Explication as a lens for the formalization of mathematical theory through guided reinvention

Realistic Mathematics Education supports students’ formalization of their mathematical activity through guided reinvention. To operationalize “formalization” in a proof-oriented instructional context, I adapt Sjogren's (2010) claim that formal proof explicates (Carnap, 1950) informal proof. Explication means replacing unscientific or informal concepts with scientific ones. I use Carnap's criteria for successful explication – similarity, exactness, and fruitfulness – to demonstrate how the elements of mathematical theory – definitions, axioms, theorems, proofs – can each explicate their less formal correlates. This lens supports an express goal of the instructional project, which is to help students coordinate semantic (informal) and syntactic (formal) mathematical activity. I demonstrate the analytical value of the explication lens by applying it to examples of students’ mathematical activity drawn from a design experiment in undergraduate, neutral axiomatic geometry. I analyze the chains of meanings (Thompson, 2013) that emerged when formal elements were presented readymade alongside those emerging from guided reinvention.     

Mathematics education and democracy

In this paper, we investigate the relationship between mathematics education and the notions of education for all/democracy. In order to proceed with our analysis, we present Marx’s concept of commodity and Jean Baudrillard’s concept of sign value as a theoretical reference in the discussion of how knowledge has become a universal need in today’s society and ideology. After, we engage in showing mathematics education’s historical and epistemological grip to this ideology. We claim that mathematics education appears in the time period that English becomes an international language and the notion of international seems to be a key constructor in the constitution of that ideology. Here, we draw from Derrida’s famous saying that “there is nothing beyond the text”. We conclude that a critique to modern society and education has been developed from an idealistic concept of democracy.     

A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms.     

Mathematics learning and tools from theoretical, historical and practical points of view: the productive notion of mathematics laboratories

In our research work, we have looked at the way in which artefacts become, for teachers as well as for students, instruments of their mathematical activity. The issues related to the use of tools and technologies in mathematical education are now widely considered. A look to history highlights the different ways in which the same questions have been studied at different times and in different places. This suggests that the contribution of artefacts to mathematics learning should be considered in terms of various contexts. Our “visits” to these contexts will be guided by the coordination of two main theoretical frameworks, the instrumental approach and the semiotic mediation approach from the perspective of mathematics laboratory. This journey through history and schooling represents a good occasion to address some questions: Are there “good” contexts in which to develop mathematical instruments? Are there “good” teaching practices which assist students’ instrumental geneses and construct mathematical meanings? How is it possible to promote such teaching practices? Some study cases are discussed.     

Williams college student course survey form: Instructor modified version

Opening a copy of The Mathematical Intelligenceryou may ask yourself uneasily, “What is this anyway—a mathematical journal, or what?” Or you may ask, “Where am I?” Or even “Who am I?” This sense of disorientation is at its most acute when you open to Colin Adams’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be harmless.     

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.     

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.     

Young students’ subjective interpretations of mathematical diagrams: elements of the theoretical construct “frame-based interpreting competence”

During the last few decades several studies have showed that mathematical visual aids are not at all self-explanatory. Nevertheless, students do make sense of those representations spontaneously and—as a matter of course—cannot avoid their own sense-making. Further, the function of visual aids as “re-presentation” of a given structure is complemented through an epistemological function to explore mathematical structures and generate new meaning. But in which way do socially learned interpreting schemes (frames) influence children’s subjective interpretations of mathematical diagrams? The CORA project investigates which frames can be reconstructed in young pupils’ interpretations of visual diagrams. This paper presents central ideas, theoretical background and—by means of short sequences from pre- and post-interviews—first aspects of “frame-based interpreting competence”. We describe children’s subjective frames in a range between “object-oriented” (focus on the diagram’s visible elements) and “system-oriented” (focus on relation between those elements).     

Possible Student Justification of Proofs

The National Council of Teachers of Mathematics calls for an increased emphasis on proof and reasoning in school mathematics curricula. Given such an emphasis, mathematics teachers must be prepared to structure curricular experiences so that students develop an appreciation for both the value of proof and for those strategies that will assist them in developing proving skills. Such an outcome is more likely when the teacher feels secure in his/her own understanding of the concept of “mathematical proof” and understands the ways in which students progress as they take on increasingly more complex mathematical justifications. In this article, a model of mathematical proof, based on Balacheff's Taxonomy of Mathematical Proof, outlining the levels through which students might progress as they develop proving skills is discussed. Specifically, examples of the various ways in which students operating at different levels of skill sophistication could approach three different mathematical proof tasks are presented. By considering proofs under this model, teachers are apt to gain a better understanding of each student's entry skill level and so effectively guide him/her toward successively more sophisticated skill development.     

Brouwer and Euclid

We explore the relationship between Brouwer’s intuitionistic mathematics and Euclidean geometry. Brouwer wrote a paper in 1949 called The contradictority of elementary geometry. In that paper, he showed that a certain classical consequence of the parallel postulate implies Markov’s principle, which he found intuitionistically unacceptable. But Euclid’s geometry, having served as a beacon of clear and correct reasoning for two millennia, is not so easily discarded.Brouwer started from a “theorem” that is not in Euclid, and requires Markov’s principle for its proof. That means that Brouwer’s paper did not address the question whether Euclid’s Elements really requires Markov’s principle. In this paper we show that there is a coherent theory of “non-Markovian Euclidean geometry”. We show in some detail that our theory is an adequate formal rendering of (at least) Euclid’s Book I, and suffices to define geometric arithmetic, thus refining the author’s previous investigations (which include Markov’s principle as an axiom).Philosophically, Brouwer’s proof that his version of the parallel postulate implies Markov’s principle could be read just as well as geometric evidence for the truth of Markov’s principle, if one thinks the geometrical “intersection theorem” with which Brouwer started is geometrically evident.     

Don’t Touch the Button

Opening a copy of The Mathematical Intelligenceryou may ask yourself uneasily, “What is this anyway—a mathematical journal, or what?” Or you may ask, “Where am I?” Or even “Who am I?” This sense of disorientation is at its most acute when you open to Colin Adams’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be harmless.     

The mathematical ethicist

Opening a copy ofThe Mathematical Intelligencer you may ask yourself uneasily, “What is this anyway—a mathematical journal, or what?” Or you may ask, “Where am I?” Or even “Who am I?” This sense of disorientation is at its most acute when you open to Colin Adams’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be harmless.     

Partition Theorems from Creatures and Idempotent Ultrafilters

We show a general scheme of Ramsey-type results for partitions of countable sets of finite functions, where “one piece is big” is interpreted in the language originating in creature forcing. The heart of our proofs follows Glazer’s proof of the Hindman Theorem, so we prove the existence of idempotent ultrafilters with respect to suitable operation. Then we deduce partition theorems related to creature forcings.     

A subprime lending market primer

Opening a copy of TheMathematical Intelligencer you may ask yourself uneasily, “What is this anyway—a mathematical journal, or what?” Or you may ask, “Where am I?” Or even “Who am I?” This sense of disorientation is at its most acute when you open to Colin Adams’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be harmless.     

Exceptional functions wandering on the sphere and normal families

We extend Carathéodory’s generalization of Montel’s fundamental normality test to “wandering” exceptional functions (i.e., depending on the respective function in the family under consideration), and we give a corresponding result on shared functions. Furthermore, we prove that if we have a family of pairs (a, b) of functions meromorphic in a domain such that a and b uniformly “stay away from each other,” then the families of the functions a resp. b are normal. The proofs are based on a “simultaneous rescaling” version of Zalcman’s lemma.     

Boolos‐style proofs of limitative theorems

Boolos's proof of incompleteness is extended straightforwardly to yield simple “diagonalization‐free” proofs of some classical limitative theorems of logic. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Problem solving in the United States, 1970–2008: research and theory, practice and politics

Problem solving was a major focus of mathematics education research in the US from the mid-1970s though the late 1980s. By the mid-1990s research under the banner of “problem solving” was seen less frequently as the field’s attention turned to other areas. However, research in those areas did incorporate some ideas from the problem solving research, and that work continues to evolve in important ways. In curricular terms, the problem solving research of the 1970s and 1980s (see, e.g., Lester in J Res Math Educ, 25(6), 660–675, 1994, and Schoenfeld in Handbook for research on mathematics teaching and learning, MacMillan, New York, pp 334–370, 1992, for reviews) gave birth to the “reform” or “standards-based” curriculum movement. New curricula embodying ideas from the research were created in the 1990s and began to enter the marketplace. These curricula were controversial. Despite evidence that they tend to produce positive results, they may well fall victim to the “math wars” as the “back to basics” movement in the US is revitalized.     

Theories of Mathematics Education,Edited by Bharath Sriraman and Lyn English: Common Ground for Scholars and Scholars in the Making

Theories of mathematics education: Seeking new frontiers is the first book in a new Springer series, Advances in Mathematics Education. To some degree the book is based on a collection of previously published articles from special issues of ZDM—The International Journal on Mathematics Education (previously known as Zentralblatt für Didaktik der Mathematik). These articles, dealing with the role and use of theories in and about mathematics education, originally stem from various conferences and meetings such as PME and CERME. For this reason some of the articles in the book are already well known, a few may even be considered to be “modern classics” within theories of mathematics education such as Frank Lester's “On the theoretical, conceptual, and philosophical foundations for research in mathematics education.” What is new—and non-traditional—in the book, however, is its form of presentation and format, the main articles being accompanied by preludes and commentaries by established researchers and newcomers.     

Integrating algebra and proof in high school mathematics: An exploratory study

Frequently, in the US students’ work with proofs is largely concentrated to the domain of high school geometry, thus providing students with a distorted image of what proof entails, which is at odds with the central role that proof plays in mathematics. Despite the centrality of proof in mathematics, there is a lack of studies addressing how to integrate proof into other mathematical domains. In this paper, we discuss a teaching experiment designed to integrate algebra and proof in the high school curriculum. Algebraic proof was envisioned as the vehicle that would provide high school students the opportunity to learn not only about proof in a context other than geometry, but also about aspects of algebra. Results from the experiment indicate that students meaningfully learned about aspects of both algebra and proof in that they produced algebraic proofs involving multiple variables, based on conjectures they themselves generated.