Maths is the Science of Pattern

From the earliest experiences we have with maths the role of pattern plays a key role in how we learn and remember maths facts, concepts and relationships. The pattern of the number names as well as the numerals must be learned so that we can label and communicate our mathematical ideas in the same way that letter recognition and pronunciation is a prerequisite for literacy.

Even the idea of rational counting is a pattern where one more is added to get the next number or quantity. This can be extended to tens, hundreds, thousands and so on just by changing the referent being counted.

Counting by 2s and 5s is another  pattern that helps us learn and remember numerical relationships. Try counting by 5s starting at 3, instead of 0, and see what happens. The further you go the easier it gets because you very quickly see the pattern of 3s and 8s. The same can be done with fractions although it is clearly more difficult. Visualize half of a half of something. You should be seeing a quarter. Procedurally you have double the denominator to make one half into one quarter. So what is half of a third? A sixth? You double the 3 to get 6. Now you can find a half of any fraction without having to do the desperately miserable fraction multiplication algorithm. Try one third of a quarter. A twelfth, right?  You multiplied the denominator by 3. Now you can find a third of any fraction, even a third of three quarters; three twelfths. But that one is easier to do by dividing the numerator, 3, by 3 to get 1. A third of three quarters is one quarter. A third of 3 horses is 1 horse.

All of which brings us to the picture of the Sierpinski triangle fractal. Fractals are wonderful examples of patterns in maths. This triangle will go on for ever like all fractals. Look at the three pennies at the top. This group of three pennies is repeated over and over again to make bigger triangles. Here's a link to the wonderful Fractal Foundation site where pattern reigns supreme.       

What is Maths?

For the past several year I have defined maths as the science of pattern and the art of making sense. This definition has probably developed as a response to the still somewhat prevalent view that maths, at least at the elementary school level, is the study of numbers.

I have tried to operationalize this idea by having student look at maths from many different angles especially from the aesthetic and relational points of view. I have included the study of fractals in my courses because I feel they epitomize the idea of numerical patterns and relationships which are so important as we try to get children to remember concepts, ideas and factual information.

I have read much by Jo Boaler and other illuminaries who believe that maths should be taught the way I believe it should be but I have never read anything quite as clear and eloquent as this piece by John Seibert of the Math learning Center, publisher of the wonderful Bridges Math Program.

The analogy of  defining maths as the study of numbers and comparing it with literature as the study of the alphabet is absolutely brilliant in its simplicity and absurdity. Numbers, like letter, are simply some of the things we use to convey meaning and communicate with.  And this sentence, "Perhaps it is through the lens of patterns that math can transcend the procedures that have come to define it" is exactly what we should be helping young children do.  Seibert then concludes with this masterstroke, "After all, when we teach our students their ABCs we expect them to one day write their own papers and poems. In teaching our students their 123s, shouldn’t we support the same creativity?".