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.

Notice & Wonder

Here's an activity I did on 10/15 with my algebra class. Note that everyone in the class has a class notebook, so I had them draw a line down the middle and make two columns, writing "I notice..." at the top of one column and "I wonder..." at the top of the other. Then, I put this problem on the board, titled “Spare Change”.

Now, take a moment and ask yourself what I asked the students – what do you notice, and what do you wonder? Take note, as I told them, that there is no wrong answer here – there’s not even a question! We’re just after what you see, and what you think could use some clarification. Here’s what I, as a grad student in the engineering and physical science fields, would say:

I (the teacher) notice:

• Prashant needs $1.25
• Chan has only quarters and dimes
• Chan has 8 coins total
• This is exactly enough money for Prashant
• (Dimes are worth $0.10)
• (Quarters are worth $0.25)

I (the teacher) wonder:

• Is there enough information to find the number of quarters and dimes?
• Is there even a solution?
• Could there be more than one solution?
• Are there amounts of money where quarters and dimes couldn’t add up right?
• What if Chan had nickels too?

Of course, I would just about have a heart attack if a student in freshman algebra, still struggling with the very idea of word problems, wrote all this, but then that’s the point isn’t it? I gave everyone about three or four minutes to write down their noticing and wondering (hereafter, N&W), then I tried (operative word here!) to have a discussion, asking everyone to volunteer their N&Ws and making a class list. This was a big long session of what I like to call crickets, because everyone was silent until I started conscripting volunteers by calling on them.

Still, since I only ended up with about three or four N&Ws each, I got sneaky and went through their notebooks after class. I think there’s a pun involving shy and spy floating around here, but I can’t quite find the words. Anyway, below is a transcription of everything the students wrote in their class writing notebooks:

I (the class) notice:

• Where it says any it means the same thing as nothing
• You do laundry with quarters
• $1.25 is what I pay for lunch
• That it’s lunch time
• They’re at lunch
• It’s talking about money
• It’s money for lunch
• It’s a money problem
• Prashant needs $1.25
• Prashant has no money
• Chan has 8 coins
• It doesn’t say how much money Chan has in coins
• Prashant has to ask for money
• Chan only has quarters and dimes
• Chan has exactly enough money
• $1.25 / 8 = $ 0.156

I (the class) wonder:

• Who’s Prashant?
• What grade are they in?
• Why are there coins on the paper?
• How much money does Prashant need for lunch?
• What school do they go to?
• Did Prashant ever pay Chan back?
• How many quarters and dimes  each did Chan have?
• What did Prashant buy?
• How much money in coins does Chan have?
• Is he going to have enough money?
• Who’s Chan?

There is some overlap between the two lists, since one person may well have noticed the answer to another one’s wondering, but to the extent that we can imagine a class consciousness, this is a decent snapshot of that state of mind.

Look at how many stray, random, irrelevant thoughts are in there! Two thoughts on this. First, I purposefully gave some really broad, vague instructions to make sure everyone wrote something, along the lines of “Write down everything you think is important” for the noticing, and “Write down anything you don’t understand or you’re curious about” for the wondering. I only obliquely mentioned that they might try to find math-related things to notice or wonder, and I definitely didn’t tell them to try to solve the problem.

My second thought is, well, this is what your brain looks like before you’ve learned to filter out the important information in a word problem. After 15-odd years of doing math of some sort or other, I the teacher and probably you the reader are a point where we instinctively filtered most of that irrelevant stuff out. Indeed, my wonderings were pretty meta- in nature, abstracting the problem to the point of wondering about uniqueness and degeneracy of solutions (bonus points if you know what I mean by degenerate here!).

Most of the key information of the problem has been observed, at least by the class group-mind, though only a few students had all those key pieces assembled on their own, and fewer still deduced the answer to the unstated question – 3 quarters and 5 dimes. Before we’re ready to solve, we need to learn to filter all that extra stuff out.

But there’s a limit to how many new connections between neuron synapses you’re going to make in any given class, or any given blog post, so until next time, ciao!

Postscript: This post is my first made using Windows Live Writer, a free blogging composition program that can upload to Blogger.  The interface is WAY nicer than Blogger’s default editor, and includes handy things like full-screen composition and preview windows, real-time updated wordcounts and  much easier link and media content insertion.