Mathematics teaching and values education—an intersection in need of research

This paper raises issues concerning the teaching of values in the context of mathematics education. It argues that a focus on education for democracy inevitably involves educating about values. It reviews the major relevant research and theoretical perspectives and argues for more research attention to be paid to this area. Although there has been relevant research done in the affective domain, both in general and in relation to mathematics, and in social and cultural issues, there is little direct research focus on either values or valuing. Teachers are rarely aware of teaching values either explicitly or implicitly, yet values teaching clearly does take place, mostly implicitly. If there are desires to change the directions of mathematics teaching to be more attuned to life in modern democratic societies then this aspect of mathematics education needs to be better understood in order that it can be better taught.     

Pourquoi, pour qui enseigner les mathématiques?

The evolution of education in Belgium is described from 1830 to our days. The mastery of education has always been a subject of disputes between the political forces in presence. With the flow of years, more attention is paid to social considerations and the necessity of a more democratic educational system arises. Simultaneously the pedagogical ideas change and active methods of teaching are promoted, although not always used. Different schoolsystems still coexist. However, except for philosophical and religious education, the differences between them tend to diminish. This is in particular the case for mathematics teaching whose evolution is also described with some details.     

Mathematics education and democracy

In this paper, we investigate the relationship between mathematics education and the notions of education for all/democracy. In order to proceed with our analysis, we present Marx’s concept of commodity and Jean Baudrillard’s concept of sign value as a theoretical reference in the discussion of how knowledge has become a universal need in today’s society and ideology. After, we engage in showing mathematics education’s historical and epistemological grip to this ideology. We claim that mathematics education appears in the time period that English becomes an international language and the notion of international seems to be a key constructor in the constitution of that ideology. Here, we draw from Derrida’s famous saying that “there is nothing beyond the text”. We conclude that a critique to modern society and education has been developed from an idealistic concept of democracy.     

A Comparison of Preservice Elementary Teachers' Beliefs About Mathematics and Teaching Mathematics: 1968 and 1998

The study replicates Collier's (1972) work. It focuses on the beliefs of a large sample of elementary education students at four stages of teacher preparation, about both the nature of and the teaching of mathematics. The instrument measures what Collier termed a “formal‐informal” dimension of belief. The data suggest that initially the 1998 students held significantly more informal (constructivist) beliefs than did their 1968 counterparts. In both years, students moved toward more informal beliefs during the course of their programs, with the most significant changes occurring in their beliefs about how mathematics should be taught. However, apparent contradictions in belief structures were observed both at the start and at the end of their programs. Thus, it appears that though many students acquired new, more informal beliefs during the course of their programs, they did not develop robust, consistent philosophies of mathematics education.     

Proofs as bearers of mathematical knowledge

Yehuda Rav’s inspiring paper “Why do we prove theorems?” published in Philosophia Mathematica (1999, 7, pp. 5–41) has interesting implications for mathematics education. We examine Rav’s central ideas on proof—that proofs convey important elements of mathematics such as strategies and methods, that it is “proofs rather than theorems that are the bearers of mathematical knowledge”and thus that proofs should be the primary focus of mathematical interest—and then discuss their significance for mathematics education in general and for the teaching of proof in particular.     

Pourquoi et pour qui enseigner les mathématiques?

The history of reform in the teaching of mathematics in Quebec since the start of the 20th Century has been strongly influenced by a social and political context that gives meaning to its particular orientation. In this paper, we attempt to trace the important moments in the evolution of the teaching of mathematics through the roles and missions that successive governments have given to schooling and subsequent curricular reforms. Our analysis shows that the teaching of mathematics has evolved from an essentially practical role, prior to 1945, to a double role that is both practical and cultural in the 1950s, with the balance tipped in favor of the practical role. In the 1960s, the political will for universal education and, more recently, the concern over forming persons who can adapt to a constantly evolving society, are gradually pushing the Quebec curricula towards major changes. Mathematics is presented as a powerful tool that gives one a handle on reality, a complement of culture and an important language that is essential to the communication of ideas. Throughout the various reforms, these two finalities-the practical and the cultural-are constantly present, varying in importance and meaning with the successive roles attributed to schools. Our analysis shows a strong pragmatic anchoring of the first mathematics programs in Quebec, one that has not disappeared over time. The gradual curricular changes show, however, a progressive integration of the cultural formation role of mathematics     

What students value in effective mathematics learning: a ‘Third Wave Project’ research study

Considerations of how mathematics can be effectively taught and learnt may be cognitive, affective or sociocultural in approach. ??What students value in effective mathematics learning?? is a research study of the Third Wave Project, an international consortium of research teams adopting a sociocultural approach to investigate the harnessing of relevant values to optimise school mathematics teaching and learning. This paper seeks to contextualise the study, as part of the study examines what the high-achieving East Asian mathematics students value. The study is framed by knowledge relating to the relative cultural influence on effectiveness in mathematics learning, Alan Bishop??s values in mathematics education and the role of interactions in education, in particular David Tripp??s idea of critical incidents as reflecting professional judgement (and, thus, underlying values). Features of the innovative qualitative research design are also presented, which include the facilitation of photo-voice, the argument for an international collaborative team and focus group interviews for all values research, and a two-stage data analysis process aimed at clarifying both etic and emic perspectives.     

Changes in Preservice Teachers' Attitudes and Beliefs: The Net Impact of Two Mathematics Education Units and Intervening Experiences

Preservice teachers in a bachelor of education (Primary and ECE) program were surveyed about their beliefs and attitudes toward mathematics and its teaching and learning, at the beginning of their first mathematics curriculum unit and again at the end of their second mathematics education unit, approximately 18 months later. Significant differences were found for several items. However, in comparison to previously noted changes in the beliefs of similar cohorts from the beginning to end of a single unit, the changes were relatively small. Aspects of the course that students considered valuable and which may have contributed to the changes are also reported.     

Observing mathematics teaching in three different countries lessons learnt?

It has been reported that the students’ level of mathematicswhen entering university has changed in recent years. A surprisingnumber of students struggle with mathematics or have a mathematicsanxiety. Although this was widely reported within the UK, therewas some uncertainty if colleagues in other countries were experiencingthe same challenges. Therefore, lectures in three countrieswere observed and the effects different educational backgroundsand cultures have on the students’ attitudes towards mathematicswere determined. Experiences and ideas were exchanged and itbecame clear that the problems lecturers face in the UK withrespect to teaching mathematics are in fact not limited to thiscountry but that their approach to lecturing at university isdifferent. This article discusses the outcomes of analysing and observingmathematics taught in three different countries. Similaritiesand differences are presented, together with feedback on thesuccess of implementing new ideas at Birmingham.     

Negative numbers: Reflections of a learning researcher

Historically, the development of mathematics and the natural sciences can time and again be attributed to the origin of ideas that prove to be productive. If one takes a closer look at the historical context in which ideas thrive, it will soon become apparent that generally these ideas did not merely appear out of the blue. Often there was already something in the air, waiting to happen. The same also applies for didactising in mathematics education.In this article, I attempt to reconstruct the productive ideas that evolved from already existing sources for the teaching and understanding of negative numbers. Ideas that can be, and, to a certain extent, already have been, developed more fully through classroom research. A limited number of examples from developmental research that has been carried out thus far in this respect is discussed in the following. Not every aspect of the effects of researchers' learning process on the teaching and learning of negative numbers, which might at some future time become theory, can be elaborated within the confines of the current undertaking.     

Mathematics Majors' Perceptions of Conviction,Validity, and Proof

In this paper, 28 mathematics majors who completed a transition-to-proof course were given 10 mathematical arguments. For each argument, they were asked to judge how convincing they found the argument and whether they thought the argument constituted a mathematical proof. The key findings from this data were (a) most participants did not find the empirical argument in the study to be convincing or to meet the standards of proof, (b) the majority of participants found a diagrammatic argument to be both convincing and a proof, (c) participants evaluated deductive arguments not by their form but by their content, but (d) participants often judged invalid deductive arguments to be convincing proofs because they did not recognize their logical flaws. These findings suggest improving undergraduates' comprehension of mathematical arguments does not depend on making undergraduates aware of the limitations of empirical arguments but instead on improving the ways in which they process the arguments that they read.     

Introducing the Emerging Discipline of Statistics Education

Increasing attention has been given over the last decade by the statistics, mathematics and science education communities to the development of statistical literacy and numeracy skills of all citizens and the enhancement of statistics education at all levels. This paper introduces the emerging discipline of statistics education and considers its role in the development of these important skills. The paper begins with information on the growing importance of statistics in today's society, schools and colleges, summarizes unique challenges students face as they learn statistics, and makes a case for the importance of collaboration between mathematicians and statisticians in preparing teachers to teach students how to understand and reason about data. We discuss the differences and interrelations between statistics and mathematics, recognizing that mathematics is the discipline that has traditionally included instruction in statistics. We conclude with an argument that statistics should be viewed as a bridge between mathematics and science and should be taught in both disciplines.     

Impacts of inquiry pedagogy on undergraduate students conceptions of the function of proof

Mathematicians and mathematics educators agree that proof is an important tool in mathematics, yet too often undergraduate students see proof as a superficial part of the discipline. While proof is often used by mathematicians to justify that a theorem is true, many times proof is used for another purpose entirely such as to explain why a particular statement is true or to show mathematics students a particular proof technique. This paper reports on a study that used a form of inquiry-based learning (IBL) in an introduction to proof course and measured the beliefs of students in this course about the different functions of proof in mathematics as compared to students in a non-IBL course. It was found that undergraduate students in an introduction to proof course had a more robust understanding of the functions of proof than previous studies would suggest. Additionally, students in the course taught using inquiry pedagogy were more likely to appreciate the communication, intellectual challenge, and providing autonomy functions of proof. It is hypothesized that these results are a response to the pedagogy of the course and the types of student activity that were emphasized.     

Preservice Teacher Beliefs About Proofs

This article describes a study of backgrounds, beliefs, and attitudes of teachers about proofs. Thirty preservice elementary teachers enrolled in a mathematics content course and 21 secondary mathematics teachers in an abstract algebra course were surveyed. The study explored four issues: preservice teachers' experiences/exposure to proof, their beliefs about what constitutes a proof and the role of proof in mathematics, and their beliefs about when proof should be introduced in grades K-12. Results of the survey are described as a means for discussing the backgrounds and beliefs future teachers hold with regard to teaching proofs in their own classrooms. Finally, a short collection of sample explorations and questions, which could be used to encourage the thinking and writing of proofs in grades K-12, is provided. One of these questions was posed to 215 secondary students; examples of their reasoning and a discussion of the various techniques employed by the students are included.     

The Effect of a Modified Moore Method on Attitudes and Beliefs in Precalculus

As part of a study on the effects of teaching with a Modified Moore Method (MMM), a survey containing 20 items from Schoenfeld's (1989 ) investigation of attitudes and beliefs about mathematics was administered to students in undergraduate precalculus classes. The study included one section of precalculus taught with an MMM, a student‐centered and inquiry‐based teaching method, and two sections taught using traditional lecture methods. The survey was administered one week into the semester, following the drop/add date, and during the last week of classes. In this paper, we discuss the findings of the attitudes and beliefs portions of the survey and correlations with scores on a common final exam. We looked for differences between the MMM and traditional sections as well as gender differences. There were only a few differences in the attitudes and beliefs among the students, although there were more changes for females than males and all the significant differences were positive. The correlation between attitudes and beliefs and final exam scores was much stronger in the traditionally taught classes than in the MMM class. When separated by gender, only the reported attitudes and beliefs of the females in the traditional class significantly correlated with final exam scores.     

Mathematics teachers as democratic agents

This paper considers the role of schools, and particularly that of all teachers (including mathematics teachers), in the construction of social democracy. It is argued that, in the present context of economic rationalism, in which teachers are becoming progressively deprofessionalised, and in which schools are being increasingly subjected to market forces, there is an urgent need to challenge the effects of economic rationalism on education, and especially to produce better educated and more highly professionalised teachers. Further, and more significantly, it is argued that such teachers, having (i) a significantly broadened knowledge base, (ii) a deep commitment to political change leading to democratic social life, and (iii) greatly increased power enabling them to regain control of their professional discourse and the process and content of schooling, might then as teachers of subjects and children, simultaneously undertake a leading role in the difficult task of social reconstruction directed towards promoting increased automomy and real democratic participation of the citizenry.     

Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers

This mixed-methods study describes classroom characteristics and student outcomes from university mathematics courses that are based in mathematics departments, targeted to future pre-tertiary teachers, and taught with inquiry-based learning (IBL) approaches. The study focused on three two-term sequences taught at two research universities, separately targeting elementary and secondary pre-service teachers. Classroom observation established that the courses were taught with student-centred methods that were comparable to those used in IBL courses for students in mathematics-intensive fields at the same institutions. To measure pre-service teachers' gains in mathematical knowledge for teaching, we administered the Learning Mathematics for Teaching (LMT) instrument developed by Hill, Ball and Schilling for in-service teacher professional development. Results from the LMT show that pre-service teachers made significant score gains from beginning to end of their course, while data from interviews and from surveys of learning gains show that pre-service teachers viewed their gains as relevant to their future teaching work. Measured changes on pre-/post-surveys of attitudes and beliefs were generally supportive of learning mathematics but modest in magnitude. The study is distinctive in applying the LMT to document pre-service teachers' growth in mathematical knowledge for teaching. The study also suggests IBL is an approach well suited to mathematics departments seeking to strengthen their pre-service teacher preparation offerings in ways consistent with research-based recommendations.