Visual Patterns – Who Needs Them?

(An excerpt from this essay.)

Visual patterns – who needs them?  After all, very little in the world comes in the form of a neat little sequence of growing Tetris pieces. (A growing doodle, perhaps.  Windows of a rising building. Towers of children’s blocks. Apples, being laid out for display.)

Far more common in school than visual patterns are patterns that show themselves through numbers, graphs, or tables.  The L-Shape pattern that appears above could easily been presented in any of these three other forms. These other forms are more common, flexible and useful. Why bother with all this picture-pattern stuff?

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I see three types of thinking about visual patterns: recursive, relational and functional thinking. Relational thinking – that connecting of the step and a dimension of the picture – is not available when the pattern is presented numerically, or in a table or a graph. Relational thinking is this perspective that is only useful for visual patterns. It’s what makes visual patterns different.

(Don’t graphs allow for special, graphical ways of finding a step in a pattern? Graphical patterns are different, too.)

In a sense, visual patterns are easier for students than other representations of patterns. I see this most often when my students work with non-linear visual patterns. Recursive and functional thinking often doesn’t occur to them. Relational thinking, on the other hand, eventually occurs to many of my young students, and they’ll use this to make sense of patterns that would otherwise be inaccessible.

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Relational thinking is great, but it’s not broadly useful. The most powerful perspective on a pattern is functional thinking, the holy grail of many a high school course. It’s the sort of thinking that helps an expert quickly look at a pattern and make careful predictions about any step in the sequence. Many students don’t get there, though. The journey from recursive to functional thinking can be rocky. It’s hard for a lot of kids.

Relational thinking can only really be applied when the pattern is presented in a visual form. It’s certainly beautiful, but it’s not broadly useful all on its own. To the extent that relational thinking isn’t just beautiful, but also useful, it’s because relational thinking can help students gain this hard-to-obtain functional perspective. The important question, then, is how do students develop a functional perspective out of a relational one?

(For more, read here.)

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Essay: On Visual Patterns and Feedback

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can you find a pattern in every direction?

Last summer I wrote an essay about how feedback and the math that visual pattern problems can help students learn.

Looking back, I don’t think this essay ever worked entirely, as a piece of writing.As my initial excitement about the piece soured, I never got around to giving it the big edit that it needed. Still, there are some good ideas in there that it helped me to figure out.

Here’s the essay: On Visual Patterns and Feedback

Here’s an excerpt:

I knew what I wanted to help Toni see. She was looking for a pattern in the growth, but she was having trouble getting specific about it. I wanted to ask a question that would draw Toni’s attention to helpful features of the pattern’s growth and help her get specific about precisely how this shape is changing.

This would involve a bit of guessing on my part, though, since I didn’t really know what question would work!

My first question was a promising dud: “Can you see the previous step in the following step?”

To which Toni responded, “no.”

I tried again, this time directing her attention more directly: “Do you see the second picture in the third? Imagine that you were building the third picture from the second. Where would you put the extra bricks?”

Bingo. She grabbed her pencil and started sketching.

Why did that question work? I think it’s because it encouraged Toni to see the static picture on the page as a changing thing. Toni had lots of experience playing with blocks and adding on parts to existing doodles. By asking her to think of one picture in the next, I helped direct her thinking to this analogy, and she was able to see the pattern’s growth in a useful way that related to things she had lots of experience with.

Like I said, an interesting failure. Enjoy! Let me know if you find parts of this useful.

Applicable to Writing about Teaching Too?

In this instance, as in others we observed in this group, the conversational routine involved the following: (a) normalizing a problem of practice, (b) further specifying the problem, (c) revising the account of the problem (its nature and possible causes), and (d) generalizing to principles of teaching. Through a routine of normalizing, specifying, revising, and generalizing, they created an interactional space rich with opportunities to learn about teaching practice.

From: Horn and Little, Attending to Problems of Practice: Routines and Resources for Professional Learning in Teachers’ Workplace Interactions