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Topic: Drill and Kill

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Subject:   RE: Balancing the drill
Author: LFS
Date: Jun 17 2011
I am thinking perhaps I was not clear. I gave several examples that I do NOT
like in my last letter. Below is one I like.

I do NOT think "drill=rigor" or "difficulty=rigor" or "tricky questions=rigor",
although many curricula seem to think this.
Balance is important and endless conversation about this is also important as
the principal's letter illustrates.

I think rigor = correctness + completeness particularly for K-12 and even for
undergraduate mathematics.
And I think that problems that are straightforward and solved within this
(albeit extremely concise written) format make any subject - not just
mathematics - rigorous.

If you disagree with this, please give samples of problems and their solutions
and why you think they represent rigorous or non-rigorous mathematics. I am
not very good with discourse. I understand  examples.
For example, I consider this a good problem when working with Pythagoras'
==How many triangles are there with sides 14, 20 and 7?  Is this a right
triangle? Why or why not? If it is, draw it.
A rigorous solution from a teacher (student's work abbreviated but containing
the steps) would say something like:
0. All units are the same. Good.
1. If a triangle exists then by SSS, it is unique.
2. A triangle exists if it obeys the triangle inequality ... testing Is 14+20>7
yes, Is 14+7 >20 yes, Is 20+7>14 yes. So triangle exists.
Answer 1: There is exactly one such triangle.
3. Largest side is 20 so if right triangle, this is hypotenuse.
4. Triangle is right if it obeys Pythagoras ...  testing:  Is 7^2 + 14^2 = 20^2,
Is 49+196=400 no.
Answer 2: Triangle is not a right triangle since no Pythagoras.
(In place of "Is", I carefully write a question mark over each of the =, <, >
thus making true/false operators and NOT statements.)

Next question: How many triangles are there with sides 13, 1' and 5?  Is this
a right triangle? Why or why not? If it is, draw it.
The question is straightforward and makes sense.
The solution is rigorous and makes sense.
It is neither "ball-busting" nor "intellectually demanding".

These questions require logical thinking and problem solving concepts. They
reinforce previous learning (i.e. triangle inequality, SSS), require
understanding of Pythagoras' and its converse. They require you to check details
such as units and be precise.  They require basic mathematics skills. Тhey
can be solved with or without technology. (BTW: I am presenting virtually at
ISTE using these problems where students should either work or check their work
using interactive applets.)


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