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.     

Preservice teachers learning to teach proof through classroom implementation: Successes and challenges

Proof and reasoning are central to learning mathematics with understanding. Yet proof is seen as challenging to teach and to learn. In a capstone course for preservice teachers, we developed instructional modules that guided prospective secondary mathematics teachers (PSTs) through a cycle of learning about the logical aspects of proof, then planning and implementing lessons in secondary classrooms that integrate these aspects with traditional mathematics curriculum in the United States. In this paper we highlight our framework on mathematical knowledge for teaching proof and focus on some of the logical aspects of proof that are seen as particularly challenging (four proof themes). We analyze 60 lesson plans, video recordings of a subset of 13 enacted lessons, and the PSTs’ self- reported data to shed light on how the PSTs planned and enacted lessons that integrate these proof themes. The results provide insights into successes and challenges the PSTs encountered in this process and illustrate potential pathways for preparing PSTs to enact reasoning and proof in secondary classrooms. We also highlight the design principles for supporting the development of PSTs’ mathematical knowledge for teaching proof.     

Social constructivism,individual constructivism and the role of computers in mathematics education

By asking the question: “Can computers do mathematics?” this paper investigates the relationship between computers and views of mathematics from both individual and social constructivist perspectives. Although these two perspectives ask many of the same questions, they frame these questions in quite different ways of viewing the interaction between individual, subject matter, culture, and cultural tools (e.g. computers). I argue that whereas social constructivist are more likely to take the position that computers alter the way we do mathematics, individual constructivists would more likely say that computers changes the mathematics that we do. Individual constructivists, by placing mathematics itself within the actions carried out by an individual, provide a theoretical framework that allows for the richness and diversity of student constructions that can expand our understanding of mathematics beyond the bounds of any one particular culture.     

One of Berkeley’s arguments on compensating errors in the calculus

This paper addresses three questions related to George Berkeley’s theory of compensating errors in the calculus published in 1734. The first is how did Berkeley conceive of Leibnizian differentials? The second and most central question concerns Berkeley’s procedure which consisted in identifying two quantities as errors and proving that they are equal. The question is how was this possible? The answer is that this was not possible, because in his calculations Berkeley misguided himself by employing a result equivalent to what he wished to prove. In 1797 Lazare Carnot published the expression “a compensation of errors” in an attempt to explain why the calculus functions. The third question is: did Carnot by this expression mean the same as Berkeley?     

A case study of pedagogy of mathematics support tutors without a background in mathematics education

This study investigates the pedagogical skills and knowledge of three tertiary-level mathematics support tutors in a large group classroom setting. This is achieved through the use of video analysis and a theoretical framework comprising Rowland's Knowledge Quartet and general pedagogical knowledge. The study reports on the findings in relation to these tutors’ provision of mathematics support to first and second year undergraduate engineering students and second year undergraduate science students. It was found that tutors are lacking in various pedagogical skills which are needed for high-quality learning amongst service mathematics students (e.g. engineering/science/technology students), a demographic which have low levels of mathematics upon entering university. Tutors teach their support classes in a very fast didactic way with minimal opportunities for students to ask questions or to attempt problems. It was also found that this teaching method is even more so exaggerated in mandatory departmental mathematics tutorials that students take as part of their mathematics studies at tertiary level. The implications of the findings on mathematics tutor training at tertiary level are also discussed.     

A changing trajectory of proof learning in the geometry inquiry classroom

The purpose of this study is to explore how students changes through learning to construct mathematical proofs in an inquiry-based middle school geometry class in Korea. Although proof has long been considered as one of the most important aspects of mathematics education, it is well-known that it is one of the most difficult areas of school mathematics for students. The geometry inquiry classroom (GIC) is an experimental class designed to teach geometry, especially focusing on teaching proof, based on students’ own inquiry. Based on a 2-year participant observation in the GIC, this study was conducted to address the following research question: how has students’ practice of mathematical proof been changed through their participation in the GIC? The in-depth analysis of the classroom discourse identified three stages through which the students’ practice of mathematical proof was transformed in the GIC: ‘emergent understanding of proof’, ‘proof learning as a goal-oriented activity’, ‘experiencing proof as the practice of mathematics’. The study found that as learning evolved through these stages, so the mathematics teacher’s role shifted from being an instructor to a mediator of communication. Most importantly, this research showed that the GIC has created a learning environment where students develop their competence in constructing meaningful mathematical proof and grow to be ‘a human who proves’, ultimately ‘a person who playfully engages with mathematics’.     

The student experience of mathematical proof at university level

While proofs are central to university level mathematics courses, research indicates that some students may complete their degrees with an incomplete picture of what constitutes a proof and how proofs are developed. The paper sets out to review what is known of the student experience of mathematical proof at university level. In particular, some evidence is presented of the conceptions of mathematical proof that recent mathematics graduates bring to their postgraduate course to teach high school mathematics. Such evidence suggests that while the least well-qualified graduates may have the poorest grasp of mathematical proof, the most highly qualified may not necessarily have the richest form of subject matter knowledge needed for the most effective teaching. Some indication of the likely causes of this incomplete student perspective on proof are presented.     

Visualisation and proof: a brief survey of philosophical perspectives

The contribution of visualisation to mathematics and to mathematics education raises a number of questions of an epistemological nature. This paper is a brief survey of the ways in which visualisation is discussed in the literature on the philosophy of mathematics. The survey is not exhaustive, but pays special attention to the ways in which visualisation is thought to be useful to some aspects of mathematical proof, in particular the ones connected with explanation and justification.     

Pre‐Service Teachers and Mathematics: The Impact of Service‐Learning on Teacher Preparation

In spite of repeated reform efforts, there is research and data that suggest that teachers lack the needed knowledge to successfully teach elementary mathematics. Some argue that teachers lack the needed content knowledge while other argue that a lack of confidence and practice are impacting teachers’ ability to successful teach mathematics. As a result of these issues, this paper looks at the impact of a service‐learning experience on pre‐service teachers’ confidence and preparation in the area of mathematics. The service‐learning experience had a number of intended and unintended outcomes. As a result of the service‐learning experience, pre‐service teachers noted increased understanding, innovation and confidence in the area of mathematics.     

Supporting Preservice Teachers' Understanding of Place Value and Multidigit Arithmetic

This article provides an analysis of a teaching experiment conducted in the context of teacher education designed to support preservice teachers' understandings of place value and multidigit addition and subtraction. The experiment addresses the following research question: Can the results from research conducted in elementary mathematics classrooms guide preservice elementary teachers' development of conceptual understanding of the same concepts? In both cases, the students (e.g., elementary students and preservice teachers) participated in activities from an instructional sequence designed to support conceptual understanding of both place value and multidigit addition and subtraction. Analyses of the episodes from the teaching experiment document the learning of the preservice teachers and how that learning was supported by initial conjectures grounded in the research on elementary students' ways of reasoning.     

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.     

Supporting Preservice Teachers' Understanding of Place Value and Multidigit Arithmetic

This article provides an analysis of a teaching experiment conducted in the context of teacher education designed to support preservice teachers' understandings of place value and multidigit addition and subtraction. The experiment addresses the following research question: Can the results from research conducted in elementary mathematics classrooms guide preservice elementary teachers' development of conceptual understanding of the same concepts? In both cases, the students (e.g., elementary students and preservice teachers) participated in activities from an instructional sequence designed to support conceptual understanding of both place value and multidigit addition and subtraction. Analyses of the episodes from the teaching experiment document the learning of the preservice teachers and how that learning was supported by initial conjectures grounded in the research on elementary students' ways of reasoning.     

Mathematics professional development as design for boundary encounters

This theoretical paper examines a process for researchers and teachers to exchange knowledge. We use the concepts of communities of practice, boundary encounters, and boundary objects to conceptualize this process within mathematics professional development (MPD). We also use the ideas from design research to discuss how mathematics professional development researchers can make professional development the focus of their research. In particular, we examine the question: How can MPD be conceptualized and designed around research-based knowledge in ways that promote knowledge exchange about students’ mathematics and mathematics learning among researchers and teachers to improve the practices of both the research and the teaching communities? We propose that MPD is a premier space for researchers and teachers to exchange knowledge from their communities, impacting both researchers’ and teachers’ practices without reducing the importance of either.     

Slodkowski定理的Chirka证明的一些注解(英文)

通过在一个积分算子上运用Schauder不动点定理,Chirka对Slokowski的全纯运动扩张定理提供了一个优美的简单证明.本文中,作者对构造此证明的启发提供一个参考同时用此方法通过不同方式来构造全纯扩张.一个自然的问题是研究这些用不同方式得到的全纯扩张是否相同.因此对全纯扩张唯一性已有的判断法提供一个简单的综述.最后介绍在全纯扩张唯一性存在时的一个应用.     

Teaching children to add and subtract

At a 1980 conference, leading mathematics educators synthesized previous knowledge on children's early understanding of addition and subtraction and proposed central parameters for future research in these areas form a cognitive science perspective. We have, since 1980, increased our knowledge about how children learn to add and subtract, but we need to know more about the best ways for teachers to guide children as they construct knowledge of addition and subtraction.In this article, we review several studies that focus on an enhanced role for teachers in enabling children to learn addition and subtraction. These studies describe efforts that have been made to teach children to use diagrams and mediational representations, number sentences, or algorithms and procedures. The studies report improvement in children's problem-solving performance, but the impact of the efforts described on children's conceptual understanding is less clear. Thus, we analyze this research, pose questions on the relationship of instruction to children's knowledge construction, and propose a research agenda that we believe will enable us to understand how teaching can best help children learn to add and subtract.     

关于二阶常系数线性齐次常微分方程的通解的一个注记

文献[1]给出了二阶常系数线性齐次常微分方程的通解形式,但未回答除了通解形式的解是否还有其它形式的解,本文指出通解包含了所有的解,并给出一个只需要高等数学知识的证明.     

Laurent polynomials and Eulerian numbers

Duistermaat and van der Kallen show that there is no nontrivial complex Laurent polynomial all of whose powers have a zero constant term. Inspired by this, Sturmfels poses two questions: Do the constant terms of a generic Laurent polynomial form a regular sequence? If so, then what is the degree of the associated zero-dimensional ideal? In this note, we prove that the Eulerian numbers provide the answer to the second question. The proof involves reinterpreting the problem in terms of toric geometry.     

Ethnomathematics: Why, and what else?

Ethnomathematics—we all have some notions of what it is, but should it be influencing school mathematics? This paper considers a justification for the influence from a constructivist perspective, and considers the implications of enactivism and other ways of thinking on ethnomathematics. My conclusion is that the influence should be broader—that different cultural groups may have different ways of knowing, that we may be asking the wrong questions, and that we might need to consider ‘ethno-education’