Vi Hart‘s good for lots of things mathematical. I thoroughly enjoyed her deviously recursive video called, “How to Make a Video About How to Make a Video…” Even better was her recent video, “They Became What They Beheld:,” in which she discusses who she creates for and why. In it, Vi reads a fantastic quote by Edmund Snow Carpenter that I wanted to share.
“The trouble with knowing what to say and saying it clearly & fully, is that clear speaking is generally obsolete thinking. Clear statement is like an art object: it is the afterlife of the process which called it into being.”
This is why it’s so important to do real mathematics with young people. The process generates thinking. Hearing the foolproof method and mastering the technique is not enough, because it merely replicates the fait accompli. It’s the creation of the algorithm, the invention of the problem, and the mathematical process that need to be replicated, because these habits fuel creative acts. So which is the math? The object or the process?
“The problem with full statement is that it doesn’t involve: it leaves no room for participation: it’s addressed to consumer, not co-producer.”
There it is exactly. If the goal of math class is to get this stuff into those heads, then who cares about having co-producer? Is anything even to be produced? My job, as I see it, is to turn the math on in the brains of these kids. If I do it right, they come alive. They interact. They create. They control. THAT is participation. And the best part is, it infects your brain. Doing math affects how you see things, what occurs to you, what decisions you make.
See, this is what we work on in school. This is what the students and I co-produce — their minds. In math class, we make new mathematicians.
Paul Salomon
Lost In Recursion 
I loved both videos. And this post.
Thanks, Tariq!
This is an excellent point, and one I generally agree with. For the most part, the thinking is the thing when it comes to learning mathematics, and students get there by doing, not merely by observing..
That being said, while Carpenter’s idea resonates, the extension of the idea (in the video and here) fails to capture something important: even if we are most interested in the process, the final product is still of great value.
Artists and musicians draw inspiration, both broad and technical, by interacting with “finished” pieces; this, too, is a valuable learning experience. I think the same is true for students and teachers of mathematics.
I definitely agree! We should be working towards somewhat polished finished products. These can be pieces of mathematics, mathematical objects, technical mastery, etc. But all of these things lose value when isolated from the thinking and creative process.
Great post, Paul. I wish we worked together. In response to MrHonner’s comment above about products, I am inclined to believe that “doing mathematics” will inevitably result in students creating/producing something. Sometimes this is a polished algorithm and perhaps other times a murky ending leading to new questions. The issue, I suspect, is that most of the time people interpret outcome/product/creation as something that must be in line with content standards. I continue to think that their value is overrated and that by setting them as an end destination we are unavoidably missing opportunities to pursue student driven questions and other fascinating, engaging investigations. Thanks for the post.
Thanks for the comment, Bryan. I would LOVE to work with you. I have much admiration for you and your thoughts. Sometimes I feel like our thoughts are eerily in sync.
Sorry I’m a little late to this post, Paul, but to me it’s still fresh with affirmation of what I’ve always wanted teaching and learning mathematics to be. I agree with both Mr.Honner and Bryan — product is of value, but it should also be a natural student-driven-and-teacher-guided/scaffolded precipitation of doing mathematics. If there’s room for collaboration with you and Bryan, I’d love to squeeze in.
Thanks, Paul!
Right on, Fawn!
Good point. Some teachers are so excellent in explaining that you thought you understood everything he said. It was only during the exam that you realized that you understood very little.
Exactly! Thanks for your comment.
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I suggest you look at Veritasium on YouTube:
http://www.youtube.com/user/1veritasium
What I like about his process is how he takes the interviewees explanations and tests them. I’d like to see mathematics taught this way, and see if it helps people understand it better.
I don’t think I’m explaining myself clearly, probably because I haven’t thought about it enough to get at the essence of it. Something about how he interacts with people makes the process of learning more engaging. I feel that we can translate this interactivity to mathematics. In science, you’re studying a phenomenon and trying to tease apart its principles. When I learn a new topic in math, I learn it best be treating it like a phenomenon, then trying to tease apart its principles by experimenting with it.
I know veritassium some. Good channel. I think I see some of my classroom practice in what you’re saying. When students offer up a conjecture or idea for a solution, I stay away from telling them yes/no, as if I were the arbiter of truth. Instead, I prod at their ideas with questions and see how they hold up. Often that’s stress testing with wild cases, or even some of the mundane ones. Deciding together what’s true or false definitely keeps things active and demands engagement.