Of course this doesn't work.
What was funny was that as he started to explain his trick, he stopped, looked at me and announced, "I'm pretty sure I just lost you."
I looked at him and said, "no, I'm with you. I see what you are saying."
He replied, "yeah, but I lost myself".
We both laughed out loud. This is the kind of interaction you get with kids when you take the time to talk with them. In the end, Larry was able to see how this trick works for adding but not for multiplying. Eventually he understood that when multiplying the 5 and the 9, if you reduce the 5 to a 4, you are really taking away 9 things. And if you increase the 9 to a 10, you are really adding 5 things. And that is why 4 x 10 = 40 while 5 x 9 = 45. A difference of 5.
I don't know whether I've explained this very well, but let me assure you that I got goose-bumps when Larry looked up with a fully illuminated light bulb hovering over his head.
I think you've described the situation perfectly Joe. There is nothing more satisfying than 1:1 exploration of ideas & concepts with a student. It is for moments like this that I teach. Thanks for sharing.
ReplyDeleteThis is what I really hope kids get out of arithmetic - a feeling for operations - which they don't get unless they can play around and experiment (and talk it through with someone!).
ReplyDeleteIncidentally, Larry's trick works pretty well for squaring numbers:
8x8 = (8-2)x(8+2) +4 = 6x10 + 4 = 64
Mental calculators use this trick to do harder squares:
43x43 = (43-3)x(43+3) + 9 = 40x46 + 9 = 1849
Love this story. It shows how important it is to have students justify and explain their thinking and show their understanding. I am sure that Larry will remember this exchange and how you made him feel - successful! He recalled a trick but had overgeneralized it and was able to have an 'aha' moment with your support. Bravo!
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