Proof validation and modification in secondary school geometry

Proof validation is important in school mathematics because it can provide a basis upon which to critique mathematical arguments. While there has been some previous research on proof validation, the need for studies with school students is pressing. For this paper, we focus on proof validation and modification during secondary school geometry. For that purpose, we employ Lakatos’ notion of local counterexample that rejects a specific step in a proof. By using Toulmin’s framework to analyze data from a task-based questionnaire completed by 32 ninth-grade students in a class in Japan, we identify what attempts the students made in producing local counterexamples to their proofs and modifying their proofs to deal with local counterexamples. We found that student difficulties related to producing diagrams that satisfied the condition of the set proof problem and to generating acceptable warrants for claims. The classroom use of tasks that entail student discovery of local counterexamples may help to improve students’ learning of proof and proving.     

Definable predicates of standardness in internal set theory

It is shown that Nelson’s internal set theory IST has no definable predicate that is a proper extension of the standardness predicate and satisfies the carry-over, idealization, and standardization principles. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 803–809, December, 1999.     

Resolution over linear equations modulo two

We consider an extension of the resolution proof system that operates with disjunctions of linear equalities over ${\mathbb{F}}_2$; we denote this system by Res(⊕). It is well known that tree-like resolution is equivalent in behavior to DPLL algorithms for the Boolean satisfiability problem. Every DPLL algorithm splits the input problem into two by assigning two possible values to a variable; then it simplifies the two resulting formulas and makes two recursive calls. Tree-like Res(⊕)-proofs correspond to an extension of the DPLL paradigm, in which we can split by an arbitrary linear combination of variables modulo two. These algorithms quickly solve formulas that explicitly encode linear systems modulo two which were used for proving exponential lower bounds for conventional DPLL algorithms.We prove exponential lower bounds on the size of tree-like Res(⊕)-proofs. We also show that resolution and tree-like Res(⊕) do not simulate each other.We prove a space vs size tradeoff for Res(⊕)-proofs. We prove that Res(⊕) is implicationally complete and also that Res(⊕) is polynomially equivalent to its semantic version.We consider the proof system $\text{Res}(\oplus ;⩽k)$ that is a restricted version of Res(⊕) operating with disjunctions of linear equalities such that at most k equalities depend on more than one variable. We simulate $\text{Res}(\oplus ;⩽k)$ in the OBDD-based proof system with blowup $2^k$ and in Polynomial Calculus Resolution with blowup $2^{nH(2k/n)}poly(n)$, where n is the number of variables and $H(p)$ is the binary entropy; the latter result implies exponential lower bounds on the size of $\text{Res}(\oplus ;⩽\delta n)$-proofs for some constant $\delta >0$. Raz and Tzameret introduced the system R(lin) which operates with disjunctions of linear equalities with integer coefficients. We show that Res(⊕) can be p-simulated in R(lin).     

Isomorphism is equality

The setting of this work is dependent type theory extended with the univalence axiom. We prove that, for a large class of algebraic structures, isomorphic instances of a structure are equal—in fact, isomorphism is in bijective correspondence with equality. The class of structures includes monoids whose underlying types are “sets”, and also posets where the underlying types are sets and the ordering relations are pointwise “propositional”. For monoids on sets equality coincides with the usual notion of isomorphism from universal algebra, and for posets of the kind mentioned above equality coincides with order isomorphism.     

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.     

Effective rates of convergence for Lipschitzian pseudocontractive mappings in general Banach spaces

This paper gives an explicit and effective rate of convergence for an asymptotic regularity result ‖Tx_n−x_n‖→0 due to Chidume and Zegeye in 2004 [14] where (x_n) is a certain perturbed Krasnoselski-Mann iteration schema for Lipschitz pseudocontractive self-mappings T of closed and convex subsets of a real Banach space. We also give a qualitative strengthening of the theorem by Chidume and Zegeye, by weakening the assumption of the existence of a fixed point. For the bounded case, our bound is polynomial in the data involved.     

Simulating non‐prenex cuts in quantified propositional calculus

We show that the quantified propositional proof systems G_i are polynomially equivalent to their restricted versions that require all cut formulas to be prenex Σ^q_i (or prenex Π^q_i). Previously this was known only for the treelike systems G*_i. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim