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.     

The effect of quantitative reasoning on prospective mathematics teachers’ proof comprehension: The case of real numbers

We report a mixed-methods research study investigating the effect of quantitative reasoning on prospective mathematics teachers’ comprehension of a proof on real numbers. Nineteen prospective mathematics teachers engaged in quantitative reasoning while developing real numbers as rational number sequences in a series of instructional activities. All participants completed a proof comprehension assessment prior to and upon completion of the instruction. Six of the prospective mathematics teachers also participated in semi-structured interviews after the post-test. Results showed a significant difference in proof comprehension performance between the pre- and post-tests. Moreover, results from the interviews showed that prospective teachers reasoned quantitatively on the proof comprehension dimensions. Results suggest that engaging in quantitative reasoning during instruction may help to develop proof comprehension, particularly in situations involving the analysis of proofs entailing properties of the real number system. We recommend embedding quantitative reasoning in teacher education and professional development programs to facilitate mathematics teachers’ proof comprehension and proving activities.     

The opaque nature of generic examples: The structure of student teachers’ arguments in multiplicative reasoning

The study aims to explore the structural aspects of generic examples, to get better insight into what makes them potentially opaque for learners. We have analyzed 27 written arguments, for which student teachers (grades 1–10) were asked to use a generic example to prove a given statement in multiplication. Using Toulmin’s framework, we developed five categories of arguments based on their structure: examples, empirical arguments, leap arguments, embedded arguments, and other arguments. Also, we conclude that none of the student teachers provided arguments that we recognize as complete generic examples. The results bring us to a discussion about features of generic examples making them difficult to come to grips with, having implications for how teacher educators can support student teachers’ learning to prove. From this, we propose a definition of generic examples that attends to the criteria suggested in previous research, yet, emphasizing their structural nature.     

When an argument is the content: Rational number comprehension through conversions across registers

This article reframes previously identified misconceptions about repeating decimals by describing these misconceptions as limited understandings of how mathematics concepts are referenced. In particular, misconceptions about repeating decimals and their quotient of integer representations are recast as limited understandings of mathematics as a discipline that derives its content from representational systems and the denotations they provided. Under this framework, arguments (e.g., proofs) that convert repeating decimals to their quotient of integer representations provide content for “rational number,” which is represented in multiple ways, each offering distinct opportunities for mathematical activity. The notion of an argument as content is illustrated as arguments providing access to a concept. One Grade 8 student’s struggle with understanding rational number is used to illustrate this framework and its implications for teaching and learning.     

Counter-examples for refinement of conjectures and proofs in primary school mathematics

The purpose of this study is to explore how primary school students reexamine their conjectures and proofs when they confront counter-examples to the conjectures they have proved. In the case study, a pair of Japanese fifth graders thought that they had proved their primitive conjecture with manipulative objects (that is, they constructed an action proof), and then the author presented a counter-example to them. Confronting the counter-example functioned as a driving force for them to refine their conjectures and proofs. They understood the reason why their conjecture was false through their analysis of its proof and therefore could modify their primitive conjecture. They also identified the part of the proof which was applicable to the counter-example. This identification and their action proof were essential for their invention of a more comprehensive conjecture.     

A task that elicits reasoning: A dual analysis

This paper reports on the forms of reasoning elicited as fourth grade students in a suburban district and sixth grade students in an urban district worked on similar tasks involving reasoning with the use of Cuisenaire rods. Analysis of the two data sets shows similarities in the reasoning used by both groups of students on specific tasks, and the tendency of a particular task to elicit numerous forms of reasoning in both groups of students. Attributes of that task and ways that those attributes can be replicated in other domains may have implications in the teaching of early reasoning.     

Asymptotic behavior of averaged and firmly nonexpansive mappings in geodesic spaces

We further study averaged and firmly nonexpansive mappings in the setting of geodesic spaces with a main focus on the asymptotic behavior of their Picard iterates. We use methods of proof mining to obtain an explicit quantitative version of a generalization to geodesic spaces of a result on the asymptotic behavior of Picard iterates for firmly nonexpansive mappings proved by Reich and Shafrir. From this result we obtain effective uniform bounds on the asymptotic regularity for firmly nonexpansive mappings. Besides this, we derive effective rates of asymptotic regularity for sequences generated by two algorithms used in the study of the convex feasibility problem in a nonlinear setting.     

Monotone Proofs of the Pigeon Hole Principle

We study the complexity of proving the Pigeon Hole Principle (PHP)in a monotone variant of the Gentzen Calculus, also known as Geometric Logic. We prove a size‐depth trade‐off upper bound for monotone proofs of the standard encoding of the PHP as a monotone sequent. At one extreme of the trade‐off we get quasipolynomia ‐size monotone proofs, and at the other extreme we get subexponential‐size bounded‐depth monotone proofs. This result is a consequence of deriving the basic properties of certain monotone formulas computing the Boolean threshold functions. We also consider the monotone sequent expressing the Clique‐Coclique Principle (CLIQUE) defined by Bonet, Pitassi and Raz [9]. We show that monotone proofs for this sequent can be easily reduced to monotone proofs of the one‐to‐one and onto PHP, and so CLIQUE also has quasipolynomia ‐size monotone proofs. As a consequence of our results, Resolution, Cutting Planes with polynomially bounded coefficients, and Bounded‐Depth Frege are exponentially separated from the monotone Gentzen Calculus. Finally, a simple simulation argument implies that these results extend to the Intuitionistic Gentzen Calculus. Our results partially answer some questions left open by P. Pudlák.     

On arithmetical completeness of the logic of proofs

In this paper, we establish a stronger version of Artemov's arithmetical completeness theorem of the Logic of Proofs ${\mathsf{LP}}_0$. Moreover, we prove a version of the uniform arithmetical completeness theorem of ${\mathsf{LP}}_0$.     

A Rate of Asymptotic Regularity for the Mann Iteration of κ-Strict Pseudo-Contractions

In this article, we apply methods of proof mining to obtain a uniform effective rate of asymptotic regularity for the Mann iteration associated to κ-strict pseudo-contractions on convex subsets of Hilbert spaces.