Understanding what a proof is: a classroom-based approach

Being unaware of the assumptions underlying a deductive argument is widespread among learners and is a major stumbling block to their understanding of proof. Thus, the basic idea of the present paper is that at some points in the course of secondary education there should be classroom-based interventions addressing this difficulty and making the axiomatic organization of mathematics a theme. Students should be made aware that there are axioms in mathematics, what their role is and how mathematicians come to agree about which axioms should be accepted. An axiom which is not yet accepted is simply a hypothesis. A hypothesis is evaluated by deductively drawing consequences and by investigating whether these consequences agree with experience or should be accepted for other reasons. The teaching intervention discussed in this paper exemplifies this idea by way of the example of ancient attempts at modelling the path of the sun, the so-called “anomaly of the sun”. It is investigated to what extent the teaching intervention fostered students’ understanding of the conditionality of mathematical/astronomical statements, that is, of the fact that the truth of these statements is dependent on the initial hypotheses.     

Distinguishing Features of Mechanical and Human Problem-Solving

Over the years, research in mathematical problem-solving has examined expert/novice problem-solving performance on various types of problems and subjects. In particular, DeFranco examined two groups of Ph.D. mathematicians as they solved four mathematics problems and found that although all were content experts, only one group were problem-solving experts. Based on this study, this article posits the notion that one distinguishing feature between experts and novices is that experts tend to look for special features of a problem and use algorithms only as a “fail-safe” system while novices act like “machines” relying on algorithms to solve the problems. Why? The article explores the idea that novice problem solvers learned to solve problems the way they learned proof, that is, in a formal, abstract and mechanizable way. Beliefs about proof and the culture in which it is practiced help frame a mathematician's view of the discipline and ultimately impacts classroom practice. The authors believe that current classroom instruction tends to create a culture that fosters algorithmic proficiency and a “machine-like” approach to the learning of mathematics and problem-solving. Further, they argue that mathematicians need to be aware of the distinction between knowing a proof is true and explaining why it is true. When these distinctions are appreciated and practiced during classroom instruction, then and only then will students begin to acquire the mathematical knowledge to become better problem solvers.     

Contributions of Lo Yang to Value Distribution Theory

Professor Lo Yang is a world famous mathematician of our country. He made a lot of outstanding achievements in the value distribution theory of function theory, which are highly rated and widely quoted by domestic and foreign scholars. He also did a lot of work to develop Chinese mathematics. It can be said that Professor Yang is one of the mathematicians who made main influences on the mathematical development in modern China. This paper briefly introduces Professor Yang’s life, mainly discusses his academic achievement and influence, and briefly describes his contributions to the Chinese mathematics community.     

Pure mathematics applied in early twentieth-century America: The case of T.H. Gronwall,consulting mathematician

Thomas Hakon Gronwall (1877–1932) was a Swedish-American mathematician with a broad range of interests in mathematical analysis, physics, and engineering. Though he was primarly known for his results in pure mathematics, his career as a “consulting mathematician” in America from 1912 to his death in 1932 provides a backdrop against which one can discuss contemporary issues involved in the increasing application of mathematics to engineering, industrial, and scientific problems. This paper attempts a summary of his major mathematical contributions to industrial, governmental, and academic institutions while relating his often difficult life during these years.     

Olga Taussky,a torchbearer for mathematics and teacher of mathematicians

Olga Taussky-Todd's mathematical and personal life (1906-1995), her achievements and obstacles, her scientific reasoning and teaching all have served as inspiration to many mathematicians. We describe her role in the mathematics world of the previous century as a torchbearer for mathematics and mathematicians, bearing the “torch of scientific truth” that burns inside of mathematics and its applications. Besides her many deep math contributions – too many to elaborate – she excelled at distilling and presenting mathematical concepts and ideas in her work and gave us many visionary papers and math talks. By sharing her mathematical vision freely she has inspired many of us. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Impacts of inquiry pedagogy on undergraduate students conceptions of the function of proof

Mathematicians and mathematics educators agree that proof is an important tool in mathematics, yet too often undergraduate students see proof as a superficial part of the discipline. While proof is often used by mathematicians to justify that a theorem is true, many times proof is used for another purpose entirely such as to explain why a particular statement is true or to show mathematics students a particular proof technique. This paper reports on a study that used a form of inquiry-based learning (IBL) in an introduction to proof course and measured the beliefs of students in this course about the different functions of proof in mathematics as compared to students in a non-IBL course. It was found that undergraduate students in an introduction to proof course had a more robust understanding of the functions of proof than previous studies would suggest. Additionally, students in the course taught using inquiry pedagogy were more likely to appreciate the communication, intellectual challenge, and providing autonomy functions of proof. It is hypothesized that these results are a response to the pedagogy of the course and the types of student activity that were emphasized.     

The establishment of functional analysis

This article surveys the evolution of functional analysis, from its origins to its establishment as an independent discipline around 1933. Its origins were closely connected with the calculus of variations, the operational calculus, and the theory of integral equations. Its rigorous development was made possible largely through the development of Cantor's “Mengenlehre,” of set-theoretic topology, of precise definitions of function spaces, and of axiomatic mathematics and abstract structures. For a quarter of a century, various outstanding mathematicians and their students concentrated on special aspects of functional analysis, treating one or two of the above topics. This article emphasizes the dramatic developments of the decisive years 1928–1933, when functional analysis received its final unification.     

Theorems that admit exceptions, including a remark on Toulmin

It is a plausible assumption that proof-novices try to make sense of the meaning of mathematical proof out of the perspective of every day thinking. In every day thinking, however, the domain of objects to which a general statement refers is not completely and definitely determined. Thus the very notion of a “universally valid statement” is not as obvious as it might seem. The phenomenon of a statement with an indefinite domain of reference can also be found in the history of mathematics when authors spoke of “theorems that admit exceptions”. Without having understood and accepted the theoretical nature of the idea of a universally valid statement the logical distinctions between, for example an implication and its converse loose their meaning for the learner. This might explain some disappointing findings of empirical research. Following a proposal by Inglis, Mejia–Ramos and Simpson it is suggested that in modelling mathematical thinking in proof situations the full scheme of Toulmin should be used including qualifications and rebuttals rather than a reduced version as is frequently done.     

Probability and exams: The work of Antonio Bordoni

The Italian mathematician Antonio Bordoni is mainly known for his adherence to the Lagrangian approach to the foundations of calculus and for his role in creating an important school of mathematics. In this paper, I consider his less known work on the application of probability to design exams and analyze their outcomes. Within this framework, he obtained in 1837, as Mondésir and Poisson, the result that would lead Catalan to formulate his “new principle” of probability (Jongmans and Seneta, 1994). Moreover, in 1843, Bordoni also gave an early complete proof of the finite rule of succession.     

What do mathematicians mean by proof? A comparative-judgement study of students’ and mathematicians’ views

We present a study in which mathematicians and undergraduate students were asked to explain in writing what mathematicians mean by proof. The 175 responses were evaluated using comparative judgement: mathematicians compared pairs of responses and their judgements were used to construct a scaled rank order. We provide evidence establishing the reliability, divergent validity and content validity of this approach to investigating individuals’ written conceptions of mathematical proof. In doing so, we compare the quality of student and mathematician responses and identify which features the judges collectively valued. Substantively, our findings reveal that despite the variety of views in the literature, mathematicians broadly agree on what people should say when asked what mathematicians mean by proof. Methodologically, we provide evidence that comparative judgement could have an important role to play in investigating conceptions of mathematical ideas, and conjecture that similar methods could be productive in evaluating individuals’ more general (mathematical) beliefs.     

Progress and stagnation of gender equity: contradictory trends within mathematics research and education in Sweden

During the last decade women in Sweden have reduced men’s lead in participation in mathematics education and in professional careers as mathematicians. However, the development is uneven and slow overall. In some areas and at the highest levels women have increased their participation only marginally. Why, one may ask, is progress so slow after almost 20 years of active work from the Women and Mathematics movement in Sweden and within a society in which gender equity is highly valued at the societal and political levels? The development is described in quantitative measures going back 20 years. Several concrete and successful initiatives from the last decade intended to “de-gender” mathematics and to involve women and men alike in mathematics are described. In contrast a gender-blind position or a view of women as problems in mathematics seems to reign within some influential bodies.     

Cut-Elimination Theorem for the Logic of Constant Domains

The logic CD is an intermediate logic (stronger than intuitionistic logic and weaker than classical logic) which exactly corresponds to the Kripke models with constant domains. It is known that the logic CD has a Gentzen-type formulation called LD (which is same as LK except that (→) and (?–) rules are replaced by the corresponding intuitionistic rules) and that the cut-elimination theorem does not hold for LD . In this paper we present a modification of LD and prove the cut-elimination theorem for it. Moreover we prove a “weak” version of cut-elimination theorem for LD , saying that all “cuts” except some special forms can be eliminated from a proof in LD . From these cut-elimination theorems we obtain some corollaries on syntactical properties of CD : fragments collapsing into intuitionistic logic. Harrop disjunction and existence properties, and a fact on the number of logical symbols in the axiom of CD . Mathematics Subject Classification : 03B55. 03F05.     

Proofs as bearers of mathematical knowledge

Yehuda Rav’s inspiring paper “Why do we prove theorems?” published in Philosophia Mathematica (1999, 7, pp. 5–41) has interesting implications for mathematics education. We examine Rav’s central ideas on proof—that proofs convey important elements of mathematics such as strategies and methods, that it is “proofs rather than theorems that are the bearers of mathematical knowledge”and thus that proofs should be the primary focus of mathematical interest—and then discuss their significance for mathematics education in general and for the teaching of proof in particular.     

The relationship between mathematicians’ pedagogical goals,orientations, and common teaching practices in advanced mathematics

Despite mathematics educators’ research into more effective modes of teaching, lecture is still the dominant mode of instruction in undergraduate mathematics courses. Surveys suggest this is because most mathematicians believe this is the best way to teach. This paper answers a call by mathematics education researchers to explore mathematicians’ needs and goals concerning teaching. We interviewed eight mathematicians about findings in the mathematics education research literature concerning common pedagogical practices of instructors of advanced mathematics classes: “chalk talk,” the presentation of formal and informal content, and teacher questioning. We then analyzed the responses for resources, orientations, and goals that might influence the participants to engage in these practices. We describe how participants believed common lecturing practices allowed them to achieve their goals and aligned with their orientations. We discuss these findings in depth and consider what implications they may have for researchers that aim to change mathematicians’ teaching practices.     

灵活的应用数学技术

E．E．Daivd指出“当今被称颂的高技术实质上是一种数学技术”，H．Neunzert称“数学是关键技术的关键”．这是人类对数学的新认识一数学既是科学又是技术．数学技术主要是指应用数学技术．它的高难度突出地体现在以解决实际问题为目标的研究上．这就必须灵活地运用数学思想和方法，抓住事物内在最本质的数学结构，提炼其特殊的数学模型，给出精巧的好算法并解决之．本文简述了华罗庚应用数学技术的特色、近期发展及其某些思想与新概念等．     

Prototype Proofs in Type Theory

The proofs of universally quantified statements, in mathematics, are given as “schemata” or as “prototypes” which may be applied to each specific instance of the quantified variable. Type Theory allows to turn into a rigorous notion this informal intuition described by many, including Herbrand. In this constructive approach where propositions are types, proofs are viewed as terms of λ‐calculus and act as “proof‐schemata”, as for universally quantified types. We examine here the critical case of Impredicative Type Theory, i. e. Girard's system F, where type‐quantification ranges over all types. Coherence and decidability properties are proved for prototype proofs in this impredicative context.     

Is the definition of mathematics as used in the PISA assessment Framework applicable to the HarmoS Project?

The project known as the “Harmonisation of the Obligatory School”, or in its shortened form as “HarmoS”, has a high priority for Switzerland's educational policy in the coming years. Its purpose is to determine levels of competency, valid throughout Switzerland, for specific areas of study and including the subject of mathematics. The general theoretical basis of the overall HarmoS Project is constituted by the expertise written under the direction of Eckhard Klieme and entitled “Zur Entwicklung nationaler Bildungsstandards” (Klieme 2003) [i.e. “On the Development of National Education Standards”]. The proposal announcing the HarmoS partial project devoted to Mathematics includes references to the results and subsequent analysis of PISA 2003. It thus seems appropriate for us to begin our work on HarmoS with a critical consideration of the definition of mathematics and mathematical literacy as they are used in the PISA Study. In a first part, we want to describe the core ideas of HarmoS. In a second part, we will address the meaning of general educational goals for the development of competency models and education standards to the extent that it is necessary to properly locate our problem. In a third part we will analyse the concept of mathematics which is at the basis of the PISA Study (OECD 2004) and more precisely defined in the publication “Assessment Framework” (OECD 2003) In the fourth and last part, we will try to provide a differentiated answer to the question posed in the title of this paper.