Although understanding is the object of a growing literature in epistemology and the philosophy of science, only few studies have concerned understanding in mathematics. This essay offers an account of a fundamental form of mathematical understanding: proof understanding. The account builds on a simple idea, namely that understanding a proof amounts to rationally reconstructing its underlying plan. This characterization is fleshed out by specifying the relevant notion of plan and the associated process of rational reconstruction, building in part on Bratman's (...) theory of planning agency. It is argued that the proposed account can explain a significant range of distinctive phenomena commonly associated with proof understanding by mathematicians and philosophers. It is further argued, on the basis of a case study, that the account can yield precise diagnostics of understanding failures and can suggest ways to overcome them. Reflecting on the approach developed here, the essay concludes with some remarks on how to shape a general methodology common to the study of mathematical and scientific understanding and focused on human agency. (shrink)

The BankXX system models the process of perusing and gathering information for argument as a heuristic best-first search for relevant cases, theories, and other domain-specific information. As BankXX searches its heterogeneous and highly interconnected network of domain knowledge, information is incrementally analyzed and amalgamated into a dozen desirable ingredients for argument (called argument pieces), such as citations to cases, applications of legal theories, and references to prototypical factual scenarios. At the conclusion of the search, BankXX outputs the set of argument (...) pieces filled with harvested material relevant to the input problem situation.This research explores the appropriateness of the search paradigm as a framework for harvesting and mining information needed to make legal arguments. In this article, we describe how legal research fits the heuristic search framework and detail how this model is used in BankXX. We describe the BankXX program with emphasis on its representation of legal knowledge and legal argument. We describe the heuristic search mechanism and evaluation functions that drive the program. We give an extended example of the processing of BankXX on the facts of an actual legal case in BankXX's application domain — the good faith question of Chapter 13 personal bankruptcy law. We discuss closely related research on legal knowledge representation and retrieval and the use of search for case retrieval or tasks related to argument creation. Finally we review what we believe are the contributions of this research to the understanding of the diverse disciplines it addresses. (shrink)

Scientific and mathematical concepts are significantly different from everyday concepts and are notoriously difficult to learn. It is shown that particular instances of such concepts can be identified or generated by different possible modes of concept interpretation. Some of these modes use formally explicit knowledge and thought processes; others rely on less formal case‐based knowledge and more automatic recognition processes. The various modes differ in attainable precision, likely errors, and ease of use. A combination of such modes can be used (...) to formulate an “ideal” model for interpreting scientific concepts both reliably and efficiently. Comparisons are made with the actual concept interpretations of expert scientists and novice students. The discussion elucidates some cognitive and metacognitive reasons why the learning of scientific or mathematical concepts is difficult. It also suggests instructional guidelines for teaching such concepts more effectively. (shrink)

In practice, mathematical proofs are most often the result of careful planning by the agents who produced them. As a consequence, each mathematical proof inherits a plan in virtue of the way it is produced, a plan which underlies its “architecture” or “unity”. This paper provides an account of plans and planning in the context of mathematical proofs. The approach adopted here consists in looking for these notions not in mathematical proofs themselves, but in the agents who produced them. The (...) starting point is to recognize that to each mathematical proof corresponds a proof activity which consists of a sequence of deductive inferences—i.e., a sequence of epistemic actions—and that any written mathematical proof is only a report of its corresponding proof activity. The main idea to be developed is that the plan of a mathematical proof is to be conceived and analyzed as the plan of the agent(s) who carried out the corresponding proof activity. The core of the paper is thus devoted to the development of an account of plans and planning in the context of proof activities. The account is based on the theory of planning agency developed by Michael Bratman in the philosophy of action. It is fleshed out by providing an analysis of the notions of intention—the elementary components of plans—and practical reasoning—the process by which plans are constructed—in the context of proof activities. These two notions are then used to offer a precise characterization of the desired notion of plan for proof activities. A fruitful connection can then be established between the resulting framework and the recent theme of modularity in mathematics introduced by Jeremy Avigad. This connection is exploited to yield the concept of modular presentations of mathematical proofs which has direct implications for how to write and present mathematical proofs so as to deliver various epistemic benefits. The account is finally compared to the technique of proof planning developed by Alan Bundy and colleagues in the field of automated theorem proving. The paper concludes with some remarks on how the framework can be used to provide an analysis of understanding and explanation in the context of mathematical proofs. (shrink)

Work on analogy has been done from a number of disciplinary perspectives throughout the history of Western thought. This work is a multidisciplinary guide to theorizing about analogy. It contains 1,406 references, primarily to journal articles and monographs, and primarily to English language material. classical through to contemporary sources are included. The work is classified into eight different sections (with a number of subsections). A brief introduction to each section is provided. Keywords and key expressions of importance to research on (...) analogy are discussed in the introductory material. Electronic resources for conducting research on analogy are listed as well. (shrink)

A study of a group of elementary school students learning to control a computer‐implemented Newtonian object reveals a surprisingly uniform and detailed collection of strategies, at the core of which is a robust “Aristotelian” expectation that things should move in the direction they are last pushed. A protocol of an undergraduate dealing with the same situation shows a large overlap with the set of strategies used by the elementary school children and thus a marked lack of influence of classroom physics (...) training on this student's naive physics. The data from these two studies are pooled and elaborated into a “genetic task analysis” of how one might come to understand Newtonian dynamics as a more or less natural evolution from the naive state. (shrink)

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 61students 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 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 evaluate their validity. (shrink)