Beliefs and beyond: hows and whys in the teaching of proof

In this paper, we report on a study aimed at describing the way secondary school teachers treat proof and at understanding which factors may influence such a treatment. This study is part of a wider project on proof carried out for many years. In our theoretical framework, we combine references from research on proof with those from research on teachers in relation to their beliefs. The study was carried out through interviews with secondary school teachers aimed at learning how they describe their work with proof in the classroom, and to elicit beliefs and other factors that shape this work. Through the interviews we were able to detect reasons behind teachers’ choices in planning their work in the classroom. In the present paper, we concentrate on four cases that, among other factors, offer elements suitable to unravel the problem of inconsistencies using the construct of leading beliefs, i.e., beliefs (whose nature may vary from teacher to teacher) that seem to drive the way each teacher treats proof.     

The role of prediction in the teaching and learning of mathematics

The prevalence of prediction in grade-level expectations in mathematics curriculum standards signifies the importance of the role prediction plays in the teaching and learning of mathematics. In this article, we discuss benefits of using prediction in mathematics classrooms: (1) students’ prediction can reveal their conceptions, (2) prediction plays an important role in reasoning and (3) prediction fosters mathematical learning. To support research on prediction in the context of mathematics education, we present three perspectives on prediction: (1) prediction as a mental act highlights the cognitive aspect and the conceptual basis of one's prediction, (2) prediction as a mathematical activity highlights the spectrum of prediction tasks that are common in mathematics curricula and (3) prediction as a socio-epistemological practice highlights the construction of mathematical knowledge in classrooms. Each perspective supports the claim that prediction when used effectively can foster mathematical learning. Considerations for supporting the use of prediction in mathematics classrooms are offered.     

Australian teachers’ views of effective mathematics teaching and learning

Thirteen Australian teachers who had been nominated by their professional mathematics teachers’ associations as excellent teachers of elementary school mathematics were interviewed on their beliefs about mathematics, mathematics learning and mathematics teaching. In particular, they were asked to discuss the characteristics of effective teachers of mathematics and excellent mathematics lessons. In spite of their differences in location, experience and teacher education, the teachers displayed a lot of consistency in their responses and in their lists of characteristics. While this group of teachers cannot be claimed to be representative of Australian teachers, they have provided a snapshot of what is regarded as effectiveness in mathematics education in Australian elementary schools.     

Appropriate use of a CAS in the teaching and learning of mathematics

If the use of a computer algebra system (CAS) is to be meaningful and have an impact on students, then it must be grounded in good pedagogy and have some clearly defined goals. It is the authors' belief that an important goal for teaching mathematics with the CAS is that courses be designed so that students can become active participants in their learning experience, planning the problem-solving strategies and carrying them out. The CAS becomes an important tool and a partner in this learning process. To this end, here the authors' have linked the use of the CAS to an existing classification scheme for Mathematical Tasks, called the MATH Taxonomy, and illustrated, through concrete examples, how the goals of teaching and learning of mathematics can be set using this classification together with the CAS.     

Hong Kong teachers’ views of effective mathematics teaching and learning

Twelve experienced mathematics teachers in Hong Kong were invited to face-to-face semi-structured interviews to express their views about mathematics, about mathematics learning and about the teacher and teaching. Mathematics was generally regarded as a subject that is practical, logical, useful and involves thinking. In view of the abstract nature of the subject, the teachers took abstract thinking as the goal of mathematics learning. They reflected that it is not just a matter of “how” and “when”, but one should build a path so that students can proceed from the concrete to the abstract. Their conceptions of mathematics understanding were tapped. Furthermore, the roles of memorisation, practices and concrete experiences were discussed, in relation with understanding. Teaching for understanding is unanimously supported and along this line, the characteristics of an effective mathematics lesson and of an effective mathematics teacher were discussed. Though many of the participants realize that there is no fixed rule for good practices, some of the indicators were put forth. To arrive at an effective mathematics lesson, good preparation, basic teaching skills and good relationship with the students are prerequisite.     

United States teachers’ views of effective mathematics teaching and learning

This study investigates US teachers’ cultural beliefs concerning effective mathematics teaching using semi-structured interviews with 11 experienced teachers. For US teachers, effective teaching is student-centered. Cognitively appropriate mathematical content should be understood through many hands-on activities that allow students to explore by themselves the relationship between mathematical knowledge and their life experiences. Correspondingly, the US teachers view an effective teacher as a facilitator who is sensitive to student social and cognitive needs and is skillful at organizing collaborative learning. The result of this study helps researchers and educators understand the student-centered learning model in US classrooms.     

Differentiation of teaching and learning mathematics: an action research study in tertiary education

Diversity and differentiation within our classrooms, at all levels of education, is nowadays a fact. It has been one of the biggest challenges for educators to respond to the needs of all students in such a mixed-ability classroom. Teachers’ inability to deal with students with different levels of readiness in a different way leads to school failure and all the negative outcomes that come with it. Differentiation of teaching and learning helps addressing this problem by respecting the different levels that exist in the classroom, and by responding to the needs of each learner. This article presents an action research study where a team of mathematics instructors and an expert in curriculum development developed and implemented a differentiated instruction learning environment in a first-year engineering calculus class at a university in Cyprus. This study provides evidence that differentiated instruction has a positive effect on student engagement and motivation and improves students’ understanding of difficult calculus concepts.     

Connecting the learning of advanced mathematics with the teaching of secondary mathematics: Inverse functions,domain restrictions,and the arcsine function

Prospective secondary mathematics teachers are typically required to take advanced university mathematics courses. However, many prospective teachers see little value in completing these courses. In this paper, we present the instantiation of an innovative model that we have previously developed on how to teach advanced mathematics to prospective teachers in a way that informs their future pedagogy. We illustrate this model with a particular module in real analysis in which theorems about continuity, injectivity, and monotonicity are used to inform teachers’ instruction on inverse trigonometric functions and solving trigonometric equations. We report data from a design research study illustrating how our activities helped prospective teachers develop a more productive understanding of inverse functions. We then present pre-test/post-test data illustrating that the prospective teachers were better able to respond to pedagogical situations around these concepts that they might encounter.     

It matters which class you are in: student-centred teaching and the enjoyment of learning mathematics

The 2007 Trends in International Mathematics and Science Survey highlighted how attitudes to mathematics had declined sharply for students in many of the high attaining countries in the survey, England being no exception. There is a notable drop in positive attitudes to mathematics between 9 and 14, as well as a remarkable decline for 14 year olds over time. This paper explores survey data collected from over 3000 11-year-olds in 16 schools during 2008 with the goal of exploring possible factors that might be contributing to this attitudinal decline. The association between student-centred teaching and enjoyment of learning mathematics is reported as part of a multi-scale analysis that shows the extent to which student experiences differ between schools and between classes within schools.     

Differences in teaching and opportunities for learning in primary mathematics classes

The widespread acceptance in the view that learning is an active constructive process requires teaching that is fundamentally different from classical pedagogy. It is generally accepted that teaching must consist of highly interactive and discursive situations. However, these differences in teaching are not well understood. In this paper, examples of teaching and the ways these distinctions influence children’s opportunities for learning.     

Issues in theorizing mathematics learning and teaching: A contrast between learning through activity and DNR research programs

By continuing a contrast with the DNR research program, begun in Harel and Koichu (2010), I discuss several important issues with respect to teaching and learning mathematics that have emerged from our research program which studies learning that occurs through students’ mathematical activity and indicate issues of complementarity between DNR and our research program. I make distinctions about what we mean by inquiring into the mechanisms of conceptual learning and how it differs from work that elucidates steps in the development of a mathematical concept. I argue that the construct of disequilibrium is neither necessary nor sufficient to explain mathematics conceptual learning. I describe an emerging approach to instruction aimed at particular mathematical understandings that fosters reinvention of mathematical concepts without depending on students’ success solving novel problems.     

Theories in practice: mathematics teaching and mathematics teacher education

Whilst research on the teaching of mathematics and the preparation of teachers of mathematics has been of major concern in our field for some decades, one can see a proliferation of such studies and of theories in relation to that work in recent years. This article is a reaction to the other papers in this special issue but I attempt, at the same time, to offer a different perspective. I examine first the theories of learning that are either explicitly or implicitly presented, noting the need for such theories in relation to teacher learning, separating them into: socio-cultural theories; Piagetian theory; and learning from practice. I go on to discuss the role of social and individual perspectives in authors’ approach. In the final section I consider the nature of the knowledge labelled as mathematical knowledge for teaching (MKT). I suggest that there is an implied telos about ‘good teaching’ in much of our research and that perhaps the challenge is to study what happens in practice and offer multiple stories of that practice in the spirit of “wild profusion” (Lather in Getting lost: Feminist efforts towards a double(d) science. SUNY Press, New York, 2007).     

Kaizen teaching and the learning habits of engineering students in a freshman mathematics course

Which are the teaching methods that actually contribute to the learning of mathematics? The answer to this certainly is the holy grail of didactic and pedagogy, and should be supported by large scale statistical evidence. Our article aims at providing an initial step into this direction by first illustrating a teaching paradigm that is suited for the generation of large scale data sets: based on industry best practice quality assurance standards we introduce the Kaizen teaching paradigm which enforces Kolb’s reflective learning cycle on the students’ side. Second, we present and analyze the data we obtained through our pilot implementation at a engineering freshman mathematics course in the Sultanate of Oman. These emphasize the effectiveness of Kaizen teaching and once again show the necessity of continuous learning. A practice that seems to be forgotten in traditional university engineering courses due to the mere size of the audience. In particular it seems that a Markovian estimator for students’ performance may have to be considered.     

Chinese (Mainland) teachers’ views of effective mathematics teaching and learning

This study investigates Chinese teachers’ cultural beliefs concerning effective mathematics teaching through semi-structured interview with nine experienced teachers. For the Chinese teachers, an effective teacher should always be passionate and committed to the teaching profession. She should not only understand the knowledge in the textbook thoroughly but also be able to carefully craft the knowledge from the textbook for teaching by predicting possible students’ difficulties. Although Chinese teachers emphasize student participation and flexible teaching, they tend to see the teacher’s ability to design and lead coherent lessons as the key for facilitating students’ understanding. The result of this study helps researchers and educators understand the teacher-designed and content-oriented teaching model in Chinese classrooms.     

Mexican high school students' social representations of mathematics,its teaching and learning

This paper reports a qualitative research that identifies Mexican high school students’ social representations of mathematics. For this purpose, the social representations of ‘mathematics’, ‘learning mathematics’ and ‘teaching mathematics’ were identified in a group of 50 students. Focus group interviews were carried out in order to obtain the data. The constant comparative style was the strategy used for the data analysis because it allowed the categories to emerge from the data. The students’ social representations are: (A) Mathematics is…(1) important for daily life, (2) important for careers and for life, (3) important because it is in everything that surrounds us, (4) a way to solve problems of daily life, (5) calculations and operations with numbers, (6) complex and difficult, (7) exact and (6) a subject that develops thinking skills; (B) To learn mathematics is…(1) to possess knowledge to solve problems, (2) to be able to solve everyday problems, (3) to be able to make calculations and operations, and (4) to think logically to be able to solve problems; and (C) To teach mathematics is…(1) to transmit knowledge, (2) to know to share it, (3) to transmit the reasoning ability, and (4) to show how to solve problems.