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.     

The Mathematics in Society Project: a new conception of mathematics(1) †

The Mathematics in Society Project (MISP) began in 1980 as an international association of mathematics educators in three continents. The main purpose of MISP is the writing of innovative secondary school mathematics courses based on a new conception of mathematics itself. The starting point for MISP was the fact that mathematics in school is found to be difficult and unpleasant by the great majority of pupils, but mathematics in society is widely diffused and used implicitly by most people. MISP therefore sees mathematics as a ‘living body’ representing all its uses (implicit and explicit) in society, in contrast to the ‘skeleton’ concept of mathematics which has led to such failure in schools. The gradual development of this new conception is informally analysed in the style of Kuhn and Lakatos.

     

A DNR perspective on mathematics curriculum and instruction. Part II: with reference to teacher’s knowledge base

Two questions are on the mind of many mathematics educators; namely: What is the mathematics that we should teach in school? and how should we teach it? This is the second in a series of two papers addressing these fundamental questions. The first paper (Harel, 2008a) focuses on the first question and this paper on the second. Collectively, the two papers articulate a pedagogical stance oriented within a theoretical framework called DNR-based instruction in mathematics. The relation of this paper to the topic of this Special Issue is that it defines the concept of teacher’s knowledge base and illustrates with authentic teaching episodes an approach to its development with mathematics teachers. This approach is entailed from DNR’s premises, concepts, and instructional principles, which are also discussed in this paper.     

Mathematics teachers’ views about the limited utility of real analysis: A transport model hypothesis

In the United States and elsewhere, prospective teachers of secondary mathematics are usually required to complete numerous advanced mathematics courses before obtaining certification. However, several research studies suggest that teachers’ experiences in these advanced mathematics courses have little influence on their pedagogical practice and efficacy. To understand this phenomenon, we presented 14 secondary mathematics teachers with four statements and proofs in real analysis that related to secondary content and asked the participants to discuss whether these proofs could inform their teaching of secondary mathematics. In analyzing participants’ remarks, we propose that many teachers view the utility of real analysis in secondary school mathematics teaching using a transport model, where the perceived importance of a real analysis explanation is dependent upon the teacher’s ability to transport that explanation directly into their instruction in a secondary mathematics classroom. Consequently, their perceived value of a real analysis course in their teacher preparation is inherently limited. We discuss implications of the transport model on secondary mathematics teacher education.     

Being a mathematics teacher in times of reform

Within research on mathematics teachers and/or their professional development, the concept of identity emerges as a critique of views of how teaching practice is related to teachers’ ‘internal states’ of knowledge and beliefs. Identity relates teachers’ professional lives to teaching practices and to the contexts in which the teaching and/or professional development occurs. However, what might count as the context still needs in-depth discussion. In order to contribute to the development of a theoretical framework for understanding mathematics teachers’ professional lives, we will draw on one remarkable teacher’s identity as a primary mathematics teacher in relation to one political, sociocultural, and pedagogical context. We use this teacher’s experience to discuss how education policies that create what Ball (2003) called ‘terrors of performativity’ tend to impede the formation of a balanced teacher identity.     

Mathematics capital in the educational field: Bourdieu and beyond

Mathematics education needs a better appreciation of the dominant power structures in the educational field: Bourdieu's theory of capital provides a good starting point. We argue from Bourdieu's perspective that school mathematics provides capital that is finely tuned to generationally reproduce the social structures that serve to keep the powerful in power, while ensuring that less powerful groups are led to accept their own failure in mathematics. Bourdieu's perspective thereby highlights theoretical inadequacies in much mathematics education research, insofar as it presumes a consensus about a ‘what works agenda’ for improving achievement for all. Drawing on one case where we manufactured awkward facts, we illustrate a Bourdieusian interpretation of mathematics capital as reproductive, and the crucial role of its cultural arbitrary. We then criticise the Bourdieusian concept of ‘mathematical capital’ as the value of mathematical competence in practice and propose to extend his tools to include the contradictory ‘use’ and ‘exchange’ values of mathematics instead: we will show how this conceptualisation goes ‘beyond Bourdieu’ and helps explain how teaching-learning might (ideally) produce ‘cultural use value’ in mathematical competence, while still recognising the contradictions teachers and learners face. Finally, we suggest how critical education research generally can benefit from this theoretical framework: (1) in exposing the interest of the dominant classes; but also (2) in researching critical pedagogic alternatives that challenge orthodoxy in educational policy and practice both in mathematics education and more generally.     

Conceptualizing inquiry-based education in mathematics

The terms inquiry-based learning and inquiry-based education have appeared with increasing frequency in educational policy and curriculum documents related to mathematics and science education over the past decade, indicating a major educational trend. We go back to the origin of inquiry as a pedagogical concept in the work of Dewey (e.g. 1916, 1938) to analyse and discuss its migration to science and mathematics education. For conceptualizing inquiry-based mathematics education (IBME) it is important to analyse how this concept resonates with already well-established theoretical frameworks in mathematics education. Six such frameworks are analysed from the perspective of inquiry: the problem-solving tradition, the theory of didactical situations, the realistic mathematics education programme, the mathematical modelling perspective, the anthropological theory of didactics, and the dialogical and critical approach to mathematics education. In an appendix these frameworks are illustrated with paradigmatic examples of teaching activities with inquiry elements. The paper is rounded off with a list of ten concerns for the development and implementation of IBME.     

Identifying and supporting teachers’ robust understanding of proportional reasoning

This case study uses the Framework for Teachers’ Robust Understanding of Proportional Reasoning for Teaching (Weiland et al., 2020) to characterize how 51 mathematics teachers solved a comparison proportional problem. We found 50 of the 51 teachers productively drew upon four knowledge resources: (1) proportional situation, (2) ratios as part: part or part: whole, (3) unit rates, and (4) ratio as measure. This study details these and teachers’ less commonly used knowledge resources, as well as counterproductive statements related to the knowledge resources. We analyze the structure of the comparison proportion problem and suggest why teachers drew on particular knowledge resources. Lastly, we highlight how counterproductive statements highlight areas of focus for mathematics teacher educators and extends the operationalizing of the robust proportional reasoning framework for mathematics education researchers.     

‘The unplanned impact of mathematics’ and its implications for research funding: a discussion-led educational activity

‘The unplanned impact of mathematics’ refers to mathematics which has an impact that was not planned by its originator, either as pure maths that finds an application or applied maths that finds an unexpected one. This aspect of mathematics has serious implications when increasingly researchers are asked to predict the impact of their research before it is funded and research quality is measured partly by its short term impact.

A session on this topic has been used in a UK undergraduate mathematics module that aims to consider topics in the history of mathematics and examine how maths interacts with wider society. First, this introduced the ‘unplanned impact’ concept through historical examples. Second, it provoked discussion of the concept through a fictionalized blog comments discussion thread giving different views on the development and utility of mathematics. Finally, a mock research funding activity encouraged a pragmatic view of how research funding is planned and funded.

The unplanned impact concept and the structure and content of the taught session are described.     

Mathematics teaching is democratic education

It is a commonly held belief that mathematics teaching has no political effects. Astonishingly, however, the fact is that the style of argument now used in mathematics everywhere was not developed originally to do mathematics. Originally its function was to counteract the teaching by the early Greek sophists of rhetoric. Their training gave the rich and privileged such an advantage in public speaking that democracy was threatned. Making respectable a new form of argument, in which evidence and logical structure predominated, was a very radical act of enlightened democratic education. Mathematics teaching in the form of open critical dialogue between teacher and taught remains a powerful form of education in democratic attitudes. Ambitions to produce political ideas as infallible as mathematics have a modern origin. In the early part of this century, mathematics education was again becoming universal throughout Europe. In the same period the belief arose that mathematics could eventually be completed as a single structure of truth. This transformed mathematics into a paradigm of democracy in which unorthodoxy must necessarily be eliminated. Communicated to people everywhere by universal education, this belief increased respect for similar political ideas. Gödel’s proof that mathematics can never be completed came too late to correct these political effects, but modern teachers can again use mathematics as a proof of the value and success of democratic attitudes and ideas. Whilst mathematics itself is ethically neutral, the ethical principles which produced both democracy and mathematics and which can be converyed in mathematics teaching are highly relevant to the modern world, and should be understood and taught by teachers everywhere.     

To be or not to be constructive,that is not the question

In the early twentieth century, L.E.J. Brouwer pioneered a new philosophy of mathematics, called intuitionism. Intuitionism was revolutionary in many respects but stands out – mathematically speaking – for its challenge of Hilbert’s formalist philosophy of mathematics and rejection of the law of excluded middle from the ‘classical’ logic used in mainstream mathematics. Out of intuitionism grew intuitionistic logic and the associated Brouwer–Heyting–Kolmogorov interpretation by which ‘there exists $x$’ intuitively means ‘an algorithm to compute $x$ is given’. A number of schools of constructive mathematics were developed, inspired by Brouwer’s intuitionism and invariably based on intuitionistic logic, but with varying interpretations of what constitutes an algorithm. This paper deals with the dichotomy between constructive and non-constructive mathematics, or rather the absence of such an ‘excluded middle’. In particular, we challenge the ‘binary’ view that mathematics is either constructive or not. To this end, we identify a part of classical mathematics, namely classical Nonstandard Analysis, and show it inhabits the twilight-zone between the constructive and non-constructive. Intuitively, the predicate ‘$x$ is standard’ typical of Nonstandard Analysis can be interpreted as ‘$x$ is computable’, giving rise to computable (and sometimes constructive) mathematics obtained directly from classical Nonstandard Analysis. Our results formalise Osswald’s longstanding conjecture that classical Nonstandard Analysis is locally constructive. Finally, an alternative explanation of our results is provided by Brouwer’s thesis that logic depends upon mathematics.     

A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms.     

Poverty,inequality and mathematics performance: the case of South Africa’s post-apartheid context

South Africa’s recent history of apartheid, its resultant high levels of poverty and extreme social and economic distance between rich and poor continue to play-out in education in complex ways. The country provides a somewhat different context for exploring the relationship between SES and education than other countries. The apartheid era only ended in 1994, after which education became the vehicle for transforming society and a political rhetoric of equity and quality education for all was prioritized. Thus education focused on redressing inequalities; and major curriculum change, with on-going revisions, was attempted. In this sense engagement with SES and education became foregrounded in policy, political discourse and research literature. Yet for all the political will and rhetoric little has been achieved and indicators are that inequality has worsened in mathematics education, where it is particularly pronounced. This paper proposes that continued research confirming poverty–underachievement links, which suggest an inevitability of positive correlations, is unhelpful. Instead we should explore issues of disempowerment and agency, constraints and possibilities, and the complex interplay of factors that create these widely established national statistics while simultaneously defying them in particular local contexts. Such research could shift the focus from a discourse of deficit and helplessness towards a discourse of possibilities in the struggle for equity and quality education for all.     

A false belief about fractions – What is its source?

This paper presents the results of interviews with 174 participants solving a problem of elementary mathematics, connected with the part–whole approach to fractions. The motive for the investigation was a specific kind of difficulty observed during a case study conducted to verify the elementary school student's understanding of the concept of fractions. The authors decided to examine the problem in a broader population of mathematics learners at different levels of education: from elementary school to university students and graduates of science majors. Approximately 65% of respondents reported the wrong answer immediately after reading the fraction problem taken from the fourth grade of elementary school. Detailed analysis of the respondents’ performance showed that the source of many wrong answers was a false belief about fractions: The only way to get 1/n of a given whole is to divide this whole into n equal parts, not yet described in educational literature.     

Epistemological beliefs concerning the nature of mathematics among teacher educators and teacher education students in mathematics

Beliefs constitute a central part of a person’s professional competences as beliefs are crucial to the perception of situations and as they influence our choice of actions. The present article focuses on epistemological beliefs about the nature of mathematics among future teachers and their educators at university and post-university teacher-training institutions in Germany. The data reported are part of a larger sample originating from the MT21 study [supported by the National Science Foundation through a grant to W. S. Schmidt and M. T. Tatto (REC-0231886). MT21 started in 2003] which explores and compares mathematics teacher education in Bulgaria, Germany, Mexico, South Korea, Taiwan, and the United States. In this article, we examine the structure and level of beliefs concerning the nature of mathematics for teacher education students in Germany both at the beginning (n = 368) and the end of their education (n = 286) as well as their educators (n = 77) in three academic disciplines (mathematics, mathematics pedagogy and general pedagogy). In the first part of the article, the literature on epistemological beliefs and their structure will be reviewed. In the empirical part, analyses on the level and the structure of beliefs for our samples and subsamples will be presented. Relations between educators’ and students’ beliefs will be explored.     

Problematizing mathematics and its pedagogy through teacher engagement with history-focused and classroom situation-specific tasks

We explore the conjecture that engaging teachers with activities which feature mathematical practices from the past (history-focused tasks) and in today’s mathematics classrooms (mathtasks) can promote teachers’ problematizing of mathematics and its pedagogy. Here, we sample evidence of discursive shifts observed as twelve mathematics teachers engage with a set of problematizing activities (PA) – three rounds of history-focused and mathtask combinations – during a four–month postgraduate course. We trace how the commognitive conflicts orchestrated in the PA triggered changes in the teachers’ narratives about: mathematical objects (such as what a function is); how mathematical objects come to be (such as what led to the emergence of the function object); and, pedagogy (such as what value may lie in listening to students or in trialing innovative assessment practices). Our study explores a hitherto under-researched capacity of the commognitive framework to steer the design, evidence identification and impact evaluation of pedagogical interventions.     

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.     

Charting the microworld territory over time: design and construction in mathematics education

The study discusses the development of theoretical ideas and constructs related to digital microworlds within the mathematics education community and their implications for interpretations of mathematics learning. Starting from Papert’s introduction of the concept during ICME 2 in 1972, we trace the evolution of theoretical approaches concerning the essence of the idea in an attempt to situate the notion of constructionism in the light of contemporary frameworks. We argue that microworlds, and the search for a learnable mathematics, have a continued relevance to mathematics education, but that the lens research attention has shifted over time, with the current foci on communal design, situated and embodied approaches and artefacts whose use crosses boundaries between different practices. To illustrate these shifts and the challenges that still remain, we present examples from our current work involving the use of microworlds for learning and teaching through communication, design and construction.