when he can think, "find the z score," without thinking of the separate algorithmic steps for doing so.
Cifiareli,1988 The second level of reflective abstraction is representation. Representation is the level where a student becomes able to run through a problem mentally and is able to anticipate potential sources of difficulty and promise. Solvers who operate at this level are more flexible in their thinking and are not only able to recognize similarities between problems, they are also able to notice the differences that might cause them difficulty if they tried to repeat a previously used method of solution. Such solvers could imagine using the methods and could even imagine some of the problems they might encounter but could not take the results as a given. At this level, the subject would be unable to think about potential methods of solution and the anticipated results of such activity.
Piaget and Beth as summarized in Dubinsky (1991) 2) Pseudoempirical abstraction is intermediate between empirical and reflective abstraction. A 1tol relationship of sets is intermediate, because the external properties are used to form the internal abstraction of correspondence.
Pirieand Kieren, 1994 The third level is image having is the ability to work through processes and imagine the results without actually carrying out the operation. The fourth level is property noticing, in which one can manipulate and combine aspects of one's images to abstract new, context specific properties. The fifth level, normalizing, is the abstraction of a method or common quality from the previous image. It is a chunking of noticed properties. A person at this level is able to reflect on and coordinate formal activity. This is called observing.
11
The third stage is the completion stage. It is characterized by the existence
of the representation as an object. The concept stands on its own. Connections with
earlier representations are relational and not supportive. The representation may be
supporting newer, less developed representations. The representation does not
depend on other representations for its completion, rather the concept is on an equal
footing with others. The concept of significance testing in statistics is seen as
connected to confidence intervals, but not dependent on them. The concept is a
unit, and may be used in the formation of new concepts. When the idea of sampling
distribution becomes a unit, it can be used in the creation of other concepts, such as
standard error. The five other authors' descriptions of growth levels corresponding
to the completion stage is given in Table 3.
12
Table 2.3 Descriptions of Growth Corresponding to the Completion Stage
Goldin, 1988 An autonomous stage: The new system of representation separates from the old. It can stand in symbolic relationships with systems different from the one that was the template driving its development. As these new possibilities occur, the transfer of'meaning' (or, in the case of internal representations, of competencies) from old to new domains becomes possible.
Sfard, 1991 Reification is the stage where the solver can conceive of the mathematical concept as a complete object with characteristics of its own. Concepts that have been reified can be thought of in relation to other concepts and can be placed in hierarchies. The first two processes occur gradually, but reification occurs quickly, in one sudden shift. The learner sees a familiar object in a new light.
Ciffareli, 1988 The third level of reflective abstraction is structural abstraction. Structural abstraction is said to occur when the student evaluates solution prospects based on mental runthroughs of potential methods as well as methods that have been used previously. The student is able to discern the characteristics that are necessary to solve die problem and is able to evaluate the merits of a solution method based on these characteristics. This level evidences considerable flexibility of thought.
The final level of reflective abstraction described is structural awareness. A solver operating at this level is able to anticipate the results of potential activity without having to complete a mental runthrough of the solution activity. The problem structure created by the solver has become an object of reflection. The student is able to consider such structures as objects and is able to make judgments about them without resorting to physically or mentally representing methods of solution. The levels of reflective abstraction described above indicate that as solvers attain the higher levels they become increasingly flexible in their thinking.
Piaget (and Beth) as summarized in Dubinsky (1991) Reflective abstraction has a source internal to the subject. It is a general coordination of actions. Generalizations and property isolation are examples. Once properties have been isolated, internal manipulation of the object is possible. New objects are constructed by the conjunction of abstractions. This is a synthesis of abstract properties into a new mental object Interiorization is a type of reflective abstraction. When a reader states 'I can understand each step of the proof; but not the big picture' she is expressing the need for interiorization. Interiorization is the assembly of separate objects into a whole. This unit may then be contained in, or operated on by other units. Coordination is similar to interiorization. Instead of multiple objects being seen as a whole (interiorization), multiple objects are seen as operating in concert to form a new process. The last form of reflective abstraction is the generalization, in which properties, once learned, are applied to novel situations.
Pirieand Kieren, 1994 Structuring occurs when the student attempts to synthesize her observations into a theory. A person at this level is aware of how theorems are interrelated, and call for argument or proof. Inventizing_is the stage at which die learner has a fully developed concept and is able to use that concept as a unit to think about other topics, or to extend the given topic.
13
The three levels of growth used in the current project are a synthesis of the
growth frameworks summarized in Tables 1 to 3. The levels are hierarchical in that
the requirements for higher levels contain the requirements of the lower levels. The
growth framework used in the current project is summarized in Table 2.4.
14
Table 2.4 Growth Framework for Current Project
Growth Level Characteristics of Level
Initial The initial stage is characterized by struggling to make new combinations, or construct new internal representations from external stimuli. This is a creative stage. Representations are seen in terms of their parts rather than as a unit. Representations may be incorrect, or that in logical conflict with other representations. Connections may not exist. Connections are supportive and not relational. The learner cannot imagine working through the process to arrive at a conclusion without actually doing so, and will therefore be unable to make a reasonable estimate of the outcome of a calculation. The learner will also have difficulty in recognizing unreasonable answers to problems.
Development The second stage is the development stage. It is characterized by transition from the initial to the complete stage. The concept is not yet a fully formed object. The process is beginning to be seen as an object. The process is seen as made from elements of previous learning, and is still connected to those elements. Students will not be able to think of inputs and outputs of the concept without thinking of the process itself. Connections with other representations are forming, but are not complete.
Completion The third stage is the completion stage. It is characterized by the existence of the representation as an object. Connections with earlier representations are relational and not supportive. The representation does not depend on other representations for its completion; rather the concept is on an equal footing with others. The concept is a unit, and may be used in the formation of new concepts.
15
Research Questions
The formation of useful representations is an important goal of mathematics
instruction (National Council of Teachers of Mathematics, 2000). It would be
helpful to have knowledge about how representations are formed and used by
students. Understanding the mental state of students could aid teachers in assessing
learning and in planning instruction. Accordingly, I propose the following research
questions:
1. What representations do students construct in an introductory
statistics class?
2. What changes occur in these representations over time?
3. Is there a relationship between the representations formed by
students and their success in solving problems?
These questions were explored by examining the external representations
produced by students in the process of problem solving, and by inferring the
existence of internal representations through clinical interviews and observation of
problem solving.
Overview of the Methodology
Six student volunteers were the subjects in this study. They were
interviewed during five problem solving sessions. They were asked to report on
16
their thinking, the reasons for applying methods, and their preferences for alternate
methods. The interviews were transcribed and examined for evidence of the use of
representations. Level of growth, conflict or connection with other representations,
and success at problem solving were all noted when they occurred.
Structure of the Dissertation
Chapter 1 describes the purpose, the conceptual framework, and an overview
of the methodology. Chapter 2 is a review of the literature. Chapter 3 describes the
methodology, including criticisms of the interview method of gathering data, and
answers to those criticisms. The five prompts and the solution to the problems in
those prompts is also in Chapter 3. Chapter 4 is the analysis of the interviews.
Chapter 5 contains a summaiy of the findings, suggestions for teaching, and
suggestions for further research.
17
CHAPTER 2
REVIEW OF THE LITERATURE
This chapter presents a review of the idea of representation in the cognitive
science and mathematics education literature. Also included are reviews of papers
that criticize and defend the idea of representations in both cognitive science and
mathematics education. The sources of the operational definitions of representation,
categories of representations, and growth of representations are presented here. The
operational definitions themselves are in chapter 1, and are repeated here with the
accompanying references and explanations.
Representations in Mathematics learning Theory
When learners interact with external mathematical (and other) physical
objects, mental objects are formed and stored in memory. These objects are
variously referred to as representations, schemas, and frames (Davis, 1984; Kaput,
1987; Schoenfeld, 1987). There is a lack of clarity in the literature about the precise
definitions and dividing lines between these objects. Generally, schemas and
frames are defined as more complex objects with several parts. They can be thought
of as being made of several representations. The present study focuses on the idea
of representations, especially as used in the mathematics education literature. I will
18
be using the definitions proposed by Goldin and Kaput (1996). Theirs is a
comprehensive set of definitions and is published in mainstream mathematics
education literature.
Goldin and Kaput (1996, p. 400) defined a representation as "a configuration
of some kind that, as a whole or part by part, corresponds to, is referentially
associated with, stands for, symbolizes, interacts in a special manner with, or
otherwise represents something else." These representations may be either external
or internal.
External Representations
External representations are typically on paper or other media and include
such objects as graphs, tables, and symbolic formulas that represent functions,
numerals that represent numbers, maps that represent geographical relations, and
figures that represent geometric relations. They can also be manipulative objects
such as colored counting chips used to teach numeration and computer based
microworlds used to simulate mathematical concepts. "... we use the term external
representation to refer to physically embodied, observable configurations such as
words, graphs, pictures, equations, or computer microworlds. ...Of course, the
interpretation of external representations as belonging to structured systems, and the
interpretation of their representing relationships, is not 'objective* or 'absolute' but
depends on the internal representations of the individual(s) doing the interpreting."
19
(Goldin & Kaput, 1996). Some forms of external representation were of particular
interest in this study. They are the written work produced during the interviews,
verbal and body language communication, and the built in statistical functions of the
programmable graphing calculators used in solving the problems. These were the
external representations used to infer the existence of the student's internal
representations
Internal Representations
The present study is concerned with internal representations. Internal
representations exist in the mind and are private. Goldin and Kaput (1996)
succinctly defined internal representations. "We use the term internal
representation to refer to possible mental configurations of individuals, such as
learners or problem solvers." They may be inferred from selfreports or interviews
with the subject. Goldin and Kaput (1996) distinguished between self reports of
mental activity, which are objectively observable, and the mental objects
themselves, which are at best only observable by the subject, and are susceptible to
error in reporting. The purpose of the present study is to use interviews, written
work, and calculator activity as external representations to infer the existence of and
describe the student's internal representations.
Internal representations have been categorized by Goldin and Kaput (1996)
into five types. These are:
20
a) Verbal/syntactic: These are representations stored as sentences or words. They
include wordtoword correspondence, including synonyms and other dictionary
type information. There may be memorized statements such as "a squared plus
b squared equals c squared." Such statements may or not be related to
information about right triangles and the ability to do computations. It should be
the goal of instruction to make the connections between the verbal
representation and the imagistic and computational representation so that the
verbal codings are useful in solving problems.
b) Imagistic systems: These include visual/spatial, auditoiy/rhythmic, and
tactile/kinesthetic systems. Imagistic systems are necessary for the meaningful
interpretation of verbal representations. The verbal encoding "a squared plus b
squared equals c squared" should be connected to an imagistic visual/spatial
representation of a right triangle in order to be useful. Visual/spatial
representations include quasipictorial information such as maps and geometric
figures. When the learner brings these representations into working memory
they come as pictures. Internal auditory/rhythmic representation is observed
when children learning to count or recite multiplication facts do so in rhythm.
Many adults remember "6 times 8 is 48" as a rhyme. Students learning to count
learn rhymes such as "one two buckle my shoe," and more sophisticated
counters pause or inflect their voices at regular intervals, frequently at tens.
Tactile/kinesthetic representation refers to imagined physical actions by or on
21
the person. For example, when solving an algebraic equation, a student may
leam to see the V moving across the equal sign and changing sign as it does so.
I have observed students push their palms together when describing the
transformation of a sampling distribution due to increasing sample size.
c) Formal notational systems: These are usually seen as external representations,
for example, equations. The competencies used to manipulate these equations
are internal representations. These are observed when a learner talks about the
rules for operating within a formal system. The notion of dividing both sides of
an equation by a quantity is a rule for use in algebra. The competency for doing
so is an internal representation, although when the rule is printed in a book or
demonstrated by a student on paper it is an external representation.
d) A system of planning, monitoring, and executive control: These are the heuristic
processes used to monitor and control the problem solving process. The
problem solver may be thinking about what comes next, monitoring the
reasonableness of intermediate answers, or deciding to change the notation
system used.
e) A system of affective representation: This refers to the affective aspects of
problem solving. These representations are dynamic and rapidly changing.
They include curiosity, surprise, frustration, anxiety, fear, encouragement,
pleasure and satisfaction.
22
The Connection Between External and Internal Representations
The relationship between external and internal representations is important
in mathematics education. Students use external representations of mathematical
ideas, including text, teacher demonstrations, and manipulative objects in order to
form internal representations. Goldin (2001) claims that one of the primary goals of
mathematics education is the development of representations as tools for creating,
storing, and transmitting mathematics to others. One facet of this goal is to facilitate
the translation between various representations.
Of special importance are the twoway interactions between internal
and external representations. Sometimes an individual externalizes
in physical form through acts stemming from internal structures, that
is, acts of writing, speaking, manipulating the elements of some
external concrete system, and so on. Sometimes the person
internalizes by means of interactions with the external physical
structures of a notational system, by reading, interpreting words and
sentences, interpreting equations and graphs, and so on. (Goldin,
1996, p.401).
The relation between the representation and the represented concept is bi
directional and is context and situation related. This idea has been referred to as the
signifiedsignifier relation. For example, a cartesian graph may help a student form
an internal representation of a particular function, or it may aid the same student in
reporting to others what internal representations she has formed. The present study
explored these interactions through clinical interviews of students as they solved
problems in inferential statistics.
23
Development of Representations
Representational systems are constructed in stages. Goldin and Kaput
(1996) proposes three stages. These are an inventive or semiotic stage, a structural
development stage, and an autonomous stage. This model is reported here because
it is the only model in the literature that is specific to representations. The other
models summarized in Chapter 1 refer to cognitive development in general. The
three stages of Goldin and Kaput are similar to the stages used in this dissertation.
During the inventive or semiotic stage new characters are created or learned
and are used to symbolize aspects of a prior representational system or systems. An
example could be the development of the idea of the normal distribution from
experience with the binomial. In this phase the symbols and algorithms for using
them are separate, in the sense that student cannot look at a symbol or formula and
imagine what it represents without performing an algorithm.
During the period of structural development the development or construction
is driven principally by structural features of the earlier system. This process makes
use of the symbolization that was established in the first stage. In this way
configurations are built up from characters, and a syntax for the new system is
constructed. In this stage rules for describing the new system are learned. In the
example of the normal distribution, students learn to describe points on the
24
distribution and the basic properties of the distribution, but they do not see the
distribution as an autonomous object that can be included as a unit in other systems.
The autonomous stage is characterized by separation. The new system of
representation, now mature, separates from the old. It can at this point stand in
symbolic relationships with systems different from the one that was the template
driving its development. As these new possibilities occur, the transfer of meaning
(or, in the case of internal representations, of competencies) from old to new
domains becomes possible. These stages are similar to the three stages proposed
for the present study.
This framework for describing the growth of representations is comparable
to Piaget's idea of reflective abstraction (Piaget, 1952) and Sfard's idea of reification
(Sfard, 1994). These two ideas are summarized in Chapter 1 and discussed later in
this chapter.
Representation in the Context of Learning Theories
Two theories of learning have been widely used and discussed in the last
halfcentury. These are constructivism and behaviorism. For the purpose of this
study of representation, these two views are broadly categorized as theories held by
those who are willing to infer mental processes and structures from indirect
evidence, and those who are not willing. Behaviorists would reject the idea of
inferring mental activity or cognitive structures. Rather they claim that what is
25
knowable and therefore a subject of inquiry is observable behavior. Skinner was
especially critical of cognitive science, and its attempts to infer mental activity.
"We can talk about love and will and ideas and memories and feelings and states of
mind, and no one will ask us what we mean; no one will raise an eyebrow"
(Skinner, 1984, p.164). In the 1980's behaviorism was challenged by the ideas of
constructivism.
Within constructivism all knowledge is seen as constructed from the
individual's experience of the world (von Glasersfeld, 1991). Radical constructivists
tend to reject the idea of objectivity. They would claim that mathematical truth is a
result of negotiated meaning in a social context, or the result of the coherence of
meanings held by the learner. Von Glasersfeld (1987), a radical constructivist (his
own description), has criticized representational ideas on the ground that the only
reality accessible to the learner is internal. Hence there is no external reality to
represent.
Goldin stated that a representational approach offers a bridge between these
conflicting philosophies.
It involves explicit focus on both the external and the
internal, with the utmost attention given to the interplay between
them. Through interaction with structured external representations
in the learning environment, students' internal representational
systems develop. The students can then generate new external
representations. Conceptual understanding consists in the power and
flexibility of the internal representations, including the richness of
the relationships among different kinds of representation. Thus the
research on representation achieves a synthesis of the other two
26
perspectives, drawing on the best insights each offers without
dismissing the contributions of the other.
This lends itself to a more inclusive educational philosophy 
one that values skills and correct answers as well as complex
problem solving and mathematical discovery, without seeing these as
contradictory. The goal is high achievement in mathematics, for the
vast majority of students, through a variety of different
representational approaches. (Goldin & Shteingold, 2001, p. 8)
Behaviorists focus on observable outcomes of learning. They typically
reject all attempts to characterize what is going on in the mind. Although most of
the writings of Skinner are earlier than the first mentions of representations, it seems
likely that they would have no objection to the idea of an external representation.
Skinner (1984) did write about his view of the effects of cognitive science on
American education. He is most critical of what he sees as the nebulous nature of
the subject of cognitive science. He claims that this nebulous nature comes from
trying to describe things that cannot be directly observed. Since internal
representations cannot be observed directly, but can only be inferred, Skinner would
object to their use in describing learning and doing mathematics.
Representations are the organizing idea of the present study. The
representations are not directly observable. They will be inferred from observation
and interviews with students and their written work and calculator use as they solve
problems in inferential statistics. The criticism of Skinner and others is that these
are not legitimate objects of scientific study because of their internal, unobservable
nature. The clinical interview is a probe of these unobservable objects in the same
27
way that sound waves in the ocean are probes that can infer the existence of
submarines. The fact that a sonar operator is only observing waves, and not the ship
directly, does not prevent the discovery of the ship. This point is addressed further
in the section on clinical interviews.
Constructivists in mathematics have criticized the notion of representation
and representational view of the mind. Von Glasersfeld (1987) and Cobb, Jackel
and Wood (1992) have argued that we should not be talking about representation
but rather presentation, since it is the internal construction of knowledge that is
primary. They would claim that the primary object is internal and the
representational act comes from trying to express that object externally.
Cobb, Jackel and Wood (1992) set up the representational view as one in
which learning is a process in which students modify their internal mental
representations to construct mathematical relationships or structures that mirror
those embodied in external instructional representations. They argue that this view
is flawed in three ways. They argue on the basis of cognitive, anthropological, and
pedagogical considerations.
The first argument of Cobb, Jackel and Wood is on cognitive grounds. It is
that mathematics learning cannot be viewed as a process in which learners recognize
relationships presented in instructional representations. They argue that if this were
true, it would require that the mental structure of the learner already be as complex
as the presented representation. Rather, they claim, learning is a process in which
28
students construct knowledge as they try to make sense of their world by connecting
the newly learned with previous knowledge, and as a result of social interaction with
other students and teachers.
This argument does not attack the type of representations described in the
current study. There is nothing in the definitions of representations, presented
earlier in this chapter, which prevents them from being built up gradually. Indeed,
the descriptions of growth of representation explicitly acknowledge the deliberate
growth of representations as learners gain sophistication with the concepts being
studied.
The second argument of Cobb, Jackel and Wood (1992) is on
anthropological grounds. The use of the word "anthropological" stems from the
idea of shared beliefs. They discuss the use of manipulatives as external
representations of mathematical ideas. These representations are transparent to
teachers and other experts in the particular subject. These experts often assume that
students will perceive the same correspondence between these manipulatives and
the more abstract ideas as do the experts. This is often not the case. Students
frequently find the manipulatives to be just as difficult to understand as the abstract
concept and fail to connect the two. If the student truly shared the understanding of
the teacher, there would be little point in continuing the lesson. Cobb, Jackel and
Wood (1992) used the term "taken as shared" to describe the meanings attributed by
29
individuals to representations. That we are able to converse about mathematical
subjects is evidence, but not proof that we have shared meaning.
Cobb, Jackel and Wood (1992) also pointed out that when children are
noticing the correspondence between external representations and mathematical
concepts, they arrive at consistent correspondences. Cobb, et al. attributed this to
the social construction of the correspondence, as opposed to the representational
view which sees the correspondence as coming from the external representations.
Their argument is that if the representational view were correct, learners would not
consistently focus on the relationships that are considered to be most important by
teachers and other experts. Rather, the consistent focus must be the result of social
interaction.
The preceding argument is an attempt to refute an extreme view of
representation that is not required in the present study. Many believe that
mathematical concepts are socially negotiated. This belief dates to at least the early
twentieth century and the work of Vygotsky (1978). We believe that concepts are
socially negotiated because concepts differ somewhat from culture to culture. This
is not incompatible with the definitions of representation proposed above. Students
can negotiate meaning with their peers or teachers, and these socially negotiated
concepts can become internal representations.
Cobb, Jackel, and Wood (1992) use a third argument against the idea of
representation. It is on pedagogical grounds. They state that if we adopt the
30
representational view, students will make a separation between their mathematical
activity in school and their use of mathematics outside of school. That is, internal
representations will correspond only to the external representations of the school
setting, and not to authentic settings. If teachers take their concept of an external
representation as an end point of learning, then they are in danger of treating
mathematics as a series of algorithms to be learned.
The pedagogical argument is about policy. It does not claim that
representations are poor ways of studying the mental state of learners. Rather the
argument is that focusing on particular representations is a poor way to teach. There
is no argument made against using representations as an organizing concept for
studying the development of concepts in learners of statistics.
Pedagogical Considerations Regarding Representations
Teachers use representations to aid in concept development and students use
them to observe, store and process knowledge. Problems arise when students are
given representations that they are not prepared to use, or when they are asked to
create representations and are unable to do so, or create faulty representations.
DufourJanvier, Bednarz, and Belanger (1987) ask five questions regarding
the use of representations in mathematics classrooms. These are: (1) What are the
motives for using external representations, (2) what are the expected outcomes of
the use of representation, (3) are these expectations being met, (4) what is the
31
accessibility of these representations to students, and (5) to what extent can we use
students' own representations in their learning?
The motivation for using some form of external representation is obvious.
Mathematics could not be done at all without using symbols for numbers, geometric
shapes, and other higher concepts. Mathematicians use these representations as
tools to store and communicate ideas, and they manipulate them using formal
notational systems (another type of representation) to produce new mathematics.
One goal in mathematics education is to have students also learn to use these
external representations as tools. Another motivation for teachers is that the
representations are concretizations of mathematical ideas. These may be multiple
representations, as when graphs, tables, and equations are used to represent
functions. The hope is that students will be able to translate among these
representations and abstract the common properties to form the intended internal
representation. Representations are also used to make mathematical concepts more
accessible to students. Diagrams, graphs and manipulative objects are examples of
these external representations. The question arises as to whether this goal is being
met in schools.
Unfortunately, there is evidence that students do not use representations as
tools to solve problems and these representations frequently do not aid in concept
development. Students frequently see school math and their needs outside of school
as completely separate. Greer and Harel (1998) referred to this disconnection as the
32
problem of isomorphism. That is, students do not see the problem that is presented
as isomorphic with the representations that they have been learning in school, unless
the problem is presented closely in time and is nearly identical with the
representations studied. One frequently cited example is from sampling theory.
Kahneman and Tversky (1982) reported a study in which psychology graduate
students who were supposedly able to use statistical techniques and were using them
in their research were asked the following question:
A certain town is served by two hospitals. In the larger hospital
about 45 babies are bom each day, and in the smaller hospital about
15 babies are bom each day. As you know, about 50% of all babies
are boys. The exact percentage of baby boys, however, varies from
day to day. For a period of one year, each hospital recorded the days
on which more than 60% of the babies bom were boys. Which
hospital do you think recorded more such days: the larger hospital,
the smaller hospital, or about the same?
Sampling theory tells us that variance should decrease with increasing
sample size. These students failed to apply this knowledge. Less then 40% of them
gave the correct answer, with most of the incorrect responses being that there would
be no difference. Kahneman and Tversky claim that the same students were able to
answer similar questions in the context of their statistics classes but were unable to
perceive the isomorphism of the hospital problem when presented outside of their
class and in a setting in which the use of statistical concepts was not explicitly
requested.
33
There is also a body of research under the title of "ethnomathematics" in
which groups of people using math in the context of their trade or business invent
their own algorithms (representations) for solving problems. Tailors in Liberia
(Reed & Lave, 1981), carpenters in South Africa (Millroy, 1992), street vendors in
Brazil (Carraher, Carraher, & Schliemann, 1985), and grocery shoppers in the
United States (Lave, 1988) have been observed solving arithmetic problems. The
common characteristic is that they do not use school taught representations, yet
arrive at correct solutions. This seems to indicate that representations provided to
students are not internalized by them, and are not used in situations outside of
school math situations.
The success of people in solving problems without using school taught
methods raises the question of whether students can create their own representations
for use in problem solving. DufourJanvier, et al. (1987) claimed that this is both
possible and desirable. When students are given representations by teachers, the
danger is that these representations are just as abstract to the student as the
mathematical concepts which they are meant to explain. Students are then left to
manipulate the representation according to some rules or algorithms without
understanding, and without grasping the isomorphism or making a connection
between the representation and the concept.
34
Other Related Concepts in Cognitive Science
Reification is a concept used by some in cognitive science to describe
concept development. It is associated with Anna Sfard of the Hebrew University in
Israel. According to Sfard (1991), concept development occurs in three stages.
These are interiorization, condensation, and reification.
Interiorization is the stage in which the learner performs operations on lower
level mathematical objects. While in this stage the learner must perform an
operation in order to think about it. As an example, when learning to perform the
long division algorithm, a person in the interiorization stage must consciously think
about the steps of the algorithm. When the learner has completed this stage, the
process becomes automatic. When the learner divides without thinking about the
sequence of operations the process has been interiorized.
Condensation is the stage during which a complex process in reduced into a
form that is easier to use and think about. It is an idea that is similar to the idea of
subroutines in computer programming. Sfard (1991) used the term "squeezing" to
describe the process of reducing long algorithms to more manageable units. While
in this stage the idea is tied to a procedure. The learner is becoming able to
generalize and make comparisons, and is increasingly able to move between various
representations of the concept. For example, the learner is beginning to see the
fraction as a complete object, and no longer sees it as an instruction to take 2 of 3
35
parts. The learner begins to see the connection between decimals, common
fractions, and percents. In statistics, the learner has attained condensation when he
can think "find the z score" without thinking of the separate algorithmic steps for
doing so.
Reification is the stage where the solver can conceive of the mathematical
concept as a complete object with characteristics of its own. Concepts that have
been reified can be thought of in relation to other concepts and can be placed in
hierarchies. Sfard (1991) believes that the first two processes occur gradually, but
that reification occurs quickly, in one sudden shift. The learner sees a familiar
object in a new light. The normal distribution may be viewed as reified when the
learner sees it in relation to other distributions, can grasp it as a legitimate object
without thinking of particular distributions of measurements.
In the case of the normal distribution, many students, including bright and
motivated students, leave their university statistics courses stuck in the
interiorization stage. Wilinsky (1997) cites several examples of doctoral
psychology and education students who had taken several statistics courses to
support their research, were able to do computations correctly, but were
uncomfortable with the results because they did not understand what they were
doing. They were unable to make connections between the tables and the graphs of
the normal distribution. They had to follow stepbystep algorithms to arrive at
36
answers, and could not explain to an interviewer why they were performing
particular algorithms.
Literature on TaskBased Interviews
The task based interview is sometimes called a clinical interview. Credit for
legitimizing the method as a data gathering tool in cognitive development is often
given to Jean Piaget (Opper, 1977). Piaget used the clinical interview to examine
cognitive development of children in a variety of subjects including number,
measurement, and probability (Piaget, 1953; Piaget & Inhelder, 1975). In these
interviews, the child was typically presented with a task to complete and the
interviewer probed the child for the modes of thinking used in the process. These
interviews resulted in a large body of data about the cognitive development of
children.
Konold (1981) has proposed some standardized vocabulary and some theory
of interviews as a data gathering tool. Three types of problembased interviews are
proposed. In the "thinking aloud" interview subjects are given problems and asked
to solve them with a minimum of reflection, backtracking, or additional probing.
The "indepth interview", also called Piagetian or clinical interview, probing is
much more flexible, and is meant to delve into thinking in the most depth. In the
"tutorial" interview correct solutions are sought, so interviewers ask leading
questions.
37
The present study incorporated elements of the second two, the clinical and
the tutorial interviews. Probing was used to elicit responses about the
representations of interest. If students used formal notation exclusively to solve a
problem, they were asked if they formed any graphical images during the solution.
There were also aspects of tutorial interviews. These students were enrolled in a
class taught by this researcher. It would be unethical to allow incorrect beliefs
about statistical concepts to persist if they could be corrected during the interview.
Because the research questions focus on types of representation and not on learning
correct solution methods, the research conclusions were not compromised by
elements of tutoring during the interview.
Three types of protocol analysis have been proposed by Konold (1981).
Coded analysis involves identifying key elements of interest and marking their
location in the transcript of the interview. In descriptive analysis, the researcher is
interested in providing a clear restatement of what the subject did and said during
the interview. In interpretive analysis, inferences are made about the structure of
the subjects reasoning process.
The analysis in the present study was interpretive. I used the results of the
interview to infer the existence and the stage of development of internal
representations used by students in solving statistics problems.
38
Interviews for the Purpose of Inferring
Internal Representations
Given the widespread use of the idea of representation, there is surprisingly
little published work on the use of internal representations by math learners. Most
of the published work on representations, including most of the work cited in this
dissertation, is theoretical or descriptive of types of representations. There is little
work that involves observing learners and inferring representational use from the
observation. Zang (1995) studied the use and development of internal
representations in 9 to 12 year old children. He used task based interviews and
studied development over a two year period. He was able to infer the existence and
character of the representations through the use of the interviews. Taking a
constructivist perspective, he concluded that children form their own representations
through activity, and that they do not need to be given representations by a teacher.
Schoenfeld (1993) studied a problem solving session involving the use of
algebraic graphing software. A 9th grade girl was observed and interviewed in
several sessions over a three month period. Schoenfeld was able to infer the internal
structure of the girl's representations of some algebraic ideas through the use of the
interviews.
39
Summary
Representations have been described in the literature of cognitive
psychology and in the mathematics education literature. Representations can be
categorized as imagistic, formal, syntactic, executive control, and affective. Several
frameworks for describing growth in mathematics learning have been described.
The growth framework used in the present study is a synthesis of these other
published frameworks. Representations have been placed in the context of
mathematics learning, and the relation of representations to behaviorism and
constructivism have been described. Reification is a concept in cognitive science
that is closely related to the idea of growth of representations. Task based
interviews are the way that the existence of internal representations were inferred in
this study.
40
CHAPTER 3
METHODOLOGY
This chapter includes the research questions, the subset of statistical ideas
being examined, a description of the subjects, and a discussion of the issues of
reliability and validity.
There are three research questions being investigated. They are:
1. What internal representations in inferential statistics are constructed by
students in an introductory high school AP Statistics class.?
2. What changes occur in these representations over time?
3. What is the relationship between the representations formed by students
and their success in solving problems?
Approach and Rationale
Data collection was based on clinical interviews. The research design was
descriptive and comparative. That is, representations used are described and those
used by successful and unsuccessful problem solvers are compared. It is
longitudinal in that the subjects are observed over the course of five interviews
during the portion of the school year from November to early May.
41
The interview method of data collection is suited to this topic because it
allows for students to report their internal state and for the interviewer to follow up
on reports while the state still exists or is fresh in the memory of the student. It
allows for the interviewer to probe for additional information based on the progress
of the interview.
The Role of the Researcher
I am the teacher of the subjects, the designer of the questions used as
prompts, the interviewer, and the analyzer of the data. It is possible for this
conjunction of roles to cause problems with research. I claim that any problems
were minimal or nonexistent. Possible problems are addressed later in this chapter.
Data Collection Methods
Problembased interviews were used for data collection. These are
sometimes called clinical interviews or Piagetian interviews. Students were given
problems to solve. These problems were meant to elicit thinking about aspects of
the statistics curriculum and to show student use of specific representations. The
students were asked to think aloud as they were solving the problem. The
interviewer was listening for evidence of the representational structure used by the
student, and probed for more information when such evidence was heard. The
interviews were transcribed, and evidence of representation use was coded. Since
42
one of the research questions involves the relationship between type of
representation used, and success at problem solving, the relative success of the
student was judged and recorded.
Prompts
The area of statistics learning being investigated is inferential statistics. In
an introductory course, this branch includes notions of randomness, regression,
sampling methods and sampling distributions, parameter estimation, hypothesis
testing, and confidence intervals. These ideas are applied to several types of data,
including univariate means and proportions, difference of means for paired data, and
regression for paired data.
There were five initial prompts, which were used by each of the subjects.
The questions were given one at a time over the course of the study, which began in
November, 2002 and ended in May, 2003. The questions were timed to coincide
with the subject being studied by the student.
The students were shown the problem and allowed to read it. They were
asked about understanding of the question. The students were asked to verbalize
thought processes as they worked on the problem. When the students were done
with the problem, the interviewer requested further self reports on learning. For
example, if a student used a formal representation for solving the problem, the
interviewer asked if the student also had available an imagistic representation of the
43
problem. The goal was to determine all of the representations available to the
student, and the ranking of those representations by the student.
All interviews had the audio portion recorded, and all material used by the
student, including scratch paper, final answer, tables or other reference material was
collected and preserved. The interviewer made written or audio notes sufficient to
identify the written material produced by the student and to key that material to the
transcript of the audio portion. Each interview was transcribed in a timely manner,
so that there was the maximum likelihood of recalling inaudible portions of the
recording and linking the written material to the transcript.
The five prompts were: (I) A problem concerning regression and residuals,
(II) a problem about sampling and the Central Limit Theorem, (III) an inference
problem with one proportion, (TV) an inference problem with one mean, and (V) an
inference problem with two means. The prompts and suggested satisfactory
responses follow.
Prompt I Residuals and Regression. The first page of prompt I is from the
1998 AP Statistics Exam, Free Response Question 4. The second page was
produced for the present project to give students a chance to answer the question
using a different form of the residual plot.
44
Figure 3.1
Prompt I
1998 AP STATISTICS 9
4. In a study of the application of a certain type of weed killer, 14 fields containing large numbers
of weeds were treated. The weed killer was prepared at seven different strengths by adding
1, 1.5, 2, 2.5, 3, 3.5, or 4 teaspoons to a gallon of water. Two randomly selected fields were
treated with each strength of weed killer. After a few dayB, the percentage of weeds killed on
each filed was measured. The computer output obtained from fitting a least squares regression
line to the data is shown below. A plot of the residuals is provided as well.
Dependent variable is: percent killed
R squared = 97.2% R squared (adjusted) = 96.9%
s = 4.505 with 14 2 = 12 degrees of freedom
Sum of Mean
Source Squares df Square Fratio
Regression 8330.16 1 8330.16 410
Residual 243.589 12 20.2990
Variable Coefficient s.e. of Coeff tratio Prob
Constant 20.5893 3.242 6.35 <0.0001
No. Teaspoons 24.3929 1.204 20.3 <0.0001
4 .
A
19 3 o m
3
06
4
m
20 40 60
Predicted
(a) What is the equation of the least squares regression line given by this analysis? Define any
variables used in this equation.
(b) If someone uses this equation to predict the percentage of weeds killed when 2.6 teaspoons
of weed killer are used, which of the following would you expect?
O The prediction will be too large.
O The prediction will be too small.
O A prediction cannot be made based on the information given on the computer output.
Explain your reasoning.
45
Figure 3.1 (continued)
Response to Prompt 1. This question calls for the interpretation of residuals.
It was given to students after they had studied the subject of regression in the
course. The plot of residuals indicates that the relation is not linear, but perhaps a
power or an exponential relation. The question calls for a determination of the
direction of error in an estimate if the estimate is based on the linear regression line.
46
Question a asks for the equation of the least squares regression line. The
coefficients are given as part of the computer output. Students only need to read the
coefficients and insert them into the regression equation. The correct response is
y = 20.5893 + 24.3929*.
Question b asks for the direction of the error if someone were to use the
equation to predict the percentage of weeds killed for a concentration of 2.6
teaspoons of weed killer. It is possible to do this without computation. Students
could find the two plotted points that correspond to a concentration of 2.5
teaspoons, note that both of the measured values have a negative residual, and
estimate that the residuals for a concentration of 2.6 teaspoons would also be
negative, and the prediction would therefore be too high.
Beyond the written prompts, students were asked to draw an estimated least
squares regression line on a scatterplot and to draw a graphical representation of the
residual for one of the plotted points. The graphical representation of the residual is
a vertical line segment connecting the plotted point with the regression line. This
provides opportunity for use of imagistic representations.
Prompt II Central Limit Theorem. The first part of prompt II is from the
1998 AP Statistics Exam, Free Response Question 1. The second part is from
Kahneman and Tversky (1972).
47
Part 1: Consider the sampling distribution of a sample mean obtained
by random sampling from an infinite population. This population
has a distribution that is highly skewed toward the larger values.
a. How is the mean of the sampling distribution related to the
mean of the population?
b. How is the standard deviation of the sampling distribution
related to the standard deviation of the population?
c. How is the shape of the sampling distribution affected by the
sample size?
Part 2: In a certain town there are two hospitals, a small one and a
large one. In the small one there are, on the average, about 15 births
a day. In the large one there are, on the average, about 45 births a
day. The two hospitals have kept track of the days during which
more than 60% of the births were boys. Is it more likely that:
a. There were more days in which 60% of the births were boys
in the large hospital.
b. There were more days in which 60% of the births were boys
in the small hospital.
c. The number of days in which 60% of the births were boys
was about the same in both hospitals.
Response to Prompt II. Part 1 of this question was included to bring out
representations concerning sampling distributions and the Central Limit Theorem.
The correct response to question a is that the mean of the sampling distribution is
the same as the mean of the population. The correct response to question b is that
the standard deviation of the sampling distribution is smaller than the standard
48
deviation of the population, and is smaller by a factor of j=. The correct response
yjn
to question c is that the shape of the sampling distribution approaches normal as the
sample size increases.
Part 2 of this question was included to determine whether students could use
the properties of the Central Limit Theorem that they had just stated in the first part
of the prompt. The correct response is that there would be more days in the small
hospital with more than 60% boys because the sample size is smaller and the
standard error is therefore larger. If the standard error is larger, there is more of the
distribution of sample proportions that is greater than 60%.
Prompt HI Inference With One Proportion. This prompt was included to
observe representations used while computing and explaining a oneproportion z
confidence interval and a oneproportion z test.
A researcher is studying whether teens get enough sleep. She used a
test to determine if lack of sleep affected attention in school. The
researcher used an appropriate method for selecting a random
sample, and tested this sample. It was determined that of the 35
students tested, only 11 had enough sleep to perform well in school,
and 24 had not had enough. She decided that 90% was an
appropriate confidence level.
a. Find a confidence interval for the proportion.
b. Test the claim that the population proportion is less than 0.5
49
Response to Prompt HI. The correct response to a is to use 11/35 as a
proportion and to construct a confidence interval around that proportion. The
endpoints of the interval are .185 and .443.
The correct response to question b is to perform a onesided oneproportion
z test. The null hypothesis is that the proportion is equal to .5 and the alternate is
that the proportion is less than .5. The results of the test are z = 2.2 and p = .014.
Students should state the conclusion that if the population proportion were 0.5,
results as extreme as 11/35 or less could be expected to occur by chance only about
1 % of the time. If the researcher thought that a confidence interval of 90% was
appropriate, she would probably select an alpha level of 0.1 for a two sided test, or
.05 for a onesided test. The null would therefore be rejected and the alternate
accepted.
Prompt IV Inference With One Mean. This prompt was included to
observe representations used while computing and explaining a onesample t
confidence interval and a onesample t test. It was written by the investigator for
the present project.
A researcher is investigating the possibility that high school
seniors do not get enough sleep in order to be effective learners in
school. There is some historical data that shows that in the
nineteenth century, before the widespread use of electric lights and
entertainment, the mean hours of sleep per night for 16 to 18 year
olds was 7.8 hours. The researcher has information from a random
sample of 37 students. Their mean hours of sleep was 7.55 with a
sample standard deviation of 0.8 hours. The researcher has chosen
to use an alpha of 0.05 and 95% for confidence intervals.
50
Response to Prompt IV. An explicit question was deliberately not included.
Students were expected to compute the interval and perform the test without further
prompting. The endpoints of the confidence interval are 7.283 and 7.817. The t test
has a null hypothesis of ^ = 7.8 and an alternate hypothesis of ^ < 7.8. The
results of the test are t = 1.9 and p = 0.033. The correct interpretation is that if the
population mean were 7.8, and sample mean of 7.55 or smaller could occur by
chance about 3% of the time. Since alpha is .05, the null should be rejected and the
alternate accepted.
Prompt V Inference With Two Means. This prompt was included to
observe representations used while computing and explaining a twosample t
confidence interval and a twosample t test of means. It was written for the current
project.
A researcher is studying the sleep deficits of high school
seniors. Sleep deficit is defined as the amount of sleep the subjects
actually get subtracted from the amount that they should get. The
hypothesis proposed by the researcher is that males have a larger
sleep deficit than females.
The researcher has 32 male subjects with a mean deficit of
.59 hours and a sample standard deviation of .45 hours and 31
female subjects with a mean deficit of .41 hours and a sample
standard deviation of .39.
Response to Prompt V. A specific question was intentionally not included.
Students were expected to propose a reasonable confidence level and an alpha,
calculate a twosample t confidence interval and perform and interpret a twosample
t test. The endpoints of the 95% confidence interval are .032 and .392, and the
51
endpoints for 90% are .0029 and .357. The t test has a null hypothesis of
and an alternate hypothesis of //, > //2 The results are t = 1.698 and p =
0.0473. The correct interpretation is that if the two population means are the same,
the sample means could differ by chance by the amount of the sample difference or
more about 4.7% of the time. The null hypothesis should be rejected and the
alternate accepted.
Subjects
There were six subjects who completed the prompts. There were originally
eight subject who had volunteered. Two of these dropped the course in the middle
of the year. It is common for about 20% of students to drop this course at the end of
the first semester. These two students completed the first prompt, but their results
are not included because they did not complete any other prompts. Of the six who
were in the study for the full year five completed all five prompts and one (RM)
completed four. The subjects were all White and middle class. There were four
females (LM, LW, SH, and CR) and two males (SP and RM). All six had
completed three years of college preparatory high school math. One (LW)
completed second year algebra, one (LM) completed a three semester course in AP
Calculus, and the other four had completed a precalculus course. They represented
the range of ability and motivation in the class. They were all in middle to high
tracks in their high school mathematics curriculum, thus they were a reasonably
52
homogenous group. This and the small sample size limit the generalizability of the
study.
Site Selection
The site of the study is a high school in Denver, Colorado. The school is the
one in which I teach. Site selection was based on convenience and ease of entry.
Since the site is an urban high school and the class is an AP statistics class, the
results of this study may not be applicable to college classes or secondary classes
taught at a different level from the AP classes.
Data Analysis
All of the interviews were transcribed in a timely manner. Doing the
transcription soon after the interview maximized recollection of events such as
gestures and expressions and aided in the annotation of the transcript.
The transcript was studied for evidence of the use of representations, growth
in representations, and conflicts and connections between representations. These
are reported in chapter 4.
Clinical Interviewing
There are special considerations when clinical interviews are used as the
primary means of data gathering. Kvale (1994) lists 10 objections commonly made
53
to the interview as a data gathering technique. He then defends the clinical
interview in the context of each of these objections. The ten points are:
1. It is not scientific, but only common sense. Kvale's counter is to ask just what is
science. He claimed that there is no accepted definition, but there are some
common characteristics including that it produces new knowledge that is
systematic and this new knowledge is obtained methodically. Kvale argued that
interviews can share these characteristics. In the sense that the interviews in this
study produce knowledge about the representations used by students, and that
this knowledge is obtained methodically, and that the knowledge obtained is
refutable by other investigators, and potentially replicable, it can be called
scientific.
2. The qualitative interview is not objective, but subjective. This study is
necessarily subjective in that I use the words and written work of subjects to
make interpretations about their cognitive state. However, these interpretations
are explained, and the basis of the interpretations will be available to readers.
3. It is not trustworthy, but biased. Sources of bias are deliberate lying and
unwitting bias. It may be that the subjects believed that one sort of
representation is more desirable than another, and therefore made false reports
either deliberately or unwittingly. This is a matter where interview skills and
interview protocol can minimize objectionable results. I do not have reason to
54
suspect that students reported falsely. There is evidence of some false
confidence as an affective representation. This is reported in chapter 4.
4. It is not reliable, but rests upon leading questions. Leading questions would
result in the sort of bias discussed in 3 above. The questions, protocols, and
interview transcripts are all be available to readers. The protocols and prompts
have been examined by others prior to use to check for content validity.1
5. It is not intersubjective; different interpreters find different meanings. If
different interpreters make different interpretations, then the instrument is not
reliable. As a form of triangulation the two other persons cited in number 4
above have examined a subset of the data and their interpretations compare to
mine. There were no disagreements about the type of representation being used
or the level of growth of the representations. I believe this was due to the
explicitness of the operational definitions of representations and levels of
growth.
6. It is not a formalized method; it is too persondependent. While interviewing in
general is not formalized, there is literature on the clinical interview. The
interviews in this study follow protocols and the interpretations are based on
1 The two persons who assisted in the triangulation were Dr. William Jurascheck
and Dr. Cathy Martin. William Jurascheck is a professor in the School of Education
at the University of Colorado at Denver. His specialty is mathematics education.
Cathy Martin is a curriculum specialist with Denver Public Schools.
55
operational definitions that are available to the reader. Any reader may read the
interview and use my definitions to compare his interpretation to mine.
7. It is not scientific hypothesistesting; it is only explorative. In the case of this
study, the purpose is indeed explorative. I want to describe the types of
representations used, describe how they change over time, and explore the
relationship between success at solving problems, and the types of
representations used. There are no hypotheses being tested.
8. It is not quantitative, only qualitative. This is true. It is the point of this study to
be descriptive in a qualitative manner
9. It does not yield generalizable results; there are too few subjects. There will be
six subjects in this study. The lack of large numbers (as well as the selection
process) means that the characteristics of large random samples cannot be
claimed. The advantage of small samples is that the subjects thinking can be
explored in greater depth.
10. It is not valid, but rests on subjective impressions. It is true that interpretations
are being made, and that this is a threat to validity, however, the interpretations
are open, in that the rationale will be explained, and triangulation by other
competent persons has been employed. There are no unresolved discrepancies
to report.
56
Reliability
Reliability refers to the consistency of an instrument over time and among
users. The present study involved the growth over time of understanding by
beginning students. It is not possible to duplicate this growth with the same
students, and since the study was over the course of one year, it will not be repeated
with a second group of students. The students were available to this researcher as
his students in his class. Interrater reliability was measured indirectly as described
below. Because of these issues, reliability in the current project is an issue of
concern.
Reliability of the interview method can be judged by the consistency of
responses by students. The representations used by students were reasonably stable
over time. Instruction and other experience by the student affected representations
as students learned and practiced, but the changes were gradual. In fact, the lack of
growth was a surprising finding of this project.
Threats to Reliability
The status of the researcher is a possible threat to external reliability. In this
study the researcher is also the teacher of the subjects. Because the interviews are
not expected to elicit controversial material, researcher status was not considered to
be a serious threat to reliability. To lessen this threat, participants were assured that
57
their participation would not affect their grade in any way, and that the interviews
would not be shared with others without deleting identifying information.
Informant choice is the process of selecting participants. It can affect
reliability if the participants are not representative of the population of interest. My
participants were volunteers from a class. It is possible that those who volunteered
are indeed different in some way from those who did not. This issue would be more
likely to affect reliability if the research questions were more affective in nature. It
is more likely that volunteers would have different feelings about learning than that
they would have different cognitive processes and structures. This threat is noted,
but no obvious differences between the volunteers and the nonvolunteers were
detected. There was a range in ability in the volunteers from the lowest to the
highest in the class.
Poorly defined analytic constructs and premises are a threat to reliability.
Ideas about representations, in the case of this study, should be consistent with those
used in the literature. My definitions of representations and the categories of
representation have been taken from mainstream mathematics and cognitive
psychology literature. The operational definitions and conceptual framework are
explained in chapter 1.
Internal reliability refers to the extent to which multiple observers make the
same interpretations within one study. Triangulation is a method for addressing this
threat.
58
There were two facets to the triangulation employed in the current project.
Cathy Martin examined a subset of the analysis. She was provided with the original
transcripts from one of the six subjects, and a set of 24 representations that had been
identified and described by the researcher. She was asked to evaluate the
descriptions in terms of type of representation and level of growth. There was one
disagreement affecting three of the 24 representations. This disagreement was the
result of a loose definition of type of representation. During the interviews, the
student stated a belief that a problem with means as the data type required a t test.
This is described in Chapter 4 as an if means then t executive control
representation. Since this is a use of language to describe a concept it could also be
considered to be a syntactic representation. The definitions of these two
representations were clarified as a result of this disagreement. No disagreements
were left unresolved.
The second facet of triangulation was an observation of two interviews by
William Juraschek. Dr. Juraschek observed one interview each with two of the
subjects. He then read the transcripts and interpretations of the transcripts. There
were no disagreements.
Validity
An research method has validity if it fulfills the function for which it is being
used. The validity of the interpretation of the data will come from two sources. The
59
first is content validity. Content validity has its source in the agreement by persons
familiar with the subject matter, in this case introductory statistics, that the interview
protocols are likely to elicit responses that are capable of revealing the
representations used by students. Each of the interview protocols was reviewed by
two other persons who are capable of making reasoned judgments about their
content validity. These were the same two persons mentioned in the reliability
section above. There were no unresolved disagreements about content validity.
Validity involves the question of whether the methods used actually measure
the constructs of interest. In the case of this study, the question of validity is: Do
the problems posed and the interviews surrounding these problems actually elicit the
representations held and used by students? Threats to validity include internal and
external threats.
Internal Threats to Validity
LeCompte and Goetz (1982) list history and maturation, observer effects,
selection and regression, mortality and spurious conclusions as threats to internal
validity.
Histoiy and maturation have effects on long term studies. The extent to
which results are different at the end and at the beginning of the study is a measure
of this threat. Informants mature and become habituated to the instruments being
used. The current study involves students over the course of most of two semesters,
60
or about seven months. The instruments were used only once on each student. It is
unlikely that history and maturation have affected validity.
Observer effects refer to the differences in behavior practiced by informants
when the observer is present versus when he is absent. In the case of interviews and
close observation of problem solving, it will be necessary for the researcher to be
present. Effects can be minimized by avoiding a relationship in which the
informants perceive that the interviewer has authority in the context of their class, or
the grade that they will receive. It was made clear to the volunteers that their
participation was completely separate from the grade in the class they were taking.
Selection effects are due to the necessity of selecting a subset of the students
in the class. It may be the case that the volunteers do not represent the frill range of
students taking an introductory statistics class. This effect cannot be controlled, but
the participants can be described along relevant dimensions so that readers can make
judgements about the representativeness of the participants. In particular, the
subjects are Caucasian middle class high school students in an AP Statistics class.
The results may not be generalizable to other high school students or to college
undergraduates due to age, class, and context differences.
Mortality refers to the changing participants in a group over time. Although
two of the original volunteers dropped the course midyear and did not complete the
study, mortality in the traditional sense is not seen as a significant problem. The six
61
students left were representative of the entire enrollment in the class in ability and
motivation.
Spurious conclusions are those in which a researcher infers the existence of
relationships that do not exist. In this study, I will be interviewing subjects in the
process of problem solving and attempting to infer the existence of cognitive
objects. As long as there is no method for directly observing cognition, these
inferences will necessarily remain suspect. Triangulation can minimize the
potential for spurious conclusions.
External Threats to Validity
External validity refers to the extent to which the research is generalizable
beyond the participants in the study. It was not be possible to select a random
sample of the population of students taking an introductory statistics class for the
first time. Volunteers were solicited from one class in one school. Threats to
external validity include selection effects, setting effects, history effects, and
construct effects.
Selection effects refer to the fact that some constructs cannot be compared
across groups, and therefore, results cannot be generalized to groups outside of the
study group. Setting effects are the changes to the culture that necessarily occur due
to the presence of the researcher. History effects refer to the unique experiences of
the participants. The participants in this study have had particular teachers using
62
particular curricula leading up to the statistics course, and will have one teacher
using one curriculum during the study. This researcher is familiar with the teachers
and curriculum experienced by the subjects in the three years prior to the study.
There is no reason to believe that there is anything unusual about that experience.
Construct effects are those due to loose or changing definitions of key ideas.
The ideas of representation are indeed used in different ways by different writers.
These effects can be minimized by clearly identifying and defining all terms.
Clarification of terms used in the present study is in Chapter 1.
63
CHAPTER 4
RESULTS OF INTERVIEWS
This chapter contains the information gathered from the interviews and
inferences of the representations employed by the subjects. The chapter is
organized by representational category, in the order imagistic, formal, syntactic,
executive control, and affective. Each instance of use of any of the types of
representation by the students is described, the level of the representation is given,
and any conflict or connection with other representations is described.
In the quotations from the interview transcripts, SCC refers to the researcher.
There are references to Fathom. Fathom is statistical learning software (Finzer,
2002). There are also references to calculator functions and menus. All of these
apply to either the Texas Instruments TI83 or the TI89 calculators. The complete
transcripts are in Appendix B.
There are three research questions being addressed in this study. They are
1. What internal representations in inferential statistics are constructed by students
in an introductory high school AP Statistics class.?
2. What changes occur in these representations over time?
3. What is the relationship between the representations formed by students and then
success in solving problems?
64
Representation 1 Imaeistic
Imagistic representations were the most frequently observed type of
representation in this study. The prompts all called for the interpretation of graphs,
the construction of graphs, the use of graphs to support problem solving, or to
support interpretation of results. Also, a kinesthetic representation was used by
some students in describing the effects of the Central Limit Theorem. Although
graphs are external representations, the competencies and preferences associated
with using them are internal. The interviews were used to infer these competencies
and preferences. There are three distinct types of problem in the five prompts (See
Chapter 3 for detail.) and they are presented separately. Prompt I is a problem with
a scatterplot. Prompt II concerns the Central Limit Theorem and sampling
distributions. Prompts ID, IV and V are all inference problems that require
knowledge of the Student's t or z tests and confidence intervals with means or
proportions.
Prompt I was from the 1998 AP exam. It asked students to interpret
computer output from a linear regression from a hypothetical experiment with
different concentrations of weed killer applied to a field. The two variables were
concentration of weed killer and the percentage of weeds killed. The output
included a plot of residuals. The residual plot had the predicted values ( y) on the x
axis. This is common in statistics reporting, but the students had not seen this form
65
before. The form that they were used to seeing had the independent variable (in this
case the concentration of weed killer) on the horizontal axis. The prompt therefore
lead students to go through an intermediate step. They used the regression equation
to change the axis from predicted or fitted values to predictor. Then, interpretation
of an imagistic representation was required. It was also been possible to do this
problem without this transformation. A subject could have used the fact that the
specified concentration of 2.6 was about in the middle of the range of concentration
values, and that the predicted value would also be about in the middle, and therefore
that the residual was negative.
Whether or not they were able to interpret the first residual plot, they were
presented with a second page with a more standard scatterplot and residual plot.
The axes of the two plots are parallel. In the second residual plot, the independent
variable (concentration of weed killer) is on the horizontal axis. They were asked to
draw a best fit line through the scatterplot without calculation. They were asked to
describe the differences in how they used the two residual plots, and state which
was more difficult and why. They were also asked to represent a residual as a line
segment on the scatterplot.
In general, all students had difficulty with the residual plot with the response
variable on the horizontal axis, all were able to make basic interpretation of the
residual plot with the predictor variable on the horizontal axis, and all were able to
draw a reasonable regression line. They were also generally able to determine
66
residuals from the scatterplot. Only two of the students were able to draw a vertical
line segment on the scatterplot as a graphical representation of the residual.
Students drew sampling distribution and population distribution curves for
Prompts II through V. They were asked to use these graphs to show the effect of
sample size on the shape, mean, and spread of the distribution. They were also
asked to show the relationship of the measurement, the standard scale (t or z), and
the cumulative probability.
Most of the students drew normal and t curves that were inadequate in that
the drawings did not represent some important properties of these curves. Both the
Student's t and the normal distribution curves should be symmetric, have inflection
points, and show asymptotic behavior. Because the sample size was adequate in all
of the prompts, Students t should look approximately like the normal curve, and
have inflection points located such that the proportion of the area under the curve
between the two inflection points is about 0.68. The curves drawn by the students
typically had symmetry, although a few did not. They often lacked inflection points
or had the inflection points too far from the mean. They also often failed to
represent the asymptotic behavior of the curves. The curves that failed to show
asymptotic behavior were either shaped like truncated parabolas, with the curve
intersecting the axis, or they had asymptotes well above the horizontal axis. These
properties are important in using the graphs to support calculations or to stand alone
in solving problems. In order to use the graphs to solve problems, the proportion of
67
area between points on the horizontal axis should approximate the actual
probabilities.
The Student's t and normal density functions can be drawn with any of three
scales on the horizontal axis. The horizontal scale could be in terms of the standard
t or z statistic, the measurement under consideration, or the cumulative probability.
Of the six students in the study, only LM was able to show graphically the relation
of the measurement (x or p) and the standard score (t or z), and she was not
consistent in her ability. All of the students were able to do this using formal
representations, but they were not able using imagistic representations. The
students also generally failed to connect the graphical representation of the
confidence interval with the algorithm for computing it, in that they were unable to
correctly name and plot the center of the interval.
Prompt II was explicitly written to get information about the students
concept of sampling distributions and the Central Limit Theorem. There were two
parts to this prompt. In the first part, students were asked about the three parts of
the Central Limit theorem. They were asked about the mean, standard deviation,
and shape of the sampling distribution of a population with a strongly right skewed
distribution. They were expected to state that the mean of the sampling distribution
is the same as the mean of the population, the standard deviation of the sampling
distribution, or the standard error, is equal to the standard deviation of the
68
population divided by the square root of the sample size, and that the shape of the
sampling distribution approaches normal as the sample size increases.
Part two was written to investigate the ideas about the effect of sampling
size on standard deviation. Subjects were asked whether a large or a small hospital
was more likely to have a day on which 60% of the births were boys. The correct
response is that the small hospital is more likely to have 60% boy births because the
standard deviation of the sampling distribution is larger, and therefore the
probability of being far from the population mean is greater.
Most of the students were able to state the three parts of the Central Limit
Theorem, although some required additional prompting. All of the students had
difficulty with the second part. Most were unable to identify the population and the
sample from the information given. Most failed to apply what they had just stated
about the Central Limit Theorem. Some used the availability heuristic (Tversky &
Kahneman, 1974) and stated that the larger hospital was more likely to have a day
with 60% boy births because there were more boys altogether.
All of the students used graphical representations of sampling distributions
to aid their reasoning.
Prompts HI, IV, and V are similar in that they all asked for standard
hypothesis tests. Prompt III was a test of a proportion, prompt IV was a test of a
mean, and prompt V was a test of two means. All tests had the opportunity for
students to use graphs of the normal or Student's t distribution to support reasoning
69
or calculations. If graphs were not used, the interviewer asked the students to draw
graphs and include multiple scales on the horizontal axis. Typically, the graphs
were inadequate representations in that they failed to represent all of the
characteristics of the distributions, and therefore could not be used to estimate
probabilities. There was little growth observed in the quality of the graphs, and the
quality of the graphs was not related to ability in problem solving. The students
with the best graphs were not the best at solving the problem.
Students had problems with the scales on the axes of the graphs, and points
on the confidence intervals. Most students were unable to correctly place the three
scales on the horizontal axis of the distributions. If they were able to place the scale
of Student's t values on the axis, they were usually unable to place the measurement
scale's equivalent points on the axis, although they were typically able to find
equivalent points using formal methods. Students were also typically unable to
name the center of a confidence interval, even though they had used the sample
mean to create the confidence interval. The three scales were not connected within
the imagistic representational system, and the imagistic representations of sampling
distributions and confidence intervals were not connected to the formal
representations.
Growth of imagistic representations was in the initial stage if they were seen
as a group of parts, were not connected to other representations, or were connected
incorrectly to other representations. The imagistic representations were in the
70
completion stage if they were seen as a whole, with their parts connected by logical
necessity, did not require support from other representations, and were useful in
obtaining correct answers to problems. The development stage is intermediate
between initial and completion.
Subject 1 LM
LM used imagistic representations in all of the prompts. She used graphs of
residual plots, scatterplots, and probability density functions, and worked with axes.
The growth level of these ranged from initial to complete. There was some
evidence of growth in the use of axes, but this growth was uneven, showing decline
from the fourth to the fifth prompt.
Scatterplots and Residual Plots. In prompt I, LM was not able to interpret
the first residual plot without considerable help, although she could use the
scatterplot, draw a regression line, use the second regression plot, and draw a
graphical representation of the residual. She was one of two subjects who were able
to draw a graphical representation of the residual as a line segment.
In prompt I, LM knew that a pattern to the residuals was an important
characteristic of the graph, but it is unclear that she could go further with the
interpretation. She was not asked about the implication of the pattern, as this was
not the goal of the interview. She did not volunteer that a linear interpretation was
inappropriate.
71
SCC Can you describe what you see on the graph, it has the axes labeled.
LM I see a pattern to the residuals.
Although she saw a pattern, she did not use this pattern to answer the
question about the prediction. It is likely that she was revealing a syntactic
representation that was connected to the idea of transformations to achieve linearity
of data in a scatterplot. This pattern could have been used to answer the question in
the prompt, but it was not. The representations connected to the idea of
linearization were not used to solve the problem that was stated in the context of
residuals.
LM did not use the first residual plot to determine that the residuals are
negative and that the prediction would be too high. She needed to change the axes
to predictor variable on the horizontal axis before she could make an interpretation.
SCC Ok, this one has predicted on the xaxis. Would you expect the
predicted to be higher or lower than the actual value? Can you tell
from there?
LM Um, no.
SCC What do you need to do?
LM I need to have the predictor on the xaxis.
After computing the response from the given value of the explanatory
variable, LM was still not able to interpret the graph without significant prompting.
She was able to determine the polarity of the residual and answer the question,
although not without additional prompting. She answered the question incorrectly,
relating negative residual with "the predicted would be lower." She was also
distracted by the presence of two plotted points corresponding to 2.5 teaspoons.
72
LM
SCC
LM
SCC
LM
SCC
LM
SCC
LM
SCC
LM
SCC
LM
SCC
LM
SCC
LM
SCC
LM
Is it one of these two points? (She points to the two points with
predicted = 40. The plotted points are distracting from the task)
Those would be right on 40. You notice that the teaspoons went in
increments of 0.5? So 2.6 would be in between those two. So, it is
not going to be one of those points.
It would be right on this line there. (She draws a line at predicted =
42)
OK, so, would you expect your predicted value to be higher than the
actual value or lower?
Well, I dont know where it oh 1 dont know where it falls on this
line.
Where would it probably fall on that line?
If it fit the pattern ... (Pause as she struggles with next step.)
If it fit the pattern where would the actual measurement be? Or,
where would the residuals be on that line?
Ok, so to figure out how it fits from the actual? Ok, so the actual
is... Didn't I just find the actual?
No, you found the predicted.
Ok.
The only way to find the actual would be to go into the field and
measure it.
Wait, so, it would be, so I cant, I dont know the... oh.
What would the residual be, roughly, on that line? What would you
expect to get for a residual?
Around 10, or, what would I expect to get for a residual? Well, if it
fit the pattern, it would be around here. (She puts a dot on the
vertical line about half way between the two points on 40. It is
unclear where she obtained the estimate of 10.)
And what is that value?
About minus four.
Ok, if you have a residual of negative 4, would the predicted value be
higher or lower than the measured?
The predicted would be lower.
LM drew a regression line on the scatterplot, and used visual clues to evaluate
the quality of the line.
SCC Ok, let's look at this one. It is more like what you are used to, right?
Without doing any calculation, could you eyeball a straight least
squares line on this graph?
73
LM Should I draw it in?
SCC Yeah, when you think you have it. (draws line)
LM Its a little off. It should be down more.
The fact that LM could not interpret the residual plot directly indicates that it
is not a complete representation. She could interpret the residual plot after taking
the intermediate step of computing the response (42.84% of the weeds killed) from
the predictor (2.6 teaspoons), and even after that transformation, she needed help in
interpreting the graph. She was eventually able to state that the residual would be
negative, but then stated "the predicted would be lower." This syntactic statement is
in conflict with the imagistic representation of the residual. Her use of this graph is
in the initial stage. She can be observed struggling with the basic form of the graph.
She does not use the given external stimuli (the graph and the parameters provided)
to make an interpretation. The graph is not connected with, and in fact, is in conflict
with the formal and the syntactic representation of the residual. The fact that "there
is a pattern to the residuals" seems to be related to the problem of transforming data
to achieve linearity and is not transferred to the problem of interpreting residuals.
When locating the probable measured value on the residual plot LM was
distracted by the presence of plotted points on the graph. Her attention was drawn
to these points, and instead of using them to indicate a trend along which the value
in question should lie, she was drawn to place the requested point on the vertical
line through these two points. This distraction is further evidence that the imagistic
74
representation of the residual plot is in the initial stage. The parts of the graph are
not being used together.
The scatterplot was in the completion stage. LM was able to use the data
represented on the scatterplot to draw a regression line, and was able to draw a
representation of the residual as a vertical line on the scatterplot She was also able
to connect this representation to a formula and a verbal statement of the definition
of the residual. The scatterplot and the representation of the residual as a line
segment are complete in that they were drawn confidently after the request to do so,
and in that they are connected in a relational and not a supportive way to syntactic
and formal representations, and to each other.
Sampling Distributions. In prompt II, the sampling distribution prompt, LM
was asked to draw a distribution "highly skewed to the larger values". She at first
drew a partial curve that began about 2 cm above the axis and a little to the left of
the maximum. This curve began at the cusp that is visible in the sketch on the
prompt. After being questioned about the hanging part of the curve, she sketched
the left side down to the horizontal axis, without showing inflection. The right side
of the curve has an inflection and an indication that it approaches the axis
asymptotically. In the second part of the prompt, she drew two sampling
distributions. Both have inflection and asymptote indications, although both have
the inflection points farther out from the mean than correct. The second curve has a
poorly shown inflection on the right side.
75
LM had a good concept of what it means for the axes to be transformed by
dilation due to changes in sample size. This concept was reported as an imagistic
representation in the quote below. She was able to make statements about the
distribution of the area of the curve and how that area changes with the scaling of
the axis. She was able to use the scaling on its own to make statements about
probability.
LM This has a larger standard deviation, so more of it is in the center.
SCC You are pointing to the large hospital distribution, saying that it has a
large standard deviation
LM I'm excuse me, the smaller standard deviation, so more of it is in the
center, so that means that less would be on the outside.
LM related a formal representation ("the smaller standard deviation") to an
imagistic one ("more of it is in the center") to make a judgement about probability.
She was using these representations to answer the question about which of the two
hospitals was more likely to have 60% boy births on a given day. The connection
between the two is relational and not supportive, and the representation was used to
correctly solve the problem. The imagistic representation of the dilation is therefore
complete.
LM used the connection between standard error (she uses the term "standard
deviation") and the shape of the sampling distribution. She was able to use this
imagistic representation to make judgements about how the standard error affects
the relative probabilities of an event in the two hospitals. She used this image
without support from other representations, such as the formal or syntactic. LM
76
used both an imagistic and a formal representation to answer the question in the
prompt, and used both correctly. Her drawings lacked some important
characteristics, but she used them to support her thinking in the solution of the
problem. Even though she used this image with confidence, later in this same
prompt she expressed a preference for the formal.
SCC Which one are you more comfortable with? You did the graphical
one first. Is it more convincing to see it graphically or more
convincing to see the numbers?
LM More convincing to see the numbers, but, we have done enough
learning how the numbers work that I can picture it in a graph.
The sketches of the sampling distributions are imagistic representations and
are in the development stage. They are used to support the problem solving without
using other forms, and therefore do not rely on supportive connections with other
representations, but they lack important properties of the curve, and they could not
be used to estimate probabilities. This lack prevents the sketches from being
complete.
Density Function Sketches. In prompt HI, LM's curve is similar to the one in
prompt II. It has inflection and asymptotic properties, but the inflection point is too
far from the mean. The sketches in prompts IV and V are improved in shape,
although in both, the inflection points are still too far from the mean.
There is growth in these imagistic representations over time. In the first
drawings, they are in the development stage. The fact that the inflection points are
not present or in the wrong place, and the fact that the asymptotic behavior is not
77
represented means that the images cannot be used for the purpose of estimating
probabilities from area. The images are not connected with the tables, formulas, or
calculator. Although her drawing in prompt V has inflection and asymptotic
behavior, the location of the inflection points means that the graph still cannot be
used to determine probabilities. The graph could be used to support other problem
solving, or other representations. Although the sketches have improved, they
remain in the development stage because the important properties of the curve are
not shown.
In prompt IV, LM has forgotten to use the standard error of the sampling
distribution, and instead uses the standard deviation of the population. This causes
the spread of the distribution of samples to be much too large. She uses imagistic
clues to determine that an error has occurred. In this case, the imagistic
representation is well connected to the formal representation. She understands the
implications of placing the Student's t value of 1.9 under the hours value of 7.55.
The fact that the imagistic representation has a clear error clues her that there must
be an error in the formal representation, because they are connected. In this context,
her imagistic representation is in the completion stage. The imagistic representation
is not supported by others, rather, the imagistic is used as a clue that there is an error
in her formal representation.
SCC What was the t, 1.9?
LM Negative 1.9
SCC Where would that be on the graph?
78
LM Won't it be where the data is? (She has 7 and 8.6 at inflection points.
When she says "where the data is", points to 7.55. The 7 and the 8.6
are at inflection points because she has used the standard deviation
instead of the standard error to draw the curve.)
SCC Is that 1.9 standard deviations, or standard errors?
LM No, so something is wrong with my picture.
SCC So, how come it is not .8
LM So, it should be about 2 standard deviations away.
SCC 1.9 is.
LM Yeah. So, maybe .8 isn't the standard deviation of the historical
sample.
LM has 7 and 8.6 at inflection points. When she says "where the data is,"
she points to 7.55. She has connected the inflection points with one standard error,
a connection that is not present in any of the other opportunities in the course of this
investigation.
Axes and Other Graphical Representations. In prompt V LM is able to
sketch a curve and place both the t and measurement scales in about the right
locations. She was the only one of the six students who was able to do this on any
of the prompts. The graphical representations of the t and hour axes are connected.
In this prompt, LM is able to connect the tscore with the measurement scale in
hours of difference and with the probability scale. Some aid was required for her to
do this. She had at first placed the 18 hours at. 18 on the tscale. There is an
unlabeled mark where she did this. After prompting, she was able to correct the
location. She has also not placed t=l under the inflection point. These are evidence
of regression since prompt IV. She has difficulty in placing 0 in the center. The
79
fact that when comparing two means, the null should be that the difference in the
means is zero was not explicitly brought out by instruction.
SCC Ok, so where would 1.7 be on that graph? Ok, so you put it right
with the shaded.
LM Right.
SCC That is the t scale. You have 0 and 1 and 1.7. What about a scale of
differences in the deficits? What would be right in the middle of that
graph, in terms of difference of hours?
LM I don't know if I would put the mean for the males or the mean for
the females?
SCC What does the zero mean. If you had a t of zero, what would that
mean?
LM That would mean that there is no difference.
SCC So, what would go right in the middle in terms of hours? If there is
no difference. I am asking for a scale of difference of sleep deficits.
LM So one would go in the middle. Or, zero. Hold on. So, if I am using
the differences, then I would not actually be putting .59 or .41 on the
graph. I would use the difference.
SCC What is the difference? Maybe that would help.
LM Difference of what?
SCC Deficits for males and females. How much bigger is the deficit for
males?
LM .18
SCC Do you know where. 18 would go on the graph?
LM In the middle.
SCC You told me earlier that we were measuring deficits. And that what
does a t of zero mean?
LM That there is no difference. So then it would be 18 right there. (She
puts. 18 at t=. 18. See mark on paper.)
SCC What difference would go under t equals zero?
LM The difference would be zero.
SCC Ok, put a zero there. Where would the 18 go?
LM Would it go right here?
SCC Where?
LM Where the shaded part is.
SCC Where under the shaded part, exactly?
LM Where it starts.
80
LM's drawings of distributions improved over time, although they did not
reach a complete level. The drawings did not consistently include the needed
characteristics of the distributions, especially inflection points in the correct
locations and correct asymptotes. They were adequate for supporting her thinking
and problem solving, but not for estimating probabilities. She related the three
scales on the axis of the distributions, and was able to use the connection between
the graphs and the formal to correct an error in the formal. The fact that her
drawings lacked some or all of the important characteristics of the curves did not
prevent her from using them to supporting reasoning during problem solving. Her
imagistic representations of the axes of the sampling distributions are in the
completion stage. This is inferred because the formal results of calculations are
placed on the axis in approximately correct positions, and because these placements
do not rely on the support of other representations.
LM used residual plots, scatterplots, sketches of sampling distributions and
probability density functions, and multiple axes to support her problem solving, to
detect errors, and to explain results. Her sketches of distributions were poor, but she
was the best of the six subjects at problem solving. Quality of sketches were not
related to ability at problem solving in here case. There was some conflict between
syntactic and imagistic forms detected in the context of using the residual plot.
There was more connection detected in the later prompts. An imagistic
81
representation was used to detect a formal error and to lead to a correction. This
connection was related to her ability to solve problems.
Subject 2 LW
LW used imagistic representations in all of the prompts. She used graphs of
residual plots, scatterplots, and probability density functions, and worked with axes.
The growth level of these ranged from initial to complete.
Scatterplots and Residual Plots. LW was not able to use the first residual
plot to determine the polarity of the residual and the direction of the error in using
the regression line to estimate a value of the independent variable. She needed
some scaffolding to do this. She was able to interpret the scatterplot, draw an
estimated regression line, represent a residual with a line segment on the scatterplot,
and interpret the second residual plot.
LW knew that the residual plot could be used to answer the question about
the relative error of the estimate, but her response was confused, and she was not
able to answer the question without further prompting.
LW Well, the residuals, I guess I could find that out by looking at the
graph, right?
SCC Ok
LW So, if it is one of these, that means that... like, what did that look
like before that turned into that? Kind of like how that looks. I dont
know how to use that.
82
When LW says "one of these", she is indicating the two points at a predicted
value of about 40. She is assigning unwarranted importance to these points, and
they are distracting her from the task. When she says "what did that look like before
that turned into that?" she is trying to imagine an inverse operation that would turn
the residual plot into a scatterplot. She states that "I don't know how to use that."
But, with some prompting she is able to state a correct answer to the prompt.
SCC Ok, how much would they predict using your equation for 2.6
teaspoons?
LW (calculates) 42.8 percent of weeds are killed. And then ... can I use
any of this? No.
SCC Well, these are residuals. What would the residual be at 42.8?
Would it be negative or positive?
LW It would be way up here. It would be positive. I think. (Plaintive
and questioning. She is treating the 42.8 as a residual and pointing
to the y axis, stating that the residual will be high up on that axis.).
Because it is a positive number. Positive.
SCC Where is your 42.8 on the graph?
LW Here's 50, so around here, right? (She marks on horizontal axis.)
SCC Ok, you have a plot of residuals, you can see what the residuals are at
42.8.
LW So, well...
SCC What are they, positive or negative?
LW The closest ones are negative, like these ones. Oh, ok. ("ok" shows
satisfaction)
SCC What's that mean? Would the measured point be below the predicted
point or above it?
LW Below it. The predicted would be too small. (More assertively)
The measured value is below the predicted, but that makes the predicted too
high. LWs imagistic representation is in conflict with her syntactic representation.
Syntactically, below and too small related, but she failed to use them correctly.
83
She was able to interpret the residual plot, but only with some help. She then
looked at the second residual plot and found it much easier to use.
SCC These graphs are more like what you are used to. Can you answer
the question using this graph?
LW Ok, so, they say that it will be 42.8. And then the teaspoons will be
2.6, and the closest ones to the teaspoons, they are all negative.
Yeah, that is a lot easier to use. So, those are negative.
She stated that the second residual plot is easier to use. It is not apparent whether
this was because the first one was unfamiliar, or because of the perceived need to
change the axis to the predictor variable
LW was able to draw a regression line on the scatterplot, but at first she tried
to use the equation parameters to draw the line. She struggled with where to put the
yintercept because it is not in the part of the graph given to her.
LW So, yintercept equals negative 20.5
SCC Well, where would you put that?
LW Am I doing it on this one right here? Um, I dont have one, so it
would be like down here. So it would be really, like... And then 20.
And then where would I put that? Its negative, well, its not that.
Thats number of teaspoons.
SCC Do you have one point that you could put on the graph?
LW I have my.. .that I could just add to the graph, right? Yeah, I have 2.6
and 42.8. So, 42.8 and 2.6. And like so, it would be right there,
right?
LW plotted this point approximately correctly. When asked to just plot the
line without using the coefficients, she was able to do so quickly and confidently.
She appeared to have a preference for using formal methods over imagistic to plot
the regression line. When asked to draw the line, instead of trying to estimate its
84
position, she tried to use the slope and intercept to plot the line. She drew the line
quickly and with confidence when using informal imagistic methods.
When asked to represent the residual graphically, LW was able to do so by
drawing a vertical line segment for the data point at 2.0 teaspoons.
LWs imagistic representation of the residual plot was in the initial stage.
The residual plot was not used as an object. It was used in terms of its parts, such as
axes, scales, and plotted points, and then only after prompting. This is inferred
because some parts (the plotted points at 2.5 teaspoons) distracted her from using
other parts (the scaled and labeled axes). She used the axes incorrectly, first trying
to place the value of the predicted (42.8) on the vertical axis, and only placing it
correctly after prompting from the interviewer. She expressed a preference for the
second residual plot, with the predicted on the horizontal axis.
LW was able to draw a line to represent the regression equation. She did
this informal line quickly and with confidence, and the line was reasonably accurate.
She struggled with using the slope and yintercept because the intercept was not part
of the graph. Her representation of the regression equation as a line was in the
completion stage. She was able to represent a residual as a line segment on the
scatterplot.
85
Sampling Distributions. In the first part of prompt II, LW drew seven
different graphs of distributions. Her three normal distributions are symmetric and
mound shaped. They all lack the asymptotic property. One of the graphs does not
have an axis, but it appears that LW meant to draw the curve to the axis and then
follow the axis horizontally. They all have inflection points but in two of the graphs
the inflection points are too far from the center. One of the graphs is approximately
correct, but it is not clear that she understood the role of asymptotic behavior. The
graphs that are meant to show the skewed distribution either have a hanging
maximum, or a vertical line from the high point to the axis.
In the second part of prompt II, LW drew three normal curves. The first of
them had the inflection points approximately correct, but did not show asymptotic
behavior. The other two lack asymptotic behavior and had inflection points that are
barely visible and too far from the center.
In prompt II, LW drew a normal curve for the population and a normal curve
for the sample. She drew the population distribution with a smaller standard
deviation, and a symmetric shape. The interviewer challenged her about the shape
of the population distribution. When she reread the prompt she drew a "hanging"
curve skewed left. This is the third curve on her prompt. It is skewed towards the
smaller values, contrary to the information in the prompt.
LW had conflicting representations of the standard error of a sampling
distribution.
86
LW How is the standard deviation of the sampling distribution related to
the standard deviation of the population? (reading) Um, hum, it is
not the same. I know that, well, I don't know, how's it?
SCC Bigger or smaller?
LW Oh, smaller, no it will be bigger, because when you have the whole
population, you have more people, so the normal population is drawn
in towards the mean, um, so it should be it should be bigger, yeah.
LW drew the population and sample mean distributions with the wrong
order of spread, and, even though she could state the formula for the standard error
of the a proportion, she did not use it to correct her misconception. Her imagistic
representation is in conflict with the formal and syntactic representations. She
appeared to have a kinesthetic representation of the sampling distribution being
"drawn in towards the mean". She later stated a preference for the imagistic
representation of the shape of the sampling distribution. She stated that her
knowledge came from watching demonstrations using computer simulation with
graphical output, but has failed to retain the correct conclusions from these
demonstrations.
LW How is the shape of the sampling distribution affected by the sample
size? (reading) So, I guess as the sample size gets bigger ... so it will
still be ... skewed.. .but it would normalize more, wouldn't it like as
the sample size gets bigger it gets more normal, but this isn't a
normal distribution.
SCC What do you remember that tells you that, pictures, equations and
words in your text?
LW You showed us that on Fathom. That every time you made more and
more, like larger samples that it became more and more normal.
Because it goes more towards the mean. So, I guess it would be less
skewed each time, wouldn't it? Like it would be more towards the
mean, like let's say the mean would be right there.
87
However, later in the prompt, she stated that the graphs were not very important to
her.
SCC You have relied less on the pictures and more on the calculations
than the other kids. How do you think those drawings have
supported your calculations. Could you have done it without the
drawings? Do you think they were important to support your
calculations?
LW I could have done it without the drawings. (Very confidently.)
SCC So, how important were the drawings?
LW Not very. It helps me to see what I am doing, but I know that if I just
multiply the standard deviations times two, and add it to the mean,
then I can get the answer. Or else if I just take if I want a 95%
confidence interval then I know it is something like 1.6 or 1.64, then
I can just take that and multiply that times the mean.
Density Function Sketches. In prompt HI, the first of the curves is
approximately correct. The other two are not symmetric, have poorly drawn
inflection points, and lack asymptotic behavior. In prompt IV, there are three well
drawn graphs, and in prompt V, there are two that are are also well drawn, although
the second has tails that are too high above the axis.
There was growth in the drawing of the curves. By the fourth and fifth
prompts the curves have the appropriate properties and could be used to estimate
probabilities. They therefore could stand alone, and are in the completion stage.
In prompt HI, LW used the draw function on the calculator and immediately
knew that something was wrong when most of the distribution was shaded. The
imagistic representation (as well as the high p) told her that an error had occurred.
88
Her exclamation of "whoa" indicated that she was surprised by the results. She
went back to the calculator and fixed the inputs.
LW Test the claim that the population proportion oh ok, so Ho, so the
null hypothesis is that not equal, so it would be equal to .5 and the Ha
is that not equal to .5. Ok then, so the null hypothesis is that the
population proportion equals .5, so that would be that half of the
students get enough sleep to do well in school. And then the
alternate hypothesis is that the population proportion isn't equal to .5
and so it is either more or less. It is just not equal to .5 of the
students get enough sleep. Ok, so it is a not equal one, and it is
drawing it now. Whoa. So, ok, my p = .95 so, like with the drawing,
it is really easy to get this. So, you have a 95% chance to get a
population oh, shoot. I put the wrong thing in, I put .31 in, I need to
put .5 in, escape, escape.
SCC What told you that something was wrong?
LW Because, over here when I drew this thing, it is like all the way over
here and when I did the confidence interval with the population
proportion in the middle it was just, you don't have a very good
chance, it looks like, of getting this. So I have to do it again. P isn't
.95
LW used imagistic vocabulary in describing the confidence interval, using
the words "in there", "outside of' and "inside there".
LW .5, and .5 is outside of the interval. So, when I put .31 in there it said
with like 95% chance of having that in there, but with like my
confidence interval .5 isn't inside there, and so it seems like the
chance of having that would be a lot smaller.
In prompt IV, LW had another high p because of a wrong direction to the
alternate hypothesis. She used the shade option on the calculator, and saw the large
p, but did not stop and correct.
LW And then another sample and then another. Or a sample and then
another mean and a sample and another mean, so, I see what is going
on now, so it is not yeah its like // and the null jj would be 7.8
89
hours and the alternate ok, that is better. Ok, so the p is .96 and the t
= 1.9 so that means ... I need to do a confidence interval too, or
should I explain this?
SCC Explain that first.
LW This says that there is a 96% chance to get a /i, like a sample mean
of 7.55 when you have a distribution with 7.8 as the mean. And so,
that is a really good chance, I think.
SCC Anything unusual about that number?
LW It is big. Why, is it too big?
SCC Did you do the shade?
LW Yeah. Like a lot of that distribution is filled.
This happened in previously in prompt HI, and caused a greater reaction than in this
prompt. After stating "Like a lot of that distribution is filled." she continued to try
to use the .96 value to make decisions. There was no growth as a result of the error
in prompt HI.
In prompt V, LW drew curves and labeled the axes, but did not use them to
support her work.
The sketches and graphical output of the calculator were used to aid in
problem solving. The quality of the sketches improved over time. They began in
the initial stage and ended in the complete stage. The last drawings had all of the
important characteristics of the density functions. The sketches drawn on the
calculator included shading. The shading sometimes caused an alert, but did not
lead her to correct the problem. Because the shading alerted her to a problem, but
did not lead to a correction, it is judged to be only partially connected to other
representations, and is therefore in the development stage.
90
Axes and Other Imaeistic Representations. In prompt HI, LW assigns
importance to the fact that 0 is not in the confidence interval. The prompt asks for a
confidence interval in part a, without stating a null hypothesis. She assumes that 0
is the important point.
LW Second, one proportion z interval. There were 11 successes and 35
and she wanted a 90% confidence interval, so, the confidence
interval is 1852 to .4434 and so that does not have zero in it, so that
means that you reject the null.
SCC What would it mean if it had zero in it?
LW It would mean that you could say with 90% confidence that there is
no difference.
Also in prompt IE, LW was able to put the .31 in the center of the confidence
interval.
SCC Ok, you just drew what looks like a normal distribution curve with
zero in the middle. If you were going to put the confidence interval
on there, what should go in the middle?
LW What would I put in the middle? I would put .31
But, later in the same prompt, she assigned some importance to the fact that .31 is in
the interval. She seemed to be unaware that the sample proportion (0.31) was used
to construct the interval and was therefore required to be in the center of the
interval.
SCC You told me when you were doing the confidence interval,
something about zero not being in the interval. In order to use this
confidence interval, to check on whether your test is working right,
what would you be looking for in the interval?
LW .31
SCC .31 is in that interval, isn't it?
LW Right.
SCC It is in the middle of that interval, isn't it?
91
