“Mathematics, in the common lay view, is a static discipline based on formulas…But outside the public view, mathematics continues to grow at a rapid rate…the guide to this growth is not calculation and formulas, but an open-ended search for pattern.” - Lynn Arthur Steen, from On the Shoulders of Giants
The more I think about this quote, the more I like it. Note that Steen isn’t saying that calculation and formulas are not part of math or that they’re not necessary. I’m sure he’d agree that they are a very necessary part of math. But what drives math, he is saying, is the open-ended search for patterns.
The open-ended search for patterns. Two things strike me about this phrase. First, growth in math, whether we’re talking about the growth of the discipline itself or the growth of students learning math (which is what I’m focusing on), occurs when students are encouraged to explore math ideas, play with them, discover the underlying concepts and then apply them in practical situations. Of course they need the calculation and formulas that Steen mentions, but if we confine students only to calculations and formulas, if we don’t encourage divergent thinking and exploration, then our students will be mechanical solvers rather than creative problem solvers.
The other thing I like is his emphasis on patterns. Math has been called the science of patterns and patterns are everywhere in math, often in unexpected places. For example, take this problem I present to third graders:
Kim has 3 hats and 3 jackets. How many different combinations of a
hat and a jacket can she wear?
The simplest way for young children to solve this is with a diagram: connect the hats and coats and then count the connections:
The answer, of course, is 9. But what’s important here, and what hooks students every time, is the discovery of a very cool visual pattern. They begin to understand that math is not a morass of discrete algorithms, procedures, rules, and formulas but, rather, a unified and integrated whole that is logically consistent and makes sense to them.
Another good example of math patterns is Pascal’s Triangle:
In this triangle, if you add the numbers in each row you get consecutive powers of 2. Each row (other than the first) is a multiple of 11. If you look at the diagonals, each one presents us with a pattern: the counting numbers or triangular numbers for example. The diagonal next to the triangular numbers is a pattern where the pattern grows by triangular numbers. And then there’s the hockey stick pattern. (See if you can find it before going online for the answer.)
Finally, type in ‘Math Doodles’ in your search engine and you’ll be rewarded with short videos that explore math patterns in a delightful way. Here’s one of my favorite sites to explore.
Mathematician in Residence