Knots for Tots

My son told me yesterday that math in school is boring, but math with me is fun. “I just don’t like doing all of those addition problems,” he told me. What could I say? I agree completely. I know that it is important for kids to practice doing addition and other basic arithmetic at some point, but…I really don’t like the way it is done in schools. I hated it as a kid. Everyone hates it. So I told him that sometimes we have to practice, even though that’s not what math is about, “like doing scales on your violin.” To which he replied “what are scales? Stop confusing me, Dad.” Oh, yeah; my wife doesn’t have him play scales on the violin. Why not? Well, it’s boring and not really what playing the violin is all about, so we don’t want that to be his first introduction to it. He’s not ready for scales anyway.

Which brings us back to the math.

So what is it that he likes to do with me? “Drawing knots,” is what he told me he likes. Here are some examples:

But honestly what is the value of having a kindergartener draw knots? Could drawing knots be anywhere as valuable as practicing addition? After all, it’s not like he’s actually learning any knot theory, is he? He’s certainly not learning any math! Or is he?

According to Keith Devlin (see Devlin’s Angle, April 2005) there are nine mental capacities that humans use when doing mathematics. They are (and I’m quoting here):

1. Number sense. This includes, for instance, the ability to recognize the difference between one object, a collection of two objects, and a collection of three objects, and to recognize that a collection of three objects has more members than a collection of two. Number sense is not something we learn. Child psychologists have demonstrated conclusively that we are all born with number sense.

2. Numerical ability. This involves counting and understanding numbers as abstract entities. Early methods of counting, by making notches in sticks or bones, go back at least 30,000 years. The Sumerians are the first people we know of who used abstract numbers: Between 8000 and 3000 B.C. they inscribed symbols for numbers on clay tablets.

3. Spatial-reasoning ability. This includes the ability to recognize shapes and to judge distances, both of which have obvious survival value for many animals.

4. A sense of cause and effect. Much of mathematics depends on “if this, then that” reasoning, an abstract form of thinking about causes and their effects.

5. The ability to construct and follow a causal chain of facts or events. A mathematical proof is a highly abstract version of a causal chain of facts.

6. Algorithmic ability. This is an abstract version of the fifth ability on this list.

7. The ability to handle abstraction. Humans developed the capacity to think about abstract entities along with our acquisition of language, between 75,000 and 200,000 years ago.

8. Logical-reasoning ability. The ability to construct and follow a step-by-step logical argument is another abstract version of item 5.

9. Relational-reasoning ability. This involves recognizing how things (and people) are related to each other, and being able to reason about those relationships. Much of mathematics deals with relationships among abstract objects.

Devlin is certainly recognized as an expert in this field and all of these capacities certainly make sense to me. Still, I’m not sure that he got all of them. Putting my doubt aside, let’s think about whether drawing knots helps my son develop any of these abilities. Glancing over the list, it occurs to me that drawing knots helps develop 3. Spatial reasoning ability and 9. Relational-reasoning ability. After all knots are curves or strings in space and the essence of a knot is how it relates to itself and the space around it.

And it doesn’t have to be knots. My daughter prefers to draw maps and tell stories about them. This develops even more of the abilities. Here is one of her maps:

Now, how about those sheets of addition… which of the nine capacities does that develop? You may be tempted to think that doing mass amounts of addition develops 1. number sense and 2. numerical ability. However, remember that number sense is something that we (and at least some other animals) are born with. It is not developed. Doing addition problems may help with 2. numerical ability, but I tend to think that these addition sheets are less about understanding numbers as abstract objects and more about honing one’s capacity as a calculator. Being an efficient calculator does not make you a good mathematician.

How about you? Have you tried drawing any knots lately? Sit down and do it. You might discover some things. For instance, it’s not so easy to draw knots! Even for knot theorists it can be very difficult. In fact I believe that tying knots is a much more natural and easy activity than drawing knots. When you are drawing a knot you have to think ahead: “oh, wait I am going to want to cross over this strand later.” Tying a knot is much more of a step-by-step-”go here then go under there and around here”-process. That’s why I usually like to draw my knots without any crossings first and then go back and trace over my initial drawing on another sheet of paper, putting the crossings in as I go, like my daughter is doing here:

So go ahead and draw some knots just for fun. Don’t be intimidated. Draw some knots and then crumple them up before anyone else sees, but draw some knots. It’s fun, lots more fun than addition.