Summary
In his article, Borwein who is a proponent of computer technology talks about its influence on math proof (deduction). He defines experimental math (induction) as a mathematical method via the use of the computer which facilitates math discoveries, imagination, development of insight and intuition, hypothesis-testing, and lengthy computations. In contrast, proof exhibits the aesthetics of math and de-emphasizes computation. He states that one advantage of computation is to help one see the result of a conjecture in advance to determine whether it merits any proof. As modern computer software generates better graphics and sounds, it makes proof more accessible and computation more accurate. For example, proving the square root of 2 to be irrational pictorially with inscribed triangles shows how easy it is to prove or disprove conjectures on a computer. However, Borwein points out a drawback to the high efficiency of computation which may hinder students and researchers' ways of examining math structure when they can thoughtlessly do enormous computations accurately on a computer. He concludes that computer technology is a good tool for mathematicians but is not a good substitute for proof.
Reflections
I think proof and computation work hand in hand. Proofs are needed to help us make sure that we put theory into practice correctly. They also help us check our understanding and reasoning. At the same time, computation is necessary to bring us back on course when we make mistakes in our proofs. For example, in Pre-calculus 12, when my students and I prove trigonometric identities together, occasionally we make minor mistakes in our proof which prevent us from completing it. Then, we use a calculator to substitute a value for the angle in each step until we find and correct the mistakes. Most of my students are interested in proving identities. However, it comes to formal proofs of math theorems, my students ignore them and go straight to the worked examples. I think because the formal logic which serves as the basis for proofs is not emphasized in high school, students are uninterested in math structure. Time is a major factor for de-emphasizing math proofs in schools. This is also the reason why I cannot emphasize any simple proofs in my learning centre. Besides that, parents expect tutors to teach their children how to do computations for the purposes of school tests.
I've always wondered about whether one should encourage students to use a theorem if they don't yet have the background to understand the proof behind it. I always try to find a proof (or at least a demonstration) to explain to students why we are using a certain formula or process to do computations. I know they don't always fully understand, but I hope it will make it more meaningful for them.
ReplyDeleteComputers indeed are changing the ways we both do and teach mathematics. In either case, we need to be sure that we're using them correctly. Students and mathematicians alike can be convinced of answers that actually bear no grounding in truth, simply due to a typo. That said, I think becoming adept at using a computer to do some basic fact checking of whatever math you are doing is certainly a skill more students could benefit from. You can avoid headaches and proceed with more confidence - something I wish we could give to all students.
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