Over three years ago, I wrote a post to try to address a fallacy that is used to refute the idea of novel ways of teaching mathematics and science. That fallacy basically says that mathematics and the way people learn it have not fundamentally changed in hundreds if not thousands of years, and therefore the methods of teaching that have “worked” up to this point in history don’t need changing. Or more colloquially, “*We were able to put a man on the moon with the way we’ve taught math for hundreds of years, so we shouldn’t change it now.*” I sometimes refer to this as the “man on the moon” fallacy because of that second interpretation.

To understand why I think this is a fallacy, read the post above – or better yet, read this long quote from a 1988 paper by Edsger Dijkstra, one of the great scientific minds of the last 100 years and one of the authors of modern computer science. The paper is called “On the Cruelty of Really Teaching Computer Science” and has a lengthy section addressing the notion of “radical novelty”. In it, he has a few choice words about novelty-avoidance among mathematicians:

For instance, the vast majority of the mathematical community has never challenged its tacit assumption that doing mathematics will remain very much the same type of mental activity it has always been: new topics will come, flourish, and go as they have done in the past, but, the human brain being what it is, our ways of teaching, learning, and understanding mathematics, of problem solving, and of mathematical discovery will remain pretty much the same. Herbert Robbins clearly states why he rules out a quantum leap in mathematical ability:

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Nobody is going to run 100 meters in five seconds, no matter how much is invested in training and machines. The same can be said about using the brain. The human mind is no different now from what it was five thousand years ago. And when it comes to mathematics, you must realize that this is the human mind at an extreme limit of its capacity.”My comment in the margin was “so reduce the use of the brain and calculate!”. Using Robbins’s own analogy, one could remark that, for going from A to B fast, there could now exist alternatives to running that are orders of magnitude more effective. Robbins flatly refuses to honour any alternative to time-honoured brain usage with the name of “doing mathematics”, thus exorcizing the danger of radical novelty by the simple device of adjusting his definitions to his needs: simply by definition, mathematics will continue to be what it used to be. So much for the mathematicians.

If “doing mathematics” is taken *axiomatically* to mean, “the way we’ve done mathematics for five thousand years” then logically any alternative to this is not going to be “doing mathematics”. But this is not an argument in favor of a certain definition of mathematics – it’s a tower-defense game in which a small cadre of gatekeepers fend off invaders to keep mathematics pure. Dijkstra points out that mathematics itself is the loser in this game.

The man-on-the-moon fallacy is the same thing applied to teaching. *We put a man on the moon using the teaching methods from the last 300 years, so we shouldn’t go changing the way we do teaching*. This does not admit that there could be improvements in our understanding of how people learn, changes in how instruction can be done (especially with technology), and – I would say especially – the profound changes in the contexts in which mathematics is situated that have taken place over the last few decades. Calculus may not have changed much since Newton and Leibniz – although I believe this is debatable – but the problems that need calculus, the options available for solving those problems, the people who need the solutions to those problems, and the people doing the solutions have changed almost completely since the 1960s, when we actually did put a man on the moon.

I’m hopeful that we in the mathematics community will begin to see this fallacy for what it is, be more open-minded about how mathematics is taught, and avoid some future Dijkstra saying, “So much for math education.”