(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?
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.
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.)