I'm not experienced in university maths (I did science), but from what I do know I want to say that there is a continuity, and not just out of respect for the efforts of young children.
Of course, Colin is correct: there are an awful lot of sums in schools, hopefully less than there were when we were at school, but still, many people do think that sums are the thing. And it must get boring getting thought of as a difficult sum doer!
But, but... there is another side to school maths, and one that I would like to see take over as people's picture of school maths.
Take this Kindergarten Interlude that Joe Schwartz posted on the other day.
Here are children in a comfortable situation, getting to know number, its constancy, its patternability, its many possibilities. I like how Joe says, he is there to "poke, push and experiment".
Now this is well-trodden ground, but children are building up their knowledge here, step by step, tentatively at times, in leaps at other times. Sometimes the knowledge will be certain: however you arrange three bears, they will always be three. Other times it will be a hunch: you need four bears before you can have a pattern.
Or take Kristin's post on Articulating Claims in Math, The elementary kids are doing "sums" here, but they're doing a lot more, they're making generalisations about the way sums work:
Now you might argue that proof involves more than this. That it involves formal generalised description. Or that it must be a shared and socially verified knowledge.
But I would say the exploration comes first, then the hunches, then more exploration, then the knowledge, then the sharing.
Consider Polya's treatment of Nicomachus's theorem. How would you first discover this? Polya writes:
Mathematics presented with rigor is a systematic deductive science, but mathematics in the making is an experimental inductive science.
And that is what is happening in many excellent maths classrooms, and could be happening in even more. I like Polya's example, not least because the nine year olds in my class had a go at exploring this last year.
A different kind of activity comes before proof. You get a sense of it in this video:
And proof is not about formal language either. There's the famous Bhaskara proof:
|Bhaskara's only word: "See!"|
So, I want to make the case, here as I do in my teaching, that there is a continuity, there is in the best classrooms a relationship with what kids are doing and what mathematicians do.
What do you think?