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1. Vol. 5
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31st
annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
193
A CONCEPTUAL CHANGE LENS ON THE EMERGENCE OF A NOVEL STRATEGY
DURING MATHEMATICAL PROBLEM SOLVING
Mariana Levin
University of California, Berkeley
levin@berkeley.edu
This paper reports on an analytic case study of a pre-algebra student who makes a surprising
and significant mathematical discovery over the course of several episodes of problem solving.
The research reported in this paper is motivated by the goal of understanding how and why the
student’s strategies shifted from a simple, yet purposeful, guessing and checking approach to a
sophisticated approach based on linear interpolation. The paper illustrates how a conceptual
change framework developed in the science education literature can provide useful analytic tools
for understanding shifts in problem solving strategies in terms of underlying conceptual
refinements and reorganization.
Introduction
The phenomenon of interest in this paper is how new strategies emerge during mathematical
activities, such as problem solving. Microgenetic analyses of strategy change (See Siegler, 2006
for a review) have focused on developing techniques for tracking shifts in strategy usage at a fine
grain level of detail. While we share this attention to fine-grained analyses, our focus in the line
of research reported in this paper will ultimately be on the processes by which an individual
constructs a novel strategy from existing conceptual resources as opposed to the processes by
which individuals come to reliably activate and use one strategy over another competitor
strategy. In other words, the approach proposed by this research project is to re-frame analyses
of strategy change in terms of underlying conceptual change. We will illustrate how an analytical
approach known as “knowledge analysis” (diSessa, 1993; Sherin, 2001) for studying growth and
change of conceptual structures can provide useful analytic tools for understanding the shifts in
problem solving strategies that come about due to underlying conceptual refinements and
reorganization.
To illustrate the potential of this approach, we will explore a case study of a pre-algebra
student, Liam, who largely independently re-invents a deterministic and essentially algebraic
problem solving strategy, known as linear interpolation, through the activity of solving algebra
word problems using a purposeful guessing and checking strategy. Previous research has
documented that students use informal problem solving approaches such as guessing and
checking prior to instruction with algebraic solving techniques (Johanning, 2004, 2007; Kieran,
Boileau & Garançon, 1996; Nathan & Koedinger, 2000; Stacey & MacGregor, 2000). However,
the conceptual nature of students’ guessing strategies and what kind of mathematical ideas can
potentially be developed as a consequence had not previously been objects of extensive study.
One reason for this is that the prior studies of students’ pre-algebraic problem solving approaches
were based primarily on written records and hence did not offer access to the richness and
learning potential of students’ informal strategies.
The case shared in this paper offers a surprisingly clear demonstration of how important
algebraic ideas such as function, co-variation and rate of change can emerge and be developed
through the successive refinement of informal problem solving strategies. Such potential for the
development of algebraic reasoning is discussed in Levin, 2008. An important contribution of

2. Vol. 5
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31st
annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
194
the current research is the explicit identification of key knowledge resources that are activated
and used as the student constructs the linear interpolation strategy. Thus, the specifics of the case
of Liam are new to the literature on the development of algebraic thinking, but beyond that, the
case of Liam is an interesting site to begin elaborating theoretical and analytical tools for
studying the growth and change of knowledge (i.e., conceptual change) in mathematics, and is
thus of more general interest.
Theoretical Framework
In this paper, the “knowledge in pieces” epistemological perspective proposed by diSessa,
(diSessa, 1993) is adapted to analyze the conceptual underpinnings of an observed strategy shift
in the case of Liam. In this theoretical perspective, an important underlying assumption is that
individual knowledge can be thought of as a complex system comprised of many knowledge
elements of diverse types. As individuals learn and gain in expertise, activation of relevant
knowledge elements becomes more appropriately context sensitive and coordinated as ensembles
of elements. DiSessa, 1993 and Sherin, 2001 argue that it is fruitful analytically to engage with
the complexity of individual knowledge systems by defining a base vocabulary of sub-
conceptual primitive knowledge elements. One reason this is argued to be useful analytically is
because important features of expertise may manifest themselves only at the level of primitives.
Data Collection and Methodology
The data corpus for the Liam case study includes video and written work collected over the
course of six individual semi-structured tutorial sessions with a researcher, each approximately
one hour in length. Liam was one of six pre-algebra students participating in this study aimed at
analyzing students’ emergent understanding of variable and letter-notation using a curricular
tool, a Guess and Check chart, as suggested by a widely-used algebra curriculum (Sallee, Kysh,
Kasimatis & Hoey, 2002). The Liam data corpus was selected for extended analysis because of
the unexpectedly rich conceptual development that occurred during the sessions. The analysis of
video and transcripts of problem solving in this study allowed access to students’ real-time
reasoning as they solved problems. Video data was transcribed for analysis, annotated with
relevant details such as students’ gestures, and coordinated with written work artifacts.
Research Questions
1. How can we characterize Liam’s conceptual understanding in a way that will make
tracking moment-by-moment shifts in understanding analytically feasible?
2. What conceptual understandings did Liam develop between two contrasting episodes that
may have allowed the observed change in problem solving strategy to occur?
Background and Context
Liam is a pre-algebra student who initially approached solving word problems using a
purposeful guessing and checking approach, which he devised (as research shows that many
students do) without any previous instruction about how to solve such problems. Over the course
of several individual sessions, he refined his purposeful guessing and checking approach,
organized in tabular form, to an essentially algebraic algorithm (linear interpolation) for solving
word problems. The linear interpolation strategy we will examine in this paper emerged naturally
over the course of the sessions with Liam, and was not something that was explicitly designed to
be part of the sessions with the tutor/researcher.

3. Vol. 5
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31st
annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
195
Data excerpts illustrating the approach taken by Liam at the beginning and ending of the
sessions are given below. One can see that while Liam’s initial approach was based on
purposeful guessing and checking, his later approach is deterministic, building off the linear
structure underlying the problem contexts, and in fact no longer involves “guessing” at all.
Episode One: Systematic and Purposeful Guessing and Checking
In this first focal episode, one observes that Liam is using a purposeful, systematic guessing
and checking approach to solve the given word problem. This problem was the first in the series
of sessions where the tutor had suggested that Liam organize his guessing and checking strategy
in a Guess and Check chart. Previously, Liam had used the invented strategy of “guessing and
checking,” (though not arranged in a chart).
The base of a rectangle is three more than twice the height. If the perimeter of the rectangle is
sixty inches, find the height and the base of the rectangle.
Below is a reproduction of the chart Liam constructed, along with excerpts from the
transcript coordinated with his activity with the chart.
Figure 1. Transcript from Liam’s problem solving approach in episode one in which Liam used
“successive approximation” to find the solution to the word problem. The chart is a typed
reproduction of Liam’s work.
Already in episode one, Liam is already making very purposeful choices about the sequence
of trial values he constructs. Certainly, his choices of guesses are far from “random.” In fact, he
already appears to have an approximate sense for how the input/output pairs he generates co-vary
linearly. One can also notice that he is making inferences in terms of both “scalar” judgments
(“a little too high”) and also “proportional” judgments (“it’s a little less than twice” the target
value).

4. Vol. 5
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31st
annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
196
Episode Two: Leveraging Linearity to Solve Problems
Later in the series of sessions (session 5 out of 6), Liam had refined his strategy from merely
“purposeful guessing and checking” to “linear interpolation.” In this data excerpt, one can see
Liam deploy his newly constructed linear interpolation strategy to solve a problem of a similar
underlying (linear) form as in focal episode one. The problem he was working on this episode
was:
The sum of three consecutive integers is 414. Find the three integers.
In solving this problem, Liam continues to organize his work in a “guess and check” chart (a
typed reproduction is pictured below). After having solved the problem and when asked to
explain his solution strategy in this later, contrasting episode, Liam says
“I took 408 and 423 [see chart below]. I have the difference between those [between 408 and
423] which is 15. The difference between these two [between 135 and 140] is 5. And 15 divided
by 5 is three. So that means that for every one this changes [indicates the first column], this one
[indicates the sum column] changes by 3. So, then I took 423 and I subtracted that [moved hand
up to problem statement to indicate the target value of the sum: 414]; the difference was 9. 3
times 3 is nine. So, I knew that it would have to be three less than this [indicates 140].”
Figure 2. Transcript from episode two in which Liam used the “linear interpolation” method he
constructed. The chart is a typed reproduction of Liam’s work. Though quotations are presented
separately (to highlight the multiple steps involved in Liam’s strategy), this constitutes one
uninterrupted utterance by Liam.

5. Vol. 5
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31st
annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
197
To give a quick recap of Liam’s activity in the second episode, we see that after Liam has
finished the computations with two trial input values, Liam forms the ratio of the difference
between the two outputs to the difference of the two trial inputs. This allows him to figure out
the rate of change (in this case the constant of proportionality, or the slope) of the underlying
(linear) function. Liam explicitly interprets the ratio he has formed as the unit worth of one
guess: the amount the output will change corresponding to a change in one of the input. Liam
then takes the output corresponding to one trial input he has selected as a reference and figures
out how far that output is away from the target output. He then uses the unit worth of one guess
to figure out how much he should change the input by in order to produce the change in output
he just computed.
In episode two, Liam has refined his sense of how inputs and outputs co-vary. He has now
found a way to quantify and explicitly leverage his intuitions about the underlying linear
relationship that all the input-output pairs satisfy. Notice that the idea that a given input is
“worth” a fixed amount in terms of its effect on the output is a refinement of the earlier
qualitative versions of proportionality Liam used in episode one.
Discussion of the Two Contrasting Focal Episodes
An important point of contrast between the two focal episodes is that in episode one, Liam’s
solution method is highly dependent on his inferences about a particular guess. However, in
episode two, Liam realizes that his solution method is general, and depends only on determining
the rate of change between any two input-output pairs. Further, he purposefully uses two trial
values not for the purpose of converging to the solution to the problem, but for the purpose of
determining an invariant (the rate of change) of the underlying functional relation which all
input/output pairs must satisfy. Once he has determined this invariant, he uses it to deduce the
unknown value that solves the problem.
Analytical Framework
The key analytical move and insight made in this paper involves reframing the “strategy
change” observed between episode one and two in terms of “conceptual change.” To understand
strategy change as conceptual change, we need to go deeper than a top-level description of the
contrasting features of strategy one and strategy two. The task before us now is to find a way to
describe the relevant shifts in conceptual understanding that allowed the strategy change to take
place. Of course, we recognize that Liam has many other forms of resources that could
potentially contribute to the construction of a new problem solving strategy (epistemological,
meta-representational, etc.) in addition to the conceptual resources we will discuss in this paper.
We focus on shifts in the activation and coordination of knowledge resources in this paper
because even in this limited arena, there is significant analytical work to be done.
The first strand of analysis in this study involves recognizing that the approaches in the two
focal episodes are qualitatively different and giving a characterization of some of the important
dimensions of this difference. Some aspects of difference were discussed in the previous
sections where it was noted that the move from “qualitative” to “quantitative” formulations of
proportionality over the course of the sessions was particularly noteworthy as an underlying
conceptual shift.
The focus of a second analytic strand is to give a characterization of and provide an argument
for a set of relatively primitive and elemental knowledge resources, which allow will one to track
processes of change in fine-grained detail. The underlying assumptions of the knowledge in

6. Vol. 5
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31st
annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
198
pieces epistemological framework directly guide how the conceptual resources are identified in
our analysis. In this analysis, we seek to identify the knowledge that was relevant to Liam and
that he was drawing upon in solving the problems. To do this, one can consider the justifications
he makes concerning his choices for trial values. This line of analysis has resulted in the
identification of several conceptual resources that Liam activates and uses over the course of the
sessions. Examples from the two focal episodes presented in this paper are given below.
Candidate
knowledge resource
Description Examples of resource activation in focal
episodes
Monotonicity Larger inputs (in reference to
previous inputs) result in larger
outputs and smaller inputs result
in smaller outputs.
All guesses in focal episode one fit this
pattern. If a particular input resulted in an
output that was too high, the next input was
chosen to be a number lower than the
previous input. Likewise, if a particular input
resulted in an output that was too low, then
next input was chosen to be higher than the
previous input. (Focal episode one)
Sandwiching/In-
betweeness
If an input yields an output that
is too high and another input
yields an output that is too low,
then the true input must be in
between these two inputs.
“Well it was actually definitely 9 if this
[result for 8] was too low and this [result of a
guess of 10] was too high. Unless it was a
decimal number.”
(Focal episode one)
Qualitative
formulation of
proportionality
Small changes in input
correspond to small changes in
output.
This was “a little too high.” [then he chooses
a next guess that is two integer values lower].
(Focal episode one)
Medium changes in input
correspond to medium changes
in output.
N/A in focal episodes one and two.
Large changes in input
correspond to large changes in
output
“This is way too much” [and he follows up
by choosing a guess that is a lot lower than
the previous guess] (Focal episode one)
Half as a reference
point
If an input yields an output that
is about twice (or exactly twice)
as much as the target output,
then the next guess should be
about half (or exactly half) as
much.
This is “almost twice too much.” [then he
chooses a next guess that is nearly half as
much] (Focal episode one)
Unit
worth/Quantitative
formulation of
proportionality
A change of one in the input
corresponds to a fixed change in
output.
“So that means that for every one this one
changes [notes the input column], this one
[notes the corresponding output column]
changes by three.” (Focal episode two)
Figure 3. A summary of “knowledge resources” identified in the analysis of Liam’s
justifications for choices of next trial values.
Discussion and Findings
The main goal of this paper has been to illustrate how the emergence of a novel strategy in
episodes of problem solving can be productively framed in terms of underlying conceptual
reorganization. As we have seen, the landscape of the knowledge resources that students draw
upon in employing informal problem solving methods is surprisingly rich. Through a

7. Vol. 5
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31st
annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
199
preliminary analysis with the data from the case of Liam, we have seen that something as
apparently simple as students solving word problems using informal strategies like guessing and
checking actually can yield a striking complexity under analysis. The main contribution of this
paper is an analytic framework that re-positions observed strategy changes for solving problems
in terms of underlying conceptual reorganization. In the case elaborated in this paper, explicit
candidate knowledge resources have been named that should allow one to track the dynamics of
change between the two contrasting episodes discussed in this paper.
One of the challenges of tracking strategy change in terms of underlying conceptual
reorganization is that the conceptual reorganizations are likely to be of a small scale and highly
situated to the task at hand. In the data excerpts, we saw evidence that Liam had “invented linear
interpolation” in a tabular context. Without being explicitly taught about functions, Liam
implicitly recognized the “guess and check” chart he was generating as he solved problems as a
tabular representation of a function. In inventing linear interpolation in this context, he
discovered the tabular version of what might be stated in graphical terms as “two points
determine a line” and the fact that once you have a point and the slope you can get to any other
point on a line.
Since Liam was not familiar with “symbolic” or “graphical” representations of functions at
the time of the sessions, one would not expect that he would spontaneously recognize and apply
his linear interpolation approach in these other representational contexts. Hence, his
understanding of “linear interpolation” is only a projection into the tabular representational
context of a mature understanding of “linear interpolation.” Accordingly, the sub-conceptual
grain-size posited by the “knowledge in pieces” framework is particularly well adapted to the
goals of the analytic work in this line of research. A fine-grained and situated characterization of
knowledge will be required to make sense of the emergence of Liam’s strategy in the tabular
representational context.
Future Research
Future analytic work grounded in this case study and other replication case studies will be
needed to continue to identify other potentially relevant knowledge resources used by students.
Preliminary analyses of a complementary classroom data corpus and data from the interviews
with the five other pre-algebra students give evidence that the knowledge resources we have
discussed in this paper are sometimes but not always activated in the solution strategies of other
students. This certainly does not mean that students don’t “have” such resources. It may only
indicate that they do not see them as relevant to the task at hand. One would hypothesize that
classroom practices shape in fundamental ways what knowledge a student sees as relevant to
activate as they engage with solving problems.
The analytic framework sketched in this paper naturally extends into at least two other
additional strands. The work in the first two strands of analysis presented in this paper focused
on (1) documenting that there was a change in the organization of Liam’s knowledge and (2)
generating a vocabulary with which to describe that change. As suggested throughout this paper,
the natural next analysis would focus on giving a genetic account of knowledge growth and
change using the specific resources identified in this paper. Certainly, this would involve looking
at the episodes in between focal episodes. Further, it is natural, both from the perspective of
giving accounts of learning processes and from the perspective of designing instruction informed
by such accounts, to ask what factors influence the process of conceptual change and by what
mechanisms. Accordingly, a fourth analytic strand would focus on going beyond a description of

8. Vol. 5
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31st
annual meeting of the
North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA:
Georgia State University.
200
the dynamics of change over the sessions in terms of the resources to proposing likely
mechanisms that drove the process of conceptual reorganization forward. There are several
potential candidates for significant mediators of the constructive process. Some examples of
potential mediators include the role of the representational form in organizing the data obtained
from individual trials, social interactions and questioning in the learning setting (both by the
researcher and the student), the role of the activity of solving a problem in driving the inquiry,
and the role of a students’ prior knowledge and understanding. In addition, extending the
analysis to other kinds of resources, such as epistemological resources and beliefs (Hammer,
2000), could lead to future insights about how and why strategies emerge in episodes of problem
solving.
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