This Post inspired me to blog; I had a lot I wanted to get down and clarify so I decided not to clutter up the comments. I also don’t have anything to add to the lovely arguments laid out in the comments (which you should totally read btw) but the whole post made me think of the battle that often goes on in a lot of students heads:
Intuition vs. Mathematics¹
Intuition is really important in mathematics; it is the intrinsic value that makes us spot patterns and make leaps when we solve problems. It can also cause us to struggle however, as we encounter many aspects of mathematics that break down these intuitions and replace them with a structure make more mathematical intuition from. If intuition is the hypothetical leap, then mathematical rigor is the platform for higher jumps.
Perhaps an example or two will help people follow what the hell is going on in my mind:
- Infinity: people actually have a good grasp of infinity; several young students have told me, when asked about the ‘biggest’ number that “you can’t have one, because you could just add one to it and be bigger”, but their intuition falls down that it can be treated like a number (infy – infy = ??) and that there is more than one infinity (infy many in fact)
- Continuity/Differentiability/Integrable: Mathematcians struggled when Weierstrass came up with a function that was continuous everywhere but differentiable nowhere; intuition seems to suggest that if it is continuous it shuold have a gradient function
- Koch Snowflake: [or indeed any of many fractals] The idea of a finite area bounded by an infinite perimeter blew the minds of my AS Level students when we studied this in sequences; beautifully expressed by one girl “Sure it makes mathematical sense, but it doesn’t make real sense”
- Normal Distribution: The area under the curve is 1. But the curve never touches the axis. But the area is finite. And it looks wrong. Shouldn’t it be infinite?
It occurs to me that a these examples are all really based around the idea of a limit, but I’m sure that there are some out there that I haven’t thought of that illustrate this idea better.
So what has this got to do with anything? Well the more I think about it, the more I am convinced (intuition right?) that when students encounter mathematics that challenges thier intuition, unless they trust the mathematics and allow it to change their intuition there will always be an issue with that piece of mathematics To use an example from above, some students in my class have real difficulty remembering that the total area under the standard normal distribution is 1. Is this because they don’t really believe that it is? To them, does it make mathematical sense but not real sense. Most importantly what can I do about it?
Answers, questions, examples and arguments in comments, on twitter or by carrier pigeon.
¹: If I had graphics skills this would have been a movie poster