Thank you. This was really interesting and sparked a couple of thoughts. I've taught math and science in American high schools for 23 years.
1. I think it's so important that high schools offer students a choice between Calculus and Statistics. I wish there were more math courses in high school that are not "about" alegbraic manipulation and equation solving like Algebra 2, Precalculus, and calculus are.
2. I earned a physics degree in college without ever taking a statistics course. I later taught AP Stats for many years. I still feel resentful that no one made me take stats my senior year of high school or first year in college. The class is so important to actually understanding how things work and being an informed citizen.
3. I think there is a similar argument in the humanities where high schools are deciding if Literary Analysis should be the most important aspect to English classes and if AP Literature is the capstone all other English classes should be preparing students for.
I was involved in a Twitter discussion a few days ago where the initial question was along the lines of "What did you learn in school that you never used?" The overwhelmingly popular answer was "algebra" or just "math" in general. I objected (of course I did) and pointed out several ways that at least some math is required that you can't possibly avoid in the modern world. It was nice that I got quite a bit of positive engagement on those tweets, so at least the world isn't populated entirely by philistines.
I would like to at least mention one reason for the antipathy for the standard course of instruction presented as “Calculus”. The professor (F. Brock Fuller) presenting the lectures for Calculus I at Caltech for freshmen referred once to what he was not interested in even discussing as “Cookbook Calculus”. That is what you get in a high school or introductory college calculus course: cookbook calculus. A bunch of formulas and tricks to memorize and largely forget shortly later. I only began to understand his position when as a physics grad student at Stanford, I took the advanced undergrad Analysis course.
It was a revelation how much more coherent and satisfying the ‘same’ material could be when treated as mathematics rather than a mysterious and confusing cookbook. I had a brief taste of this difference when I took an Abstract Alegebra course as an undergrad.
I think usual cookbook calculus course is a poor represntative of actual mathematics which leaves students (even those capable and willing to plow through like the freshmen at Caltech) with a distaste for “advanced math”. I think a much better representative would be a ‘gentle’ course in topology or abstract algebra (which is almost nothing like what is usually portrayed as a course in “advanced algebra”). In other words mathematics presented as a humanities subject rather than an engineering subject.
It's so weird to me how differently we treat algebra and calculus. We know that every student must understand algebra, so we push algebra concepts as far down the curriculum as we can - students are introduced to the idea that a letter could represent a number several years before they reach "solve for x," and we introduce "solve for x" before we explain why it's important. We know that only the smart students will understand calculus, so it doesn't enter the curriculum until we've weeded out the students who "will never need it." It seems like if we started introducing those concepts earlier, it would lead to a softer landing in calculus. For that matter, I don't know how much things have changed, but the landing in calculus seemed to be kind of intentionally rough, as well - "first let's prove we need to do calculus, then lets derive it from algebra, OK now that you hate this class, let's learn how to take a derivative!"
To some extent, this reflects the fact that calculus is not really an integrated (heh) part of the high school curriculum, but a bonus add-on that's supposed to be more like (and possibly take the place of) college math courses. Which are moving toward proof-driven upper-level courses, so there's a shift in emphasis toward proving things, often at tedious length. (There's a reason I'm an EXPERIMENTAL physicist...)
The best way to motivate calculus, though, really comes from other subjects-- it was invented to make doing physics easier (of course, I WOULD say that...). You could make the subject seem more useful by motivating it in the contexts of science and economics, but we don't really do an adequate job of teaching those at the K-12 level, either.
Thank you. This was really interesting and sparked a couple of thoughts. I've taught math and science in American high schools for 23 years.
1. I think it's so important that high schools offer students a choice between Calculus and Statistics. I wish there were more math courses in high school that are not "about" alegbraic manipulation and equation solving like Algebra 2, Precalculus, and calculus are.
2. I earned a physics degree in college without ever taking a statistics course. I later taught AP Stats for many years. I still feel resentful that no one made me take stats my senior year of high school or first year in college. The class is so important to actually understanding how things work and being an informed citizen.
3. I think there is a similar argument in the humanities where high schools are deciding if Literary Analysis should be the most important aspect to English classes and if AP Literature is the capstone all other English classes should be preparing students for.
I was involved in a Twitter discussion a few days ago where the initial question was along the lines of "What did you learn in school that you never used?" The overwhelmingly popular answer was "algebra" or just "math" in general. I objected (of course I did) and pointed out several ways that at least some math is required that you can't possibly avoid in the modern world. It was nice that I got quite a bit of positive engagement on those tweets, so at least the world isn't populated entirely by philistines.
I would like to at least mention one reason for the antipathy for the standard course of instruction presented as “Calculus”. The professor (F. Brock Fuller) presenting the lectures for Calculus I at Caltech for freshmen referred once to what he was not interested in even discussing as “Cookbook Calculus”. That is what you get in a high school or introductory college calculus course: cookbook calculus. A bunch of formulas and tricks to memorize and largely forget shortly later. I only began to understand his position when as a physics grad student at Stanford, I took the advanced undergrad Analysis course.
It was a revelation how much more coherent and satisfying the ‘same’ material could be when treated as mathematics rather than a mysterious and confusing cookbook. I had a brief taste of this difference when I took an Abstract Alegebra course as an undergrad.
I think usual cookbook calculus course is a poor represntative of actual mathematics which leaves students (even those capable and willing to plow through like the freshmen at Caltech) with a distaste for “advanced math”. I think a much better representative would be a ‘gentle’ course in topology or abstract algebra (which is almost nothing like what is usually portrayed as a course in “advanced algebra”). In other words mathematics presented as a humanities subject rather than an engineering subject.
It's so weird to me how differently we treat algebra and calculus. We know that every student must understand algebra, so we push algebra concepts as far down the curriculum as we can - students are introduced to the idea that a letter could represent a number several years before they reach "solve for x," and we introduce "solve for x" before we explain why it's important. We know that only the smart students will understand calculus, so it doesn't enter the curriculum until we've weeded out the students who "will never need it." It seems like if we started introducing those concepts earlier, it would lead to a softer landing in calculus. For that matter, I don't know how much things have changed, but the landing in calculus seemed to be kind of intentionally rough, as well - "first let's prove we need to do calculus, then lets derive it from algebra, OK now that you hate this class, let's learn how to take a derivative!"
To some extent, this reflects the fact that calculus is not really an integrated (heh) part of the high school curriculum, but a bonus add-on that's supposed to be more like (and possibly take the place of) college math courses. Which are moving toward proof-driven upper-level courses, so there's a shift in emphasis toward proving things, often at tedious length. (There's a reason I'm an EXPERIMENTAL physicist...)
The best way to motivate calculus, though, really comes from other subjects-- it was invented to make doing physics easier (of course, I WOULD say that...). You could make the subject seem more useful by motivating it in the contexts of science and economics, but we don't really do an adequate job of teaching those at the K-12 level, either.