I think it could be taught very differently, and with more honesty about where it is applicable. I was discussing trigonometry recently with my son, and it was clear that he had no motivation beyond learning formulas as taught. In fact, trigonometry itself becomes much clearer if you start out by developing a solid grasp of complex numbers and can understand the identities in terms of exponentials.
Case in point: How is it that when I first learned about Fourier transforms, it had to do with sine waves, but the FFT I learned in a computer science class is based on āroots of unityā? Itās trivial if you know exp(ix) = cosx + i sin x. Indeed this is taught in pre-calculus, but it is not emphasized to the point that most students really think of these as the same thing. The syllabus moves on to the next topic, and trigonometry is once again a menagerie of isolated functions and identities.
There are two questions: What is the goal in teaching this topic? Is the approach effective? An exceptional student will fill in the gaps by themselves. An excellent school, perhaps a magnet or private school, may provide effective mentoring. But for most students (unless it is applicable such as in signal processing or literally ārocket scienceā) it is hard for me to understand the emphasis on difficult and isolated skills that are soon forgotten.
I look at it more as a term that identifies a missing area in K-12 and a call to action to figure out the curriculum. Itāll take time to sort out. My understanding of it is that it includes more than just statistics, but also the presentation of data.
If we ask the question whether people need algebra (or trignometry, or any other elementary math) in their career, 90% or more of them would probably say no. Does that mean we shouldnāt teach these subjects in K-12?
In any event, calculus is so fundamental and it opens the door to all sorts of other math a student may need for many other subjects, including some branches of computer science.
Since this is getting completely out of (the boring) topic, this will be my last post.
None of the ādata scienceā areas is new. The only difference is that they are a new fad in the tech sector, after the previous ābig dataā fad. Some people that like buzzwords are trying to ride it and bring it to education. IMNSHO it is a fad and will pass. I might be wrong.
I donāt want to belabor my already verbose commentary so maybe I can try to summarize. Of course, itās great to learn many things and it stretches the mind to do so. Personally, I enjoyed calculus and still think about it. The question, though, is what do most students get out of these subjects, specifically as they are usually taught rather than in the ideal?
Hereās what I think most students get: they jump through intellectual hoops to prove themselves worthy and having done so are able to present a certificate of worthiness to interested parties. In terms of relevance, they might just as well have taken piano lessons or studied Ovid in the original Latin. These too will stretch the mind and may separate out students in terms of aptitude, ambition, and earnestness. So, pardon my cynicism. I do not believe that calculus is fundamental, except in math analysis, physics, some other physical sciences, and specific engineering disciplines. By all means, teach it, but donāt engage in false advertising about it.
Substitute ācalculusā with any other class and the same could hold true. The quality of how calculus or any other subject is taught in high school is a function of where a student attends. Some not so good. Others very good.
I think the problem at this school was the serious physics majors had already completed multivariate calculus before they got to college.
And with respect to what I was addressing, I was not saying that you could do computer science without calculus (TBH, Iāve never taken a CS course but my son studied computational and mathematical engineering and Iām sure a lot of optimization was involved) but that you could learn to code without calculus.
You forgot to mention many other disciplines, including life sciences, social sciences, business and finance, essentially anything in depth that can be modeled (or approximated) by continuous variables (the discrete world you may be dealing with is relatively small in comparison). Anything that deals with rates. Anything that needs to be optimized, etc. You previously mentioned linear programming as something that doesnāt need calculus. Well, if you broaden the ālinearā restriction to include non-linear optimization, you may find that calculus would have to be involved.
Most (good) CS jobs still only ārequireā* a bachelorās degree. It is mainly the small subset of research jobs that require advanced degrees.
*Some employers accept CS knowledge and skills from self-education, if proven during the hiring process. That is how people with unrelated degrees or no degree get into CS jobs.
Iām willing to state categorically that you can do computer science without calculus. Find the most recent texts on algorithms and data structures, operating systems, and programming language theory and show me how these depend on calculus* (there may be some mention).
It is probably true that most students likely to excel in the major have had exposure to calculus anyway, but that thatās a different question. Interestingly, some of the techniques in asymptotic analysis require a facility in doing summations that are analogous to integrals. But this subject is rarely taught before itās needed. Nor are difference equations, as opposed to differential equations. Graph theory, combinatorics, and inductive proof are also rarely taught before college, though these may be more accessible to students without much foundation.
While I donāt question that (a) calculus is a great thing to know, (b) most research-level computer scientists learn it at some point, and (c) it comes up in some subfields that fit under the wide umbrella of ācomputer scienceā, I still consider it very unfortunate if it is used to block anyone from further pursuit of a subject where it is very rarely applied.
Well, for example, math up to algebra 2 with some trigonometry (enough to get that itās a conversion between cartesian and polar coordinates and maybe not heavy on all the identities) is definitely going to help in any field that requires an ability to interpret or express yourself in formulas, as well as set up equations (whether you learn lots of techniques for solving them). This is definitely going to be useful in computer science or any field with a quantitative element.
However, calculus taught as the pursuit of the ever elusive closed form of an integral (which exist only for carefully curated specimens) is simply the playing out of a parlor game that was once great fun for minds such as Gauss or Euler, but of little interest to bored teenagers in the 21st century (indeed, it may still be great fun for a few and more power to them).
But also this detracts from developing a better facility for applying these kinds of formulas when theyāre useful because you were too busy jumping through them like an obstacle course. I think somebody mentioned statistics, and that would also be a better focus for those who have completed algebra 2.
Aside from math, just about any writing course is enormously useful no matter what you major in. High school science courses that convey both facts and an understanding of empirical research are useful. There may also be time for calculus, and I would not have wanted to skip it, but it is still hard to see how to justify it as a requirement except in the minority of technical subjects where it is used.
But note that CS major degree programs often require calculus-based probability (whether its own course or embedded in some other course). For example, Stanford CS 109 Probability for Computer Science lists multivariate calculus as a prerequisite.
You might instead argue that, instead of a general calculus course for most majors, each major may only need its own version of calculus covering what it needs. This is already commonly done for business majors. However, increasing the course divergence earlier may make it more difficult for undecided students to work toward several possible majors; such students would have to take the most comprehensive general calculus course acceptable for all majors instead of the more specialized versions. Of course, with majors like CS, it is also not assured that a frosh student choosing between comprehensive general calculus and ācalculus for CSā that covers just the needed basics for CS (including any required probability course) will know whether or not they will eventually go into a specialty that needs the more comprehensive general calculus material.
The best advise for HS students seeking to pursue any engineering degree, IMO, is to take 4 years of the math sequence offered at their HS. If they have trouble with a HS progression they most likely will have increased difficulty in college.
Iām curious at which school this is at which prospective physics majors complete MV calc before college, yet a notable portion of students also do not take any calculus during HS?
My personal experience is at Stanford, which sounds quite different. Freshmen math, physics, and CS all have different course sequences to choose from, with different levels of acceleration/rigor. Almost everyone takes calculus during HS. When I attended it was much more common that physics majors take AP Physics during HS than take math beyond AP calculus. AP Physics is a prereq for the highest level of freshman physics that is not uncommon among physics majors, although physics majors can also choose the standard physics level. Prospective physics majors whose placement test says they need to repeat HS calculus, are allowed to take the slow/less rigorous version of calculus while concurrently taking the standard not super rigorous version of freshman physics.
Calculus is not required to take basic CS classes that would be required to pass a job interview, although calculus is required for both a CS minor and CS major. A student could certainly learn to code adequately through classes without getting the minor/major, by instead attending a 2-year community college, or even without attending college at all. However, many desirable CS first jobs limit/screen hires to CS majors, rather than just being able to pass an interview with coding questions. I doubt that this calculus requirement prevents a noteworthy portion of students from the āprestigious collegesā that are the subject of this thread from pursuing CS. Iād expect things like grading in CS classes, past programming experience, social effects, and personality traits to have much more influence in which students choose to stick with the major, and which choose to switch out.
Agreed. In case it wasnāt entirely clear, my comments above relate to what you actually need to know in order to do computer science, and not just coding but a great deal of advanced research-level work. What āprestigiousā universities actually require and what top students manage without too much difficulty is a different matter. Calculus is fine, I guess, as a proxy for mathematical aptitude. However, I think its privileged status is more of a historical carry-over than a best fit for needs when it comes to computer science.
That really isnāt saying much though, right? I am always amazed at the number of attorneys you can made nervous by telling them you are going to do math. And not high level math but really just middle school math. Many people who are brilliant in the law suddenly get nevous and are clueless. Have had people accuse me of voodoo magic with spreadsheets that werenāt very complicated or sophisticated. Its like I went to law school and they told me there wouldnāt be any math here. LOL
Lawyers use Microsoft Word and hate Excel! (At least this one does).
I used to be in-house counsel, and one of my friends, whom I worked closely with, was in the finance department. He used to type his text documents in Excel, which cracked the folks up in the Legal Department.
Wouldnāt this at least hurt their SAT scores? (Assuming they attended a āprestigious collegeā) It bothers me a lot when people try to reach decisions without numbers, because the right choice often comes down to a quantitative comparison.
Thereās a world of difference between āSounds really unlikely.ā and deriving a p-value to attempt to quantify the probability of an outcome happening by chance (though this can be done badly or abused). Often your hunch is not borne out by the analysis. I also remember reading about the difficulty mathematicians have proving gerrymandering in federal courts, simply because the justices are not used to thinking that way.
Quantitative reasoning also comes up, obviously, in questions of cost or feasibility, which is a serious problem if legislators, very often lawyers by background, are innumerate.
