Recently, I got to visit Mountain Grove schools and do a problem-solving workshop with middle school students. I don’t get to do these as often anymore, and – other than the fact that middle schoolers seem to have gained even more energy than they once had (!) – I had a great time! It is always energizing to watch students thinking and working on problems and enjoying it as well!
I had an interesting perspective-confirming experience that day, but some background is needed. Perhaps you’ll remember a Brain Teaser called “The Horse Trader” we discussed a couple of years ago. It can be stated simply: “A man buys a horse for $60, sells it for $70, buys it back for $80, and finally sells it again for $90. How much did the man make or lose in the transactions?”. If you want to take a minute to play with this, we’ll wait for you.
I love this problem and have used it for years – you can tell how old it is by the horse prices! Not only do students of all ages like it, but they almost always come up with at least 3 different answers. Interestingly all three of these seem reasonable with the right explanation! So, it makes for good discussions – the kinds of discussions good problem solvers need to have and think about.
As usual, these students were ‘grabbed’ by the problem. They immediately dove in, sometimes on their own, sometimes with neighbors, and soon we began recording answers to discuss on the whiteboard. As I had hoped/expected, there were several proposed answers, including ‘the usual suspects’, namely made $10, made $20, and made $30.
Now, here’s the deal: Before I went, I had asked the teacher to facilitate students having access to calculators if they wanted them. I was glad to see this was fine with her.
As we know, students love calculators and will grab them even when they aren’t needed – the ‘using a cannon to kill a flea’ syndrome. Do I think this practice is sometimes foolish? Of course. Do I condemn it? No. There’s nothing like discovering ON ONE’s OWN that a calculator may not be terribly helpful in situations. It’s a great perspective to learn.
Back to the story, then. As the answers to The Horse Trader problem began to appear, I also began seeing some brand new (to me) suggestions. Two of them that I remember were “$80” and “$140”. (Just those numbers – no reference to making or losing money.)
Condensed summary: All the students gradually realized that these latter ‘solutions’ represented numbers that simply showed up on calculators and were not necessarily relevant to the question at hand, namely how much money was made. For example, if you grab a calculator, start with 60 and then enter -60, +70, -80, and +90, you end up with 80. Reflection shows this probably represents the money the horse trader ended up with but is only helpful if you take the information a step or two further.
Delightfully, this in turn, led to great questions (to each other): “Wait, what is a PROFIT?” “What do I do with this number?”
Some will quickly say this anecdote reinforces that calculators are evil. I don’t view it that way. It helped get the students thinking. Look at the questions asked, and the lessons learned about reasoning. Those are ‘basic skills’! Calculators simply cannot think, and they cannot make sense of data for you. Thinking and reasoning remain skills that students must still put in their tool bags and learn to exhibit on their own.
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