Here is a video where some kids empty a cylinder of popcorn into exactly three cones with the same height and radius, demonstrating that the volume of the cone is one third the cylinder’s.
Here is an example problem from a textbook, where students use that relationship to find a cone’s volume.
One reason why people dislike math.
These are two ways of knowing the same thing—one concrete, the other abstract. One you can hold in your hand. The other you can only hold in your head.
They are both valuable in different ways, but the explicit goal of math class is to move students away from the concrete to the abstract, from the stuff you can hold to the stuff you can count to the stuff you can represent with a number and then a variable and then functions of variables. That’s the direction of math learning in every formal math education system I’m aware of.
There are people in this world who see straight through the corn in the textbook. They look at the problem and don’t think of the kernels you hold in your hand. They see a cone, its diameter, its height, the question, and they know what they’re going to do. These people make up a small percentage of the world’s population and a large percentage of the world’s math professionals.
A much larger group of people, one which includes most of the world’s math students tends to really dislike it when they are asked to disintegrate themselves, to separate and even forget the part of them that knows what corn kernels feel like when they run through your hands.
One way teachers engage students in math.
Teachers who are able to engage their students in the study of math tend to have the same skill: they can help students move up and down this ladder of abstraction.
Those teachers help students preserve a connection to concrete knowledge even while developing fluency with abstract knowledge. They help students reintegrate themselves in a discipline that is designed for disintegration.
Hayakawa described this as operating “on all levels of the abstraction ladder”:
It is obvious, then, that interesting speech and writing, as well as clear thinking and psychological well-being, require the constant interplay of higher-level and lower-level abstractions, and the constant interplay of the verbal levels with the nonverbal (“object”) levels. [..] The work of good novelists and poets also represents this constant interplay between higher and lower levels of abstraction. [..] The interesting writer, the informative speaker, the accurate thinker, [the effective math teacher? -dm] and the sane individual operate on all levels of the abstraction ladder, moving quickly and gracefully and in orderly fashion from higher to lower, from lower to higher, with minds as lithe and deft and beautiful as monkeys in a tree.
The same is true of teaching math. But in math class, you’ll often find what Wendell Johnson described as “dead-level abstracting”—people unable to operate at more than one rung of the ladder. They just live there.
This is all windup to say: Bowman Dickson is operating along the entire ladder with this math problem here.
Notice how this problem, which Bowman floated on Twitter last week, connects both the concrete and abstract. It invites the student to work with the formula for volume while also imagining the cone flipping over and water sloshing from one end to the other. Bowman even drew it in a way that suggests a sensory experience of water! You kind of want to play with it, don’t you?
This, then, is the challenge that is unique to math or at least most intensely felt there: to realize that students come to class with concrete knowledge about the day’s math that is valuable and then to conceive of math instruction as the act of fortifying, rather than replacing it.
My colleague Chris Nho alerted me to a Reddit thread where someone asked for YouTube channels or videos that can help us learn to teach math. The responses are almost 100% videos helping people learn math—explanatory videos. It occurs to me every now and again how easy it is to find videos on the internet helping us do lots of different things, but teaching is not really among them.
Perhaps that’s the nature of the discipline—less visible to the eye than cooking an omelet or changing the cabin air filter in my car . But I’m interested in finding the limits of the medium.
So here is a video that I haven’t seen before. Two knowledgeable teaching coaches discuss a teacher’s teaching as they watch it in real-time. (The teacher is me FWIW.) I’m curious if this is a model we should expand and develop.
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You should collect all your wisdoms in a book, Dan. Longing to read it!
And I loved watching the video of you teaching. So much to discuss! Something that I was thinking about afterwards was the habit we teachers have of asking students the answer to simple arithmetic questions, i.e. "What is 10 divided by 5?", while solving a problem (an equation/an inequality). I wonder if that runs the risk of actually overflowing the students' working memory (some of the students in the video seemed to struggle with even the simpler calculations), so that they don't have enough space in working memory to actually follow the argument. We're so used to asking those questions, as a means of checking if the students are with us. But for the students who do those calculations effortlessly, the question probably seems "stupid". And for the students who don't do them effortlessly, the question might deprive them of following the main ideas. What are your thoughts on this?
There are instructional videos that demonstrate “best practices” in teaching math, but the reason you aren’t finding them easily on YouTube is that researchers who are known as leaders in this area only make them available by subscription. Some subscriptions are only available to higher ed institutions. If you look up Deborah Lowenstein Ball you’ll find some material on YouTube but the really good stuff is available only to subscribers or educators who attend her Teaching Works PD sessions. Folks gotta make a living!