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Middle School Students’ Experiences with Symbolism
Introduction to the study
Background
Whenever we think something and express or communicate about it, we need
some tools. In doing algebra, symbols provide one such tool with which we can think
and represent our thoughts and ideas. Not only are symbols a tool for representation, they
have also played a critical role in developing algebra. If we consider generality as what
makes algebra most different from arithmetic, the beginning of algebra is historically
traced back to ancient Mesopotamia and Egypt. In spite of almost four thousand years of
history of algebra, the history of symbols had not begun until the 16th century. Before
then, algebraic ideas were stated rhetorically, and special words, abbreviations, and
number symbols were used as notations. It was Vieta who used symbols purposefully
and systematically after some mathematical symbols (e.g. , , ) were introduced with
letters used for unknowns (Kline, 1972). Since Vieta, algebra has rapidly developed from
a science of generalized numerical computations, to a science of universal computations
and then into a science of abstract structures thanks to symbolism (Sfard, 1995).
However, symbolism is one of the major difficulties for young students in
learning algebra even though symbolism made it possible to study abstract structures in
algebra by expressing complicated mathematical ideas succinctly. Hiebert et al. (1997)
explained that the difficulties in dealing with symbols as a learning tool were attributed to
the fact that “meaning is not inherent” in symbols (p. 55). They insisted that meaning is
not attached to symbols automatically and without meaning symbols could not be used
effectively. So students should construct meaning for and with symbols as they actively
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use them. The National Council of Teachers of Mathematics’ (2000) Algebra Standard
also encouraged using symbols as a tool to represent and analyze mathematical situations
and structures in all grade levels. Specifically, students in Grades 6 – 8 are recommended
to have
… extensive experience in interpreting relationships among quantities in a variety
of problem contexts before they can work meaningfully with variables and
symbolic expressions. An understanding of the meanings and uses of variables
develops gradually as students create and use symbolic expressions and relate
them to verbal, tabular, and graphical representations. Relationships among
quantities can often be expressed symbolically in more than one way, providing
opportunities for students to examine the equivalence of various algebraic
expressions (p. 225-226).
In this recommendation, NCTM put emphasis on using problem contexts to help students
develop meaning for symbols and appreciate quantitative relationships.
In line with the issues mentioned above, the present study is intended to provide
insight into students’ experiences with symbolism. In particular, the educational purpose
of this study is to inform mathematics educators of how students construct meaning of
symbols and learn mathematical concepts with symbols so that mathematics educators
can enhance students’ learning of mathematics with symbols.
Research questions
In the abstract development of algebra with systematic symbolism, Wheeler
(1989) argued that abstract algebra sacrificed the implicit meanings for its applicability
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unlike rhetorical and syncopated algebra (cited in Kieran, 1992). For instance,
Diophantus, as a syncopated algebraist, created numeral expressions like 10 – x and 10 +
x and multiplied them to get 100 – x2 as if they were numbers like 2 and 3 to solve his
word problem. As he denoted the letter x as an unknown but fixed value in the context of
the problem, he could keep the meaning of x for its applicability explicitly. However, he
might have not obtained the notion of variables, which abstract algebra achieved (Sfard &
Linchevski, 1994). As Kieran (1992) elaborated, symbolic language made algebra more
powerful and applicable by eliminating “many of the distinctions that the vernacular
preserves” and inducing the essences (p. 394). However, the powerful yet
decontextualized language brought difficulties for young students who were beginning to
learn algebra:
Thus, the cognitive demands placed on algebra students included, on the one hand,
treating symbolic representations, which have little or no semantic content, as
mathematical objects and operating upon these objects with processes that usually
do not yield numerical solutions, and, on the other hand, modifying their former
interpretations of certain symbols and beginning to represent the relationships of
word-problem situations with operations that are often the inverse of those that
they used almost automatically for solving similar problems in arithmetic (Kieran,
1992, p. 394).
In fact, many researchers (e.g. Kieran, 1992) have studied students’ difficulties in
manipulating symbols as mathematical objects and modifying their interpretations of
symbols. Also some studies (e.g. Stacey & MacGregor, 1997) were conducted to
investigate how meaning for symbols could be developed. Hiebert and Carpenter (1992)
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reviewed such literature and summarized that making meaning for symbols could develop
connections between symbols and other representation forms. In analyzing two primary
functions of symbols of a public function and a private function, they insisted that
connections between symbols were required for a public function, involved in
representing something known already for communication, and connections with other
representation forms were necessary for a private function, involved in organizing and
manipulating ideas as objects (p. 73-74). This analysis led to my curiosity about early
algebraic students’ experiences with symbolism. My initial curiosity included sporadic
questions like; how do young students interpret mathematical symbols?, in what ways do
they use symbols?, do they feel the need of symbols?, what do they want to represent
with symbols?, in what ways do their understanding of symbols affect learning
mathematical concepts?, and so on. Inspired by these questions, the present study will
investigate how middle school students develop algebraic reasoning with symbols while
doing mathematical activities. The following questions will guide this study:
1.
How do students make sense of symbols used in mathematical activities?
2.
How do students’ mathematical concepts develop and evolve as they use
symbols throughout mathematical activities?
I presume that students’ prevalent experiences with symbolism occur in classroom
learning situations and the learning experience includes teacher’s lecture, reading
mathematics books, doing hands-on activities, observing how teacher and other students
use symbols, and discussion with other students. Under these circumstances, we can
never simply assume that students understand symbols in the way that each activity
means to provide. So I would like to investigate how students make sense of symbols
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used in classroom activities through the first question. The second question focuses on
students’ understandings of mathematical concepts that certain symbols are intended to
represent. For instance, y = mx + b is a symbol as a whole or a symbolic representation
that describes linear relationship between two variables of x and y. Also m and b are
symbols for the slope and the y-intercept in the linear relationship, respectively.
Embedded mathematical ideas in the symbol or symbolic representation are what students
need to learn ultimately. Since I believe that students’ understanding of mathematical
concepts cannot occur once and for all, I expect students’ understanding of certain
mathematical ideas to evolve throughout various activities. So I would like to investigate
how students’ mathematical concepts develop as they engage in mathematical activities.
Theoretical framework
This section includes theoretical perspectives with which the present study will be
guided. The first part is allotted to my perspectives on learning mathematics in general,
and the second part is to show briefly what theoretical perspectives on symbolism were
considered that eventually lead to the adoption of the procept model (Tall et al., 2001).
Perspectives on learning
Perspectives on learning concern statements of subjectivity, that will inform and
affect all the activities of the present study in general, rather than establish a theoretical
framework. In particular, this section mainly includes my current personal beliefs formed
through learning and teaching mathematics and studying mathematics education as a
graduate student. Since the beliefs will provide lenses and constraints with which I
design the present study and interpret all possible phenomena the study will bring, I
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believe that it is worthwhile to state here. My perspectives on learning are quite parallel
to what radical constructivists assume as underlying principles: (1) knowledge is not
passively received but built up by the cognizing subject, and (2) the function of cognition
is adaptive and serves the organization of the experiential world, not the discovery of
ontological reality (von Glasersfeld, 1995, p. 18). Although the way I interpret the
principles might not be as identical as most radical constructivists presume, these inform
my view of mathematics learning. What I believe about learning mathematics are:

A learner does not receive mathematical knowledge passively.

A learner constructs his own knowledge.

Learning occurs through experiences.

Knowledge is primarily personal.
As an elaboration, I believe that a learner does not receive mathematical knowledge
passively. Even without taking a constructivist perspective that a learner actively
constructs mathematical knowledge, I have seen evidences of my belief. For example, I
had learned mathematics via lectures and fortunately most of my mathematics teachers
helped me understand mathematical concepts. When I discussed or worked on problems
with my classmates who had shown similar mathematical abilities and performances, I
could find differences in how we understood a certain topic and strategies we used. If we
were receivers of knowledge, we should barely find differences in knowledge or
understanding of it. I think the differences came from what we did in our own mind. My
second belief is that a learner constructs his own knowledge. What I mean by “construct”
is not necessarily the same as constructivist perspective. Construction means neither
invention like what professional mathematicians do nor isolated construction without
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help. Back in my example aforementioned, I believe that my friends and I constructed
our own mathematical knowledge mainly with the teacher’s help. We could not absorb
what the teacher told us, but instead each of us tried to make sense of mathematics around
us. In other words, we made sense of all mathematics from previous learning, textbook,
everyday life experiences, communication with classmates, and teacher’s explanation in
order to fit all of them together. Third, I believe that learning occurs through experiences.
What I mean by experiences is mostly the learner’s interaction with his/her environment,
and examples of experiences related to learning include learning activities, reading books,
communicating with others, listening to teachers, reflecting and so on. Finally, I believe
that knowledge is primarily personal. In fact, mathematical knowledge definitely had
social and cultural aspects not only because mathematics has been built historically and
culturally throughout a long period of time but also because most of the mathematical
experiences we have occur in school settings with a teacher and a class of students.
Although we develop mathematical knowledge or meaning as a community in classroom,
it is ultimately the learner who makes choices of whether the developed knowledge will
be included meaningfully into his or her knowledge structure and whether the knowledge
will be actively used for mathematical activities like solving problems.
Perspectives on symbolism
In order to study students’ experiences with symbolism, the present study needs to
decide which symbols will be focused on and how to define them. Cobb (2000) seemed
to define the term symbol broadly such as:
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…to denote any situation in which a concrete entity such as a mark on paper, an
icon on a computer screen, or an arrangement of physical materials is interpreted
as standing for or signifying something else (p. 17).
Following his definition, I could use two distinct erasers pretending they were cars in
order to explain a traffic accident that I experienced. By moving the two erasers, I could
explain how my car was hit by the other. Here the erasers were symbols since they were
concrete entities in the accident situation and stood for cars. In addition, Janvier et al.
(1992) differentiated the erasers and the car that the erasers stood for as “signifier or
referent” and “signified or referenced”, respectively. Unlike symbols, the signified or
referenced (what symbols stand for) is not limited to a concrete thing. It could be an
action, an idea or a concept, depending on context.
Although accepting Cobb’s (2000) notion of symbols and the concepts of signifier
(or referent) and signified (referenced) by Janvier et al. (1992), the present study will
focus on standard mathematical written symbols since mathematical activities considered
in this study will happen in classroom contexts. Unlike the previous example of
everyday-life symbols, it is not always clear to tell what mathematical written symbols
stand for. For example, a fraction 3/4 is a mathematical symbol, or signifier, but its
signified is not as clear as that of the erasers in the previous example. In various contexts,
3/4 could stand for a fair share of sharing 3 apples among 4 children, a ratio of 3 out of 4,
or an operator as in 3/4 of something. Even when removing a specific context, 3/4 still
could refer to the abstract concept of fraction. This example shows two faces of symbols
as process and concept. It seems the frequent case that symbols as a vehicle to signify
process are prioritized, and after the process is familiarized enough they can be used as
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objects carrying concepts. Mason’s (1980) spiral model also explained the shift of
symbol uses as:
from confidently manipulable objects/symbols,
through their use to gain a ‘sense of’ some idea involving a full range of imagery
but at an inarticulate level,
through a symbolic record of that sense,
to a confidently manipulable use of the new symbols,
and so on in a continuing spiral (cited in Mason, 1987, p. 74-75).
In particular, Mason (1980) not only explained the shift of symbol uses between as
process and as concept but also showed the continuous acquisitions of new symbol uses
with confidently usable symbols.
However, both the dichotomous uses of symbols and the acquisition of symbols as
concept after process seem artificial since I believe process and concept are mingled
together so that the demarcation is not clear and they grow together. Thus, I believe,
building on Mason’s (1980) spiral model, Gray and Tall (1991, cited in Tall et al., 2001,
p5.) introduced the notion of procept. Tall et al. (2001) elaborated as:
It [procept] is now seen mainly as a cognitive construct, in which the symbol can
act as a pivot, switching from a focus on process to compute or manipulate, to a
concept that may be thought about as a manipulable entity. We believe that
procepts are at the root of human ability to manipulate mathematical ideas in
arithmetic, algebra and other theories involving manipulable symbols. They
allow the biological brain to switch effortlessly from doing a process to thinking
about a concept in a minimal way (p. 5).
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Therefore, rather than studying whether symbols are used as process or concept, they
considered a powerful way of using symbols to switch between process and concept
flexibly. Then they developed three different performance levels showing how students
used symbols. They are the procept, process and procedure levels. To distinguish
procedure and process, they meant procedure as “a specific sequence of steps carried out
a step at a time” and process as “in a more general sense any number of procedures which
essentially have the same effect” (Tall et al., 2001, p. 7). In order to show what is
expected at each level, the following figure from Tall et al. (2001, p.8) is provided:
Spectrum of outcomes
procedural
To DO
routine
mathematics
accurately
proceptual
To perform
mathematics
flexibly &
efficiently
To THINK
about
mathematics
symbolically
Procept
Process (es)
Procedures
Progress
Process
Procedures
Procedure
Sophistication
of development
Figure 1: A spectrum of performances in the carrying out of mathematical processes (Tall
et al., 2001, p.88)
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In progressing up to the proceptual level, students can choose the most suitable process
under a given situation having more options and then “Being able to think about the
symbolism as an entity allows it to be manipulated itself, to think about mathematics in a
compressed and manipulable way, moving easily between process and concept” (Tall et
al., 2001, p. 88). So this model will guide the present study theoretically so that through
this model I can see students’ experiences with symbolism throughout their mathematical
activities.
Methods
The present study will be conducted within the activities of a NSF-funded project,
Coordinating Students’ and Teachers’ Algebraic Reasoning (CoSTAR). The project
purposely studies “teachers’ and students’ understandings of shared classroom
interactions and ways that teachers and students work together to shape the teaching and
learning of middle-school algebra” (Izsak et al., 2002, p. 2). Whereas CoSTAR
investigates both teachers’ and students’ algebraic reasoning, the present study will be
conducted from only students’ perspectives, so that some activities (such as classroom
interactions) that will be approached through both teachers’ and students’ perspectives in
CoSTAR, will be interpreted only from the viewpoint of the students in this study.
Participants in the present study will be selected from students in a rural middle
school that provides the research site for CoSTAR. It is anticipated that participants of
the present study will be students in grade 7 in the fall of 2003 and they will begin to
learn algebra with materials from the Connected Mathematics Project (CMP; Lappan et
al., 2002). Students in grade 7 will study 6 to 8 units throughout the whole academic year,
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but the first unit, “Variables and Patterns: Introducing Algebra”, will be the mathematical
content of the present study. In the unit, students will learn how to represent a changing
situation in different ways. Eventually, students will describe patterns of change relating
one variable to another verbally, in tabular form, graphically, and symbolically.
As the data collection method, a teaching experiment will be adopted since the
goal of the present study fits with this method. According to Steffe and Thompson.
(2000), researchers through teaching experiments aim to construct models of students’
mathematics and so they supposedly look behind what students say and do in order to
understand their mathematical realities (p. 269–270). In the teaching experiment, I will
play a role of teacher-researcher, who will try to understand students’ experiences with
symbolism and to build models of the students’ symbolic activity with the notion of
procept (Tall et al., 2001). Participating students will work in pairs in the teaching
experiment since they can provoke each other’s thinking through interactions and by
reflecting on their own thinking relative to that of the other. Two pairs of students are
expected to participate in the present study.
As a personnel element of teaching experiments, an observer will support the
teacher-researcher providing feedback from planning each teaching episode to the whole
process of the teaching experiment. Tasks in the teaching experiment will be designed to
understand students’ symbolism and also to extend their experiences with symbolism
based on what they do during the class activities and their written performances.
Each teaching episode will be video-recorded with two cameras, which will
provide a whole picture of interactions among the teacher-researcher and a pair of
students and a focused view of students’ work. To generate the primary data source, the
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views from both cameras will be synchronized and digitized. Then the video data will be
analyzed iteratively. Informed by iterative videotape analyses (Lesh & Lehrer, 2000),
data analyses in the present study will go through several interpretation cycles. The first
interpretation cycle will include debriefing from on-the-scene observers right after each
teaching episode. Notes and feedback from both the teacher-researcher and an observer
will be included. The second interpretation cycle will produce observation notes as
replaying videotape of a teaching episode before conducting the next episode. The third
cycle will produce written transcripts for each teaching episode. The last interpretation
cycle will produce analyses of all the teaching episodes for each pair of students and
across the pairs. Unlike the three previous interpretation cycles, the fourth cycle will be
repeated to produce interpretations of students’ activities from the theoretical
perspectives of the present study.
Analysis of Students interviews
As a part of the activities of the project CoSTAR, I had three interview sessions
with a pair of students, Ansley and Aisha (pseudonyms), who were in grade 6 in the
spring of 2003. The major mathematical content was the operations of fractions. In the
first interview session, they were mainly asked about adding and subtracting fractions
with number line representations and fraction strips. They had learned number line
representations of addition and subtraction in class just before the interview. Also
fraction strips were used when introducing fractions and when estimating fractional sums
and differences. In the second interview session, students were questioned about what
they did on the test taken about a week before. In the last interview session, the main
task was multiplying fractions in the form of ‘fraction of fraction’ with an area model. In
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order to increase an accuracy of equal partitioning and help students’ thinking, the
computer software, “Fraction Bars”, was provided for students in the third interview
session. Although the time constraint did not allow me to analyze the students interview
data through the complete interpretation cycles described in the above section, I could
find a couple of themes about students’ symbolism across the interview data. Below I
will present those assertions that I made through the data analysis.
First, students appeared to prioritize procedures with which they processed a
certain task. Although they had learned various procedures for a task, they did not seem
to consider the multiple procedures as equal choices. Instead, they would rather rank the
procedures and resort to the best procedure that they picked. For addition and subtraction
of fractions, students had learned at least three different methods such as quantitative
reasoning with fractions strips, using number line representations, and using common
denominators. However, they put the priority on the method of using common
denominators. When I had the first interview session, I asked students to add and
subtract fractions with number line representations. Also fraction strips were available
for students to partition a number line and draw a certain length of segment. When asked
to solve 1/5 + 2/5 on a number line, Aisha drew a 1/5-long-segment and a 2/5-long
segment using the fifth fraction strip and she answered 3/5 instantly saying, “because 2
plus 1 is 3 and you just put 5 instead of adding”. For the question of 1/2 + 1/3, she drew
a 3/6-long-segment and a 2/6-long segment with the sixths fraction bar instead of using
the halves strip and the third strip and found the answer by adding numerators.
Seemingly, Aisha was confident with adding and subtracting fractions using common
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denominators and considered number line representation as a visual form to show the
procedure rather than another procedure of operation.
As converting improper fractions into proper ones, they could process through
division, subtraction, reasoning quantitatively, and using multiplication facts, but they
thought that the division method was the most powerful. Even when converting 7/6, both
students divided 7 by 6 to obtain 1 1/6. Asked whether they had another way to convert,
Aisha told that she could use subtraction, but she added soon that subtraction method did
not always work. She appeared to think that she could only use subtraction method when
an improper fraction is less than 2 rather than considering a repeated subtraction for an
improper fractions bigger than 2.
Then I reminded them of another method as showing a video clip where their
teacher drew a picture of 7/6 as one whole and 1/6. For the picture, Aisha explained to
Ansley that she knew 7/6 is more than 1 and so she had a leftover of 1/6. However, her
explanation was not quite quantitative reasoning such as 7/6 means seven 1/6s and
because six 1/6s make a whole 7/6 is the same as one whole and a 1/6, that is, 1 1/6. So
Dr. Olive provided a problem of 1/6 oz. bags of spice and asked a series of questions
about how many bags were needed to make 1/2 oz., 5/6 oz., and 7/6 oz. And he finally
asked how much of ounces seven 1/6-ounce spice bags were. After answering correctly,
Ansley said, “I think that one is easier than the other ways”, but she continued, “it is
easier to understand but division is quicker”.
Then I showed Ansley’s conversion of 4 and 13/10 into 6 and 1/10 using division
on her test (see Picture 1.).
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Picture 1: Ansley’s conversion of 4 and 13/10
She did a long division of 13 divided by 10 correctly and answered 6 and 1/10 mistakenly.
She told in the interview that she did not know why she did so. This might show her that
division method is not as easy as some other methods. However, both students did not
change their preference of division method for conversion even after admitting another
method was easier to understand and the burden of long division.
Second, both students showed a strong preference of symbolic manipulations for
operating with fractions. So when they were required to use other procedures, they
tended to make sense of them and process them by referring to their results that they
obtained through symbolic manipulations. In the second interview, I asked Aisha to
explain her number line representation of 3/5 – 5/10 on the test. Her written work
showed a correct symbolic manipulation as below but her number line representation was
not supportive (see Picture 2.).
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Picture 2: Aisha’s number line representation of 3/5 – 5/10
Based on her comment, “I solve first and draw it, draw it backward”, I inferred that she
found the answer 1/10 first and tried to represent involving fractions of 3/5, 5/10, and
1/10 somehow on the number line. Instead realizing that a 3/5-long segment is a
combination of a 5/10-long segment and a 1/10-long segment, she put 5/10 on the
segment beginning at 5/10 and ending at 6/10 on the number line. Then the interview
revealed that she did not consider the length of the segment. Even when she learned what
number line representations meant, she failed to think flexibly as she could take a 5/10long segment from a 3/5-long segment and have a 1/10-long segment left either in the
front or at the end of the 5/10-long segment.
Other supporting evidence was found in the third interview when they worked on
fractions of fractions. In the third interview session, the main task was;
Greg bought 2/5 of a square pan of brownies that had only 7/10 of the pan left.
a. Draw a picture of brownie pan before and after Greg bought his brownies.
b. What fraction of a whole pan did Greg buy?
They had the task as homework after learning how to find fractions of fractions with
pictorial representations. One representation was with thermometers partitioning
horizontally or vertically and the other was with brownie pans partitioning horizontally
and vertically. Unlike my expectation that the context and pictorial representations
would help them think, they had a difficulty to understand the context and draw the
pictures. It took more than half of the interview session for them to get a brownie pan
with each 1/10 of 7/10 of a brownie pan partitioned into fifths (see Picture 3.).
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Picture 3: A brownie pan with partitioning each 1/10 of 7/10 into fifths
Then Aisha finally told Ansley to color 2 small pieces in each 1/10 bar. Ansley seemed
to pay little attention to what she was doing and followed Aisha’s direction. When asked
to explain, Aisha said, “I thought the answer’s 14/50. I solved and I started counting
down, which gave me 5 and across 7. So I just colored 2 bars. 7 times 2 is 14” and
added, “I get my answer first and then showed how I got it”. So apparently Aisha
understood or remembered that the answer to a fraction of a fraction was obtained by
multiplying numerators and denominators of the two fractions, and tried to make sense of
the pictorial representation.
As a finding of the brief data analysis, I concluded that both students had not yet
reached the process level in the spectrum in Figure 1. Although they had learned various
procedures for each operation as adding and subtracting fractions, converting improper
fractions, and multiplying fractions, they did not perform the mathematical processes
flexibly and efficiently by selecting a best procedure under a specific problem setting.
Instead, they adhered to a certain procedure for a mathematical process. Commonly both
students preferred symbolic manipulation for each operation, but the preference does not
mean that they have reached the proceptual level either so that they can think about
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mathematics symbolically. So, the findings informed me that both students were in
between the procedural level and the process level since they could perform at least a
procedure for each mathematical process confidently and they could understand and
perform some other procedures although they did not use them flexibly and efficiently.
This suggested that the procept model might have room for elaboration or modification to
explain how students perform a mathematical process. Therefore, in the proposed study,
I anticipate elaborating the procept model in students’ learning of variables and patterns
or modifying the model if it is necessary.
Interview Tasks
In this section, I will describe the interview schedules and the purpose of each
interview task. The interview schedules will be set according to how students’ learning
in class progresses and the interview tasks will be based on the unit to be investigated in
the present study, “Variables and Patterns: Introducing Algebra”. The unit has 5
investigations, but the last investigation will not be included in the interview tasks since
the investigation is about how to use a graphing calculator to generate a table or a graph,
not to use a graphing calculator as an investigating tool for the mathematical topic. For
each of four investigations, two interview sessions will be conducted; one in the middle
of the investigation and the other at the end of the investigation. The first interview
session in each investigation will mainly focus on how students make sense of symbolism
in class activities and the second one will focus on students’ mathematical concepts
through problem solving activities. The problems to be used in the second interview
session of each investigation will be adopted from those in the “Application, Connections,
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Extensions” sections in CMP. Below, I describe the purpose of tasks in each interview
session with example problems for each second interview session.
1. Investigation 1: Variables and coordinate graphs
The main tasks are to describe a situation with two variables, to identify variables in a
situation, to identify which variable is independent and which is dependent, and to plot
the graph of two variables.
Example: The convenience store across the street from Metropolis School has been
keeping track of their popcorn sales. The table below shows the total number of bags
sold beginning at 6:00 A. M. on a particular day.
a. Make a coordinate graph of these data. Which variable did you put on the x-axis?
Why?
b. Describe how the number of bags of popcorn sold changed during the day.
Explain why these changes may have occurred.
Time
Total bags sold
Time
Total bags sold
6:00 A. M.
0
1:00 P. M.
58
7:00 A. M.
3
2:00 P. M.
58
8:00 A. M.
15
3:00 P. M.
62
9:00 A. M.
20
4:00 P. M.
74
10:00 A. M.
26
5:00 P. M.
83
11:00 A. M.
30
6:00 P. M.
88
noon
45
7:00 P. M.
92
2. Investigation 2: Graphing changes
Students are here to make a table and/or a graph after reading a narrative of a changing
situation, to interpret data given in a table and a graph, and to compare tabular, graphic
and narrative representations of the situation.
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Example: Make a table and a graph of (time, temperature) data that fit the following
information about a day on the road:

We started riding at 8 A. M. The day was quite warm, with dark clouds in the sky.

About midmorning the temperature dropped quickly to 63F, and there was a
thunderstorm for about an hour.

After the storm, the sky cleared and there was a warm breeze.

As the day went on, the sun steadily warmed the air. When we reached our
campground at 4 P. M. it was 89F.
3. Investigation 3: Analyzing graphs and tables
The interview tasks are to search for patterns of change in a graph and a table, to describe
a situation with verbal rules, and to predict a change.
Example: Recall that the perimeter of a rectangle is the sum of its side lengths.
a. Make a table of all the possible whole-number values for the length and width of a
rectangle with a perimeter of 24 meters.
b. Make a coordinate graph of your data from part a. Put length in the x-axis and
width on the y-axis.
c. Describe what happens to the width as the length increase.
d. Would it make sense to connect the points in this graph? Explain your reasoning.
4. Investigation 4: Patterns and Rules
Students are to show their understanding of the relationship between rate, time, and
distance, to identify and represent rates in a table and a graph, to express patterns in
symbols.
Jeonglim Chae
Prospectus
22
Example: Sean just bought a new CD player and speakers from the Audio Source for
$315. The store offered Sean an interest-free payment plan that allows him to pay in
weekly installment of $25.
a. How much will Sean still owe after one payment? After two payments? After
three payments?
b. Using n to stand for the number of payments and A for the amount still owed,
write an equation for calculating A for any given value of n.
c. Use your equation to make a table and a graph showing the relationship between n
and A.
d. As n increases by 1, how does A change? How is this change shown in the table?
On the graph?
e. How many payments will Sean have to make in all? How is this shown in the
table? How is this shown on the graph?
In particular, the first interview sessions in each investigation will provide an answer to
the first research question of the present study – how do students make sense of symbols
used in mathematical activities?, and the second interview sessions in each investigation
will mainly provide evidences of how students’ mathematical concepts develop and
evolve as they use symbols throughout mathematical activities. Specifically, a series of
thought process maps adopted from Tall et al. (2001, p.11) will show students’
development and evolvement of their mathematical concepts. As an example of a
thought process map, Figure 2 shows how Aisha solved 1/2 + 1/3 on a number line.
Jeonglim Chae
Prospectus
To find a
common
denominator
of 2& 3
Solve
1/2 + 1/3
on a number
line
To find equivalent
fractions with
denominator 6
6
1
2
To find the
sum with
symbol
manipulation
3/6 + 2/6
= 5/6
1/2 = 3/6
1/3 = 2/6
3
4
ooooo
strategic sub-goal
symbol manipulation
link to the goal
1, 2, 3, …
Draw two segments of
lengths 2/6 and 3/6
successive sub-goals
Figure 2: Aisha’s strategies for solving 1/2 + 1/3 on a number line
Refer to
symbol
manipulation
23
Jeonglim Chae
Prospectus
25
Reference
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Prospectus
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