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Yes, all the time. I've heard it for physics, I've heard it for CS and I've heard it for programming, all with good reason: the way you teach somebody deeply invested in your field is different than the way you teach somebody deeply invested in a different field, learning yours as a supplement. Or maybe a better way of putting it is that the way you teach somebody interested in a field in and of itself is different than the way you teach somebody who has other interests and motivations, but still needs to learn.

And hey, maybe before college, almost everyone is a non-mathematician. There's the small majority of students who'd love learning group theory because it's fun and beautiful, and then there's everyone else who need practical applications and real-world examples. But the way you teach them is ultimately just like the way you'd teach non-mathematicians in college or further on in life, when there's a clearer delineation.

This situation actually started a pretty nasty divide at my university. Engineering and Math are in separate organizational units which gives each a degree of independence. The politics worked out that most of the operational budget of the Math department came from teaching students from other disciplines. Over the years Engineering was dissatisfied that after going through the prerequisite Math courses their students had plenty of theoretical knowledge but couldn't actually do the Math -- even those that took the applied variants. So after Engineering came into more money they hired engineers to teach "Calculus/LinAlg/DiffEq For Engineers" courses that focused on application and topics that more directly applied to their other curriculum.

It was a success overall for the Engineers but Math wasn't happy. The engineers working on advanced coursework needed higher level math courses that were only available in Math. Not only were they constantly failing which angered Engineering but the professors teaching them had to devote more time to the Engineering students which took away from the math students and angered Math.

The sort of uneasy truce that they eventually came to was Engineering students take the regular Math courses and their Physics/Engineering curriculum supplements what they're taught in Math. This annoys double major Math/Eng students because there's far too much repeated material in their curriculum but it's the best they could do.

Absolutely agree with this comment. If you get beyond the title, what Gowers is talking about specifically is "If everyone were compelled to take 2 more years of mathematics, what should it be?" It is very different from what we would teach undergraduates getting engineering degrees; it would also be very different from what we would teach math majors. If we were to replace "mathematics" by, say, "programming," it would also not be surprising that one might come up with a list of topics differing from what we would want an aspiring computer scientist to know.

I did not find anything really surprising in the first part of Gowers's essay, but I thought the list of questions was great, independent of whether or not it is really appropriate for a required course.

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