Notice & Wonder, Part 2

If you missed the previous discussion, on 10/15 Steve’s freshman algebra class came up with a whole mess of observations on this problem.  On 10/19 I returned to put the same problem up on the board (to much groaning and general consternation -- “We already DID this one, Mr. McDonald!”), followed by a Powerpoint slide of all the N&Ws that I scraped from their notebooks into the previous post.

Last post, I made a tongue-in-cheek allusion to saturating the number of synapse connections you can make in a given time in the last post. I think this is a legitimate concern, and maybe just a science-ey way of saying something that’s common sense – you don’t go from ABCs to astrophysics overnight.

<pseudoscience>
My basic understanding of learning is that you are forming new synapses between neurons (brain cells), and that by repeated exposure to a new concept or way of thinking that you use and strengthen these new connections.  Given that framework (and any biology or biomed types who want to clarify this, feel free), I suspect you pretty quickly reach a limit to how many new synapses you can form in a given time.  Put simply, try to string too many neurons together too fast, and the chain breaks.
</pseudoscience>

Now, keep in mind that setting up and solving a system of equations like

Q + D = 8
0.25*Q + 0.10*D = 1.25

is really a rather smartypants and esoteric way of solving this spare change problem, especially for kids who aren’t yet skilled at seeing patterns, doing systematic trial and error, or indeed very good at abstracting mathematical concepts in general.  Mosquito, meet cannonball.  Rather than waste oxygen trying to write that sort of thing on the board, my goal is to get to a point where we are comfortable pulling information out of a word problem, testing out possible solutions, and noticing patterns and shortcuts to the answer.

With that in mind, we spent this next class period talking about all the things we noticed and wondered about the Spare Change story.  I put up the version of the problem with the question at the end,

then went through every last one of the questions they wondered about and answered them all – as it happens, Chan and Prashant are the names of two buddies of mine, so they’re college students.  Knowing Prashant, Chan probably didn’t get paid back.  Chan is nice because he’s just that kind of guy.

We briefly… discussed would be a stretch, let’s say I coerced replies from my captive audience about what types of wonderings were most likely to help answer the problem – that would be things involving money, not so much the background details like what school they went to, or how old they were.  I wanted to plant the seed of distinguishing between math and non-math details of the problem.

Finally, I spent a few moments asking what sorts of answers we could expect: would –4 be a good answer?  No?  You can’t have a negative number of quarters?  Okay, how about 9?  Well, we only had 8 coins, so maybe 2 would be better since we don’t even have 9 whole coins, much less quarters.  But the question asks how much of the money was in quarters, so that means the answer is an amount of money, not just a number of quarters.  Then I started asking them if we could have an answer of $0.20, or $.45 or $.55.  Eventually, I led the horse/class to water/the realization that the answer had to be a multiple of $0.25, since that’s how much a quarter is worth.

As a closing thought, I put up this new problem (below), had a student read it aloud for the class, and had them put their new N&Ws in their notebook for next time.