I am going to start with a question, what is this?
2 x 3 = 6
We’ll come back to that later… The first part of this post is going to be devoted to a bit of background about my journey in mathematics and where the title came from. No, you didn’t mis-read, I do think that mathematics is a language, with its own culture and literature, but seemingly inaccessible to the non-speakers. One of my earliest memories of mathematics comes from being taken aside by my class teacher and asked to explain Napier’s Bones to the Headmaster. I was around 8 or 9. I thought this was a bit strange, because the mechanical application is quite straightforward and most of the rest of the class was able to use them to multiply, and fewer of them could also divide. I didn’t know why I had been chosen to ‘explain’ to the Headmaster. So, I demonstrated. He asked me some questions, gave me a few ‘sums’ to do, and then said “Do you know why this works?” I said I did and went on to explain about the placeholder system. Now I look back, I can see why I was probably singled-out: I had grasped the concept behind the tool. It still seems perfectly natural to me, and I really don’t know what it’s like not to get the concept – wherein lies the problem or challenge!
Those of you, like me, who ‘get it’ and, more to the point, understand and enjoy teaching mathematics probably feel the same frustration as a tourist in the Brazilian rainforest trying to describe a 4G phone to an Amazonian who knows what telephones are, but doesn’t really see how they can be useful in hunting tapir… At least, if you’re anything like me you will!
Another ‘oddity’ about maths teachers is that they tend to congregate. At virtually every event I’ve been to – plays, concerts, operas, races, conferences – when I’ve got into a long conversation with someone, rather than just the social niceties and small talk, it has also turned out to be another maths teacher. Teachers do tend to gravitate towards other teachers, but it always seems that maths teachers form their own little subset. Is it because we listened to Mozart in the womb, or is it because nobody else really understands the language of maths teachers…?
There is or seems to be a thrust towards (re-)introducing problem-solving in early years rather than the so-called “drill-and-kill” methods. This may be a good thing, especially if young children are encouraged to use their creativity, imagination and natural curiosity to explore applications of number. Yes, application of number rather than mathematics, because if you believe Piaget’s developmental stage theory the ability to grasp the abstract concepts behind mathematics is probably not going to happen until the formal operational stage. Personally, I am with Vygotsky in thinking that language plays a greater rôle than Piaget assigned it in cognitive development, and my long-term research interest is in the impact the language used in teaching has on the learning of the subject.
So, back to the opening question, what is this?
2 x 3 = 6
Student response: “What? That’s a stupid question! Why are you asking this?”
Parent response: “It’s sums, isn’t it?” or “Hmm, maybe a formula or an equation, not sure…”
Maths teacher response: “Well, it depends. It is, of course, a symbolic representation and although primarily a basic arithmetic fact, you know that arithmetic is what most people mistakenly call mathematics, it has been taken out of its context and….blah, blah, blah”
I exaggerate, I hope! But if you are really honest, isn’t it a symbolic representation of:
two times three equals six,
two threes are six,
dos por tres es igual a seis,
zwei mal drei gleich sechs
deux fois trois font six
два раза три равна шесть….
I think you get the point. Maybe if we have a mindshift, as maths teachers, towards thinking of our subject as a language, the language of abstraction (perhaps the most difficult language of all to learn), with its own culture and literature, and give our students the opportunity to see where and how number is used and is useful, there will be less “phobia” and more “philia” of something we already love, know and enjoy!