Mathematics Teaching and Learning as a Mediating Process: The...

Mathematical Thinking and Learning, 10: 374–406, 2008Copyright © Taylor & Francis Group, LLCISSN 1098-6065 print / 1532-7833 onlineDOI: 10.1080/10986060802291642

HMTL1098-60651532-7833Mathematical Thinking and Learning, Vol. 10, No. 4, September 2008: pp. 1–53Mathematical Thinking and Learning

Mathematics Teaching and Learning as a Mediating Process: The Case

of Tape Diagrams

Mathematics Teaching And Learning As A Mediating ProcessMURATA Aki MurataStanford University School of Education

This article examines how visual representations may mediate the teaching andlearning of mathematics over time in Japanese elementary classrooms. Using theZone of Proximal Development Mathematical Learning Model (Murata & Fuson,2006; Fuson & Murata, 2007), the process of mediation is explicated. The tapediagram, a central visual representation used in Japanese mathematics curriculum,is explored for its roles and the student learning that is intended to be mediated overtime, illuminating aspects of the process. The study argues that the consistent andcoherent use of one representation can bridge student understanding over time,focusing on mathematical relationships and problem-solving processes. The studyalso suggests different instructional approaches between U.S. and Japanese curriculathat are reflected in the uses of representations.

This article examines how a visual representation may mediate the mathematicsteaching and learning process when it is used over time. The process is explicatedusing the Zone of Proximal Development (ZPD) Mathematical Learning Model(Murata & Fuson, 2006; Fuson & Murata, 2007), based on Vygotskiian sociocul-tural theory (1978, 1999) to highlight the connections between social experiencesof the learner and his/her cognitive development. In analyzing the learning process,the model helps bring forward the role of the representations (in the social learn-ing experience) in student learning (cognitive development). This article focuseson a particular representation—tape diagrams—as they are uniquely used formultiple mathematics topics and grade levels in the Japanese curriculum, unlike

Correspondence should be sent to Aki Murata, Stanford University School of Education,520 Galvez Mall, Stanford, CA 94305. E-mail: [email protected]

MATHEMATICS TEACHING AND LEARNING AS A MEDIATING PROCESS 375

other representations that are used more exclusively for certain topics (and gradelevels). Thus, tape diagrams provide the case for the mediation process over time.Mathematics textbooks are compared and contrasted cross culturally (betweenthe United States and Japan) for their representation uses to bring forward theimplicit assumptions that are not visible in their own cultural context. Textbooksprovide the framework for instruction (in any culture; Mayer, Sims, & Tajika,1995; Remillard, 2005), and the analysis of the textbooks reveals unique aspectsof the taken-as-shared cultural teaching practices.

COGNITIVE DEVELOPMENT THROUGH SOCIAL INTERACTIONS AND SEMIOTIC MEDIATION

When tools are used consistently in a cultural activity, they function to frameparticipants’ ongoing thinking and to support their understanding. In a learningactivity, tools work to support learners’ cognitive development. Vygotski (1978)described cognitive development as a process of forming the mind; the twoprimary influences on such development are social interactions and semioticmediation (van de Veer & Valsiner, 1991; Vygotski, 1978). Tools are first usedto communicate with others and to mediate contact with our social worlds, thento mediate our interactions with ourselves and to internalize their uses (Moll,1990; Vygotski, 1978; Wertsch, 1991).

In a social interaction, a learner actively participates in the process ofconstructing his/her own meaning. Meanwhile, the social partner (often a moreexperienced member of the culture) adjusts his/her expectations to maintain thelearning process for the learner. Concepts are acquired in this social process asthe learner and the partner co-construct meanings in their use (e.g., Lerman,1996; Rogoff, 1990). Partners develop intersubjectivity through repeated sharedexperience, and in such processes, cultural tools1 mediate semiotic meaning tosupport cognitive development. Tools are an important part of cultural practices,and the mind is formed in interaction in practice with the tools (Luria, 1973;Wertsch, 1991). For the present analysis, representations function as tools. Toolsnot only represent cultural meanings within the activity but also become a neces-sary part of the practice because they provide meanings. Sfard (2000) argued thattools are part and parcel of the act of communication and, therefore, of cognition.While our intra- and inter-psychological planes interact through social experi-ences, the tools in the interactions take on social meanings in our cognition(Vygotski, 1978).

In discussing the stages in the development of mental acts, Gal’perin (1969)conceptualized schemas (represented in the form of diagrams, outlines, models,etc.) as being generated through the materialization of properties and relationshipsof concrete materials that expand the learner’s ability to retain the important

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conditions for systematic formation of new mental activity. As learners experi-ence the stages of mental acts that transform understanding based on concretematerial objects to more abstract and cognitive acts, the materialization of therelationships in the activity context through representations is critical. (This isdiscussed in detail in the following sections.) As learners come to use particularrepresentations in learning activities, the representations help guide the learningprocess and become a part of the learners’ cognition. The ways tools are integratedin specific cultural practices and the ways learners make sense of tools in theactivity frame the learners’ understanding of the concept. How tools are intro-duced, how frequently tools are used, how connections are made between the usesof particular tools, the implicit concepts that tools represent, and how ownershipof the use of tools is transferred to the learners all influence learners’ thinking inclassrooms. Tools and representations occupy an intermediate position betweenpseudoconcepts (everyday knowledge) and real concepts in learners’ minds(Vygotski, 1978).

Mathematics is a human activity of problem solving with the help of tools thatare invented to organize fields of experience in a schematic way (Freudenthal1973, 1978, 1991). With tools, mathematics concepts and structures come to holdmeanings for learners. Mathematics as a discipline can be considered as a collectionof related symbolic systems that present our world in a particular way, and thesocialization of the mathematics meanings for these symbol systems may beconsidered part of the social and semiotic mediation (Lerman, 2000). If a certainrepresentation is consistently used with instruction, this representation willbecome a part of students’ mathematical thinking and the foundation for theirfuture understanding. Students who primarily learn mathematics with abstractsymbols will have a different mental schema from students who learn it throughmanipulation of concrete objects and social examples (Lave, 1988).

ZPD MATH LEARNING MODEL

Figure 1 shows the ZPD math learning model from prior work (Murata & Fuson,2006; Fuson & Murata, 2007). This model embeds learning in the interaction ofdevelopment of fluency and increasing understanding that are supported byconceptual instructional conversation and teaching supports. Presented as fourphases of learning, this model of instruction suggests that valuing students’ priorknowledge, gradually connecting their spontaneous ideas with a mathematicallyvalued method, and using various forms of “practices” will help to develop stu-dents’ understanding and fluency with a mathematically valued method. The bottomhalf of the model illustrates the ZPD as originally interpreted and modeled byTharp and Gallimore (1988, 1990), and the top half shows the mathematicaldevelopment of fluency and increasing understanding in the model.


The ZPD model was originally developed by Tharp and Gallimore (1988) andGallimore and Tharp (1990) with the view that teaching and learning occur whenassistance is offered at points of ZPD in which performance requires assistance.Assistance is only provided when needed; decreasing amounts of assistance areprovided as students progress through their own learning paths in any giventopic. Figure 1 shows the four stages: Stage I is the stage with assistance pro-vided by more capable others; Stage II is the stage with assistance provided bythe self (as the means of assistance of others is internalized into speech-for-self);Stage III is the stage with internalization-automatization-fossilization; and Stage IVis the stage with de-automatization with recursion through the stages as perfor-mance that was once mastered slips away over time. This often occurs as backingup one stage at a time until performance can be recovered and involves processessuch as folding back described by Pirie and Kieren (1994). Decreasing assistanceover time is part of responsive assistance, a term that emphasizes the need forcreating intersubjectivity between the assister and assistee, and for giving assis-tance adapted to the assistee. This underscores Vygotski’s view of learning as aconstructive activity by the learner so that the internalization process across thesestages does not involve rote copying of behavior.

The phases of the ZPD math learning model correspond with these stages.With model teaching, the teacher draws out and works with the preexisting

FIGURE 1 Zone of Proximal Development Math Learning Model.

Developing Fluency, Kneading Knowledge

Zone of Proximal Development

Stage I Assistance provided by more-capable others

Speech-for-others

Stage II Assistance provided

by self

Speech-for-self External Internal

Stage II I

Automatization, abbreviation,

“fossilization”

Stage IV

De-automatization: recursiveness through

prior stages

Re-View: Recursive mo ve, more rapid than or iginal lear ni ng

Phase 1 (Guided introduction): a. Teacher helps create interests and

accessibility, b. Students solve problems, c. Students share and discuss solutions

(Teacher facilitates levels and errors)

Increasing Understanding

Phase 2 (Learning unfolding, analysis): a. Accessible and general meth od(s) is/are described and

explained, b. Errors are anal yz ed and explained, c. Discussion of various methods continue

Phase 3 (Kneading knowledge through practice): Developing u nder standing and fluency with accessible and gene ral me thod(s) through various pr actice s

Phase 4 (Integration of fluency and understanding): Delayed practice and feedback

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understanding of students in Phase 1. This requires teacher knowledge of studentlevels of understanding and a meaningful learning setting from which to elicit studentideas. In Phase 2, the teacher focuses on or introduces mathematically desired meth-ods and helps students move to these methods while building networks of knowl-edge. With effective questioning and facilitation, if students generate incomplete yetconceptually sound approaches and methods, teachers focus all the students’ atten-tion on the mathematics behind the approach. If such ideas are not generated by stu-dents, teachers may make a bridge between student ideas and the core mathematicalideas of the teaching at this phase. In Phase 3, the teacher helps students gain fluencywith the desired methods so everyone moves along his/her learning path. And inPhase 4, the teacher facilitates remembering by occasional delayed practice withfeedback and relates other topics as they arise. Phase 1 situates itself in Stage I of theZPD level in which assistance is given by the teacher. Phases 2 and 3 start in Stage Ibut move to take advantage of students’ self-assistance as they internalize the ideas.The end of Phase 3 and Phase 4 utilize students’ automatization and abbreviation ofthe concept, and Phase 4 may require a recursive move as they re-learn andstrengthen the understanding of the concept with delayed practice. In the process, stu-dents continue to develop fluency and increase their understanding.

Gal’perin’s (1969) model of development of mental acts closely correspondswith the ZPD model (Figure 1). In Gal’perin’s model, he distinguishes five levelsof development: “(1) familiarization with the task and its conditions; (2) an actbased on material objects, or their material representations or signs; (3) an actbased on audible speech without direct support from objects; (4) an act involvingexternal speech to oneself; and (5) an act using internal speech” (p. 250). In thecurrent ZPD model, because of the focus on classroom teaching and learningcontexts, the model separates physical and cognitive activities in an attempt toclearly articulate the learning process. While the top of Figure 1 outlines whatphysically goes on in classroom settings (in terms of teacher actions), the bottomof the figure explains cognitive stages. It is, however, important to note that thecognitive stages develop and are supported by the physical activities, and Figure 1shows the interactions between the two while Gal’perin’s model integrates theminto a linear progression. Tharp and Gallimore’s work (1988, 1990) and my exten-sion of that work incorporate classroom settings in which multiple learners inter-act and learn together; thus Figure 1 extends to the classroom setting. Instructionalconversation as a means of assistance for performance and understanding providesa context for students to share their ideas, understand multiple approaches, andmake sense of culturally valued approaches under the guidance of a teacher.

In the instructional conversation, students share their ideas and make connec-tions, and visual representations work as a tool to support the development ofunderstanding and to carry potential meanings from one context to the next. Thetool brings the new concept close to students at the initial stage and later provides“self-assistance” when the primary responsibility for learning is shifted from the


teacher to the students much in the way of self-speech and self-reminding. Overtime, the tool has come to present mathematical meaning to its user, and it func-tions to connect different, new, more complex, and related concepts together.Students then continue to use the tool under different circumstances to developdeeper understanding and build increased fluency.

JAPANESE MATHEMATICS CURRICULUM

With recent attention to international studies, we are increasingly becomingaware that mathematics is not taught as it is in other countries (e.g., Third Inter-national Mathematics and Science Study, 2003; Programme for InternationalStudent Assessment, 2003). The Third International Mathematics and ScienceStudy (National Center for Education Statistics, 2003) compared differentaspects of mathematics education in different cultures to examine the reasons forachievement differences (e.g., video study of teaching); one of the core studiesexamined mathematics curricula (Schmidt et. al., 1997). The study suggested thatalthough some common mathematics topics are taught, the organization and thepresentation of these topics varied among different countries.

Japan is one of the high-scoring countries in the international studiesmentioned above. The Japanese curriculum focuses on a few core topics andthere is little repetition and re-teaching of these topics. Concepts are typicallyintroduced as an extension of students’ prior learning to make the connectionsamong their learning experiences stronger. Moreover, a long time is spent oneach major topic to create successful learning by all students. These curricularapproaches minimize the need for re-teaching. Topic presentation is carefullythought out with common visual representations to connect core ideas acrosstopics and across grades: students’ mathematics experiences are centered onsupportive representations and situations to help students build meanings (seeother studies for discussion such as Mayer, Sims, & Tajika, 1995). For example,the National Center for Education Statistics (2003) found, in comparing videos ofteaching as a part of Third International Mathematics and Science Study, that theJapanese instruction made two to four times more use of visual representationsthan the instruction of other countries.

This article examines how Japanese elementary mathematics curricula (grades1–6) use a common visual representation—tape diagrams—over time to helpstudents learn mathematics. The National Council of Teachers of Mathematics(NCTM, 2000) defines representation as the “act of capturing a mathematicalconcept or relationships in some form and to the form itself” (p. 67). Representa-tions are an essential part of and are an effective tool for learning and doing math-ematics (NCTM, 2000). Tape diagrams are tape-like representations that visuallyillustrate relationships among quantities in a problem (see examples in Figure 2).

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FIGURE 2 Examples of the Types of Tape Diagrams.

Diagram type Chapter Names (Grade level)

Examples

Linear representation of objects (pre-tape)

Subtraction II (G1) Example 1:

Hiroshi

Akiko

Problem: Hiroshi and Akiko picked up leaves. Hiroshi picked up 9 leaves and Akiko picked up 13 leaves. Who picked up more and how many more?

Addition and subtraction I (G2)

Example 2: Total number of sheets

Blue paper Red paper

Problem: There are 38 sheets of blue paper and 63 sheets of red paper. How many sheets of paper are there?

Single tape

Addition and subtraction II (G2)

Example 3:

Total passengers: 34

Number of passengers Number of passengers who

Problem: 27 passengers were on a bus. More passengers got on, so now there are 34 passengers in all. How many more passengers got on?

(This diagram accompanies a picture of a child saying, “Addition and subtraction have the opposite effect,” and showing how this problem can be solved by addition and by

Double tape Addition and subtraction (G3)

Example 4: Red roses

Difference

Blue roses

Problem: There are 245 red roses and 138 blue roses. Which color of roses are there more of and by how many? [Additive Comparison]

Single tape with a number line

Multiplication with decimal numbers (G5)

Example 5:

Weight

Length 0 1 2 3 4 (meter)


Their role in Japanese mathematics textbooks is special: unlike other representa-tions, they are used consistently across grade levels to support student learning ofdifferent mathematics topics. (Different types of tape diagrams will be discussedin detail later.) While other representations (e.g., ten frames, base ten blocks) areused for certain mathematical topics to model the concept (e.g., base-ten numbersystem), tape diagrams are versatile in the way that they extend across multipletopics.

Tape diagrams are not unique to Japanese textbooks. The recent attention toSingapore mathematics education in the United States after Singapore students’high mathematics achievement in TIMSS (National Center for Education Statis-tics, 2003) revealed that their textbooks also use “strip” diagrams in a consistentmanner to support students’ mathematical problem solving (Beckman, 2004). Thesediagrams are also found in U.S. textbooks (e.g., Charles et al., 2004); however, their

FIGURE 2 (Continued).

Problem: There is a wire that is 1 meter long and weighs 2.3 grams. How many grams does 4 meter of this wire weigh?

Calculation with multiples (G4)

Example 6:

Length of his jump

His height

Problem: Hiroshi is 135 centimeters tall. He jumped 270 centimeter. How many times did he jump to his height?[Multiplicative Comparison]

Double tape with a number line

Proportion (G6) Example 7:

Average 30 meters

Ratio 0 1 7/5

Problem: Takeshi and his friends threw softballs. The average distance was 30 meters. Takeshi was 7/5 of the average distance. How many meters did Takeshi throw the softball? (Grade 6)

Double number line

Multiplication and division with fractions I (G6)

Example 8:

Distance

Time 0 1 4/3 (hour)

Problem: It takes 4/3 hours to drive 60 kilometer by car. What is the speed?

Unit quantity Total quantity

270 centimeters

135 centimeters

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usage is not consistent and is often limited to particular topics or particular gradelevels (this will be discussed later).

Beginning with a cross-cultural comparison of U.S. and Japanese textbookseries in terms of their visual representation uses, the sections that follow explorethe roles that tape diagrams may play in students’ cognitive development as wellas the mediation process they can provide, as they support mathematical learningpaths coherently over time in Japanese curriculum.

METHODS

The Japanese textbooks2 Study with Your Friends Mathematics (Gakkotosho,2005) and the U.S. textbooks Scott Foresman–Addison Wesley Mathematics(Charles et. al., 2004) were analyzed for comparison in terms of different repre-sentation uses and curricular approaches. These two textbook series were chosenbecause they each were identified as one of the most commonly used textbooksin their respective countries (Japanese Ministry of Education, 2006; HorizonResearch, 2002). Each unit in each of grades 1–6 volume for the Japanese curric-ulum and grades 1–5 for the U.S. curriculum (due to different elementary schoolsystems) was first analyzed for the use of different representations.

The five most frequent visual representations in the two textbook series acrossall grades were identified: pictures, tape diagrams, number lines, ten frames, andbase-ten blocks. The units in the two series that used these representations werefurther analyzed for their uses in terms of contextual problems (story problems)and non-contextual problems (numerical problems), as the differences betweentwo curricula in terms of their treatment of representations for these problemsbecame apparent in the analysis. The units in the U.S. and Japanese textbooksthat had similar instructional goals (for addition and subtraction) were thenpulled out and contrasted for the differences in terms of instructional steps andfoci. Teachers’ instructional manuals were also examined for additional informa-tion in terms of instruction and homework assignments. Addition and subtractionconcepts are considered to be the most fundamental in elementary mathematics;thus, part of the instruction focuses on grounding students in learning how torepresent mathematical ideas. Therefore, the topics were chosen to investigatethe mediating process for this study.

When the prevalent and consistent uses of different types of tape diagramswere found in Japanese textbooks, their particular uses were analyzed as well ashow the lesson structures supported their uses over grade levels. Different varia-tions of the tape diagrams and uses of the diagrams were found and noted. Thesetypes of diagrams were then analyzed in terms of their uses, the topics for whichthey are used, how they relate to each other, and especially how they change andevolve as the grade levels progress and mathematics topics become more complex.


To provide a more detailed and focused analysis, curricular units that supportedthe learning of addition and subtraction were carefully examined for teachingapproaches that were supported by tape diagrams.

Teachers’ instructional manuals that accompanied the textbooks were alsoanalyzed for additional explanations of the use of tape diagrams in instruction.These manuals provided information on different aspects of the diagrams andhow they are used as a part of instruction, students’ mathematical thinking of thetopics shown by the diagrams, and how the diagrams can provide support as stu-dents learn key mathematics topics. The textbook editors and original curriculumdesigners were interviewed for their ideas and insights, with a particular focus onthe roles of diagrams in student learning. Follow-up phone and e-mail discussionswere conducted to further articulate their ideas on the roles of tape diagrams in themathematics instruction presented in the textbooks.

COMPARISON OF REPRESENTATION USES BETWEEN U.S. AND JAPANESE TEXTBOOKS

Prior studies that compared U.S. and Japanese textbooks characterized andagreed on several key differences (e.g., Mayer, Sims, & Tajika, 1995; Stevenson& Bartsch, 1991; Stigler, Fuson, Ham, & Kim, 1986): U.S. textbooks are largeand glossy with colorful pictures that are often unrelated to the mathematicscontent; content is fairly simple and low-level, and many topics are covered with-out depth; and representations are used to aid simple procedures. In comparison,studies have found that Japanese textbooks are small and frugal; content is high-level and addresses a few important topics more deeply rather than many at a sur-face level; and they contain many complex “worked-out” examples with relatedillustrations for students to analyze problem-solving processes. Mathematicalideas are supported by pictures and diagrams, and materials are shown in asuccinct manner. My analysis found similar characteristics with the two textbookseries examined, and instead of repeating the same findings, the followingsections focus on representation uses.

Representations for Addition and Subtraction

Table 1 contrasts representation uses in the U.S. and Japanese textbooks forgrades 1–3 addition and subtraction units. The meaning-making foundation forthe use of representations is built at this stage because much of students’ earlyexperiences with representations focus on addition and subtraction conceptdevelopment.

In the units concerned with addition and subtraction for both textbook series,frequent representations were chosen for the analysis. The units were examined if

384 MURATA

they used a certain representation at least once for either contextual or non-contextualproblems. For both textbook series, pictorial representations (actual pictures ofobjects) are the primary representation for the early part of their learning (grade 1and early grade 2); ten frames are used as students work with numbers less than 10;and base-ten block representations are used for multi-digit addition and subtraction.

While many units in both textbook series use visual representations, their usesdiffer in an important way. The U.S. series is more likely to use representationsto accompany problems that are not contextual. For example, pictures of flowersaccompany solutions to the numerical problems such as 3 + 5. In contrast, theJapanese series uses representations for contextual (story) problems. On average(of five frequent representations), the U.S. series uses the representations 18% ofthe time for contextual problems and 46% of the time for non-contextual problems(number problems). In contrast, the Japanese series uses the representations 41%of the time for contextual problems, and only 10% of the time for non-contextualproblems.

Zooming in further, Figure 3 contrasts examples of guiding steps and ques-tions used for problem solving using tape diagrams in the two textbook series(for contextual problems). The Japanese example was taken from the grade 2 uniton addition and subtraction; the U.S. example was taken from the grade 3 unit onaddition and subtraction number sense. These two units were chosen because

TABLE 1Percentage of Addition-Subtraction Units in Grade 1–3 that Used Visual

Representations for Contextual and Non-contexual Problem Solving in Two Textbook Series

Types of Representations

Scott Foresman–Addison Wesley Mathematics

Study with Your Friends Mathematics

Contextual Non-context Contextual Non-context

Pictorial 50% 65% 88% 6%Tape diagram. 15% 5% 53% 0Number line 10% 50% 0 18%Ten frame 0 35% 12% 6%Base-ten blocks

and cubes15% 75% 53% 18%

Average percentage of representation use

18% 46% 41% 10%

Note. Only the units in each textbook series that concerned addition and subtraction conceptswere analyzed (e.g., geometry and measurement chapters were not analyzed). There were 20 suchunits in Scott Foresman–Addison Wesley Mathematics, and 17 units in Study with Your FriendsMathematics for their grades 1–3 volumes. When pictorial representation is used but does notsupport actual problem-solving process (e.g., a large picture of an apple is shown for a problem thatdiscussed grocery shopping), it was not considered to be a representation that supports problem solving.


they introduced the use of tape diagrams for the first time to students. In theU.S. example, the problem was given at the beginning (Mitch bought a capand a pennant. How much did he spend altogether?), the necessary informa-tion followed (a table summarizing the prices of the items), and students arethen guided by a set of questions (in student books) to solve the original prob-lem. In the Japanese example, the context is given at the beginning (There are38 blue sheets of paper and 63 red sheets), and questions follow to examinethe given situations that accompany different tape diagram representations.The questions guided students’ attention to different aspects of the additionand subtraction relationships in the given context and how the tape diagramcould show the relationships. The U.S. approach utilizes the representation to

FIGURE 3 Examples of Guiding Steps and Questions used for Problem-solving with TapeDiagrams in Two Textbook Series (as Shown in Student Books).

Scott Foresman Addison Wesley Mathematics Grade 3, chapter 2 (early Grade 3); Addition and subtraction number sense

Study with Your Friends Mathematics Grade 2, chapter 7 (mid Grade 2); Addition and subtraction (1)

Problem: Mitch bought a cap and a pennant. How much did he spend altogether?

Guiding steps: 1. Identify the main idea. 2. Draw a picture to show the main idea*. 3. Describe the two parts in this problem. 4. Write a number sentence for the

problem.

Problem: There are 38 blue sheets of paper and 63 red sheets.

Guiding questions: 1. How many sheets of colored paper are

there?

2. Which one is how many more?

3. She uses 25 red sheets, how many will be left?

Note. * In teachers’ manuals, the sample picture for this steps is shown as

Eagles Baseball Souvenirs Cap . . . . . . . . $12 T-shirt . . . . . . $18 Pennant . . . . . $4 Poster . . . . . . $3

?9

24

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solve a problem while the Japanese approach focuses on analysis of the con-text using representation.

While representing quantities, the tape diagrams in the Japanese textbooks andthe U.S. textbooks differ in another significant way. As Figure 3 shows, when thediagram is used in the U.S. textbook to show three quantities, it shows threequantities independently of one another as different tapes. When the Japanesetextbook shows the similar situation, it shows two quantities with tapes andlabels these tapes in three different ways (as two smaller quantities and as thetotal). And, these differences are consistent between the textbook series acrossgrade levels. The Japanese representation is truer to the problem situation whenthere are two quantities and the third is the total of the two. The U.S. representa-tion presents the total as an additional quantity.

Lesson Structures and Goals that Support the Use of Representations

U.S. and Japanese lesson structures are also different in the two textbook series.A U.S. lesson starts with a simple problem (often non-contextual) that shows astep-by-step solution to the problem. Following the example, several problemsfollow that require students to take the same steps to find the answers. Picturesand representations guide the problem solving at this point, but they are typicallyshown in an identical manner from one problem to the next so students will focuson finding the patterns with the set of problems and answers. There may be oneor two story problems embedded in the lesson while the focus of the lesson isclearly on learning how to follow the given steps and finding answers to a set ofsimilar problems.

The Japanese lesson structure is different. A lesson begins with one problemand for an entire class period students take the time to represent, solve, and discussthe problem and their approaches to solving the problem. Problems in Japanesetextbooks are often “worked out” and relevant illustrations support the entireproblem-solving process (Mayer, Sims, & Tajika, 1995). Representations are usedfor students to analyze and understand the problem and to generate a space forinstructional conversation. Each instructional unit includes a set of non-contextualproblems, but they are not the main focus of the learning.

There may be different purposes for these instructional approaches that frametheir own uses of the representations. If the goal of the instruction is to supportcompetent mathematics students who can follow steps and learn to solve numericalproblems quickly and accurately, the U.S. approach of focusing on simple stepsand solving multiple problems of the same kind will make sense. The Japaneseapproach may be more closely aligned to the NCTM’s goal of supporting stu-dents’ understanding of the aspects of mathematical situations by analyzing anddiscussing the problem context and deepening the understanding of how themathematics works in context (NCTM, 2000). Gal’perin (1969) discussed three


types of task orientation: (1) trial and error, (2) strict adherence to all the direc-tions, and (3) analysis of a new model. The U.S. approach found in the analysis issimilar to type (2) while the Japanese is similar to (3). Gal’perin discusses howlearners’ future acts are determined to a great extent by their orientation at thebeginning; thus, these different instructional approaches set the stage for stu-dents’ future learning of mathematics.

USE OF TAPE DIAGRAMS ACROSS GRADES IN JAPANESE TEXTBOOKS

Table 2 summarizes the use of tape diagrams in Study with Your Friends: Mathe-matics for Elementary School for grades 1 to 6 (Gakkotosho, 2005). Tape dia-grams are used across grade levels with different mathematics topics. Before theofficial “tape” diagram is used in grade 2, grade 1 students use and becomefamiliar with linear representations of concrete objects to assist their thinkingabout problem situations (see Table 2). The instructional path that consistentlyuses the tape diagrams shows that they are important tools for mathematicalproblem solving in Japanese instruction.

The types of the diagrams become more complex as mathematical topicsincrease in complexity and as students consider multiple quantities and steps insolving a problem. Figure 2 shows examples of various types of tape diagramsformed by combinations of tapes and number lines. These examples wereselected from the units identified in Table 2 to represent different grade levelsand different mathematics topics in a balanced manner so that examples wouldnot concentrate on a few grade levels and/or a few topics. The tape representa-tions gradually change as students advance in grade levels, from using actualconcrete objects within a tape, to semi-concrete objects (e.g., circles) in a tape(since young students need to see units within an amount), to a single-tape repre-sentation with no embedded objects, and to more complex combinations of tapeand number line combinations with multiple quantities.

In upper grades, tapes and number lines are used together (Figure 2). In grades 3and 4, tapes are often used to show quantities while a number line accompanies thetapes to show relationships between the quantities. Ratio and proportion are topicstaught with tape and number line representation in grades 5 and 6 because propor-tional relationships can be shown visually with the representation (Murata, 2008).Tape representations are also effectively used for division of decimals to support stu-dents’ understanding by illustrating divisible/multiplicative relationships betweenquantities. In late grade 6, double number lines are used to show two quantities sideby side and to show how a change with one quantity affects the other. This conceptcan be represented with two tapes and a number line model, too, but the double num-ber line model helps show the corresponding changes of two quantities more directly.

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unk

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tal (

in te

en n

umbe

rs)

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nsC

hapt

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trac

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(2)

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trac

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h un

know

n re

sult

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ns2

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g ab

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ge b

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rete

pic

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pres

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mer

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equ

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a d

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stor

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now

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sult

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2-di

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ith a

ctua

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ects

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embe

dded

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pter

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ddit

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pare

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how

two

quan

titie

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ithi

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e la

rge

quan

tity

3C

hapt

er 1

: Add

itio

n an

d su

btra

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o sh

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elat

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hips

in in

trod

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g 3-

digi

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arat

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pare

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ence

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arat

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pare

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mpa

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ity)

3-di

git

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e di

agra

ms

to s

how

two

quan

titie

s w

ithi

n on

e la

rge

quan

tity,

dou

ble

tape

to s

how

di

ffer

ence

4C

hapt

er 3

: Div

isio

nT

o sh

ow d

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ible

rel

atio

nshi

ps in

intr

oduc

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divi

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oble

ms

(mea

sure

men

t div

isio

n)Si

ngle

and

2-d

igit

Tap

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agra

ms

wit

h tw

o un

know

n qu

antit

ies

Cha

pter

4: D

ivis

ion

with

si

ngle

-dig

it nu

mbe

rT

o si

tuat

e di

visi

on/m

ultip

licat

ion

rela

tions

hip

in

intr

oduc

ing

diff

eren

t sto

ry p

robl

ems

(mul

tiplic

atio

n an

d pa

rtiti

ve d

ivis

ion)

Sing

le a

nd 2

-dig

itT

ape

wit

h nu

mbe

r li

ne

389

Cha

pter

14:

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ctio

nT

o sh

ow r

elat

ions

hips

bet

wee

n th

e w

hole

and

par

tsD

oubl

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pe re

pres

enta

tion

s (o

ne fo

r th

e w

hole

, ano

ther

for

the

part

s)5

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nkin

g ab

out c

alcu

lati

on*

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show

the

mul

tiplic

ativ

e re

latio

nshi

p be

twee

n qu

anti

ties

(m

ulti

plic

atio

n)D

ecim

al x

sin

gle-

digi

t num

ber

Tap

e (o

rigi

nal q

uant

ity/

sets

) an

d a

num

ber

line

(sho

w m

ultip

les)

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pter

3: M

ultip

licat

ion

of

deci

mal

num

bers

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show

the

rela

tion

ship

bet

wee

n de

cim

al a

nd w

hole

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mbe

rs (

mul

tipli

catio

n, r

atio

)D

ecim

al x

sin

gle-

digi

t num

ber

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rigi

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uant

itie

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nd

num

ber

line

(mul

tiple

s)T

hink

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cula

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le-

digi

t num

ber

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rigi

nal q

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ity)

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num

ber

line

(par

titio

ns)

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pter

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ivis

ion

of

deci

mal

num

bers

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show

the

divi

sion

rel

atio

nshi

p (p

artit

ive

divi

sion

, pr

opor

tion)

Dec

imal

/ si

ngle

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git n

umbe

rT

ape

(ori

gina

l qua

ntity

) an

d nu

mbe

r lin

e (p

arti

tions

, pro

port

ion)

Cha

pter

9: F

ract

ion

To

show

the

rela

tion

ship

bet

wee

n w

hole

and

par

ts

(par

titiv

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visi

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rigi

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num

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ctio

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ropo

rtio

n)C

hapt

er 1

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atio

To

show

rel

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nshi

ps in

intr

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ratio

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tidig

it nu

mbe

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(ori

gina

l qua

ntit

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nu

mbe

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e (r

atio

, pro

port

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6C

hapt

er 6

: Per

uni

t mea

sure

To

show

uni

t am

ount

whe

n ra

tio

betw

een

two

diff

eren

t qu

anti

ties

are

cons

ider

edM

ultid

igit

num

bers

Tap

e (o

rigi

nal q

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and

num

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line

(uni

t am

ount

)C

hapt

er 7

: Mul

tiplic

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d di

visi

on o

f fr

actio

ns (

1)T

o sh

ow m

ultip

lica

tive

rela

tions

hips

with

fra

ctio

nFr

actio

ns x

frac

tions

or

/ w

hole

num

bers

Tap

e (o

rigi

nal q

uant

ity)

and

a

num

ber

line

(fra

ctio

n)C

hapt

er 8

: Mul

tiplic

atio

n an

d di

visi

on o

f fr

actio

ns (

2)T

o sh

ow r

elat

ions

hips

in f

ract

ion

mul

tiplic

atio

n an

d di

visi

on p

robl

ems

Frac

tions

x fr

actio

ns

or /

who

le

num

bers

Dou

ble

num

ber l

ine

to s

how

two

orig

inal

qua

ntiti

es, c

hang

e on

e of

th

e qu

antit

y an

d fi

nd th

e un

know

n am

ount

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cula

tion

of m

ulti

ples

*T

o sh

ow m

ultip

les

and

ratio

rel

atio

nshi

psW

hole

and

dec

imal

nu

mbe

rsD

oubl

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pes

(qua

ntiti

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and

a nu

mbe

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e (r

atio

)C

hapt

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1: D

irec

t pr

opor

tion

To

show

pro

port

iona

l rel

atio

nshi

ps (

used

as

one

repr

esen

tatio

n ne

xt to

tabl

es a

nd g

raph

s)W

hole

and

dec

imal

nu

mbe

rsD

oubl

e nu

mbe

r lin

es to

sho

w

diff

eren

t qua

ntiti

es

*Ind

icat

es in

depe

nden

t sho

rt s

ecti

on to

exa

min

e a

part

icul

ar to

pic.

390 MURATA

ZPD MATH LEARNING MODEL AND TAPE DIAGRAMS: ADDITION AND SUBTRACTION3

This article has discussed that Japanese curriculum has particular structures thataim to support student learning of mathematics mediated through the consistentuse of different types of tape diagrams. In this section, the mediation process isdescribed by using the phases of the ZPD model discussed earlier (Figure 1) usingexamples from the textbooks. In the ZPD math learning model, the representations(along with other learning supports) become the foundation and support the devel-opment of conceptual discussion that assists understanding throughout the phases.The analysis focuses on the early elementary grade levels (grades 1–3) to under-stand how the representation provides the important basis for problem solving byfocusing on relationships among quantities in the problem context.

Phase 1: Teacher Draws Out and Works with the Preexisting Understandings of Students

When students are first introduced to tape diagrams in the unit of Addition andSubtraction (1) in grade 2 (Table 2), the representation is considered as one of themany representations to show the problem (see Figure 4). Students are familiarwith the addition concept at this point, and by taking advantage of that familiar-ity, the focus of the activity is to elicit students’ ideas and to present them withdifferent representations to highlight the relationship among quantities includingthe tens and ones in the quantities. Discussing the similarities and differencesamong the representations is the core activity (instructional conversation);students are encouraged to evaluate representations for their strengths and weak-nesses. The unit progresses with story problems of different kinds (join, change,compare, and order4), for which quantities are represented with tape diagramsthat show one or more unknown quantities. At this phase, teachers elicit students’ideas and approaches and they connect students’ ideas with the tape-diagramrepresentations. Meanwhile, students may use different approaches to solve prob-lems, but the use of the tape diagram is increasingly supported.

Phase 2: Teacher Focuses on (or Introduces) Mathematically Desired Methods and Helps Students Move to One while Building Networks of Knowledge

Unit 16 of Addition and Subtraction (2) in grade 2 (Table 2) starts by showingadditive quantitative relationships in four different situations with tape diagrams(Figure 5) to help students think about the similarities and differences amongthese situations. The activities in this unit are different from the activities in theearlier grade 2 Unit 7 (Addition and Subtraction (1), discussed above) because


now students are asked to show different relationships using tape diagrams andthen to express the relationships using numbers and symbols such as +, =, etc.There is no discussion of different representations, and tape diagrams are the rep-resentation used. There is a six-month interval between the units Addition andSubtraction (1) and (2), and students may not yet be fluent with the use of thisrepresentation. While learning to use tape diagrams, the diagrams work to pro-vide assistance for students in understanding mathematical relationships in prob-lems. Without teachers giving direct and explicit instruction, tape diagrams workto guide student learning because students may not yet be able to see quantitativeand mathematical relationships right away from the words in a word problempresentation. The tape diagram representations provide needed assistance andstructure for the students’ thinking. In Figure 1, on the bottom, in the Zone ofProximal Development, students provide assistance to themselves as their speechbecomes internal. Tape diagrams work to assist the internalization processes,similar to the way that “self-speech” does, as both self-speech and tape diagramsprovide certain structures to students’ thinking paths that help focus their atten-tion to the core concept (Vygotski, 1978, 1999).

FIGURE 4 Grade 2 Textbook page Example 1.

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FIGURE 5 Grade 2 Textbook page Example 2.


Phase 3: Teacher Helps Students Gain Fluency with Desired Methods so Everyone Moves Along a Learning Path

In the latter part of this unit (grade 2, Unit 16), students solve different storyproblems by following the process of (1) identifying and understanding the keyaspects of the problem, (2) representing the context using a tape diagram, (3)writing an equation using � (empty square for unknown quantity), and (4) solv-ing the equation and finding an answer. Students are then given a tape diagramwith quantities already specified and they are instructed to create their own storyproblems. This part of the unit gradually shifts students to Phase 3 of the ZPDMath Learning Model, with different kinds of practice that require them to usetape diagrams and to think about mathematical situations represented by thediagrams. Students begin to gain fluency in these exercises.

The grade 3 Unit 1 of Addition and Subtraction (1) is the first unit when theschool year starts, a month after Addition and Subtraction (2) in grade 2. In thisunit, students are introduced to three-digit addition and subtraction problemswith a strong emphasis on understanding place value. Tape diagrams are used toextend the initial problem context similar to the way students used them in grade 2but the ideas presented with the diagrams are then represented with three-columnplace value representations (Figure 6) before numerical equations are con-structed. The primary role of the tape diagrams in this unit shifts from helpingstudents see the relationships in the process of creating the diagram representationto providing an extra step (showing the relationship) before students reconstructthe problem with a new focus (seeing the quantities in terms of place value quan-tities). Relationships are considered to exist now in the diagrams, and they areused to construct a consistent analysis of the problem. In this way, the tool isbecoming part of students’ cognition; the diagrams are used to become founda-tional to the way that students think about mathematics problems.

Phase 4: Teacher Facilitates Remembering by Occasional Delayed Practice with Feedback and Relating Other Topics as they Arise

Delayed practice also happens later in the students’ learning (in upper grades, withdifferent mathematics topics; see Table 2). The use of the diagram potentiallybecomes automatic, and students are expected to use diagrams naturally as part oftheir thinking and problem-solving process. In the future when students encounter astory problem they will recall using a tape diagram to solve this kind of problemand will be likely to practice to gain fluency and increase understanding by solvingthe problem with the diagram. The delayed practice can also be embedded instudents’ future learning when more advanced and complex problem situationsrequire students to use tape diagrams to represent familiar situations embedded inthe new problems to think about the new mathematical relationships. Through

394 MURATA

FIGURE 6 Grade 3 Textbook page Example.


various experiences, students’ understanding increases, and fluency with the tooldevelops over time as students progress through different grades through the years.

TOOLS AND LEARNING

When students come to classrooms with their own meaningful yet less-structuredideas, concepts, and solution methods, representations such as tape diagrams canhelp present the parts of the less-structured ideas to become more coherent in aproblem-solving process. As Vygotski (1978) mentioned how learners’ abstractideas are made concrete in the learning process, one of the affordances of visualrepresentations is to give structures to students’ ideas (to make them meaning-fully visible and concrete) so that students can focus on core aspects of the problemand engage in their own sense-making process. Especially when the ideas origi-nally come from students and teachers identify key points of students’ ideas byplacing them in the representation (therefore highlighting the relationships), thesense-making process may be experienced seamlessly by the students.

After the concepts are understood more concretely in the representation, how-ever, this understanding can be lifted from the concrete form to be made moreabstract in classroom discussion and students’ future learning. In the case of tapediagrams, once the parts of a student’s solution method are made visible withthe diagram, the solution may be expressed using numbers and symbols that aremore formal and abstract. The new abstract representation (with numbers andsymbols) will create another way to understand the concept, which can beextended to different problem situations in the future.

Representations are indeed artifacts until users assign meanings (Guin, Ruthven, &Trouche, 2006; Vergnaud, cited in Andersen, 2007; Verllion & Rabardel, 1995)and tape diagrams are no exception. When curricula and teachers take students’ideas in a meaningful instructional context and place them in a diagram, parts ofstudents’ ideas that might not have been clear so far come to present meaningscoherently, often in relationships. Once students assign meanings to the diagram(artifact), it then becomes a tool for their thinking, and when the tool is used in dif-ferent problem contexts, it creates a common foundation for thinking. Thus, theconsistent use of one representation, such as tape diagrams, can provide ongoingsupport for student thinking and problem solving by being a cognitive tool.

MEDIATED MEANINGS THROUGH TAPE DIAGRAMS

In this section, I discuss how tape diagrams as a teaching tool can mediate meanings(Table 3). The mediating support criteria that follow were generated throughextensive discussion with curricular writers and textbook editors in the revision

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process of textbooks and teachers’ manuals. Examples are selected from the text-books to balance and to show the range of different topics and grade levels thatexemplify particular meanings. Tools themselves do not hold meanings, but themeanings are constructed and communicated when they are used in practice.Examining how tape diagrams are used will illuminate the processes of mediation.

Meaning Support 1: Relating Quantities in a Problem

Tape diagrams are designed to bring forward the relational meanings of thequantities in a problem by showing the connections in context. By clarifying the

TABLE 3Meaning Support of Tape Diagrams

Meaning Support

1 Relating different quantities (assigning related meanings)• To grasp the whole problem context and understand what is going on in the problem• To see what the quantities presented mean and how they relate in the context• To prepare students for the use of continuous number line in the future

2 Allowing space to understand the problem context before using abstract (numerical) representation (Meaning-making)• To grasp the whole problem context and understand the meaning behind calculation• (Used in introductory lessons) to situate a new concept in a contextual problem to make sense• To allow time to understand the problem (different aspects of the problem), think about the

mathematical operations to be used, and make connections between two3 Relating different mathematical operations

• To understand different underlying mathematical concepts and solutions• To understand the relationships among different mathematical operations

4 Freezing the problems in time• To clarify the relationship among different quantities in the problem situation, represent all the

elements together by suspending the time element in problems that outline event occurrence chronologically

5 Providing continuity across mathematical topics6 Experiencing mathematics as a subject of systematic relationships

• By focusing students’ attention to how different quantities (and mathematical concepts) relate in the problem situation, support students to experience mathematics as a subject of quantitative and logical relationships

7 Supporting discussion (Community and Instruction)• To discuss how quantities relate to one another• To maintain the meanings in the discussion of solution methods• To discuss the relationships among different operations• To discuss a mathematical concept across different problem presentations• To discuss solution and methods grounded in the problem context• To discuss formula and mathematical equation in relationship to the problem context• To ground the discussion of the methods and solution in a problem context• Used in introductory lessons to make sense of a new concept in a problem context and relate

through the common representation to earlier concepts and problem types


relationships among different quantities, students can grasp problems both holis-tically and analytically and can understand and portray what is actually happeningin each problem.

Example 7 in Figure 2 presents a complex situation for which the two quantities(30 meters and 7/5) are related. The diagram helps students understand that theyare not only manipulating two quantities directly to find the answer but also therelationships between the quantities that are the core idea of the problem. Fromthe relationship (ratio), the answer to the problem can be found. If this problem istaught procedurally as a task of multiplying fractions, the particular quantitativerelationship may be overlooked and missed entirely (even though students canexplain the meaning of operations and how to solve the problem mechanically).

Even in lower elementary grades in Japan, understanding relationships amongquantities in a problem situation is considered important. Example 4 in Figure 2supports grade 3 students in understanding the relationships between the quantitiesthrough additive comparison. This presentation will help students assign situa-tional meanings to quantities and understand the operation through relationships.

Meaning Support 2: Relating Different Mathematical Operations

Examples 2 and 3 in Figure 2 are taken from a grade 2 textbook. By showing theadditive relationship among three quantities (two smaller quantities within atotal), the diagrams intend to support students’ understanding of how these quan-tities relate. This understanding also becomes the foundation for the students’thinking about addition and subtraction operations in the future. These operationsare just different ways to express the same relationship, and tape-diagram repre-sentations uniquely highlight the connections among different operations in away that other representations do not. Using the representation, it is more naturalfor students to grasp the relationship first and then to assign more abstract andmathematical operational and computational definitions.

Young children tend to focus on contextual differences in story problems andtherefore understand the addition and subtraction differently. For instance,Example 3 is more naturally understood as an addition situation by children aspeople “got on” the bus, while adults may quickly have a subtraction equation inmind. It is a challenge for children to see different situations (e.g., changing andcomparing) as the same “subtraction” problems (Carpenter et. al., 1999; Fuson,2003). In similar ways, research shows how students of minority backgroundstend to become engaged more with contextual aspects of word problems andfind it hard to connect the abstract concepts to them (Boaler, 1994a; Cooper &Dune, 1998; Means & Knapp, 1991; Lubienski, 2000). Visual representations,such as tape diagrams, can help highlight the mathematical characteristicsembedded within problem contexts for students who tend to pay attention to thecontextual aspects of problems and who may need extra space for generalizing

398 MURATA

and making connections between the context and mathematical ideas in theinitial meaning-making process.

Tape diagrams help students visualize mathematical concepts in the problemcontext. Example 6 in Figure 2 shows how the relationships between the twoquantities (270 centimeters and 135 centimeters) are presented side by side forstudents to interpret the relationship.

In the process of interpretation, it will become clear that Hiroshi’s height is ½of the length of his jump, and that the jump was twice as long as his height. Themultiplicative relationship helps students see how to find the answer by usingdivision or multiplication (270/135 = � or 135 × � = 270). In the process, therelationships between different mathematical operations (multiplication and divi-sion) become clearer through this comparison context.

Meaning Support 3: Experiencing Mathematics as a Subject of Systematic Relationships

In typical U.S. instruction, elementary school mathematics is not considered as asubject to systematically study a set of different quantitative and logical relation-ships; this does not happen until students are older and in higher-level mathematicscourses. However, elementary school students can also enjoy and analyze mathe-matics if the topics are relevant and developmentally appropriate and if opportu-nities are given. Since the use of diagrams requires students to analyze a problemcarefully, they can engage in careful analysis of different aspects of a problemand the quantitative relationships in the problem. The use of tape diagrams pavesthe path for students to consider mathematics as a subject of systematic relation-ships and helps students prepare for their future experiences.

Meaning Support 4: Providing Continuity across Mathematics Topics

Tape diagrams used across grade levels and with different mathematics topicscan help situate a new mathematics concept in a familiar problem representationand thus assist student learning. When used in introductory lessons, tapediagrams situate a new concept to help students make sense of the concept withsomething they already know. Thus, they provide continuity across mathematicaldomains. For example, when the idea of multiplication is presented using a tapediagram, students’ attention may be drawn to the additive relationship of thegroups and the whole. From that, a new concept (multiplication) may be intro-duced to consider groups with same number of objects as subunits within thewhole.

The Meaning Supports 5, 6, 7 that follow apply to representations in generaland are not limited to tape diagrams. For example, base-ten block representationwill also allow space for students to understand the problem before they attempt


to solve a multidigit addition problem with regrouping (Meaning Support 5).I focus my discussion of the supports in terms of tape diagrams since it is thepurpose of the section, but different representations may also provide similarsupports in instruction.

Meaning Support 5: Allowing Space for a Problem to be Understood before Using Abstract (Numerical) Representation

Representations are used to create a thinking time and space for students betweenword presentation and abstract numerical representation of the problem. It isthe meaning-making space that representations such as the tape diagrams providewhere students make sense of the situation. In mathematics classrooms, since oneimportant and shared goal of learning is to find an answer through numericalmethods, students and teachers may tend to hurry and move to use numbers andequations (Boaler, 1994a; Brown, Colins, & Deguid, 1989). While this goalshould be maintained, all students (especially young students and students ofdiverse backgrounds) do require extra time to understand the problem situationbefore moving on to use abstract representation (Lubienski, 2000). This extratime and space is particularly valuable when students are trying to make sense ofdifferent parts of the problem and to grasp the whole problem context. A goodmeaning-making space can guide students to understand the reasons behind for-mulas and the abstract representations that follow.

Meaning Support 6: Freezing the Problem in Time

Tape diagrams work to suspend the time element in the process by showing theproblem as a one-time snapshot picture. This reduces the additional complexityso that students can readily focus on the quantitative relationships at hand. This isparticularly true in the case with problems that present chronological events. It isoften difficult and confusing for students to keep track of different events in oneproblem (Fosnot & Dolk, 2001; Fuson 2003). While constructing a tape diagramrepresentation, students follow the flow of the problem as presented in the story.When all the necessary information is placed, the final diagram represents,together, the steps embedded in the problem. (This is not critical for problemswithout the time element.)

Example 3 in Figure 2 presents the changes in the problem that occurred withtime as passengers got on the bus. Seeing the problem situation presented in thetape diagram, however, the time is frozen in the representation. This helpsstudents focus only on quantitative relationships and suspend the time elementfor the problem-solving process: the additive relationships among quantitiescome forward, and this helps students to concentrate on the mathematical task athand.

400 MURATA

Meaning Support 7: Supporting Discussions

Representation is inherently a social activity (Pape & Tchoshanov, 2001) thatsupports students’ collaborative meaning making through discussion in class-room community and instruction. Tape diagrams, as other representations, pro-vide and facilitate students discussing their ideas based on relationships (amongdifferent quantities, operations, mathematics domains) in the problem. It helps tomaintain the meanings grounded in the original problem in the discussion assolution methods are presented. Discussion is more likely to be meaningful ifstudents stay focused in the problem situation.

In classroom discussion, students and teachers at first sometimes find itharder to explain their multistep thinking processes than simply to state answers(Hufferd-Ackles, Fuson, & Sherin, 2004). When we shift the focus of mathemat-ics teaching and learning from finding the right answer quickly to understandingconnections among different ideas, we all need to learn to explain our thinkingclearly in our classrooms to help everyone learn. Discussing our thinking can bedifficult because we tend not to pay close attention to small steps of our thinkingas we solve problems; thus, it is hard to break down the steps in our explanation(Forman, Mccormick, & Dontato, 1997). From listeners’ points of view, it isalso hard to follow someone’s explanation because we often do not share thesame thinking foundations. Having a common representation requires thespeaker to pay attention to different aspects of the representation as he/she con-structed it, provides a common foundation for discussion, and pulls listenerscloser to the speaker’s thinking with a shared reference. It helps to developshared understanding among members of the classroom community (Boaleret. al, 2008).

As discussed before, in constructing tape diagram representations, the “timeelement” can be put on hold in the problem context. When in discussion, how-ever, the time comes to play an important role in verbal explanation since it is notshown visually. As the speaker explains the problem and his/her thinking, time isadded back to the problem presentation (verbally) to make the context come backto life. Thus, speaker and listeners reconstruct their shared new understandingand image of the problem in discussion to further their understanding ofthe mathematics and the process together. Listener-ship (K. Inagaki, personalcommunication, April 2002) is no longer a passive process. It is a way to activelyparticipate in the meaning making.

With experience and time, students will come to make connections on theirown readily, but when first introducing the diagrams it is important for teachersto assess and confirm what students understand with the diagrams. Some explicitsharing of ideas among students would be helpful, and consistent connectionmaking between what students represent with the tool, what they discuss, andgeneral mathematics concepts will be essential.


SUMMARY

Mediating tools, such as tape diagrams, support people’s cognitive developmentby helping organize their experiences in a schematic way. They guide our atten-tion to certain aspects of the activity, and by doing so, teach us certain ways tointerpret and use these aspects (Moll, 1990; Vygotski, 1978; Wertsch, 1991). Thedevelopment of the mind is a gradual process, and people are supported over theyears across different activities to come to make sense of the world. By focusingon particular aspects in the process of the activities (e.g., relationships amongquantities), people develop ways to understand the world that are valued in thecommunity (e.g., different ideas and connections among them in mathematicslearning) (Rogoff, 1990; Saxe, 1991).

Vygotski (1978, 1999) discussed how tools are a part of cultural practices. Healso described how the ways that participants make sense of the tools in theirpractice determine what is valued in the culture. Tools shape cultural practices inessential ways as mediational means, and the ways in which a particular conceptis understood in social interactions and is supported by tools influence the devel-opment of the shared cognitive process in the community. Shared cultural valuesthen guide people’s activities in the future (in this case, how they go about doingmathematics), while the interactions between the cultural ways of thinking andthe activities continuously refine and change each other.

Learners must be assisted through the process of meaning construction so thatthey can eventually manage comprehension by themselves in the absence ofmore knowledgeable others who provide interpretations (Tharp & Gallimore,1988). A tool can work to provide the necessary support before learners can man-age concepts on their own. It then becomes a foundational cognitive structure toorganize information in related or unrelated situations. In the case of Japanesemathematics instruction, tape diagrams come to be simply “background music”so eventually students can “hear” the important relationships in their problemsmore clearly.

In terms of mathematics teaching and learning, it is not only tools but otheraspects of the instructional experiences that also influence and support the viewof learning mathematics harmoniously. The U.S. and Japanese textbook seriestook different approaches to incorporating representations to support students’learning: the U.S. series used representations to narrow students’ thinking to helpthem solve problems in the manner students previously learned with numerals;the Japanese series used representations to create space for students to analyzemathematical situations. These different approaches are also found in otheraspects of mathematics teaching in the two cultures. In traditional U.S. curricula,contextual problems are placed at the end of the lessons/units to apply what stu-dents have just learned. In Japanese mathematics curricula, contextual problemsare typically presented at the beginning of the lessons/units for students to

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analyze the situations, and mathematical ideas are extracted out of the contextslater on. The purpose of the contextual problems in U.S. curricula is application,while in Japanese curricula, it is analysis. In the application approach, studentsenter the problem situations knowing that they are to use what they have justlearned to solve them, and thus quickly move to make the contextual problemsinto numerical ones (Boaler, 1994b; Brown, Collins, & Deguid, 1989). In theanalysis approach, while students know that mathematical ideas will follow con-textual problems, space is allowed to think about the presented situations withouta known goal so that students engage in the situation in more meaningful ways.

When students come to school with varied backgrounds, providing only dis-crete and abstract (numerical) mathematical information to be memorized willnot help all students develop fluency and understanding. An individual student’scognitive development depends on how cultural and mathematical knowledge ispulled closer to where students are cognitively at present (Murata, 2004). Thisarticle described how a consistent and meaningful use of tape diagrams may helpground these different types of knowledge in practice in Japanese classrooms,and bridge the distance by providing coherent experiences to create particularmathematics experiences and understanding.

IMPLICATION AND CONTRIBUTION OF THE STUDY

This study has several implications for curriculum designers, teacher educators,professional developers, and teachers. While it is important for students to under-stand mathematical concepts and to be able to use multiple representations toshow their understanding of the concepts (NCTM, 2000), this study argued thatsome consistent and coherent use of one representation can also bridge studentunderstanding over time. Textbooks typically combine different representationsto support development of student understanding, and the process of going backand forth between different representations will push students to see the coreconcepts across different cognitive approaches. But the selection of different rep-resentations for a concept should not be random, and when possible, using acommon representation can also provide the familiarity for students who arelearning new concepts. In considering the meanings different representationsmay mediate, it is important to focus on what mathematics concept particular fea-tures of a representation (or multiple representations) help highlight and develop.We will need to pay closer attention to this aspect of curriculum design in thefuture.

When considering mathematics teaching and learning in the United States,there is a taken-as-shared cultural pattern to follow, such as showing students astep-by-step procedure, having them practice similar problems, and presentingcontextual problems for application of the concept they have learned. This article


has presented a contrasting pattern, has highlighted the key differences and theirreasons for the differences, and has examined how mathematics may be learneddifferently with a focus on representation uses. Importing a cultural practice to adifferent context by following the surface features will not produce meaningfulresults, and good ideas have been historically proven ineffective when treatedsuperficially (Lewis, Perry, & Murata, 2006). Understanding the reasons behindthe cultural pattern should assist curriculum designers, teachers, and teacher edu-cators in other nations to modify and adapt parts of the process more meaning-fully to their existing systems.

NOTES

1. Many researchers consider language as a primary tool for such instances. For this article, I amusing the term “tools” to more broadly discuss any mediating objects (concrete or non-concrete) thatare used in cultural activities. In mathematics classrooms, mathematical symbols (e.g., + and =),visual representations, and concrete manipulatives are examples of the tools.

2. Japanese schools adhere to a national curriculum. The Japanese Ministry of Education, Culture,Sports, Science, and Technology determines and publishes a Course of Study that includes generaleducation guidelines for schools (including broader national goals and guidelines for differentsubject areas and grade levels). This Course of Study is revised approximately once every tenyears. Publishing companies then closely follow the guidelines to design their textbooks. Thereare six elementary mathematics textbook publishers in Japan, and all share a common structure,organization, and presentation of mathematics topics for each grade and thus use tape diagrams.The grade 1 to 6 textbook series analyzed for this study, Study with Your Friends Mathematics(Gakkotosho, 2005) is one of the most commonly used textbook series (Japanese Ministry ofEducation, 2006). It is also available in English.

3. In the sections that follow, text at some places may seem to suggest student learning, however thisstudy did not collect empirical data on student learning. What the article intends to communicateis that the Japanese curricular approaches found in the study tend to support certain student learning.These claims on student learning are also grounded in the high achievement of Japanese studentsin international comparative studies as well as the author’s prior studies based on empirical data(Murata, 2004; Murata, Hattori, Otani, & Fuson, 2004).

4. While U.S. research identifies three major story problem types for students’ addition-subtractionlearning (join, change, compare; see Carpenter, 1999), in Japanese textbooks, there is the fourthcategory: order. In “order” problems, objects/people are typically lined up, and by using the infor-mation of the order of particular object/people in the line, students find answers: “Some childrenare standing in a line. Takeo is the fourth from the front; Yoko is the fifth person behind Takeo.What is the order of Yoko from the front?”

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