Cognitive trajectory of proof by contradiction for transition-to-proof students

History and research on proof by contradiction suggests proof by contradiction is difficult for students in a number of ways. Students’ comprehension of already-written proofs by contradiction is one such aspect that has received relatively little attention. Applying the cognitive lens of Action-Process-Object-Schema (APOS) Theory to proof by contradiction, we constructed and tested a cognitive model that describes how a student might construct the concept ‘proof by contradiction’ in an introduction to proof course. Data for this study was collected from students in a series of five teaching interventions focused on proof by contradiction. This paper will report on two participants as case studies to illustrate that our cognitive trajectory for proof by contradiction is a useful model for describing how students may come to understand the proof method.     

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.     

Argumentation and algebraic proof

This paper concerns a study analysing cognitive continuities and distances between argumentation supporting a conjecture and its algebraic proof, when solving open problems involving properties of numbers. The aim of this paper is to show that, unlike the geometrical case, the structural distance between argumentation and proof (from an abductive argumentation to a deductive proof) is not one of the possible difficulties met by students in solving such problems. On the contrary, since algebraic proof is characterized by a strong deductive structure, abductive steps in the argumentation activity can be useful in linking the meaning of the letters used in the algebraic proof with numbers used in the argumentation. The analysis of continuities and distances between argumentation and proof is based on the use of Toulmin’s model combined with ck¢ model.     

A case study in program transformation

A simple problem relating to binary trees is considered, and a recursive program is proved to solve this problem. The program may be transformed into a more efficient iterative one, usinggoto-statements and explicit stack handling. An independent correctness proof for this transformed version is given and compared with the original proof. Finally it is shown how a non-recursive conception of the problem may yield a structured iterative program, based onwhile-statements. A proof of this program is presented, reflecting its non-recursive conception, and it turns out that this proof has little similarity with the other two.     

Systolic Inequalities and Minimal Hypersurfaces

We give a short proof of the systolic inequality for the n-dimensional torus. The proof uses minimal hypersurfaces. It is based on the Schoen–Yau proof that an n-dimensional torus admits no metric of positive scalar curvature.     

A bijective proof of a limiting case of Watson’s 8 φ 7 transformation formula

Andrews gave a combinatorial proof of the Rogers–Fine identity. In this paper, we present a combinatorial proof of a special case of Watson’s ₈ φ ₇ transformation formula, which is a generalization of Andrews’ proof. A.J. Yee is an Alfred P. Sloan Research Fellow.     

An intuitionistic formula hierarchy based on high‐school identities

We revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal) propositional sequent calculi are formula (i.e., sequent) isomorphisms corresponding to the high‐school identities, we show that one can obtain a more compact variant of a proof system, consisting of non‐invertible proof rules only, and where the invertible proof rules have been replaced by a formula normalization procedure. Moreover, for certain proof systems such as the G4ip sequent calculus of Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the non‐invertible proof rules as strict inequalities between exponential polynomials; a careful combinatorial treatment is given in order to establish this fact. Finally, we extend the exponential polynomial analogy to the first‐order quantifiers, showing that it gives rise to an intuitionistic hierarchy of formulas, resembling the classical arithmetical hierarchy, and the first one that classifies formulas while preserving isomorphism.     

A changing trajectory of proof learning in the geometry inquiry classroom

The purpose of this study is to explore how students changes through learning to construct mathematical proofs in an inquiry-based middle school geometry class in Korea. Although proof has long been considered as one of the most important aspects of mathematics education, it is well-known that it is one of the most difficult areas of school mathematics for students. The geometry inquiry classroom (GIC) is an experimental class designed to teach geometry, especially focusing on teaching proof, based on students’ own inquiry. Based on a 2-year participant observation in the GIC, this study was conducted to address the following research question: how has students’ practice of mathematical proof been changed through their participation in the GIC? The in-depth analysis of the classroom discourse identified three stages through which the students’ practice of mathematical proof was transformed in the GIC: ‘emergent understanding of proof’, ‘proof learning as a goal-oriented activity’, ‘experiencing proof as the practice of mathematics’. The study found that as learning evolved through these stages, so the mathematics teacher’s role shifted from being an instructor to a mediator of communication. Most importantly, this research showed that the GIC has created a learning environment where students develop their competence in constructing meaningful mathematical proof and grow to be ‘a human who proves’, ultimately ‘a person who playfully engages with mathematics’.     

Probabilistic verification of proofs in calculuses

We define a special kind of a probabilistically checkable proof system, namely, probabilistically checkable proof calculuses (PCP calculuses). A proof in such a calculus can be verified with sufficient confidence by examining only one random path in the proof tree, without reading the whole proof. The verification procedure just checks all applications of inference rules along the path; its running time is assumed to be polynomial in the theorem length. It is shown that the deductive power of PCP calculuses is characterized as follows: (i) the decision problem for theorems is in PSPACE for all PCP calculuses; and (ii) the mentioned problem is PSPACE-hard for some of the calculuses. Bibliography: 14 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 241, 1997, pp. 97–116 This research was supported in part by the Russian Foundation for Basic Research. Translated by E. Ya. Dintsin.     

A Teacher's Conception of Communication in Geometry Proofs

Mathematical proof has many purposes, one of which is communication of the reasoning behind a mathematical insight. Research on teachers' views of the role that proof plays as mathematical communication has been limited. This study describes how one teacher conceptualized proof communication during two units on proof (coordinate geometry proofs and Euclidean proofs). Based on classroom observations, the teacher's conceptualization of communication in written proofs is recorded in four categories: audience, clarity, organization, and structure. The results indicate differences within all four categories in the way the idea of communication is discussed by the teacher. Implications for future studies include attention to teachers' beliefs about learning mathematics in the process of understanding teachers' conceptions of proof as a means of mathematical communication.     

An elementary proof of the irrationality of ζ(3)

An elementary proof of the irrationality of ζ(3) is presented. The proof is based on a two times more dense sequence of Diophantine approximations to this number than the sequence in the original proof of Apery.     

A short proof of Kneser's conjecture

In the paper a short proof is given for Kneser's conjecture. The proof is based on Borsuk's theorem and on a theorem of Gale.     

An easy proof of the Rogers-Ramanujan identities

A proof of the Rogers-Ramanujan identities is presented which is brief, elementary, and well motivated; the “easy” proof of whose existence Hardy and Wright had despaired. A multisum generalization of the Rogers-Ramanujan identities is shown to be a simple consequence of this proof.     

Newton's solution to the equiangular spiral problem and a new solution using only the equiangular property

This paper is a study of Proposition IX of Book I of Newton's Principia, the problem of determining the centripetal force for an equiangular spiral. In Newton's main proof of this proposition there is an error concerning his reason for the figure SPRQT being “given in kind,” and a very interesting technique of varying things in the neighborhood of a limit. This main proof utilized Newton's formula for the limit of SP²QT²/QR given in Corollary I to Proposition VI of the Principia. Newton also gave an alternate proof which utilized his formula for SY²PV given in Corollary III to Proposition VI. The “given” of Proposition IX was “a spiral PQS, cutting all the radii SP, SQ, &c., in a given angle.” Both the main proof and the alternate proof implicitly depend on the property of the equiangular spiral that the radius of curvature at any point is proportional to the pole distance SP. We here offer a new proof of Newton's proposition which does not depend on this implicit assumption.     

A tutorial proof that extreme points are basic feasible solutions

Given a set of m linear equations in n unknowns with the requirement that the solution space be nonnegative, a simple, heuristic proof is offered which shows that the extreme points of the set of feasible solutions are also basic feasible solutions. This proof can be used in many text treatments of Linear Programming which omit the proof on the grounds that it is too difficult to prove.     

Minkowskische Reduktionsbedingungen für positiv definite quadratische Formen in 5 Variablen

H. Minkowski stated that a positive definite quadratic form in 5 variables is reduced if a certain finite system of inequalities is fulfilled. This fact was first proved byS. S. Ryshkov in 1971. Here a new proof is given which is similar toMinkowski's treatment of forms in 4 variables. In place of the 4 different cases distinguished inRyshkov's proof only a distinction between 2 cases is necessary in the presented proof.     

The level set method of Joó and its use in minimax theory

In this paper we discuss the level set method of Joó and how to use it to give an elementary proof of the well-known minimax theorem of Sion. Although this proof technique was initiated by Joó and based on the intersection of upper level sets and a clever use of the topological notion of connectedness, it is not very well known and accessible for researchers in optimization. At the same time we simplify the original proof of Joó and give a more elementary proof of the celebrated minimax theorem of Sion.     

Nearly perfect systems and effective generalizations of Shidlovski's Theorem

An effective proof of Shidlovski's Theorem is presented. The proof utilizes partial differential operators. A number of generalizations of Shidlovski's Theorem are proven, including results about approximation at more than one point. Additionally, partial differential equations are considered. The new methods give a particularly direct proof of Shidlovski's Theorem.     

The weighted hook length formula

Based on the ideas in Ciocan-Fontanine, Konvalinka and Pak (2009) [5], we introduce the weighted analogue of the branching rule for the classical hook length formula, and give two proofs of this result. The first proof is completely bijective, and in a special case gives a new short combinatorial proof of the hook length formula. Our second proof is probabilistic, generalizing the (usual) hook walk proof of Greene, Nijenhuis and Wilf (1979) [15], as well as the q-walk of Kerov (1993) [20]. Further applications are also presented.     

A proof of strongly uniform termination for Gödel's T by methods from local predicativity

We estimate the derivation lengths of functionals in G?del's system of primitive recursive functionals of finite type by a purely recursion-theoretic analysis of Schütte's 1977 exposition of Howard's weak normalization proof for . By using collapsing techniques from Pohlers' local predicativity approach to proof theory and based on the Buchholz-Cichon and Weiermann 1994 approach to subrecursive hierarchies we define a collapsing f unction so that for (closed) terms of G?del's we have: If reduces to then By one uniform proof we obtain as corollaries: A derivation lengths classification for functionals in , hence new proof of strongly uniform termination of . A new proof of the Kreisel's classific ation of the number-theoretic functions which can be defined in , hence a classification of the provably total functions of Peano Arithmetic. A new proof of Tait's results on weak normalization for . A new proof of Troelstra's result on strong normalization for . Additionally, a slow growing analysis of G?del's is obtained via Girard's hierarchy comparison theorem. This analyis yields a contribution to two open pro blems posed by Girard in part two of his book on proof theory. For the sake of completeness we also mention the Howard Schütte bound on derivation lengths for the simple typed -calculus. Received August 4, 1995