IAFOR Journal of Education

Volume 6 – Issue 3 – Winter 2018

The Effects of Using Representations in Elementary Mathematics: Meta-Analysis of
Research
Andrzej Sokolowski
Lone Star College, Houston, Texas, USA

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Abstract
The current study provides a meta-analysis of global research on using representations to
support the learning of mathematics in Pre-K through Grade 5. A total of 13 primary studies
encompassing 1,941 subjects was analyzed. The weighted mean effect size for the 13 studies
was reported to be ES = 0.53 (SE = 0.05). A 95% confidence interval around the overall mean
– Clower = 0.42 and Cupper = 0.63 – proved its statistical significance and its relative precision.
The calculated effect size signifies strong, robust support for the use of representations in PreK through Grade 5 mathematics classes and highlights the importance of providing students
with opportunities to construct and explore transitioning between various forms ofrepresentations. Moderator analysis revealed differences among the effects due to a different
type of representation, grade levels, and concepts taught. A synthesis of moderator effects
allowed for a formulation of a general way of applying representations that produces maximum
learning effects and that the teachers can adopt in their school practice. While the effect sizes
provided a means of determining the most effective ways of applying representations,questions about how to develop students’ transitioning from one representation to anotherremain unsolved. A further discussion of the impact of the study findings beyond theboundaries of elementary mathematics classrooms follows.
Keywords: meta-analysis, mathematical representations, internal and external representations,mathematics teaching and learning, elementary school130

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Introduction
It is well documented that reading and understanding a mathematical context requires
embodying various abstract entities such as symbols, graphs, and tasks encoded symbolicallyin a language that students can comprehend. Given that elementary students are at the stage ofdeveloping their abstract thinking, designing effective teaching strategies that would allow
such communication is not an easy task for mathematics educators and curriculum designers
thus attempts are made to make the process more accessible for the learners. Research findings(e.g., Clark & Mayer, 2011) show that learning mathematical objects and the development ofcorresponding mental images are linked. On the other hand, imagination and the ability toconstruct, retrieve and explore internal representations form foundations for learning ofmathematics (Lingefjärd & Ghosh, 2016). The ultimate question that was posited in this study
was what representations are the most accessible to an elementary math student.
Researchers (e.g., Hoffler & Leutner, 2007) have determined that people learn more deeply
from words supported by graphics than from words alone. This finding corresponds with themodern view on mathematical learning, which claims that utilizing multiple representations
and making connections between graphical, symbolic, and verbal descriptions of mathematical
relationships will empower and simultaneously help students develop a deeper understandingof mathematical relationships and concepts (National Council of Teachers of Mathematics[NCTM], 2000; Porzio, 1999). Following this notion, a general agreement exists that using
different mathematical forms of representations and translating between these forms, are key
skills in mathematics (e.g., Ainsworth, Bibby, & Wood, 2002). In order to respond effectively
to learners’ perception, further research is necessary. Nistal, Van Dooren, & Verschaffel (2012)noted that there was a need for research that would focus especially on the contextual factorsthat promote flexible representation choice for students in mathematics. It was hoped that thisstudy would shed more light into this area.
Representations, especially their graphical forms, can also be perceived as learning experiencesthat are transmitted to the learner by pictorial media (Clark & Mayer, 2011). As such, they help
the learner identify meaningful pieces of information and link the information with thelearner’s prior experiences. Although the constructs of using diverse forms of representations
to enhance the development of mathematical concepts and problem-solving techniques havebeen widely researched (e.g., Jitendra, Star, Rodriguez, Lindell, & Someki, 2011; WeberRussell & LeBlanc, 2004), a formal meta-analysis in this domain was not found using standard
library search engines. Students’ early experiences with the content of mathematics might havea profound impact on their further engagement and success in the subject. Therefore this study
emerged to fill the gap and to provide a broader view of using representations to support thelearning of mathematics in elementary school.
Theoretical Background
Representations and Constructivist Learning Theory
The effect of using representations is not new to the mathematics education community.
However, it has recently attracted more attention by being supported by the constructivist
learning theory that leads contemporary research in education (Cuoco, 2001). By treating
mathematical concepts as objects, and thus embodying them with representations that can beobserved and manipulated, a construction of mental pictures in the students’ minds can beevoked (Dubinski, 1991). Such constructed mental pictures are stored in students’ long-term

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memory and are available for retrieval. Research (Zazkis & Liljedahl, 2004) suggests that oneof the ways to induce the process of converting concepts to objects is to create representations
and act on them. Clark and Mayer (2011) suggested that knowledge acquisition is based on thefollowing principles of learning: dual channel – people have separate channels for processing
visual/pictorial material and auditory/verbal material; limited capacity – people can actively
process only a few pieces of information in each channel at one time; and active processing –learning occurs when people engage in appropriate cognitive processing such as attending torelevant material and organizing the material into a coherent structure. Active learning appears
as a method that supports the linkage of external representations with internal images.
Human cognitive architecture (Paas, Renkl, & Sweller, 2003), states that the most crucial
structures affecting the rate of information processing are working memory and long-term
memory. Human working memory has limited capacity as opposed to long-term memory,where capacity is unlimited (Kintch, 1998). For the information to be stored in a learner’s longterm memory, it needs to be processed initially through its working stage. Being presented withcomplex information, the learner might feel overwhelmed, which might result in theinformation not being fully processed. This state will consequently block the information fromreaching the learner’s long-term memory and from being learned and accumulated. The
primary goals of using representations are to reduce the contextual load by converting theinformation to a visual form and to transmit such information to the learner’s visual channel.
This process in return will reduce the need for high capacity of working memory. The virtue
of using representations is rooted in their capacity to present the knowledge of conveyablegraphical embodiments supported by verbal elaborations rather than vice versa. Suchknowledge presentation creates appealing conditions for being longer retained and accessible
for future usage.
Representations in Mathematics
Representations are broadly defined as passive entities. By learner’s active engagement, they
are transformed into active semiotic resources (Thomas, Yoon, & Dreyfuss, 2009) and can be
stored in a learner’s long-term memory. Knowledge externalized by graphics is easily
retrievable for analysis and can be readily exhibited and communicated (Ozgun-Koca, 1998).
Representations as a means by which individuals make sense of situations (Kaput & Roschelle,1997) can be expressed in forms of combinations of written information on paper, physicalobjects, or a carefully constructed arrangement of thoughts. Schnotz (2002) emphasizes thedistinction between descriptive (symbolic) and depictive (iconic) representations. While
depictive representations are most useful to provide concrete information and are often
effective as specific information, descriptive representations usually express abstract
information. Duval (2006) claimed that using various representations in mathematics classes
is a necessity because only multiple external representations allow learners to utilize thedifferent advantages each representation offers. Falcade, Laborde, & Mariotti, (2007) claimed
that the link between external representations and internal representations goes beyond pureanalogy in their functioning and rests on the real tie that can be recognized between particular
tools (external representations) and particular signs (internal representations).
Each representation of a mathematical object brings some aspects to the fore, simultaneously
hiding other aspects of the object and thus affecting the way the object is seen (Laborde, 2007).
Representations can also be used to explore aspects of a context that might otherwise not beapparent to a learner; they amplify properties of mathematical structures not easily imaginable(Monk, 2003). In the process of knowledge accumulation, representations are converted into132

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internal images. Mediated by the level of entry into learners’ memory systems, representations
are categorized as external or internal. Both types of representations are interrelated in the sensethat the meaning of an internal representation stored in a learner’s long-term memory strongly
depends on the learner’s perception of its external counterpart.
External representations (see Figure 1) encompass physically embodied, observableconfigurations – such as pictures, concrete materials, tables, equations, diagrams, and drawingsof one-, two-, or three-dimensional figures or various forms of schemata (Jitendra et al., 1998).

A. Pictorial

B. Schemata

C. Concrete

Figure 1. Examples of representations using in elementary school.
(Source: https://www.google.com)
All these embodiments can be provided in the form of drawings or can be digitalized bycomputer programs. They can also be generated by the instructor as he/she introduces therepresentations to the learners. External representations also encompass dynamic graphics,
which are generated with the help of technology, for example, graphing calculators orcomputer-based simulations (Goldin & Shteingold, 2001). Ainsworth and Van Labeke (2004)
categorize external representations as time-persistent representations, time-implicit
representations, and static representations. In mathematical terms, time-persistent
representations are embodied by algebraic functions, time- implicit by relations and static
representations would encompass any drawings that students produce, not necessarily placing
their products in a coordinate system.
Eisenberg and Dreyfus (1991) noted that students might end up with an incorrect solution if
their algebraic skills are not strong; however, if they possess the skills to graphically solve theproblem or support its solution process, the graphed representation might serve as a backup or
a way of solution verification. Being exposed to mathematical representations, learners“acquire a set of tools that significantly expand their capacity to model and interpret physical,
social, and mathematical phenomena” (NCTM, 2000, p. 4). In this regard, external
representations can also serve as a means to overcome students’ misconceptions in scienceclasses (Thompson & Logue, 2006).
Internal representations encompass mental images corresponding to internal formulations ofwhat human beings perceive through their senses and as such they cannot be directly observed.
Internal representations are defined as the knowledge stored in a learner’s long-term memory.
Internal representations are formulated based on one’s interaction with the environment(external representations) and are altered throughout a lifespan. In the process of learning,
external representations prompt the emergence of internal representations. Being able toformulate concepts’ internal representations through the process of understanding theirexternal embodiments and retrieving the mental pictures plays an essential role in133

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communicating messages in mathematics. Hiebert and Carpenter (1992) maintained thatlearners establish a strong relationship between created external and internal representations
and that the strength of linking these representations determines students ‘understanding.
Furthermore, internal representations of the knowledge accumulated through experiencing
visual representations produce stronger impulses in learners’ long-term memory. Enablingthese experiences by engaging and intellectually stimulating learners through carefully
designed learning environments supported by representations deems to be a significant factor
in nurturing effective learning and developing students’ mathematical knowledge. Nitsch et al.
(2015) found that that to understand the concept of function, that is central in mathematics
curriculum, it important not only to know the different mathematical representations offunctional dependency, but also the translations between these forms of representations. For
students to develop a holistic understanding of the concept of mathematical functions, they
have to be able to identify the connecting elements and to combine these representations.
Synthesis of Prior Research
As the constructivist theory strongly supports the use of representations in the learning process,several research studies have explored the effects of using representations on students’ mathconcept understanding. These results converge with contemporary theories of cognitive load
and multimedia learning principles developed by Clark and Mayer (2011) and have practical
implications for mathematical instructional designs. A meta-analysis of 35 independent
experimental studies conducted by Haas (2005) shed light on using representations as a meansof supporting teaching methods at the secondary school level. Haas concluded that math
instruction, supported by multiple representations, manipulatives, and models, produced a high(ES = 0.75) effect size. Schemas, which are defined as generalized representations that link
two or more concepts are frequently researched at the Pre-K through fifth-grade level. For
example, Jitendra and colleagues (1998) found that having students of Grades 2-6 categorize
problems and then having them solve the problems by using schemas produced a positive
medium-size learning effect (ES = 0.45). The virtue of using representations embodied byschemas is that they are easily converted by learners into internal representations, and, as such,they can be stored in long-term memory and allow for treating diverse elements of information
regarding larger, more general units (Kalyuga, 2006). According to Pape and Tchoshanov(2001), schematic representations also lead to enhanced student problem-solving performance.
Another group of researchers investigated whether representations should be provided to
students or if the students should be the producers of representations (e.g., De Bock,
Verschaffel, Janssens, Van Dooren, & Claes, 2003; Rosenshine, Meister, & Chapman, 1996).
These scholars concluded that if representations are provided, their forms must be sufficiently
informative and detailed to be transferrable by students into mathematical algorithms. They
also emphasized that having students construct their representations benefits the learners themost. The importance of possessing the ability to transfer a given context (e.g., a story problem)
into a representation was highlighted by Jonassen (2003), who claimed that successful problem
solving requires the comprehension of relevant textual information along with the capacity tovisualize that data and transfer it into a conceptual model. Following Riley, Greeno and Heller(1984), developing students’ abilities to identify the matching representation that helps withproblem conceptual understanding should emerge as a priority of elementary mathematics
teaching.

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Representations are also used to support the introduction of new mathematical concepts. For
example, several studies (Tzur, 1999) were conducted on the development of students’
notations of fractional parts of areas, called fair sharing, which provided a meaningful
representation of dividing a whole into parts that were then easily comprehended by elementary
students. Hiebert (1988) noted that students’ understanding of new ideas strongly depends onthe degree to which the learners are engaged in investigating the relations between new
representations and the representations whose understanding is already mastered. A study
conducted by Ross and Willson (2012) not only supported the claim that mathematics students
learn more effectively when instruction focuses on using representations but, moreover, provedthat the most effective strategies for building representations are those rooted in constructivist
learning theory. The range of using representations in Pre-K through fifth grade is wide, thus
synthesizing the experimental research findings and identifying the most effective strategies
manifests as a worthy undertaking.
Challenges of Inducing Representations in Pre-K through Grade 5
Investigating the effect of using representations has recently attracted more attention due tobeing supported by the constructivist learning theory that leads contemporary research ineducation (Cuoco, 2001). Such constructed mental pictures are stored in students’ long-term
memory and are available for retrieval. Research (Zazkis & Liljedahl, 2004) suggests that oneof the ways to induce the process of converting concepts to objects is to create representations
and act on them. Sfard (1991) concluded that the process of transferring abstract mathematicalconcepts into their mental images is challenging for both the learner and the instructor, who is
to guide the learner through the transferring processes. What are the challenges faced byelementary school children as they attempt to embody mathematical structures into visual
representations?
Equations and their conceptualization are frequently investigated in K-5 mathematics research.
Swafford and Langrall (2000) noted that students generally can make use of various
representations and can identify patterns between isolated variables, but they cannot findconsistency among a larger set of variables or generalize the patterns and convert them into
mathematical forms. Dreyfus (1991) suggested four learning phases with representations: using
one representation; (using more than one representation; (making links between parallel
representations; and integrating the representations. Representations at the elementary schoollevel encompass general structures used in mathematics such as ratio, rate, percent or newly
developed schemata for problem-solving, thus pinpointing and understanding how to uncoverthese principles acts as a catalyst for selecting correct representation. According to Swafford
and Langrall (2000), the emphasis in the curriculum at the pre-algebra level should be on
developing and linking multiple representations to generalize problem situations. They
concluded that the lack of generalization skills is rooted in instruction focused on reaching only
the initial stages of problem analysis and leaving the process of generalization for the students
to formulate. A similar conclusion was reached by Deliyianni, Monoyiou, Elia, Georgiou, andZannettou (2009), who observed that first-graders restricted themselves to providing uniquesolutions even though the questions required a general pattern formulation. Other researchers(e.g., Lesh & Harel, 2003) have shown that elementary school children bring potent intuitions
and sense-making tools, yet how to mediate these intuitions with abstract math concepts toembody these concepts into representations is a challenge still facing the mathematics researchcommunity.

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Research Methods
Meta-analysis, with its quantitative methods, was used to compute the research findings. Metaanalytic techniques provide tools to assess the learning effect size of treatments, considering
a gathered pool of studies as a set of data collected within prescribed criteria. By allowing
the measurement of the effect sizes according to the population of participants in primarystudies, such undertaking allows for analyzing a larger number of studies that can vary by
population sizes and also by the conduct (Gijbels, 2005). Furthermore, meta-analysis allows
also for employing subgroup moderator analysis and extracting factors that contribute to themagnitude and direction of the mean effect size.
Research Problems
Based on the prior research, a hypothesis suggesting that using representations in mathematics
helps students comprehend abstract mathematical concepts and enhances the skills of theconcepts’ applications emerged for this study. Understanding the degree to which
representations help learners comprehend the different mathematics entities, compared to
traditional methods of instruction, constituted the main objective of this study and guided theresearch questions:
1. What are the magnitude and direction of the learning effect sizes of using
representations in Pre-K through fifth-grade mathematics when compared to
traditional teaching methods?
2. What are the possible moderators that affect students’ achievement and whatclassroom settings produce the most optimal learning effect sizes whenrepresentations are used?
3. What are the features of the most effective representations in mathematics suitablefor Pre-K through Grade 5 levels?
It is hoped that the answers to these questions will advance the knowledge of using
representations and assist math curriculum policymakers to design effective learning materials.
Data Collection Procedure
This meta-analysis sought to encompass 12 years of global research on using representationsin Pre-K through fifth-grade mathematics, with student groups ranging in age from 3 to 12, inboth public and private schools, with a minimum sample size of 15 participants. In the processof collecting the applicable research studies, ERIC (Ebsco), Educational Full Text (Wilson),
Professional Development Collection, and ProQuest Educational Journals, as well as Science
Direct, Google Scholar, and other resources available through a university library, were used
to identify relevant studies published between January 1, 2000, and December 31, 2012. While
extracting the relevant literature, the researchers used the following key terms: graphicalrepresentations, mathematics education, primary, students, and experimental research. Inorder to broaden the search, the terms graphics, visualization, and problem-solving were also
used. Such defined criteria returned 131 papers, out of which 13 satisfied the conditions formeta-analysis (13 effect sizes). Several studies, although providing valuable findings, were
rejected due their qualitative type (e.g., Castle & Needham, 2007) or due to their focus oncomparing the effects of using representations that did not contain control groups (e.g., CoquinViennot & Moreau, 2003).

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Coding Study Features
The main construct under investigation was the learning effect of using representations in PreK through fifth-grade mathematics classes. While some of the characteristics, for example,
year of study conduct, locale, or type of research design, were extracted to support the study
reliability, others, like grade level or intervention type, were extracted to seek possible
moderators. Following are the descriptions of these features that were further aggregated toapply a subgroup moderator analysis.
Grade. This variable described the grade level of the group under investigation and referred togroups ranging from kindergarten to Grade 5.
Descriptive parameters. Descriptive parameters encompassed the locale where the studies
were conducted, the date of publication, and the sample size representing the total number ofsubjects under investigation in experimental and control groups.
Publication bias. All studies included in this synthesis were peer-reviewed and published as
journal articles; thus, no additional category for publication was created.
Group assignment. This categorization refers to the mode that was used to select and assign
research participants to treatment and controlled groups. Two main groups were identified: (a)randomized, where the participants were randomly selected and assigned to the treatment andcontrol group; and (b) quasi-experimental, where the participants were assigned byadministrator selection. This categorization is aligned with Shadish, Cook and Campbell’s(2002) definitions of group assignment.
Type of research designs used in the meta-analysis. Only pretest-posttest experimental
studies with control groups were synthesized.
Intervention. The intervention (treatment approach) was classified into four categories
reflecting the type of representations used in Pre-K through fifth-grade mathematics as definedby Swing, Stoiber, and Peterson (1988) and Xin and Jitendra (1999): pictorial (e.g.,
diagramming); concrete (e.g., manipulatives); mapping instruction (e.g., schemata based); andother (e.g., storytelling, keywords).
Output assessment. This variable described the assessment instrument and indicated whether
the assessment was developed by the researcher or was standardized.
Data Analysis
The following analysis is organized deductively. It begins by describing the general study
characteristics, moves to discuss the mean effect size, and concludes by presenting subgroupmoderator computations. Such established sequence follows the order of the study researchquestions.
General Study Characteristics
The summary of the study characteristics extracted from the pool of experimental pretestposttest studies is presented in Table 1.

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Table 1. Tabularization of experimental pretest-posttest study features
Authors

Date

Locale

RD

SS

GL

IRT

Alibali, Phillips, & Fischer
Van Oers
Poland, Van Oers, &Terwel
Xin, Zhang, Park, Tom,Whipple, & Si
Booth & Siegler
Csikos, Szitányi, &KelemenGamo, Sander, &Richard
Terwel, Van Oers, Van Dijk,&Van den Eeden
Casey, Erkut,Ceder, & Young
Jitendra, Griffin, Haria,
Leh, Adams, & Kaduvettoor
Fuchs, Fuchs, Finelli, Courey,& HamlettSaxe, Taylor, McIntosh, &Gearhart
Fujimura

2009
2010
2009
2011

USA
Netherlands
Netherlands
USA

QE
QE
QE
QE

91
239
54
27

3rd & 4th
4th
2nd
4th

Pictorial
Pictorial
Schemata based
Schemata based

2008
2012
2010
2009

USA
Hungary
FranceNetherlands

R
QE
QE
R

52
244
261
238

1st
3rd
4th & 5th
5th

Pictorial
Pictorial
Schemata based
Pictorial

2008
2007

USA
USA

QE
QE

76
88

Pre-K
3rd

Other (storytelling)
Schemata based

2004

USA

R

436

3rd

Schemata based

2005

USA

QE

84

4th & 5th

Pictorial

2001

Japan

R

51

4th

Concrete

Note. SS = sample size, GL = grade level, RD = research design, QE = quasi-experimental, R = randomized,
IRT = intervention representation type.

The data revealed that there is substantial diversity in the representations used in elementary
mathematics classes, ranging from schemas supporting problem-solving to storytelling
supporting operations on fractions. The majority of the studies (nine, or 69%) were quasiexperimental, while four (31%) were randomized. Regarding grade, a dominating group of six
studies was represented by fourth grade. Because problem-solving dominates the math learning
objectives in K-12 math education, how representations help students improve their problemsolving techniques emerged as a possible moderator of the study. Considering the type ofintervention, pictorial (six studies, or 46%) and schemata based (five studies, or 38%)
dominated the pool.
The Mean Effect Size and its SignificanceThe quantitative inferential analysis in the form of a meta-analysis was performed on pretestposttest experimental studies. The outcome variable, defined as the overall effect size of using
representations in mathematics teaching was sought in this meta-analysis. Student achievementscores were further expressed as effect size computed using mean posttest scores ofexperimental and control groups and coupled standard deviation using Hedge’s (1992) formula.
For the meta-analytic methods to be applied, the responses for the experimental studies werestandardized, and the accuracy of the effect sizes was then improved by applying Hedge’s(1992) formula:x -x
g= 1 2s*
In this formula; x1 represents the posttest mean score of the treatment group, x2 represents
the posttest mean score of the control group, and s * represents pooled standard deviation.
This process allowed the elimination of the sampling bias (Lipsey & Wilson, 2001).

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The overall weighted mean effect size for the 13 primary studies (13 effect sizes) was reported
to have a magnitude of 0.53 (SE = 0.05) and a positive direction. A 95% confidence intervalaround the overall mean – Clower = 0.42 and Cupper = 0.63 – indicated a nonzero population
effect and its relative precision (Hunter & Schmidt, 1990). According to Lipsey and Wilson(2001), an effect of 0.53 is of medium size. Herein, the overall effect’s magnitude along with
its positive direction indicated that the score of an average student in the experimental groups
was 0.53 of a standard deviation above the score of an average student in the control groups.
By incorporating the U3 Effect Size Matrix (Cooper, 2010), the average pupil who was taught
mathematics structures using representations scored higher on unit tests than 70% of students
who were taught by traditional methods. Thus, it can be deduced that using representations in
the teaching of mathematics, as a medium supporting instruction, has a profound impact on
students’ math concept understanding when compared to conventional methods of teaching.
Therefore, contextualizing math ideas and letting students embed math operations in contexts
meaningful to them has a positive effect on storing the ideas in their long-term memory. Table
2 provides a summary of the individual effect sizes of the meta-analyzed studies along with
their confidence intervals and qualitative findings.
Table 2. Effect sizes of using representations in Pre-K through Grade 5
Study(FirstAuthor)ES

SE

95% CI
Lower Upper

Alibali(2009)

0.92

0.22

0.19

1.05

Van Oers(2010)

0.23

0.13

0.36

0.89

Poland(2009)

1.22

0.29

0.04

1.23

Xin
(2011)

0.60

0.39

-0.19

1.44

Booth
(2008)

0.20

0.28

0.05

1.19

Csikos
(2012)

0.62

0.13

0.36

0.88

Gamo
(2010)

0.61

0.14

0.34

0.91

Research Findings
The strategy of representing theprocess of equalizing equations
improved problem representation
techniques.
Children improved fraction
understanding when they wereallowed to construct own
representations guided by theteacher.
Introducing dynamic
schematizing improved
understanding of the concept of
the process during problemsolving.
Conceptual representations
helped students learn the processof problem solving.
Providing accurate visual
representations of the magnitudesof addends and sums increased
children’s computational skills.
Presenting word problems withdifferent types of visualization(e.g., arrows) improved
techniques of problem solving.
Mapping data into graphicalrepresentations helped students
with problems involving
fractions.

139

Source of Assessment

Researcher designed.

Researcher designed.

Researcher-createdschematizing test.

Used textbook items
adopted by the districts;
Cronbach ’s alpha = 0.70.
Wide Range Achievement
Test–Expanded (WRAT–
Expanded).
Test items adopted from
National Core Curriculum;
Cronbach’s alpha = 0.83.
Researcher designed.

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Terwel(2009)

0.41

0.13

0.36

0.88

Casey
(2008)

2.00

0.31

0.38

2.63

Jitendra(2007)

1.36

0.22

-0.12

1.07

Fuchs
(2004)

0.22

0.19

0.26

0.99

Saxe
(2005)

0.33

0.22

0.18

1.07

Fujimura(2001)

0.71

0.29

0.05

1.20

Having students learn to design
representations helped them bringmore model-based knowledge tothe structure of mathematics
problems.
Representing geometry concepts
in a story context improved mathknowledge retention.
Addition and subtraction: usedgraphics to support multiple
representations.
The applied schema for problemsolving improved students’
algorithmic outcomes.
Percent: represented fractions
with standard part-to-whole
representations.
Highlighting the idea of physicalunits in setting the proportions
improved students’ conceptual
understanding.

Researcher developed
criteria; Cronbach’s alpha= 0.76.
Used Kaufman-Assessment
Battery for Children (KABC; Kaufman &Kaufman, 1983).
Used Pennsylvania System
of School Assessment mathtest.
Researcher-developed.
Researcher-developed.

Researcher developed;
interrater agreement 97%(N = 76).

Note. ES = effect size, SE = standard error.

Calculated confidence intervals (CIs) for each effect size revealed that 11 of the effect sizes
fell within 95% confidence intervals. The researcher used Statistical Package for the Social
Sciences (SPSS) software to visualize the position of the effect sizes as well as the confidence
intervals for each study around the computed overall mean of the pool of studies. Some means
were revealed to be outside of the area of the funnel graph (see Figure 2).

Figure 2. Funnel graph for the pretest-posttest experimental studies.

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The individual effect sizes of some of the studies showed to be outside of the confidence
intervals indicating a lack of homogeneity of distributions within the study pool. This was also
depicted by the significant p-value (p < 0.001). As the purpose of a meta-analytic study is tocompute effect size (Willson, 1983), the lack of homogeneity does not undermine the validityof the calculated mean effect; rather, it explicates the characteristics of the studies, indicatingthat some of them originated from different distributions.
The highest learning effect size (ES = 2.00) was generated in a study conducted withkindergarten pupils who were exploring the creation of verbal representations of geometry
concepts (Casey, Erkut, Ceder, & Young, 2008). This study revealed that immersing mathconcepts in an environment that students can relate to their experiences and fantasies and letting
students explore the links makes the math concepts tangible and results in them being easily
stored in students’ long-term memories. Another study with a high effect size (ES = 1.22),
conducted by Poland, Van Oers and Terwel (2009), investigated the impact of dynamic
representations on kindergarten students’ math achievement. Dynamic representations
provided more opportunities for having the learners explore their structures, thus generating ahigher engagement factor and consequently higher learning effects. A positive learning effectof students’ explorations was also advocated by Lesh and Harel (2003), who concluded thatsuch situated learning enhances the processes of mathematical modeling that can play a vitalrole in developing students’ scientific curiosity and they are problem-solving beyond theprimary school level.
Analysis of Moderator Effects
The process of a further synthesis of the studies’ features allowed for identifying the following
moderators: treatment length, mode of introducing the representations, grade level, and content
standards. The moderators were further disaggregated by their levels. Where applicable, thelevels within the moderators were contrasted, and inferences on differences were made. The
following criteria were applied to disaggregate the moderators.
Treatment length. The treatment length followed a partition established by Xin and Jitendra(1999): short – less than one week; intermediate – between 1 week and one month; and long more than one month.
Mode of representation induction in the lesson cycle. This moderator followed operationalroles of representations and contained two levels: concept introduction and problem-solving.
Grade level. The large range of grades was distributed into two levels according to standard
classification (NCTM, 2000). The lower group level encompassed all students from Pre-K to
Grade 3, and the upper level included Grades 4 and 5.
Content standards. This moderator reflected general standards examined in the studies thatwere clustered into the following: number and operations, proportions, and geometry. A
summary of the weighted effect sizes is presented in Table 3.

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Table 3. Summary of subgroups’ moderator effect sizes
Moderator and Levels
Grade Level
Lower: Pre-K through 3
Upper: 4-5
Representation Type
Pictorial
Schemata based
Concrete
Other
Treatment Length
Short
Intermediate
Long
Content Standard
Numbers and operations
Geometry
Ratio and proportions
Mode of Induction
Concept Introduction
Problem-solving

N

ES

SE

95 % CI
Lower

Upper

6
7

0.60
0.47

0.08
0.07

0.45
0.33

0.76
0.60

6
5
1
1

0.45
0.49
0.71
2.00

0.06
0.09
0.29
0.31

0.32
0.31
0.05
1.38

0.57
0.67
1.20
2.63

5
4
4

0.46
0.53
0.60

0.07
0.10
0.10

0.31
0.33
0.40

0.61
0.72
0.80

10
2
1

0.45
1.61
0.71

0.06
0.22
0.29

0.34
0.17
0.05

0.56
0.24
0.20

7
6

0.68
0.49

0.07
0.08

0.54
0.34

0.82
0.64

Note. N = number of participants, ES = effect size, SE = standard error.

The computing of the mean subgroup effect sizes provided a basis for answering more detailed
research questions. When compared by grade level, the effect of using representations washigher in Pre-K through Grade 3 than in Grades 4-5. This conclusion might be supported by
the fact that as students’ progress with learning math concepts, they learn more abstract
semantics that might be difficult to embody in representations, for instance, the idea of fraction
division. Students can follow the initial and final stage of the process. However, the diversityof the methods of dividing that is embodied by the syntax of division along with the various
representations of rational expressions might not be entirely comprehended and thus it needs
more clarity.
When mediated by the type of representation, concrete and others produced the highest effectsizes; yet, their significance could not be fully apprehended because each subgroup wasrepresented by a single primary study. When pictorial representations (ES = 0.45) andschemata-based representations (ES = 0.49) were contrasted, schemata representations showeda higher impact on student learning, which supports the findings of other scholastic research(e.g., Jitendra et al., 2007; Terwel, Van Oers, Van Dijk, &Van den Eeden, 2009; Xin et al.,2011). Overall, schemata-based representations and their applications emerged as the maintype of representations supporting problem-solving. According to Owen and Sweller (1985), a
schema is a general cognitive structure that allows the learner to categorize the problem andthen apply specific tools to solve it. A moderate effect size (ES = 0.49) indicates that thislearning strategy helps students understand underlying mathematical ideas in given word
problems and solve them. Hiebert and Carpenter (1992) posited that while developing theschemas, students activate a complex network of concepts stored in their long-term memory.
Furthermore, the networks constitute the model that will be called an internal representation of
the domain embodied by an external representation. As Xin et al. (2011) suggested, instead of
telling students, for instance, the numerical magnitude of the unit rate, e.g., ten apples per142

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Volume 6 – Issue 3 – Winter 2018basket, demonstrating students a real concrete representation along with its symbolic
mathematical model will better support the conceptual understanding and its mathematicalembodiment.
How should representations be furnished to effectively develop the conceptual networks?
Learners can be provided with the representations, or the representations can be derived by thelearners under the teacher’s guidance. It is inferred from this study that providing students with
opportunities to derive the representations deems to be a more effective teaching strategy
because it allows the learners to retain the concepts longer and apply them in new situations
more frequently.
An interesting direct variation was observed when the effect sizes were contrasted withtreatment lengths. It became apparent, from examining this relation, that the longer thetreatment, the higher the effect size (ES = 0.46 for short treatments, ES = 0.53 for intermediate,and ES = 0.60 for long). This result provides support for applying representations in classes
daily.
Concerning content standards, geometry representations yielded a higher effect size (ES =1.61). This result reflects the visual nature of content in this branch of mathematics, which by
virtue is rooted in representations. The analysis of the concluding subgroup—mode of inducing
in the lesson cycle—allowed contrasting the effect sizes of using representations to support
conceptual understanding and problem-solving. It is apparent that representations are more
effective with concept introduction (ES = 0.69) than problem solving (ES = 0.49). Thus, one
could conclude that supporting concept introduction with representations builds a stronger
network of impulses in students’ long-term memory.
Summing all these findings led to the formulation of a classroom setting that would generate
the highest learning effect sizes. It seems that using concrete representations to introduce
geometry concepts in Pre-K through Grade 3 would yield the highest learning effects.
The research findings also allow for the formulation of recommendations for effective
representations. Fujimura (2001) concluded that representations should share similar features
as the target domain and must be manipulative so that children can explore and uncoverembedded math structures by themselves. He further suggested that representations should bedesigned in a way that they develop children’s creativity in constructing mathematical models.
Casey et al. (2008) found that students retain mathematical knowledge if the knowledge isembedded in a story context. Developing mathematical knowledge through sequenced
mathematics problems related to the storyline is also suggested by the researcher of the metaanalysis. Booth and Siegler (2008) highlighted accuracy and transparency of representations
as a significant factor affecting students’ mathematical learning in early grades, whereas
Poland et al. (2009) brought forth the idea of using dynamic representations to support theprocesses of arithmetic operations.
As mathematics seeks to develop students’ concise, abstract thinking, the results of this
synthesis show that it also needs to reflect on representations that students use in daily life andwhose contents are adequate to their experiences. Presenting artificially created representations
that do not adhere to students’ experiences might disconnect mathematics concepts from therealm and rather support the notion that mathematics is an abstract subject.

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General Conclusions
The findings of this study support the following hypothesis: representations help Pre-K throughfifth-grade students learn and apply abstract math concepts, especially when such
representations are applied to support new concept understanding and students’ problemsolving skills. Certain limitations and recommendations for further research emerged from thisstudy, as discussed below.
Threats to Research Validity
The primary parameter limiting the study findings was a lower-than-expected pool of primarystudies that satisfied the conditions to be meta-analyzed. The validity of the study computations
was supported by double research data coding at the initial and concluding stages of the study
process. Any potential discrepancies were resolved. Although strictly specified, the literaturesearch was undertaken with broader conceptual definitions in mind that allowed for, assuggested by Cooper (2010), adjustment of the definitions and strengthening of the literature
relevance. Thus, as the initial literature search revealed that representations in Pre-K throughGrade 5 are often used to support problem-solving, the term problem solving was then used tolocate more studies.
Schemata as a Major Type of External Representations
Among different representations (see Table 1) schemata-based representation and theirapplications emerged as the most commonly used to support problem-solving. According toOwen and Sweller (1985), a schema is a general cognitive structure that allows the problem
solver to categorize the problem and then apply certain tools to solve it. A moderate effect size
(ES = 0.49) indicates that this learning strategy helps students understand underlying mathideas in given word problems. Hiebert and Carpenter (1992) posited that in the process ofdeveloping the schemata, students’ thinking blends a complex network of concepts in one
coherent picture. Furthermore, the networks constitute a mental model that will be called aninternal representation of the domain imaged by an external representation. The conceptual
networks can be developed either by representations provided by the teacher or byrepresentations derived by the learners under a teachers’ guidance.
Is having students use schemata sufficient to have them learn the holistic picture of the meaningof this mathematical representation? Several researchers concluded that once children are
exposed to certain representations – for instance, schematic representations to solve problems– they retain those methods and apply the schemas regardless of age (Coquin-Viennot &Moreau, 2003). Some scholars noted (e.g., Castle & Needham, 2007), this idea cannot be
overemphasized; children also need some working space to analyze problems and devise theirways to solve the problems with the support of provided schemata. Thus schemata should beperceived as suggestions for mathematization of certain patterns not as fixed formulas to use.
It seems that more research should focus on having students recognize the type of scientificunderpinning of the problem that students should apply to determine the principles embedded
in a given word problem.
Hiebert and Carpenter (1992) proposed four semantic categories (schemata) for arithmeticaloperations that are: change, combine, compare, and equalize. Using these schemata to model
story problems allows certain flexibility, for example, in some cases can be perceived ascompare, or compare can suffice to combine. Emphasizing the schemata to reach the final
solution reduces learners’ opportunities to explore and be immersed in the process of analyzing

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the problem structure. There can also be cases when two or more schemata can be used insuccessions. For example, in order to compare, students might need to combine similarelements first. Thus students should be allowed to exhibit flexibility in applying the schemata
and interpret them. However, that the primary meaning of each should be consistently
executed. To illustrate that consider the following problem discussed by Jitendra et al., (2007):
Music Mania sold 56 CDs last week. It sold 29 fewer CDs last week than this week. How manyCDs did it sell this week? This problem was intended to support the idea of compare. There is
merit to use the schema of compare in this problem, but is the schema compare the mostrepresentative to mathematize the process of selling the CDs? The problem mentioned two
events happening at two different time instants referring to similar objects, can then the learner
be directed to considering rather finding the difference? Thus would the schemata of change
better describe the process and elicit its solution? It seems that referring students to compare
gears their thinking toward the output of the problem thus finding the final product, not toward
the principal process, the change that supported the process of reaching the output. By directing
students’ attention to the problem output, the phase of problem analysis is reduced. Referringto the problem context and considering the definition of change as Change = Final value –Initial value, and solving for Change, one will receive Change = This week sells – Last week
sells. Substituting the given values results in 29 = This week sells – 56, that leads further toThis week sells = 56 + 29 which leads to the conclusion that Music Mania sold 85 CDs. With
the implementation of change, the representation involved negative numbers that perhaps were
not intended in Jitendra’s study. Thus to further discuss the applicability of this problem to
Grade 3 math curriculum, the problem would have to be redesigned, however providing
students with the flexibility of exercising the underlying process that is missed is worth further
research. Zooming further change in quantity values is concluded by subtracting the initialvalue from the final value: Change = Ending – Beginning. This standard definition of change
is applied not only in mathematics to calculate, for example, instantaneous or average rate ofchange (e.g. Stewart, 2006) but also in sciences, especially in physics where the concept ofchange is often used to calculate change of temperature, or object’s displacement (e.g.,
Giancoli, 2005).
One might be interested in learning how the schemata of change are induced in the literature.
The idea of using change was introduced by Marshall (1995) see Figure 3 and was modeled by
the following problem; Jane had 4 video games. Then her mother gave her 3 video games forher birthday. Jane now has 7 video games.
+3
Video GamesViCHANGE

4
Video Games

7
Video GamesBEGINNING

ENDING

Figure 3. Representation of change inspired by Marshall (1985).
Change in quantities values is concluded by subtracting the initial value from the final value:
Change = Ending – Beginning. This standard definition of change is applied not only inmathematics to calculate, for example, instantaneous or average rate of change (e.g. Stewart,

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Volume 6 – Issue 3 – Winter 20182006) but also in sciences, especially in physics where the concept of change is often used to
calculate change of temperature, or object’s displacement (e.g., Giancoli, 2005). This equation
can be rearranged to reflect Marshal’s idea Beginning + Change = Ending. However, therearranged form is not aligned with the fundamental principle of the process of change that
seems to be the core idea of the problem. If the schema of change were to be used, then perhaps
the diagram could have been redesigned to reflect on the difference in the quantity amount that
represents the change.
These two examples were brought up to signify a need for verifying interdisciplinary
consistency of the interpretations of the fundamental concepts that are meant to support
problem-solving in K-5 mathematics. It is understood that the equations symbolizing theschemata can be rearranged and executed with a dose of flexibility. What stages were being
used would depend on individual perception, yet general foundations for problem analysis
would perhaps require more consistency. Perhaps establishing fewer schemata and letting
students manipulate on them to reflect the process of a given problem could be an alternative
avenue to pursue? The mathematical operations behind calculations of change, combine,compare, and equalize are very fundamental in sciences, and it seems that understanding theircore meanings might have a profound impact on students success on problem-solving not onlyat an elementary but also at a high school level and beyond.
Looking Ahead: Linking the Representations
Cheng (1999) proposed four learning stages that can help students in developing conceptual
understanding through using representations: domain, external representation, concept, and theinternal network of concepts. While moving from one stage to another to reach the internal
network, the learner is immersed in four processes: observation, modeling, acquisition, andintegration. Except the study conducted by Rittle-Johnson, Siegler, & Alibali (2001) andTerwel et al. (2009), the majority of the gathered research did not explicate on these processes,
focusing instead on applying fixed models without discussing their possible modifications.
It seems that possessing the right representation does not suffice for an understanding. To
confirm an understanding, one needs to be able to put this representation through its paces,
explaining and predicting novel cases. Thus, to have an understanding of a representation is to
be in a state of readiness, taking the representation as a point of departure in the solution
process, not as an unquestionable formula a representation. Terwel et al. (2009) proved thathaving students explore and modify given representations produced the highest effect size. Thiscan be supported by the effects of induced math modeling phases that allowed the students tolink representations with the constraints of real scenarios (Sokolowski, 2018).
Applying representations often creates exploratory learning environments (English & Watters,2005) that consequently can be guided by inductive or deductive inquiry processes. Thus, otherthemes worthy of a further investigation emerged; should the use of representations to beorganized inductively, as suggested by Nunokawa (2005)? How do elementary school students
perceive these two major scientific inquiries? Are these inquiries rooted in virtues ofmathematical representations, or content-domain? Having students develop principles ofrepresentations by identifying commonalities due to applications and use such representations
to model other contexts beyond the boundary of a math classroom would be an interesting
pursuit for future studies.

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Bridging Representations Used at Elementary and High School Levels
There are other questions can be generated from this study, for example; how does the use ofrepresentations evolve as students’ progress with their mathematics classes, especiallyschemata-based that dominate problem-solving in Pre-K through Grade 5. Fuchs, L., Fuchs,
D., Finelli, Courey, & Hamlet (2004) suggested using schemata more extensively for problemsolving also at the high school level, especially targeting students with learning disabilities.
Having high school students derive processes of transitioning from, for example, proportion toa linear or rational function or from percent rate to an exponential function seem like valuable
topics to explore.
Another conclusion calls for extending the idea of using schemata to sciences and othersubjects in a consistent manner that will carry out their general principles. This transition would
help broaden the meanings and consequently built a stronger image of these representations instudents’ long-term memories. Do students experience applications of similar representationsin their science classes? Should these main avenues of knowledge acquisition depend on thenature of the representation (schemata or pictorial) or their general purpose? Further researchin these areas is needed, and it is believed that this paper provides some prompts for suchactions.

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Corresponding author: Andrzej Sokolowski
Contact email: Andrzej.Sokolowski@lonestar.edu

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