Over the past several decades, many of those involved with mathematics education at the college and university level have turned their attention to the difficulties students experience in mathematics courses requiring them to write proofs. Because even mathematics majors have trouble in more advanced courses, a number of mathematicians have written "transition" books whose aim is to bridge the gap between lower-level, computationally oriented courses and upper-level classes where deductive reasoning and abstraction play a major role. See, for example, books by Smith, Eggen, and St. Andre [44], Mason [28], Lucas [27], Velleman [50], Solow [45], and Fletcher and Patty [18]. Moreover, because computer science students also need to learn to operate in a mathematically sophisticated environment, one of the goals of discrete mathematics courses is to enhance students' logical reasoning and proof-writing abilities. See, for example, Gersting [20], Johnsonbaugh [26], Rosen [38], Ross and Wright [39], Gries and Schneider [22], Epp [13], and Dubinsky and Fenton [9]. Over the same period, researchers in mathematics education have been investigating the cognitive processes that underlie students' difficulties with proof production. A summary of some of this research can be found in Tall [46]. Among recent contributors are Goetting [21], Harel and Sowder [23], Moore [30], [31], Selden and Selden [41], [42], and Thompson [49]. Many additional references and articles can be found on the Internet website "Preuve," the "International Newsletter for the Teaching and Learning of Proof" (http://www-didactique.imag.fr/preuve/). In the late 1970s, before texts for transition courses had become generally available, I started teaching a course to provide background for students who would go on to take advanced undergraduate courses in mathematics and computer science. Initially, I had assumed that the reason our students were doing so poorly in our advanced courses was that the teachers moved too quickly to "interesting mathematics" and paid inadequate attention to basic material such as sets, functions, and relations. I expected that the new course would solve this problem by giving students an adequate amount of time to focus exclusively on foundations. As I taught the course, however, I found that my students' difficulties were much more profound than I had imagined. Indeed, I was almost overwhelmed by the poor quality of their proof-writing attempts. Often their efforts consisted of little more than a few disconnected calculations and imprecisely or incorrectly used words and phrases that did not even advance the substance of their cases. My students seemed to live in a different logical and linguistic world from the one I inhabited, a world that made it very difficult for them to engage in the kind of abstract mathematical thinking I was trying to help them learn. 2. THE NEED FOR INSTRUCTION IN FORMAL REASONING SKILLS. Evaluating the truth and falsity of even very simple mathematical statements involves complex cognitive activity. ...

Over the past several decades, many of those involved with mathematics education at the college and university level have turned their attention to the difficulties students experience in mathematics courses requiring them to write proofs. Because even mathematics majors have trouble in more advanced courses, a number of mathematicians have written "transition" books whose aim is to bridge the gap between lower-level, computationally oriented courses and upper-level classes where deductive reasoning and abstraction play a major role. See, for example, books by Smith, Eggen, and St. Andre [44], Mason [28], Lucas [27], Velleman [50], Solow [45], and Fletcher and Patty [18]. Moreover, because computer science students also need to learn to operate in a mathematically sophisticated environment, one of the goals of discrete mathematics courses is to enhance students' logical reasoning and proof-writing abilities. See, for example, Gersting [20], Johnsonbaugh [26], Rosen [38], Ross and Wright [39], Gries and Schneider [22], Epp [13], and Dubinsky and Fenton [9]. Over the same period, researchers in mathematics education have been investigating the cognitive processes that underlie students' difficulties with proof production. A summary of some of this research can be found in Tall [46]. Among recent contributors are Goetting [21], Harel and Sowder [23], Moore [30], [31], Selden and Selden [41], [42], and Thompson [49]. Many additional references and articles can be found on the Internet website "Preuve," the "International Newsletter for the Teaching and Learning of Proof" (http://www-didactique.imag.fr/preuve/). In the late 1970s, before texts for transition courses had become generally available, I started teaching a course to provide background for students who would go on to take advanced undergraduate courses in mathematics and computer science. Initially, I had assumed that the reason our students were doing so poorly in our advanced courses was that the teachers moved too quickly to "interesting mathematics" and paid inadequate attention to basic material such as sets, functions, and relations. I expected that the new course would solve this problem by giving students an adequate amount of time to focus exclusively on foundations. As I taught the course, however, I found that my students' difficulties were much more profound than I had imagined. Indeed, I was almost overwhelmed by the poor quality of their proof-writing attempts. Often their efforts consisted of little more than a few disconnected calculations and imprecisely or incorrectly used words and phrases that did not even advance the substance of their cases. My students seemed to live in a different logical and linguistic world from the one I inhabited, a world that made it very difficult for them to engage in the kind of abstract mathematical thinking I was trying to help them learn. 2. THE NEED FOR INSTRUCTION IN FORMAL REASONING SKILLS. Evaluating the truth and falsity of even very simple mathematical statements involves complex cognitive activity. ...

In this chapter, we examine the relevance of and interest in including some instruction in logic in order to foster competence with proof in the mathematics classroom. In several countries, educators have questioned of whether to include explicit instruction in the principles of logical reasoning as part of mathematics courses since about the 1980s. Some of that discussion was motivated by psychological studies that seemed to show that "formal logic … is not a model for how people make inferences" (Johnson-Laird 1975 ) . At the same time, university and college faculty commonly complain that many tertiary students lack the logical competence to learn advanced mathematics, especially proof and other mathematical activities that require deductive reasoning. This complaint contradicts the view that simply doing mathematics at the secondary level in itself suffi ces to develop logical abilities.

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