Proof Frameworks: A Way to Get Started

Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to construct proofs and to prepare them for proof-based courses, such as abstract algebra and real analysis. We have developed a way of getting students, who often stare at a blank piece of paper not knowing what to do, to start writing proofs. This is the technique of writing proof frameworks, which is based on the logical structure of the statements of the theorems and associated definitions. Also, in order to unpack the conclusion and know what is to be proved, students need definitions to become “operable.”     

A case study of one instructor’s lecture-based teaching of proof in abstract algebra: making sense of her pedagogical moves

This paper is a case study of the teaching of an undergraduate abstract algebra course, in particular the way the instructor presented proofs. It describes a framework for proof writing based on Selden and Selden (2009) and the work of Alcock (2010) on modes of thought that support proof writing. The paper offers a case study of the teaching of a traditionally-taught abstract algebra course, including showing the range of practice as larger than previously described in research literature. This study describes the aspects of proof writing and modes of thought the instructor modeled for the students. The study finds that she frequently modeled the aspects of hierarchical structure and formal–rhetorical skills, and structural, critical, and instantiation modes of thought. This study also examines the instructor’s attempts to involve the students in the proof writing process during class by asking questions and expecting responses. Finally, the study describes how those questions and responses were part of her proof presentation. The funneling pattern of Steinbring (1989) describes most of the question and answer discussions enacted in the class with most questions requiring a factual response. Yet, the instructional sequence can be also understood as modeling the way an expert in the discipline thinks and, as such, offering a different type of opportunity for student learning.     

Preparing the Next Generation of Science Teacher Educators: A Model for Developing PCK for Teaching Science Teachers

Science education doctoral programs often fail to address a critical piece—the explicit attention to the preparation of future science teacher educators. In this article, we argue that, in addition to developing skills and a knowledge base for research, doctoral students must be given the opportunity to observe, practice, and reflect on the pedagogical knowledge necessary to instruct science teachers. In particular, we contend that the construct of pedagogical content knowledge (PCK) can be adapted to the context of knowledge for teaching science teachers. We use the PCK construct to propose a model for the development of knowledge for teaching science teachers, grounded in our experiences as doctoral students and faculty mentors. We end by recommending a vision for doctoral preparation and a new standard to be included in the ASTE Professional Knowledge Standards for Science Teacher Educators.     

The Role of Proof in Comprehending and Teaching Elementary Linear Algebra

We describe how elementary Linear Algebra can be taught successfully while introducing students to the concept and practice of ‘mathematical proof’.This is done badly with a sophisticated Definition–Lemma–Proof–Theorem–Proof–Corollary(DLPTPC) approach; badly – since students in elementary Linear Algebra courses have very little experience with proofs and mathematical rigor. Instead, the subjects and concepts of Linear Algebra can be introduced in an exploratory and fundamentally reasoned way. One seemingly successful way to do this is to explore the concept of solvability of linear systems first via the row echelon form (REF). Solvability questions lead to row and column criteria for a REF that can be used repeatedly to: compute subspaces, settle linear (in)dependence, find inverses, perform basis change, compute determinants, analyze eigensystems etc. If these subjects are explained heuristically from the first principles of linear transformations, linear equations, and the REF, students experience the power of a concept–built approach and reap the benefit of deep math understanding. Moreover, early ‘salient point’ proofs lead to an intuitive understanding of ‘math proof’. Once the basic concept of ‘proof’ is ingrained in students, more abstract proofs, even DLPTPC style expositions, on normal matrices, the SVD etc. become accessible and understandable to sophomore students. With the help of this gentle early approach, the concept and construct of a ‘math proof’ becomes firmly embedded in the students' minds and helps with future math courses and general scientific reasoning. This revised version was published online in July 2006 with corrections to the Cover Date.     

Using Pictures to Enhance Students' Understanding of Bayes' Theorem

Students often have difficulty understanding algebraic proofs of statistics theorems. However, it sometimes is possible to prove statistical theorems with pictures in which case students can gain understanding more easily. I provide examples for two versions of Bayes' theorem.     

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.     

Proof image

The emergence of a proof image is often an important stage in a learner’s construction of a proof. In this paper, we introduce, characterize, and exemplify the notion of proof image. We also investigate how proof images emerge. Our approach starts from the learner’s efforts to construct a justification without (or before) attempting any formal argument, and it focuses on the process by which a complete but not necessarily communicable image of that justification becomes available to the learner and provides explanation with certainty. We consider the interplay between the learner’s intuitive and logical thinking and, using the theoretical framework of Abstraction in Context, we trace the construction of knowledge that results from and enables progress of this interplay. The existence and identification of proof images and the nature of the processes by which they emerge constitute the theoretical contribution of this paper. Its practical value lies in the empirical analyses of these processes and in the potential to apply them to the design of tasks that support students in constructing their own proofs images and proofs. We believe that such processes are likely to considerably enrich students’ mathematical experience.     

Doctoral students on the university–industry interface: a review of the literature

Doctoral students are highly important in university—firm relationships, since they are significant producers of knowledge in collaborative research projects, they are an important channel for knowledge transfer between universities and firms, and are vital in network configurations between firms and universities. An increasing number of doctoral students interact with firms, but we know relatively little about the experiences of these students or how collaboration influences their training, research and subsequent careers. With this in mind, this paper presents a literature review of (1) theoretical assumptions concerning the roles doctoral students are expected to fulfill in university–industry relationships, and (2) empirical research of doctoral students’ interaction experience and outcomes of doctoral student-industry interaction. The aim of the paper is to develop hypotheses for further research on doctoral student—industry interaction.     

Multimedia resources designed to support learning from written proofs: an eye-movement study

This paper presents two studies of an intervention designed to help undergraduates comprehend mathematical proofs. The intervention used multimedia resources that presented proofs with audio commentary and visual animations designed to focus attention on logical relationships. In study 1, students studied an e-Proof or a standard written proof and their comprehension was assessed in both immediate and delayed tests; the groups performed similarly at immediate test, but the e-Proof group exhibited poorer retention. Study 2 accounted for this unexpected result by using eye-movement analyses to demonstrate that participants who studied an e-Proof exhibited less processing effort when not listening to the audio commentary. We suggest that the extra support offered by e-Proofs disrupts the processes by which students organise information, and thus restricts the extent to which their new understanding is integrated with existing knowledge. We discuss the implications of these results for evaluating teaching innovations and for supporting proof comprehension.     

Unpacking the logic of mathematical statements

This study focuses on undergraduate students' ability to unpack informally written mathematical statements into the language of predicate calculus. Data were collected between 1989 and 1993 from 61 students in six small sections of a bridge course designed to introduce proofs and mathematical reasoning. We discuss this data from a perspective that extends the notion of concept image to that of statement image and introduces the notion ofproof framework to indicate that part of a theorem's image which corresponds to the top-level logical structure of a proof. For simplified informal calculus statements, just 8.5% of unpacking attempts were successful; for actual statements from calculus texts, this dropped to 5%. We infer that these students would be unable to reliably relate informally stated theorems with the top-level logical structure of their proofs and hence could not be expected to construct proofs or validate them, i.e., determine their correctness.     

Proof constructions and evaluations

In this article, we focus on a group of 39 prospective elementary (grades K-6) teachers who had rich experiences with proof, and we examine their ability to construct proofs and evaluate their own constructions. We claim that the combined “construction–evaluation” activity helps illuminate certain aspects of prospective teachers’ and presumably other individuals’ understanding of proof that tend to defy scrutiny when individuals are asked to evaluate given arguments. For example, some prospective teachers in our study provided empirical arguments to mathematical statements, while being aware that their constructions were invalid. Thus, although these constructions considered alone could have been taken as evidence of an empirical conception of proof, the additional consideration of prospective teachers’ evaluations of their own constructions overruled this interpretation and suggested a good understanding of the distinction between proofs and empirical arguments. We offer a possible account of our findings, and we discuss implications for research and instruction.     

线性代数课程教学的探索与实践——以数学证明教育价值实现的课堂教学为例

要教好线性代数课程,其核心问题就是要通过课堂教学,使学生理解相关的数学知识;训练和培养学生的思维能力以及数学交流能力;帮助学生寻找新旧知识之间的内在联系,使知识系统化;在巩固已有知识的基础上,让学生自己去发现新知识.要实现这一教学目标,训练学生掌握"数学证明"的概念和在实践中的应用至关重要.传统的做法往往是通过"定义一引理一证明一定理一证明一推论"这种复杂的、程序化方法来进行训练的.由于在初等代数课程中,学生很少接受严格数学证明的训练,所以这种俗套的做法成效甚微.相反,如果把线性代数的主题和概念用一种完全合理的探究式方法来引入,那么数学证明的概念和架构将牢固植根于学生的头脑,并且这种思维习惯将对他们后续课程的学习和掌握公理化推理方法都会有很大帮助.     

KNOWLEDGE USE IN THE CONSTRUCTION OF GEOMETRY PROOF BY SRI LANKAN STUDENTS

Within the domain of geometry, proof and proof development continues to be a problematic area for students. Battista (2007) suggested that the investigation of knowledge components that students bring to understanding and constructing geometry proofs could provide important insights into the above issue. This issue also features prominently in the deliberations of the 2009 International Commission on Mathematics Instruction Study on the learning and teaching of proofs in mathematics, in general, and geometry, in particular. In the study reported here, we consider knowledge use by a cohort of 166 Sri Lankan students during the construction of geometry proofs. Three knowledge components were hypothesised to influence the students’ attempts at proof development: geometry content knowledge, general problem-solving skills and geometry reasoning skills. Regression analyses supported our conjecture that all 3 knowledge components played important functions in developing proofs. We suggest that whilst students have to acquire a robust body of geometric content knowledge, the activation and the utilisation of this knowledge during the construction of proof need to be guided by general problem-solving and reasoning skills.     

Software that assists learning within a complex abstract domain: the use of constraint and consequentiality as learning mechanisms

This paper describes a research project into undergraduates’ use of a software tool to learn symbolic logic—a complex abstract domain that has been shown to be intimidating for students. The software allows the students to manipulate proofs in certain ways and then calculates the consequences of their actions. A research method has been developed that allowed students’ use of this tool to be modelled, and this model was then used to identify, refine and create visual cues that provide support for students’ reasoning. The focus of this paper is the role of the software as an artefact to aid students’ visualisation of reasoning processes rather than the logic itself. The main mechanisms by which this visualisation is supported are the imposition of constraints on the actions available and the demonstration to students of the consequences of their actions. The study shows that the software encouraged experimentation with different routes to a proof, and constituted a challenge to fixated reasoning.     

等价类在等价无穷小分析中的应用

利用抽象代数学中的基本知识介绍了等价类在等价无穷小分析中的应用并给出了三个定理     

Making Explicit the Analysis of Students’ Mathematical Discourses – Revisiting a Newly Developed Methodological Framework

Sfard and Kieran [Kieran, C., Educational Studies in Mathematics 46, 2001, 187–228; Sfard, A., Educational Studies in Mathematics 46, 2001, 13–57; Sfard, A. and Kieran, C., Mind, Culture, and Activity 8, 2001, 42–76] have developed a methodological framework, which aims at characterizing the students’ mathematical discourses while they are working in groups. In this study, I focus on an important aspect of this methodological framework, namely the interactive flowcharts. The aim of this study is to suggest two complementary analyses for the construction of the interactive flowcharts: an additional analysis by means of the analytical construct of contextualization as well as an analysis of types of mathematical discourses. Based on data from a study of how four groups of Swedish engineering students collaboratively construct concept maps in linear algebra. I show that the two complementary analyses make the construction of the interactive flowcharts more coherent and transparent, and hence, more reliable. Furthermore, the two complementary analyses dramatically changed the picture as to whether the studied discourses were to be seen as mathematically productive or not. In the end of the article, I discuss the possibilities of performing the suggested additional analyses within the original methodological framework.     

Inquiry-Learning with WebLab: Undergraduate Attitudes and Experiences

The Microelectronics WebLab at MIT allows students to do actual (not simulated) laboratory research on state-of-the art equipment through the Internet. This study assesses the use of WebLab in a junior-level course on microelectronic devices and circuits in 2004–05 and 2005–06. In quantitative surveys and qualitative interviews, students and faculty reported that WebLab was effective as an instrument of learning, and grew more so with refinements of the program. WebLab allowed undergraduates to learn at their own pace and on their own schedules. It enabled them to use different processes of learning (intuitive, visual, abstract), and it gave them an opportunity to link individual and collaborative effort in creative combinations. Online laboratories on this model have broad applications in the experimental sciences and in other research-oriented disciplines.     

构建化学实验高效课堂的有效路径刍论

化学实验能使抽象的化学知识变得生动有趣,培养学生敢于追求真理的科学精神,帮助学生形成实事求是的科学态度。要进行高质量的化学实验教学,教师可开展翻转实验教学,激发学生对化学的兴趣;充分利用电化教学,提高实验教学效率;创设实验问题,提升学生探究能力。文章对构建初中化学实验高效课堂的有效途径进行概述。     

Differences in knowledge networks about acids and bases of year-12, undergraduate and postgraduate chemistry students

This study examines differences in conceptual knowledge representations among three groups of students with different levels of study of chemistry. Variation in the structural characteristics of participants’ concept maps on the topic of acid-base equilibrium were sought by indirect means. Year 12 secondary chemistry students, undergraduate chemistry majors and honours, masters and doctoral candidates participated in the study. Paired propositional links in the concept maps for the three groups were analysed by the scaling algorithms “Pathfinder” and multidimensional scaling. Results show differences among groups in the structural significance in the networks of abstract process-related nodes and matter-related nodes. Implications for theories of conceptual change are discussed.     

思维导图在大学生知识管理中的应用探究

思维导图是一种提高发散性思维的工具,正确使用有利于开启大脑潜能、厘清思维的脉络。目前大学生对知识管理认识不清,能力不强,效率不高,合理利用思维导图能为大学生知识管理提供新的方法,帮助他们学会学习知识、运用知识、管理知识,合理构建自己的知识体系,更好地完成学习任务。