I Saw a White Horse Today – Proving Algebraic Identities or Not

Reasoning Pattern I

Reasoning Pattern I

Reasoning Pattern II

Reasoning Pattern II


About jrh794

I am a sixty-five year old math instructor at Southern Oregon University. I taught at the College of the Siskiyous in Weed California for twenty-six years. Prior to that I worked as a computer programmer, carpenter and in various other jobs. I graduated from Rice University in 1967 and have a MS in Operations Research from Stanford. In the past I have hand-built a stone house and taken long solo bicycle tours. Now I ride my mountain bike and play golf for recreation.
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3 Responses to I Saw a White Horse Today – Proving Algebraic Identities or Not

  1. suszann says:

    Great little break in the day – so true.

  2. Nick says:

    Your post reminds me of something I read just yesterday regarding proof by contradiction. It strikes me that the inductive jump to “all horses are white” is only glaring to me (the teacher) because I assume my students are trying to find general truths by patterning and that they mean what they say. A statement like “the expressions are equal for any value of x” (either false or true) is not something a student would probably come up with to summarize their discovery that one or perhaps two values of x made the expressions equal, unless they were both lazy (because they didn’t check more values) and practiced in “math speak”. Do they really believe it? It’s hard to tell without further questioning. On the other hand, when asked to determine a formula for a given table of values, they may easily believe that a given formula works for all values, even though they have only checked a few of them. You ask them if their formula works, and they say “Yes. I checked the first few values.” They are satisfied that the formula is beginning to work for them — why should it stop working?

    At my college, especially with students at the lower levels, we lean on inductive reasoning a lot, particularly when they are “checking” that they have simplified an algebraic expression correctly. Maybe I could be doing more to advertise the fact that this is just lending evidence, but not credence, to the notion that they have done the right work or found the right formula. I think we have set up the framework for your “Reasoning Pattern II”, but we sometimes step over the line and promote the flawed 2nd pattern.

    Thanks for the nice post, Jim!

  3. Nick says:

    Meant to say “flawed 1st pattern”, and my link shouldn’t begin with “http//”. [I need a preview button for everything in life.]

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