Apples and…Basketballs? I recently had the opportunity to talk with Alfie Kohn, one of my favorite theorists of education. In fact, I asked him if he might have a look at some of the games on our site. It was with a sense of dread that I opened up his email later that afternoon.

Mr. Kohn (If he holds a Ph. D., he is self-effacing enough never to mention it, even on his blog, is a leader in a counterrevolution in American letters, whether kindergarten mathematical reasoning or applied political philosophy. He’s a fan of situated learning, which is to say, giving children tasks that are inherently worth doing, and working the specific fact learning in at the margins. I appreciate his point of view, and I really, really wish that American education would have taken that course when it played with going that direction in the post-McCarthy Era through the Sixties. Situated learning is associated with thinkers like John Dewey, and it brought us victory in the Space Race with the Soviets – that same empire that is piecing itself back together over the bodies of Ukrainians, and before that, Georgians (that is, the kind whose last name ends with “-vili,” not the kind that keeps winning the National League East). Clearly, setting children loose with meaningful tasks is a great way to help them think about math, and at, we mostly turn fact recall and algebraic problem-solving into a matter of rapid-fire muscle memory.

So would Alfie (as he prefers) rip me a new one, given that what we have done on has nothing in common with what he proposes as a way to educate students to be able to apply mathematical reasoning to real problems great and small? No. He said, “While in principle I think that this kind of game is counterproductive, it is so only if the game is used to replace situated learning, not to supplement it (emphasis mine).” In other words, the corporate education system is screaming for kids to learn math by rote so that they can fill in those bubbles quickly and accurately, which Alfie rejects (see the abstract from “The Schools Our Children Deserve” on his website at, but at, we don’t prepare kids to do that! We simply take advantage of the fact that anything that sucks you in and grabs your attention is going to make an impression.

Is it “worthwhile mathematics” to know by fast-twitch muscle response that 7×6=42, or equally hateful, that 7×8=56? Well, no. However, if the student who could devote an hour to Candy Crush Saga spends it instead on the mastery of math trivia to the point of not having to spend a scintilla of effort on how many sides in a hexagon, the product of 7×6 (or 7×8, for that matter), or what x makes 3x-1=20 true, might that student be free of “skill, drill, and kill” forever and be free to explore math in context of life? I think so. The second-youngest President of the United States attributed his ability to succeed at what mattered to his ability to routinize everything that doesn’t. You would never catch this man dead designing his daily workouts, figuring out what to eat for breakfast, mixing and matching his wardrobe, or any of a host of tasks in which you and I sink precious energy.

So what about comparing apples to basketballs? Last week, we discussed the fact that a highly complex math simulator, DimensionM (not a Mathnook product, alas), has been shown in at least one peer-reviewed study to increase skill, aptitude, and interest in high-school level algebra among middle-school students, family background and economics factored out. DimensionM is a highly sophisticated simulation that has more to do with James Cameron’s Avatar than with the simple designs that allow me to get so many games up so fast. Yet, I get similar results. Why? Let’s say that DimensionM is an apple, crisp, cold, and healthy. If you want a healthy math mind, you want to evangelize DimensionM and similar products as opposed to wasteful social media and mindless entertainment – the potato chips and Milky Way bars of consumer electronics. However, developing a healthy math aptitude requires a good diet and a healthy dose of exercise. Consider the basketball. Let’s play!

P. S. For those of you who want to teach oddball facts like 7×6 and 7×8, let me give you an idea. Two, actually.

1) Divide and conquer. Most people have 7×3=21 and 7×4=28 (fewer this one) committed to memory, and may even be able to produce an array of seven rows and three or four columns. Using the associative property, 7×6 = (7×3)x2, or (7×3)+(7×3) = 21+21 = 42. Similarly, 7×8=(7×4)x2, (7×4)+(7×4), 28+28, or 56.

2) Nearest square: Most people can give you 6^2=36, 7^2=49, and 8^2 (chessboard) = 64. Adding one more six to 36 depends on knowing what multiplication means, but I am all for learning the meaning before practicing the facts. Similarly, adding a seven to 49 or subtracting an eight from 64, while a little more arithmetically cumbersome, amounts to the same thing.