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Epistemological and Methodological Foundations of Creating A
Learning Trajectory of Children’s Mathematics
Nicole L. Fonger1, Amy Ellis2 and Muhammed F. Dogan3
1Syracuse University, USA; nfonger@syr.edu
2University of Georgia, USA; amyellis@uga.edu
3Adiyaman University, Turkey; mfatihdogan@adiyaman.edu.tr
This paper argues for creating learning trajectories of children’s mathematics by integrating
evidence of shifts in the mathematics of students, theory of goal-directed instructional design, and
evidence of instructional supports. We networked two theories in support of this stance: a radical
constructivist theory of learning, and Duality, Necessity, Repeated Reasoning (DNR)-based
instruction. We exemplify how our networking of theories guided methodological choices by drawing
on a program of research aimed at understanding and supporting students’ ways of understanding
quadratic growth as a representation of a constantly changing rate of change. We close by discussing
challenges for creating and sharing learning trajectories.
Keywords: Learning trajectory, constructivism, cognition, learning theories, epistemology.
Learning trajectories in mathematics education
Attention to learning trajectories and progressions remains a prominent strand of research in
mathematics education (e.g., Clements & Sarama, 2004). The influence of this domain of research is
evident in the development of mathematics standards (e.g., National Governor’s Association Center
for Best Practices, 2010; UK Department of Education, 2009), funding priorities, topics conferences
(e.g., the Third International Realistic Mathematics Education Conference), and special reports (e.g.,
Daro, Mosher, & Corcoran, 2011). Given the relatively wide-spread promulgation of trajectories and
progressions in our discourse in research and practice, there is a need to articulate theoretical and
methodological foundations of learning and teaching (Simon, 2013). This paper argues for the
importance of creating learning trajectories as research-based models of teaching and learning that
take seriously a commitment to understanding and supporting students’ mathematics.
The notion of a learning trajectory has different meanings among mathematics education researchers.
Simon’s (1995) original discussion offered a description of a hypothetical learning trajectory (HLT)
consisting of “the learning goal, the learning activities, and the thinking and learning in which
students might engage” (p. 133). An HLT constitutes a starting point for task design, and is then
modified into a learning trajectory based on empirical data, often in the form of a teaching experiment.
Clements and Sarama (2004) described a learning trajectory as an elaboration of children’s thinking
and learning in a specific mathematical domain, connected to a conjectured route through a set of
tasks. These definitions emphasize the construct as a tool for hypothesizing what students might
understand about a particular mathematical topic and how that understanding may change over time
in interaction with carefully designed tasks and teaching actions.
Building on this body of work, we argue that the main purpose of a learning trajectory is to effectively
convey the relationships between teaching, task-design, and shifts in student conceptions. We
advocate for a stance that articulates an integrated system of ways of bringing about conceptual
change in the mathematics of students in relation to theory-driven instructional support. Thus, in order
to create learning trajectories, we need mutually informing theories of learning and teaching. This
paper elaborates two such theories, the radical constructivist theory of learning and Duality,
Necessity, and Repeated Reasoning-based instruction, and identifies how these two related theories
influence our methodological approach to building learning trajectories. We provide an example
situated in a rate of change approach to quadratic function.
Theoretical and epistemological framing
A theory of learning mathematics
In discussing the theory that guides our approach to crafting learning trajectories, we address three
major issues: (a) distinguishing students’ mathematics from the mathematics of students, (b)
leveraging the epistemic student, and (c) leveraging a model of learning. Theory of instructional
design is treated in the next section.
Learning trajectories articulate students’ evolving conceptions within a particular instructional
context. In order to do this, we distinguish from our own mathematics as researchers and teachers,
our students’ mathematics, and the models we create of our students’ mathematics. The need to
distinguish our own mathematics from students’ mathematics is borne out of our epistemological
stance. We consider mathematical knowledge to develop as part of a process in which children
gradually construct and then experience a reality as external to themselves (von Glasersfeld, 1995;
Piaget, 2001). From this perspective, knowledge is considered viable if it stands up to experience,
enables one to make predictions, and allows for the enactment of desired objectives.
The term students’ mathematics refers to the models students construct to organize, comprehend, and
control their experiences – i.e., students’ knowledge. The mathematics of students is the set of models
we construct of our students’ knowledge (Steffe & Olive, 2010). Often there is little distinction
between these two notions in curricula or in standards documents. We believe, however, that the
mathematical knowledge we attribute to students in the creation of a learning trajectory must be
viewed as different from our own knowledge. Our goal is to determine how to engender and explain
students’ productive thinking. By distinguishing our mathematics from the students’ mathematics,
we recognize that students bring significant knowledge to bear when engaging in school mathematics,
and we position students as logical, coherent thinkers and doers of mathematics. The job of
establishing a learning trajectory then becomes one of explaining students’ thinking in a way that
portrays it as coherent and internally consistent (Steffe, 2004).
Individual students differ in their personal backgrounds, knowledge, and dispositions. A learning
trajectory should depict not one particular student or group of students, but rather the epistemic
student. An epistemic student is an organization of schemes that researchers build to explain students’
characteristic mathematical activity and how that activity changes in the context of teaching (Steffe
& Norton, 2014). Researchers construct epistemic students through teaching interactions with
specific students, but epistemic students are not specific to those particular interactions. Instead we
conceive them to be useful models of students’ schemes that one can leverage to describe, explain,
and predict the mathematical actions of similar students who may be operating at the same level.
Formulating an explanation of changes in students’ concepts and operations is not merely an
empirical matter. We also bring to bear a set of conceptual tools in order to interpret students’ activity
and problem solving. These tools have their origins in the radical constructivist (RC) model of
knowing (von Glasersfeld, 1995; Piaget, 2001). For the purposes of learning trajectory construction,
we rely particularly on the constructs of mental operation, scheme, assimilation and accommodation,
and abstraction. A mental operation is an internalized, reversible mental action that is an element of
a larger structure, such as a scheme, constituted by the coordination of operations. A scheme is an
organization of actions or operations which enables anticipation of results without having to engage
in mental activity. As an example, Piaget (2001) described the mental operation of combining objects
(such as addition). Several successive additions are the equivalent of a single addition (so one can
compose additions), and they can be inverted into the operation of taking away, or subtraction.
Treating new material as something already known is an act of assimilation. When assimilating, one
encounters an experience and incorporates it into a scheme. When the enactment of a scheme results
in an unexpected outcome, a learner may experience perturbation or disequilibrium. One response
can be a change in the learner’s recognition, in effect spurring a reorganization of one’s scheme. This
reorganization is accommodation, which many consider to be the source of conceptual change.
DNR-based instructional design
The theory of DNR-based instruction (Duality, Necessity, and Repeated Reasoning; Harel, 2008a;
2008b) informs our instructional design principles. Drawing on the RC theory of knowing, Harel
(2008b) noted that “any observations humans claim to have made is due to what their mental structure
attributes to their environment” (p. 894). He emphasized that researchers’ observations are merely
models of students’ conceptions; using our language, these are models of the mathematics of students
(Harel, 2013). Drawing on the mechanisms of assimilation and accommodation, Harel (2013)
characterized knowing as a developmental process that proceeds through a back-and-forth between
the two in order to reach equilibrium.
The duality principle addresses two forms of knowledge, Ways of Understanding (WoU) and Ways
of Thinking (WoT). WoU can be thought of as subject matter, consisting of students’ definitions,
theorems, proofs, problems, and their solutions (Harel, 2008a). WoT are students’ conceptual tools,
such as deductive reasoning, heuristics, and beliefs about mathematics (Harel, 2013). The Duality
Principle states that students develop WoT through the production of WoU, and, conversely, the WoU
they produce are afforded and constrained by their WoT (Harel, 2008a). We contend that the
mathematical content of a learning trajectory must be formulated in terms of both WoU and WoT.
The necessity principle states that in order for students to learn the mathematics we intend to teach
them, they must have an intellectual need for it (Harel, 2008b, 2013). We can engender intellectual
need through problematic situations that necessitate the creation of new knowledge in order to be
resolved. Finally, the repeated reasoning principle addresses the need for teachers to ensure that their
students internalize, retain, and organize knowledge (Harel, 2008a). Repeated reasoning should not
be confused with drill and practice of routine problems. Rather, it is an instructional principle that
advocates providing students with sequences of problems that require thinking through puzzling
situations and solutions; the problems must respond to students’ ever-changing intellectual need.
Methodological approach and rationale
Our methodological approach to establishing learning trajectories is a direct consequence of our
theoretical and epistemological framing. In the following sections we elaborate how our networked
theories (RC, DNR-based instruction) informed our methodologies. We describe three aspects: (a)
leveraging theory to create an HLT, (b) ongoing refinement of an HLT into an LT vis a vis enacting
a teaching experiment, and (c) finalization of an LT through retrospective analysis in order to
characterize students’ changing concepts and operations and links to tasks and instructional supports.
Creation of an HLT
We enact a form of design-based research in which our goal is to simultaneously engender and study
innovative forms of learning (Cobb & Gravemeijer, 2008). The planning phase involves creating an
HLT (Simon, 1995) informed by the networking of the RC theory of learning and the theory of DNR-
based instruction. Our HLT was a tentative progression of student concepts and associated tasks that
we hypothesized would necessitate a WoU that quadratic functions represent a constantly-changing
rate of change between two covarying quantities. Simultaneously, our aim was to support a WoT that
functions can be representations of covariation and can be explored and understood through a
covariational lens (Thompson & Carlson, 2017). Consequently, we devised a dynamic representation
of proportionally-growing rectangles in which students could investigate situations that, to us,
entailed the three continuously covarying quantities, height, length, and area (Figure 1).
The relationship between height, h, and area, A, can be expressed as A = ah2 where a is the ratio of
length to height. We wanted the students to develop the following specific WoUs: (a) the rate of
change of a rectangle’s area grows at a constantly-changing rate for each same-unit increase in height
(or length); (b) the rate of the rate of change of the rectangle’s area is constant for same-unit height
(or length) increases; (c) given a height h, the rectangle’s area could be determined by ah2; and (d)
the rate of the rate of change of area is dependent on the change in height. In order to engender these
WoUs, we devised tasks in which students had to predict the nature of growth, determine areas for
specific height values and vice versa, and decide whether given tables of values represented rectangles
that grew in proportion to one another or not. See Ellis (2011) for an elaboration of the mathematics.
Figure 1: A growing rectangle and associated table of height-area values.
Teaching experiment: Ongoing refinement of an HLT into a tentative LT
Drawing from the theories we networked, we engaged in an in-depth, 15-day teaching experiment
following the method of Steffe and Thompson (2000). We taught 15 lessons to a group of 6 middle-
grades students (ages 13−14) who were enrolled in pre-algebra (3 students), algebra (2 students), and
geometry (1 student). The second author was the teacher-researcher (TR). One purpose of a teaching
experiment is to gain direct experience with students’ mathematical reasoning, which affords the
creation and testing of hypotheses about the mathematics of students in real time. This means that our
mathematical tasks were not wholly predetermined, but instead were created and revised on a daily
basis in response to hypothesized models of students’ mathematics. Because our problem context
relied on area models, it was important to first identify the students’ existing schemes and operations
for area. After conducting pre-interviews and developing an initial model of the mathematics of
students, we created new tasks to necessitate more robust constructions of area as not dependent on
whole-unit iterations. During and between each session, we engaged in an iterative cycle of (a)
teaching actions, (b) assessment and model building of students’ thinking, and (c) task revision and
creation on an ongoing basis. In this manner, during each session, we continually revised our HLT
into a tentative, empirically based LT.
Retrospective analyses: Finalizing an LT
In addition to our ongoing analysis, we relied on retrospective analysis to inform the development of
a learning trajectory as a model of the mathematics of students (Steffe & Thompson, 2000). One
purpose of retrospective analysis is to build a model of the epistemic student and to characterize
students’ changing WoU and WoT throughout the course of the teaching experiment. A secondary
purpose, for us, was to contextualize and explain changes in students’ schemes and operations with
respect to the tasks and teaching actions they encountered. We aimed to elaborate features of tasks,
teacher moves, questioning, and socio-mathematical norms that supported the students’ scheme
accommodation. Our inclusion of instructional supports into a learning trajectory relied both on our
analyses of empirical data and on our understandings of the local instructional theories grounding our
design and enactment of the teaching experiment. Figure 2 identifies the goals, tools and constructs
we leveraged in order to create a learning trajectory. Notice how primary and secondary analytic aims
are explicitly linked to a coordination of RC and DNR-based instructional design theories.
Figure 2: Analytic aims, goals, tools, and constructs for creating a learning trajectory.
An excerpt of a learning trajectory for quadratic function
In our previous work, we identified WoT and WoU in students’ learning of quadratic function from
a rates of change perspective (Fonger, Dogan, & Ellis, 2017). Below, we provide an example of a
link between goal-directed instructional supports and a shift in student thinking (i.e., an excerpt of a
LT). We focus on one shift from a student’s WoU that (a) the rate of the rate of change of the
rectangle’s area is constant with ∆x implicit, to (b) the rate of the rate of change of area is dependent
on ∆x, which is explicit. In the excerpt, the students had already created well-ordered tables for the
growing rectangles (Figure 3). They had also attended to the constantly changing rate of change of
area as determined by finding area increases for same-unit increases in, but there was a lack of
coordination of change in area with change in height. Prompted to explain the growth in area of a
2 cm by 3 cm rectangle (Task A), one student said, “It goes 4.5 and then 7.5 and then 10.5 and
then…just keeps going.” In this manner, the students attended to the area’s growth, but did not
coordinate it with growth in height or length. In response, the TR prompted the students to draw
diagrams of the growing area, predicting that the act of drawing would necessitate a coordination of
height and area. The students’ drawing activity did necessitate a coordination, but many students kept
the change in height implicit, as evidenced by their language “every time.” For instance, Jim drew a
picture of a growing 2 cm by 3 cm rectangle and explained the rate of change of area as “how many
new squares it’s gaining every time it grows.”
In an attempt to further encourage an explicit coordination of the rate of area with a quantified change
in height, the TR asked the students to create a table for a 2 cm x 5 cm growing rectangle (Task B),
anticipating that the students would make tables with different height increments. This did occur: For
instance, Jim created a table in which ∆x was 1 cm, and Daeshim created a table with ∆x as 2 cm
(Figure 3). The different tables resulted in a conflict about whether the constantly-changing rate of
change of area should be 5 cm2 or 20 cm2 until one student, Jim, realized that the rate depended on
∆x: “I’m going up by1’s and they’re going up by evens.” After a class discussion in which the students
agreed that the rate could legitimately be either 5 cm2 or 20 cm2, depending on ∆x, Jim exclaimed,
“Your rate of growth can change no matter what!” In subsequent days the TR encouraged the students
to think about other proportionally-growing rectangles, and, ultimately, to draw diagrams relating
changes in area to the rectangles’ dimensions. These diagrams further emphasized explicit attention
to all three quantities, height, length, and area, and enabled the students to explicitly link the change
in area to the change in height. For instance, Daeshim determined that the rate of the rate of change
of area would be twice the area of the original rectangle, and Jim determined that it would be
equivalent to twice the length for any 1 cm by L cm rectangle.
Figure 3. Jim and Deashim’s well-ordered tables for a 2x5 growing rectangle.
We propose that a learning trajectory should include not only an articulation of particular WoUs and
the shifts between them, but also a hypothesized connection between these shifts and specific
instructional supports. Although space constraints limit an in-depth discussion of all of the
instructional supports in play, we see that the instructional move to prompt diagram drawing
necessitated a functional accommodation: students began to attend to and then coordinate increases
in height and length, not just area. In addition, the open structure of the task to determine rates of
change of area for a 2 cm by 5 cm rectangle further encouraged explicit attention to ∆x.
Discussion
In the domain of research on learning trajectories, attention to a theory of instructional design is
lacking. In this research we networked RC and DNR-based theories to inform task development,
pedagogical actions, and retrospective analyses. This approach and the resulting product (i.e., an
empirical LT for quadratic growth) are novel and not elaborated in the literature thus far. This paper
contributes to an understanding of theory undergirding an approach to learning trajectories research,
and how a networking of theoretical assumptions guide methodological choices in establishing
learning trajectories. We argue for learning trajectories research to be guided by theoretical lenses on
the mathematics of students as well as on a theory of instructional design. One challenge we see in
learning trajectories research is leveraging this construct as a way to not only frame a study (e.g., in
creating hypothetical learning trajectories), but to also retrospectively share learning trajectories in
ways that are consistent with the theories undergirding their creation. This research illustrates one
approach for addressing the challenge of creating learning trajectories as empirically based models
of an interweaving of shifts in students’ mathematical understandings and goal-directed, theoretically
grounded instructional practices.
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