On the idea of Learning Trajectories: Promises and Pitfalls

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The Mathematics Enthusiast The Mathematics Enthusiast
Volume 8
Number 3
Number 3
Article 6
7-2011
On the idea of Learning Trajectories: Promises and Pitfalls On the idea of Learning Trajectories: Promises and Pitfalls
Susan B. Empson
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On the Idea of Learning Trajectories: Promises and Pitfalls1
Susan B. Empson2
The University of Texas at Austin
Learning mathematics is a complex and multidimensional if not an inherently
indeterminate process. A necessary goal of research on learning is to simplify
this complexity without sacrificing the ability of research to inform teaching. This
goal has been addressed in part by researchers focusing on how to represent
research on learning for teachers and on how to support teachers to use and
generate models of students’ learning (e.g., Franke, Carpenter, Levi, & Fennema,
et al., 2001; Hammer & Schifter, 2001; Simon & Tzur, 2004; Steffe, 2004). Recently,
the idea of learning trajectories has gained attention as a way to focus research on
learning in service of instruction and assessment. It is influencing curriculum
standards, assessment design, and funding priorities. In this paper – which grew
out of my response to Michael Battista’s keynote address on learning trajectories
at the last annual meeting of the North American chapter of Psychology in
Mathematics Education (Battista, 2010) – I examine the idea of learning
trajectories and speculate on its usefulness in mathematics education.
1 Thanks to Jennifer Knudsen of SRI International for invaluable feedback.
2 empson@mail.utexas.edu


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The National Research Council (2007) described learning progressions as
“successively more sophisticated ways of thinking about a topic that can follow
one another as children learn about and investigate a topic” (p. 214). The recently
released Common Core Standards in Mathematics (CCSM) (2010), noted that the
“development of these Standards began with research-based learning
progressions detailing what is known today about how students’ mathematical
knowledge, skill, and understanding develop over time” (p. 4). The idea of
learning trajectories has a great deal of intuitive appeal and may offer a way to
bring coherence to how we think about learning and the curriculum. As research
on learning trajectories proliferates and is brought to bear on some of the most
vexing problems in teaching and learning mathematics, however, it is worth
considering what it foregrounds and what it may obscure.
In this paper, I briefly describe the origins of learning trajectories in
mathematics education and then consider three points for us to keep in mind as
we study learning and apply our findings to serve the purposes of
understanding and addressing the problems of practice.
1) The idea that learning progresses is not especially new. What do we know
about learning mathematics and how does it fit with the idea of a
trajectory?
2) Learning trajectories focus on specific domains of conceptual development
and may be limited in characterizing other valued aspects of the
mathematics curriculum.

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3) Learning in school is function of teaching. Too tight a focus on learning
trajectories may lead us to oversimplify or ignore critical drivers of
learning associated with teaching.
My goal in making these points is not to state the obvious but to foreground the
question of what the idea of learning trajectories affords us education researchers
and practitioners, and what it might obscure.
Origins of Learning Trajectories
The term learning trajectory appears to have been first used in mathematics
education in Marty Simon’s oft-cited 1995 paper, “Reconstructing Mathematics
Pedagogy from a Constructivist Perspective.” As I reread this paper, the most
important things I noticed – besides the fact that that the actual words “learning
trajectory” did not appear until 21 pages into the article – were that a) a learning
trajectory did not exist for Simon in the absence of an agent and a purpose and b)
it was introduced in the context of a theory of teaching. According to Simon, a
hypothetical learning trajectory is a teaching construct – something a teacher
conjectures as a way to make sense of where students are and where the teacher
might take them. It is hypothetical because an “actual learning trajectory is not
knowable in advance” (p. 135). Teachers are agents who hypothesize learning
trajectories for the purposes of planning tasks that connect students’ current
thinking activity with possible future thinking activity. A teacher might ask,
“What does this student understand? What could this student learn next and

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how could they learn it?” and create a hypothetical learning trajectory as a way
to prospectively grapple with these questions.
The idea of learning progressions appears to have emerged first in the context
of science education and is now virtually synonymous with learning trajectory.
In a special issue of the Canadian Journal of Science, Mathematics, and Technology
Education devoted to the topic of “long-term studies” of learning in science
education, Shapiro (2004) traced the notion of learning progression in part to
Rosalind Driver in her 1989 article, "Students' Conceptions and the Learning of
Science." In it, Driver drew attention to the increasing number of studies of the
development of children’s thinking in specific science domains that documented
patterns in what she called conceptual progressions and sequences of conceptual
progressions, which she termed conceptual trajectories (Shapiro, 2004, p. 3). In
contrast to Simon, the focus in that special issue of CJSMT was on describing
children’s learning as it had actually occurred under a given set of conditions,
rather than on a thought experiment about how it could occur. Neither of these
senses of learning trajectory – as a teacher-conjectured possible progression or a
researcher-documented progression of actual learners– predominates in current
conceptions of the notion.
Since 2004, there has been a groundswell of research that explicitly identifies
itself as concerned with learning trajectories or progressions, as reflected in
conferences and special journal issues (Clements & Sarama, 2004; Duncan &
Hmelo-Silver, 2009), reports (Catley, Lehrer, & Reiser, 2005; Cocoran, Mosher, &

TME, vol8, no.3, p .575
Rogat, 2009; Daro, Mosher, & Cocoran, 2011), and books (Clements & Sarama,
2009). A report by the Center for Continuous Improvement in Instruction (Daro,
et al., 2011) treats learning trajectories as interchangeable with learning
progressions, reflecting the general trend.
Because the metaphor of trajectory implies a sequenced path, researchers who
focus explicitly on learning trajectories have taken pains to draw attention to
their multidimensional character. For example, Clements and Sarama (2004)
defined learning trajectories as complex constructions that include “the
simultaneous consideration of mathematics goals, models of children’s thinking,
teachers’ and researchers’ models of children’s thinking, sequences of
instructional tasks, and the interaction of these at a detailed level of analysis of
processes” (p. 87). Confrey and colleagues (2009) defined them as “researcher-
conjectured, empirically-supported description[s] of the ordered network of
experiences a student encounters through instruction … in order to move from
informal ideas … towards increasingly complex concepts over time” (p. 2).
Three Points to Keep in Mind
Learning Trajectories are Not Really New – So What does the Metaphor Buy Us?
The idea that students’ learning progresses in some way as a result of
instruction is at the very heart of the enterprise of mathematics education.
Researchers have been studying students’ mathematics thinking and what it
could mean for that thinking to progress in identifiable ways since long before
the term learning trajectories was introduced. Chains of inquiry focused on

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children’s mathematics learning – we could call these research trajectories – have
stretched over decades. For example, Glenadine Gibbs’s (1956) study of students’
thinking about subtraction word problems helped to pave the way for later
researchers such as Carpenter and Moser (1984) to create frameworks portraying
the development of children’s thinking about addition and subtraction, and for
Carpenter, Fennema, and Peterson to study how teachers used this information
about children’s thinking to teach for understanding (Carpenter, et al., 1989;
Carpenter, et al., 1999). Les Steffe and John Olive’s recent (2010) book on
Children’s Fractional Knowledge detailing the evolution of children’s conceptual
schemes for operating on fractions synthesized two decades’ worth of prior
research, as did Karen Fuson’s findings on the development children’s multidigit
operations (1992). None this work mentioned learning trajectories as such, but
each focused on elucidating the development of children’s understanding and
identifying major conceptual advances.
Why then talk about learning trajectories now? The metaphor emphasizes the
orderly development of children’s thinking and draws our attention to learning
targets and possible milestones along the way.
To what extent is this kind of assumption about learning warranted? That is,
in what sense does children’s mathematics learning follow predictable
trajectories? Some domains appear to readily lend themselves to analysis in
terms of a pathway, such as the development of young children’s counting skills
(Gelman & Gallistel, 1986). The progression of children’s strategies for addition

TME, vol8, no.3, p .577
and subtraction story problems from direct modeling, to counting, to the use of
derived and recalled facts also has been well established (Carpenter et al., 1999;
Carpenter, 1985; Fuson, 1992). Yet even given such a robust progression in a
basic content domain, how and when – and sometimes whether – children come
to understand and use these strategies depends on a variety of factors differing
from classroom to classroom and from child to child. Trying to represent
research on learning in terms of trajectories quickly gets complicated, even for as
fundamental a concept as rational number (e.g., Figure 2 “Learning Trajectories
Map for Rational Number Reasoning,” in Confrey, Maloney, Nguyen, et al.,
2009) or measurement (Figure 1).
Figure 1. Battista’s (2010) representation of one student’s actual learning path in
measurement
Other research suggests that the development of much of children’s thinking
is more piece-meal and context-dependent than representations of learning
trajectories might lead us believe (DiSessa, 2000; Greeno & MMAP, 1998). For

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example, in a cross-sectional, cross-cultural study, Liu and Tang (2004) found
differences in progressions of students’ conceptions of energy in Canada and
China over several years of schooling, which they attributed to differences in
curriculum and instruction in each country. The topic of rational numbers in
mathematics has an ample research base that illustrates, in some cases
meticulously, how children’s thinking about fractions could progress (Behr,
Harel, Post, & Lesh, 1992; Davydov & Tsvetkovich, 1991; Empson & Levi, 2011;
Hackenberg, 2010; Steffe & Olive, 2010; Streefland, 1991; Tzur, 1999). Taken
collectively this research does not appear to converge on a single trajectory of
learning.
Why might this be? In practice, learning cannot be separated from tasks and
the instructional context; the “selection of learning tasks and the hypotheses
about the process of student learning are interdependent” (Simon & Tzur, 2004,
p. 93). What children learn is sensitive to the context in which they learn it – a
context that is constituted by many factors, including most immediately the
types of instructional tasks and how teachers organize students’ engagement
with these tasks.
For example, in classrooms where part-whole tasks (Fig. 2a) dominate
instruction on fractions, children learn to think about fractions in terms of
counting parts rather than as magnitudes (Thompson & Saldanha, 2003).
Students are likely to think about 5/8 as “5 out of 8” and of 8/5 as an impossible
fraction. In classrooms where teachers have students solve and discuss equal

TME, vol8, no.3, p .579
sharing tasks (Fig. 2b), children learn to think about fractions in terms of
relationships between quantities and later in terms of a multiplicative
relationship between numerator and denominator (Empson, Junk, Dominguez, &
Turner, 2005; Empson & Levi, 2011). They are more likely to think of 5/8, for
example, as 5 groups each of size 1/8, instead of “5 out of 8.” In classrooms
where teachers engage students in reasoning about multiplicative comparisons
of measures (Fig. 2c), students learn to think about fractions as a ratio of
measures (Brousseau, Brousseau, & Warfield, 2004; Davydov &Tsvetkovich,
1991; Steffe & Olive, 2010). Children learn to interpret 5/8 as a multiplicative
comparison between 5 and 8. Both of these latter types of tasks – equal sharing
and measuring – coupled with norms for engaging in tasks that put a premium
on intellectual effort and agency (Hiebert & Grouws, 2007) – appear to constitute
productive approaches to learning fractions.
How much pizza is left on the plate?
(a)
8 children want to share 10 candy bars so that each one gets
the same amount. How many candy bars can each child
have?

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(b)
How long is B compared to A? A compared to B?
(c)
Figure 2. Examples of types of tasks to teach fractions: a) part-whole, b) equal
sharing (with sample solution), and c) measurement
Within the context of documenting regularities and patterns in the
development of children’s thinking, however, it’s important to recognize
individual children’s ways of reasoning and the significant contributions this
reasoning could make to a group’s learning. To return to my research on equal
sharing, for example, we found that students frequently produced strategies for
solving problems that were, from the perspective of a trajectory, “out of
sequence” and presented rich learning opportunities for other students (e.g.,
Turner et al., in press). There was a progression in what students learned but
“deviations” were consistent and numerous, and, I am suggesting, fruitful – not
anomalies to be ignored but significant occurrences that teachers could use to
advance everyone’s learning.
Consider first a simple progression of strategies for equal sharing (Empson &

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Levi, 2011). To figure out how much one person got if 8 people were sharing 6
burritos equally, a child using a basic strategy might draw all 6 burritos, decide
to split each burrito into 8 pieces, and give each person 1 piece from each burrito
for a total of 6 pieces. A more sophisticated strategy would involve imagining
that each burrito could be split into 8 pieces and mentally combining those pieces
to conclude that one person’s share consisted of 6 groups of 1/8 burrito or 6/8
burrito. Ultimately, children come to the understanding that the problem can be
represented by 6÷8, which is the same as 6/8.
Within this simplified progression, there are several other ways to solve the
problem that do not fall into a sequence and do not appear as an inevitable
consequence of development. These other strategies were a function of specific
quantities in a problem as well as what tools children were using and children’s
prior knowledge. For example, a fifth grader solved the problem by reducing it
to an equivalent ratio involving 1 1/2 burritos and 2 children, which she easily
solved by finding half of 1 and half of 1/2 and combining the amounts (Fig. 3).
Another fifth grader used a similar strategy, but used cubes to represent each
quantity (8 total cubes for sharers, 6 total cubes for burritos), specifically
highlighting the ratio character of the strategy. These strategies were
appropriated by several children who saw them as more efficient and they
provided an opportunity for the teachers to address concepts of fraction and
ratio equivalence.

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Figure 3. A solution for 6÷8 involving the equivalent ratios 3 for 4 and 1 1/2 for 2
As students’ understanding develops and diversifies, they become more
likely to see and make connections between their ways of thinking and different
ways of thinking expressed by their fellow students. Making these connections
enriches learners’ understanding and cultivates their ability to recognize and
pursue new avenues of reasoning independently of the teacher’s direction and to
monitor their thinking. The balance in instruction between supporting students’
agentic initiative and aiming to instill specific conceptions can be difficult to
manage. Indeed, some researchers have cautioned that representing learning as
progressive sequences of content understanding could lead teachers to direct
students through the sequences at the expense of allowing students to “express,
test, and revise their own ways of thinking” (Lesh & Yoon, 2004, p. 206; Sikorski
& Hammer, 2010). At the same time, other research suggests that, at the right
level of abstraction, representations of the progressive development of students’
understanding can enhance teachers’ ability to respond to students’ thinking in
ways that open up or are generative of new possibilities (e.g., Franke et al. 2001).

TME, vol8, no.3, p .583
In either case, it’s important to recognize that research on learning in specific
mathematics domains has a long history that, while concerned with progress,
may not fit easily into the idea of a single trajectory.
Learning Trajectories Involve Specific Domains of Conceptual Development – So Their
Reach May be Limited
Researchers have made the study of mathematics learning more tractable
by focusing in particular on conceptual development in specific content domains,
represented by sets of well-defined, interrelated tasks. Steffe and Olive’s (2010)
research on the development of fraction concepts and Clements and Sarama’s
(2009) research on children’s understanding of measurement are examples of
such an approach. This work, like a great deal of the research in mathematics
education including my own, is informed by a Piagetian-like view of learning, if
not in its emphasis on levels, then certainly in its emphasis on a conceptual
trajectory, in which less sophisticated concepts give way to more sophisticated
concepts. Because this work is based on children’s thinking about specific types
of tasks, its power lies in its capacity to inform teachers’ use and interpretation of
these tasks to foster students’ conceptual development in a coherent unit of
study (e.g., Fennema et al, 1996; Simon & Tzur, 2004).
However powerful, these kinds of portrayals of learning necessarily
represent only one dimension or a small set of what we value as a field about
mathematics and wish for students to learn. Learning is a multidimensional
process, comprised of a variety of intertwined cognitive and social processes. In

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particular, since the publication of Everyone Counts (National Research Council,
1989) and the Curriculum and Evaluation Standards (National Council of Teachers
of Mathematics, 1988), mathematics educators have increasingly focused on
teaching students to engage in practices such as posing and solving problems
(Hiebert et al., 1996), constructing models (Lesh & Doerr, 2003), and making
convincing arguments (Lehrer & Schauble, 2007) – that is, to do mathematics.
Doing mathematics involves a complex and integrated set of content
understanding and disciplinary practices (Bass, 2011; Kilpatrick, Swafford, &
Findell, 2001) as well as the ability to monitor the interplay between these things
(Schoenfeld, 1992).
The ability to engage in mathematical practices such as the ones above is
as critical as content knowledge to a well-developed capacity to think
mathematically, but it is less amenable to analysis in terms of sequences of
development. For example, students engaged in mathematical modeling or
problem solving may draw on multiple content domains and work
collaboratively on tasks that have many possible resolutions such that the
solutions they produce appear to follow no predictable trajectory over time.
Examples of such tasks include creating simulations of disease spread (Stroup,
Ares, & Hurford, 2005), optimizing the occupancy of a hotel during tourist
season (Aliprantis & Carmona, 2003), and designing a template to generate a
quilt pattern (Lesh & Doerr, 2003). These kinds of tasks and thinking practices
pose considerable challenge for researchers seeking to codify and systematically

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represent learning in terms of a trajectory, because of the variety of
understanding and practices that students bring to bear in the their solutions.
Learning trajectories may be limited in what they can and cannot specify
in terms of learning mathematics over time; and in particular, they may not be
applicable to certain critical aspects of the mathematics curriculum. Catley,
Lehrer, and Reiser (2005) recognized this potential limitation when they argued
that “scientific concepts are never developed without participation in specialized
forms of practice” and “concepts are contingent on these practices” (p. 4) – such
as the ones listed in the Common Core Standards in Mathematics (2010). Among
others, these practices include making sense of problems and persevering in
solving them; using appropriate tools strategically; attending to precision; and
looking for and making use of structure (CCSM, 2010, pp. 6-8). Most, if not all,
current characterizations of learning trajectories do not address the practices that
engender the development of concepts – although it’s worth thinking about
alternative ways to characterize curriculum standards and learning trajectories
that draw teachers’ attention to specific aspects of students’ mathematical
practices as well as the content that might be the aim of that practice.
What is a reasonable unit of students’ mathematical activity for teachers to
notice? If a unit is too small or requires a great deal of inference (e.g., a mental
operation), then teachers in their moment-to-moment decision-making may not
be able to detect it and respond to it; likewise if a unit is too broad or stretches
over too long a period of time (e.g., “critical thinking”), teachers may not

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recognize it when they are seeing it. The most productive kinds of units of
mathematical activity would allow teachers to see and respond to clearly defined
instances of student’s thinking during instruction and to gather information
about students’ progress relative to instructional goals. For example, in research
in elementary mathematics, strategies and types of reasoning are productive units
because we know that teachers can learn to differentiate students’ strategies and
use what they learn about students’ thinking to successfully guide instruction
(e.g., Fennema, et al., 1996). Catley and colleagues (2005) proposed “learning
performances” as a way to represent the “cognitive processes and associated
practices linked to particular standards” (p. 5). Formative assessments that
include a variety of points of access and possible solutions and that require
students to engage in various mathematics practices could also yield rich
information about students’ understanding of and engagement in mathematics
(cf., Aliprantis & Carmona, 2003; Lesh & Doerr, 2003). The important thing is to
take into account the interplay of practices and content in students’ learning over
time.
Teaching is Integral to Learning and Learning Trajectories
Learning school mathematics depends on teaching. To support learning,
teachers need be able to “understand, plan, and react instructionally, on a
moment-to-moment basis, to students' developing reasoning” and coordinate
these interactions with learning goals (Battista, 2010). Similarly, Daro and
colleagues (2011) concluded that:

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Teachers are going to have to find ways to attend more closely and regularly
to each of their students during instruction to determine where they are in
their progress toward meeting the standards, and the kinds of problems they
might be having along the way. Then teachers must use that information to
decide what to do to help each student continue to progress, to provide
students with feedback, and help them overcome their particular problems to
get back on a path to success. (Daro et al., 2011, p. 15)
We know very little about how teachers do these things, in contrast to what
we know about children’s learning, whether it falls under the rubric of learning
trajectory research or not. As teachers interact with students and decide how to
proceed, there are many types of decisions to be made – how to gather
information about children’s thinking, how to respond to it appropriately in the
moment, how to design tasks that extend it, and even what to pay attention to.
With the right tools, teachers have access to the most up-to-date information
about each student, what they understand and are able to do, their disposition,
their history, and so on, and can make decisions based on their own informed
understanding of these things and their relationships. Good tools, such as
formative assessment frameworks in particular, enhance this knowledge and
support teachers to engage in the active, contingent process of creating
instructional trajectories informed by knowledge of actual children’s learning.
Further, learning mathematics in school takes work and depends
fundamentally on interpersonal relationships of trust and respect, which cannot
be designed into a tool or a list of learning goals. Teaching is a relational act and
the relationship between the teacher and the student is at the center of students’
learning in school (Gergen, 2009; Grossman & McDonald, 2008). These

Empson
relationships can have a profound effect on what students learn and how they
come to see themselves.
In the face of what can seem like a tidal wave of top-down mandates, I
suggest that we mathematics educators keep sight of the fact that teaching is
driven essentially by interpersonal relationships and happens from the bottom
up, beginning with the teacher and the student relating to each other and the
content. We need to be sure that teachers are equipped with knowledge of the
domain and its learning milestones without forgetting that both teachers and
students are active agents in learning.
Closing Thoughts
“Clearly … the trajectories followed by those who learn will be extremely
diverse and may not be predictable” (Lave & Wenger, 1991)
In choosing to focus on learning trajectories, we embrace a metaphor that, for
all its appeal, implies that learning unfolds following a predictable, sequenced
path. Everyone knows it is not that simple; researchers and educators alike
acknowledge the complexity of learning. As Simon (1995) emphasized, learning
trajectories are essentially provisional. We can think of them as the provisional
creation of teachers who are deliberating about how to support students’
learning and we can think of them as the provisional creation of researchers
attempting to understand students’ learning and to represent it in a way that is
useful for teachers, curriculum designers, and test makers.

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I firmly believe that a critical part of our mission as researchers is to produce
something that is of use to the field and serves as a resource for teachers and
curriculum designers to optimize student learning. No doubt this includes
creating, testing, and refining empirically based representations of students’
learning for teachers to use in professional decision-making and, further,
investigating ways to support teachers’ decision-making without stripping
teachers of the agency needed to hypothesize learning trajectories for individual
children as they teach. This focus would add a layer of complexity to our
research on learning and invite us to think seriously about how to support
teachers to incorporate knowledge of children’s learning into their purposeful
decision-making about instruction. Further, I suggest we consider, in the end,
“Whose responsibility is it to construct learning trajectories?” (Steffe, 2004, p.
130). If we researchers can figure out how to supply teachers with knowledge
frameworks and formative assessment tools to facilitate their work, teachers will
be able to exercise this responsibility with increasing skill, professionalism, and
effectiveness.
Because of the growing popularity of learning trajectories in education circles,
it is worth thinking hard about the role of learning trajectory representations in
teaching, and in particular, whether a learning trajectory can exist meaningfully
apart from the relationship between a teacher and a student at a specific time and
place. Simon’s (1995) perspective on teaching and learning suggests not. As the
field moves forward with research on learning trajectories and strive for

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coherence in learning across the grades, I would like to remain mindful of both
the affordances and constraints this particular type of representation offers for
teachers and students alike.

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