The ostensive dimension through the lenses of two didactic approaches

The paper presents how two different theories—the APC-space and the ATD—can frame in a complementary way the semiotic (or ostensive) dimension of mathematical activity in the way they approach teaching and learning phenomena. The two perspectives coincide in the same subject: the importance given to ostensive objects (gestures, discourses, written symbols, etc.) not only as signs but also as essential tools of mathematical practices. On the one hand, APC-space starts from a general semiotic analysis in terms of “semiotic bundles” that is to be integrated into a more specific epistemological analysis of mathematical activity. On the other hand, ATD proposes a general model of mathematical knowledge and practice in terms of “praxeologies” that has to include a more specific analysis of the role of ostensive objects in the development of mathematical activities in the classroom. The articulation of both theoretical perspectives is proposed as a contribution to the development of suitable frames for Networking Theories in mathematics education.     

Interaction analysis of mathematical communication in primary teaching: The epistemological perspective

Communication between students and teacher in the mathematics classroom is a form of social interaction which focuses on a specific topic:mathematical knowledge. This knowledge cannot be introduced into classroom interaction “from the outside”, but grows through the communicative process, in the course of interactive exchanges between the participants of discussion. Although mathematical communication must be seen and analysed in the same way as any other form of communication, the particularity of interactive constructions of mathematical knowledge and its specificsocial epistemology within the context of teaching processes has to be taken into consideration. Also, the institutional influences of school institutions and those of teaching (analysed in the frame of general socio-interactive research approaches) must be considered. An epistemology-oriented interaction research approaches the specificity of amathematical classroom and communication culture in its analyses.     

Embodying mathematics and science: Microworlds as representations

This paper examines the category of open-ended exploratory computer environments that have been labeled “microworlds.” The paper reviews the ways in which the term “microworld” has been used in the mathematics and science education communities, and analyzes examples of specific computer microworlds. Two definitions of microworld are proposed: a structural definition that focuses on design elements shared by the environments, and a functional definition that highlights commonalties in how students learn with microworlds. In the final section of the paper, the notion that computer microworlds can be said to “embody” mathematical or scientific ideas is addressed, within the context of a re-evaluation of the general concept of representation.     

Traveling down the road: from cognitive neuroscience to mathematics education … and back

In this commentary paper to the special issue on “Cognitive Neuroscience and Mathematics Education”, we reflect on the connection between cognitive neuroscience and mathematics education from an educational research point of view. The current issue highlights that cognitive neuroscience offers a series of tools, methodologies and theories to investigate cognitive processes that take place during mathematical thinking and learning. This might complement and extend our knowledge that has been obtained on the basis of behavioral data only, the common approach in educational research. At the same time, we note that the existing neuroscientific studies have investigated mathematical performance in relative isolation from the educational context. The characteristics of this context have, however, a large influence on mathematical performance and its correlated brain activity, an issue that should be addressed in future research. We contend that traveling back and forth from cognitive neuroscience to mathematics education might yield a better understanding of how mathematical learning takes place and how it can be influenced.     

From how to why: A quest for the common mathematical meanings behind two different division algorithms

During their education cycle in mathematics, students are exposed to algorithms as early as primary school. Several studies show how students frequently learn to perform these algorithms without controlling the mathematical meanings behind them. On the other hand, several National Standards have highlighted the need to construct meanings in mathematics from the first cycle of education. In this paper we focus on division algorithms, investigating to what extent 6th graders can be guided to understanding the whys behind an algorithm, through the comparison of two different algorithms for integer division. Our results suggest, on the one hand, that “it could work!”, and on the other hand, that the exposure to different algorithms for the same mathematical operation seems particularly significant for bringing out the whys behind such algorithms, as well as for capturing the difference between a mathematical operation and algorithms for calculating the result of such an operation.     

Teachers' Understandings of Realistic Contexts to Capitalize on Students' Prior Knowledge

The theory of realistic mathematics education establishes that framing mathematics problems in realistic contexts can provide opportunities for guided reinvention. Using data from a study group, I examine geometry teachers' perspectives regarding realistic contexts during a lesson study cycle. I ask the following. (a) What are the participants' perspectives regarding realistic contexts that elicit students' prior knowledge? (b) How are the participants' perspectives of realistic contexts related to teachers' instructional obligations? (c) How do the participants draw upon these perspectives when designing a lesson? The participants identified five characteristics that are needed for realistic contexts: providing entry points to mathematics, using “catchy” and “youthful” contexts, selecting personal contexts for the students, using contexts that are not “too fake” or “forced,” and connecting to the lesson's mathematical content. These characteristics largely relate to the institutional, interpersonal, and individual obligations with some connections with the disciplinary obligation. The participants considered these characteristics when identifying a realistic context for a problem‐based lesson. The context promoted mathematical connections. In addition, the teachers varied the context to increase the relevance for their students. The study has implications for supporting teachers' implementation of problem‐based instruction by attending to teachers' perspectives regarding the obligations shaping their work.     

Research, practice, uncertainty and responsibility

Three issues concerning the relationship between research and practice are addressed. (1) A certain ‘prototype mathematics classroom’ seems to dominate the research field, which in many cases seems selective with respect to what practices to address. I suggest challenging the dominance of the discourse created around the prototype mathematics classroom. (2) I find it important to broaden the school-centred discourse on mathematics education and to address the very different out-of-school practices that include mathematics. Many of these practices are relevant for interpreting what is taking place in a school context. That brings us to (3) socio-political issues of mathematics education. When the different school-sites for learning mathematics as well as the many different practices that include mathematics are related, we enter the socio-political dimension of mathematics education.On the one hand we must consider questions like: Could socio-political discrimination be acted out through mathematics education? Could mathematics education exercise a regimentation and disciplining of students? Could it include discrimination in terms of language? Could it include sexism and racism? On the other hand: Could mathematics education bring about competencies which can be described as empowering, and as supporting the development of mathematical literary or a ‘mathemacy’, important for the development of critical citizenship?However, there is no hope for identifying a one-way route to mathemacy. More generally: There is no simple way of identifying the socio-political functions of mathematics education. Mathematics education has to face uncertainty, and this challenge brings us to the notion of responsibility.     

Justifying the methods of OR

Operational research practitioners use mathematical, statistical, scientific, and other methods to structure and analyse issues in order to advise and assist their clients. In doing so they apply values, follow rules and use methodologies. The paper examines the justification of these methods, values and methodologies. Starting with a conceptual model drawn from the philosophy of science, a justification framework is developed for operational research (OR). Making a distinction between OR academic research and OR practice helps to clarify the issues. OR research is similar to scientific, mathematical and social science research; OR practice, as technology, is closer to engineering. While OR academic researchers will seek justification in the academic discipline within which they choose to work, it is argued that the justification of OR practice lies in its usefulness. For academic OR, justification lies in the justification of mathematics, statistics, science and social science; for practice, it is practitioners who decide what usefulness means in their context.     

Expanding Notions of “Learning Trajectories” in Mathematics Education

Over the past 20 years learning trajectories and learning progressions have gained prominence in mathematics and science education research. However, use of these representations ranges widely in breadth and depth, often depending on from what discipline they emerge and the type of learning they intend to characterize. Learning trajectories research has spanned from studies of individual student learning of a single concept to trajectories covering a full set of content standards across grade bands. In this article, we discuss important theoretical assumptions that implicitly guide the development and use of learning trajectories and progressions in mathematics education. We argue that diverse theoretical conceptualizations of what it means for a student to “learn” mathematics necessarily both constrains and amplifies what a particular learning trajectory can capture about the development of students’ knowledge.     

Computer use in mathematics education – present situation,intentions, and problems in the Federal Republic of Germany†

Computers, and computer‐related thinking structures, are only gradually influencing mathematics education. On the one hand, there is a discrepancy between involved teachers who already have changed their own classroom teaching to a great extent, and a majority of mathematics teachers who have not yet taken notice of the computer for teaching purposes. On the other hand, knowledge of the computer and of algorithms is frequently merely added to the mathematical subject matter. As opposed to that, the authors argue that it is necessary to genuinely integrate such subject matter, and to include general topics such as social impact and changed attitudes toward application. With regard to implementation, they develop concrete ideas which are aligned in a differentiated manner to the specific situation and the opportunities offered in the Federal Republic of Germany. The rationale for that is that only such reference to a specific situation will provide an opportunity for readers abroad to usefully apply approaches and ideas to the situation given in their own cultural environment.

An abbreviated version of this paper for cursory reading or other purposes has been marked by bold lines on the margin.

     

Rhetoric,Reality, and Possibilities: Interdisciplinary Teaching and Secondary Mathematics

The National Council of Teachers of Mathematics has proposed a broad core mathematics curriculum for all high school students. One emphasis in that core is on “mathematical connections” both among mathematical topics and between mathematics and other disciplines of study. It is suggested that mathematics should become a more integrated part of all students' high school education. In this article, working definitions for the terms curriculum, interdisciplinary, and integrated and a model of three categories of curriculum design based on the work of Harold Alberty are developed. This article then examines how a “connected” mathematics core curriculum might be situated within the different categories of curriculum organization. Examples from research on interdisciplinary education in high schools are presented. Issues arising from this study suggest the need for a greater emphasis on building and using models of curriculum integration both to frame and to give impetus to the work being done by teachers and administrators.     

The role of the researcher’s epistemology in mathematics education: an essay on the case of proof

Is there a shared meaning of “mathematical proof” among researchers in mathematics education? Almost all researchers may agree on a formal definition of mathematical proof. But beyond this minimal agreement, what is the state of our field? After three decades of activity in this area, being familiar with the most influential pieces of work, I realize that the sharing of keywords hides important differences in the understanding. These differences could be obstacles to scientific progress in this area, if they are not made explicit and addressed as such. In this essay I take a sample of research projects which have impacted the teaching and learning of mathematical proof, in order to describe where the gaps are. Then I suggest a possible scientific programme which aspires to strengthen the research practice in this domain. Eventually, I make the additional claim that this programme could hold for other areas of research in mathematics education.     

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.     

Adolescent girls’ construction of moral discourses and appropriation of primary identity in a mathematics classroom

This qualitative study examines the way three American young adolescent girls who come from different class and racial backgrounds construct their social and academic identities in the context of their traditional mathematics classroom. The overall analysis shows an interesting dynamic among each participant’s class and racial background, their social/academic identity and its collective foundation, the types of ideologies they repudiate and subscribe to, the implicit and explicit strategies they adopt in order to support the legitimacy of their own position, and the ways they manifest their position and identity in their use of language referring to their mathematics classroom. Detailed analysis of their use of particular terms, such as “I,” “we,” “they,” and “should/shouldn’t” elucidates that each participant has a unique view of her mathematics classroom, developing a different type of collective identity associated with a particular group of students. Most importantly, this study reveals that the girls actively construct a social and ideological web that helps them articulate their ethical and moral standpoint to support their positions. Throughout the complicated appropriation process of their own identity and ideological standpoint, the three girls made different choices of actions in mathematics learning, which in turn led them to a different math track the following year largely constraining their possibility of access to higher level mathematical knowledge in the subsequent schooling process.     

Moving toward shared realities through empathy in mathematical modeling: An ecological systems theory approach

The mathematics education community has routinely called for mathematics tasks to be connected to the real world. However, accomplishing this in ways that are relevant to students’ lived experiences can be challenging. Meanwhile, mathematical modeling has gained traction as a way for students to learn mathematics through real-world connections. In an open problem to the mathematics education community, this paper explores connections between the mathematical modeling and the nature of what is considered relevant to students. The role of empathy is discussed as a proposed component for consideration within mathematical modeling so that students can further relate to real-world contexts as examined through the lens of Ecological Systems Theory. This is contextualized through a classroom-tested example entitled “Tiny Homes as a Solution to Homelessness” followed by implications and conclusions as they relate to mathematics education.     

Distinguishing Features of Mechanical and Human Problem-Solving

Over the years, research in mathematical problem-solving has examined expert/novice problem-solving performance on various types of problems and subjects. In particular, DeFranco examined two groups of Ph.D. mathematicians as they solved four mathematics problems and found that although all were content experts, only one group were problem-solving experts. Based on this study, this article posits the notion that one distinguishing feature between experts and novices is that experts tend to look for special features of a problem and use algorithms only as a “fail-safe” system while novices act like “machines” relying on algorithms to solve the problems. Why? The article explores the idea that novice problem solvers learned to solve problems the way they learned proof, that is, in a formal, abstract and mechanizable way. Beliefs about proof and the culture in which it is practiced help frame a mathematician's view of the discipline and ultimately impacts classroom practice. The authors believe that current classroom instruction tends to create a culture that fosters algorithmic proficiency and a “machine-like” approach to the learning of mathematics and problem-solving. Further, they argue that mathematicians need to be aware of the distinction between knowing a proof is true and explaining why it is true. When these distinctions are appreciated and practiced during classroom instruction, then and only then will students begin to acquire the mathematical knowledge to become better problem solvers.     

The mathematics of war

During the Desert Storm, the Gulf war, it was possible to read in the newspapers words such as: “Inmathematical terms, was is becoming more and more electronically controlled and, as a result, it is moving away from the battlefield. Then, when war comes down to earth, it becomes bloody, it loses its mathematical asceticism” Reading the newspapers in those days, one had the impression that modern warfare is based on mathematics, as if it were not men but computers that decided where to carry out “surgical operations”. By contrast, the volume published a few years before the Gulf war conceived as a didactic unit to be used in schools with a guide for the teacher with the titleLa matematica della guerra (The Mathematics of War) published by Gruppo Abele in Turin begins with the words “Mathematics, like any other discipline, lends itself to building several paths towards education for peace”. The volume, written by a group of teachers belonging to an anti-violence organisation forming part of the “education for peace” project, highlights the power or ambiguitiy of mathematical models used to simulate war or conflict situations and demonstrates that in some cases the use of mathematics leads to a better understanding of the situation, but in other cases, the mathematical model itself can lead to conclusions which are either wrong or morally unacceptable.     

The state-of-art in mathematical creativity

Creativity is a topic which is often neglected within mathematics teaching. Usually teachers think that it is logic that is needed in mathematics in the first place, and that creativity is not important and learning mathematics. On the other hand, if we consider a mathematician who develops new results in mathematics. we cannot overlook his/her use of the creative potential. Thus, the main questions are as follows: What methods could be used to foster mathematical creativity within school situations? What scientific knowledge, i.e. research results, do we have on the meaning of mathematical creativity?     

Trends in researching the socioeconomic influences on mathematical achievement

We introduce the topic of socioeconomic influences on mathematical achievement through an overview of existing research reports and articles. International trends in the way the topic has emerged and become increasingly important in the international field of mathematics education research are outlined. From this review, there is a discussion about what appears to be neglected in previous work in this area and how the papers in this issue of ZDM provide information about some of these neglected areas. The main argument in this article is that socioeconomic influences on mathematical achievement should not be considered as a taken-for-granted fact that is accepted uncritically. Instead, it is suggested that the relationship between multiple socioeconomic influences and various understandings of mathematical achievement are historically contingent ways of understanding exclusions and inclusions in mathematics education practices. Research is not simply “evidencing” the facts of these relationships; research is also implicated in constructing the ways in which we think about these. Thus, mathematics education researchers could devise more nuanced approaches for understanding the social, political and historical constitution of these relationships.     

A comprehensive theory of representation for mathematics education

Representation is a difficult concept. Behaviorists wanted to get rid of it; many researchers prefer other terms like “conception” or “reasoning” or even “encoding;” and many cognitive science resarchers have tried to avoid the problem by reducing thinking to production rules.There are at least two simple and naive reasons for considering representation as an important subject for scientific study. The first one is that we all experience representation as a stream of internal images, gestures and words. The second one is that the words and symbols we use to communicate do not refer directly to reality but to represented entities: objects, properties, relationships, processes, actions, and constructs, about which there is no automatic agreement between two persons. It is the purpose of this paper to analyse this problem, and to try to connect it with an original analysis of the role of action in representation. The issue is important for mathematics education and even for the epistemology of mathematics, as mathematical concepts have their first roots in the action on, and in the representation of, the physical and social world; even though there may be a great distance today between that pragmatical and empirical source, and the sophisticated concepts of contemporary mathematics.