On an Intuitionistic Logic for Pragmatics

We reconsider the pragmatic interpretation of intuitionistic logic [21] regarded as a logic of assertions and their justications and its relations with classical logic. We recall an extension of this approach to a logic dealing with assertions and obligations, related by a notion of causal implication [14, 45]. We focus on the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on polarized bi-intuitionistic logic as a logic of assertions and conjectures: looking at the S4 modal translation, we give a denition of a system AHL of bi-intuitionistic logic that correctly represents the duality between intuitionistic and co-intuitionistic logic, correcting a mistake in previous work [7, 10]. A computational interpretation of cointuitionism as a distributed calculus of coroutines is then used to give an operational interpretation of subtraction.Work on linear co-intuitionism is then recalled, a linear calculus of co-intuitionistic coroutines is dened and a probabilistic interpretation of linear co-intuitionism is given as in [9]. Also we remark that by extending the language of intuitionistic logic we can express the notion of expectation, an assertion that in all situations the truth of p is possible and that in a logic of expectations the law of double negation holds. Similarly, extending co-intuitionistic logic, we can express the notion of conjecture that p, dened as a hypothesis that in some situation the truth of p is epistemically necessary

Algebraic Reasoning and Conceptual Understanding: A Mixed Methods Comparison of In-Person and Virtual Classroom Strategies

Algebraic reasoning is the beginning school of thought to critical thinking. Employers are looking for this 21st century skill. The purpose of this research was to investigate equity in mathematics education using the NCTM Teaching and Learnings Beliefs Survey. Four area were studied: the number of years in education, the degree earned, the grade level taught in education, and the number of years in an educator’s teaching position. A mixed methods inventory was used. Most results were not rejected in this study. Two statements in the survey warranted a discussion. Recommendations were made for further research

Instructional strategies in explicating the discovery function of proof for lower secondary school students

In this paper, we report on the analysis of teaching episodes selected from our pedagogical and cognitive research on geometry teaching that illustrate how carefully-chosen instructional strategies can guide Grade 8 students to see and appreciate the discovery function of proof in geometr

Calculational Proofs in ACL2s

Teaching college students how to write rigorous proofs is a critical objective in courses that introduce formal reasoning. Over the course of several years, we have developed a mechanically-checkable style of calculational reasoning that we used to teach over a thousand freshman-level undergraduate students how to reason about computation in our "Logic and Computation" class at Northeastern University. We were inspired by Dijkstra, who advocated the use of calculational proofs, writing "calculational proofs are almost always more effective than all informal alternatives, ..., the design of calculational proofs seems much more teachable than the elusive art of discovering an informal proof." Our calculational proof checker is integrated into ACL2s and is available as an Eclipse IDE plugin, via a Web interface, and as a stand-alone tool. It automatically checks proofs for correctness and provides useful feedback. We describe the architecture of the checker, its proof format, its underlying algorithms, its correctness and provide examples using proofs from our undergraduate class and from Dijkstra. We also describe our experiences using the proof checker to teach undergraduates how to formally reason about computation

Exploring the episodic structure of algebra story problem solving

This paper analyzes the quantitative and situational structure of algebra story problems, uses these materials to propose an interpretive framework for written problem-solving protocols, and then presents an exploratory study of the episodic structure of algebra story problem solving in a sizable group of mathematically competent subjects. Analyses of written protocols compare the strategic, tactical, and conceptual content of solution attampts, looking within these attempts at the interplay between problem comprehension and solution. Comprehension and solution of algebra story problems are complimentary activities, giving rise to a succession of problem solving episodes. While direct algebraic problem solving is sometimes effective, results suggest that the algebraic formalism may be of little help in comprehending the quantitative constraints posed in a problem text. Instead, competent problem solvers often reason within the situational context presented by a story problem, using various forms of "model-based reasoning" to identify, pursue, and verify quantitative constraints required for solution. The paper concludes by discussing the implications of these findings for acquiring mathematical concepts (e.g., related linear functions) and for supporting their acquisition through instruction