A Methodology for Teaching Logic-Based Skills to Mathematics Students (original) (raw)
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Occasionally one hears in mathematics departments that logic courses are not very helpful to mathematics students learning to make and understand proofs. This seems to happen when a logic course emphasizes a symbolic, procedural approach with little attention to applications, and the proofs are in the sparse style of natural language deductive reasoning that undergraduate mathematics students normally see. In this paper we will suggest an explanation for this situation. We will first analyze the customary style in which mathematics proofs are written and then discuss ways in which logic is used in these proofs. Mathematics proofs are often read in a complex, interactive way which we call validation. During validation mathematicians establish, for themselves with great reliability, the correctness of a proof -- while also enhancing their conceptual understanding. Although the style in which proofs are written has evolved within the mathematics community mostly without plan or design, it is solidly established and has a number of aspects which seem useful for minimizing validator reasoning errors: (1) Moment-to-moment cognitive load during validation is kept to a minimum (at the expense of lengthening the validation process). For example, neither the overall proof structure nor the definitions, theorems, or logic used is often explained within the body of a proof. (2) Whenever possible, symbols represent specific nonchanging objects, and statements involving universal quantifiers are avoided. (3) Reasons for assertions are only occasionally included and, if so, are only briefly indicated. A validator may have to produce what amounts to an independent subproof to establish a single assertion. This is not normally seen as a defect in the proof, because mathematical proofs are not argued from foundations, but rather are written for (fairly expert) idealized validators, while the domain knowledge of actual validators varies. These aspects lengthen the time validation takes, but shift control to the validator, while removing potential distracting passages from a validator's view, thereby aiding in an attempt to gain an overview. Sometimes, a validator repeatedly reads (or mentally rehearses) the proof and, in the latter readings, skims over (chunks) previously established sections. This allows a validator to minimize the possibility of failing to see errors, by separating out subsidiary arguments. Typical validations expected of undergraduate mathematics students seem to need very few of the properties and procedures of symbolic logic. Rather, what is needed seems to be more pattern-matching and substitution. For example, (1) When a logical result, such as the equivalence of an implication and its contrapositive, is needed, this must be recognized as a pattern within the context of the natural language setting of mathematical proofs. Since much learning seems to be context bound, this may present quite an obstacle for students who have learned logic very abstractly. (2) Proofs are normally written in a rather concrete (let x be a number) style, in contrast to the more general quantified (every differentiable function is . . .) style of definitions and theorems. This leads to a need for substitution, especially multi-level substitution, which appears to be much less transparent for students than is generally supposed. (3) Finally, unpacking the logical structure of an informally, or memorably, stated theorem and relating this to the overall structure of its proof is remarkably difficult for students. [See our paper, "Unpacking the logic of mathematical statements," Educational Studies in Mathematics 29 (1995), 123-151.] In addition to the above, the psychological way logic is used is important. Some uses of logic are conscious, at the level of internalized speech or vision, and some are below that level. There are situations in which one simply recognizes a pattern of statements and "automatically" knows the subsequent statement is justified (or not) without any awareness of mental activity. One might think of this situation as a schema being activated, and it appears to lack the kind of error control associate with conscious applications of logic. This is rather like the "automated" way a beginning high school student might know that 5/10 is 1/2, rather than consciously applying (ab)/(ac) = b/c using internalized speech. In validating proofs, much of the logic used seems to be on this schema level, which raises the question of where such schema come from and whether a logic course might influence schema construction. The paper will expand on these ideas and illustrate them by analyzing a validation of a beginning calculus proof. In it, we draw on our experiences as mathematicians and on a conceptual framework from mathematics education and cognitive psychology. We hope this analysis will lead to more research on students' (and others) use of logic in proofs.
Guiding principles for teaching mathematics via reasoning and proving
Le Centre pour la Communication Scientifique Directe - HAL - Inria, 2022
Against the backdrop of policy documents and educational researchers' vision of proof as an essential component of teaching mathematics across content areas and grade levels, teaching of reasoning and proof in mathematics classrooms remains an elusive goal. Teachers and the type of teaching they enact in classrooms are crucial for achieving this goal. This theoretical paper builds on the concept of proof-based teaching and suggest a set of guiding principles for what we call teaching mathematics via reasoning and proving. These principles were developed as a part of a multi-year design based project, and implemented in an undergraduate course Mathematical Reasoning and Proving for Secondary Teachers. We illustrate these principles using examples from proof-oriented lessons plan developed by prospective secondary teachers.
Elementary logic as a tool in proving mathematical statements
2008
Declaration viii Acknowledgements ix List of Figures x List of Tables xi Contents Chapter 1 Introduction 1.1 The influence of Mathematics on Economic Development 1 1.2 Background to the study 1 1.3 The Aim of the study 12 1.4 The Main Research Question 13 Chapter 2 Literature Review 2.1 Introduction 2.2 Effect of logic on proving at school level 14 2.3 Effect of logic on proving at tertiary level 2.4 Literature review summary Chapter 3 Theoretical Framework 3.1 Chapter 4 The influence of Emotion, Confidence, Experience and Practice on the learning process 4.1 Introduction 35 4.2 The influence of emotion 4.3 The influence of Confidence 4.4 The influence of Experience 4.5 The influence of Practice 4.6 Negotiating the learning process 39
2019
The study aimed at describing university students' ability in constructing mathematical proofs using logical arguments and constructing an effective teaching-learning map. The study adopted a correlational research design. An open-ended mathematics assessment involving questions on mathematical logic and construction of mathematical proofs was administered to fifteen University students of mathematics (control group) at Mountains of the Moon university. The students' responses were analyzed and their ability to construct valid mathematical proofs was measured and correctness of their solutions was discussed. A concept map for teaching-learning mathematical logic and methods of proof was developed and shared with a different group of fifteen students of mathematics (assumed to have same level of mathematical knowledge). The group was assessed using the similar assessment tool. The mean level of knowledge of mathematical logic and method of proof was from to be 2.2 (between mo...
1999
Mathematics departments rarely require students to study very much logic before working with proofs. Normally, the most they will offer is contained in a small portion of a "bridge" course designed to help students move from more procedurally-based lower-division courses (e.g., calculus and differential equations) to more proof-based upper division courses (e.g., abstract algebra and real analysis). What accounts for this seeming neglect of an essential ingredient of deductive reasoning? 1 We will suggest a partial answer by comparing the contents of traditional logic courses with the kinds of reasoning used in proof validation, our name for the process by which proofs are read and checked.
Mathematical Reasoning and Proving for Secondary Teachers
2021
The Mathematical Reasoning and Proving for Secondary Teachers course aims to enhance preservice secondary teachers\u27 (PSTs\u27) knowledge and disposition for integrating reasoning and proof into classroom practices. Through content, methods, and clinical practice, PSTs are able to learn about the importance of the logical aspect of proof and apply it to pedagogical implications. The four aspects of proof utilized for the course are (1) quantification and the role of examples in proving, (2) conditional statements, (3) direct proof and argument evaluation and (4) indirect reasoning. The course spends about three weeks on each of these modules. Within this time PSTs are engaged in opportunities to integrate proof and reasoning into the traditional US curriculum, provided class time to lesson plan, and are shown modeled types of proof tasks for support. Lessons are planned and implemented in Middle or Secondary classrooms. For each lesson PST\u27s record, reflect, receive feedback, a...
Technical Report: The Role of Logic in the Validation of Mathematical Proofs
Mathematics departments rarely require students to study very much logic before working with proofs. Normally, the most they will offer is contained in a small portion of a "bridge" course designed to help students move from more procedurally-based lower-division courses (e.g., calculus and differential equations) to more proof-based upper division courses (e.g., abstract algebra and real analysis). What accounts for this seeming neglect of an essential ingredient of deductive reasoning? 1 We will suggest a partial answer by comparing the contents of traditional logic courses with the kinds of reasoning used in proof validation, our name for the process by which proofs are read and checked. First, we will discuss the style in which mathematical proofs are traditionally written and its apparent utility for reducing validation errors. We will then examine the relationship between the need for logic in validating proofs and the contents of traditional logic courses. Some topics emphasized in logic courses do not seem to be called upon very often during proof validation, whereas other kinds of reasoning, not often emphasized in such courses, are frequently used. In addition, the rather automatic way in which logic, such as modus ponens, needs to be used during proof validation does not appear to be improved by traditional teaching, which often emphasizes truth tables, valid arguments, and decontextualized exercises. Finally, we will illustrate these ideas with a proof validation, in which we explicitly point out the uses of logic. We will not discuss proof construction, a much more complex process than validation. However, constructing a proof includes validating it, and hence, during the validation phase, calls on the same kinds of reasoning. Throughout this paper we will refer to a number of ideas from both cognitive psychology and mathematics education research. We will Þnd it useful to discuss short-term, long-term, and working memory, cognitive load, internalized speech and vision, and schemas, as well as reßection, unpacking the meaning of statements, and the distinction between procedural and conceptual knowledge.