The case for ending calculus requirements for science majors

I am currently trying to tutor my daughter through college-level calculus. She wants to be a civil engineer. Civil engineers, obviously, need to know calculus.

But calculus is not being taught to her by engineers. It is being taught by mathematicians. And as part of teaching it, they are attempting to instill a "mathematical mindset". Which means, in practice, that it is an entry to proofs.

Every proof has a trick. If you can see the trick, you can generally figure out how to do the proof. If you can't see the trick, you can't. The same thing is true with all of the very tricky limits they are making her learn to solve.

I hold a PhD in aerospace engineering with a minor in applied mathematics. I specialized in control theory. I use calculus every day of my life. In twenty years of engineering practice I have never once had to take a difficult limit, and can probably count on one hand the number of times I've had to take a limit of any kind.

I had a heart-to-heart with her about this today. I told her that what she actually needed to understand about calculus was how to use it to solve engineering problems, not how to manipulate polynomial fractions, chain trig identities, and do substitutions of variables. And she was not going to learn that in this class, but that she nevertheless needed to learn the tricky algebra and epsilon-delta proofs simply because this was a hoop she needed to jump through to get to the actual engineering classes.

There was a not inconsiderable amount of cursing, but I understood because I hit the same wall, just thirty years prior. Nothing has changed.

The author claimed that calculus is required to biology major in many colleges. I would argue the opposite, I thought at two different universities and talked to many people. They always find it strange that it is not required for most biology fields. There is even an American invention called algebra based physics. This for people in biology who never saw calculus and your try to teach them physics and mechanics without calculus (Good luck with that, that is at best lost opportunity to understand physics) I thought in algebra bases physics classes many times and it was always horrible experience for both sides. Also calculus is not mandatory in US pre-college.

In general, you don't understand physics without calculus, period.

But regardless, while I disagree with the article’s argument it’s valid to debate this topic. My undergrad background is in engineering so I’m very biased here (and work as an engineer and have never used calculus for work). I do think a math major should take biology as much as a biology major should take math (calculus)! Folks often talk about the goal of a liberal arts education being learning a “breadth of topics” or the “classical teachings” and that’s fair and important. But I also think the same for individuals studying a STEM field. Maybe bio majors don’t need the level of calculus rigor required for a physics major but I personally think it’s such a fundamental topic they should learn it.

Sometimes the math is needed for courses in the other major; for example, if an earth science major is taking a fluid dynamics course in their major, they'll need 3 terms of calc, and maybe differential equations and linear algebra. This is not to say that all majors in a department will use all the math they're required to take. I once asked a colleague in chemistry how often they actually needed to do derivatives or integrals; she replied that many chemists (e.g. experimental chemists) would rarely need calculus, but people in theoretical chemistry would. The simplest thing is often to require all majors in a given field to take the same math courses, even if only a few will use all of it.

The CS department at the school I retired from seems to have shifted its math requirements from 2 terms of calc to one term, plus stuff like statistics and linear algebra. (I don't recall all the details.)

Another reason for math requirements being what they are is that certain math courses are required for other departments to have their programs accredited. A while ago we were revising the syllabus for the calc course which was taken by business majors. I asked the business department chair if they had any comments or suggestions; he thought everything was okay, but wanted to verify that certain topics were in there (I think maxima and minima was one). He said their program accreditors wanted to see it.

I think other departments gradually adjust math requirements for their majors; it's in their interest to ensure their majors are prepared for jobs or grad school So if you graduated a while ago and you feel that the math you took wasn't that useful (and you have suggestions for alternatives), take a moment and mail your old department and let them know - they'll appreciate it, and you'll be helping future students.

Perhaps a better discussion is how to teach calculus more effectively?

I feel really sorry for the young woman mentioned in the article having to take what was for her an unnecessary class. As a non-academic, my casual four years to get a BS was probably really good for me.

The perfect solution, assuming you happen to be wealthy enough to hire mathematicians without worry.

because sure, p hacking, but i’ll tell you this for free: you will not get far in any discipline before you are whacked in the face with a differential equation based model of a system, and if you cannot grok that you are totally hosed.

I think my experience suggests that even outdoor types who are drawn to field work should learn calculus in some functional way if they want to do science. I did an undergrad in environmental sciences and biology, but understanding the role of modeling in theory and practice is key to those fields and makes heavy use of calculus. I went on to get a PhD in ecology, but I used calculus explicitly and implicitly routinely. There is great satisfaction in writing out ideas and hypotheses about the world and then translating them into math and then code so that you can play with those ideas. Sometimes that just requires algebra, but it usually pulls in calc.

The other risk is that by not learning broadly about the fields that are foundational to conducting science you simply limit your options. Maybe that field tech job loses its luster after a while and you start to want to understand how plate tectonics affected how the dirt got this way. Need calc. Personally, I switched to industry and do a combination of data science and engineering and calculus continues to be useful to me in practice and as a way of reasoning about the world.

On the flip side, you can always learn it later when it does become explicitly needed. But again I think the problem is teaching geologists and biologists to do epsilon-deltas but not showing them how to specify and simulate a model based on rates of change of growth or propagation of waves or whatever. A lot of coursework is not thoughtfully conceived or delivered and the concept of weeder courses is super lame. But thats not the fault of calculus.

+1 for requiring statistics too but virtually all of the above applies there too.

Where I went to school, "real" stats and experimental design topics were (mostly) offered as graduate courses. Psychology majors were given enough stats to read a psych paper. Physicists and Chemists were given enough stats to understand stat mech / pchem. Biologists and Geologists were given a much less rigorous intro to stats.

I believe the OP is advocating for more stats, not less calculus. Or at least a more consistent approach to teaching stats & experimental design at the undergrad level.

Interesting question, why should I be required to pass biology?

There are a lot of good points addressed thoroughly in other threads here, but (while calculus may fill a bit of a different role) education to a little bit of depth in a variety of areas does seem to be part of the role of a college (at least at present). I think requiring statistics to be taken more broadly would be great for our society, but calc (especially when taught well) definitely touches on areas of thinking that other subjects don't cover.

(Obviously slightly biased perspective from a math major.)

So: you need better syllabi and teachers.

Do they teach measure theory and the Radon-Nykodim theorem in those Statistics courses? And the central limit with proofs? No.

So: it is not the Calculus. It is the teaching of it.

I like calculus because it allows us to put a measure on how things in our world change, it greatly simplifies our calculations of the rate at which these changes occur than we would otherwise be without it.

Things in this universe are rarely fixed, unchanging and immutable and the fact that they're not is why the universe works as it does, and it's why we exist and are around to actually study it. Calculus is a principal tool in helping us make observations about how the universe works.

From my experience, probably the single biggest problem with calculus is its frightening reputation. There's likely nothing better to give someone a lifelong phobia of calculus than for a teacher to say to him or her as a kid 'you must be very good at arithmetic if you are ever to be good at that advanced subject calculus'.

When I was a kid calculus had a frightening and awesome reputation for being difficult so by the time we'd reached highschool and were confronted with learning it many kids already had preconceived notions that they weren't going to do well in the subject.

This notion about calculus being difficult isn't new and it goes back a long way. Only a few days ago I read a HN story about physicist Richard Feynman learning calculus as a teenager and that he learned it from an early type of 'teach yourself' book titled Calculus Made Easy (1914) by Silvanus Thompson. I'd not seen this book previously so out of curiosity about how the great man came to learn the subject I downloaded a copy and it was an eye-opener.

Its Chapter I titled To Deliver You from the Preliminary Terrors, only confirmed the fact that not only my generation of kids had been forewarned of calculus' 'terrors' but also so had previous generations of kids - even those long before the author's time. Thompson, an electrical engineer, professor of physics and educator, was well aware of its 'terror' factor ipso facto the chapter's title. In only one half pages of the most delightfully written prose for a mathematics text he attempts to alleviate the reader's fears with simple straightforward explanations. Therein he explains the dreaded 'd' as simply meaning 'a little bit of' so dx means a little but of x. (In my opinion all educators of the subject would do well to read this chapter.)

Despite Thompson's best efforts to dispel fears of calculus they still remain with many of us today. Perhaps the reason why the students referred to in the article failed calculus is that they've carried this preconceived notion of its difficulty with them from their earliest days and that the level of (or the subject material) wasn't appropriate to their courses.

I'm not in favor of removing calculus from courses because I believe it is necessary to understand what it teaches us about the world, as it explains the underpinnings of almost everything we do, especially so the physical world, and we need to have some knowledge of how that happens.

If I could I would go even further than Thompson and teach calculus to primary school kids. Get in early enough and kids wouldn't have time to develop a fear of the subject.

For skeptics who say we can't teach anything meaningful about calculus at that age then they ought to think again.

For instance, getting kids to learn Simpson's Rule by showing them how to work out the area under a curve by having them cut out rectangles of cardboard and best-fitting them to the area is well within their capabilities. Couple this with a stress on how important this is in real world siutations and we'd be well on the way.

Then there's the old fable about the frog on a stone in the middle of a pool who wants to jump to the outside bank. At first he jumps half way and then a quarter and then an eighth and so on as he tires. Every kid knows this story and that the poor hapless frog never makes it.

Now teach kids the frog really doesn't perish after all - because our hero Calculus says his feet are too big. Happy ending.

(Apologies to any mathematicians whose sensibilities I've offended.)

___

Edit: I should have stressed that in professions that have a less rigorous dependence on mathematical calculations, their calculus courses should place more emphasis on the meaning of what it teaches us rather than the manipulation of symbols and equations per se.

(Humans have always had difficulties in recognizing small and large rates of change - exponential growth etc. - especially in their early stages (when often still manageable).

We should always emphasize to students why calculus is an essential tool for processing the mathematics of rates of change. In many cases understanding the underlying reasons is more important than doing calculations (if one actually understands the problem then one can always call on someone with better mathematical knowledge - after all, at times even the best did this - Einstein for instance).

Haha, that's what it means when you fail these days.

"lost"

Wow

Or you need to stop calling biology, geology and other majors forms of science or STEM if you eliminate calculus.

There is a case for different education perspectives for different majors (trying to teach in a more familiar way for people with different backgrounds).

But dropping it entirely? Why are you in that major to begin with? It's like going into compsci and not expecting math.

https://www.ncbi.nlm.nih.gov/books/NBK25500/