The most scary thing to me is that the most mathematically talented students are often turned off by what they see in math classes, even at the undergraduate and graduate levels. Math serves as a backbone for the sciences, so this may badly undercutting scientific innovation at a societal level.
I honestly think that it would be an improvement on the status quo to stop teaching math classes entirely. Thurston characterized his early math education as follows:
I hated much of what was taught as mathematics in my early schooling, and I often received poor grades. I now view many of these early lessons as anti-math: they actively tried to discourage independent thought. One was supposed to follow an established pattern with mechanical precision, put answers inside boxes, and “show your work,” that is, reject mental insights and alternative approaches.
I think that this characterizes math classes even at the graduate level, only at a higher level of abstraction. The classes essentially never offer students exposure to free-form mathematical exploration, which is what it takes to make major scientific discoveries with significant quantitative components.
I distinctly remember having points taken off of a physics midterm because I didn’t show my work. I think I dropped the exam in the waste basket on the way out of the auditorium.
I’ve always assumed that the problem is three-fold; generating a formal proof is NP-hard, getting the right answer via shortcuts can include cheating, and the faculty’s time is limited. Professors/graders do not have the capacity to rigorously demonstrate to themselves that the steps a student has written down actually pinpoint the unique answer. Without access to the student’s mind graders are unable to determine if students cheat or not; being able to memorize and/or reproduce the exact steps of a calculation significantly decrease the likelihood of cheating. Even if graders could do one or both of the previous for a single student, they are not 30x or 100x as smart as their students, making it impractical to repeat the process for every student.
That said, I had some very good mathematics teachers in higher level courses who could force students to think, and one in particular who could encourage/demand novelty from students simply by asking them to solve problems that they hadn’t yet learned to solve. I didn’t realize the power of the latter approach until later (and at the time everyone complained about exams with a median score well under 50%), but his classes were always my favorite.
I’ll be writing more about this later.
The most scary thing to me is that the most mathematically talented students are often turned off by what they see in math classes, even at the undergraduate and graduate levels. Math serves as a backbone for the sciences, so this may badly undercutting scientific innovation at a societal level.
I honestly think that it would be an improvement on the status quo to stop teaching math classes entirely. Thurston characterized his early math education as follows:
I hated much of what was taught as mathematics in my early schooling, and I often received poor grades. I now view many of these early lessons as anti-math: they actively tried to discourage independent thought. One was supposed to follow an established pattern with mechanical precision, put answers inside boxes, and “show your work,” that is, reject mental insights and alternative approaches.
I think that this characterizes math classes even at the graduate level, only at a higher level of abstraction. The classes essentially never offer students exposure to free-form mathematical exploration, which is what it takes to make major scientific discoveries with significant quantitative components.
I distinctly remember having points taken off of a physics midterm because I didn’t show my work. I think I dropped the exam in the waste basket on the way out of the auditorium.
I’ve always assumed that the problem is three-fold; generating a formal proof is NP-hard, getting the right answer via shortcuts can include cheating, and the faculty’s time is limited. Professors/graders do not have the capacity to rigorously demonstrate to themselves that the steps a student has written down actually pinpoint the unique answer. Without access to the student’s mind graders are unable to determine if students cheat or not; being able to memorize and/or reproduce the exact steps of a calculation significantly decrease the likelihood of cheating. Even if graders could do one or both of the previous for a single student, they are not 30x or 100x as smart as their students, making it impractical to repeat the process for every student.
That said, I had some very good mathematics teachers in higher level courses who could force students to think, and one in particular who could encourage/demand novelty from students simply by asking them to solve problems that they hadn’t yet learned to solve. I didn’t realize the power of the latter approach until later (and at the time everyone complained about exams with a median score well under 50%), but his classes were always my favorite.