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ORIGINAL ARTICLE
Learning trajectories: a framework for connecting standardswith curriculum
Jere Confrey • Alan P. Maloney • Andrew K. Corley
Accepted: 21 May 2014
� FIZ Karlsruhe 2014
Abstract Educational Standards provide a statement of
educational competency goals. How to integrate such goal
statements with the instructional core, in ways that promote
curricular and instructional coherence and continuity of
student learning, is a perennial challenge. In the United
States, the Common Core State Standards for Mathematics,
or CCSS-M, have been widely adopted, and are claimed to
be based on research on learning in general and on learning
trajectories in particular. The relationships, however, are
tacit and incompletely, and sometimes controversially,
articulated. This paper describes a body of work that
associates the first nine grades of Standards (K-8) to
eighteen learning trajectories and, for each learning tra-
jectory, unpacks, interprets, and fills in the relationships to
standards with the goal of bringing the relevant research to
teachers (TurnOnCCMath.net). The connections are made
using a set of descriptor elements, comprised of conceptual
principles, coherent structural links, student strategies,
mathematical distinctions or models, and bridging stan-
dards. A more detailed description of the learning trajec-
tory for equipartitioning (EQP) shows the detailed research
base on student learning that underpins a particular learn-
ing trajectory. How curriculum materials for EQP are
designed from the learning trajectory completes the ana-
lysis, illustrating the rich connections possible among
standards, descriptors, an elaborated learning trajectory,
and related curricular materials.
Keywords Learning trajectories � Mathematics
standards � Mathematics curriculum � Equipartitioning �Descriptors
1 Introduction
Adoption of the Common Core State Standards for
Mathematics (CCSS-M) (CCSSI 2010) by 45 states and 5
territories (out of a total of 56 states and territories) in the
U.S. provides an opportunity to organize curriculum
development around the concept of learning trajectories
(or learning progressions). A set of grade-by-grade edu-
cational goal statements for student competencies in
Mathematics, the CCSS-M, has been described as a means
to promote increased rigor, focus, and coherence among
the standards across the various states in the U.S. The
CCSS-M document, developed by two associations of
state officials, thus represents rarely seen consensus on
specific education policy among the majority of States, a
necessity due to Constitutional restrictions on the role of
the federal government in education. While these partic-
ular organizational relationships are peculiar to the United
States, the question of how standards can be related to
research on learning and then made visible and useful to
practitioners is a problem shared internationally.
The writers of the CCSS-M had a mandate to base the
standards on established research. A meeting hosted at
North Carolina State University brought writers of the
standards together with researchers to discuss the potential
of learning trajectories (LTs) research to act as an evidence
base for the standards (Daro et al. 2011). Researchers
The term curriculum can refer to a framework of standards, a scope
and sequence of detailed objectives, or a set of organized materials.
The term in the title refers to the full array of meanings; other uses in
the paper specify the reference.
J. Confrey � A. P. Maloney (&) � A. K. Corley
North Carolina State University, Raleigh, NC, USA
e-mail: [email protected]
URL: http://www.gismo.fi.ncsu.edu
123
ZDM Mathematics Education
DOI 10.1007/s11858-014-0598-7
subsequently submitted trajectory examples to the writers.1
Learning trajectories or progressions in mathematics edu-
cation are research-based frameworks developed to docu-
ment in detail the likely progressions, over long periods of
time, of students’ reasoning about big ideas in mathemat-
ics. The writers used this research base among other
sources of evidence. They also acknowledged the need for
continued research, writing, ‘‘One promise of common
state standards is that over time they will allow research on
learning progressions to inform and improve the design of
standards to a much greater extent than is possible today’’
(CCSSI 2010, p. 5). Thus, by using research on learning
trajectories and also calling for continued work, the writers
helped to establish standards as living documents with
clear expectations of change and further research over
time. This stance has international implications, recogniz-
ing that learning trajectories are dynamic; they are
responsive to tasks, forms of instruction, culture, and tools.
Different researchers will propose different sequences;
with evolving methodologies, our understanding of them
will change over time. The purpose of this paper is to
describe how to relate standards, research on learning,
specific learning trajectories, and the development of cur-
ricular materials. Similar approaches can be developed in
any country for which explicit standards exist.
Once the CCSS-M had been widely adopted, pressure
increased to help teachers interpret and implement them.
To assist, our research group divided the Standards into 18
learning trajectories and set about unpacking them. We
named what we wrote, descriptors of the standards. As we
wrote more of them, we identified key elements of those
descriptors that generalized across the learning trajectories.
Thus, our use of the term learning trajectory broadened to
refer to clusters and sequences of standards and their
related descriptors. While these ‘‘learning trajectories’’ fit
our initial meaning for a learning trajectory, they tended to
be less precisely structured, because we now sought to
communicate a range of research results, ordered to align
roughly to the Standards. We say roughly because the
process is probably best thought of as ‘‘fit,’’ because we
accepted that the Standards were the policy instrument for
organizing instruction by grade level, but by their own
admission the Standards were not a ‘‘curriculum.’’ (Note:
the authors of the Standards also did not define their
meaning of this term.) By necessity and design, the Stan-
dards represent only what competencies should be
accomplished at grade levels, but not in what order within
grade levels, or with indication of how to connect them
across grade levels. Thus, our process was to order the
Standards within learning trajectories, to help teachers
understand the related research, and to make their intro-
duction to the Standards more efficient and compact.
As we were performing this service to the field, we also
continued to conduct research in-depth on a particular
learning trajectory, equipartitioning. This work represents
what a learning trajectory looks like when developed
independent of Standards. In the context of this work, we
created curriculum materials and an assessment system to
allow us to study how students progressed in the learning
trajectory at the upper levels. When it came time to unpack
the equipartitioning learning trajectory in relation to the
Standards, we worked diligently to link our empirical
work, our design research on curriculum materials and
assessment, and the descriptors and the Standards.
In this paper, we distill this process into a description of
how a broad synthesis of a ‘‘learning trajectory as standards
and descriptors’’ can be connected to an ‘‘empirically
derived learning trajectory’’ that is itself linked to a set of
curriculum and assessment materials. We do this in six steps:
1. Analyze standards into learning trajectory clusters;
2. Develop ‘‘learning trajectories as standards and
descriptors’’;
3. Specify the ‘‘empirically based learning trajectory’’ as
proficiency levels;
4. Design related curriculum and assessment materials;
5. Study implementation;
6. Revise the learning trajectory and curriculum materials,
based on empirical evidence (including assessment).
In addition, after we describe the ‘‘empirically based
learning trajectory’’ (Sect. 3.3.1), we add a section dis-
cussing how that previously derived empirically based
learning trajectory was linked to the CCSS-M ‘‘learning
trajectories as standards and descriptors.’’ This section is
important to demonstrate how it was necessary to ‘‘fit’’ the
different types of learning trajectories together, based on
how they evolved in real time.
Thus, while we present the six steps in a sequence, we
note that they often inform each other, are fitted together,
and should be revised and refined over time. We suggest
that articulating the six steps is useful in describing the
overall iterative process of progress toward coherence
among policy, research, design and development.
2 Definitions
2.1 What are standards?
Relating standards to curriculum writ large via learning
trajectories is a means to bring research into mainstream
1 Author Confrey served on the National Validation Committee for
the Common Core State Standards. She analyzed drafts of the CCSS-
M for alignment with the learning trajectories research base and
provided corresponding feedback to the CCSS-M writers.
J. Confrey et al.
123
educational practice. It requires careful effort to align
policy, research, and practice, and will only succeed to the
degree to which these factors, along with teacher prepa-
ration, curriculum development, and assessment, are
aligned. Iterative design and development of these com-
ponents are required.
The CCSS-M, first and foremost, are goal statements for
learning. They summarize what students are intended to
accomplish and the practices they are intended to adopt to
collectively become an educated citizenry, intellectually and
economically successful in the 21st century global economy.
Standards are also, necessarily, compromises among
communities of experts. They contain a mixture of math-
ematical and psychological language. While some might
argue for simple propositional statements of skills or con-
cepts, devoid of pedagogy, common educational standards
inevitably draw in pedagogy through content emphasis and
timing. The strong emphasis in the CCSS-M on the use of
the number line as a representation of the number system,
for example, expresses in part a pedagogical preference,
not a mathematical obligation. Likewise, expressing the
Standards by grade level, along with the intent of ensuring
coordination across grades, implies certain pedagogical
dispositions. Finally, the Standards describe when students
are held accountable for learned ideas or skills; for ideas
that develop gradually, over extended periods of time,
scaffolding must be put in place earlier if students are to
meet accountability schedules.
The CCSS-M are in fact underspecified, which is nec-
essary to facilitate flexible implementation of educational
approaches that are continually being developed and
revised as we collectively learn more about how students
learn. However, they do identify sequences and linkages of
ideas:
…the ‘‘sequence of topics and performances’’ that is
outlined in a body of mathematics standards mus-
t…respect what is known about how students learn.
As Confrey (2007) points out, developing
‘‘sequenced obstacles and challenges for stu-
dents…absent the insights about meaning that derive
from careful study of learning, would be unfortunate
and unwise.’’ In recognition of this, the development
of these Standards began with research-based learn-
ing progressions detailing what is known today about
how students’ mathematical knowledge, skill, and
understanding develop over time (CCSSI 2010, p. 4).
2.2 What are learning trajectories?
We use the terms trajectories and progressions inter-
changeably here, with the following working definition for
learning trajectory:
… 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 (Confrey et al. 2009a).
This definition, similar to others in mathematics edu-
cation (cf. Clements et al. 2004), emphasizes that learning
trajectories are conjectures, based on empirical study of
student learning and of how student ideas develop from
naı̈ve conceptions to learned ideas (domain goal under-
standings). Classroom instruction is assumed to play a
central role, including all forms of teacher support,
appropriate tasks and tools, peer-to-peer discourse, and the
language necessary to specify and build ideas.
Research aimed at developing learning trajectories/pro-
gressions in K-12 mathematics, including curriculum based
on such learning trajectories, includes early childhood
mathematics (Clements and Sarama in press), early algebra
reasoning (Blanton and Knuth 2012), geometric and spatial
thinking (Battista 2007), length and area measurement
(Barrett et al. 2012), distribution and spread (Leavy and
Middleton 2011) and data modeling (Lehrer et al. in press),
fractions, percentages, decimals, and proportions (van
Galen et al. 2008), linear measurement and geometry (van
den Heuvel-Panhuizen and Buys 2005), early numeracy
(Van den Heuvel-Panhuizen 2008) and early foundational
work on children’s learning of fractions and ratio (Streef-
land 1991), and developmental progressions for aspects of
probabilistic (Watson and Kelly 2009) and statistical
(McGatha et al. 2002; Watson 2009) reasoning. This list is
by no means exhaustive.
Learning trajectories are not a stage approach (Piaget
1970), which delineates developmental stages that must be
mastered before passage to later stages. Rather, they are
probabilistic statements that claim that, given rich tasks
and tools carefully sequenced to build from prior knowl-
edge, students tend to exhibit predictable ranges of
behaviors, including their responses to the tasks and their
ways of speaking about or explaining their reasoning.
Among the features of these empirically based claims is
that students’ beliefs may be productive at an early point,
but become dysfunctional or incorrect later, requiring stu-
dents to modify or replace them. As an example from
equipartitioning, young students may claim that when a
single whole is equipartitioned, the parts must be of the
same size and shape to qualify as fair shares. This student
belief is often productive in facilitating their early rea-
soning about fair shares. But when children later encounter
two identical rectangles—one split in half vertically and
A framework for connecting standards with curriculum
123
the other split in half diagonally—and are asked which
piece, if any, is larger, they come to recognize that two
shapes do not need to be congruent to be equal in size
(Confrey and Maloney 2012).
Learning trajectories are also not logical analyses based
on disciplinary prerequisites. They are, instead, the result
of empirical investigation of what children recognize,
value, and perceive, and how distinctions evolve from
those experiences (Confrey and Maloney in press; Cle-
ments and Sarama in press). In the afore-mentioned
example, of children first believing that equal-sized parts
must have the same shape to have the same size, and later
encountering the realization that equal parts must be the
same size but not necessarily the same shape, the founda-
tions for the original belief and its modification are not
logical but experiential.
3 Process of linking standards, learning trajectories,
and the intended and enacted curriculum
3.1 Analyzing standards into learning trajectory
clusters
The CCSS-M and its corresponding documentation estab-
lish an intended connection between the Standards and
learning trajectories, but fully delineated relationships were
not developed. Several groups, such as the team at Illus-
trative Mathematics (www.illustrativemathematics.org),
provide illustrations of the range and types of mathematical
work that students experience in a faithful implementation
of CCSS-M. We wished, however, to lay out more fully
articulated relationships between empirical study of student
learning and the Standards, and to distinguish such
analyses from (1) logical/mathematical or propositional
analyses based on thought experiments of possible
sequences, and (2) the treatment of topics that one would
find in textbooks for elementary mathematics or pedagog-
ical methods. We sought to interpret in detail the intention
of CCSS-M to support deep, conceptual mathematical
reasoning via learning progressions. The impetus for this
work was two-fold: (1) to provide a resource that interprets
the more conceptually oriented CCSS-M from the stand-
point of student learning, and to lend more coherence to the
potential curricular implementation of the Standards; and
(2) to take the opportunity for the development of the
Standards to bring mathematics education research to the
attention of teachers by relating the Standards systemati-
cally to the research.
The result of these efforts is TurnOnCCMath.net:
Learning Trajectories for the K-8 Common Core Math
Standards (Confrey et al. 2012b), in which we delineated
18 learning trajectories (Table 1), focusing on ‘‘big ideas,’’
that embedded all of the K-8 Standards. A big idea is ‘‘a
statement of an idea that is central to the learning of
mathematics, one that links numerous mathematical
understandings into a coherent whole’’ (Charles 2005).
Here, the general process will be described first, and then
the specific steps and components of that process will be
elaborated by example of the equipartitioning learning
trajectory.
Numerous design decisions are embedded in the Turn-
OnCCMath resources. Alternate models were considered;
for instance, one that would emphasize the multiple
meanings of Standards in relation to different LTs. At first
glance, our placement of particular clusters of Standards
into particular LTs can seem somewhat unconventional.
For example, we constructed a Division and Multiplication
Table 1 Learning trajectories encompassing the K-8 CCSS-M
Learning trajectory Grades Number (Fig. 1) Learning trajectory Grades Number (Fig. 1)
Counting K–2 1 Time and money K–3a 9
Addition and subtraction K–4 2 Elementary data and modeling K–5 7
Shapes and angles K–7 6 Early equations and expressions K–7 10
Length, area, and volume K–8 5 Place value and decimals K–8 3
Equipartitioning 1–5 4 Division and multiplication 2–6 8
Fractions 3–5 11 Integers, number lines, and coordinate planes 5–8 14
Ratio and proportion, and percents 6–8 12 Linear equations, inequalities, and functions 6–8 13
Rational and irrational numbers 6–8 15 Variation, distribution, and modeling 6–8 17
Chance and probability 7 18 Triangles and transformations 7–8 16
The grade spans reflect both CCSS-M and bridging standards. Number (Fig. 1) column lists the number of the learning trajectory on the hexagon
map in Fig. 1, belowa Limiting time and money to grades K-3 represents a decision by the writers of CCSS-M to limit attention to these topics as mathematical topics
in their own right, rather than as applications of other topics (use of numbers, decimals, etc.). Kamii and Russell (2012) argue that the CCSS-M
treat time simply as a system of representations, implicitly understating the complexity of student learning about time by failing to account for
children’s need to conceptualize temporal relationships and to coordinate hierarchical units
J. Confrey et al.
123
LT as well as a Fractions LT. Multiplication and division
of fractions were placed at the top level of the Division and
Multiplication LT rather than within the Fractions LT,
while addition and subtraction of fractions remained within
the Fractions LT (and not in the Addition and Subtraction
LT). Our rationale was that division and multiplication
create fractions, and fractions become critical multiplica-
tive operators, so fractional division and multiplication are
considered an extension of the meaning of those opera-
tions, whereas addition and subtraction, well established
prior to the introduction to fractions and combining
(additive reasoning) with fractions, are treated as a com-
ponent of a fractional system of numeration.
A hexagon map (Fig. 1; Confrey et al. 2012b) embeds
the K-8 CCSS-M within these learning trajectories, with
the LTs proceeding generally from the lower left to the
upper right, to visually reinforce the notion of increasing
sophistication in student reasoning and complexity of
content across time. Each hexagon represents a single
Standard. The gray-scale view shown here depicts the
Standards by LT across grade levels; an alternate view (not
shown) color-codes the Standards by grade level. As far as
was practical, if the main content of one Standard followed
directly from that of a previous Standard, those Standards’
hexagons were positioned contiguously, upwards and to the
right; when content related to different Standards could be
taught in parallel, those Standards were positioned roughly
in parallel along the general direction of the LT.
3.2 Developing ‘‘learning trajectories as standards
and descriptors’’
A ‘‘learning trajectory’’ implies gradual learning overtime
based on predictable patterns in student thinking. As we
analyzed each learning trajectory cluster and wrote up the
various research findings, we found that those findings
were comprised of a number of common elements. The
research team formalized these ideas into detailed
descriptors comprised of five ‘‘elements’’ (Confrey 2012).
These can be regarded as a set of lenses through which to
view the kinds of intermediate understandings through
which students move from the initial levels of trajectories
to target understandings. Each is briefly described below:
1. Coherent structure a recurring framework or structure
for reasoning, which can be fostered through
Fig. 1 Hexagon map of the K-8
Common Core State Standards
for Mathematics embedded in
18 learning trajectories
(Confrey et al. 2012b). Number
code of the LTs is shown in
Table 1; some LTs composed of
non-adjacent Standards (e.g. 4,
10). See www.TurnOnCCMath.
net for color version of map and
complete descriptions of the 18
learning trajectories
A framework for connecting standards with curriculum
123
instruction to support student investigation and reflec-
tion from lower through upper levels of a learning
trajectory.
2. Underlying conceptual or cognitive principles ‘‘big
ideas’’ within the learning trajectory, including the
target understanding. These may begin with a cogni-
tive primitive, an action from which embodied oper-
ations and schemes are built.
3. Students’ strategies, inscriptions and representations,
misconceptions how students make their reasoning and
intermediate understandings visible. Students invent,
adapt, or adopt strategies and representations as they
solve challenges, demonstrating their ways of thinking
and, often, revealing misconceptions that must be
addressed instructionally. However, misconceptions
often have a kernel of ‘‘right thinking’’ (Confrey
1990), so they must be elicited and then refined into
alternative conceptions or valid intermediate steps on
paths to more sophisticated thinking.
4. Meaningful distinctions and multiple models emergent
mathematical distinctions and models support increas-
ingly sophisticated and nuanced building of the big
ideas. In developing intermediate proficiencies, stu-
dents often invent or adopt generalizations or language
for different categories, or express models that corre-
spond to different schemes that support recognition of
different mathematical situations.2
5. Bridging standards The hexagon map’s progression of
Standards comprises an ‘‘abridged’’ learning trajectory.
Bridging standards represent learning targets that may
be needed instructionally to support greater continuity
of student learning than might be discerned from the
major intellectual targets encapsulated in the CCSS-M
alone.
In Sect. 3.3.2, we describe how depicting the equipar-
titioning learning trajectory as ‘‘standards and descriptors’’
illustrates each of these elements.
3.3 Specifying the ‘‘empirically based learning
trajectory’’ as proficiency levels
3.3.1 Equipartitioning proficiency levels
Equipartitioning3 (EQP) is the construct of ‘‘cognitive
behaviors that have the goal of producing equal-sized
groups (from collections) or equal-sized parts (from con-
tinuous wholes), or equal-sized combinations of wholes and
parts, such as is typically encountered by children initially
in constructing ‘‘fair shares’’ for each of a set of individu-
als’’ (Confrey et al. 2009a, Confrey et al. 2009b). Early
research conducted by Pothier and Sawada (1983) identified
four levels of student partitioning capabilities during
engagement with tasks of partitioning wholes: sharing,
algorithmic halving, evenness, and oddness. Pepper and
Hunting (1998) identified distinct performance differences
in students’ strategies for sharing collections of items,
classified by the extent of systematicity of students’ strat-
egies and whether they resulted in equal shares. Our work
identified three cases of fair sharing in the literature (A—
collections; B—single wholes; C—multiple wholes) and
synthesized student-centered research involving these into
the single construct of equipartitioning (Confrey et al.
2009a). We refined these three cases into a set of profi-
ciency levels. Proficiency levels delineate skills and
understandings that students develop, from less to more
sophisticated, and leading to the capstone level of gener-
alization, the target understanding of the learning trajectory.
Additional research on students solving problems with
different numbers of people sharing and different geo-
metric shapes being shared led to the identification of task
classes. Task classes incorporated different parameter
values for problems. For instance, empirical observations
demonstrated that equipartitioning increases in difficulty
somewhat systematically for both collections and wholes,
from 2-splits (creating two equal-sized groups), to splits of
higher powers of two, other even splits, and then odd splits.
Similarly, splitting rectangular wholes is often easier for
children than splitting circular wholes. A matrix of the
proficiency levels and relevant task classes for equiparti-
tioning is shown in Fig. 2.
Levels 1–5 of the EQP LT include accomplishing
equipartitioning of collections and wholes, justifying
strategies and results, naming the fair share in relation to
the size of the whole (or collection) as ‘‘1/nth of,’’ and,
conversely, identifying the size of the whole in relation to a
single fair share as ‘‘n times as many/much.’’
2 Educators recognize the importance of prior knowledge and of
identifying clear targets for learning. A major challenge, however, lies
in identifying and evaluating intermediate states of proficiency and
understanding their role in moving students forward in their thinking.
To describe these intermediate states, teachers and researchers must
recognize or invent meaningful distinctions; vocabulary terms for
these tend to exhibit properties that are both cognitive and mathe-
matical, such as partitive vs. quotative division, which later simply
collapse to ‘‘division.’’ We refer to these as ‘‘meaningful distinc-
tions.’’ In addition, ‘‘big ideas’’ (conceptual principles within a LT)
may depend on earlier models corresponding to different schemes
governing recognition of situations in the real world. These are
typically captured as a ‘‘generalization’’ that encapsulates multiple
meanings for experts, but tends to obscure the distinctions and models
that students need to grapple with as they learn. Students need ample
opportunity to explore such distinctions and models before moving to
a generalization, to understand implications and adaptability of a
generalization for its many referents and applications.
3 Confrey (2009) introduced equipartitioning as a more general term
for the operation of splitting, in which equal shares are produced. It is
distinct from partitioning, which can refer to breaking or segmenting
into unequal parts, such as a room partition.
J. Confrey et al.
123
The middle levels (6–11) include various emergent
relations and properties related to equipartitioning of single
wholes (rectangles and circles), including composing splits
(anticipating multiplication), compensation of number of
shares and size of shares, and recognizing the theoretical
possibility of sharing a whole for any number of people
(Continuity Principle). The upper levels (12–16) focus on
multiple wholes, and draw on the proficiencies that stu-
dents have developed in working with collections and
single wholes. There are two parallel, main types of strat-
egies: (1) co-splitting (Corley et al. 2012; Corley 2013),
coordinating splits of both the objects and sharers in fair-
sharing problems, resulting in equivalent multiplicative
changes in the values of each quantity (anticipating ratio),
and (2) strategies for dealing, splitting, and distributing
multiple wholes to sharers. Each of these types of strategies
then leads students to the claim that for any a objects
shared among b sharers, the fair share will be a/b objects
per sharer (target understanding, level 16).
3.3.2 Connecting the empirically based LT to the CCSS-M
In this section, we circle back to show how the
empirically based learning trajectory was linked to the
learning trajectory as standards and descriptors post
hoc. This was necessitated by the timing of the work.
The matrix representation in Fig. 2 was developed as a
tool for the construction and mapping of assessment
items to proficiency levels, and developing diagnostic
assessments and construct validity studies. Such repre-
sentations are more relevant to researchers than to
educational practitioners. How then to communicate to
most educators the essential aspects of student pro-
gression from a cognitive primitive (binary split of a
collection or single whole), to articulating foundations
for multiplicative reasoning, including the groundwork
for reasoning about ratio, fractions, division and mul-
tiplication, and do so in relation to the CCSS-M?
Adapting this learning trajectory for the CCSS-M
required designing a way to represent the equiparti-
tioning LT in a way that is both faithful to the research
(to articulate the progression of student reasoning about
equipartitioning), and recognizes the policy significance
of the Common Core Standards. We illustrate how this
was done by describing the application of the descriptor
elements to the ideas from the proficiency matrix.
In TurnOnCCMath, the Common Core Standards and
bridging standards for equipartitioning are organized in
three sections to reflect the case structure of the equipar-
titioning LT (Table 2). We first discuss the six CCSS-M
Fig. 2 Equipartitioning
learning trajectory matrix, with
proficiency levels and task
classes (Confrey and Maloney
2012, updated 2013). ‘X’
represents task classes most
relevant to particular
proficiency levels
A framework for connecting standards with curriculum
123
Standards, and then show how the bridging standards fill in
gaps. The first three relevant CCSS-M Standards occur in
first and second grade; all three concern sharing a contin-
uous whole (LT Section 2). Standard 1.G.A.3 (grade 1)
mentions student understanding for two and four shares,
anticipates the naming of those shares both by relative size
(unit fractions 1/2 and 1/4) and relationally (‘‘one-half of’’
and ‘‘one-fourth of’’), and introduces qualitative compen-
sation between the number of sharers and the size of the
corresponding share. The LT emphasizes describing the
relationship between the reassembled whole and the single
shares as ‘‘times as much as’’ a single share. We interpreted
the Standards’ request to describe the whole in terms of the
number of shares to allude to reassembly within the LT
structure. Standard 2.G.A.3 simply builds on 1.G.A.3 by
including thirds, and adds the student understanding that
fair shares do not need to be congruent (Property of
Equality of Equipartitioning, or PEEQ—Level 10 of the
EQP LT). The third Standard in the LT (2.G.A.2) describes
partitioning a rectangle into rows and columns; in the LT,
as elaborated by the TurnOnCCMath descriptors, the bi-
directional equipartitioning of a rectangle is included as
composition of splits (multiplicative reasoning), even
though the Standard explicitly suggested that verifying the
number of (equi)partitions is a counting exercise (additive
reasoning).
Two-third-grade Standards were incorporated into the
equipartitioning LT: 3.G.A.2—generating unit fractions
from shapes, and 3.NF.A.1—generating fractional quanti-
ties, a/b, by combining a unit fractions (1/b) of an equi-
partitioned whole. The final Standard identified for the
equipartitioning LT, from fifth grade (5.NF.B.3), equates
fractions of the form (a/b) with whole-number division
statements (a 7 b), and indicates that problems involving
whole-number division can have outcomes stated as frac-
tions. This Standard was embedded into the top level of the
learning trajectory (Generalization—i.e., that fair sharing
of multiple wholes (a) among multiple sharers (b), can be
Table 2 Common Core and bridging standards in the equipartitioning learning trajectory
Section 1: Equipartitioning evenly divisible collections
1.EQP.A Equipartition evenly divisible collections into fair shares among up to four sharers, and name the results as a number of objects and
as part of a whole evenly divisible collection (e.g., one-half, one quarter, or one-third)
1.EQP.B Reassemble the evenly divisible collection and describe the whole collection as two, three, or four times as many as in a single share
2.EQP.B Use fair sharing of evenly divisible collections to create 2 9 2 boxes in which one column is labeled as number of objects and the
other as number of people sharing, filling in two rows with 1) the original size of collection, number of people sharing and 2) the size of fair
share, one person
2-3.EQP.E Predict that equipartitioning collections into larger (smaller) numbers of shares results in smaller (larger)-sized shares
(qualitative compensation), and that doubling, tripling, quadrupling, etc., the number of sharers halves, thirds, quarters, etc. the size of the
share (quantitative compensation)
2-3.EQP.F Reallocate shares from departing sharers among remaining sharers
Section 2: Equipartitioning wholes
1.G.A.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and
use the phrases half of, fourth of, and quarter of. Describe the whole as two of or four of the shares. Understand for these examples that
decomposing into more equal shares creates smaller shares
2.G.A.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third
of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the
same shape
2-4.EQP.C Understand that any single whole (e.g., a circle, a square, a rectangle) can be equipartitioned into any n equal shares
2.G.A.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them
2.EQP.D Predict the resulting number of parts from equipartitioning a rectangle using vertical and horizontal cuts
3.G.A.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole
3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b
as the quantity formed by a parts of size 1/b
Section 3: Equipartitioning multiple wholes
2-4.EQP.A Solve and record problems involving fair sharing of collections of indivisible items or of multiple wholes, using co-splits
4-5.EQP.A Equipartition multiple wholes by dealing, splitting, and distributing
5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (a/b = a 7b). Solve word problems involving division of
whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent
the problem
CCSS-M Standards are listed in plain text, and labeled by the CCSS-M nomenclature; bridging standards are listed in italics, and labeled by a
similar nomenclature, such that ‘‘EQP’’ indicates the learning trajectory (equipartitioning), the first number indicates grade level, and the last
letter indicates an alphabetical order
J. Confrey et al.
123
expressed as fraction a/b and as a division statement a 7b).4
The LT’s proficiency levels are only partially encom-
passed by the Common Core Standards themselves. The
LT includes equipartitioning of collections, and other
major concepts and distinctions necessary for students’
progression through foundations for early multiplicative
reasoning pertaining to ratio and fraction. Children apply
strategies for fairly sharing collections even before they
enter school (Hunting and Sharpley 1988). A coherent
structural element, the three criteria for equipartitioning,
emerges from students sharing both wholes and collections
(Confrey and Maloney 2010; detailed below). Student
reasoning about multiple wholes and fractions (a/
b) requires students to leverage understanding of fair
sharing of both collections and single wholes, and both
should be reflected in early instruction. We therefore added
a number of bridging standards (Fig. 2) to the EQP LT, as a
kind of instructional ‘‘connective tissue’’ for the typically
sparse goal statement skeleton of the CCSS-M.
Bridging standards 1.EQP.A and 1.EQP.B (Sect. 1)
introduce equipartitioning of evenly divisible collections,
reassembly of the created parts, and relational naming.
Bridging standard 2-4.EQP.C (Sect. 2) spans several grade
levels, to address the Continuity Principle of equiparti-
tioning, which represents an anticipatory schema (Steffe
1994). A bridging standard that spans grades reflects that
the particular understandings develop gradually, with
increasingly complex situations that students are likely to
encounter over several years.
The second-grade bridging standard 2.EQP.B introduces
the ‘‘fair share box’’ (Fig. 3) early in the LT, for use ini-
tially with collections, and later for use with both single
wholes and multiple wholes. This simple, two-by-two
tabular device supports children in recording the pairs of
values produced for each of the two quantities (original
amount shared among the original number of people, and
size of the fair share for one person) in fair sharing situa-
tions (Confrey 2012; Confrey et al. 2012b). It visually
establishes that multiplicative relationships are four-valued
(Vergnaud 1994), mediating student understanding of the
inverse relationship of equipartitioning and reassembly
(times-as-many or times-as-much), and later the inverse
relation between division and multiplication.
For equipartitioning collections, bridging standards
2–3.EQP.E and 2–3.EQP.F include compensation (which
later can be extended to form a basis of inverse variation),
and address reallocation, the efficient redistribution of extra
shares in the case that a number of sharers depart, which
sets up distributivity.
Standard 2.G.A.2 (Sect. 2, single wholes) introduces
shapes partitioned into equal-area parts. Bridging standard
2.EQP.D focuses on multiple ways of equipartitioning such
shapes—in particular, the composition of horizontal and
vertical splits as a model for generating such equipartitions
(of a rectangle). This focus helps teachers support students
to develop multiplicative reasoning, and avoid defining
multiplication solely as repeated addition. While repeated
addition may be useful for justifying whole-number multi-
plication, it is thoroughly inadequate for all other rational
numbers, and actually undermines students’ learning about
the multiplicative structure of other numbers (Devlin 2008).
Co-splitting (Sect. 3), which supports subsequent
development of ratio, was added as a grade-spanning
bridging standard, 2–4.EQP.A. Researchers have recog-
nized for decades that ratio reasoning evolves far earlier
than it is introduced in the Standards (Confrey and Scarano
1995; Resnick and Singer 1993; Streefland 1984, 1985).
Other pre-6th grade foundations for ratio, from Standards
that support covariation and correspondence in patterns in
tables (4th- and 5th-grade), are embedded in the Early
Equations and Expressions LT.
Finally, bridging standard 4–5.EQP.A emphasizes how
students gradually combine and extend equipartitioning
proficiencies for collections and single wholes, to multiple
wholes—using strategies such as deal-and-split, split into
benchmark fractions and deal, and split-all—to gradually
approach the general understanding that for any number of
objects and any number of sharers, a objects shared among
b people is the same as a/b objects per person.
Fig. 3 Fair-share box, in this case showing the numerical relation-
ships among the total number of items in a collection, the number of
people sharing, and the number of items in a single share, for a single
person. Pairs of directional arrows indicate corresponding changes in
number of people and number of objects
4 In our professional judgment, fractions and division should be
instructionally related earlier than 5th grade. However, mutual
accommodation of empirically derived LTs and LTs as Standards
and descriptors required compromises. Nonetheless, we would
recommend that teachers build the link earlier.
A framework for connecting standards with curriculum
123
Combining CCSS-M Standards and bridging standards,
examples of the descriptor elements can be identified:
• Coherent structure The three criteria for equipartition-
ing, used for all instances of equipartitioning and
justification: (1) creating the desired number of shares,
(2) creating equal-sized shares, and (3) exhausting the
whole or collection.
• Underlying cognitive principle The proto-operation of
splitting, identified as a cognitive primitive distinct
from counting (Confrey 1988).
• Strategies, representations, and misconceptions Strate-
gies for fairly sharing collections, single wholes, and
multiple wholes, the fair-share box (representation),
and the (initially productive) misconception that fair
shares of congruent single wholes must be congruent.
• Meaningful distinctions and multiple models Compen-
sation, PEEQ, co-splitting, and multiple models for the
notation a/b (as embedded in the LT’s target
understanding).
3.4 Designing related curriculum materials
To conduct research to further specify and validate the
EQP LT, we were required to develop relevant curriculum
materials (grades two to five), because students currently
lack systematic instructional exposure to equipartitioning.
We have now conducted a total of four teaching experi-
ments using versions of the curriculum materials (Confrey
et al. 2012a; Corley 2013). The materials comprised seven
paper-based curricular ‘‘packets’’ mapped onto the learning
trajectory matrix, as shown in Fig. 4, showing the LT
proficiency levels that are explicitly covered by each cur-
riculum packet and measured by assessment items corre-
sponding to each packet. Assessment items were
incorporated in digital, tablet-based diagnostic assessments
developed specifically for each packet. From early
assessment item studies, we recognized that assessing
individual proficiency levels was neither practical (would
require far too many items for the entire LT) nor desirable
(student reasoning about individual items typically incor-
porated multiple proficiency levels). Furthermore, profi-
ciency level overlap between curricular packets reflects the
applicability of some proficiency levels to multiple cases –
Justification and Naming, for instance, were recognized as
distinct practices germane to development of early math-
ematical reasoning, and were therefore included as distinct
proficiency levels.
The instructional model was to present tasks and pro-
mote group discussion intended to elicit the main ideas in
groups of proficiency levels, build student agreement on
task strategies, representations, and promote mathematical
practices such as justification and argument. Tasks
systematically used the cases and the task classes relevant
to individual rows of the EQP LT matrix, varying the
anticipated item difficulty at those levels. For example,
tasks for sharing collections used the context of pirate
treasure (see Wilson et al. 2012), and varied the number of
pirates and the gold coins. The tasks typically began with a
challenge for students, requiring sharing of strategies,
justifications, and results, and concluded with introduction
or recapitulation of relevant terms and vocabulary.
3.5 Studying implementation
Curricular and assessment tasks generated evidence about
what topics came easily to students and which were more
novel or produced unexpected results (Confrey et al.
2012a; Corley 2013). We provide three examples here to
illustrate how those studies of implementation of the cur-
riculum shed light on subsequent development of the LT,
its relation to the standards, and implied revisions to the
curriculum itself, or our approach to assessment.
Example 1 (reassembly of shares and relational naming).
The first teaching experiment involved students who had
completed 2rd or 3rd grades (from 7 to 10 years of age).
They quickly generated fair shares of collections by deal-
ing (distributing one-by-one). Observing one another, they
quickly appropriated methods of systematic dealing and
using composite units, and readily named fair shares as a
count of the number of objects. When fair sharing for two
persons, they could express the idea that each person would
‘‘get half’’ of the collection. However, relational naming of
shares and collections was much more difficult to engen-
der. When sharing 12 coins among 3 people, they described
the fair share as 4 coins or 4 coins per person, but
describing the size of one share in relation to the size of the
collection was challenging.
We adapted the instructional plan to teach them how to
describe a part of a collection relationally, and vice versa
(for the above example, how to reason the whole collection
as three times as many as a single share). This involved the
students essentially re-unitizing a collection of n objects to
a single whole, and a fair share comprising multiple objects
as one share out of three total shares. It emerged that
helping students view the whole collection, when shared
among n people, as n times as many as a single share,
required helping them first view a single share, then two
shares as twice as many as the single share, then three
shares as three times as many, and so on. The most suc-
cessful strategy involved visually bracketing drawings of
shares in consecutive groups of groups, each as a form of
‘‘times as many’’ as a single share. We argue that intro-
ducing the ‘‘times as many’’ concept as the inverse to
splitting is necessary to (1) establish reversibility of
J. Confrey et al.
123
equipartitioning, so that, subsequently, division and mul-
tiplication are viewed as inverse operations, (2) avoid
making multiplication explicitly dependent on, or derived
from, repeated addition, and (3) establish a recursive rather
than iterative foundation for multiplication. This points to a
broader foundation for multiplicative reasoning across a
variety of models, such as scaling, Cartesian products, and
area, as well as the use of repeated groups.
These insights engendered discussion of reordering
some of the levels of the learning trajectory. Naming of
shares and justifying strategies were initially integrated in
the levels for sharing a collection and sharing a whole.
Subsequent experience led us to make naming and justi-
fying individual proficiency levels, distinguish such
mathematical practices from mere accomplishment and
basic strategies of fair sharing, while recognizing that they
transcended individual cases. Naming, justifying, and
reassembly for sharing collections and then wholes
reflected our conjecture that through treatment of collec-
tions and single wholes in parallel we could (1) introduce
early multiplicative reasoning foundations to offset the
traditional bias towards repeated addition, and (2) help
establish and reinforce the practices.
Example 2 (criteria for equipartitioning). In sharing a
single whole (cake), students often exhibited typical mis-
conceptions. Our curricular approach to elicit students’
understandings (including misconceptions) was based on
students creating their own inscriptions (Lehrer et al. 2007)
and to promote class identification and discussion of
features.
Student inscriptions for sharing a rectangular cake for 4
persons (Fig. 5) fell into several categories (allowing for
imprecision of drawing). Students sorted the rectangles into
groups, with spirited discussion about how the represen-
tations were similar and different. Bottom row rectangles:
same-size shares but rectangle not entirely shared (the
leftmost three) or has an incorrect number of shares
(rightmost). Middle row: correct number of (unequal-size)
pieces. Top row: correct number of (approximately) equal-
size shares, and rectangles shared entirely.
In helping students to distinguish among the rectangles,
the researchers facilitated them in becoming accustomed to
and proficient in articulating the three criteria for equi-
partitioning; the three criteria soon became a refrain when
students justified their fair sharing outcomes.
Example 3 (student application of early levels of the LT
to higher levels). Corley (2013) conducted a design study
of the development of co-splitting, the action of simulta-
neously splitting objects and sharers into groups that
maintain the same fair share per person as in the original
collections of objects and sharers. Some tasks were derived
from Streefland (1991): for a number of pizzas and a
number of people, distributing people and pizzas among
multiple tables in order to maintain the fair share of pizza
among the people. The student participants, who had
completed 3rd, 4th, or 5th grades (ages 8 through 11), had
already been taught multiplication and division in their
schooling. Corley examined the degree to which students
relied on the lower levels of the learning trajectory as they
learned co-splitting (students first had a brief (4-day) series
of instruction on equipartitioning collections and single
wholes). In the co-splitting tasks, the students used the
lower level proficiencies without prompting, providing
further validation of the value of a sequenced curricular
Fig. 4 Mapping of curriculum
packets to EQP LT proficiency
levels
A framework for connecting standards with curriculum
123
progression based on the learning trajectory. The study also
linked the development of co-splitting to the depth of
understanding of multiplication and division, suggesting
that at higher levels and in more advanced learning tra-
jectories, interactions and relations among learning trajec-
tories become evident and important.
Design studies of implemented curricula can help to
improve instruction on targeted topics, and can be empir-
ically linked to student learning on subsequent topics. By
pragmatically incorporating awareness of descriptor ele-
ments into the curricular design, curricular materials can be
adjusted to support different levels of understanding, from
fundamental cognitive principles, to strategies and repre-
sentations, to emergent properties. Students can be sup-
ported through instruction to enhance the likelihood of
making mathematical distinctions, thereby supporting their
ability to apply increasingly sophisticated ‘‘big ideas’’ and
arrive at mathematical generalizations.
3.6 Revising the learning trajectory and curriculum
materials, based on empirical evidence (including
assessment)
The teaching experiments also included a diagnostic
assessment component. We had concurrently designed and
implemented a prototype software, LPPSync, that had a
practice mode and a diagnostic assessment mode for peri-
odically examining students’ progress on the LT profi-
ciency levels (Confrey et al. 2012a). The central priority in
this diagnostic approach was to generate specific evidence
of student progress (or lack thereof) in direct relation to the
underlying theory of learning—that is, the learning tra-
jectory itself (Confrey et al. 2011). The assessment system
was designed for deployment on tablets, browser based,
and built in HTML5. Student response data were processed
in a centralized server-based database. The diagnostic
assessments comprised six items (two each of easy, med-
ium, and difficult items) per packet, covering several levels
of an LT; items were themselves randomly generated from
a parameterized sample space. The digital diagnostic
assessment environment employed virtual manipulatives
and text- and number-entry, including statements of justi-
fication, and supported analysis of student strategies for the
three equipartitioning cases. The prototype system pro-
vided teachers and students data on students’ learning
within the LT in real time, both individually and accu-
mulated across all students. The system is still relatively
primitive, but for one lesson, it led to the creation of
assignments tailored to the topics in sharing a whole cus-
tomized for individual students. The approach showed
promise in helping students become partners in assessing
and addressing, through their own practice, their own
understanding and growth.
Our teaching experiments, incorporating the curriculum
materials, LPPSync system, and ongoing review of class-
room interactions around instruction, led to revisions of the
LT. We offer two examples. (1) Fig. 4 already reflects the
importance of the co-splitting construct, initially discerned
in the initial teaching experiments based on work of
Streefland (1984, 1985), and extended in Corley’s (2013)
research. Co-splitting has now been substantiated as an
early proto-construct that helps students link reasoning
about splits of collections of indivisible items to splits of
collections of divisible wholes, and revealed how readily
students’ recognize the utility in what is later referred to as
a unit ratio. (2) Analysis of the first two teaching experi-
ments led us to further reconsider relational naming for
early fraction sense. We initially conjectured that students
would readily recognize the equivalence of a share of an
equipartitioned collection of items (e.g. from sharing 12
items for 4 people) and the same equipartition of a single
whole (which they readily named ‘‘one-fourth’’), by rec-
ognizing the share of three items as one share or group out
of four shares or groups. Analysis of their reasoning, and
further interviews trying out various juxtaposition of ma-
nipulatives, then pointed us toward modifications of the
curriculum that we believe will promote students to
Fig. 5 Student rectangles (one
rectangle per student) shared
fairly for four persons. Figure is
a black and white representation
of students’ original work done
on colored construction paper
J. Confrey et al.
123
identify this version of equivalence of relational naming of
shares and wholes, which could be incorporated earlier in
instruction than is currently typical in most curricula.
4 Summary
We suggest that learning trajectories have a major role to
play as boundary objects (Star and Griesemer 1989) that
connect standards, curriculum, assessment, and profes-
sional development (Confrey and Maloney in press; Con-
frey 2011; Confrey and Maloney 2012), and can act as a
unifying framework for models of teaching (Sztajn et al.
2012). From those perspectives, we recounted here the
development of a broadly usable set of learning trajectories
that embed the entire K-8 CCSS-M in the US, and provide
a framework for interpreting those Standards from the
standpoint of research on student learning of mathematics.
We then depicted our own work on equipartitioning as an
example of the level of detail that comprises development
of a fully articulated learning trajectory based primarily on
direct study of student learning, that otherwise may not
have been clear or fully articulated and realized by the
Standards alone. We described curriculum materials
developed specifically with respect to the LT and then used
in teaching experiments—a next phase of research to refine
the LT from a standpoint of construct and instructional
validity. We briefly described a browser-based diagnostic
assessment system developed as a prototype for formative
diagnostic assessment use and as a vehicle for field-testing
items in relation to LT’s proficiency levels, task classes,
and instruction. We then returned to recount how we
incorporated CCSS-M standards and new bridging stan-
dards into a CCSS-M-based, modified learning trajectory
that illustrates progression of student learning of the
equipartitioning construct that reflects empirical research
while remaining true to the intent of the Common Core
project.
Awareness of student learning enables teachers to better
recognize students’ conceptual growth through intermedi-
ate understandings, and thus to facilitate students’ overall
progress towards the higher-level generalizations or overall
learning targets (Wilson et al. 2013). We believe that the
descriptor elements provide specific scaffolding to support
teachers in anticipating student mathematical reasoning
and building instructional coherence over time as a means
of implementing the CCSS-M in the US, or standards-
driven curriculum in any country.
The equipartitioning LT was used to illustrate the
forging of research-based links to standards and to develop
an early version of a LT-based curriculum. Our own
research and development agenda during the past 10 years
has been similar to work led by Clements and Sarama
(early number, shape, and geometry), Lehrer and Schauble
(data modeling), Battista, Barrett, and Clements (shape,
length and area measurement), and others internationally:
to develop learning trajectories for various major concep-
tual areas (in our case, rational number or multiplicative
reasoning).
Identified as a distinct topic only recently, equiparti-
tioning has yet to be incorporated as such into conventional
curricula. Relevant tasks are included in many curricula,
but foundational competencies of the learning trajectory
(e.g., naming fair shares in relation to a collection, or
developing relational naming of fractions through equi-
partitioning) are lacking and insufficiently covered. When
taken as a whole, these competencies lay the groundwork
for student reasoning about division and multiplication,
fractions, and ratio. Conventional curricula treat multipli-
cation solely as repeated addition, which is a major
impediment to developing an early sense of multiplicative
reasoning as a foundation for fraction and ratio. We believe
that the equipartitioning LT, with its cognitive principles
related to contexts familiar to students (splitting, sharing)
and its unifying case-based structure, provides teachers a
framework with which to focus instructionally on student
behaviors that promote early multiplicative reasoning
competencies.
The learning trajectories-based analysis in this paper
suggests how the other two overarching goals of the CCSS-
M—improved coherence and focus—can be enhanced
through the linkage of standards to more in-depth learning
trajectories and thoughtfully related curriculum and
assessment. Coherence and focus imply not merely inten-
tion, but careful curricular, instructional, and assessment
design, and subsequent interpretation of implementation in
relation to achieved outcomes. Learning trajectories, more
than any other approach we are aware of, seem to have
tremendous potential as an underlying principled frame-
work for aligning curriculum, instruction, and assessment
(both formative and summative) into an internally coherent
whole that would improve student learning in any educa-
tional system. The six-step process outlined in this paper
provides one prototypical path for accomplishing this goal.
In conclusion, we offer some final observations that we
hope are relevant to the international community. (1)
Mathematics educational standards should be designed to
incorporate careful attention to research on student learn-
ing. (2) The descriptor elements we describe here provide
overlapping but distinct lenses for practitioners to use in
improving their awareness—and support—of their own
students’ learning. (3) Because standards seem inevitably
to have variations in topic or competency ‘‘grain size,’’ and
because they are not curricula, the bridging standards are a
valuable tool to fill gaps between standards and to
emphasize intermediate understanding and transition
A framework for connecting standards with curriculum
123
between major topics. Bridging standards provide a
reminder to practitioners that teaching merely standard by
standard is likely to embed inappropriate sequences of, and
transitions between, topics, and undermine, rather than
amplify, student learning. (4) Learning trajectories offer a
systematic way for research to be communicated to
teachers and other practitioners for integration into their
own instructional practice. We remind, however, that
learning trajectories are not linear or fixed (e.g., research is
ongoing; innovative instructional technologies can lead to
major transformation in how students approach learning
and how they learn concepts), that student learning is not
assumed to be uniform across cultures or regions, and that
student learning depends heavily on instruction. (5)
Finally, we propose that learning trajectories offer an
opportunity to synthesize existing research from the entire
international community (our first synthesis of rational
number reasoning (Confrey 2008) drew on the work of
researchers from more than thirty countries), and in so
doing, enrich our understanding of differences in student
learning as grounded in different cultures and countries.
Acknowledgments This material is based upon work supported by
the National Science Foundation (DRL-0758151 and DRL-073272)
and Qualcomm. Any opinions, findings, conclusions, or recommen-
dations expressed in these materials are those of the authors and do
not necessarily reflect the views of the National Science Foundation
or other funders. The authors wish to acknowledge the contributions
of K. Nguyen, S. Varela, N. Monrose, Z. Yilmaz, and L. Neal to the
development of the LPPSync software, curriculum writing, data
analysis, and/or the conduct of the teaching experiment itself.
References
Barrett, J., Clements, D., Sarama, J., Cullen, C., McCool, J.,
Witkowski-Rumsey, C., et al. (2012). Evaluating and improving
a learning trajectory for linear measurement in elementary
grades 2 and 3: a longitudinal study. Mathematical Thinking and
Learning, 14(1), 28–54.
Battista, M. T. (2007). The development of geometric and spatial
thinking. In F. K. Lester (Ed.), Second handbook of research on
mathematics teaching and learning (pp. 843–908). Charlotte,
NC: Information Age Publishing Inc.
Blanton, M., & Knuth, E. (2012). Developing algebra-ready students
for middle school: Exploring the impact of early algebra. Paper
presented at the Project Poster presented at 2012 DRK-12
Principal Investigators Meeting, Washington, DC.
CCSSI (2010). Common Core State Standards for Mathematics, from
http://www.corestandards.org/assets/CCSSI_MathStandards.pdf.
Charles, R. (2005). Big ideas and understandings as the foundation for
elementary and middle school mathematics. Journal of Mathe-
matics Education Leadership, 7(3), 9–24.
Clements, D. H., Wilson, D. C., & Sarama, J. (2004). Young
children’s composition of geometric figures: a learning trajec-
tory. Mathematical Thinking and Learning, 6(2), 163–184.
Confrey, J. (1988). Multiplication and splitting: Their role in
understanding exponential functions. In Behr, M., Lacampagne,
C. and Wheeler, M.M. (Eds.), Proceedings of the tenth annual
meeting of the North American chapter of the International
Group for the Psychology of Mathematics Education (PME-NA).
Dekalb, Northern Illinois University, IL, pp. 250–259.
Confrey, J. (1990). A review of the research on student conceptions
in mathematics, science and programming. In C. Cazden (Ed.),
Review of research in education (Vol. 16, pp. 3–56).
Confrey, J. (2007). Tracing the evolution of mathematics content
standards in the United States: Looking back and projecting
forward. Conference on K-12 Mathematics Curriculum Stan-
dards. Washington, DC. February 5–6, 2007.
Confrey, J. (2008). Synthesis of the research on rational number
reasoning. Plenary address to the XI International Congress in
Mathematics Education. Monterrey, Mexico, July 7.
Confrey, J. (2011). Engineering [for] effectiveness in mathematics
education: Intervention at the instructional core in an era of
common core standards. Paper presented at the Highly Success-
ful STEM Schools or Programs for K-12 STEM Education: A
Workshop, Washington, DC.
Confrey, J. (2012). Articulating a Learning Sciences Foundation for
Learning Trajectories in the CCSS-M. Paper presented at the
Thirty-Fourth Annual Meeting of the North American Chapter of
the Internationl Group for the Psychology of Mathematics
Educaiton, Western Michigan University, Kalamazoo, MI.
Confrey, J., Hasse, E., Maloney, A. P., Nguyen, K. H., & Varela, S.
(2011). Executive summary: Recommendations from the confer-
ence ‘‘Designing Technology– Enabled Diagnostic Assessments
for K-12 Mathematics.’’ North Carolina State University,
Raleigh, NC, USA
Confrey, J., & Maloney, A. (2010). Defining and Implementing
Learning Trajectories as Research Tools. Paper presented at the
Research Presession of the Annual Meeting of the National
Council of Teachers of Mathematics, San Diego, CA.
Confrey, J., & Maloney, A. P. (2012). Next generation digital
classroom assessment based on learning trajectories in mathe-
matics. In C. Dede & J. Richards (Eds.), Steps toward a digital
teaching platform (pp. 134–152). New York: Teachers College
Press.
Confrey, J., Maloney, A., Nguyen, K. H., & Corley, A. K. (2012a). A
Design Study of a Wireless Interactive Diagnostic System Based
on a Mathematics Learning Trajectory. Paper presented at the
Annual Meeting of the American Educational Research Asso-
ciation, Vancouver, BC, Canada.
Confrey, J., Maloney, A. P., Nguyen, K. H., Mojica, G., & Myers, M.
(2009a). Equipartitioning/splitting as a foundation of rational
number reasoning using learning trajectories. In Tzekaki, M.,
Kaldrimidou, M. & Sakonidis, H. (Eds.), Proceedings of the
33rd Conference of the International Group for the Psychology
of Mathematics Education (Vol. 2, pp. 345–352). Thessaloniki,
Greece.
Confrey, J., Nguyen, K. H., Lee, K., Panorkou, N., Corley, A. K., &
Maloney, A. P. (2012b). Turn-On Common Core Math: Learning
Trajectories for the Common Core State Standards for Mathe-
matics, from http://www.turnonccmath.net.
Confrey, J., & Scarano, G. H. (1995). Splitting reexamined: Results
from a three-year longitudinal study of children in grades three
to five. In Owens, D., Reed, M. and Millsaps, M. (Eds.),
Proceedings of the Seventeenth Psychology of Mathematics
Education-NA (Vol. 1, pp. 421–426). Columbus, OH
Confrey, J., Wilson, M., Penuel, W., Maloney, A. P., Draney, K., &
Feldman, B. (2009b). Collaboration and the interplay among
design, policy contexts, and rigor: Building valid, student-
centered mathematics assessments. Paper presented at the annual
J. Confrey et al.
123
meeting of the American Education Research Association, San
Diego, CA.
Corley, A. K. (2013). A Design Study of Co-splitting as Situated in the
Equipartitioning Learning Trajectory. (Ph.D. Doctoral), North
Carolina State University.
Corley, A., Confrey, J. & Nguyen, K. (2012). The co-splitting
construct: Student strategies and the relationship to equiparti-
tioning and ratio. Paper presented at the Annual Meeting of the
American Education Research Association, Vancouver, Canada.
Daro, P., Mosher, F. A., & Corcoran, T. (2011). Learning Trajectories
in Mathematics: A Foundation for Standards, Curriculum,
Assessment, and Instruction. Consortium for Policy Research
in Education.
Devlin, K. (2008). Multiplication and Those Pesky British Spellings.
Devlin’s Angle Retrieved October, 2010, from http://www.maa.
org/devlin/devlin_09_08.html.
Hunting, R. P., & Sharpley, C. F. (1988). Fraction knowledge in
preschool children. Journal for Research in Mathematics
Education, 19(2), 175–180.
Kamii, C., & Russell, K. A. (2012). Elapsed time: why is it so difficult
to teach? Journal for Research in Mathematics Education, 43(3),
296–315.
Leavy, A. M., & Middleton, J. A. (2011). Elementary and middle
grade students’ constructions of typicality. Journal of Mathe-
matical Behavior, 30(3), 235–254.
Lehrer, R., Kim, M.-J., & Schauble, L. (2007). Supporting the
development of conceptions of statistics by engaging students in
measuring and modeling variability. International Journal of
Computers for Mathematical Learning, 12(3), 195–216.
McGatha, M., Cobb, P., & McClain, K. (2002). An analysis of
students’ initial statistical understandings: developing a conjec-
tured learning trajectory. Journal of Mathematical Behavior,
16(3), 339–355.
Pepper, K. L., & Hunting, R. P. (1998). Preschoolers’ Counting and
Sharing. Journal for Research in Mathematics Education, 29(2),
164–183.
Piaget, J. (1970). Genetic epistemology. New York, NY: W.W.
Norton & Company Inc.
Pothier, Y., & Sawada, D. (1983). Partitioning: the emergence of
rational number ideas in young children. Journal for Research in
Mathematics Education, 14(5), 307–317.
Resnick, L. B., & Singer, J. A. (1993). Protoquantitative Origins of
Ratio Reasoning. In T. P. Carpenter, E. Fennema, & T.
A. Romberg (Eds.), Rational Numbers: An Integration of
Research (pp. 231–246). Hillsdale: Lawrence Erlbaum.
Star, S. L., & Griesemer, J. T. (1989). Institutional ecology,
‘translations’ and boundary objects: amateurs and professionals
in Berkeley’s museum of vertebrate zoology, 1907–39. Social
Studies of Science, 19(3), 387–420.
Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel & J.
Confrey (Eds.), The development of multiplicative reasoning in
the learning of mathematics (pp. 3–40). Albany, NY: State
University of New York Press.
Streefland, L. (1984). Search for the Roots of Ratio: some Thoughts
on the Long Term Learning Process (Towards… A Theory)…Part I: reflections on a Teaching Experiment. Educational
Studies in Mathematics, 15(4), 327–348.
Streefland, L. (1985). Search for the roots of ratio: some thoughts on
the long term learning process (Towards… A Theory)… Part II:
The outline of the long term learning process. Educational
Studies in Mathematics, 16(1), 75–94.
Streefland, L. (1991). Fractions in realistic mathematics education: A
paradigm of developmental research. Dordrecht, The Nether-
lands: Kluwer Academic Publishers.
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 den Heuvel-Panhuizen, M. (Ed.). (2008). Children learn math-
ematics: a learning-teaching trajectory with intermediate attain-
ment targets for calculation with whole numbers in primary
school. Rotterdam: Sense Publishers.
van den Heuvel-Panhuizen, M., & Buys, K. (Eds.). (2005). Young
Children Learn Measurement and Geometry: A Learning-
Teaching Trajectory with Intermediate Attainment Targets for
the Lower Grades in Primary School. Rotterdam, The Nether-
lands: Sense Publishers.
van Galen, F., Feijs, E., Figueiredo, N., Gravemeijer, K., van Herpen,
E., & Keijzer, R. (2008). Fractions, percentages, decimals and
proportions: A leaning-teaching trajectory for grades 4, 5 and 6.
Rotterdam: Sense Publishers.
Vergnaud, G. (1994). Multiplicative conceptual field: What and why?
In Harel, G. & Confrey, J. (eds.) The development of multipli-
cative reasoning in the learning of mathematics. State University
of New York, Albany, NY, USA, pp. 41–60.
Watson, J. M. (2009). The development of statistical understanding at
the elementary school level. Mediterranean Journal of Mathe-
matics Education, 8(1), 89–109.
Watson, J. M., & Kelly, B. A. (2009). Development of student
understanding of outcomes involving two or more dice. Interna-
tional Journal of Science and Mathematics Education, 7, 25–54.
Wilson, P. H., Meiers, M., Edgington, C., & Confrey, J. (2012). Fair
shares matey, or walk the plank. Teaching Children Mathemat-
ics, 18(8), 482–489.
Wilson, P. H., Mojica, G. F., & Confrey, J. (2013). Learning
trajectories in teacher education: supporting teachers’ under-
standings of students’ mathematical thinking. Journal of Math-
ematical Behavior, 32, 103–121.
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