Misconceptions Reconceived: A Constructivist Analysis of Knowledge in Transition

This article uses a critical evaluation of research on student misconceptions in science and mathematics to articulate a constructivist view of learning in which student conceptions play productive roles in the acquisition of expertise. We acknowledge and build on the empirical results of misconceptions research but question accompanying views of the character, origins, and growth of students' conceptions. Students have often been viewed as holding flawed ideas that are strongly held, that interfere with learning, and that instruction must confront and replace. We argue that this view overemphasizes the discontinuity between students and expert scientists and mathematicians, making the acquisition of expertise difficult to conceptualize. It also conflicts with the basic premise of constructivism: that students build more advanced knowledge from prior understandings. Using case analyses, we dispute some commonly cited dimensions of discontinuity and identify important continuities that were previously ignored or underemphasized. We highlight elements of knowledge that serve both novices and experts, albeit in different contexts and under different conditions. We provide an initial sketch of a constructivist theory of learning that interprets students' prior conceptions as resources for cognitive growth within a complex systems view of knowledge. This theoretical perspective aims to characterize the interrelationships among diverse knowledge elements rather than identify particular flawed conceptions; it emphasizes knowledge refinement and reorganization, rather than replacement, as primary metaphors for learning; and it provides a framework for understanding misconceptions as both flawed and productive.     

A Critical Examination of Three Factors in the Decline of Proof

Proof seems to have been losing ground in the secondary mathematics curriculum despite its importance in mathematical theory and practice. The present paper critically examines three specific factors that have lent impetus to the decline of proof in the curriculum: a) The idea that proof need be taught only to those students who intend to pursue post-secondary education, b) the view that deductive proof need no longer be taught because heuristic techniques are more useful than proof in developing skills in reasoning and justification, c) the idea that deductive proof might profitably be abandoned in the classroom in favour of a dynamic visual approach to mathematical justification. The paper concludes that proof should be an essential component in mathematics education at all levels and compatible with both heuristic techniques and dynamic visual approaches.     

Experiencing equivalence but organizing order

The notion of equivalence relation is arguably one of the most fundamental ideas of mathematics. Accordingly, it plays an important role in teaching mathematics at all levels, whether explicitly or implicitly. Our success in introducing this notion for its own sake or as a means to teach other mathematical concepts, however, depends largely on our own conceptions of it. This paper considers various conceptions of equivalence, in history, in mathematics today, and in mathematics education. It reveals critical differences in the notion of equivalence at different points in history and a meaning for equivalence proposed by mathematicians and mathematics educators that is at variance with the ways that learners may think. These differences call into question the most popular view of the subject: that the mathematical notion of equivalence relation is the result of spelling out our experience of equivalence. Moreover, the findings of this study suggest that the standard definition of an equivalence relation is ill-chosen from a pedagogical point of view but well-crafted from a mathematical point of view.     

MATHEMATICS EDUCATORS’ VIEWS ON THE ROLE OF MATHEMATICS LEARNING IN DEVELOPING DEDUCTIVE REASONING

This study examines the views of people involved in mathematics education regarding the commonly stated goal of using mathematics learning to develop deductive reasoning that is usable outside of mathematical contexts. The data source includes 21 individual semi-structured interviews. The findings of the study show that the interviewees ascribed different meanings to the above-stated goal. Moreover, none of them said that it is possible to develop formal logic-based reasoning useful outside of mathematics, but for different reasons. Three distinct views were identified: the intervention–argumentation view, the reservation–deductive view, and the spontaneity–systematic view. Each interviewee’s view was interrelated with the interviewee’s approach to deductive reasoning and its nature in mathematics and outside it.     

李善兰的数学哲学思想——纪念李善兰诞生200周年

李善兰(1811-1882)是19世纪我国最重要的数学家,他的数学成就在我国近代科学史上具有十分独特的地位。他的数学成就与他的数学哲学思想是分不开的。他的数学哲学思想主要之点是:数学是富国强兵的工具;研究数学的方法和步骤是"深思"、"通其法"和"解明之";要继承,更要创新;会通中西数学;以研究数学为主,会通相关学科。     

Learning opportunities and negotiating social norms in mathematics class discussion

In this paper, we describe our current understanding of the relationship between learning opportunities and the negotiation of social norms in mathematics class discussion. With supportive social norms, a student non-acceptance can lead to constructive discussion in which students compare, contrast, elaborate and refute mathematics assertions just as mathematicians do. Under this circumstance, opportunities for learning are ample because students are more likely to find a situation problematic during inquiry activities. However, incompatible interpretations often cause emotional responses. Negative emotional responses, when not dealt with appropriately, tend to discourage students from participating in inquiry activities. Therefore, teachers need to take an active role in helping students interpret such situations through negotiating social norms. The negotiation of social norms makes possible the negotiation of mathematical meaning.     

A Review of the Literature: Fraction Instruction for Struggling Learners in Mathematics

Fractions are an essential foundational skill for future mathematics success (NMAP, 2008). The purpose of this article was to review current instructional practices for teaching fractions to struggling learners and to examine the quality and effectiveness of contemporary research with a view to indicating directions for future research. A comprehensive search of literature published between 1990 and 2008 resulted in the identification of 10 empirical studies that targeted fraction skills for struggling learners. Results indicated that three interventions, found to be effective for improving outcomes in mathematics for struggling learners, were also effective for teaching fractions: graduated sequence, strategy instruction, and direct instruction. In addition, explicit instruction was identified as necessary for improving students’ performance in fractions. Overall, this review highlighted the paucity of research in this critical mathematical content area.     

探讨高等数学教学中的创新教育

提出在高等数学的课堂教学时，引入“抽象模式、数学结构”教学法，采取课堂讨论等措施，在高等数学教学中体现创新教育理念。运用督促检查法、网上评阅答疑法、作业评改小组法等方法，改进高等数学作业的评阅方式。采用多渠道评价方式代替单一考试评价方式。对高等数学的这种创新教学的效果进行了调查和分析。     

Analysis of the cognitive unity or rupture between conjecture and proof when learning to prove on a grade 10 trigonometry course

We present results from a classroom-based intervention designed to help a class of grade 10 students (14–15 years old) learn proof while studying trigonometry in a dynamic geometry software environment. We analysed some students’ solutions to conjecture-and-proof problems that let them gain experience in stating conjectures and developing proofs. Grounded on a conception of proof that includes both empirical and deductive mathematical argumentations, we show the trajectories of some students progressing from developing basic empirical proofs towards developing deductive proofs and understanding the role of conjectures and proofs in mathematics. Our analysis of students’ solutions is based on networking Boero et al.’s construct of cognitive unity of theorems, Pedemonte’s structural and referential analysis of conjectures and proofs, and Balacheff and Margolinas’ cK¢ model, while using Toulmin schemes to represent students’ productions. This combination has allowed us to identify several emerging types of cognitive unity/rupture, corresponding to different ways of solving conjecture-and-proof problems. We also show that some types of cognitive unity/rupture seem to induce students to produce deductive proofs, whereas other types seem to induce them to produce empirical proofs.     

Prospective teachers’ skills of attending,interpreting and responding to content-specific characteristics of mathematics instruction in classroom videos

This study explores prospective teachers’ skills of attending, interpreting and responding to content-specific characteristics of mathematics instruction in classroom videos. Prospective teachers analyzed the mathematics instruction of two teachers through four video clips and proposed alternative instructional ways to support the teaching and learning of mathematics. The results indicated that as prospective teachers examined the teachers’ instructional practices, they increased their level of attending and interpretation to content-specific aspects of instruction rather than focusing on generic dimensions of the instruction. When they watched and compared different characteristics of teachers’ mathematics instruction, they provided more detailed and mathematical instructional suggestions.     

Metacognition and Middle Grade Mathematics Teachers: Supporting Productive Struggle

Communicating mathematical ideas through writing, listening, and verbalizing allows students to think about how they “think” about mathematics. By focusing this communication on a reflection of how one thinks about mathematics, metacognitive writing engages students as mathematicians and learners. In this article we describe a professional development that we implemented with middle grade mathematics teachers focused on metacognitive writing as a tool to support productive struggle in the mathematics classroom. Thus, this practitioner article adds to the knowledge base on how to develop middle grade teachers metacognitive writing through engagement in productive struggle. Recommendations for practice are included.     

对普通高中数学课程标准“基本理念7”的思考

数学的“形式化”与“本质”的争论由来已久,至今不绝．综合多个视角考察可知,数学的本质是经验性与演绎性在实践基础上的辨证统一．如果把数学的本质看成一枚硬币,演绎性,即“形式化”是其一面;经验性．即“经验化——非形式化”是其另一面．由是观之,普通高中数学课程标准“基本理念7”的表述存在让人误解与含糊之处．     

数学开放性问题和平行任务的设计:处理数学学习差异的一种途径

普及教育推行以来,处理课堂中学生的学习差异一直是一个世界性的课题。在数学科,按照数学问题或习题的难度进行分层教学是一贯的做法,期望不同能力的学生逐阶而上。然而,激发学生的数学思考,提供不同学生更多的学习机会,是处理差异化教学的关键。在面对混合能力的大班教学中,运用数学开放问题,结合平行练习任务进行分层教学或是一条有效的处理学生数学学习差异的途径。     

Fostering Instructor Knowledge of Student Thinking Using The Flipped Classroom

Abstract

In this article, we share a model of flipped instruction that allowed us to gain a window into our students’ mathematical thinking. We depict how that increased awareness of student thinking shaped our mathematics instruction in productive ways. Drawing on our experiences with students in our own classrooms, we show how flipped instruction can be used to design experiences that help students make sense of mathematics during class sessions.     

Reification of instructional materials as part of the process of developing problem‐based practices in mathematics education

This teacher development study closely examined a teacher's practice for the purpose of understanding how she selected and implemented instructional materials, and correspondingly how these processes changed as she developed her problem‐based practice throughout a school year. Data sources included over 20 hours of planning and analysis meetings with the teacher and 27 video‐taped lessons with discussions before and after each lesson. Through qualitative analysis we examined the data for: students' cognitive demand for curricular materials the teacher selected and implemented; teacher's beliefs and practices for students' engagement in mathematical thinking; and teacher's and students' communication about mathematics during instruction. We found that the teacher shifted her views and use of instructional materials as she changed her practice towards more problem‐based approaches. The teacher moved from closely following her traditional, district‐adopted textbook to selecting problem‐based tasks from outside resources to build a curriculum. Simultaneously, she changed her practice to focus more on students' engagement in mathematical thinking and their communication about mathematics as part of learning. During this shift in practice, the teacher began to reify instructional materials, viewing them as instruments of her practice to meet students' needs. The process of shifting her views was gradual over the school year and involved substantial analysis and reflection on practice from the teacher. Implications include that teachers and teacher educators may need to devote more attention and support for teachers to use instructional materials to support instruction, rather than materials to prescribe instruction. This use of instructional materials may be an important part of transforming practice overall.     

数学焦虑的成因及对策

数学焦虑是由数学问题和情境引发的一系列自我强迫和紧张,接近精神病学中的焦虑症状.在深入了解我国中小学数学教育现状的基础上分析其成因,着重讨论教师观念上的错误和教学行为的不当,提出帮助学生克服数学焦虑的若干建议.     

Mathematics and mathematics education: Content and people, relation and difference

A broad view of mathematics education takes it as the study of how people learn and do mathematics. Starting with this view, the actual and potential relationships of mathematics education as are search discipline to mathematics as a field of knowledge and activity and to the mathematicians carrying out that activity are analyzed. This leads to the picture of a gulf between the two scientific communities which are based in different cultures of thinking and research. A (meta-)study of mathematics and all its facets termed here mathematicology is proposed. It could serve as common ground for cooperative studies by mathematicians and mathematics educators. Thereby the gulf will not necessarily become narrower but abridge over the differences and mutual misunderstandings could be built.     

Imagining the mathematician: young people talking about popular representations of maths

This paper makes both a critical analysis of some popular cultural texts about mathematics and mathematicians, and explores the ways in which young people deploy the discourses produced in these texts. We argue that there are particular (and sometimes contradictory) meanings and discourses about mathematics that circulate in popular culture, that young people use these as resources in identity making as (non-)mathematicians, negotiating their meaning in ways that are not always predictable, and that their influence on young people is diffuse and nevertheless important. The paper discusses the discourses that prevail in some of the popular cultural images of mathematics and mathematicians that came up in our research. We show how mathematics is represented as a secret language, while mathematicians are often mad, mostly male and almost invariably white. We then explore how young people negotiate these discourses, positioning themselves in relation to mathematics. Here we draw attention to the fact that both those who continue with mathematics after it ceases to be compulsory and those who do not, deploy similar images of mathematics and mathematicians. What is different is how they respond to and negotiate these images.     

Unpacking foreshadowing in mathematics teachers’ planned practices

This paper provides an empirical exploration of mathematics teachers’ planned practices. Specifically, it explores the practice of foreshadowing, which was one of Wasserman’s (2015) four mathematical teaching practices. The study analyzed n?=?16 lessons that were planned by pairs of highly qualified and experienced secondary mathematics teachers, as well as the dialogue that transpired, to identify the considerations the teachers made during this planning process. The paper provides empirical evidence that teachers engage in foreshadowing as they plan lessons, and it exemplifies four ways teachers engaged in this practice: foreshadowing concepts, foreshadowing techniques, foregrounding concepts, and foregrounding techniques. Implications for mathematics teacher education are discussed.