I added a new page to my Web site. It’s a visual proof that the integral of y=x^{2} from 0 to b is indeed b^{3}/3.

Some interesting things about this proof:

- It was discovered by Jacob Regenstein, a high school student.
- It does not involve any algebraic manipulation.
- It shows a dramatic example of how integration increases the degree of a polynomial by 1: we are starting with 2-D squares, and we end up with part of a 3-D cube.
- It is based on pedagogical tools that are ubiquitous in my teaching and in our department‘s program: manipulatives and technology.

This last point is particularly interesting. Some people think that such tools are the enemies of mathematical understanding, which in those people’s mind can only result from someone who knows more math explaining something to someone who knows less math. However real students in real classrooms have no choice but to construct their own understanding, and merely listening to explanations only works with very few of them. This is the overwhelming conclusion of massive amounts of research on the teaching and learning of math. (And I may add, you can experience this for yourself next time someone explains some hard math to you without giving you a chance to sort it out in your own mind, probably with a pencil in hand and with your own diagrams and calculations.)

Others object to pedagogical tools because they are “crutches” which supposedly prevent students from standing on their own theoretical feet. Certainly tools, whether manipulative or electronic, are not magical, and they can be incorporated into bad teaching in any number of ways. I have written a lot about the uses and abuses of tools — see for example For a Tool-Rich Pedagogy, and follow some of the links therein.

Used well, pedagogical tools can be a powerful part of math education. Jacob’s proof confirms some of the things I’ve said about them. To quote myself:

- They can make some challenging ideas accessible to more students, across a range of learning styles, while at the same time offering an opportunity for strong students to make connections they may otherwise miss.
- In particular, they can help bridge the visual and the symbolic, a connection that is fundamental to all mathematics and science.

But Jacob’s proof takes the argument one step further: it shows that pedagogical tools can be mathematical tools, and can make it possible for both students and teachers to do some creative mathematics.

–Henri

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