Structure of Solving Visual Pattern Problems, II

Let’s give this a first shot.

Following Lesh, let’s think about what needs to happen with a constant rate of change visual pattern in terms of the modeling cycle.

  • Description: Articulate a “way of seeing”/model that allows one to describe the growth through the given steps.
  • Prediction: Turn that “way of seeing” into a mode of calculation
  • Verification and Modification: See if the model works. Tinker.

There is important thinking, variation in strategy, and levels of sophistication at each of these stages. (Another yiddish word that we lost the chance to incorporate into the broader culture is chochmah. There is a chochma involved in each of these components.)

Description:

One way or another, the point is to identify what’s changing in each step. Here are some levels of this thinking, roughly inspired by CGI.

1. Imagining the action – At an early level of experience with these problems, it is important for students to imagine being able to “add on” the new elements of the step at each stage. Young students can be given manipulatives to act this out. This sort of thinking is, by definition, recursive.

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2. Relational model – Students begin to see growth that relates the step number to the shape of the pattern elements.

There is a ladder of sophistication that students show in their representations of a relational model. This is where, for many students on their way to algebraic models, the magic happens. At an early stage, students will fail to draw a picture that exposes their way of seeing. They might be thinking about the given stages of the pattern with an “acting it out” model and not making the jump into a relational perspective.

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Other students will draw out fully specified sketches of the “jump ahead” step.

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Other students make good use of a “block” model that doesn’t draw out each individual object, but instead exposes the groupings that they see in the pattern.

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Even students with fairly sophisticated representations are not sensitive to under- and over-counting in their models, counting in full groups of objects that “naturally” fall together in rows, columns, lines or other familiar shapes.

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With time, though, students become sensitive to compensating or overcounting and develop ways of adjusting their models.

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3. Algebraic relation – Students, with experience, will begin to selectively drop their visual models when determining “jump ahead” steps. They’ll sometimes very quickly abstract an algebraic model.

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At times, students will attempt this and result in error.

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There’s more to do, of course, since we haven’t talked about prediction or verification. But let’s identify in a nice list all the models above.

  • Acting it out
  • Relational Model – Detailed drawing of the step
  • Relational Model – Schematic drawing of the step
  • Relational Model – Unsensitive to over- or under-compensating
  • Relational Model – Sensitive to over- or under-compensating
  • Arithmetic Model 
  • Algebraic Model

I’ll end this drafty post by trying to draw out some of the feedback/hints/goals that each of these models suggests.

Goal come quickly: use of each of these models would serve as a perfectly sensible mathematical goal for a group of students that hasn’t attained them or their more sophisticated descendants. If my class has never worked with patterns before, maybe my goal is for them to solve visual patterns using, at least, an acting it out model. If I’m working with students that use relational models pretty well, maybe my goal for the day is for them to be able to use arithmetic models of the pattern.

Feedback and hints can help with this, and our articulation of the models can help us come up with sensible hints and feedback.

  • Acting it out

“Can you act this out with some bricks?”

“How would you build this?”

“Can you imagine building this next step with your hands?”

“Which blocks were added on?”

“If you’re not sure how a pattern is growing, imagine yourself building each step. Maybe you’ll realize something new.”

  • Relational Model – Detailed drawing of the step

“Show me a picture of this step.”

“Why does it make sense that this row is 5 long and 4 up?”

“What instructions could you write someone for drawing out this step?”

“Drawing a picture sometimes helps us get clearer about how we’re seeing the pattern.”

  • Relational Model – Schematic drawing of the step

“Do you have to draw each box?”

“Why does it make sense that you counted 10 in this row?”

“If you’ve got a big step, it can take a long time and be hard to count all those pieces. That’s when you might try making a drawing that doesn’t count each little thing.”

  • Relational Model – Sensitive to over- or under-compensating

This gets sort of close to verification and that’s a whole other set of thinking tools. I want to focus just on the model, though, and getting to this way of seeing a pattern.

“Why doesn’t it make sense to count this square twice?”

“When we’re working with patterns, we sometimes have to worry about counting things too many times. Can you find the part of your diagram that you counted twice?”

  • Arithmetic Model 

“What about the 89th step? Or the 103rd? Could you figure those out too?”

“Why does it make sense that this turned out to be 10+10 – 1?”

“If you feel that you understand a pattern really well, it’s often helpful to look for ways of calculating that only use numbers and not pictures, because that gets us closer to finding a rule that always works. Can you try writing this in terms of only numbers?” (I feel somewhat weird about this hint, but I think that it would be entirely appropriate, depending on our mathematical goals for the student.)

  • Algebraic Model

“What about the millionth step?”

“Can you write some instructions that anybody could use to calculate the number of squares in a step?”

“When you’re looking for an algebraic rule for a pattern, it’s often helpful to try calculating a few “jump ahead” steps, because we can often just tinker our arithmetic and end up with a rule. Try to find a rule that always work using this strategy.”

Like I said, first draft. I feel this is promising, but has issues that will become easier to notice on a reread.

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