## Thursday, September 23, 2010

### For an apple and I'll let you paint this fence

So this'll be fun.  If you were to write a 10-minute-ish individual assessment on pattern sniffing for...ah, you pick the age (or tell me it doesn't matter)...what would it look like?

Here's my most recent definition of pattern sniffing:

• On the lookout for patterns
"Ok.  We’ve been working on this staircase problem and it seems that you can’t write powers of two as a sum of consecutive whole numbers."
• Looking for and creating shortcuts
"It would be nice if there were a faster way to find the greatest common factor of 2 numbers other than listing all the factors. Think we can find a way?”
Challenge question: How would you grade this little quiz of yours?

EDIT:  I should be clear.  The end goal is for students to be able to apply these habits in unfamiliar problems without a big neon sign saying *Use Pattern Sniffing*.  This is in no way an assessment of that end goal, but the first step in what I suspect will be a relatively long road.

1. What's the 10th number in this sequence?

1,2,4, ...

Give as many different answers as you can, and an explanation for each.

(Questions about Cuisenaire rods and trains are also very useful here as ones that lead to natural pattern-huntin'.)

2. The key part is "on the lookout for patterns." I'd start with that.

Give the students a whole bunch of..stuff..to look at.

I don't know the term to use here, in science we'd say phenomena or observations, "problems" has the wrong connotation for what I picture. Number sequences, like Bowen has. Or geometric shapes. Or figures. Data tables. Graphs. Pictures. Standard problems as well, like the how many rectangles on a chessboard one that Dan Finkel had on his blog.

I don't know. All sorts of weird things that are open to questions that might be approached in a mathematical way. Sorry if this is unclear, I don't have the vocab for all this math talk.

Then I think a question like, "For which of these things would looking for a pattern be a good way to start? Explain your reasoning."

I'm not even sure "explain your reasoning" is necessary here since all of the answers will likely be "it kinda looked like there might be a pattern" and that's pretty much what you're going for anyway.

As you develop more of the habits through the year the question would evolve to "Which habit/strategy would be the best starting point?" or something like that.

This would give you a lot of good info and wouldn't scream "hey, this is a pattern!"

3. "Which of these might be solved/described mathematically?" is also a good question all by itself.

4. I feel like specific open-ended problems lend themselves really well to pattern sniffing. For example, you could show students Pascal's Triangle and ask them to work together to find interesting characteristics about it. What I love about Pascal's Triangle is that the patterns are so varied (symmetry, powers of 11, triangular numbers (and n-angular numbers), hockey stick numbers, combinatorics, fractals, even the Fibonacci Sequence can be derived from Pascal). They can be simple things to notice immediately or patterns which must be fleshed out with an advanced understanding of mathematics.

To assess this, I would have the students write about their line of thinking: anything they knew about Pascal's Triangle before starting, how they began with the problem, the different ways they looked at the problem, the interesting things they noticed, how they knew what they noticed was true, etc.

5. Actually I think a big neon *Use Pattern Sniffing* sign could work... put it at the front of the class. Eventually retinal burn will cause them to see this phrase at all times.

6. A good one is: how many squares can you find on a chessboard (or a 4 by 4 chessboard, if you want to keep it a little simpler, technically).

I do like the open ended ones for creativity, and for generating patterns. This is a good one for sniffing out existing patterns. I also think it would be pretty straightforward to evaluate... did the student organize their thinking? Did they start with a smaller case and build up? Did they find patterns? Could they the patterns they found to arrive at the answer?

Math for Love