# The royal road to calculus

In the early 1900s Felix Klein lay out his vision for secondary mathematics curriculum. He wanted schools to teach calculus, so that universities would not be burdened by it. And at the core of the curriculum was to be the notion of function, which was conceptually necessary for the foundations of calculus. (I know I make a very messy story short here, but I think the basic points are more or less correct.)

All of this was over 100 years ago, and to this day we teach according to Klein's vision. My question has to do with whether now, given the current state of the art, we would chose to design the mathematics curriculum in the same way. Specifically:

1) Do you think functions should be the backbone of the curriculum. (Is it too abstract for students? Do the students leave secondary school understanding why functions is central?)

2) Do you think calculus is the appropriate "end" of the high school curriculum. If not, what would you suggest in its place.

• The discussion following the question Is Calculus Necessary? partially addresses your second question. – Joseph O'Rourke Dec 11 '18 at 11:55
• A society in which calculus is not taught in high school will succeed in training well almost no scientists and engineers. There will be some anyway, in spite of the schooling, but much potential will be wasted, to the society's detriment. – Dan Fox Dec 12 '18 at 14:42
• Arthur Benjamin's brief talk on Teach Statistics Before Calculus might be insightful for your 2nd question. – ruferd Dec 12 '18 at 18:43
• @Manya: The content is fundamental. Calculus is a basic tool in all of physics and engineering. One can't even formulate mechanics, differential equations, electromagnetism, waves, signal processing, continuous probability, etc. without calculus. Many a competent electrical engineer or physicist knows little graph theory, but all know calculus thoroughly. Calculus is also a superb tool for teaching basic reasoning skills. A student that can argue from Rolle's theorem that a nontrivial degree $k$ polynomial has at most $k+1$ real roots has learned something well beyond the result itself. – Dan Fox Dec 17 '18 at 15:48
• The goals of training future mathematicians/statisticians/physicists/engineers/etc. and the of general mathematical education for those who will not work in such technical fields are largely divergent, and what is the appropriate place of the function concept or calculus in the curriculum surely depends greatly on the intended audience. The difficulty of coordinating these needs is complicated by one-size-fits-all curricula that pretend to serve simultaneously the needs of the future scientist/engineer and the needs of the great mass of students aiming to meet minimal educational requirements. – Dan Fox Dec 28 '18 at 19:16

this is an excellent question. I would prefer to offer a different interpretation however. My dissertation was in mathematics education, and I was interested in the history of the function concept in schooling. Klein was not the only one who espoused an interest in utilizing the function concept as the backbone of the school curriculum, and it has been claimed as such for more than a century now. If you look at the current New York State high school mathematics curriculum, it is called A Story of Functions, CCSS identify functions as backbone, etc. etc.

What is important however, is to pay closer attention to what calculus means for Klein in comparison to the nature of the discipline as it has subsequently developed in school and curricula. Klein was German, and valued a Leibnizian view of calculus. There is a difference here, and approximation methods play a more pronounced role in this vision. If you notice, Klein later demands a more prominent role for "The Finite Calculus", this includes things like finite differences and interpolation; ideas which never made their way into the standard curriculum.

I think Klein's demands for a more humanistic and approximate style to the calculus resonates louder today with the rise of computing and statistics in culture, society, and mathematics.

While you can find something like a recursive sequence as a function in the CCSS, curricula and teaching still have a long way to go to valuing the finite and approximate. Too often we see these as specialized topics for later study rather than ways to fill out existing concepts like linear functions that are such mainstays in the curriculum. Some examples do exist however, maybe you'd be interested in the COMAP Mathematical Modeling Our World materials, or some older textbooks made for North Carolina high schools before the CCSS when they were teaching vectors, matrices, and discrete mathematics topics in classes like Algebra I.

So, yes, functions should be the backbone -- but a function is not a function is not a function. Functions have multiple interpretations through history, and we should aim for a less rigid notion where we study relationships in a variety of contexts as opposed to questions around function theory like "is this a function".

Yes, calculus should be the end game -- but calculus can be much more than what is presented in the AP Curriculum or MIT's first course. People arguing for statistics to replace calculus ignore the fact that statistics is just an application of calculus, particularly when you value topics like finite calculus and the probability calculus in an introductory calc setting.

• What is the source for the claim that "Klein ... valued a Leibnizian view of calculus"? – Dan Fox Dec 16 '18 at 19:58
• See his lectures on elementary mathematics from an advanced standpoint, and his lectures on secondary mathematics – jfkoehler Dec 16 '18 at 20:08
• I am afraid I don't see any signs that Klein's view is taking hold. – Peter Saveliev Dec 18 '18 at 17:04
• Depends. I prefer his approach, try to realize it in my own classroom. I think the rise of computing will make our traditional approach obsolete in the coming decades...large scale curricular change is an incredibly slow moving train, but you are free to rebel at will in the meantime! – jfkoehler Dec 29 '18 at 17:56
1. I suspect there is an implicit thinking about what the curriculum for a nascent math Ph.D. should be. But you need to realize that less than 1% of high school students will do this. And you don't know who they are ahead of time. So thinking in this mode is really not the right way to optimize the curriculum.

2. I got a functions module in pre-calc (domain/range emphasis). Think it was fine to have it at that time. Did not hurt me from learning the quadratic formula or trig or logs to not have it earlier. Be careful about the impulse to want to move everything at the beginning. You can't have everything at the beginning. Also be careful about learning categorical frames of reference before important content (logs and antilogs, sin and arcsin, etc.). Just because it makes sense in retrospect does not mean it makes sense when you are fresh to topics. Also learning too much abstract theory before getting some math muscles can be distracting.

3. For a very bright or accellerated student, there is no reason why they can't go beyond calc 1/2 within high school. The logical next step is calc 3 and ODEs because it is most common next step in a general STEM career. But conceivably if school is big enough or student has an interest they could look at other topics (e.g. linear algebra). I understand stats is in vogue nowadays in high schools also. In addition, if a kid has tapped out the math classes at his school, he could look at a comp sci course (always useful). But the ability to go on to next classes after calc for individual students is crushingly obvious. But they are outliers and can go to a juco or self study or the like.

This question about "should calc be the end" shows the same mental gap as in (1). Calc isn't even the end of MOST high school students. I will admit that it is the end step for serious college prep students. But don't act like the other bodies in the school are irrelevant. They are citizens also. Get out of the Yale or jail mindset. Lots of great people never go to college.

Even for the college prep kids, I would say it is only the upper half (maybe even upper third) that really ace calculus and place out of it in college (say a 4 or 5 on the AP). A vast amount of kids who are not in that upper tranche are going to repeat calculus anyways in college. Even for the kids that pass the AP, most do it as seniors (and were accelerated a year to even do that). So worrying about accelerating further (or having extra classes for the outliers that already accelerated so much) is penny wise, pound foolish.

1. I'm sure the real analysis lovers will be here to say that calc course should be made more theoretical, etc. But they have a tendency to miss points above. They are super great at finding the hidden counterexamples and tracking epsilons. But not so good at comparative analysis (points 1-3). And education is a human activity with costs, benefits, limits, probabilistic outcomes, etc. Any strategy for how education should be designed needs to think about it this way. It's not about the best individual outcome possible, but about maximizing overall benefit for given limits in cost, IQ, time, etc. [Also the vast majority of kids taking calculus for physics, engineering, chemistry and life sciences, not to mention nursing, business, etc. will never need to work their way through Rudin.]
• "For a very bright or accellerated student, there is no reason why they can't go beyond calc 1/2 within high school." In many systems this is explicitly impossible within the standard school day. In a country like Spain there is essentially no elective freedom in high school education (modulo choosing the science track or the humanities track). This is probably more the common situation than not. – Dan Fox Dec 28 '18 at 19:17
• If you learn a course over the summer, than you accelerate. Surely this is the normal way to accelerate (versus during the year). If schools prevent you from doing that, then that is another matter. But it doesn't really have to do with hours during the school year. – guest Dec 29 '18 at 4:13

There's certainly a tradition of dissent from the "ladder" approach to mathematics education. As examples I would recommend A Mathematician’s Lament [PDF] and In Defense of Mathematics and its Place in Anarchist Education [PDF].