Mathematics is often seen as completely separated from the daily life. This is why people need some connecting bridge. Many said that mathematical representation serves the bridge role for people to understand and express mathematical ideas. Representation consists of internal and external representation. However, representation term that researchers used mostly refers to only external part. Real-world problems can be represented using formula, visual, concrete, etc. This paper aimed to review results of researches related to mathematical representation. This paper will emphasize the role of using multiple representations in mathematics learning, the challenges that students and teachers face related to representation and teachers' role in promoting students' mathematical representation.

Concrete, pictorial and abstract representations on simple subtraction problems (Miller & Hudson, 2006 : 29).
. Mathematics instruction using representation according to Mitchel

Figures - uploaded by Aulia Ul Fitrah Samsuddin

Author content

All figure content in this area was uploaded by Aulia Ul Fitrah Samsuddin

Content may be subject to copyright.

ResearchGate Logo

Discover the world's research

  • 20+ million members
  • 135+ million publications
  • 700k+ research projects

Join for free

Journal of Physics: Conference Series

PAPER • OPEN ACCESS

Mathematical representation: the roles, challenges and implication on

instruction

To cite this article: A F Samsuddin and H Retnawati 2018 J. Phys.: Conf. Ser. 1097 012152

View the article online for updates and enhancements.

This content was downloaded from IP address 101.203.171.2 on 23/10/2018 at 08:19

1

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution

of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Published under licence by IOP Publishing Ltd

1234567890 ''""

ICRIEMS 5 IOP Publishing

IOP Conf. Series: Journal of Physics: Conf. Series 1097 (2018) 012152 doi :10.1088/1742-6596/1097/1/012152

Mathematical representation: the roles, challenges and

implication on instruction

A F Samsuddin1, H Retnawati1

1 Mathematics Education Department of Graduate School, Yogyakarta State

University Jl. Colombo No.1, Sleman, D.I. Yogyakarta, Indonesia 55281

Corresponding author: auliaulfitrah.2017@student.uny.ac.id

Abstract. Mathematics is often seen as completely separated from the daily life. This is why

people need some connecting bridge. Many said that mathematical representation serves the

bridge role for people to understand and express mathematical ideas. Representation consists of

internal and external representation. However, representation term that researchers used mostly

refers to only external part. Real-world problems can be represented using formula, visual,

concrete, etc. This paper aimed to review results of researches related to mathematical

representation. This paper will emphasize the role of using multiple representations in

mathematics learning, the challenges that students and teachers face related to representation and

teachers' role in promoting students' mathematical representation.

1. Introduction

If the real world is assumed as a land and mathematics is another land separated by a river, different and

strange to each other. A bridge is needed to link those lands. Representation served the role of such

bridge connecting the abstract mathematics concept with daily life context. In line with that, Matteson

[1] analogued learning mathematics as learning foreign language and representation is the key elements

for those who wants to understand and express mathematical ideas conveniently. In line with that

Kilpatrick [2] stated that representation is a complex system and is like a foreign language for students

since students learn to use representation while simultaneously using it to learn othe things. Duval [3]

added mathematical objects (ideas, concepts and relations) can only be accessed through representation

and its activities are affected by how the representation is used.

This paper aims to present the readers a deeper insight on mathematical representation from literature

and empirical study. As the beginning, representation will be elaborated as well as the role representation

has in mathematics learning. The discussion will be continued with definition and role of using multiple

representation in mathematic s. The paper will be completed with several challenges on using multiple

representation and the possible strategies which teachers can implement to minimize or overcame the

challenges faced by students.

2. Representations: What and Why?

Mathematical entities can be represented in multiple ways. For example, the number six can be

represented by collection of real objects, such as six beads, six apples or six straws, by iconic pictures,

such as images of six circles, or by abstract symbols like 6 or VI. These representations are observable.

Generally, mathematical representation consists of two inseparable parts, namely: (1) external

2

1234567890 ''""

ICRIEMS 5 IOP Publishing

IOP Conf. Series: Journal of Physics: Conf. Series 1097 (2018) 012152 doi :10.1088/1742-6596/1097/1/012152

representation, one that physically exist and observable, such as graphic, pictures, equations and table;

(2) internal representation, model, scheme or concepts which is mental or cognitive and cannot be

directly observable [4]. Hereinafter, the term representation refers merely to external representation.

Miller & Hudson [5] classified representation into three types, namely: (1) concrete, (2)

representational, and (3) abstract. This classification is similar to representation mode by Bruner in

Hebert & Powell [6] which are (1) enactive, (2) iconic, (3) symbolic. Suppose a teacher is using plastic

cube to represent problems on basic operations to primary students. Another teacher is asking his or her

students to walk around the edge of the classroom to introduce the concept of perimeter. Both teachers

are actually utilizing concrete representation. In the representational stage teacher can use picture

representing the plastic cube, instead of the real one. Gradually, sttudents can use only number or

abstract representation. Figure 1 showed simple subtraction problem represented concretely,

pictorial/representationally and abstractly.

Figure 1 Concrete, pictorial and abstract representations on simple subtraction problems

(Miller & Hudson, 2006 : 29).

Drawn from numbers of literature, the roles of representation in mathematics learning are : (1)

representation helps students in making sense mathematics tasks and concepts [7], (2) representation

facilitates students' learning process [8], (3) representation helps students in managing and expressing

their thinking as well as making mental model of their mathematical ideas [9], (4) representation helps

students understanding abstract mathematical concepts [10], as well as (5) being used to mathematics

problems with multiple representation helps students in analyzing problems [10].

Representation, whether it is physical, spoken or written, is necessary to communicate about

mathematical numbers and operations [1]. Plenty sources suggested the need for employing multiple

representation in mathematics instruction. This is because mathematical ideas are enhanced through

multiple representation. The multiple representation possesses roles not only as the instructional tricks

but also as a source of mathematical reasoning [2]. Having experiences with multiple representation

(graphic, table and algebreic) helps students construct a broader synergy related to Mathematics and

enables them to minimize their difficulties in a certain topics, Function, for instance [11]. Another

evidence was provided by Cai's findings [12] about Chinese students showing a better result than

students in America on process-restricted problems, but on non routine questions that does not have a

standard algorithm, American students showed a better performance than those in China. Cai & Cifarelli

[13] tried to dig deeper about this issue and found that there is a correlation between multiple

representation in problem solving and students' performance in a various assesment task.

Ainsworth [14] stated functions of using multiple representation in instructions, such as : 1) Conveying

different information. Sometimes, single representation is not enough to carry all the informations to be

represented or would be too complicated for students, 2) minimalizing possible misinterpretations of a

representation or domain, and 3) promoting deeper understanding of ideas being represented.

3

1234567890 ''""

ICRIEMS 5 IOP Publishing

IOP Conf. Series: Journal of Physics: Conf. Series 1097 (2018) 012152 doi :10.1088/1742-6596/1097/1/012152

3. Challenges in using Representations in Mathematics Instruction

Since there is an urge to use multiple representation in mathematics learning, one should be able to

choose and translate among representations. The ability to choose the best representation for

mathematical ideas is necessary since algorithms depend upon representation [2]. This is in line with

what Retnawati et al [15] found when investigating the difficulties students faced in solving geometry

problems in National Examination in Indonesia. One of factors found is students' lack of mathematical

representation. Unfortunately, behind the broad advantages representation has, researches showed that

students face difficulties related to mathematical representation. These hardships due to the lack of

diagrammatic knowledge needed in representation, ability to interpret representation by connecting it to

the real world, or being unable to translate among representations in the same domain [16]

Another distress happened when the use of representation is not accompanied by students'

mathematical comprehension. Students are forced to follow teachers' preferred procedures without

opportunity to reflect the activities and assistance to link representation with the underlaid mathematical

ideas [17].

Difficulties can also be emerged because of the double yet not compatible conditions in Mathematics.

Students have to use representation for abstract mathematical objects, and at the same time they have to

understand those objects [3]. Based on Leikin's findings [18] it can be implied that working in

mathematics with representation can foster students' speed in solving the problem, but unfortunately it

is not the case with their accuracy. This is explained by the split-attention effect representation has.

It is a challenge as well when students regard representation and the concept its represented as two

separated things. This is validated by Adu-Gyamfi & Bosse [19] who conducted research on 8 high

school students majoring pre-calculus in south-eastern United States. It is found that most students

answer correctly questions measuring representation skill generally. However, when students are asked

to identify function displayed in graph, table and equation among relations (not function), also

represented in multiple representation, the results is not in harmony with the first result

Stylianou [20] found that representation is also affected by teachers' perception about representation.

Teachers who perceive representation as another learning topic or concept in mathematics, not as a tool

to understand the concept itself, usually does not consider representation has a central role in

mathematics learning. A teacher said "you still have to teach them all the other things". Another teacher

expressed similar idea which is "in the State exam practices representation is not required." Stylianou

also stated that teachers who have such opinion believe that representation is more suitable to high-

performed students and will only make the other students confused.

According to Ainsworth [14] students face several learning demands of multiple representation,

which are: a) students need to learn the format and operators of a representation. In the case of graph,

the format would be lines, labels and axes, while the operators are how to find gradient of lines,

intercepts, etc, b) students have to learn the relation between the representation and the concept

represented and c) students have to learn the relation between the representations. Students who fail to

satisfy one of the preceding demands are predicted not to be able to benefit fully from multiple

representation advantage.

Adu-gyamfi [21] classified three types of error that students made in translation process (convert

from one representation mode to another): a) Implementation error, b) Interpretation error, and c)

Preservation error. Implementation error is related to computational or algorithmic mistakes.

Interpretation error occurs when students fail to understand the characteristics or properties of either the

source or the target of representation. Preservation error occurs when students are able to maintain the

semantic congruence of some, not all the attributes of representation

Teachers' content knowledge is also a central issue related to the success of mathematics learning

with representation. How can a teacher implement a successful instruction, in top of that with multiple

representation, if the teacher does not have sufficient knowledge about the topics. An example is

provided by Retnawati et al [22]) that showed teachers in the research have difficulties with problems

related to function, sub-topic with multiple possible representation.

4

1234567890 ''""

ICRIEMS 5 IOP Publishing

IOP Conf. Series: Journal of Physics: Conf. Series 1097 (2018) 012152 doi :10.1088/1742-6596/1097/1/012152

4. Implication in Instruction

As stated before, one of the challenges in mathematics using representation has to do with the teachers.

Teachers should have a clear concept about representation so that they can employ it in the classroom.

Based on NCTM [23], learning standards expected to be satisfied for level pre-kindergarten to K-12 are:

1. Create and use representations to organize, record, and communicate mathematical ideas

2. Select, apply, and translate among mathematical representations to solve problems

3. Use representations to model and interpret physical, social, and mathematical phenomena.

NCTM elaborated in detail teachers' role in promoting students' mathematical representation ability

based on their school level. In kindergarten to K-2, teachers' role is mainly to create learning situation

in which students can use multiple representations. Beside that, teachers need to motivate students to

communicate their preferred representation since written work ususally cannot reveal students' whole

thinking process. In level K-2 to K-5, teachers may begin to invite students discussing why there is

representation more effective than others in a certain context. For students in K-6 and K-8, teachers can

help students developing their self-belief and competence in making their own representation in a certain

context or problems. Teachers can help students linking the use of representation with daily life

situation. Table 1 documented Mitchell's role of teachers related to the use of representation generally

(regardless school level). Representation brings positive influence towards problem solving. This

applies conversely that by conducting mathematics instruction based on problem will enhance

mathematical representation of students [24]

Table 1. Mathematics instruction using representation according to Mitchel

Mitchel's Role of Teachers

Representing and solving

problem/carrying out mathematical

operations

Recognizing and abiding by the representations'

conventions

Using representations as a means to illuminate certain

mathematical ideas involved in a procedure

Employing appropriate language and notation when using

representations

Decomposing and unpacking mathematical rules and

operations through careful use of representations

Selecting representations that lend themselves to

explaining a mathematical procedure

Creating a context for connecting multiple

representations

Identifying similarities and differences between

representations

Using one representation to help students make sense of

another

Creating a context for generalizing

procedures

Using representations to build generalizations and help

students move to a more abstract level

Selecting and sequencing examples to support student

ability to generalize

Using multiple representations to help students make sense

of the underlying meaning of a mathematical procedure

Scaffolding student work on

representations and the mathematics

Using representations to surface student misconcceptions

and emphasize important mathematical ideas

5

1234567890 ''""

ICRIEMS 5 IOP Publishing

IOP Conf. Series: Journal of Physics: Conf. Series 1097 (2018) 012152 doi :10.1088/1742-6596/1097/1/012152

Using representations to trigger and remediate student

misconceptions

Flexibly moving between representations to support

student understanding

Providing a balance between explaining the representation

conventions and allowing students the space and time to

make meaning of the representations and the mathematical

ideas they are intended to illuminate

Examining whether students correctly follow the

representations' conventions and ascribe meaning to the

representations' manipulations

Pressing students to articulate the mathematical meaning

they are making out of using representations

Listening to students and unpacking their (promising)

productions around using representations

Differentiating the scaffolding provided to students

depending on

(a) the anticipated level of transparency of a given

representation and

(b) students' differential needs and their progress toward

abstracting the underlying mathematical ideas the

representation are intended to illuminate.

Bosse [25] outlined reccomendation for teachers to facilitate students' mathematics learning in respect

to translation among multiple representation, which are:

1. Teachers should realize that their beliefs regarding which representation mode students can do,

cannot do, and hould be able to do may affect instructional plans

2. Teachers should recognize which translations are more difficult than others. For instance, students

may find translation from symbolic to tabular representation easier than translation from

graphical to symbolic translation.

3. Teachers have to assure that students learn all the representations with the translation, particularly

those which are more difficult.

4. Teachers need to take everything that support students' translation to consider in the learning

process, for instance, teachers' questioning techniques.

5. Teachers should use the assessment type where students are asked to apply multiple translations

and not only translation that teacher believe students can perform.

6. Teacher can use real world contexts which are familiar to students.

7. Teacher can use a rich-tasks to engage students.

5. Conclusion

Representation has a central role in mathematics instruction. Representation fasilitates students in

understanding abstract mathematical concepts or ideas. Generally representation can be classified into

three modes, concrete (real manipulative object or activity), pictorial/representational (pictures, graph,

etc) and abstract representation (mathematics equation). Literatures suggested the need to use multiple

representation in mathematics instruction. However, in the reality there are challenges in conducting

6

1234567890 ''""

ICRIEMS 5 IOP Publishing

IOP Conf. Series: Journal of Physics: Conf. Series 1097 (2018) 012152 doi :10.1088/1742-6596/1097/1/012152

mathematics learning using representation. One of the case is when students perceive representation and

the mathematics concept being represented are two separated things. It is followed by students know

how to use multiple representation (graph, table and equation) with function, for example, but they

cannot understand the function itself. Another challenge is when teachers as a learning facilitator see

representation as a product only, not as a process in understanding mathematics. If it happens, teachers

will face difficulties in implementing mathematics learning with representation. Teachers' roles in

mathematics classroom with representation are Representing and solving problem/carrying out

mathematical operations, Creating a context for connecting multiple representations and for generalizing

procedures, and Scaffolding student work on representations and the mathematics.

6. References

[1] Matteson S M 2006 Mathematical literacy and standardized mathematical assessments Read.

Psychol. 27 no. 2– 3 pp. 205–233

[2] Kilpatrick J, Swafford J, and Findell B 2001 Adding It Up

[3] Duval R 2006 A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics

Author (s): Raymond Duval Source : Educational Studies in Mathematics , 61 , No . 1 / 2

Semiotic Perspectives in Mathematics Education : A PME Special Issue 2006 Educ. Stud.

Math. vo l. 61 no. 1 pp. 103–131

[4] Goldin G and Kaput J "A joint perspective on the idea of representation in learning and doing

mathematics," Theor. Math. Learn., no. September, pp. 397 – 430, 1996

[5] Miller S P and Hudson P J 2003 Techniques for Program Support Helping Students With

Mathematics Means

[6] Hebert M A and Powell S R, 2016 Examining fourth-grade mathematics writing: features of

organization, mathematics vocabulary, and mathematical representations Read. Writ., vol. 29

no. 7 pp. 1511–1537

[7] Mitchell R, Charalambous C Y, and Hill H C 2014 Examining the task and knowledge demands

needed to teach with representations J. Math. Teach. Educ. 17 37

[8] Martin L C 2008 Folding back and the dynamical growth of mathematical understanding:

Elaborating the Pirie-Kieren Theory J. Math. Behav. 27 64

[9] Schwarz B B, Kohn A S, and Resnick L B, 1994 Positives About Negatives: A Case Study of an

Intermediate Model for Signed Numbers J. Learn. Sci. 3 37

[10] Kang R and Liu D 2018 The Importance of Multiple Representations of Mathematical Problems:

Evidence from Chinese Preservice Elementary Teachers' Analysis of a Learning Goal Int. J.

Sci. Math. Educ . 16 125

[11] Even R 1998 Factors involved in linking representations of functions J. Math. Behav. 17 105

[12] Cai J 2000 "Mathematical Thinking Involved in U.S. and Chinese Students' Solving of Process-

Constrained and Process-Open Problems Math. Think. Learn. 2 309

[13] Cai J and Victor C 2004 Mathematical Thinking Involved in U.S. and Chinese Students' Solving

of Process-Constrained and Process-Open Problems in How Chinese Learn Mathematics:

Perspectives from Insiders pp. 71–106.

[14] Ainswroth S E 1999 Designing Effective multirepresentational Learning Environments ESRC Cent.

Res. Dev. Instr. Train. Univ. Nottingham vol. technical, p. PhD thesis and technical report

number 47

[15] Retnawati H, Arlinwibowo J, and Sulistyaningsih E, 2017 The Students' Difficulties in Completing

Geometry Items of National Examination Int. J. New Trends Educ. Their Implic.8 28

[16] Bock D D, Dooren W V, and Verschaffel L, 2015 Students' understanding of proportional

inverse proportional, and affine functions: two studies on the role of external representations Int.

J. Sci. Math. Educ. 13 47

7

1234567890 ''""

ICRIEMS 5 IOP Publishing

IOP Conf. Series: Journal of Physics: Conf. Series 1097 (2018) 012152 doi :10.1088/1742-6596/1097/1/012152

[17] Stein M K and Bovalino J W 2001 Manipulatives: One piece of the puzzle Math. Teach. Middle

Sch.

[18] Leikin R, Leikin M, Waisman I, and Shaul S 2013 Effect of the Presence of External

Representations on Accuracy and Reaction Time in Solving Mathematical Double-Choice

Problems By Students of Different Levels of Instruction Int. J. Sci. Math. Educ. vol. 11 no. 5

pp. 1049–1066

[19] Adu-Gyamfi K and Bossé M J 2014 Processes and reasoning in representations of linear functions

Int. J. Sci. Math. Educ. vol. 12 no. 1 pp. 167–192

[20] Stylianou D A 2010 Teachers' conceptions of representation in middle school mathematics

" J. Math. Teach. Educ. vol. 13 no. 4 pp. 325 – 343

[21] Adu-Gyamfi K, Bossé M J, and Stiff L V, 2012 Lost in Translation: Examining Translation Errors

Associated With Mathematical Representations Sch. Sci. Math. vol. 112 no. 3 pp. 159–170

[22] N C of T. of Mathematics 2000 Principles and standards for school mathematics

[23] NCTM 2000 Principles and standards for school mathematics

[24] Farhan M and Retnawati H, 2014 Keefektifan PBL Dan IBL Ditinjau dari Prestasi Belajar,

Kemampuan Representasi Matematis, dan Motivasi Belajar J. Ris. Pendidik. Mat

[25] Bossé M J, Adu-Gyamfi K, and Cheetham M R 2011 Assessing the difficulty of mathematical

translations: Synthesizing the literature and novel findings Int. Electron. J. Math. Educ. vol. 6

no. 3 pp. 113–133

... Representation is also affected by teachers' perception about representation [22]. Teachers' content knowledge is also a central issue related to the success of mathematics learning with representation [23]. ...

... The use of representation will support the student in creating more concrete mathematical ideas since the complex problem can be simplified by using appropriate mathematical representation [8], [11], [12]. In addition, different uses of representations can be useful support to enable students in learning as can be seen in numerous studies (see [13], [14]& [15]). In the previous studies, the students were reported to be aware in using visual and symbolic representations in their mathematical problem solving [16], [17]& [18]. ...

... Hal ini sejalan dengan pendapat Kashuba et al (2017) bahwa siswa lebih memilihi menggunakan perhitungan bentuk aljabar ketika bertemu dengan masalah matematis. Hal ini didukung dengan pendapat Samsuddin & Retnawati (2018) bahwa hanya anak yang berani dan percaya diri yang akan menggunakan representasi visual, selain itu anak yang memiliki kemampuan visual-spasial rendah akan kesulitan menggunakan gambar untuk menyelesaikan masalah matematis. Akibatnya anak akan menggunakan representasi aljabar untuk menyelesaikan masalah. ...

Abstrak: Penelitian kualitatif deskriptif ini bertujuan untuk mendeskripsikan kemampuan representasi matematis khususnya repesentasi visual ditinjau dari bakat musik anak dalam meyelesaikan masalah matematis. Siswa dikelompokkan menjadi dua kategori, yaitu anak yang berbakat di bidang musik dan anak yang tidak berbakat di bidang musik. Dua subjek dipilih berdasarkan diskusi dengan ahli dan pemberian soal tes matematis. Subjek yang dipilih berada pada jenjang pendidikan yang sama yakni kelas VIII. Hasil penelitian ini menunjukan bahwa anak yang berbakat di bidang musik memiliki kemampuan representasi visual yang lebih baik dibandingkan anak yang tidak berbakat di bidang musik. Anak yang berbakat di bidang musik melakukan model matematika bergambar untuk menyelesaikan masalah. Sedangan anak yang tidak berbakat di bidang musik menggunakan model matematika dalam bentuk ekspresi matematis untuk menyelesaikan masalah. Abstract: This descriptive qualitative research aims to describe mathematical representation ability, especially visual representation in terms of children's musical talent in solving mathematical problems. Students are grouped into two categories, namely children who are gifted in music and children who are not gifted in music. Two subjects were chosen based on discussions with experts and the provision of mathematical test questions. The subject chosen was at the same level of education, namely grade VIII junior high school students. The results of this study indicate that children who are gifted in music have better visual representation abilities than children who are not gifted in music. Children who are gifted in music do pictorial mathematical models to solve problems. Whereas children who are not gifted in music use mathematical models in the form of mathematical expressions to solve problems.

  • Michael Hebert
  • Sarah R Powell Sarah R Powell

Increasingly, students are expected to write about mathematics. Mathematics writing may be informal (e.g., journals, exit slips) or formal (e.g., writing prompts on high-stakes mathematics assessments). In order to develop an effective mathematics-writing intervention, research needs to be conducted on how students organize mathematics writing and use writing features to convey mathematics knowledge. We collected mathematics-writing samples from 155 4th-grade students in 2 states. Each student wrote about a computation word problem and fraction representations. We compared mathematics-writing samples to a norm-referenced measure of essay writing to examine similarities in how students use writing features such as introductions, conclusions, paragraphs, and transition words. We also analyzed the mathematics vocabulary terms that students incorporated within their writing and whether mathematics computation skills were related to the mathematics vocabulary students used in writing. Finally, we coded and described how students used mathematics representations in their writing. Findings indicate that students use organizational features of writing differently across the norm-referenced measure of essay writing and their mathematics writing. Students also use mathematics vocabulary and representations with different levels of success. Implications for assessment, practice, and intervention development are discussed.

  • Michael J. Bossé
  • Kwaku Adu-Gyamfi Kwaku Adu-Gyamfi
  • Meredith R Cheetham

Students perennially demonstrate difficulty in correctly performing mathematical translations between and among mathematical representations. This investigation considers the respective difficulty of various mathematical translations based on student activity (defining mathematical errors during the translation process, teacher beliefs and instructional practices, student interpretive and translation activities, and the use of transitional representations) and the nature of individual representations (fact gaps, confounding facts, and attribute density). These dimensions are synthesized into a more complete model through which to analyze student translation work and delineate which mathematical translations are more difficult than others.

  • Rui Kang Rui Kang
  • Di Liu

This article describes a study of how Chinese preservice teachers unpacked a learning goal pertaining to adding fractions and understanding the concepts underlying the operation. Based on work in the USA by Morris, Hiebert, and Spizter (Journal for Research in Mathematics Education, 40(5), 491–529, 2009), 50 Chinese preservice teachers completed a task, anticipating an ideal student response, in the context of four representations: (1) fraction pieces, (2) graph paper, (3) common denominator algorithm, and (4) pennies. Like the US-based study, this study shows that Chinese preservice teachers' ability to unpack a learning goal was highly influenced by how the problem was represented. The pennies and graph paper problems provided more supportive contexts for unpacking the learning goal; the algorithm problem provided the least supportive context. The main difference between the preservice teachers from these two countries was the US preservice teachers chose the pennies and graph paper problems as having the most potential for revealing students' understanding of the learning goal; while the Chinese preservice teachers chose the algorithm problem despite the problem's unsupportive context. Chinese preservice teachers' preference suggests that they privileged algorithmic/symbolic representations over pictorial/concrete representations. Based on our results, we argue that it is time for cross-cultural comparative research to refocus on the common barriers, challenges, and benefits as related to mathematics teacher preparation and professional development. A cooperative instead of competitive orientation will lead to richer and deeper dialogues among mathematics educators.

  • Jinfa Cai Jinfa Cai
  • Victor V. Cifarelli

Based on the findings from a number of cross-national comparative studies of US and Chinese students, we provided a retrospective review of these studies and presented a profile of Chinese learners' mathematical thinking in problem solving and problem posing. In particular, we identified several characteristics of Chinese learners' mathematical thinking in problem solving as well as pointed out some future directions to refine and extend this list of characteristics of Chinese learners. This chapter not only helps us understand the nature of Chinese students' mathematical thinking from a cross-national comparative perspective, but also provides information to refine instructional programs so that Chinese students' mathematical thinking can be better nurtured and developed.

We investigated students' understanding of proportional, inverse proportional, and affine functions and the way this understanding is affected by various external representations. In a first study, we focus on students' ability to model textual descriptions of situations with different kinds of representations of proportional, inverse proportional, and affine functions. Results highlight that students tend to confuse these models and that the representational mode has an impact on this confusion: Correct reasoning about a situation with 1 mathematical model can be facilitated in a particular representation, while the same representation is misleading for situations with another model. In a second study, we investigate students' ability to link representations of proportional, inverse proportional, and affine functions to other representations of the same functions. Results indicate that students make most errors for decreasing functions. The number and nature of the errors also strongly depend on the kind of representational connection to be made. Both studies provide evidence for the strong impact of representations in students' thinking about these different types of functions.

Translation errors and conceptual misunderstandings made by students translating among graphical, tabular, and symbolic representations of linear functions were examined. The study situated student errors in the context of the "Translation-Verification Model" developed specifically for the purpose of explaining student behavior during the process of translating relationships from one mathematical representation to another. Three distinct error types were identified to explain student performance. An examination of the error types revealed that specific translation errors tend to occur at different stages of the translation process. Translation errors are also related to "attribute density," the amount of information inherently encoded in a given representation. The findings of the study have implications for teaching linear relationships—student weaknesses and strengths are identified.

  • Rebecca Mitchell
  • Charalambos Y. Charalambous Charalambos Y. Charalambous
  • Heather C. Hill

Representations are often used in instruction to highlight key mathematical ideas and support student learning. Despite their centrality in scaffolding teaching and learning, most of our understanding about the tasks involved with using representations in instruction and the knowledge requirements imposed on teachers when using these aids is theoretical. In this study, we examine the task and knowledge demands for teaching integer operations with representations by analyzing teaching practice. Teaching integer operations is used as an intensity case, as integer operations are challenging for students, and teachers are often required to employ several representations to teach this topic. Following a practice-based approach while also taking prior literature into consideration, we first generate a list of tasks entailed in teaching with representations and then discuss the knowledge demands imposed on teachers to successfully undertake this work. We highlight these tasks and knowledge demands by analyzing and discussing an integer addition and an integer subtraction episode for each of two teachers, Bonita and Karen. Based on our analysis, we organize the generated knowledge components using the Mathematical Knowledge for Teaching framework. We conclude by drawing implications for teacher educators and curriculum developers.

  • Kwaku Adu-Gyamfi Kwaku Adu-Gyamfi
  • Michael J. Bossé

This study examined student actions, interpretations, and language in respect to questions raised regarding tabular, graphical, and algebraic representations in the context of functions. The purpose was to investigate students' interpretations and specific ways of working within table, graph, and the algebraic on notions fundamental to a conceptualization of linear functions. Through a case study method which investigated individual representations and student articulations within them, the study revealed that students can make a transition from a given representation of linear function to another and yet demonstrate limited understanding of linear functions.

This study explores the effects of the presence of external representations of a mathematical object (ERs) on problem solving performance associated with short double-choice problems. The problems were borrowed from secondary school algebra and geometry, and the ERs were either formulas, graphs of functions, or drawings of geometric figures. Performance was evaluated according to the reaction time (RT) required for solving the problem and the accuracy of the answer. Thirty high school students studying at high and regular levels of instruction in mathematics (HL and RL) were asked to solve half of the problems with ERs and half of the problems without ERs. Each task was solved by half of the students with ERs and by half of the students without ERs. We found main effects of the representation mode with particular effect on the RT and the main effects of the level of mathematical instruction and mathematical subject with particular influence on the accuracy of students' responses. We explain our findings using the cognitive load theory and hypothesize that these findings are associated with the different cognitive processes related to geometry and algebra.