(T/E) on the Logical Form of Mathematical Language*

1998

This article discusses some methods of describing and referring to mathematical objects and of consistently and unambiguously signaling the logical structure of mathematical arguments.

PRIMUS, 1998

Logic and Logical Philosophy

This paper discusses a generalization of logical expressivism. It is shown that, in the wide sense defined here, the expressivist approach is neutral with respect to different theories of inference and offers a natural framework for understanding logical forms and their function. An expressivist strategy for explaining the development of logical forms is then applied to the analysis of Frege’s Begriffsschrift, Gentzen’s sequent calculus and Belnap’s display logic.

Springer eBooks, 2009

Educational Studies in Mathematics, 1995

This study focuses on undergraduate students' ability to unpack informally written mathematical statements into the language of predicate calculus. Data were collected between 1989 and 1993 from 61 students in six small sections of a "bridge" course designed to introduce proofs and mathematical reasoning. We discuss this data from a perspective that extends the notion of concept image to that of statement image and introduces the notion of proof framework to indicate that part of a theorem's image which corresponds to the top-level logical structure of a proof. For simplified informal calculus statements, just 8.5% of unpacking attempts were successful; for actual statements from calculus texts, this dropped to 5%. We infer that these students would be unable to reliably relate informally stated theorems with the top-level logical structure of their proofs and hence could not be expected to construct proofs or validate them, i.e., determine their correctness.

instructional practices are being transformed in many mathematics classrooms as a result of reform efforts in the United States and Europe, For example, students are being asked to take a more active role in explaining their problem solution strategies. In order to achieve these changes in student activity, teachers are being encouraged to restructure the communication patterns in their classrooms. The goal of this article is to examine, in detail, the discourse of one teacher and three of her middle school students during a single lesson on area measurement. Some concepts from sociocultural theory (e.g., authoritative versus internally persuasive discourse) are used to organize and interpret our analysis, The teacher observed in this study. tried to share responsibility and authority for explaining and evaluating mathematical problems with her students. We found that two of the three students initially tried to present explanations that differed from that used by the teacher on a previous day. Each time these students offered an alternative solution strategy, the teacher overlapped her speech with theirs. Eventually both students employed the teacher's explanation. The third student who used the teacher's explanation from the beginning did not encounter her overlapping speech. As a result of our analyses, we concluded that the teacher did achieve one of her goals for this lesson, the sharing of responsibility for providing explanations, but not the goal of sharing authority. The theoretical and instructional implications of our analysis for research on educational reform are addressed.

Psychonomic bulletin & review, 2017

The existence of multiple modes of explanation means that a crucial step in the process of generating explanations has to be selecting a particular mode. The present article identifies the key conceptual, as well as some pragmatic and epistemological, considerations that license the use of the formal mode of explanation, and thus that enter into the process of selecting and generating a formal explanation. Formal explanations explain the presence of certain properties in an instance of a kind by reference to the kind of thing it is (e.g. That has four legs because it is a dog). As such, this mode of explanation is intrinsically tied to kind representations and is applicable domain-generally. Although it is possible for formal explanation to apply domain-generally, for any given kind it is selective in its application, in that it can explain some, but not all, properties of the instances of a kind. It also appears that different types of properties can receive formal explanations acr...