Building on mathematical events in the classroom

Mathematical events from classrooms were used as stimuli to encourage mathematical discussion in two groups of mathematics teachers at the secondary level. Each event was accompanied by an analysis of mathematics that would be useful to the teacher in such a situation. The Situations, mathematical events and analyses, were used originally to create a framework describing the Mathematical Proficiency for Teaching at the Secondary Level, and then they were used with both Prospective and Practicing teachers to validate the framework. Teachers involved in the validation research claimed that the process was instructional. The process is explained, and teachers’ quotes provide evidence that the experience provoked changes in teachers’ understanding of mathematics. This process, which builds on mathematical events from the classroom, holds potential as a professional development experience that helps teachers expand their expertise in teaching mathematics.     

Toward the problem-centered classroom: trends in mathematical problem solving in Japan

In this paper, I summarize the influence of mathematical problem solving on mathematics education in Japan. During the 1980–1990s, many studies had been conducted under the title of problem solving, and, therefore, even until now, the curriculum, textbook, evaluation and teaching have been changing. Considering these, it is possible to identify several influences. They include that mathematical problem solving helped to (1) enable the deepening and widening of our knowledge of the students’ processes of thinking and learning mathematics, (2) stimulate our efforts to develop materials and effective ways of organizing lessons with problem solving, and (3) provide a powerful means of assessing students’ thinking and attitude. Before 1980, we had a history of both research and practice, based on the importance of mathematical thinking. This culture of mathematical thinking in Japanese mathematics education is the foundation of these influences.     

Promoting productive mathematical classroom discourse with diverse students

Productive mathematical classroom discourse allows students to concentrate on sense making and reasoning; it allows teachers to reflect on students’ understanding and to stimulate mathematical thinking. The focus of the paper is to describe, through classroom vignettes of two teachers, the importance of including all students in classroom discourse and its influence on students’ mathematical thinking. Each classroom vignette illustrates one of four themes that emerged from the classroom discourse: (a) valuing students’ ideas, (b) exploring students’ answers, (c) incorporating students’ background knowledge, and (d) encouraging student-to-student communication. Recommendations for further research on classroom discourse in diverse settings are offered.     

Comparative readings of the nature of the mathematical knowledge under construction in the classroom

The present work reports on an attempt to empirically identify the epistemological status of mathematical knowledge interactively constituted in the classroom. To this purpose, three relevant theoretical constructs are employed in order to analyze two lessons provided by two secondary school teachers. The aim of these analyses was to enable a comparative reading of the nature of the mathematical knowledge under construction. The results show that each of these three perspectives allows access to specific features of this knowledge, which do not coincide. Moreover, when considered simultaneously, the three perspectives offer a rather informed view of the status of the knowledge at hand.     

From convergence to divergence: the development of mathematical problem solving in research, curriculum, and classroom practice in Singapore

Following the movement of problem solving in the US and other parts of the world in the 1980s, problem solving became the central focus of Singapore’s national school mathematics curriculum in 1990 and thereafter the key theme in research and practice. Different from some other countries, this situation has largely not changed in Singapore mathematics education since then. However, within the domain of problem solving, mathematics educators in Singapore focused more on the fundamental knowledge, basic skills, and heuristics for problem solving till the mid 1990s. In particular, problem solving heuristics, especially the so-called “model method”, a term most widely used for problem solving, received much attention in syllabus, research, and classroom instruction. Since the late 1990s, following the national vision of “Thinking Schools, Learning Nation” and nurturing modern citizens with independent, critical, and creative thinking, Singapore mathematics educators’ attention has greatly expanded to the development of students’ higher-order thinking, self-reflection and self-regulation, alternative ways of assessment and instruction, among other aspects concerning problem solving. Researchers have also looked into the advantages and disadvantages of Singapore’s textbooks in representing problem solving, and the findings of these investigations have influenced the development of the latest school mathematics textbooks.     

Understanding authority in small-group co-constructions of mathematical proof

Authority becomes shared in mathematics classrooms when perceived sources of valid mathematical knowledge extend beyond the teacher/textbook and allow both students and disciplinary modes of reasoning to hold authority. The goal of this research is to better understand classroom situations that are intended to facilitate shared authority over proof, namely small-group episodes where students are granted authority (Gerson & Bateman, 2010) to co-construct mathematical proofs. We sought to better understand the content of undergraduate students’ attention during group proving and the sources of legitimacy for students. Using Stylianides’ (2007) definition of proof as an analytical frame, we found that student discourse focused primarily upon the mode of argumentation, followed by the mode of argument representation, and then the set of accepted statements. We identified four themes with respect to the sources of authority students relied upon in their group proving: (1) the course rubric, (2) peers’ confidence, (3) form and symbols, and (4) logical structure. Implications for research and practice are presented.     

The concept of proof in the light of mathematical work

ZDM – Mathematics Education - Our article aims to show how illuminating mathematical work as a concept from didactics of mathematics is useful in understanding issues relating to proving and...     

Teaching mathematical modelling through project work

The paper presents and analyses experiences from developing and running an inservice course in project work and mathematical modelling for mathematics teachers in the Danish gymnasium, e.g. upper secondary level, grade 10~12. The course objective is to support the teachers to develop, try out in their own classes, evaluate and report a project based problem oriented course in mathematical modelling. The in-service course runs over one semester and includes three seminars of 3, 1 and 2 days. Experiences show that the course objectives in general are fulfilled and that the course projects are reported in manners suitable for internet publication for colleagues. The reports and the related discussions reveal interesting dilemmas concerning the teaching of mathematical modelling and how to cope with these through «setting the scene» for the students modelling projects and through dialogues supporting and challenging the students during their work. This is illustrated and analysed on the basis of two course projects.     

Using mathematical programming heuristics in a multicriteria network flow context

In this paper, we propose a local search procedure to test the robustness of a specific ‘satisfying point’ neighbourhood. It consists of the following steps: (1) build an indifference area around the satisfying point in the criteria space by using thresholds (this takes into account the possible uncertainty, vagueness and/or inaccuracy of data); (2) find some points in the satisfying point neighbourhood and the corresponding solutions in the decision variables space; (3) test the quality of these solutions from the point of view of user preference. The indifference area is defined by adding constraints to the network model. This approach, which allows us to verify the adequacy of the model, has been applied to a set of multicriteria network flow problems. A heuristic method, based on Lagrangian duality and subgradient techniques, exploits the combinatorial structure of network flow problems in order to find certain feasible points.     

Instances of mathematical thinking among low attaining students in an ordinary secondary classroom

This paper is a report of a classroom research project whose aim was to find out whether low attaining 14-year-old students of mathematics would be able to think mathematically at a level higher than recall and reproduction during their ordinary classroom mathematics activities. Analysis of classroom interactive episodes revealed many instances of mathematical thinking of a kind which was not normally exploited, required or expected in their classes. Five episodes are described, comparing the students’ thinking to that usually described as “advanced.” In particular, some episodes suggest the power of a type of prompt which can be generalized as “going across the grain.”     

Studying teachers’ mathematical argumentation in the context of refuting students’ invalid claims

This study investigates teachers’ argumentation aiming to convince students about the invalidity of their mathematical claims in the context of calculus. 18 secondary school mathematics teachers were given three hypothetical scenarios of a student's proof that included an invalid algebraic claim. The teachers were asked to identify possible mistakes and explain how they would refute the student's invalid claims. Two of them were also interviewed. The data were analysed in terms of the content and structure of argumentation and the types of counterexamples the teachers generated. The findings show that teachers used two main approaches to refute students’ invalid claims, the use of theory and the use of counterexamples. The role of these approaches in the argumentation process was analysed by Toulmin's model and three types of reasoning emerged that indicate the structure of argumentation in the case of refutation. Concerning the counterexamples, the study shows that few teachers use them in their argumentation and in general they underestimate their value as a proof method.     

Understanding the role of the teacher in emerging classroom practices: searching for patterns of participation

The relationship between acquisitionism and participationism is a challenge in research on and with teachers. This study uses a patterns-of-participation framework (PoP), which aims to develop coherent and dynamic understandings of teaching as well as to meet the conceptual and methodological problems of other approaches. The paper presents PoP theoretically, but also illustrates its empirical use. It presents a novice teacher, Anna, who often engages with mathematics and with aspects of ‘the reform’ in ways that link well with how she builds relationships with her students and positions herself in her team of teachers. However, in other situations her engagement with mathematics is overshadowed by her involvement in other practices. The study suggests that there is some potential in PoP in spite of methodological difficulties.     

The development of children's understanding of mathematical patterns through mathematical activities

In this paper we report how children (aged 8) developed their mathematical understanding through number tasks based on the Fibonacci sequence (Bamboo numbers) used in the context of a Substantial Learning Environment (SLE), which is designed to be mathematically rich, have a clear purpose and give opportunities to utilise mathematical thinking. The flexible nature of the SLEs makes it possible for teachers and children to explore various mathematical patterns. To capture children's activities when working within SLEs, we make particular reference to Pegg and Tall's work in 2005, and consider a theoretical framework based on the SOLO taxonomy (Biggs and Collis 1982) and the developmental process of understanding mathematical concepts. It was found that the key progression to be made through learning using our Bamboo number-based SLEs is from Multi-structural to Relational levels. It was also suggested that it is difficult for many children to understand the structural aspects of number patterns.