Mathematical Induction in School: An Illusion of Rigor?

An Innovative Approach to Teaching Structural Induction for Computer Science

“…In addition, they suggest that some students perform proofs by induction without a conceptual understanding of the process involved. This latter finding refines the results of Baker (1995) and the claim of Lowenthal & Eisenberg (1992) and Thompson (1996) that students tend to apply induction in a "mechanical way". According to Thompson (1996), sometimes "students appear to be approaching the proof process very algorithmically, having memorized a process instead of understanding its origin" (p. 479).…”

“…Proofs by induction are no exception. Students are exposed to proofs by induction in many courses of their curricula, and yet it is well documented in the literature that in general, students have difficulties understanding and performing them Wu Yu, 2000;Sheard, 1998;Thompson, 1996;Baker, 1995;Movshovitz-Hadar, 1993;Lowenthal & Eisenberg, 1992;Dubinsky, 1989;Dubinsky 1986;Dubinsky & Lewin, 1986;Ernest, 1984;Brumfiel, 1974). Such difficulties include difficulties understanding the steps involved in a proof by induction, the substance of the proof (seeing the proof as a convincing argument and not as a procedure to be followed), how to prove an if-then statement (which results in difficulties understanding the logic behind the induction step), the necessity to include the base case, and difficulties performing proofs by induction on problems that are not similar to the ones students encountered before.…”

“…Even though students are exposed to proofs by induction in many courses of their curricula, they still have difficulties understanding and performing them-a fact well documented in the literature Wu Yu, 2000;Sheard, 1998;Thompson, 1996;Baker, 1995;Movshovitz-Hadar, 1993;Lowenthal & Eisenberg, 1992;Dubinsky, 1989;Dubinsky 1986;Dubinsky & Lewin, 1986;Ernest, 1984;Brumfiel, 1974). "Indeed, if you question students-even those who have had several mathematics courses-although almost all of them will have heard of induction, not many of them will be able to say anything intelligent about what it is, much less actually use it to solve a problem.…”

“…It has been documented in the literature that students have difficulties understanding and performing such proofs, even after repeated instruction in different courses of their curricula Wu Yu, 2000;Sheard, 1998;Thompson, 1996;Baker, 1995;Movshovitz-Hadar, 1993;Lowenthal & Eisenberg, 1992;Dubinsky, 1989;Dubinsky 1986;Dubinsky & Lewin, 1986;Ernest, 1984;Brumfiel, 1974). Even though induction is omnipresent in the field of computer science, and the idea of proofs by induction is reinforced in many courses of the computer science curriculum, it seems that the seeds are not there for students' understanding to grow.…”

Semantic Structure of Classroom Discourse Concerning Proof and Proving in High School Mathematics

“…One teacher gives a wrong explanation to students, making them even more confused. Knuth (2002), and Lowenthal and Eisenberg (1992) observed that some teachers suffer the same inabilities and misconceptions as students. Teachers did not understand or were unconfident about understanding generality of proven statements and could not reliably distinguish between valid and invalid arguments (Knuth, 2002).…”