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Running Head: LEARNING TRAJECTORIES

Learning Trajectories, Learning Progressions, and Hypothetical Learning Trajectories

Kimberly Morrow-Leong

Analysis for Portfolio #3

August 2014

George Mason University

LEARNING TRAJECTORIES

Teachers’ in-the-moment decisions and resulting actions are key factors in the

advancement of the student learning. A classic article in a teacher practitioner journal

entitled “Isn’t that Interesting!” Buschman (2005) describes a teacher’s intentional decision

to step away from judging student responses and ask questions about their thinking

instead. The anecdotes from the classroom demonstrate the broad range of

misunderstandings, pre-understandings, and exceptionally diverse thoughts that students

share when their line of thinking is pursued. On the other hand, I reviewed a study guide

that outlined a professional development activity centered on this article (not in print).

The study guide author apparently misunderstood the author’s intent, as the professional

development guide focused on teachers’ strategies to help students correct their

misconceptions. Not all teachers have a habit of listening carefully to what students are

saying and using this information to assess their thinking.

Students’ growth and development can take surprising twists and turns, sometimes

following the curriculum set for them and at other times meandering around important

ideas without achieving the objective of the day, week, or even of the year. Nevertheless,

teachers still hold the key to guiding students toward a more robust understanding of

mathematics. With the introduction of the Common Core State Standards (2010) a close

examination of learning paths may be more feasible, as much of the population is following

the same curricular sequence. However, a sequence of content presented by grade is no

guarantee that each level of content is appropriate for the students in that grade to learn.

Much research is needed in order to conclude definitively that, for example, the division of

fractions in 6th grade is appropriately accessible for every student in the United States. In


the meantime, learning trajectories and progressions may become key tools for assisting

teachers as they learn more details about the means by which mathematics students learn.

Search of the Literature

In this paper I will review the history of research on curricular sequences such as

learning trajectories and learning progressions in mathematics education and conclude

with definitions that will show how each is valuable in its own contexts. Furthermore, I

will also describe learning progressions and learning trajectories in specific domains in

order to not only summarize some of the important known information, but also to

compare and contrast the LT and the LP when used in practice. Finally, I will present

reflections and recommendations for further areas of research and advice for practicing

teachers for using these tools.

To begin the search, I entered the following search terms in the Academic Search

Complete database: “learning trajectory(ies)” and “learning progression(s),” initially

specifying “mathematics” in the search. This yielded several hundred entries, which I

narrowed down by sorting chronologically. After scanning the first few dozen most recent

articles, I downloaded those that appeared relevant to the search topic. For the second

phase of my search, I looked up what I knew to be the seminal article (Simon, 1995) and

selected the link that led to all articles that cited that piece. When this yielded a large group

of articles, I narrowed the search to articles within the field of mathematics only. The

combined collection of articles was sorted into groups, as I was attempting to identify

research teams that were working on different trajectory projects. This phase of the search

yielded six or seven teams of researchers working on projects focused in different


mathematical domains. It also yielded a collection of published pieces that hinted at

individual research projects that utilize the frameworks designed by other teams. For

example, one article describes a study that contrasted the use of learning trajectories in the

lesson planning process by veteran and early career teachers (Amador & Lamberg, 2013).

Finally, after examining the evolution of the learning trajectory and the learning

progression within the field of mathematics, Batttista (2004) and others referred back to a

document published in reference to science education learning progressions (Corcoran,

Mosher, & Rognat; 2003).

Sequences provided for teaching within content area domains is not a uniquely modern

phenomenon: any teaching or training activity must include some form of a plan that leads

students from ignorance to a mature understanding of the domain content and skills. Lave

(2011) describes the process as apprentice tailors enter the community of practice

(Wenger, 1998) and begin to engage. The strict sequence of mastery skills and assessment

routines followed by the male tailors within the Vai and Gola community of Liberia are

notable. The sequence of skills that are learned on the periphery include sewing on a

button, but eventually move to full participation when the apprentice earns the

opportunity to tailor a suit. Students learning mathematics are also on an apprenticeship

path. The Common Core specifically identifies that path by stating that the goals of the

standards are “college and career readiness” (CCSSO, 2010). It is beyond the scope of this

paper to argue whether this is an appropriate set of standards and goals for every student

in the United States, or whether these standards truly do move students toward the stated

goals, however an understood set of goals, tasks, and a predictable sequence of expected

competencies guide the master’s efforts to mold the apprentice.


Big Picture on Student Learning

The unfolding of topics and content that students learn in a classroom is not an obvious

sequence, and opinions about this progression vary greatly. One of the key considerations

for describing the sequence students follow is the focus and intent of the sequence. One of

the outcomes of the research done by Confrey and colleagues as part of the development of

the equi-partitioning learning trajectory is a clear distinction between a sequence of

content based on mathematical ideas that originate from an idealized version of

mathematics as an end goal, and a view of mathematics as something that is created within

the individual mind (Confrey, 2012; Wilson, Mojica, Confrey, 2013). In other words, a

mathematics learning sequence can be derived as a top-down model, where the content

that students must learn is derived from a vision of what mastery of that content looks like.

Because this vision of mathematical learning is conceived by those who have already

achieved mastery, the content looks much like the mature version. On the other hand, a

developmental view of mathematics learning builds from the earliest conceptions of

number and space and assumes that learning will unfold in fairly predictable ways, given

that students have exposure to productive mathematical tasks. In this view, student

thinking and learning unfolds in “stages” or “levels” that have been shown empirically to be

predictable. Of course these approaches are overly generalized, but it is useful to adopt a

critical stance when examining any curriculum, learning progression, sequence, or learning

trajectory and decide how the writers conceptualize mathematics learning. Sztain, Confrey,

Wilson, and Edgington (2009) base their work with learning trajectories on a

developmental view of mathematics and present evidence of empirically based movement

through a series of conceptual ideas related to rational numbers.


Origins of Learning Trajectories

When Simon (1995) published this article he casually mentioned the hypothetical

learning trajectory. A careful reading shows that he may well have preferred that the

mathematics education community instead elect to study the model of the iterative

teaching decision-making process that he called the Mathematics Teaching Cycle. Beginning

with a reasonably educated guess about the interactions between a lesson’s goal, the

activities of the lesson, and the thinking and learning in which the student might engage,

the teacher lays out how a lesson might unfold: Simon called this part of the iterative

design cycle the hypothetical learning trajectory (HLT). The rest of the cycle is concerned

with the changes in lesson design both live in the class and also for future instruction. But

this is not what caught fire.

It is clear that Simon’s focus in 1995 was on the individual teacher’s work within a

single classroom, even within a short time frame. He specifically refers to “the teacher’s

prediction as to the path by which learning might proceed” (Simon, 1995, p.135). The HLT,

as an instructional planning tool has evolved considerably since this original statement.

The H has been dropped, and more definitive statements about the learning trajectory for

certain content domains have emerged. Others retain the H but have shifted the

hypothetical aspect to refer to a proposed global hypothesis about student learning within

a content domain, rather than to the context of a single teacher or classroom. Other entities

have adopted the term “learning progression,” distancing themselves from anything

remotely hypothetical. Few writers retain what I believe is Simon’s original meaning

within the context of this paper, including Simon himself (Simon and Tzur, 2004). In this


piece, the authors present the HLT as a “vehicle” for instructional planning. The

importance of tasks and a focus on students’ understanding remain, but in this piece the H

seems less hypothetical. For example, the lesson shared in Simon’s original piece begins as

an exploration of the “multiplicative relationships involved, not to teach about area”

(emphasis in the original) (Simon, 1995, p. 123), but eventually he admits that “Although

my primary focus was on multiplicative relationships, not on area, it seemed clear that an

understanding of area was necessary in order for students to think about constituting the

quantity (area) an evaluating that quantity.” In this responsive shift in focus, the

hypothesized learning trajectory for the group of students changes significantly, even in

some sense shifting domain, and the hypothesis about the students’ learning path changes.

But in the 2004, Simon and Tzur (2004) use the word “vehicle” to describe the elaborated

hypothetical learning trajectory conveys the idea that the HLT “carries” the lesson instead

of driving it. The agency appears less in the hands of the teacher and is more dependent on

the sequence of activities designed to elicit activity-effect in students. Despite this subtle

but noticeable shift in emphasis from the teacher as the author of the HLT to a research-

determined HLT, the authors do acknowledge that as knowledge of student learning

processes grows, learning trajectories in general can become more precise and predictive

than they have been in the past. However the HLT, as originally presented, has evolved

significantly in the intervening years.

Currently, Confrey and her colleagues (Confrey et al., n.d.) have outlined learning

trajectories for all of the mathematical domains presented in the Common Core standards,

building on current empirical research on student learning, particularly in the area of

rational number understandings. They use the term “learning trajectory” in a definitive


way. The LT is a “researcher-conjectured, empirically-supported description of the ordered

network of constructs a student encounters through instruction (i.e., activities, tasks, tools,

and forms of interaction), in order to move from informal ideas, through successive

refinements of representation, articulation, and reflection, towards increasingly complex

concepts over time” (Confrey, Nguyen, Mojica, & Meyers, 2009, p.347). Where the Common

Core omits intermediary standards they deem necessary, they have added “bridging

standards” to complete the connections (Maloney, 2013). In all cases, this LT is not an

individual teacher construction, but rather a compilation of extant research in the domain.

Like Confrey, Battista also excludes the “hypothetical” aspect of the learning trajectory

and illustrates in detail Simon’s distinction between the general case of a learning

trajectory and the actual case (Battista, 2011). In a series of case studies he describes

students’ acquisition of understanding of linear measurement progressing from the use of

non-numerical comparison to the abstraction of length, apart from iterated units. Using a

mountain analogy, he illustrates a student’s idealized path up the mountain as an efficient,

but not necessarily direct, path through the big ideas and concepts that characterize

progress toward a sophisticated understanding of linear measurement. This is the

prototypical trajectory. He also presents the more typical, or actual, learning trajectory

that meanders between levels and sublevels, often doubling back and retracing steps.

Altogether the actual learning trajectory could be far more disjoint than one might expect.

A fifth grader’s particularly complex progression through a collection of 34 measurement

tasks is illustrated in Figure 1. Beginning with an accurate but non-numerical comparison

of lengths on level N1, the student’s path through five other levels of thinking seems to

settle in M1, showing that he is most likely to incorrectly iterate lengths of units. This


actual trajectory is admittedly more complex than most children’s. It does, however,

illustrate that the nature of a learning trajectory is different when applied to a student’s

particular case, a particular class’ case, than when it refers to the general case.

Figure 1: RC's learning trajectory

Battista’s conception of learning trajectories broadened and extended Simon’s original

ideas. While Simon’s original description referred to the teacher’s individual process of

preparing for instruction by anticipating how students will respond to tasks, Battista’s, and

indeed Simon and Tzur’s more recent work as well (Simon, 2014), lays out a empirically

informed prediction of how students will respond to different tasks. Perhaps it is

inevitable that a continued focus on student thinking and a cataloging of their responses to

typical assessment items will yield “prototypical” results that can be sequenced and

mapped. The difference here is agency. Whereas the original conception of the HLT

included the teacher’s routine reflection, planning, and response to student thinking and

learning, which is a model more akin to a constructivist orientation (von Glasersfeld, 1995),

the elimination of the “hypothetical” aspect in practice lessens teacher agency.

This is not the case across the board. Gravemeijer (2003) still holds the teacher in the

instructional designer role, adhering to a design research brand of planning for teaching


that expects the teacher to imagine the path students will take during a lesson. Moreover,

Smith and Stein (2011) also include an expectation that teachers will include an

anticipation phase in the lesson planning process. They define anticipation as “actively

envisioning how students might mathematically approach the instructional task” (Stein &

Smith, 2011, p.8). However, they make no mention of the source of information that

teachers might draw upon for their anticipation work. Finally, Ellis, Weber, & Lockwood

(2014) acknowledge the challenges involved in planning instruction that is responsive to

students’ anticipated approaches to tasks. In their review of the literature related to

learning trajectories and learning progressions they conclude by stating that the most

honest reality is that attending to student thinking and building their own interpretations

of student learning trajectories are skills rightfully acquired over time. Teachers’

anticipation of students’ thinking and learning processes is clearly of utmost importance,

however teacher agency does not necessarily form a part of all theoretical frameworks

related to learning trajectories.

The Nature of Learning Trajectory Paths

Linear Paths

The Oxford dictionary defines a trajectory as “the path followed by a projectile flying or

an object moving under the action of given forces” (OED, 1914). If we adopt the trajectory

as a metaphor for learning, it’s important to consider the shape of the learning paths as

well as the forces that influence movement along the path. Figure 2 shows five possible

conceptualizations of the learning trajectory phenomenon. In the linear model, learning

follows a straight path, unmarred by detours or complexities. Gagné’s (1977) Task


Analysis model is typical of this learning path. The Task Analysis model begins at the end

of the learning process and identifies the goal of the task and then the most desirable

sequence of learning events. The task is then classified by the kind of activity that best

demonstrates student success, and the designer then maps out a detailed path to achieve

this goal. One of the necessary steps in the process is the establishment of prerequisite

knowledge, which are all of the skills that must be in place in order for the student to move

forward toward the target goal. Prerequisite knowledge is a concept well-known in the

tradition of lesson planning, as it is an important piece of establishing a starting point for

instruction. In reality, the Task Analysis model is more accurately called an instructional

design model rather than a model of learning because the design describes strategies that

can be used to guide student learning, but it does not describe actual student learning in

any fashion.

Linear Meandering MultipleLandscape/

NetworkBranching

Figure 2: Learning Trajectory Paths

It is important to review the work of Gagne in the discussion of learning trajectories

and learning progressions, because many of the constructs proposed by Gagne are contrary

to the nature of what a learning trajectory is. Gagne’s Task Analysis model is linear, and it is


certainly a “top-down” model of instruction in that the instructor begins with the end

target knowledge goal in mind and then scrutinizes the incremental steps that lead up to

that goal, developing a carefully constructed sequence of tasks and skills that lead directly

to the end goal. By contrast, the learning trajectory. as well as the hypothetical learning

trajectory, all begin with the learners’ most immature understandings and charts the

progression of learning toward a mature understanding. In a later section, we will explore

the subtle differences between the learning trajectory model and the learning progression.

Origins

Recently Simon (2014) outlined his most current iteration of the HLT theory supporting

a task design process for teaching skills and concepts that are not readily learned in the

problem solving context. He specifically separates this form of teaching from a problem

solving approach and has only employed these task sequences within the context of a

teaching experiment. During the teaching experiment, the interviewer elicits a student’s

responses to tasks for which the student already has some facility. The subtly sequenced

and crafted tasks allow students to move toward reflective abstraction, a state which

allows them to complete a task without the accompanying physical action. At that point the

student has constructed enough understanding to complete the task without intervening

steps. The task sequences employed in this line of research are very carefully constructed

and based on the painstakingly detailed sequence of fraction schemes outlined by Steffe

and colleagues (Steffe & Olive, 2010).

Steffe and his colleagues do not use the term “learning trajectory” when discussing their

teaching experiment results, most likely because their work is not designed for classroom


use, and the learning trajectory specifically includes the instructional task component.

Nevertheless, minus the real classroom tasks, this group has generated a highly detailed,

linear, and fine grained description of how a typical student will progress through their

understanding of rational numbers. One key component of the overall theory is the

reorganization principle. As students demonstrate incremental changes in the

instructional tasks with which they engage, they assimilate the new information (Piaget &

Inhelder, 1973) and reorganize their current undrestandings. In this minute manner, Steffe

and Olive hold that students will not be subjected to interference from the natural number

operations: the tasks are designed to subtly but surely extend the students’ understanding

of natural numbers to include the set of non-natural rational numbers (Steffe & Olive,

2010). The fraction schemes they outline are detailed and specific, including arcane

acronyms for fraction schemes that are not only anticipated and predicted, but which also

follow each other in a nearly universal order.

Despite the fact that the term learning trajectory is not used to describe the fraction

scheme, the sequence still retains some of those properties in that they present a projected

and expected sequence of concepts that will elicit disequilibrium in students. As these

states of disequilibrium are anticipated, instructional tasks are already prepared that are

ideally suited to assist students to move toward assimilation of the new ideas. For

example, Norton and Wilkins (Norton & Wilkins, 2010, 2012) conducted detailed teaching

experiments with seventh grade students in order to determine the most accurate model

for students’ acquisition of the fraction scheme. For example, a student who possesses the

partitive unit fraction scheme (PUFS) can partition a continuous whole into n number of

parts, such as a stick partitioned into 8 equal pieces. But the PUFS scheme also means that


the student can remove one piece, iterate it back against the unit whole and name the

corresponding unit fraction (1/8). PFS, the scheme hypothesized to follow the PUFS

includes the understandings required for the child to have acquired PUFS. The significant

difference is the students’ capacity to recognize fractions besides the unit fraction (1/8),

and reconstitute the unit whole. Using the 1/8 example, the student would recognize 3/8

as three iterations of 1/8 and then reconstitute the 8/8 or 1 whole. This change to more

multiplicative thinking also permits the students to make sense of the improper fraction.

9/8 is no longer perceived as “nine parts out of eight,” but rather as nine iterations of 1/8,

which itself is one out of eight partitions of the unit whole. While this scheme still retains

additive properties, it is an example of how Steffe envisioned an understanding of fractions

that is not in conflict with whole number operations, but rather part of a continuous

growth in understanding of number. And while not strictly a learning trajectory, the

schemes named and researched by Steffe and colleagues (Wilkins & Norton (2010),

Hackenberg, 2007; Olive & Steff, 2002) still retain the property of an assessable, and

imminently teachable set of touchpoint achievements and teaching tasks that are extremely

linear in nature. Simon, in his own teaching experiments is using this sequence of schemes

to formulate an HLT that can eventually be teachable (Simon, 2014). The combination of

expected student acquisition of mathematical knowledge combined with the instructional

tasks that are highly linear, put the combined work of Steffe and Simon in the category of

the linear trajectory.

The van Hiele levels of geometric thinking are certainly linear in nature (van Hiele,

2004). In order to see the linearity, it is useful to outline the nature of each of the levels


and explore how the language of understanding and the mathematical point of view do not

just describe the students operating at that level, but also define their actions at that level.

Students at level 0 focus on the overall vision of a geometric figure, which Clements

refers to as a “Gestalt-like” view (Clements, 2004). The students’ language at this stage may

sound like they understand. For example, the student may be able to count four sides to a

square and tell you that it has four sides, but not recognize this property as a defining

property of a square. A less knowledgeable teacher will be satisfied at the depth of the

student’s knowledge, but not see the student’s immature thinking. At Level 1 the student is

mindful of the properties belonging to the geometric figure. They respond to verbal

indications of properties that may not be present in the actual figural representation. A

circle is a circle because all radii are the same length, and they will accept its “circle-ness” if

given this property. This is distinctly different from level 1, where the student would

accept equal radii as an interesting, but non-essential piece of information. At Level 2 this

detail is salient to the student. Level 2 is the beginning of mastery over definitions of

geometric figures, and at Level 3 the student is fully able to engage in abstract deductive

reasoning.

The van Hiele levels of geometric thinking are imminently satisfying, as they appear

discrete, understandable, and reflect situations common to all mathematics teachers. A

teacher may wonder about the “levels” of understanding inherent to other mathematical

domains which can help them interpret the mathematical understandings of their students.

There are criticisms of the van Hiele levels, particularly Battista’s critique of the absolute

nature of the levels, which imply that when students achieve that level they are at that level


for all figures in all cases (Battista, 2011). Battista found that this is generally not the case,

in that student thinking may embody more than one level at a time depending on the task

at hand. His group has even gone so far as to identify percentages, as in 25% at Level 1,

58% at Level 2, etc.

Meandering Path

The meandering trajectory is best represented by Battista’s descriptions of the learning

progression for which he borrows inspiration from both Simon (1995) and the CPRE

science learning progressions document (Corcoran, Mosher, & Rogat; 2003). While the

CPRE document refers only to the learning progression, Battista clearly makes the case that

the learning progression in science is equivalent to the learning trajectory in mathematics.

While in mathematics “learning progression” refers solely to the development of students’

understandings of ideas, the learning trajectory also specifies the instructional activities

and expected student thinking that results from engaging in the activities. Battista’s

distinction between the LT and the LP is not universal and the differences on a more global

scale will be addressed again in a later section. What is important to recall is Battista’s

mountainside model for mathematics learning. The student’s learning path up to the

mathematical goal is not straight, but it is direct. Of course this is the average path for a

class, or a population, but in this model instruction follows the same approximate path.

The reality in the classroom looks more like the Multiple or branching path illustration.

Landscape or Network Path

The learning trajectory descriptions summarized so far are all distinctly linear, even if

the path meanders or if students travel along different paths. Fosnot and Dolk (2002)


describe their work analyzing students’ strategies, schematizing, their big ideas, as well as

their modeling processes. They pointedly reject the linear model (learning line) because in

their view, it is much too linear to represent children’s learning processing. Instead they

adopt the landscape analogy for student learning, and put the teacher’s focus on the

horizon, which is the targeted mathematical learning goal. Assisting students as they make

their individual journey through the landscape of the domain is the teacher’s role. Fosnot

and Dolk, however, present their model of teaching within this broad and varied landscape:

their landscape is not specifically about the unfolding of student learning other than to

acknowledge that children will travel on different paths. In this sense, they are not

discussing a learning trajectory, however the landscape/networking analogy is still apt.

Confrey and colleagues’ definition of a learning trajectory is “a researcher-conjectured,

empirically-supported description of the ordered network of constructs a student

encounters through instruction (i.e., activities, tasks, tools, forms of interaction and

methods of evaluation), in order to move from informal ideas, through successive

refinements of representation, articulation, and reflection, towards increasingly complex

concepts over time” (Wilson, Mojica, & Confrey, 2013). The idea of an “ordered network of

constructs” recalls the landscape that Fosnot and Dolk describe in their series of

practitioner text resources (Fosnot & Dolk, 2002). The inclusion of “activities, tools, and

forms of interaction” in the above definition is also consistent with Simon’s inclusion of

instructional tasks in his introduction of the learning trajectory. The work of the

turnonccmath.net program, developed by Confrey’s research team (Confrey, et al. n.d.), has

produced a model of eighteen learning trajectories organized into a series of tessellating

hexagonal paths that reveal the details of the trajectories (see figure 3). Based on the


Common Core standards, the turnonccmath.net trajectories attempt to meld the current

standards with the empirically supported understandings about student learning

(trajectories). When the standards do not correspond directly with the known trajectories

of student learning, the research team has created “bridging” standards that complete the

sequence of understandings. Interestingly, many of these bridging standards are part of

the body of research about student learning that were either left implied in the Common

Core or have been established since the publication of the CCSS. For example, the following

bridging standard connects 6.RP.1 to 6.RP.2 (CCSSO, 2010), elaborating on the big idea of

the unit ratio, a prominent idea in the 6th grade standards:

6.RP.1 Understand the concept of a ratio and use ratio language to describe a

ratio relationship between two quantities.

6.RPP.B Understand the concepts of ratio unit and unit ratio. Relate these concepts

for a given table of values and show them on a graph.

6.RP.3.b Solve unit rate problems including those involving unit pricing and

constant speed.

While standard 6.RP.1 introduces ratio as a relationship between two quantities, standard

6.RP.3 requires that students solve a full range of unit rate problems. One of the bridging

standards connects the ratio concept to the problem solving aspect by including the use of

both the ratio unit and the unit rate. The unit rate is any ratio relationship that includes a

unit of one as one of the values, and is a key focus in the CCSS. Recognizing that these unit

ratios are reciprocals of each other helps students increase their flexibility with the unit


ratio. On the other hand, the ratio unit is the ratio that includes the least possible whole

number value for both numbers in the ratio. For example, 3/2 : 10 has a unit ratio of 3 : 20.

The bridging standard in this instance elaborates on a big idea, parsing out the key

mathematical ideas that are inherent to mastery in the next standard. While many of the

bridging standards represent key understandings, in some cases the bridging standards

that Confrey’s research team presents are essentially elaborations on strategies and

representations. Another of the bridging standards between 6.RP.1 and 6.RP.3 indicates

that students should be able to use a ratio table and a graph to show the relationship

between co-varying quantities. While there is little dispute over the validity of using and

connecting multiple representations, the use of such strategies is not necessarily a

standard.

Figure 3: turnonccmath.net

Branching

I have included the branching form of learning trajectory as separate from the

network/landscape model primarily because Baroody’s use of the term to describe his

conception of the student learning trajectory differs from others’ in substantive ways


(Baroody, 2011). While many of the authors cited previously have very direct paths from

students’ beginning thinking to the final learning goal, Baroody’s thinking is much more

flexible. He clearly states that there is no singular learning path, and that learning can veer

off in different directions when a student’s understanding changes. He recalls Lesh’s

words, who claimed that learning is not about “ladders” but rather about “branching”

(Lesh, as cited in Baroody, 2011, p.240). In contrast to other trajectories, Baroody’s is

focused on what he calls “big ideas,” an amorphous idea that is best described as not about

details. While the expectation is that every child will move toward the bigger, broad idea, it

is also expected that no single path is the best path, and that every path will be different.

He asserts that even tens of thousands of cases of similar learning paths does not imply that

the same path is true for all students. Interestingly, Baroody is best known for his work

explicating children’s understanding of early counting and addition (Baroody, 1987),

important understandings that informed future work in the Cognitively Guided Instruction

program (Fennema, Franke, Carpenter, & Carey; 1993) and even for the Common Core.

Grain Size

In the previous section we explored the nature of learning trajectory paths, detailing the

degree to which the proposed paths progress linearly or in a more networked fashion.

Another feature that categorizes various learning trajectories is the level of detail

presented. Baroody’s 2011 review of the contemporary state of research on learning

trajectories issued a criticism of Steffe and Battista’s learning trajectory models as too

detailed and complex to be useful for classroom instruction. The intricate and fine-grained

nature of Steffe’s fraction schemes is quite detailed. The detail is remarkable because as it


is presented (see Norton & Wilkins; 2010, 2012) it is not useful to curriculum planners, let

alone classroom teachers. This is not a criticism of the quality and nature of their work but

rather a criticism of its accessibility. But Baroody’s criticism was not leveled at the

inaccessibility, but rather at the incredibly fine-grained nature of their work because it

tended to remove the teacher from the development process. As a matter of fact, as the

HLT moved away from Simon’s original presentation as a planning tool for teachers and

became a map of student learning, teacher agency was even further reduced. Ironically,

Simon now bases his teaching experiment work on the intricate sequence of scheme

development that was produced by this very same group. In Simon’s recent talk at PME

(2014), he outlined a plan for providing focused instruction including tasks that move

students along the trajectory toward mastery.

A broader-grain learning trajectory than Steffe’s may provide less guidance, but allows for

more flexible paths to learning. For example, Gravemeijer (1999) intentionally retains the

role of the teacher as the leader of classroom activities and learning trajectories are tools to

help teachers interpret student thinking and manage student progress.

Source of the Learning Trajectory

Discipline Logic

The sequence of mathematical tasks and content that is included in curricular materials

has a long tradition: multiplication in third grade, proportional relationships in 7th grade,

etc. have been ingrained in their respective grades for many teachers and curriculum

developers. However, each inclusion is a decision that is made at either the standards or at

the curriculum level. Often the sequence of standards that emerges from tradition has been


derived from a target goal, which has been parsed into less and less complex skills. As the

composite skills are broken down into intermediary skills and then parsed further until

there exists a long list of sequence lower order skills. A good example of this is the end goal

of graphing an inequality on the coordinate plane. The list of skills that students need in

order to be able to accurately complete this task is long: graphing an equation on the

coordinate plane, distinguishing greater than or equal to from greater than, substitution of

a value for a variable in an expression and evaluating, etc. The topics mentioned here are

merely the middle school level topics, and they are also only the skills necessary – the list

does not include the understandings that are also prerequisites. This hierarchy of

subordinate knowledge theory is best captured by the work of Gagne, who inspired the

strategies for decades of curricular scope and sequence models. Inherently this strategy is

“top-down,” with the end mathematical goals guiding the creation of preceding objectives

(Battista, 2011).

Another byproduct of “top-down” content is the primacy of discipline logic, a term

specifically coined by Stzajn, Confrey, Wilson, and Edington (2012) that was inspired by the

Consortium for Policy Research in Education document on learning progressions within

science education (Corcoran, Mosher, & Rogat; 2003). The direct statement is that learning

progressions “are based on research about how students’ learning actually progresses—as

opposed to selecting sequences of topics and learning experiences based only on logical

analysis of current disciplinary knowledge and on personal experiences in teaching. Ellis

and colleagues found that such a curricular approach was prevalent in curricula worldwide

(Ellis, Weber, & Lockwood; 2014). Furthermore, the discipline logic-focused curriculum

also engendered a classroom focus on “right” and “wrong” answers rather than on a


process-oriented environment. In this setting, evaluative teacher listening (Davis & Simmt,

2003) guides the teachers’ actions in the classroom. Battista (2011) was not describing

evaluative thinking in the following anecdote, but the incident does capture a perfect

example of evaluative listening. Student X is completing a task determining a distance that

is sketched on a grid structure, and the teacher is evaluating the student’s understanding of

linear measurement by interpreting her work. The student work ambiguously shows a “1”

right next to a corner (see figure 4). One teacher observes the “1” and concludes that the

student is just counting one unit when there should have been two units. Another teacher

concludes that the student is confusing area with perimeter because she is counting boxes

instead of side lengths. While the student’s answer is indeed incorrect, neither teacher

asks the student what their thought process might have been. If the student is indeed

counting squares instead of side lengths, that is an entirely different mental concept, and

stage on the learning trajectory, as the student who may not recognize that the inside

corner includes two adjacent sides. The consequences of the interpretation are great

because the student’s response indicates what direction instruction should take at that

point.

Figure 4

When teachers listen evaluatively they are looking for correct and incorrect answers,

and they view the students’ responses as related more to the teacher’s line of questioning

rather than to the students’ line of thinking. In contrast, interpretive listening is


characterized by information seeking intentions. The listener’s goal is to open spaces

between themselves and the student where they are able to establish a common language

and genuinely understand the student’s thinking and as a result, locate their thinking along

a possible learning trajectory. One of the problems with the evaluative approach is that the

teacher can be deceived by incorrect thinking that nevertheless yields a correct response.

A discipline logic focused standard or trajectory does not engender interpretive listening

because the intent is to evaluate responses as right or wrong so that the responses conform

more tightly to the end learning goal.. Moreover, the discipline logic-centered learning

trajectory may also tend to include a fixed instructional sequence. However, one of the

benefits of discipline logic-centered learning is that the instructional sequence is clear and

ordered. In other words, it is less messy.

Developmental Logic

A learning trajectory that is focused on development logic instead of discipline logic is

focused tightly on the thinking and understandings of the students in the class. Ellis

provides a summary table that well represents the different conceptions that are at the

heart of these differences (Ellis, Weber, & Lockwood; 2014).

Learning Progression Learning Trajectory

Construct Concept

Learning Goals Characterizations

Evidence Examples

Tasks Activities


Consider the difference between learning goals and characterizations. “Understanding the

meanings of the power in an exponential expression” is a typical learning goal given in a

learning progression (Ellis, Weber, Lockwood; 2014). A learning trajectory

characterization has a different tone while it is still focused on the same content. For

example, a characterization of student understanding might show that the student can

“coordinate the ratio of any two y-values for any gaps between two x-values.” This

characterization shows that the student is able to understand the values of y and the values

of x and co-relate them, particularly within determined intervals on the graph. This

coordination requires an understanding of power that goes beyond repeated

multiplication. Not only does a characterization represent a deeper level of understanding,

it also provides activities that are designed to directly address particularly challenging

developmental targets.

Learning trajectories, recalling Simon’s (1995) definition, include learning goals,

learning activities, and the thinking and learning in which students might engage. Under a

developmental logic frame, this focus on students in the instructional process is a logical

outgrowth of the learning trajectory structure. One of the key elements is the important

role of instructional tasks in moving students forward in their learning. The van Hieles

(2004) assert that students do not naturally move from one level of geometric thinking to

another- they must have extensive experiences under specific kinds of learning conditions

that give them opportunities to restructure their understandings. Battista also puts great

import on the value of instructional tasks to move through levels, even if the movement

through levels is somewhat erratic and varies according to the task (Battista, 2011).


The developmental logic approach to learning paths is not a haphazard response to

student learning based on anecdotal evidence from a sequence of tasks. The

developmental touchpoints within a learning trajectory are based on empirical evidence

gathered from the research experiences of the writers, other researchers’ accrued

knowledge, as well as from knowledge of how students in the classroom respond to

teaching activities. Nevertheless, the learning trajectory is a “bottom up” point of view, in

that student understandings create the structure of the learning, and the LT itself provides

steps along the way for teachers to look for. But, an interesting recent development is the

apparent reification of learning trajectories that seems to have occurred as each “camp”

produces more evidence that verify their hypothetical learning trajectories. Simon’s recent

presentation at a conference couches a sequence of learning activities and resultant

concepts within the frame of a concept-development set of tasks, contrasting this teaching

strategy with the classic problem solving lesson. He even hinted that this structured series

of tasks is designed to provide the exact right sequence needed to assist students who

struggle with learning to work with fractional rational numbers (Simon, 2014). One came

away with the distinct impression that following this prescribed series of activities would

lead students directly through the projected learning path: the HLT would become the

actual LT! It was a surprising turn.

Confrey’s bridging standards serve a similar role in that they fill in the empty spaces

between widely spaced CCSSM standards, standards that permit multiple access points and

paths (Confrey et al., n.d.). However, Battista credits learning trajectories with predictive

power: an LT will not tell you exactly what a student might do or understand, but it does

engage with probable outcomes (Battista, 2011). In another article Confrey’s team also


asserts that learning trajectories make reasonable predictions of what student

understanding of a concept might be (Sztajn, et al., 2012), which provides guidance for the

teacher to select tasks that will lead students to the next conceptual idea.

In the end, the process of instruction must include both orientations to learning. The

development perspective acknowledges that individuals do not adopt new ideas and

information exactly how they are presented (von Glasersfeld, 1995). Instead they re-create

mathematical understandings that build on their current understandings. That individuals

follow even remotely similar learning paths is surprising if one holds a radical

constructivist point of view. However, no learning would be valuable if none of this

information became taken-as-shared (Davis & Simmt, 2003) and agreed upon by the local

mathematics classroom community. Furthermore, none of the taken-as-shared

information is valuable unless it begins to look more like the language and symbolism of

the outside mathematics community. In this direction, the discipline logic orientation

appears more sensical. Sztajn et al. (2012) state directly that negotiation between the

discipline logic, which is the global taken-as-shared knowledge and norms, and the

developmental logic must come together to form a coherent trajectory. This trajectory

accounts for individuality, but also welcomes the standardized norms that define the

mathematics field.

Learning Trajectory – Learning Progression – Hypothetical Learning Trajectory

This review of research began as an exploration into the differences between the LT

and the LP, and the seemingly random use of all three terms in the literature. The

hypothetical learning trajectory was introduced in 1995 by Marty Simon in a piece focused

on an instructional model that was largely ignored. Instead the HLT has become an


acronym that retains little to none of the meaning with which it was originally imbued.

Hypothetical, in his estimation at that time, referred to the teacher’s conjecture at the

beginning of lesson planning of what students would think and learn as they engaged in a

set of learning tasks directed toward a determined goal. The scope of the HLT was narrow,

limited only to the classroom setting.

The term HLT served a need in the field because it was soon adopted by several other

research teams to describe students’ learning processes. Clements and Sarama, Baroody,

Battista, Confrey, and many more later adopted the learning trajectory term to describe

their new understandings of student knowledge growth. To varying degrees the

“hypothetical” was kept or discarded (ex. Baroody and Gravemeijer). When it was kept, it

ceased to refer to the teacher’s conjectures about hypothetical paths of learning students

might taken. Instead it came to qualify the learning trajectory as “conditional” on further

research, or as a means to make claims and express reasonable hesitation toward

generalizations to entire populations. But, it no longer held the original meaning that was

tied to individual teachers.

Others stopped using the “hypothetical” qualifier and their research findings suddenly

took on a more definitive tone. The learning trajectory emerged as a formidable series of

levels (or schemes, or touchpoints, or nodes, or steps) that students are likely to follow as

they learn within a specific domain.

Finally, it is important to discuss the learning progression, primarily for two reasons.

Much of the cited information given to support the validity of the learning trajectory is

traced back to a single document published by Corcoran, Mosher, & Lockwood (2003). The

document concerns itself with students’ processes as they learn about the big ideas of


science, a discipline which also can be riddled with profound misconceptions. They

adopted the National Research Council’s definition of a learning progression as “Learning

progressions in science are empirically-grounded and testable hypotheses about how

students’ understanding of, and ability to use, core scientific concepts and explanations and

related scientific practices grow and become more sophisticated over time, with

appropriate instruction (NRC, 2007).” This definition more closely aligns to the definition

of learning trajectory currently used in the mathematics education field, and many have

accepted it as their preferred usage.

Secondly, there is a continuously evolving set of documents (McCallum, Daro, & Zimba;

2014) being written for the purpose of elaborating on the standards that form the CCSSO

(2010). These learning progressions more closely resemble a discipline logic approach to

instruction. The CCSS standards certainly have more in common with the discipline logic

as well, but the learning progressions profess to do the same work that Confrey’s hexagon

model (Confrey et al., 2014) does as well: fill in necessary gaps between the broad and far-

reaching standards and begin to unpack the mathematical understandings that underlie the

new standards.

The effort to support the given standards is admirable, but as always with a discipline

logic-based system, conflicts with students’ learning achievements call into question the

appropriateness of the CCSS standards at particular grade levels. There are examples

where the scope and sequence of the national standards is not consistent with students’

potential capacity to learn the standards. One example is the coordination of three levels of

units as students learn to divide fractions. This is a standard that falls within the 6th grade


in the Common Core. At the 2012 NCTM Research Pre-session a panel of researchers

presented a set of empirical findings showing that the skills required to achieve this

particularly objective were not acquired by a significant percentage of the sample

population of the same age in their study (Steffe, Norton, Hackenberg, Thompson, &

Empson; 2012). The generalized conclusion is that if it is not possible, or probable, that the

majority of sixth graders will have achieved a level of mathematical development

sophisticated enough to meet the sixth grade standard, the standard is therefore

problematic. Their suggestion is to reorganize the standards to be more developmentally

appropriate and accessible to most if not all students. This is an example where an

emphasis on the structure of the mathematics, or discipline logic, results in a sequence of

learning goals that are not attainable by a significant portion of the population.

Cautions and Criticisms

Empson wrote a leveled critique of the enthusiasm surrounding the potential of

learning trajectories in guiding curriculum and learning (Empson, 2011). Some learning

domains, she says, are quite naturally prone to follow such predictable paths and cites

early counting skills (Gelman & Gallistel, as cited in Empson, 2011) and multi-digit

multiplication (Fuson, as cited in Empson, 2011) as examples. On the other hand, she

suggests that the learning trajectory research work on rational number understandings by

Steffe & Olive (2010), Behr, Harel, Post, & Lesh; 1992), and others do not suggest one

coherent learning trajectory, which renders their conclusions tenable. One reason may be

the extensive dependence of the LT on the learning tasks that form the structure of the

trajectory. A learning trajectory assessed and therefore driven by a certain set of tasks


looks different from another learning trajectory. In either case, these are cautions for

adopting any learning trajectories in their entirety.

Another of Empson’s (2011) criticisms is the reduction of teacher agency in the current

understanding of the learning trajectory. As the LT models become more tightly defined,

they risk becoming more structured and limiting and less able to account for individual

differences. Doerr (2006) echoes this concern when she expresses concerns that the

stringent demands of the standards environment already limit the expression of students’

generation of models of mathematical understanding. Doerr’s interpretation of the

learning trajectory more closely mirrors Simon’s original interpretation, and she credits

the learning trajectories with providing teachers with opportunities to grow along with

their students and learn about the mathematics as students do the same. Another moderate

and level view is Ellis’ statement: “A more efficacious approach may be one that attends to

the variation in students’ conceptual development, building trajectories of the students’

understanding over time (Ellis, Weber, & Lockwood, 2014, p.3-3).

Conclusion

The topic of learning trajectories offers great promise for teaching practice. The

greatest benefits of the continually evolving research on learning trajectories, as well as

learning progressions, are the guidelines that they provide to teachers as they learn more

about the sequence of students’ learning. Previous curriculum standards and textbooks

employ a discipline logic when structuring their content scope and sequence, and current

teachers are accustomed to this focus. The tasks they develop reflect this orientation and

when engaging with open-ended tasks with a higher cognitive demand, they often are not

able to anticipate the responses their students offer for these tasks. Because students


exhibit unexpected mathematical solutions, teachers are more inclined to abandon these

tasks because they can not yet interpret the responses. More understanding and

experience with learning trajectories that describe students’ mathematical thinking will

give them to the skills and confidence they need to interpret their own students’ work.

However, it is important that the learning trajectories retain a predictive role rather than a

prescriptive role. As the list of expected and anticipated conceptual developments

becomes more rigid and detailed, there is less room for individual variation. In this sense a

learning trajectory that resembles the network/landscape model is most flexible as it offers

a variety of interpretations for examining and interpreting student work in the classroom.

Finally, there is great need for professional development for teachers as they begin to make

sense of the new details of learning trajectories, particularly in those cases where the

trajectory has a developmental orientation rather than a discipline logic orientation.


References

Amador, J., & Lamberg, T. (2013). Learning Trajectories, Lesson Planning, Affordances, and

Constraints in the Design and Enactment of Mathematics Teaching. Mathematical

Thinking and Learning, 15(2), 146–170.

Baroody, A. J. (1987). The Development of Counting Strategies for Single-Digit Addition.

Journal for Research in Mathematics Education, 18(2), 141.

Battista, M. T. (2004). Applying cognition-based assessment to elementary school students’

development of understanding of area and volume measurement. Mathematical


Battista, M. T. (2011). Conceptualizations and issues related to learning progressions,

learning trajectories, and levels of sophistication. Montana Mathematics Enthusiast,

8(3).

Behr, M. J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. In D.

Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 296–

333). New York: Macmillan.

Buschman, L. (2005). Isn’t that interesting! Reflect and discuss. Teaching, 12, 34–38.

Clements, D. H. (2004). Perspective on “The Child’s Thought and Geometry.” In Classics in

Mathematics Education Research (p. 60). Reston, VA: National Council of Teachers of

Mathematics.

Confrey, J. (2012). Articulating a learning sciences foundation for learning trajectories in

the CCSS-M. In Proceedings of the 34th conference of the North American Chapter of

the International Group for the Psychology of Mathematics Education (pp. 2–22).

Kalamazoo, MI.


Confrey, J., Maloney, A., Nguyen, K., Mojica, G., & Myers, M. (2009).

Equipartitioning/splitting as a foundation of rational number reasoning using LTs.

In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), Proceedings of the 33rd

conference of the International Group for the Psychology of Mathematics Education.

(pp. 345–352). Thessaloniki, Greece: PME.

Confrey, J., Maloney, A., Nguyen, K., Lee, K. S., Panorkou, N., Corley, A., … Gibson, T. (2014,

August). Learning Trajectories for the K-8 Common Core Math Standards . Retrieved

from http://www.turnonccmath.net/

Corcoran, T., Mosher, F. A., & Rogat, A. (2003). Learning progressions in science: An

evidence-based approach to reform. Consortium for Policy Research in Education,

337–676. Retrieved from

http://www.sciencedirect.com/science/article/pii/S0375947403018098

Council of Chief State School Officers. National Governors Association Center for Best

Practices. (2010). Common Core State Standards for Mathematics. Washington, DC.

Davis, B., & Simmt, E. (2003). Understanding learning systems: Mathematics education and

complexity science. Journal for Research in Mathematics Education, 137–167.

Doerr, H. M. (2006). Examining the tasks of teaching when using students’ mathematical

thinking. Educational Studies in Mathematics, 62(1), 3–24.

Ellis, A. B., Weber, E., & Lockwood, E. (2014). The case for learning trajectories research . In

Proceedings of the Joint Meeting 3 - 1 of PME 38 and PME-NA 36 (Vol. 3, pp. 1–8).

Vancouver, Canada.

Empson, S. B. (2011). On the idea of learning trajectories: Promises and pitfalls. The

Mathematics Enthusiast, 8(3), 571–596.

http://www.sciencedirect.com/science/article/pii/S0375947403018098

http://www.turnonccmath.net/


Fennema, E., Franke, M. L., Carpenter, T. P., & Carey, D. A. (1993). Using Children’s

Mathematical Knowledge in Instruction. American Educational Research Journal,

30(3), 555–583.

Fosnot, C. T., & Dolk, M. (2002). Young mathematicians at work: Constructing fractions,

decimals, and percents. Portsmouth, NH: Heinemann.

Gagne, R. M. (1977). Analysis of objectives. In Instructional Design: Principles and

Applications (pp. 115–148). Englewood Cliffs, NJ: Educational Technology

Publications.

Gravemeijer, K., Bowers, J., & Stephan, M. (2003). Chapter 4: A hypothetical learning

trajectory on measurement and flexible arithmetic. Journal for Research in

Mathematics Education. Monograph, 12, 51–66. doi:10.1016/j.jmathb.2007.03.002

Hackenberg, A. J. (2007). Units coordination and the construction of improper fractions: A

revision of the splitting hypothesis. Journal of Mathematical Behavior, 26, 27–47.

Heritage, M. (2008). Learning progressions: Supporting instruction and formative

assessment. Retrieved from Council of Chief State School Officers website:

http://www.ccsso.org/Resources/Programs/Formative_

Assessment_for_Students_and_Teachers_%28FAST%29.html

Lave, J. (2011). Apprenticeship in critical ethnographic practice. Chicago, IL: The University

of Chicago Press.

Maloney, A. (2013, October 31). personal conversation.

http://www.ccsso.org/Resources/Programs/Formative_


McCallum, W., Daro, P., & Zimba, J. (n.d.). Progressions documents for the Common Core math

standards. The University of Arizona. Retrieved from

http://ime.math.arizona.edu/progressions/

National Research Council (2007). Taking science to school: Learning and teaching science

in grades K-8 . Committee on Science Learning, Kindergarten through eighth grade.

R.A. Duschl, H.A. Schweingruber, & A.W. Shouse (Eds.). Washington DC: National

Academies Press.

Norton, A., & Wilkins, J. L. M. (2010). Students’ partitive schemes. Journal of Mathematical

Behavior, 29, 181–194.

Norton, A., & Wilkins, J. L. (2012). The splitting group. Journal for Research in Mathematics

Education, 43(5), 557–583.

Olive, J., & Steffe, L. P. (2002). The construction of an iterative fractional scheme: The case

of Joe. The Journal of Mathematical Behavior, 20, 413–437.

Oxford English Dictionary. (1914), “trajectory“.

Piaget, J., & Inhelder, B. (1973). The psychology of the child. New York: Basic Books.

Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective.

Journal for Research in Mathematics Education, 26, 114–145.

Simon, M. A. (2014). An emerging theory for design of mathematical task sequences:

Promoting reflective abstraction of mathematical concepts. In Proceedings of the

Joint Meeting of PME 38 and PME-NA 36 (Vol. 5, pp. 193–200). Vancouver, Canada.

Simon, M. A., & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual

learning: An elaboration of the hypothetical learning trajectory. Mathematical


http://ime.math.arizona.edu/progressions/


Steffe, L. P., Norton, A., Hackenberg, A., & Thompspn, P. (2012, April). Students’ fraction

knowledge and the Common Core State Standards (CCSS)q. Presented at the NCTM

Research Presession, Philadelphia, PA.

Steffe, L. P., & Olive, J. (2009). Children’s Fractional Knowledge. Springer.

Stein, M. K., & Smith, M. (2011). 5 practices for orchestrating productive mathematics

discussions . Reston, VA: National Council of Teachers of Mathematics.

Sztajn, P., Confrey, J., Wilson, P. H., & Edgington, C. (2012). Learning trajectory based

instruction: Toward a theory of teaching. Educational Researcher, 41(5), 147–156.

van Hiele, P. M. (2004). The child’s thought and geometry. In Classics in Mathematics

Education Research (pp. 60–67). Reston, VA: National Council of Teachers of

Mathematics.

von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. London:

Routledge.

Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. Cambridge

University Press.

Wilson, P. H., Mojica, G. F., & Confrey, J. (2013). Learning trajectories in teacher education:

Supporting teachers’ understandings of students’ mathematical thinking. The

Journal of Mathematical Behavior, 32(2), 103–121.

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