FROM ONE LANGUAGE TO ANOTHER: A SEMIOTIC INTERPRETATION OF THE TRANSLATION OF MATHEMATICAL WORDS

The teaching and learning of Primary school mathematics in Malta involves the use of code-switching between the local language Maltese, and English Mathematical terms themselves are usually retained in English and teachers may use various strategies to share the meaning of these words with their pupils. One strategy that may be used in a bilingual situation is translation from one language to another. In this paper I explore how a teacher used this strategy to teach her 7 to 8-year-old pupils mathematical vocabulary related to the topic'Money and Shopping'. While Maltese equivalents for these words exist, it is the English versions that form part of the school mathematics register. I develop a semiotic model where a mathematical word is considered to be a sign, and the process of translation is viewed as a chain of signification from one language to another.     

Language and communication in mathematics education: an overview of research in the field

Within the field of mathematics education, the central role language plays in the learning, teaching, and doing of mathematics is increasingly recognised, but there is not agreement about what this role (or these roles) might be or even about what the term ‘language’ itself encompasses. In this issue of ZDM, we have compiled a collection of scholarship on language in mathematics education research, representing a range of approaches to the topic. In this survey paper, we outline a categorisation of ways of conceiving of language and its relevance to mathematics education, the theoretical resources drawn upon to systematise these conceptions, and the methodological approaches employed by researchers. We identify four broad areas of concern in mathematics education that are addressed by language-oriented research: analysis of the development of students’ mathematical knowledge; understanding the shaping of mathematical activity; understanding processes of teaching and learning in relation to other social interactions; and multilingual contexts. A further area of concern that has not yet received substantial attention within mathematics education research is the development of the linguistic competencies and knowledge required for participation in mathematical practices. We also discuss methodological issues raised by the dominance of English within the international research community and suggest some implications for researchers, editors and publishers.     

Chinese and English Mathematics Language: The Relation Between Linguistic Clarity and Mathematics Performance

In Study 1, 48 judges rated the clarity of Chinese, English, and "Chinglish" (Chinese words translated into English) mathematical words-for example, the Chinglish version of the Chinese word for quadrilateral is "four-side-shape." Native Chinese-speaking judges achieve greater agreement on the relative clarity of Chinese words than do native English-speaking judges on the relative clarity of English words. More Chinese words are rated clear than are English. Chinglish mathematical words tend to be rated more clear than English. The inherent compound word structure of the Chinese language seems well suited to portray mathematical ideas.

In Study 2, we examined the relations among the clarity of Chinese mathematical terms, U.S. urban junior high school students' Chinese reading ability, and their mathematics performance. There is a strong correlation between Chinese reading ability and performance on test items with mathematics words rated clear by Chinese judges. The relative clarity of mathematical terms in the Chinese language may contribute to Chinese-speaking students' understanding of mathematics and to superior mathematics performance.     

Teaching teachers mathematics through programming

There are two main arguments underlying the claims for the value of interactive computer programming used by students to model mathematical ideas. One is concerned with mathematical content, i.e. with mathematics as an object of study. The other is concerned with mathematical activity, i.e. doing mathematics, or ‘Mathematicking’ [1]. Both content and activity include processes and these provide the main links with programming. Examples of processes in the content of mathematics are addition, transformation and integration, and these can be described by instructions in a computer program. Examples of process in the activity are problem‐solving, proof generation and pattern finding which can be described by analogy to program building and debugging. We assess the arguments for programming, in relation to the training of teachers, and describe a pilot‐study in which student teachers with mathematical difficulties were taught the programming language LOGO. Observation of the students, learning the language and using it to manipulate computer models of mathematical ideas, which they had not understood previously, highlights both advantages and disadvantages in this approach. The problem of the representation of mathematical ideas within programming projects is discussed.     

A theorical point of view of reality,perception, and language

It is possible to view the relations between mathematics and natural language from different aspects. This relation between mathematics and language is not based on just one aspect. In this article, the authors address the role of the Subject facing Reality through language. Perception is defined and a mathematical theory of the perceptual field is proposed. The distinction between purely expressive language and purely informative language is considered false, because the subject is expressed in the communication of a message, and conversely, in purely expressive language, as in an exclamation, there is some information. To study the relation between language and reality, the function of ostensibility is defined and propositions are divided into ostensives and estimatives. © 2013 Wiley Periodicals, Inc. Complexity 20: 27–37, 2014     

Chinese and English Mathematics Language: The Relation Between Linguistic Clarity and Mathematics Performance

In Study 1, 48 judges rated the clarity of Chinese, English, and “Chinglish” (Chinese words translated into English) mathematical words-for example, the Chinglish version of the Chinese word for quadrilateral is “four-side-shape.” Native Chinese-speaking judges achieve greater agreement on the relative clarity of Chinese words than do native English-speaking judges on the relative clarity of English words. More Chinese words are rated clear than are English. Chinglish mathematical words tend to be rated more clear than English. The inherent compound word structure of the Chinese language seems well suited to portray mathematical ideas.

In Study 2, we examined the relations among the clarity of Chinese mathematical terms, U.S. urban junior high school students' Chinese reading ability, and their mathematics performance. There is a strong correlation between Chinese reading ability and performance on test items with mathematics words rated clear by Chinese judges. The relative clarity of mathematical terms in the Chinese language may contribute to Chinese-speaking students' understanding of mathematics and to superior mathematics performance.     

MATHEMATICS,LANGUAGE AND DERRIDA

Abstract

Derrida's revolutionary work in the study of language has seriously challenged the way in which we see words being attached to meanings. This paper makes tentative steps towards examining how his work might assist us in understanding the way in which our attempts to describe or capture our mathematical experiences modify the experience itself. In doing this we draw on the work of Jacques Derrida and John Mason in locating possible frameworks through which to conceptualise the relationship between language and mathematical cognition. It concludes that mathematical meaning never stabilises since it is caught between the individual's ongoing experience and society's ongoing renewal of its conventions. That is, mathematics, language and the human performing them are always evolving in relation to each other.     

Exploring informal mathematical products of low achievers at the secondary school level

This article examines the notion of informal mathematical products, in the specific context of teaching mathematics to low achieving students at the secondary school level. The complex and relative nature of this notion is illustrated and some of its characteristics are suggested. These include the use of ad-hoc strategies, mental calculations, idiosyncratic ideas, everyday rather than mathematical language, non-symbolic explanations, visual justifications and common-sense based reasoning. The main argument raised in the article concerns the challenge of valuing informal mathematical products, created by low achievers, and using them within the mathematics classroom as means for advancing such students. The data draws from several research and design projects conducted in Israel since 1991. Selected examples of students’ products, gathered from low-track mathematics classrooms involved in these projects, are presented and analyzed. The analyses highlight various features of such products, and portray the possible gains of teaching approaches that legitimize, and build onwards from, informal products of low achievers.     

Mathematics education: Procedures, rituals and man's search for meaning

An attempt is made to analyze mathematical behavior from more general psychological perspectives. The mathematical language is a special case of the human language, which is a form of expression. Many people use common language in a meaningless way. The same is true about the mathematical language. Rituals are other forms of expression. Many people identify rituals in many mathematical contexts (procedures, argumentation). Thus, quite often, they behave in a meaningless way as required by many rituals. On the other hand, the community of mathematics education struggles for meaningful learning. This can be regarded as a special case of man's search for meaning. The general claims will be illustrated by some examples from various mathematical contexts.     

Cognitive Demands and Second-Language Learners: A Framework for Analyzing Mathematics Instructional Contexts

The issues involved in teaching English language learners mathematics while they are learning English pose many challenges for mathematics teachers and highlight the need to focus on language-processing issues related to teaching mathematical content. Two realistic-type problems from high-stakes tests are used to illustrate the complex interactions between culture, language, and mathematical learning. The analyses focus on aspects of the problems that potentially increase cognitive demands for second-language learners. An analytical framework is presented that is designed to enable mathematics teachers to identify critical elements in problems and the learning environment that contribute to increased cognitive demands for students of English as a second language. The framework is proposed as a cycle of teacher reflection that would extend a constructivist model of teaching to include broader linguistic, cultural, and cognitive processing issues of mathematics teaching, as well as enable teachers to develop more accurate mental models of student learning.     

Cognitive Demands and Second-Language Learners: A Framework for Analyzing Mathematics Instructional Contexts

The issues involved in teaching English language learners mathematics while they are learning English pose many challenges for mathematics teachers and highlight the need to focus on language-processing issues related to teaching mathematical content. Two realistic-type problems from high-stakes tests are used to illustrate the complex interactions between culture, language, and mathematical learning. The analyses focus on aspects of the problems that potentially increase cognitive demands for second-language learners. An analytical framework is presented that is designed to enable mathematics teachers to identify critical elements in problems and the learning environment that contribute to increased cognitive demands for students of English as a second language. The framework is proposed as a cycle of teacher reflection that would extend a constructivist model of teaching to include broader linguistic, cultural, and cognitive processing issues of mathematics teaching, as well as enable teachers to develop more accurate mental models of student learning.     

Meaning and Formalism in Mathematics

This essay is an exploration of possible sources (psychological, not mathematical) of mathematical ideas. After a short discussion of plationism and constructivism, there is a brief review of some suggestions for these sources that have been put forward by various researcher (including this author). These include: mental representations, deductive reasoning, metaphors, natural language, and writing computer programs.The problem is then recast in terms of the relation between meaning and formalism. On one hand, formalism can be seen as a tool for expressing meaning that is already present in an individual's mind. On the other hand, and the discussion of this point is the main contribution of this paper, it is not only possible, but a standard activity of mathematicians, to use formalism to construct meaning and this can also be a source of mathematical ideas.Although using formalism to construct meaning is a very difficult method for students to learn, it may be that this is the only route to learning large portions of mathematics at the upper high school and tertiary levels. The essay ends with an outline of a pedagogical strategy for helping students travel this route.This revised version was published online in September 2005 with corrections to the Cover Date.     

Problem solving heuristics,affect, and discrete mathematics

It has been suggested that activities in discrete mathematics allow a kind of new beginning for students and teachers. Students who have been “turned off” by traditional school mathematics, and teachers who have long ago routinized their instruction, can find in the domain of discrete mathematics opportunities for mathematical discovery and interesting, nonroutine problem solving. Sometimes formerly low-achieving students demonstrate mathematical abilities their teachers did not know they had. To take maximum advantage of these possibilities, it is important to know what kinds of thinking during problem solving can be naturally evoked by discrete mathematical situations—so that in developing a curriculum, the objectives can include pathways to desired mathematical reasoning processes. This article discusses some of these ways of thinking, with special attention to the idea of “modeling the general on the particular.” Some comments are also offered about students' possible affective pathways and structures.     

Centripetal and centrifugal language forces in one elementary school second language mathematics classroom

Research on the learning and teaching of mathematics in contexts of language diversity has highlighted a number of common tensions that arise in a variety of contexts. These tensions can be explained by Bakhtin’s characterization of two sets of forces that are present in any utterance: centripetal forces represent the drive for unitary language, standardisation and linguistic hegemony; centrifugal forces represent the presence of heteroglossia, stratification and decentralisation. In this paper, I use this theoretical perspective to examine ethnographic data from a study of a second language mathematics classroom in Canada, in which the students are almost all speakers of Cree, one of the original languages of Canada. My analysis highlights three situations in which the tension between centripetal and centrifugal forces is particularly salient: the students’ use of Cree; working on mathematical word problems; and producing mathematical explanations.     

Linguistic indexicality in algebra discussions

In discussion-oriented classrooms, students create mathematical ideas through conversations that reflect growing collective knowledge. Linguistic forms known as indexicals assist in the analysis of this collective, negotiated understanding. Indexical words and phrases create meaning through reference to the physical, verbal and ideational context. While some indexicals such as pronouns and demonstratives (e.g. this, that) are fairly well-known in mathematics education research, other structures play significant roles in math discussions as well. We describe students’ use of entailing and presupposing indexicality, verbs of motion, and poetic structures to express and negotiate mathematical ideas and classroom norms including pedagogical responsibility, conjecturing, evaluating and expressing reified mathematical knowledge. The multiple forms and functions of indexical language help describe the dynamic and emergent nature of mathematical classroom discussions. Because interactive learning depends on linguistically established connections among ideas, indexical language may prove to be a communicative resource that makes collaborative mathematical learning possible.     

数学建模对高中生构建数学底层思维的作用及其教学实施建议

数学底层思维即用数学的眼光观察世界、用数学的思维分析世界以及用数学的语言表达世界,是人们面对自然和社会中纷繁多样的现象和问题时,所展现的自发的、不依赖监督的、融汇数学学科核心素养的思维方式.作为国家高中新课程标准中数学六大核心素养之一的数学建模,是培养学生数学底层思维的良好载体,对人才培养和社会发展均起到良好的促进作用.本文主要阐述了数学建模对高中生构建数学底层思维的作用,并结合教学实例给出教学实施建议.     

On a Possible Continuous Analogue of the Szpilrajn Theorem and its Strengthening by Dushnik and Miller

The Szpilrajn theorem and its strengthening by Dushnik and Miller belong to the most quoted theorems in many fields of pure and applied mathematics as, for instance, order theory, mathematical logic, computer sciences, mathematical social sciences, mathematical economics, computability theory and fuzzy mathematics. The Szpilrajn theorem states that every partial order can be refined or extended to a total (linear) order. The theorem by Dushnik and Miller states, moreover, that every partial order is the intersection of its total (linear) refinements or extensions. Since in mathematical social sciences or, more general, in any theory that combines the concepts of topology and order one is mainly interested in continuous total orders or preorders in this paper some aspects of a possible continuous analogue of the Szpilrajn theorem and its strengthening by Dushnik and Miller will be discussed. In particular, necessary and sufficient conditions for a topological space to satisfy a possible continuous analogue of the Dushnik-Miller theorem will be presented. In addition, it will be proved that a continuous analogue of the Szpilrajn theorem does not hold in general. Further, necessary and in some cases necessary and sufficient conditions for a topological space to satisfy a possible continuous analogue of the Szpilrajn theorem will be presented.      

A study of early career teachers’ practices related to language and language diversity during mathematics instruction

The role of language in mathematics teaching and learning is increasingly highlighted by standards and reform movements in the US. However, little is known about teachers’, and especially early career teachers’ (ECTs) practices and understandings related to language in mathematics instruction. This multiple case study explored the language-related understandings and practices of six ECTs in diverse elementary classrooms. Using iterative cycles of analysis, we found that all ECTs regularly attended to students’ mathematical vocabulary use and development. Yet, there was variability in ECTs’ focus on how to teach mathematical vocabulary, expectations for students’ precise use of mathematical terminology, and the use of multiple languages during instruction. These findings indicate that ECTs need more targeted support during teacher preparation and early career teaching in order to better support all students’ language development in the mathematics classroom.     

Preservice Secondary Mathematics Teachers' Conceptualizations of Equity: Access and Power as Seen Through Vignette Responses

Attention to equity in the mathematics education field has been growing in recent years. We have evidence that many novice secondary mathematics teachers do not feel prepared to teach in regards to diverse populations. We need to know more about how secondary preservice mathematics teachers (PSMTs) conceptualize equitable environments. This study investigates 30 secondary PSMTs' proposed responses to two hypothetical vignettes from mathematics department conversations regarding calculator usage and mathematical discourse, respectively, utilizing two of Gutiérrez's four dimensions of equity: Access and Power. Results suggest these PSMTs considered equity, equality, and creating a classroom that invites participation among other factors when thinking of an equitable approach with respect to calculator usage. When considering mathematical discourse, PSMTs cited the need to “model” proper use of mathematical language as well as to allow students to themselves verbalize it. Implications mathematics education and teacher education more broadly are to integrate equity and equality discussions in methods courses and to include strategies to facilitate productive discourse.