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Students think that they are assuming what they are trying to prove
Partly as a result of an increase in computer technology, iterative and recursive methods are becoming more common. If taught well, mathematical induction can help students' understanding of these methods grow. Principles and Standards for School Mathematics recommends that "students should learn that certain types of results are proved using the technique of mathematical induction" (NCTM 2000, p. 345).
When students are learning mathematical induction, they usually begin by proving such summation formulas as
formula omitted
and similar formulas, This standard way of teaching induction may be common in both high school and college classrooms because these types of problems appear in textbooks. Unfortunately, Baker (1996) found that several difficulties arise with this type of learning. Students often have little experience with summation notation, so before they even begin, the traditional instruction presents a stumbling block. Another problem frequently arises during the inductive step of the proof because students think that they are assuming what they are trying to prove. Even students who have experience in logic and proof, which most high school students lack, may have this belief. Another danger is that students may become proficient at induction algorithms while gaining little or no understanding of why induction works. In particular, students can learn to verify summation formulas by induction without understanding what they are doing or why.
This article presents my experience in a classroom setting using a new approach to teaching induction that is being developed by the Mathematical Methods in High School project. I recorded my detailed observations and an assessment in journal form. The basic idea behind the new approach is to use induction to prove that two formulas, one in recursive form and the other in a closed or explicit form, will always agree for whole numbers.
I taught the unit on induction for five days in a precalculus class at Delavan High School, in Delavan, Illinois. It is a rural high school with a total enrollment of 171 students. Precalculus is the highest-level mathematics class offered at the school. No class at Delavan has an "honors" designation, but the students are predominantly college bound. The class consisted of ten girls and eight boys; six...