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.
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2 comments:
Mike,
Some of the students notice and wonder items seem more like free association than brainstorming. I think you want the students to notice and wonder about the information in the word problem. One suggestion is that you and your faculty affiliate role play how this could be done. Perhaps a student could play the role of posing the problem. You and your faculty affiliate could then come up with notice and wonder lists. Perhaps you could even ask the class to define the problem and then determine which of your notice and wonder items are most useful in solving the problem. This would help the students understand the logic of the process.
Another idea is to use the items the students generated. Step back and ask if they can define the problem. Once they agree on the problem, ask students to try to solve the problem either alone or with a partner. Check around to find students with the correct answer. Ask them which items were useful to help them solve the problem. Have them explain why they were useful. If no one can solve the problem, go back to the notice and wonder process and ask them to generate more notice and wonder items with the idea of ferreting out notice and wonders that will help them solve the problem.
I personally favor modeling for students the process you hope they will use. This helps them learn your expectations and focus. Let me know if any of these ideas work.
Carol Cramer
Carol,
Indeed, you've hit things on the mark. At this stage, the N&Ws are indeed more stream-of-consciousness than anything else. However, that's to be expected and even part of the design at this stage.
Keep in mind that I only have about 10-15 minutes on any given day to work on such a problem. I'm working more on a timeline of 2 weeks to work on this, which at about 4 sessions of 10-15 minutes (if I'm lucky and no tests are in the way) is roughly one class session per problem.
Also, at this point I'd only been in the room perhaps twice, so I wanted to a) gauge student level of mathematical sophistication, b) start fostering an environment where it's okay not to know the answer (achieved here by making sure there wasn't even a question), and c) get a wide variety of responses to facilitate precisely the discussion you alluded to, distinguishing the relevant details of a problem.
I'll be posting in the next couple of days on the 3-4 sessions since this one, and indeed there is progress in the direction you've outlined.
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