I don't think she is talking about Math majors. She is talking about some requirements that other majors have to complete as part of finishing up their degree. Having said that, I know people in say computer science who struggled through Calculus not really caring for the subject. I can't imagine how things were for people in less technical disciplines. While Calculus is certainly important for CS majors (well probably about as much as say randomized algorithms), I can't help but ask why these classes are not replaced by a class in first order logic, especially if the end goal is to train people to think mathematically.
I agree that formal logic is probably a more relevant math-y course than calculus for the average CS student. But how to design these requirements is a longstanding problem, partly due to disagreements over what should be in a "technical core", and partly due to staffing and issues of uniformity.
Calculus is probably more relevant than symbolic logic for many kinds of engineering, for example, so the course is already being offered. It's also seen by (most) mathematicians as a more central subject, so it's what goes in the core if the mathematics department is given responsibility for designing the core. Additional complications arise over what to do with things like statistics (in some places it's in the math department, while in others it's a separate department).
Oh, I am not disagreeing with the importance of Calculus for engineering. However, calculus as a core tool for say biology majors? While, I know that there are a ton of biological models (say in population dynamics) that require knowledge of calculus, I feel like for most people, this becomes one of those hate it and get through it classes and we really need to rethink its importance.
No disagreement there, and especially the way it's normally taught now, which can feel more like a course on special-case methods for performing symbolic integration. There is some understanding you gain by knowing how to work symbolic integrals (the chain rule, integration by parts, etc., etc.), but I wouldn't put it near the top of the list of things all students must know. The actual integration can be done fine by Mathematica or Maple; what's important is what you do with that.
Spot on. All I know is the situation at Berkeley, but I imagine it's similar at other universities. The lower-division math curriculum (particularly intro calculus and linear algebra) is under pressure from a bazillion different departments to cover such a wide range of topics that 90% of the material is useless to 90% of the students.
For example, a friend of mine was a forestry major, and the only reason she needed calculus was for population modeling. She knows exponentials and logistics better than I do! But she had a hell of a time passing calculus because she had trouble with the other 90% of the course. Keep in mind that this is a discipline that until maybe a decade ago required zero math.
Maybe at least at the lower-division level, applied math needs to be a bit more targeted...? But that would require us letting go of the notion that math is intrinsically edifying, and that people "ought to", "must", or "should" know it.
I think the problem itself comes from Math departments. Typical undergrad math curricula involves throwing a mix of math at students in the hope of training them to get mathematical literacy. From personal experience, I suspect it works because of the pressure and the massive amounts of math you are exposed to over a short period of time. This clearly is not true for people in other disciplines which is why you have these haphazard courses like Linear Algebra and Calculus thrown at them in the hope that they "get math" and also "learn something useful". It is an ugly situation where a lot of people just end up hating math (which considering that their sample size is small is clearly a bad thing to do but then again, they are probably never going to learn statistics to figure that out :).
>> Calculus is certainly not important for CS majors. Compared to, say, linear algebra.
I dunno. I am trying hard to refrain from saying X is not important for C.S. because the nature of this site is that you are going to find a ton of C.S. people who point out that they would be crippled with the lack of X :-)
Having said that, I read Steve's article many many years ago and upon reflection, I have realized that Steve is way better in Math than he lets on. Some of the techniques there are designed not for you to "get" Math but to get some sort of cultural awareness of Math (which of course is a fun pursuit on its own). The reason why casual foraging of a ton of math has never worked for me when I started learning math is because one of the reasons Math is incredibly hard is that it is inherently a personal journey. Getting math sometimes requires working at it patiently for weeks before you finally understand how to wrap your brain (which really means making connections with how you think about something that you previously learned and this new math) around it. This requires hard work mostly in the form of rather mind numbing problems which at first glance can be boring and feel like you are being told to "wash on wash off" (to use a Karate Kid reference). However, the subconscious is a powerful tool and helps by developing weird connections. As I got more and more trained in math, I realized these connections became much much easier. I suspect this is why math researchers are able to look at paper and quickly grasp the intuition behind it.
Re: to get some sort of cultural awareness of Math, aka Liberal Arts degree in mathematics
The Princeton Companion to Mathematics is the best tool I know for this job. This is a bedtime reading, no problem solving needed. A bit like reading, say, Python Modules of the Week. You'd need to actually use that module in your code to "get" it, but just knowing that some module exists is very valuable. Same goes for math.
That said, it's absurd to make a dichotomy between the two. Both are essential. Maybe not at first glance at a simple CRUD app, but both get surprisingly handy once you want a bit more out of it.
Machine learning algorithms, at least the ones that I used, needs convex optimization to fully understand them which needs real analysis which is just calculus with more mathematical rigor.
