Our visit to the Arithmeum got me thinking. Sometimes we are so used to the number system that we use all the time that we forget how new it is. And our education system (both as a set of institutions called schools, and as a set of practices for helping people learn things) is so focused on memorizing math facts and being able to use them in calculations that we lose sight of the fact that the algorithms we use for calculations are merely short cuts for what can be accomplished with a number line.
In fact, although we sometimes begin teaching about addition and subtraction with a number line, it is often seen as a sign of weakness to continue to use one. This is one of the beliefs that those with right-brain-dominant children have to learn to overcome. And hanging out with some of these folks online has given me new insights into how we can do math. I typed up the multiplication tables for Tigger and printed them on a 5” x 8” index card that I let her refer to when she works on her math problems. It seems to me that the concepts and the strategies for accomplishing multiple digit multiplication and division are much more important than the memorization of the multiplication facts. (And by using them, she remembers more and more.)
To return to the Arithmeum and what it got me thinking about, I started to think about how we came to have a base-10 system and what that meant. When we learned about bases in school, I never quite got it, but all of a sudden it came to me. All “base-10” means is that we have 10 symbols in our system of numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). And it is easy to forget that they are only symbols. Some of the historic number systems discussed at the Arithmeum included one which had 64 symbols and another with 16. I can’t even begin to imagine how you calculate in a base 64 system. (Actually, before the geeks chime in, I can begin to imagine, because 64 is a multiple of 2.)
So the other day, Tigger and I sat down and we talked about bases. Instead of focusing on converting things from other bases into base 10 (which is what I recall about how I learned them), we talked about how a number system with a different base would work. I began by pointing out that base 10 merely meant that we had 10 symbols. I also reminded her of the importance of a concept of 0 and a system of place value for doing calculations on paper, as we had learned at the museum. We then went on to experiment with other bases.
I started with base-5 (a number I chose randomly) and I made up a number line — 0, ∆, , ◊, ♥. We counted and filled in our number line up to a few 3-digit numbers. We made sure we had a zero because we had learned how important that was. Then we did some addition by jumping along the number line (just like you do when first teaching children to add). Once we could do simple addition like that, I created a few multi-digit problems and Tigger used her normal strategies for adding multi-digit numbers (with redistribution) to solve them. She was fascinated. Knowing the math facts is absolutely not an issue here, and she was amazed by the fact that as long as you have a concept of zero, you can make up your own number system and do all the things we do with our base-10 system.
My purpose in using completely different symbols was to circumvent confusion with the numbers that she knows. When you are looking at a 10 it is hard to think that it might mean what we normally think of as 6 objects (as it does in base-5). My strategy worked and once she had grasped the concept, it was possible to use the same symbols that we normally use without confusion. We moved on to a base-16 (using A, B, C, D, E, F to extend the number of symbols). And then I showed her binary (base-2).
She had a great time. I think that playing with number systems like this can be engaging for kids, whether right-brain-dominant or not. The concepts behind the arithmetic become clearer as you work out how to use a number line to perform calculations. And the concept of place value can be learned through manipulation rather than accepted at face value and memorized. This conceptual learning can then be transferred back to the base-10 system we normally work in, giving children conceptual tools that they can use even when they forget the math facts and algorithms. For right-brain-dominant kids (and I’m not convinced Tigger fits that description though many strategies I’ve learned from those whose kids are much more right-brained work well with her), this could be really exciting, playing to their strengths in conceptual and visual thinking and not really requiring the memorization of math facts that they often struggle with.
After playing with bases in this way, I noticed that one of the chapters in Zaccharo’s Becoming a Problem Solving Genius was about bases. We had a look the following day. Although it started with the idea that Mrs. Mouse was writing a science fiction book in which the people of her fictional world worked in base 7, the explanations then went quickly into converting between base 7 and base 10 (something it seems to me those fictional people would only need to do to communicate with us). We worked through some of the explanation and I watched Tigger get less and less excited about something that had made her quite happy the day before. We stopped.
And then we talked a bit about the different ways of presenting the concept of bases. And we played with base 7, which had been introduced in Zaccharo, but in the way we had played with other bases the day before. We made a number line. We did some multi-digit addition and subtraction. Then we constructed a multiplication table together. And did some multiplication. She texted* her uncle: “26 x 21 = 606?” (He’s a mathematician, but he got off on the wrong track and they went back and forth a bit. Finally she told him that it was base-7. He called her a cheat but was also impressed. Then he told her she should learn more about base-16 and base-2 which he uses all the time in his work.)
That evening, after a day walking along the Rhine and hanging out in Starbucks (which is where we go online) she sat down and worked out base-6 while her dad cooked dinner. As if it was just a fun puzzle to play with. And that is how math ought to be. I’m sure we’ll get to conversion between different bases at some point. But it seems much more fun to just play around with making up number systems and doing calculations in them as if we live on some fictional planet where they count in 7s. She’s 10. There is plenty of time for useful later.
(And then this morning, she just decided to make times tables for the bases she had. We’re in the middle of base-16.)
*Do North American’s say that? Or do they say “SMSed”? Leaving aside the whole question of whether text/SMS messaging is the downfall of literate society, of course.