根与系数关系证明图式的研究

证明图式是个体对命题正确性产生疑惑和消除疑惑的心理对应物.学完一元二次方程根与系数关系证明之后的大多数学生同时具有一般式证明和举例证明图式;面对不同的问题情境时,他们会使用不同的图式.在进行一元二次方程根与系数关系的证明教学时,应引导学生观察、归纳得出猜想,然后进行证明,把证明作为探索活动的自然的延续和必要的发展.     

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.     

The Features and Relationships of Reasoning,Proving and Understanding Proof in Number Patterns

Three issues about students reasoning, proving and understanding proof in number patterns are investigated in this paper. The first is to elaborate the features of junior high students reasoning on linear and quadratic number patterns. The second is to study the relationships between 9th graders justification of mathematical statements about number patterns and their understanding of proof and disproof. The third is to evaluate how reasoning on number patterns is related to constructing proofs. Students in this study were nationally sampled by means of two stages. Some new findings which have not been discovered in some past researches are reported here. These findings include (1) checking geometric number patterns appears to have different positions between the tasks of the linear and the quadratic expressions; (2) proof with the algebraic mode is easy to know but hard to do; (3) disproof with only one counterexample is hard to know but easy to do; (4) arguments with empirical mode or specific symbols were hard for students to validate but very convincing for them; and (5) reasoning on number patterns is supportive for proving in number patterns, and reasoning on number patterns and proof in algebra should be designed as complementary activities for developing algebraic thinking.     

The Interplay of Teacher and Student Actions in the Teaching and Learning of Geometric Proof

Proof and reasoning are fundamental aspects of mathematics. Yet, how to help students develop the skills they need to engage in this type of higher-order thinking remains elusive. In order to contribute to the dialogue on this subject, we share results from a classroom-based interpretive study of teaching and learning proof in geometry. The goal of this research was to identify factors that may be related to the development of proof understanding. In this paper, we identify and interpret students' actions, teacher's actions, and social aspects that are evident in a classroom in which students discuss mathematical conjectures, justification processes and student-generated proofs. We conclude that pedagogical choices made by the teacher, as manifested in the teacher's actions, are key to the type of classroom environment that is established and, hence, to students' opportunities to hone their proof and reasoning skills. More specifically, the teacher's choice to pose open-ended tasks (tasks which are not limited to one specific solution or solution strategy), engage in dialogue that places responsibility for reasoning on the students, analyze student arguments, and coach students as they reason, creates an environment in which participating students make conjectures, provide justifications, and build chains of reasoning. In this environment, students who actively participate in the classroom discourse are supported as they engage in proof development activities. By examining connections between teacher and student actions within a social context, we offer a first step in linking teachers' practice to students' understanding of proof.     

How accurately can students evaluate the quality of self-generated examples of declarative concepts? Not well,and feedback does not help

Students are commonly asked to learn declarative concepts in many courses. One strategy students report using involves generating concrete examples of abstract concepts. If students have difficulties evaluating the quality of their generated examples, then instructors will need to provide students with appropriate scaffolds or feedback to improve judgmentaccuracy. No prior research has investigated if students can accurately evaluate the quality of the examples they generate, which was the first aim of the current research. The second aim of this research was to investigate the extent to which providing feedback while students evaluate their generated examples can improve the accuracy of their example-quality judgments. In two experiments, students generated examples for declarative concepts from social psychology and then judged the quality of their examples. When making judgments, students received no feedback (in which they were only given the key term), full definition feedback (in which they were shown the definition of the declarative concept) or idea unit feedback (in which they first evaluated if they represented each idea unit of the definition within their example). Outcomes showed that students were overconfident when judging the quality of their examples, specifically for commission errors (i.e., examples that were entirely incorrect). Surprisingly, full definition and idea unit feedback did not help students improve the accuracy of their example-quality judgments. Thus, until scaffolds are discovered to reduce student overconfidence, instructors will need to assist in evaluating generated examples as students use this strategy to learn declarative concepts.     

Providing written feedback on students’ mathematical arguments: proof validations of prospective secondary mathematics teachers

Mathematics teachers play a unique role as experts who provide opportunities for students to engage in the practices of the mathematics community. Proof is a tool essential to the practice of mathematics, and therefore, if teachers are to provide adequate opportunities for students to engage with this tool, they must be able to validate student arguments and provide feedback to students based on those validations. Prior research has demonstrated several weaknesses teachers have with respect to proof validation, but little research has investigated instructional sequences aimed to improve this skill. In this article, we present the results from the implementation of such an instructional sequence. A sample of 34 prospective secondary mathematics teachers (PSMTs) validated twelve mathematical arguments written by high school students. They provided a numeric score as well as a short paragraph of written feedback, indicating the strengths and weaknesses of each argument. The results provide insight into the errors to which PSMTs attend when validating mathematical arguments. In particular, PSMTs’ written feedback indicated that they were aware of the limitations of inductive argumentation. However, PSMTs had a superficial understanding of the “proof by contradiction” mode of argumentation, and their attendance to particular errors seemed to be mediated by the mode of argument representation (e.g., symbolic, verbal). We discuss implications of these findings for mathematics teacher education.     

First‐year secondary school mathematics students’ conceptions of mathematical proofs and proving

The aim of this study is to investigate students’ conceptions about proof in mathematics and mathematics teaching. A five‐point Likert‐type questionnaire was administered in order to gather data. The sample of the study included 33 first‐year secondary school mathematics students (at the same time student teachers). The data collected were analysed and interpreted using the methods of qualitative and quantitative analysis. The results have revealed that the students think that mathematical proof has an important place in mathematics and mathematics education. The students’ studying methods for exams based on imitative reasoning which can be described as a type of reasoning built on copying proof, for example, by looking at a textbook or course notes proof or through remembering a proof algorithm. Moreover, they addressed to the differences between mathematics taught in high school and university as the main cause of their difficulties in proof and proving.     

Does generating examples aid proof production?

Many mathematics education researchers have suggested that asking learners to generate examples of mathematical concepts is an effective way of learning about novel concepts. To date, however, this suggestion has limited empirical support. We asked undergraduate students to study a novel concept by either tackling example generation tasks or reading worked solutions to these tasks. Contrary to suggestions in the literature, we found no advantage for the example generation group on subsequent proof production tasks. From a second study, we found that undergraduate students overwhelmingly adopt a trial and error approach to example generation and suggest that different example generation strategies may result in different learning gains. We conclude by arguing that the teaching strategy of example generation is not yet understood well enough to be a viable pedagogical recommendation.     

Institutional and personal meanings of mathematical proof

Although studies on students’ difficulties in producing mathematical proofs have been carried out in different countries, few research workers have focussed their attention on the identification of mathematical proof schemes in university students. This information is potentially useful for secondary school teachers and university lecturers. In this article, we study mathematical proof schemes of students starting their studies at the University of Córdoba (Spain) and we relate these schemes to the meanings of mathematical proof in different institutional contexts: daily life, experimental sciences, professional mathematics, logic and foundations of mathematics. The main conclusion of our research is the difficulty of the deductive mathematical proof for these students. Moreover, we suggest that the different institutional meanings of proof might help to explain this difficulty. This revised version was published online in July 2006 with corrections to the Cover Date.     

Exemplifying definitions: a case of a square

In this study we utilize the notion of learner-generated examples, suggesting that examples generated by students mirror their understanding of particular mathematical concepts. In particular, we explore examples generated by a group of prospective secondary school teachers for a definition of a square. Our framework for analysis includes the categories of accessibility and correctness, richness, and generality. Results shed light on participants’ understanding of what a mathematical definition should entail and, moreover, contrast their pedagogical preferences with mathematical considerations.     

Actions speak louder with words: The roles of action and pedagogical language for grounding mathematical proof

Theories of grounded and embodied cognition posit that situated actions are central constituents in cognitive processes. We investigate whether grounding actions influence reasoning, and how pedagogical language influences the action–cognition relationship. Undergraduate students (N = 120) generated proofs for two mathematical tasks after performing either grounding or non-grounding actions. Grounding actions facilitated key mathematical insights for both tasks, but did not lead to superior proofs. Pedagogical language in the form of prompts (prospective statements) and hints (retrospective statements) accompanying grounding actions enhanced proof performance on one task but not the other. Results from transfer tasks suggested that participants learned to apply their mathematical insights to new contexts. The findings suggest that relations between action and cognition are reciprocal: actions facilitate insight, while pedagogical language strengthens the influence of task-relevant actions for proof production. Pedagogically supported grounding actions offer alternative ways of fostering mathematical reasoning.     

The Notion of Proof in the Context of Elementary School Mathematics

Despite increased appreciation of the role of proof in students’ mathematical experiences across all grades, little research has focused on the issue of understanding and characterizing the notion of proof at the elementary school level. This paper takes a step toward addressing this limitation, by examining the characteristics of four major features of any given argument – foundation, formulation, representation, and social dimension – so that the argument could count as proof at the elementary school level. My examination is situated in an episode from a third-grade class, which presents a student’s argument that could potentially count as proof. In order to examine the extent to which this argument could count as proof (given its four major elements), I develop and use a theoretical framework that is comprised of two principles for conceptualizing the notion of proof in school mathematics: (1) The intellectual-honesty principle, which states that the notion of proof in school mathematics should be conceptualized so that it is, at once, honest to mathematics as a discipline and honoring of students as mathematical learners; and (2) The continuum principle, which states that there should be continuity in how the notion of proof is conceptualized in different grade levels so that students’ experiences with proof in school have coherence. The two principles offer the basis for certain judgments about whether the particular argument in the episode could count as proof. Also, they support more broadly ideas for a possible conceptualization of the notion of proof in the elementary grades.     

Re-negotiating the discourse of lower track high school students

The purpose of this study was to examine how lower track science students would understand shifts in standard classroom discourse patterns. The researcher videotaped his daily efforts to renegotiate the lower track classroom environment to become more representative of a scientific community. This paper is an analysis of the implicit obstacles inherent in shifting class discussions to classroom arguments examining tentative hypotheses. Students have inserted struggles for social status into classroom arguments about scientific ideas making it difficult to separate in a group discussion when the evidence convinced a student or whether the social politics of the class had persuaded her. As a result of changing classroom rules for participation, engagement, and collaborative inquiry, students' abilities to argue scientifically were changed. Despite these shifts students continued to insert their own interpretations of argumentation, social norms, and strategies for active re-negotiation of the teacher's agenda for the construction of scientific classroom discourse.     

Differences in student reasoning about belief-relevant arguments: a mixed methods study

This mixed methods study investigated high school students’ evaluations of scientific arguments. Myside bias occurs when individuals evaluate belief-consistent information more favorably than belief-inconsistent information. In the quantitative phase, participants (n?=?72 males) rated belief-consistent arguments more favorably than belief-inconsistent arguments; however, they also rated strong arguments more favorably than weak arguments, which indicated they did not evaluate the arguments exclusively on whether they were belief-consistent. In the follow-up qualitative phase, we conducted interviews with purposefully-sampled students who showed either higher or lower levels of myside bias. Results indicated that students in both groups applied normative evaluation criteria to the arguments. However, students who showed little or no myside bias applied the same evaluation criteria to arguments independent of whether they were belief-consistent, whereas students who showed high levels of myside bias applied different evaluation criteria to belief-inconsistent arguments. These findings suggest that procedural and conceptual metacognition may play a role in the extent to which individuals reason independent of their beliefs.     

Meaningful Understanding and Systems Thinking in Organic Chemistry: Validating Measurement and Exploring Relationships

The purpose of this study was dual: First, to develop and validate assessment schemes for assessing 11th grade students’ meaningful understanding of organic chemistry concepts, as well as their systems thinking skills in the domain. Second, to explore the relationship between the two constructs of interest based on students’ performance on the applied assessment framework. For this purpose, (a) various types of objective assessment questions were developed and evaluated for assessing meaningful understanding, (b) a specific type of systemic assessment questions (SAQs) was developed and evaluated for assessing systems thinking skills, and (c) the association between students’ responses on the applied assessment schemes was explored. The results indicated that properly designed objective questions can effectively capture aspects of students’ meaningful understanding. It was also found that the SAQs can elicit systems thinking skills in the context of a formalistic systems thinking theoretical approach. Moreover, a significant relationship was observed between students’ responses on the two assessment strategies. This research provides evidence that students’ systems thinking level within a science domain is significantly related to their meaningful understanding of relative science concepts.     

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.     

On fractions and non-standard representations: Pre-service teachers' concepts

The present study examined the reasoning strategies and arguments given by pre-service school teachers as they solved two problems regarding fractions in different symbolic representations. In the first problem, the pre-service school teachers were asked to compare between two different fractions having the same numerical representation, and in the second problem, they were asked to compare between different notational representations of the same fraction. Numeration systems in bases other than ten were used to generate various representations of fractions. All students were asked to provide justifications to their responses. Strategies and arguments relative to pre-service teachers' concepts of fractions and place value were identified and analyzed based on results of 38 individual clinical interviews, and written responses of 124 students. It was found that the majority of students believe that fractions change their numerical value under different symbolic representations.     

初中生对几何证明的认识

直观实验和几何证明是几何教学中常用的两种重要方法.学生对几何证明的本质还缺乏认识.学生认为直观实验不能算是几何证明,一方面因为直观实验存在误差,另一方面直观实验只能从不完全归纳中得出结论.     

Fostering empirical examination after proof construction in secondary school geometry

In contrast to existing research that has typically addressed the process from example generation to proof construction, this study aims at enhancing empirical examination after proof construction leading to revision of statements and proofs in secondary school geometry. The term “empirical examination” refers to the use of examples or diagrams to investigate whether a statement is true or a proof is valid. Although empirical examination after proof construction is significant in school mathematics in terms of cultivating students’ critical thinking and achieving authentic mathematical practice, how this activity can be fostered remains unclear. This paper shows the strength of a particular kind of mathematical task, proof problems with diagrams, and teachers’ roles in implementing the tasks, by analysing two classroom-based interventions with students in the eighth and ninth grades. In the interventions, the tasks and the teachers’ actions successfully prompted the students to discover a case rejecting a proof and a case refuting a statement, modify the proof, properly restrict the domain of the statement by disclosing its hidden condition, and invent a more general statement that was true even for the refutation of the original statement.