Developing students’ ability to solve word problems through learning trajectory-based and variation task-informed instruction

Abstract

Solving word problems is challenging in elementary schools, both for the teacher in teaching students to solve word problems and for the student in learning to solve them. This paper examines how the ideas of learning trajectory and variation pedagogy could be integrated as an instructional principle for teaching this content in the context of solving additive comparison problems. Based on research literature, a learning trajectory for solving additive comparison problems was identified. Informed by variation pedagogy and using a lesson study approach, a research team explored how to teach solving comparison word problems based on this learning trajectory. Data included lesson plans, videotaped research lessons, students’ pre- and post-tests, and students’ interviews. A fine-grained analysis of the data demonstrated that the lessons unfolded through exploration of a series of deliberate tasks along the learning trajectory, focusing on the structure of comparison problems and targeted at objects of learning. Purposefully constructed patterns of variation and invariance provided students with necessary conditions to discern and experience the objects of learning. Students were actively engaged in making sense of comparison problems and articulating their thinking using multiple representations. While the post-test and interview data show students’ understanding of key aspects of solving additive comparison problems to be at various levels, students’ gains in overall performance from pre- to post-test were statistically significant. Implications for teaching comparison word problems are discussed.

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https://doi.org/10.1007/s11858-018-0983-8
ORIGINAL ARTICLE
Developing students’ ability tosolve word problems throughlearning
trajectory-based andvariation task-informed instruction
RongjinHuang1· QinqiongZhang2· Yu‑pingChang2· DovieKimmins1
Accepted: 31 July 2018
© FIZ Karlsruhe 2018
Abstract
Solving word problems is challenging in elementary schools, both for the teacher in teaching students to solve word problems
and for the student in learning to solve them. This paper examines how the ideas of learning trajectory and variation pedagogy
could be integrated as an instructional principle for teaching this content in the context of solving additive comparison prob-
lems. Based on research literature, a learning trajectory for solving additive comparison problems was identified. Informed
by variation pedagogy and using a lesson study approach, a research team explored how to teach solving comparison word
problems based on this learning trajectory. Data included lesson plans, videotaped research lessons, students’ pre- and
post-tests, and students’ interviews. A fine-grained analysis of the data demonstrated that the lessons unfolded through
exploration of a series of deliberate tasks along the learning trajectory, focusing on the structure of comparison problems
and targeted at objects of learning. Purposefully constructed patterns of variation and invariance provided students with
necessary conditions to discern and experience the objects of learning. Students were actively engaged in making sense
of comparison problems and articulating their thinking using multiple representations. While the post-test and interview
data show students’ understanding of key aspects of solving additive comparison problems to be at various levels, students’
gains in overall performance from pre- to post-test were statistically significant. Implications for teaching comparison word
problems are discussed.
Keywords Additive comparison word problems· Lesson study· Variation theory· Learning trajectory
1 Introduction
Developing procedural fluency through conceptual under-
standing in mathematics instruction has been promoted in
curriculum standards over decades (e.g., NCTM 2000; MOE
2011). Fluency in addition and subtraction of whole num-
bers is the foundation for developing strong numeracy for
elementary school students (Baroody and Purpura 2017).
Great efforts have been made to explore effective ways of
teaching and learning whole number concepts and opera-
tions (Baroody and Purpura 2017; Fuson 1992; Fuson and
Murata 2007; Fuson etal. 2014; Verschaffel etal. 2007). To
make sense of addition and subtraction of whole numbers,
various models of joining, part–part–whole, taking away,
missing addend, and additive comparison have been used to
interpret these operations (e.g., Verschaffel etal. 2007; Van
de Walle etal. 2016). Among these models, understand-
ing of the additive comparison model has been documented
as the most difficult (Fuson etal. 1996; Nunes etal. 2016;
Stern 1993; Van de Walle etal. 2016; Verschaffel 1994).
Yet, it is largely unexplored how teachers can effectively
teach additive comparison word problems (CWP, hereafter)
in classrooms based on research findings (Fuson etal. 1996;
Nunes etal. 2016). Through a lesson study approach, Huang
etal. (2017) found that a learning trajectory-based instruc-
tional approach could help develop students’ understanding
of CWP, but they were left wondering whether the same
instructional principle could be used to develop understand-
ing and fluency in solving CWP. To this end, this study was
designed to explore how the combined instructional prin-
ciples of learning trajectory (Simon 1995) and variation
pedagogy (Gu etal. 2017; Marton 2015) could be used to
teach CWP in a way that promotes students’ understanding
and fluency as well.
* Rongjin Huang
hrj318@gmail.com; Rongjin.Huang@mtsu.edu
1 Middle Tennessee State University, Murfreesboro, USA
2 Wenzhou University, Zhejiang, China

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2 Research background
This section includes four parts. First, an overarching
instructional principle for this study is described. Second,
studies on learning and teaching of CWP are summa-
rized. Third, the ways of dealing with CWP in China are
described. Finally, a framework for this study is illustrated.
2.1 The overarching instructional principle
Two major theoretical frameworks guide the design and
implementation of teaching in the current investigation.
One is learning trajectory (Clements and Sarama 2004;
Sztajn etal. 2012), and the other is variation pedagogy
(Gu etal. 2017; Marton 2015).
2.1.1 Learning trajectory-based instruction
This study adopted the notion of hypothetical learning
trajectory (LT) developed by Simon (1995) and other
scholars. In his seminal work, Simon (1995) suggested
the hypothetical LT as the pathway on which students
might proceed as they advance their learning towards the
intended goals. Three components of a LT include the
learning goal that defines the direction, the learning activi-
ties, and the hypothetical learning process—a prediction
of how the students’ understanding will evolve in the con-
text of the learning activities (Simon 1995). Clements and
Sarama (2004) further defined a LT as a “description of
children’s thinking and learning in a specific mathemati-
cal domain, and a related conjectured route through a set
of instructional tasks designed to engender those mental
processes or action hypothesized to move children through
a developmental progression of levels of thinking” (p.83).
From synthesizing studies on key aspects of classroom
instruction, Sztajn and colleagues (2012) proposed a learn-
ing trajectory-based instructional model which places LT
at the center of instruction and incorporates four critical
aspects of classroom instruction, including mathematical
knowledge for teaching, task analysis, discourse facilita-
tion, and formative assessment. This instructional model
promotes students’ learning by students’ actively engag-
ing in deliberate mathematics tasks in light of a learning
trajectory. Research shows that the use of LTs can support
teachers’ knowledge growth and instructional decision-
making, allow teachers to focus on students’ thinking, and
eventually improve students’ achievement (Clements etal.
2011; Wilson etal. 2013, 2015).
2.1.2 Variation pedagogy
Variation pedagogy (VP) here refers to the core ideas of
teaching and learning mathematics through creating pat-
terns of variation and invariance (Gu etal. 2017; Marton
2015; Pang etal. 2017). Based on the exploration of effec-
tive mathematics teaching practice over decades, a group
of scholars in China developed a theory of teaching with
variation (Gu etal. 2004, 2017) focusing on two types of
variation: concept-oriented and process-oriented variation
(Gu etal. 2017). The key idea is to construct patterns of
variation and invariance for creating necessary conditions
for mathematics teaching and learning.
Other scholars (Marton 2015; Marton and Pang 2006;
Pang etal. 2017) developed a variation theory of learning
based on the assumption that the essence of learning is to
develop new ways of seeing something, and specific com-
parisons allow one to discern critical features of the object
being studied. According to Marton (2015), there are two
types of patterns of variation and invariance, namely sepa-
ration and fusion. Separation can be further classified as
contrast separation or generalization separation. When one
critical aspect of an object of learning varies while other
aspects remain invariant, this pattern is called separation.
The aspect to be varied is perceptually separated from the
other aspects and becomes salient to the learner. If two or
more critical aspects vary simultaneously and are brought
to the learner’s awareness at the same time, then the pat-
tern of variance or invariance is called fusion. There are
two ways to help separate one critical aspect of an object
of learning from others. One is to contrast instance against
non-instance by focusing on differences; the other is to
generalize the common feature across different instances
by focusing on similarities.
Although there are differences between the Chinese the-
ory of teaching with variation (Gu etal. 2017) and the var-
iation theory of learning (Marton 2015), the theories are
complementary (Pang etal. 2017). Fundamentally, both
notions of teaching and learning through variation agree
that “new meaning can be generated when one discovers
similarities among things that are conventionally taken as
different, or differences among things that are convention-
ally taken as the same” (Pang etal. 2017, p.64). Thus,
both Marton (2015) and Gu etal. (2017) emphasize the
importance of creating certain patterns of variation and
invariance for student learning in classrooms, by examin-
ing what varies and what is invariant (Marton and Pang
2006). Pedagogically, Lo and Marton (2012) suggested,
“when learners need to discern more than two or more
critical features, the most powerful strategy is to let the
learners discern them one at a time, before they encounter
simultaneous variation of the features” (p.11).

Developing students’ ability tosolve word problems throughlearning trajectory-based and…
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Researchers developed a framework to gauge students’
learning from the variation theory of learning (Marton
and Pang 2006; Marton 2015), including intended object
of learning, enacted object of learning, and lived object of
learning. An object of learning is “a specific insight, skill,
or capability that the students are expected to develop
during a lesson or during a limited sequence of lessons”
(Marton and Pang 2006, p.194). An enacted object of
learning is described as how intended objects of learning
are enacted through experiencing the patterns of variation
and invariance that are co-constructed by the teacher and
the students. The patterns of variation consist of the neces-
sary conditions for the appropriation of the enacted object
of learning. From the students’ answers to the written and
oral questions after the lesson, we can characterize the
lived object of learning, i.e., the object of learning that is
experienced by the learners.
2.1.3 A combination oflearning trajectory andvariation
pedagogy
While the notion of LT focuses on the route of how stu-
dents learn and develop a specific concept and how associ-
ated tasks could be used to engage the learner in exploring
the concept, it does not pay specific attention to designing
and presenting the tasks to help students move forward at
higher levels. Variation pedagogy emphasizes how to use
variation tasks to help students discern critical features of
the object of learning (concept and/or problem solving)
through experiencing appropriate patterns of variation and
invariance. Variation pedagogy alone lacks attention to
learning progressions and the big ideas around the route
to the intended goals of learning. Moreover, inappropriate
use of variation tasks may lead to focusing on compli-
cated problem solving techniques rather than big ideas
of concepts and skills (Gu etal. 2004). Thus, integration
of the complementary notions of variation pedagogy and
learning trajectory may help students develop conceptual
understanding and procedural proficiency simultaneously.
Huang etal. (2016) conducted an exploratory study on
teaching division of fractions based on the combina-
tion of these two theories through lesson study in China,
and found that research lessons were improved toward
promoting students’ deep understanding. Furthermore,
Huang etal. (2017) explored how to teach CWP based
on the same instructional design principle in the US.
They found that the lesson involving CWP solving was
greatly improved regarding sense making and representa-
tion flexibility, but it failed to develop problem solving at
higher cognitive demand levels. These empirical studies
motivated the researchers to explore the ways for effective
teaching of CWP.
2.2 Studies onteaching andlearning ofsolving
CWP
According to the Common Core State Standards for Math-
ematics 1A (CCSSI 2010), first grade students should learn
to use addition and subtraction within 20 to solve 1 step
word problems involving situations of adding to, taking
from, putting together/taking apart, and comparing by using
objects, drawings, and equations. There are three models
for interpreting subtraction including taking away, missing
addend, and comparing (Van de Walle etal. 2016). The first
two are based on the part–part–whole model, which could
be visually presented by using a part–part–whole diagram.
However, the comparison model involves two distinct quan-
tities and the difference between them, which could be rep-
resented visually by counters or cubes, but “it is not imme-
diately clear to students how to associate either the addition
or subtraction operation with a comparison situation” (Van
de Walle etal. 2016, p.177). Of the four basic categories of
one-step addition and subtraction word problems, compari-
son problems are the most difficult (Briars and Larkin 1984;
Fuson etal. 1996; Riley and Greeno 1988). Comparison
problems describe a static situation in which three sets are
involved: two sets that are compared (the compared and the
referent set) and a set referring to the difference between
them (the difference set). Over the last decades, researchers
have explored the difficulties associated with solving com-
parison problems and effective ways of learning to solve
them (Fuson etal. 1996; Riley and Greeno 1988).
In CWP, the unknown quantity could be the difference
set; or the compare set; or the reference set. In a consistent-
language problem (e.g., “John has 8 more marbles than does
Mary. If Mary has 7 marbles, how many marbles does John
have?”), the question relates to the compare set; the known
variable (i.e., Mary’s) is the object of the relational sentence,
and the relational term (i.e., more than or less than) is con-
sistent with the required arithmetic operation (i.e., addition
or subtraction). In an inconsistent-language problem (e.g.,
“John has 8 more marbles than does Mary. If John has 15
marbles, how many marbles does Mary have?”), the question
relates to the reference set: the known variable (i.e., John’s)
is the grammatical subject of the relational sentence, and
the relational term (i.e., more than or less than) is in conflict
with the required arithmetic operation (i.e., subtraction or
addition). In this regard, a number of empirical studies have
shown that inconsistent-language problems are consider-
ably more difficult than consistent language problems (e.g.,
Hegarty etal. 1995; Pape 2003; Stern 1993; Verschaffel
1994; Verschaffel etal. 1992).
The difficulty of understanding and solving CWP has
been attributed to several factors including difficulties
interpreting comparative terms such as more than and less
than, acting out the static relation, and inconsistent language

R.Huang et al.
1 3
(Briars and Larkin 1984; Lewis and Mayer 1987; Pape 2003;
Múñez etal. 2013). Students must learn not only that addi-
tion can be conceived as the inverse of subtraction and vice
versa, but also that the relation “more than” can be con-
ceived as the inverse of “less than” and vice versa (Verschaf-
fel 1994). Researchers identified three levels of difficulty of
comparison word problems (Briars and Larkin 1984; Riley
and Greeno 1988): level 1 problems with unknown differ-
ence set, level 2 problems utilizing consistent language, and
level 3 problems utilizing inconsistent language.
In addition, researchers explored ways of addressing the
difficulties of problem solving in the realm of CWP (Mwangi
and Sweller 1998; Stern 1993). Mwangi and Sweller (1998)
documented that children presented with integrated worked
examples (with diagrams and equations) outperformed those
presented with examples including either diagrams or equa-
tions, but not both. Similarly, Múñez etal. (2013) found that
external representations (e.g., adding a graph to compare
problems) might help problem-solvers build a mental model
of the described problem situation. Students solved prob-
lems faster and were less error prone when CWP was pre-
sented together with external representations, in particular,
the inconsistent-language problems. Stern (1993) concluded
that in order for students to accurately solve inconsistent lan-
guage problems, they must understand that “x more than y is
z” and “x less than z is y” are the same statements. The lack
of access to flexible language makes CWP with an unknown
reference set difficult. Since all proposed solution methods
depend on understanding which quantity is bigger and which
is smaller, helping solvers decide which quantity is bigger
seems to be crucial for finding correct answers (Fuson etal.
1996). Empirical data demonstrated that students who were
provided specific intervention to help them understand who
has more in inconsistent language problems performed bet-
ter than control groups (Lewis 1989).
2.3 Chinese ways ofdealing withcompare problem
solving
In the new curriculum in China, students need to mentally
add and subtract proficiently in first grade. More specifically,
they need to solve five types of word problems: finding the
sum, finding an amount that remains, finding an unknown
which is a given amount larger than a known number, find-
ing an unknown which is a given amount smaller than a
known number, and finding a difference within 20 (MoE
2011). In China’s traditional teaching of CWP, teachers usu-
ally focus on helping students understand the meaning of
CWP using daily situations and then focus on solving three
types of problems: “finding the difference”, “finding the big-
ger number”, and “finding the smaller number”. Distinguish-
ing between different types of CWP and developing effective
strategies to solve each type has been a long tradition in
teaching CWP in China. For example, Li (1992) compared
the effect of using different structures (part–part–whole vs.
difference between bigger and smaller numbers) to represent
CWP, and found that students using the difference model
outperformed students using the part–part–whole model in
solving CWP.
2.4 A framework forteaching CWP
Researchers (Van de Walle etal. 2016) suggested that repre-
senting numbers being compared with bars and discussing
the difference between the bars to generate the third num-
ber may help students to generate a subtraction equation for
“how many more questions”. Research further suggests that
using physical actions to match concrete objects in a one-
to-one fashion leads to the use of visualized objects being
matched in the same manner, and finally to the transitional
development of number sentences using mathematical sym-
bols. This progression leads students to transition from a
concrete perception to an abstract understanding of com-
parison subtraction (Zhou and Peverly 2005). These studies
suggest that a matching conceptualization of comparison
could be developed through physical actions and concrete
manipulatives. In addition, these activities could further help
students decide which is bigger, or smaller, or what the dif-
ference is. Based on the matching conceptualization of com-
parison and difficulty levels proposed by Riley and Greeno
(1988), an overall learning trajectory of solving comparison
word problems follows.
Level 1 Solve problems with unknown difference set;
Level 2 Solve problems with unknown big or small sets
using consistent language;
Level 3 Solve problems with unknown big or small sets
using inconsistent language.
To lay a strong foundation for this learning trajectory, we
identified four sub-levels within each level:
1a Physically acting out the comparison subtraction
model within a daily context;
1b Using manipulatives to model comparison subtraction
concretely;
1c Drawing diagrams to represent the relationship visu-
ally; and
1d Creating mathematical equations.
Fig. 1 The structure of comparison word problems and its quantita-
tive relationship

Developing students’ ability tosolve word problems throughlearning trajectory-based and…
1 3
With this learning trajectory in mind, variation pedagogi-
cal principles are applied to create various tasks surrounding
the invariant structure and quantitative relationship illus-
trated in Fig.1.
By further considering the content presentation in the
curriculum and the classroom teaching tradition in China,
this study is designed to explore how adapting the theories
of LT and VP could help students develop conceptual under-
standing and proficiency in solving CWP. In this context, the
study addresses the following research questions:
1. How could a theory-informed lesson design be imple-
mented in classrooms?
2. What do the students learn from the theory-informed
lesson implementation?
3 Methods
In this section, we describe the setting for the study, how the
research lessons were developed, the data collected, and how
the data set was analyzed.
3.1 The setting andthelesson study group
The study took place in an elementary school in a city in
southeastern China. This elementary school is typical
regarding economic and academic backgrounds of students.
Each grade has 6 classrooms with approximately 40 students
in each, and there are 81 teachers and 1620 students. Each
school in China has a school-based teaching research group,
which is part of an institutionalized, nationwide initiative
for improving teaching and teacher professional develop-
ment in China (Wang 2013). Chinese lesson study, a core
component of teaching research activities, is structurally
similar to Japanese lesson study (LS) (Lewis 2016), but it
differs in its emphasis on: (1) content or teaching strate-
gies, (2) knowledgeable others’ involvement throughout the
entire process of LS, and (3) repeated teaching of the same
lesson to different groups of students in order to develop an
“exemplary lesson” (Huang and Han 2015). In mathematics
education, teaching with variation is adopted in daily math-
ematics teaching either intentionally or implicitly (Gu etal.
2004). Moreover, any instructional theories could be adopted
as guiding principles for designing, enacting and analyzing
lessons during Chinese lesson study (Yang and Ricks 2013).
To explore how to implement research-based teaching
of CWP based on LT and VP, the first author discussed
the goals of the lesson study and previous experiences and
materials related to the research lesson with the second
author. The second author formed an expert team includ-
ing two university mathematics educators and one district
teaching research specialist. The expert team contacted
a local school to form a lesson study group consisting of
four first grade mathematics teachers and the leader of the
mathematics teaching research group in the school. One
of the teachers volunteered to teach the research lessons.
She earned an associate degree in elementary mathematics
education and a bachelor’s degree in education through the
continuous education program. She, with a professional
title of “primary school senior (Xiaoxue Gaoji) teacher”,
is regarded as an above average teacher by the researchers
and other teachers. She has 19years of experience and
has won several awards for excellence in teaching from
city educational administrations. The teacher taught the
research-based lessons of CWP three times through a typi-
cal Chinese lesson study approach (e.g., planning, teach-
ing/observing, debriefing/revising) (e.g., Huang and Han
2015).
3.2 Development ofthelessons
Comparison word problems appear toward the end of the
first grade textbook, and therefore they are taught near the
end of the year. Students being taught the research lessons
had learned relevant prerequisite knowledge including add-
ing and subtracting within 10, determining how many more
or how many less with concrete objects and pictorial repre-
sentations without number sentences, and solving one step
addition (adding to and part–part–whole) and subtraction
(taking away) word problems with pictures rather than num-
ber sentences.
Building on the exploratory study (Huang etal. 2017),
the research team drafted a learning trajectory and associ-
ated mathematics tasks to help the teacher develop lesson
plans for the two research lessons. The first lesson focused
on finding the difference between two given quantities. The
second was about finding the other quantity when given the
difference and one of the two quantities. With support of
the research team, the lessons were taught three times to
different groups of students through a lesson study approach.
The lessons were revised based on rehearsal teaching and
debriefing feedback. Through the process of enactment and
reflection, the research team attempted to address the follow-
ing three issues which arose during the lesson study process.
First, teachers need to be aware of students’ levels of under-
standing of various meanings of subtraction; students may
understand subtraction only as missing addend or taking
away. Second, at this early age, using manipulative activi-
ties and pictorial diagrams is critical for understanding the
structure of comparison problems. Third, getting answers by
drawing diagrams and by arithmetic equations are equally
important. These understandings about content and students’
learning informed the development of the final exemplary
lessons.

R.Huang et al.
1 3
3.3 Data collection
All lesson plans were collected. The final and third teach-
ing of the 2 lessons with 41 students was videotaped, and
students’ work was collected. To evaluate the effectiveness
of the teaching, an assessment instrument was developed,
which included seven relevant CWP. To minimize the effect
of the pre-test on the post-test, the corresponding problems
were reworded slightly regarding their context and numbers,
and the order of items was changed.
The pre-test was administered the day before the first les-
son, and the post-test was administrated the day after the sec-
ond lesson. Thirty-nine (39) students completed the pre-test,
and 41 completed the post-test. In addition, five purpose-
fully selected students (three above average, one average,
one below average based on pre-test) were interviewed to
elicit their interpretation of their solutions to the post-test.
3.4 Data analysis
The data set was analyzed in light of objects of learning.
First, the intended objects of learning were identified by
analyzing the learning goals of the lesson plans. Second,
the enacted objects of learning were analyzed through exam-
ining patterns of variation in the lessons. The patterns of
variation were identified by focusing on what was invariant
vs. what varied when exploring variation tasks. Finally, the
lived object of learning was analyzed based on post-tests and
students’ interviews.
Based on the objectives of instruction as indicated
explicitly in lesson plans, the intended objects of learning
are shown in Table1. For example, in lesson 1 (finding the
difference when comparing two given quantities), the first
object of learning (L1O1) is to make sense of finding “how
many more, and how many less” in terms of finding the
difference which involves finding the same part of the two
given quantities, (using manipulative activity). For another
example, in the second lesson (finding the other quantity
when given the difference and one quantity), the second
object of learning (L2O2) is to understand the structure
of comparison problems (i.e., “the smaller” + “the differ-
ence” = “the bigger”), and understand by using pictorial and
symbolic representations that finding the difference means
to identify the same part (i.e., the smaller quantity) of the
two quantities.
To identify the patterns of variation and invariance, all
tasks (including games, worked examples, and practice prob-
lems) were identified and the episodes surrounding explora-
tion of these tasks were analyzed. The patterns of variation
vs. invariance were identified with regard to the discern-
ment of critical features of the intended objects of learning.
Finally, 11 occurrences of patterns of variation vs. invari-
ance were identified as shown in Table2. For example, in
lesson 1 (V12), the teacher presented three interconnected,
varied tasks (T12a.b.c) for students to explore sequentially:
Task 12a. Yangyang1 (YY) has 6 stars; Lele (LL) has 5
stars. How many more does YY have than LL?
Task 12b. YY has 6 stars; LL has 4 stars. How many more
does YY have than LL ?
Task 12c. YY has 6 stars, LL has 3 stars. How many more
does YY have than LL?
The contextual situations of the three tasks, the bigger
numbers (comparison set), and relational terms (how many
more) are invariant, while the smaller numbers (referent
set) vary; and the difference co-varies with the referent set.
Thus it is possible for students to discern the quantitative
relation: How many more (HMM) = bigger (B) − smaller
(S) through experiencing these patterns of variation and
invariance (e.g., generalization). Based on the variation
Table 1 Intended objects of learning in the two consecutive lessons
Lesson 1: Finding the difference given two quantities being compared Lesson 2: Finding the other quantity given the difference and one
quantity
L1O1: Understand that finding “how many more, or how many less”
means to find the difference, and finding the difference means to com-
pare and ignore the same part (by using manipulatives)
L2O1: Understand that finding the difference is to compare and ignore
the same part of the two quantities and find the extra part; understand
the structure of the comparison problem (through a clapping hand
game)
L1O2: Understand that finding the difference of two quantities means
to split the bigger quantity into two parts: the same part and the extra
(different) part, then compare and ignore the same part, and finally get
the different part (by drawing diagrams)
L2O2: Understand the relationship (how many more or less) of
comparison problems and explore solution methods by drawings and
equations
L1O3: Create an abstract arithmetic equation to represent the solution
of a comparison problem, and fully understand what each part of the
equation means based on the structure of a given comparison problem
L2O3: Address learning difficulties in solving inconsistent language
comparison problems; understand when addition or subtraction
should be used to find a solution through practicing with variation
problems
1 The Chinese names of Yangyang (YY), Lele(LL), Xiaohong (XH),
Xiaoliang (XL) will be represented as their acronym YY, LL, XH and
XL hereafter in this article.

Developing students’ ability tosolve word problems throughlearning trajectory-based and…
1 3
theory of learning, experiencing this pattern enacts the fol-
lowing object of learning: understanding the structure of
finding how many more when given the bigger and smaller
numbers (L1O2).
In addition, the lived object of learning was investigated
by analyzing the students’ post-lesson learning outcomes
and selected students’ interviews. Moreover, to judge the
effectiveness of the lessons, a paired t-test between pre- and
Table 2 Mathematical tasks and patterns of variation
Mathematical tasks Pattern of variation
T11. Game: Comparing 4 stars and 3 stars V11: (Separation) Invariant: the same situation
Varied: Question (how many more, how many less, the difference), use
of manipulative or pictorial representation
T12a. YY has 6 stars. LL has 5 stars. How many more does YY have
than LL has?
T12b. YY has 6 stars. LL has 4 stars. How many more does YY have
than LL has?
T12c. YY has 6 stars; LL has 3 stars. How many more does YY have
than LL has?
V12: (Generalization) Invariant: the same situation and the same bigger
number, the same relational term (how many more)
Varied: Smaller numbers; verbal and pictorial representations
T13a: Ms. Wang has 3 stars. I have 6 stars. How many less does Ms.
Wang have than I do?
T13b: Ms. Wang has 5 stars. I have 6 stars. How many less does Ms.
Wang have than I do?
T13c: Ms. Wang has 7 stars, I have 6 stars. How many less do I have
than Ms. Wang?
V13: (Generalization) Invariant: the same situation, bigger number; the
same relational term (how many less)
Varied: size of smaller number; representations (verbal, pictorial, or
numerical)
T14: There are three types of pencils: 5 green, 10 red, and 8 blue (in
pictures). How many more red pencils than green pencils? Three
equations are provided: (1) 10 − 5 = 5, (2) 10 − 8 = 2, (3) 8 − 5 = 3;
Select a correct equation corresponding to the problem
V14: (Contrast) Invariant: structure of finding difference
Varied: finding the problem given an equation vs. finding equation
given the problem
T21: YY has 6 stars. LL has 5 stars. Fill in the following blanks: ( )
has ( ) many more than ( ); ( ) has ( ) many less than ( ); ( ) is the
difference between ( ) and ( )
V21: (Separation) Invariant: the same situation
Varied: different questions and answers
T22a: Ms. Wang clapped 6 times. I clapped 2 more than Ms. Wang.
How many times did I clap?
T22b: Ms. Wang clapped 6 times. I clapped 2 less than Ms. Wang.
How many times did I clap?
V22: (Contrast) Invariant: the same situation, the same referent set
Varied: More than vs. less than; representations (Physical, pictorial)
T23a: Ms. Wang clapped 6 times. I clapped 2 more than Ms. Wang.
How many times did I clap?
T23b: Ms. Wang clapped 6 times. I clapped 3 more than Ms. Wang.
How many times did I clap?
T23c: Ms. Wang clapped 6 times. I clapped 4 more than Ms. Wang.
How many times did I clap?
V23: (Generalization) Invariant: the same situation, the same smaller
number; how many more
Varied: Difference, diagram and equations
T24a: Ms. Wang clapped 6 times. I clapped 2 less than Ms. Wang.
How many times did I clap?
T24b: Ms. Wang clapped 6 times. I clapped 3 less than Ms. Wang.
How many times did I clap?
T24 c: Ms. Wang clapped 6 times. I clapped 4 less than Ms. Wang.
How many times did I clap?
V24: (Generalization) Invariant: the same situation, the same bigger
number; how many less
Varied: Size of smaller numbers, diagrams, and equations
T25a: Ms. Wang clapped 5 times. I clapped 2 more than Ms. Wang.
How many times did I clap?
T25b: Ms. Wang clapped 5 times. I clapped 2 less than Ms. Wang.
How many times did I clap?
V25: (Contrast) Invariant: the same situation with same referent set
Varied: How many more vs. how many less
T26: Given two equations: 5 + 2 = 7 and 5 − 2 = 3, and a situation:
there are 5 chickens, there are 2 more ducks than chickens, and 2 less
geese than chickens. Match the equations to the questions: (1) How
many ducks are there? (2) How many geese are there?
V26: (Fusion) Invariant: the same situation with same numbers
Varied: How many more vs. how many less; addition vs. subtraction
T27a: XH got 5 stars; XL got 2 more stars than XH. How many stars
did XL get?
T27b: XH got 7 stars; XH has 2 more stars than XL. How many stars
did XL get?
T27c: XH got 7 stars, XH has 2 less stars than XL. How many stars
did XL get?
V27: (Fusion) Invariant: the same situation, partly the same number
Varied: how many more vs. how many less; smaller vs. bigger

R.Huang et al.
1 3
post-test scores was performed to detect student gains. Each
item was worth 4 points based on the correctness of both the
diagram (2 points) and equation (2 points). If a diagram rep-
resented two quantities correctly, but failed to align so that
the same quantity matched, 1 point was given. If an equation
was partially correct in that the three numbers were correct,
but the equation did not correspond to the relationship pre-
sented in the problem, 1 point was given. Two graduate stu-
dents worked together to grade the pre- and post-tests. Any
disagreements were resolved through extensive discussion.
4 Results
The results are presented in three sections. The first focuses
on intended objects of learning of teaching CWP. The sec-
ond focuses on the enacted objects of learning (through
presenting the patterns of variation and invariance in the
lessons) and on major features of the two lessons. The third
presents lived objects of learning.
4.1 Intended objects oflearning ofthelessons
The intended objects of learning were identified by analyz-
ing the instructional objectives stated in the lesson plans.
These are summarized in Table1.
Thus, the overall goals of learning of these two lessons
are to help students develop understanding of how to solve
comparison problems of three types: finding the difference
given two quantities, finding the bigger quantity given the
difference and smaller quantity, and finding the smaller
quantity given the difference and bigger quantity. Three key
strategies are suggested, the first of which is using multiple
representations (physical, pictorial and symbolic). The sec-
ond is using the equivalence among the three statements:
A is x many more than B; B is x many less than A; and the
difference between A and B is x. The third is identifying
which quantity is bigger and using the invariant structure of
bigger = smaller + difference.
4.2 Enacted objects oflearning ofthetwo lessons
The results are presented in three parts. First, the major
phases of the two lessons are described briefly. Second, the
major patterns of variation and invariance are summarized.
Third, the relation between patterns of variation and invari-
ance and intended objects of learning is mapped.
4.2.1 Major phases ofthetwo lessons
The two lessons, each lasting about 46min, followed a
similar structure: (I) introduction to the new topics, (II)
exploration of the major topics through variation tasks
using multiple representations, (III) practice with variation
tasks, and (IV) reflecting and summarizing. While the major
phases and activities of the two lessons are described below,
all tasks used in the lessons are listed and analyzed in the
next section in detail.
Lesson 1 (L1): Learning trajectory Level 1 and sublevels
Playing a game (T10)(12’). Students were asked to com-
pare 3 stars and 4 stars through physical matching (three
pairs and one extra) in order to see “1 more” or “1 less”
and “1 difference” representing the same comparison situ-
ation. Matching to form a one-to-one correspondence was
emphasized.
Exploration of the structure and methods of finding the
difference problems (29’). Manipulatives and drawings were
used to solve the first set of three problems (T11) which
required students to find how many more (HMM) when
given the bigger comparison set (which remained invari-
ant) and the smaller referent set (which varied). The aim
was to help students see the structure of HMM = B − S (see
Fig.1) and the method of identifying the same smaller part
by matching and finding the extra part (difference). After
students showed their understanding of the structure and
method pictorially, they were asked to write and explain an
arithmetic equation to represent their solution. To enforce
the equivalence of how many more and the difference, while
keeping the situations and corresponding diagrams and equa-
tions invariant, students were asked to determine whether the
diagrams and equations would need to be changed if the
questions were reworded as “what is the difference”.
Practice with variation tasks (5’). A set of three ques-
tions (T12) asked students to determine how many less
when given two quantities. Students were once again asked
to physically match and explain, and then to write the equa-
tion. Once again the process was intended to emphasize that
the larger number was being split into two parts, namely,
the part that was the same as the smaller one, and the extra
part (the difference). Finally, a practice task (T13) was given
in which students matched a word problem with a relevant
solution equation.
Closing (1’): Students were asked to summarize what
they learned.
Lesson 2 (L2): Learning trajectory levels 2 and 3 and
sublevels
Review and introduction (9’). The teacher presented
Task 21 by rephrasing Task T12a in L1 to check students’
understanding from L1. The teacher then organized a game
called clapping hands in which the teacher clapped 6 times
and students were asked to determine the number of claps
they needed to have under these conditions: the same as the
teacher, 2 more than the teacher, and 2 less than the teacher.
Specific attention was paid to the same part and extra part.
Exploration of the structure and method for finding the
other quantity when given the difference and one quantity

Developing students’ ability tosolve word problems throughlearning trajectory-based and…
1 3
(22’). Based on the clapping hands game, a set of three prob-
lems (T23 a,b,c) was devoted to the exploration of tasks of
the following types. Given the smaller number (6 claps) and
how many more (2, 3, and 4), find the bigger number. The
focus was on drawing two rows of circles (a circle represent-
ing a clap) and a vertical line to split the extra two circles
and align the other 6 circles. After showing the diagram on
the screen, students were asked to write an arithmetic equa-
tion to represent the solution (6 + 2 = 8), and explain what
each number in the equation represented. After students
explained why all were solved using addition, the teacher
launched another set of three problems (T24 a, b, c) in
which students were given the bigger number (6 claps) and
how many less (2, 3, 4), and were asked to find the smaller
number. Similarly, students were asked to draw all 6 circles
in a row (representing claps) and cross out the two circles
representing 2 less, revealing 4 remaining circles. A verti-
cal line split the two parts: two extra circles and four same
circles. Again, students were asked to write the equation
(6 − 2 = 4) and explain the meaning of each number. Finally,
all the three problems, diagrams and equations were shown
on screen. Then, a lively discussion about why subtraction
was used to solve these three problems, but addition was
used to solve the previous three problems, was orchestrated.
Practice with variation tasks (16’). Students were asked
to solve two variation questions (T25 a, b) similar to those
just discussed (one needing addition and the other subtrac-
tion) by drawing diagrams, writing equations, and justify-
ing their answers. The teacher presented a sorting problem
(T26): identifying and justifying a solution equation or a
diagram to match with a given word problem. After the sort-
ing task, the students were asked to create a word problem
that could be represented by a given diagram and to write
the arithmetic equation.
Finally, a set of three problems (T27) was arranged to
flexibly apply various learned methods.
Closing (1’). Students were asked to summarize what
they learned.
4.2.2 Major enacted patterns ofvariation constituted
inlessons
Enacted objects of learning are described through the pat-
terns of variation that provide possible learning opportuni-
ties for students. A pattern of variation is constituted by
what varies and what is invariant. We identified patterns of
variation in the classroom based on the instructional tasks
and their implementation. These patterns of variation and
associated mathematical tasks are displayed in Table2.
Table2 shows how a pattern of variation and invari-
ance could be created through exploring a set of deliber-
ate tasks. For example, through exploration of the three
tasks with the same situation, the same bigger number, the
same relational term (how many more), while the smaller
numbers varied and difference co-varied, using multiple
representations (diagrams and equations) (V12), it is pos-
sible for students to discern the underlying structure of this
type of CMP: HMM = B − S (L1O2). Similarly, through
exploration of another pattern of variation V13 where
three tasks with the same situation, the same bigger num-
ber, the same relational term (how many less) while the
smaller numbers varied, and difference co-varied, using
multiple representations (diagrams and equations), it is
possible for students to discern the underlying structure of
this type of CMP: HML = B − S (L1O3). By merging these
two patterns, it is possible for students to discover that
finding the difference is equal to the structure between the
two quantities: D = B − S (L1O2&3). In the next section,
the relations between patterns of variation and intended
objects of learning are illustrated.
Fig. 2 The relationship between
objects of learning and pat-
terns of variation. B Bigger, S
smaller, D difference, HMM
how many more, HML how
many less L1O1:
Matching
physically,
pictorially;
Equivalence:
HMM
=HML
=D
L1O2:
Finding
D=B-S
Visually
L1O3:
Finding
D=B-S
visually/
symbolically
L2O1:
Matching
method;
HMM=
B-S;
HML=B-S
Physically/
pictorially
symbolically
L2O2:
HMM
=B-S;
HML
=B-S
visually/
symbolically
L2O3:
B=S+HM
M;
S=B-HML
Inconsistent
language
V11V26V27V12V25
V13V14V22V23
V21V24

R.Huang et al.
1 3
4.2.3 Alignment betweenpatterns ofvariation
andintended objects oflearning
The connections between the patterns of variation and the
intended objects of learning are shown in Fig.2.
This figure shows that the patterns of variation created
in these two lessons provided appropriate opportunities for
students to explore the intended objects of learning. In the
first lesson, the pattern V11 was constructed to reactivate
relevant previous knowledge for learning L1O2. V12 and
V13 were created to address L1O2&3, respectively. V14 was
constructed to consolidate the learned structures and meth-
ods. In the second lesson, V21 was created to help students
recall what they learned in the first lesson, paving the road
for addressing L2O1. V22 was created to address L2O1,
while V23 and V24 were created to address L2O2. V25 and
V26 were created to deepen the object of L2O2, while V27
was constructed to address L2O3. Thus, these enacted pat-
terns of variation were created to achieve objects of learning
sequentially and coherently.
In summary, the lessons demonstrated four salient fea-
tures. First, to build connections between what students had
learned and what they are going to learn, two patterns of var-
iation and invariance (V11 and V21) were created. Second,
to explore the meaning and numerical relations of CWP, the
teacher paid great attention to using multiple representations
including physical matching (clapping hands), verbal rep-
resentations, manipulative tools (stars), pictorial diagrams
(drawings), and symbolic equations. Third, to discern and
highlight the structure of each type of comparison problem
(e.g., V12, V13, V23, and V24), deliberatively designed var-
iation tasks were used. Fourth, these two lessons unfolded
based on the student learning trajectory: from finding the
difference of two quantities being compared, to finding
the bigger number given the smaller and the difference, to
finding the smaller number, and finding the other number
when given one number and the difference.
4.3 Lived objects ofstudent learning
The students’ performance on the post-test is displayed in
Table3.
The table shows that the three items with the top scores
are items 2, 3 and 4. These items are about finding the
smaller number given the bigger number and how many less,
or about finding how many more given two numbers. The
item with the lowest score is item 5, which is about finding
the difference (how many more or how many less) given
two numbers.
Overall, this group of students averaged 66% of the total
score (18.22 out of 28). It is very encouraging that the group
of students achieved 65% correctness (2.45 out of total score
4) on item 7 which research indicates (e.g., Verschaffel etal.
1992) is the most challenging. However, it is a surprise that
the lowest score was on item 5, which is about finding the
difference given two numbers because this was greatly
emphasized in the two lessons.
The interviews with five students provided further infor-
mation about their performance. Based on the pre-test, three
(S1, S2 and S3) were above average; one was average (S4);
and one (S5) was below average. Students missing points on
item 7 were S3 and S5. In the interview, S3, who scored 0
on this item, was not able to figure out her error: 11 − 8 = 3.
However, S5, who scored 3 on this item, gave clear explana-
tions of her drawing, but was not sure about the correspond-
ing equation(8 − 5 = 3 or 8 − 3 = 5). Students missing points
on item 5 were S4 and S5. Both students said they did not
understand what the difference means. However, when the
interviewer explained that it meant how many more or how
many less, they got the correct answer.
Table 3 Frequencies and means of items on post-tests (N = 39)
No. Item Frequencies of scores on items
(0–4)
0 1 2 3 4 Mean (std.)
1 In the figure below, there are 4 boys and 1 girl, how many more boys are there than girls? 1 14 5 7 12 2.38 (1.33)
2 In a fishpond, we caught 8 fish yesterday. Today, we catch 1 less than what we caught yesterday. How
many fish do we catch today?
1 2 8 10 18 3.08 (1.06)
3 Xiaoming made 7 paper boats. He made 3 less paper airplanes than boats. How many paper airplanes did
Xiaoming make?
0 6 6 11 16 2.95 (1.11)
4 There are 11 apples and 15 pears; which number of apples or pears is more? How many more? 1 5 7 15 11 2.77 (1.09)
5 Xiaoming ate 2 pieces of chocolate; Liang ate 5 pieces of chocolate. What is the difference between
them?
2 12 8 9 8 2.23 (1.25)
6 Xiaoming has 3 more post-it notes than Xiaohong. Xiaohong has 4 post-it notes. How many post-it notes
does Xiaoming have?
3 5 8 17 6 2.46 (1.14)
7 There are 8 apples. There are 3 more apples than pears. How many pears are there? 6 6 5 8 14 2.46 (1.50)

Developing students’ ability tosolve word problems throughlearning trajectory-based and…
1 3
Moreover, the paired t-test (2-tailed) showed a statisti-
cally significant increase from pre- to post-test in means on
Items 1 (t = 4.12, p = 0.00), 2 (t = 8.84, p = 0.00), 3 (t = 2.91,
p = 0.06), 4 (t = 7.18, p = 0.00), 5 (t = 2.25, p = 0.03), and the
total score mean score for all items was also statistically sig-
nificant (t = 8.61, p = 0.00). However, the increased scores on
item 6 (t = 1.62, p = 0.11) and on item 7 (t = 0.81, p = 0.41)
were not significant.
In summary, students gained a significant improvement
of understanding of comparison word problems through the
two lessons, and demonstrated their understanding at vari-
ous levels. Interestingly, the inconsistent language problem
is not the big obstacle of solving CWP, but understanding
the equivalence of rewording of how many more, how many
less and the difference is still confusing.
5 Discussion andconclusion
This study demonstrated that the combination of learn-
ing trajectory and variation pedagogy can be used as an
instructional guiding principle for designing and teaching
CWP. More specifically, it revealed that if the lesson design
is based on a specific learning trajectory of CWP and sys-
tematic variation tasks surrounding the objects of learning,
and is implemented through students’ active engagement
in making sense of mathematics problems and articulating
their thinking using multiple representations, it can promote
students’ understanding of the meanings and structures of
CWP, and develop their procedural fluency as well. In par-
ticular, we further discuss the three unique contributions of
this study.
5.1 Learning trajectory asahypothetical route
ofstudents’ learning
Based on literature, originally the design of teaching com-
parison word problems was based on three broad levels of a
learning trajectory (Riley and Greeno 1988). However, based
on the teaching tradition in China, the learning trajectory
was refined by including more levels such as the following:
Level 1 Finding the difference given the two quantities.
Level 2 Finding the bigger number given smaller number
and how many more (consistent language).
Level 3 Finding the smaller number given the bigger num-
ber and how many less (consistent language).
Level 4 Finding the other quantity (smaller or bigger)
given one of the quantities and the difference (inconsistent
language).
The refined trajectory may draw learners’ attention to
the following: (1) Identifying which quantity is bigger or
smaller; (2) Understanding the equivalence of the three state-
ments: “A is x more than B”, “B is x less than A”, and “The
difference between A and B is x”; and (3) Overall structure
of comparison problems: The bigger = The smaller + the dif-
ference. Each of these aspects is crucial for understanding of
and effective solution of comparison word problems (Briars
and Larkin 1984; Fuson etal. 1996; Stern 1993). The teacher
in the study explicitly and repeatedly stressed these aspects
throughout the lessons.
5.2 Variation tasks alongwiththelearning
trajectory asvehicles
Teaching with variation has long been a mathematics teach-
ing tradition that promotes students’ meaningful learning
in large classrooms in China (Gu etal. 2017). Regarding
variation tasks, there are systematic strategies of varying
problems (Sun 2011). Exploration with variation tasks has
the goal of conceptual understanding and developing inter-
connected knowledge structures and problem solving ability.
One critical feature of using variation tasks is to explore and
discern mathematical connections and structure (Gu etal.
2017). In these lessons, a hierarchical approach was adopted
to explore the structure. First, to explore the individual struc-
ture of comparison word problems, a set of variation tasks
was explored purposefully to reveal the feature of that struc-
ture. For example, in lesson 2, the set of variation task T23
(see Table2) kept the same situation and the same smaller
number, as well as the invariant structure of the comparison
problem (smaller number + how many more = larger num-
ber), while varying the larger number. Second, to discern
the general structure of comparison word problems, a new
set of variation tasks was presented for students to explore
in lesson 2, task T27 (see Table2). In this set of problems,
the problem situation remained the same, and the numbers
of quantity and difference were invariant, but the word-
ing of difference and positions of unknown variable was
changed. Exploration of this set of problems intentionally
leads students to focus on the general structure of a com-
parison problem. The key is to determine which quantity is
bigger or smaller, what is the difference, and what operation
is applicable.
5.3 Multiple representations andflexibility ofusing
representations
Use of multiple representations is widely emphasized in
mathematics learning and teaching (NCTM 2000; Lesh etal.
1987). Research shows that US teachers put more emphasis
on pictorial representations than Chinese counterparts, while
Chinese teachers value symbolic representations more (Cai
and Lester 2005). Realizing the limited use of representa-
tions by Chinese teachers, the teacher in this study put forth
great effort to use and connect various representations (Lesh
etal. 1987). In particular, the connection and translation

R.Huang et al.
1 3
between different representations was purposefully empha-
sized. For example, in lesson 1 in task T11, students were
asked to play, draw diagrams, and articulate their thinking
and solutions, but in task T12, students were asked to write
mathematical equations based on their previous experiences
with the physical, visual and oral representations. Some-
times students were not allowed to use manipulatives, rather
they were asked to imagine and draw the diagram and write
relevant equations. At the later stage, students were asked
to translate between different representations, finding cor-
responding problems given equations, and creating prob-
lems given a structural diagram. Throughout the two lessons,
verbal, visual and symbolic representations were always
presented simultaneously. This may help develop student’s
flexibility in using multiple representations.
5.4 Limitation andsuggestions
This study demonstrated the feasibility and positive effects
of adopting the notions of learning trajectory and variation
pedagogy. But it should be noted that these lessons are not
normal class teaching; rather they are the product of Chi-
nese lesson study guided by a team of university faculty
and a teaching research specialist. Chinese lesson study is
a system-wide, job-embedded professional development
approach (Huang and Han 2015). Thus, improving teaching
and developing teachers through Chinese lesson study is rou-
tine. This study also showed that students’ performances in
different types of comparison problems are disparate. A deep
analysis of underlying reasons needs to be conducted. In
addition, exploration of the implementation of this approach
at scale and in different countries should be an interesting
endeavor to pursue.
5.5 Conclusion
This study showed that teaching CWP based on the notions
of learning trajectory and variation pedagogy could help
students understand the structure of CWP and develop prob-
lem solving skills. The study not only supports the existing
research findings about the difficulty of learning CWP and
strategies for addressing these difficulties, but also makes a
significant contribution to the area. First, the learning trajec-
tory of CWP was tested and refined, which could be used for
further research studies. Second, a set of systemic problems
and various teaching strategies illustrated in the study could
be valuable resources for practicing teachers.
Acknowledgements We extend our thanks for the strong support from
the participating school, Qiaotou no. 2 Elementary School of Yongjia
County, City of Wenzhou, and especially to teaching research spe-
cialist Ms. Yuxiao Nan and mathematics teacher Jing Huang for their
intellectual contribution to improving the design and teaching of the
comparison word problem lessons.
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