The Role of Logic in Teaching Proof

Abstract

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. ...

A research on students constructions of formulas with logical implications and its contra-position

“…Наравно, не постоји универзална сагласност како би то требало радити. Сусана Еп (Susanna S. Epp) у тексту (Epp, 2003) …”

Matematikämnet och stadiebytet mellan grundskolan och gymnasieskolan : En enkät- och klassrumsstudie

“…Den grekiska studien visar alltså bland annat på en ökad matematisk stringens efter stadiebytet samt att eleverna inte klarar av eller ens inbjuds att delta i utvecklandet av generella matematiska resultat. Dessa iakttagelser påminner om dem som beskrevs ovan gällande undervisning för nybörjar-studenter i universitetsmatematik (Epp, 2003;Gueudet, 2008;Winsløw, 2008;Winsløw m.fl., 2014). Det antyder att en ökad stringens i det matematiska innehållet och problem för studenter att anamma matematiken på det högre stadiet inte är isolerat till ett visst stadiebyte, utan förekommer vid övergångar på olika nivåer.…”

Mathematicians’ perspectives on the teaching and learning of proof

“…Professors 1 and 3, in particular, had long debated the nature of the course, with Professor 3 believing strongly in a systematic, rule-based approach and Professor 1 believing the approach should be more investigational, but feeling compelled to introduce guidelines in response to student errors. Epp (2003) discusses similar debates. Each professor, however, had seriously considered the arguments for the opposing view, and the other professors also viewed this as a complex issue.…”

“…. ," (Dubinsky & Yiparaki, 2000;Epp, 2003;Zepp, Monin, & Lei, 1987) and that this can interfere with students' abilities to make use of logical structures in constructing and validating proofs (Selden & Selden, 1995, 1999. It is worth noting that effective structural thinking is "syntactic" in the sense in which we use the term in Weber and Alcock (2004) -to refer to correct reasoning based on form and logic -and not as this term is sometimes used to refer to the meaningless manipulation of symbols or ritual use of a certain format.…”

“…Suggestions for ways to improve the teaching of proof tend to reflect one or other of these routes. Some suggest a systematic approach based on a solid grounding in logic and its associated linguistic expressions (Epp, 1998(Epp, , 2003Selden & Selden, 1999), or on a presentation or sequence of problems that can lead students to more easily see the structure of certain proof types (Harel, 2002;Leron, 1985). With a similar focus on structure, empirical studies have specified skills students need to produce simple proofs in a given area (Gholamazad, Lijedahl, & Zazkis, 2003;Weber, 2006), and have investigated technological environments that can focus students' attention on the axiomatic development of a mathematical theory (Cerulli & Mariotti, 2003).…”