Should calculus be taught in high school?

[Angry Citizen]

In my opinion, part of the problem is that mathematics is taught in 'blocks'. For instance, an entire course on geometry, or an entire course on precalculus mathematics, or an entire course on algebra, or an entire course on differential calculus. I think that's a mistake. I think attempting to integrate the various 'branches' of mathematics may be useful to link various related concepts. I see no reason why a course which is predominately geometry based cannot introduce the concept of an integral as a Riemann sum. I also see no reason why the derivative cannot be introduced when defining the slope of a line.

Mathematics education in high school seems to be about teaching algebraic techniques to apply to functions or expressions. I think that removes the intuition that is key to understanding mathematics properly. For instance, I daresay many high school kids would be overwhelmed by an application of basic kinematics if they were forced to derive an equation without a certain variable. Kids don't even realize that if you define 'y' to be some function, then even in other equations you can substitute 'y' as that function. That betrays a fundamental misunderstanding of the equals sign for God's sakes!

Obviously, I support removing calculus from high school programs and implementing a rigorous algebra course that is integrated with some basic calculus techniques. Introduce it early. Make them think like the early mathematicians who had no idea what a limit was when they defined the integral. Following the thought processes of the originators is the only way to reproduce the logic in the student's minds.

But take this as you will; I'm just barely into calculus II right now, so this is a student's perspective.

 

[G037H3]

High school kids shouldn't take calculus, as they don't have the right prerequisites. I agree with mathwonk.

 

[mathwonk]

I take it for granted that in this discussion "take calculus" means taking a standard course from a standard book. I do not object to anyone who truly understands something trying in some way to convey those ideas to very young people. E.g I myself am guilty of teaching euler's characteristic to 3rd graders. I do not say I taught a course of topology, I merely handed out cardboard models of polyhedra with colored sides and asked them to count the numbers of edges, faces and so on, and then reflect on the results. One little 7 year old girl noticed euler's formula. she later became an aeronautical engineer.

similarly one can illustrate "cavalieri's" principle (known to archimedes) in a geometry class, or just by shoving a deck of cards over at a slant. pappus theorem is also quite intuitive. On the contrary, in a course on "calculus", concern with rigorous proof often means omitting Pappus theorem even from a standard such course, whereas it could have found place easily in a discussion of ideas.

So I think one must distinguish somewhat between a "calculus course" and the ideas of the great thinkers who created it.

Of course people who understand things sometimes teach also their standard courses this way. One of my university colleagues teaches geodesics in differential geometry to undergraduates by handing out hard boiled eggs and having them draw "shortest curves" on them.

 

[Klockan3]

I think the problem is that the faster you throw new stuff at the students the more will they rely on memorizing rather than understanding, but at the same time we can't make them go too slow so we have to do a compromise. A some are ready for calculus in high school while others aren't, prerequisites doesn't really matter you have to take the hard steps sooner or later and you can take them before/during/after the calculus course it doesn't really matter.

No matter how we do it most won't understand what they do when they do maths. The only way to alleviate this is to put more good people into education, but that can be said about everything in the world, we would need more good people everywhere in our society and we have an abundance of bad people so of course that will be the case of our teachers as well.

 

[DR13]

Nothing should be taught to students who are not ready for it. Calculus requires prerequisite understanding of polynomial algebra, trig, geometry, and preferably logic. Hence most high school students should not be offered it. But since it is considered a political coup for a high school to offer calculus, most of them have made room for it by deleting their previous Euclidean geometry courses, replacing those by phony precalculus courses. This is ludicrous. To make room for a calculus class by deleting its proper prerequisite? Duhhh. I agree with post 2, teach it if you will, but do not deceive students by offering college credit for it. I guarantee you if you take my college class it will not be the same as your high school class, unless you went to Bronx high school of science or maybe Andover, and maybe not then. I have been told by the high school coordinator of one of the top private schools in the state that anyone who has had a college class in calculus is qualified to teach it in his high school.

1. I knew lots of kids from my graduating class who were plenty prepared for calculus. I took AP Calc BC junior year (then calc 3 and diff eq senior year). I don't even know what I would have done if I had to wait until college to take calculus! I would have been bored out of my mind!
2. Why not offer college credit? I am happy that I got the credit for taking the course. I had a thurough knowledge of the subject so I deserved the credit. Plus, if I had to go back to calc 1 when entering university I would be repeating 2 years of math! That would be a huge waste of time.
3. You don't need to go to a fancy private school to get a good math education. I went to a public high school and the calc 3 class I am taking at university is exactly the same as what I learned in high school. (By the way, I took my calc 3 and diff eq at my high school, not dual enroll).

 

[mathwonk]

Well i don't know you and it is of course quite possible that everything you say is correct, DR13. If so however, then you are very different from almost all the students I have met in my professional career lasting over 40 years. In all that time i have seldom met any students at all, who understand even a modicum of calculus, no matter what score they obtained on BC/AP tests and classes.

They were often deceived into thinking they understood college level calculus however because most colleges have had to dumb down their courses to accommodate these AP students. Hence although high school AP students do not understand calculus at what used to be a college level, colleges have lowered the level of their classes so as to prevent all these students from failing.

In the present day curriculum, we now offer three or four different college classes in calculus, at different levels. For the most gifted students, the best advice I can give them is to take calculus again from the beginning in college, but take it in a high level honors class, so as to get the deepest experience of it. I.e. to take a "Spivak style" class.

The reason a student should not take college credit for AP calculus and then begin in sophomore calculus is that he will have moved himself from an honors level high school course to a non honors level college course. I.e. there are almost no (you could of course be another of the one or two exceptions I have met in 30 years) graduating high school AP students who are qualified to begin college in a second honors level calculus class, i.e. a course from say Apostol volume 2, or from Loomis and Sternberg.

The few exceptions tend to wind up at Harvard or MIT, and have prepared by taking genuine college level classes in high school from real colleges, or from super high schools like Exeter and Andover, or the Bronx high school of science.

Hence NOT starting in a first year honors level college calculus class in college is usually doing yourself a disfavor, and lowering the level of your education. I.e. if you take the regular second year course you are likely qualified for, you will never again be able to enter the honors level work and you will never learn it, and you will likely never achieve the level of preparation needed to become a professional mathematician, if that is one of your possible goals.

Here is a little test for you: did you learn to prove that a continuous function on a closed bounded interval has a maximum in your high school AP class? This was covered in the first semester of my college class when I was a student, and I taught it in my first semester honors class at an average state university, not the higher level first semester Spivak class, just the class for people who had done well in AP courses.

Easier: can you state and prove the fundamental theorem of calculus? I teach this even in my non honors classes in college. Of course if you can really do these things, then indeed you have learned a lot in your high school classes and your preparation is unusual. But very few students at my university have this preparation from high school. It is certainly not included in the usual AP syllabus or tests I have seen.

 

[Sankaku]

...you will never again be able to enter the honors level work and you will never learn it, and you will likely never achieve the level of preparation needed to become a professional mathematician...

Wow - that is a pretty severe statement. Just because someone doesn't enter the course-stream where YOU think they should curses them forever to not understand math? Give me a break!

There are many routes to the same goal. Just because you have one sanctioned path for the blessed does not mean it is the only one.

 

[mathwonk]

i am apparently not being clear. There is not just one stream, there are several streams. At Harvard when I was a student there, one could take the 3 non honors calculus courses math 1,20, 105, or one could take the honors courses math 11, 55, and then go right into graduate courses.

In the non honors courses the subject was taught in the traditional way, old fashioned approach more common in physics, more numerical, less conceptual. This is paralleled at UGA today by our non honors sequence math 2250,2260,2500, and perhaps 4100, or our honors sequence 2400, 2410, 3500, 3510, and then maybe 4200, 4210, or also 4100.

In the honors sequences the material is taught in a more modern way, with more use of linear algebra, more topology, the way a practicing mathematician uses it.

But a student only takes one sequence, not both. Thus the students who take the non honors sequence never see the modern approach at all, and usually by the time they finish, they do not have time to go back and do it all over again, or maybe not even the mental flexibility, having already learned to think in the old way.

Thus I noticed at Harvard that students who knew more than I did, and seemed smarter than me, were nonetheless not learning the more powerful approach to the material that I was, and ironically it was precisely because they had not started at the beginning at Harvard, but had gotten their start in prep school.

Thus not only did they get a lower level version of the math, they also had contact with less strong students, non honors students, and they also did not get the most stimulating professors who tended to teach the honors courses.

So they never got the same perspective on the material. Moreover this lack of stimulation caused some of them to begin to find the subject uninteresting, and eventually to drop out. They were not getting the stimulating viewpoint, the stimulating professors, nor contact with the most stimulating peer group.

This eliminates them from consideration for admission to top grad schools, although it is true there are other schools where they can perhaps slowly come up to speed.

You are of course correct it is possible to take many different paths to ones goal, but it is harder, and fewer people find it, especially people who do not have the wisdom to listen to their elders.

 

[DR13]

To mathwonk:

Trust me, I am no genius. I am in the non-honors calc sequence (by choice). And even in that class I am no genius. I would say that I am in about the 75th percentile (pretty solid but nowhere near genius). There were kids at my high school that were much smarter than me. If I do not have a problem starting with calc III then these kids definitely will not. Also, I am in engineering so to be honest I do not want to take an ultra-rigorous calc class based on proofs and what-not. I do like math but I do not love it.


One note on what Astronuc said: I definitely agree that there is too much redundancy in pre-high school math

 

[Fragment]

I think a fundamental issue here is determining how rigorous of an understanding one wants. Surely engineering students do not need a Spivak-style course, where some math students do. However, this is not a rule, having taken a few courses in higher mathematics myself, and having seen the average student in those classes, I can attest that you need not know the basic of anything in order to pass. Regurgitation and memorization is still a wide-spread means of passing courses, especially in mathematics.

 

[mathwonk]

Well I could be wrong, I am just reacting to a lifetime of trying to teach students who went from high school AP to one of my own college classes. They never had the same background from high school that I would have given them in college. Now many of my friends did lower the level of their classes to accommodate this high school preparation but I found it hard to do that.

My attitude was that this was bad because the reason they took AP classes in high school was because they were smart honors students. Thus it made no sense to me that the best honors level high school students should be funneled into the non honors college classes.

You may be the best judge of what course you should take to meet your own goals provided you understand the options, but if you are undervaluing yourself, and not giving yourself the most challenging and useful background you deserve, then it is the job of your college counselor to suggest you try something else.

But for someone who does want to be a mathematician, and who thinks that a top student proves it by skipping calc 1, and going into non honors calc 2 or 3, I am just trying to explain to those people what they are really doing.

One of the saddest group of people I see is the first semester incoming class who have signed up for my calc 2 or calc 3 class, thinking they are prepared from an AP class in high school.

It puts me in a bind because I have to choose between losing a lot of them, or else dumbing down the course below what a college class should be. Both choices harm my students. I just wish the weaker ones would listen to me and retake calc 1 but on an honors level, or in some cases the non honors calc 1. These students are seduced by the offer of free college credit. Colleges know that AP classes are not really worth college credit but they feel pressured to offer it because the students will go elsewhere where they can get the credit. So in some cases AP credit is a sort of dishonest bribe to bring in strong students.

Unfortunately i admit the choice is even harder because not all professors are the same and many have chosen to dumb down their college classes. Taking one of those calc 1 classes would be mistake. This is why I emphasize interviewing the professor first to be sure you are in the right course.

There is no one size fits all program, but non honors college classes in calculus are often not intended for future mathematicians, and it seems unfortunate to me for the profession, if most honors high school AP students wind up in those without realizing this fact.

 

[thrill3rnit3]

I feel you with your concern mathwonk, but if the CollegeBoard tried to increase the level of difficulty of the AP Calculus courses, then there wouldn't be enough teachers qualified to teach it. Then the schools would just drop AP Calculus altogether and stick to the "PreCalculus" course that is nowhere to what it should really be.

I think if they are going to make a change, it should start early on because changing it in the high school level, I feel, would be too abrupt and there wouldn't be a continuity from the math courses they were used to taking into this all-new "rigorous" style of mathematics.

Again, I feel that the issue lies with the lack of proper instruction due to the lack of qualified teachers, and with the course already hard as it is (to a regular student), increasing the difficulty would need a simultaneous increase in motivation.

I really envy the students that have/had the chance to go to schools that have a really good mathematics programs. I'm a senior right now, and my school would probably fit with the 98% of the schools in the country in terms of mathematics curriculum.

 

[mathwonk]

Maybe I am giving the wrong advice here (although I think most of my colleagues agree with it.) But maybe I should just warn people that although AP courses and tests are a fait accompli in high schools, and that many college professors teach the same course as in high school, nonetheless there are some old dinosaurs like me, who still teach the way they were taught in the 1960's, believing that challenging courses are more useful than ones in which an average decent student is guaranteed an A.

Since these courses are much harder than most AP courses (based on the questions on AP tests I have seen) a student needs to be careful to understand what type of course he is getting into by either interviewing the professor or some of his previous students, and perhaps looking at some materials from the course in the recent past.

Along those lines, although this may be irrelevant since I am now retired, and maybe few other people teach like me, here are some of the tests I gave in an honors level calc 2 course a while back, that was taken by first semester students which had only high school AP preparation. As I recall, they largely felt it was the hardest course they had ever had, and at least one student dropped out after test 1 because she only got an A-.

 

[mathwonk]

heres another one.

 

[DR13]

So I looked through the tests (didnt actually do them but I was honest with my self). The first one was super easy. The second one would trip me up a bit as of this moment but if I actually studied for it then it would not be a problem. The third one would be the only one to give me real trouble as we did not cover series that well in my AP calc BC class.

 

[thrill3rnit3]

Lipschitz continuity in a Calc 2 class? I like it

edit: It's an honors class so I think it's cool. I thought it was a regular Calc 2 class.

And I take it that you never use LaTex typeset mathwonk?

 

[mathwonk]

this was an ordinary honors class, not a high level spivak honors class. this class is taken by anyone in our general "honors" program.

 

[mathwonk]

DR13. I am curious as to your answer to III i in test one. (Hint: It is a theorem of Newton.)

 

[mathwonk]

Well I thought it interesting to generalize the fundamental theorem of calculus. I.e. if f is continuous then an indefinite integral G of f is characterized by being differentiable everywhere with derivative equal to f. So I thought it should be interesting to describe an indefinite integral G of any riemann integrable function f. It turned out to be any G which is differentiable wherever f is continuous with derivative equal to f at those points, but also G is lipschitz continuous.

I.e. I was fully aware that every indefinite integral of any integrable function was continuous but i did not realize they were also lipschitz continuous and that without this one cannot nail down a G which computes the integral of f. I.e. if f is only integrable, there can be continuous G, which are differentiable with derivative equal to f wherever f is continuous, and yet G(b)-G(a) does not equal the integral of f. Whenever I teach a course I rethink all the material and try to introduce something new, so it is not the same every year. I am trying to pass on the tradition of discovery whenever I teach anything. This is something I like to think a researcher should bring to a course.

 

[DR13]

DR13. I am curious as to your answer to III i in test one. (Hint: It is a theorem of Newton.)

I would say that even if the function does not exist at 1 the integral of f over [0,1] is the same as the integral of f over [0,1) because the integral of any f over [1,1] equals 0.

(Hopefully this is right and I didnt make myself look like a fool. Its been a couple of years since I went over the rules that make a function differentiable)

 

[mathwonk]

Good. That is correct. You can ignore the value at a single point in determining integrability. But it still requires a hypothesis to show it is integrable. In this case the function is not continuous nor even piece wise continuous, so the usual theorems in most books do not apply.

Here the function is integrable because it is "monotone" decreasing. Newton already proved before Riemann that all monotone functions have a definite integral. It is certainly not a sign of foolishness to forget or not have seen this. I am just indicating the level of subtlety that a college course may contain that is not usual in high school.

In most books the theorem that all continuous functions are integrable is stated but not proved. It seemed to me that since most functions people actually encounter are piecewise monotone (e.g. all polynomials, rational functions, exponential, log, and trig functions), it would be nice to give the actual proof that such functions are integrable, since the proof is very easy. I then can also give the proof of the Fundamental theorem of calculus (FTC) for monotone continuous functions, since that is also easy and conveys the full idea behind the FTC.

I.e. to me it is a sign of the carelessness with which many books are written that they give the wrong impression that what they are doing is hard. They just haven't thought about it enough to realize that what they are doing is easy if done right. They just seem to copy the same stuff from one book to another year after year.

How can the student be expected to understand the material if the author does not even think about it deeply? This theorem on integrability of all monotone and hence all piecewise monotone functions does appear in the excellent college level honors book by Apostol. It also appears, with credit to Newton, in the excellent book by Michael Comenetz. That latter book also conveys very carefully the physical intuition behind the concepts of derivative and integral.

It is usual by the way for my students to do fairly well on my test one, which covers mostly material they have seen in high school, and then to bomb on test 2, which requires actually learning something new that has been presented in my course, and learning it rather more quickly than in high school. I.e. the one or two years of high school usually lasts about 3 weeks into the college course.

In this second semester honors course I also presented the L1, L2, and sup norms on the metric space of bounded continuous functions, and proved that sup norm convergence is preserved by taking indefinite Riemann integrals over a bounded interval. This was used to deduce convergence of the derived series of a power series by the usual trick of integrating back and using the FTC.

This sort of thing is sometimes not seen until a senior analysis course in most non honors programs, and essentially never in most high schools.

The concept of lipschitz continuity was presented in order to answer the question: suppose f is Riemann integrable but not continuous on [a,b]. Then we can still define a function H = definite integral of f from a to x, and we will have the integral of f over [a,b] equalling H(b)-H(a). But how do we recognize such an H? I.e. how d we a recognize an "antiderivative" function G for f in this case such that the integral of f over [a,b] must equal G(b)-G(a)?

The answer is that G should be any lipschitz continuous function which has a derivative equal to f at those points where f is continuous. (Since f is integrable, it must be continuous at most points, as Riemann himself showed.)

This sort of thing is probably not done in any high school course anywhere.

In the first semester of the honors course I proved that all locally bounded functions on [a,b] are globally bounded there, in particular all continuous functions on [a,b] are bounded, the main result usually not proved in first semester calculus. Then one derives the mean value theorem and hence the main corollary that a differentiable function is determined on an interval up to a constant by its derivative.

I thought through the usual proofs and remade them into more elementary arguments using infinite decimals instead of abstract axiomatic arguments, to render them easier and more concrete. I have not seen such arguments in any books.

My experience is that even strong high school AP calc students are challenged by my first semester honors course, and that is where I advise most of them to begin. The rare student who is beyond that level is advised to take the first semester spivak style "super honors" course. Hardly anyone is recommended to take a later (second semester or higher) course. The honors level ones are too hard, and the non honors level ones may be as well, but they also run the risk of falling below the honors level of challenge that an AP student deserves.

However it could be reasonable for a student who does not want to be a mathematician, but is interested in engineering, or another application of calculus, and who has the desired level of computational skill in calculus, to begin in a later non honors course. This is provided they are not interested in learning calculus at a theoretical level and are happy in a non honors class as a means to a practical end, and may not be as intellectually challenging.

 

[deluks917]

Do your students find the idea of metric confusing? How much of your honors class is spent doing the standard epsilon delta proofs? L1, L2 is somewhat advanced for calc2. Everything in your post seems like it would make for a great class but it does seem pretty hard.

 

[mathwonk]

I think they found the class hard. One of them said it was the most challenging math course he had ever taken. Then idea that courses should be easy is not one I adopt.

 

[DR13]

I think they found the class hard. One of them said it was the most challenging math course he had ever taken. Then idea that courses should be easy is not one I adopt.

Just curious. How do you curve your course? Do you curve the class average to some grade or alter your grading scale or go curveless?

 

[mathwonk]

well when the class is hard I try to give higher grades. I am trying to challenge but also encourage people. If they do poorly I adjust, and feel somewhat responsible for their poor preparation. In a really hard honors class like the one I described I try to give mostly A's and B's. In that class a C means they did not really get much of it. I think all those who stayed got A or B.

 

[kramer733]

well when the class is hard I try to give higher grades. I am trying to challenge but also encourage people. If they do poorly I adjust, and feel somewhat responsible for their poor preparation. In a really hard honors class like the one I described I try to give mostly A's and B's. In that class a C means they did not really get much of it. I think all those who stayed got A or B.

Is it true that colleges have exams backing up to their earlier days in the library? Can you give us an example of 1960s level course load and questions from the exam?

 

[Kindayr]

I think its incredible that Ontario schools hardly even begin to touch the concept of Calculus. The content is shared in a 1 semester course in grade 12 with an introduction to vectors. So you have grade 12 students going into university with 2.5 months of elementary Calculus (half of which is taken up with limits) and 2.5 months on elementary linear algebra. (we barely learn derivatives, let alone begin to discuss anything past or including integration).

Going into university, students have only briefly been introduced with 2.5 months of "calculus" before a 2 month break. This forces university calculus to have to waste time on simpler concepts due to the failures of our education system.

The worst of it is you have math majors, like myself, put a complete disadvantage without even knowing it until the damage has been done. That is, we waste university credit on learning what would be considered high school concepts to our US counterparts.

I'm going into my third year of university and I legitimately feel that I've wasted time and money in my first two years of school.

 

[Ryker]

The worst of it is you have math majors, like myself, put a complete disadvantage without even knowing it until the damage has been done. That is, we waste university credit on learning what would be considered high school concepts to our US counterparts.

I though the Canadian high school system in general is better than the one in US. Is that not the case then? Or is it just Ontario that's so bad comparatively? I know many countries have a better high school system than Canada, but I didn't think it was even worse than in the US. I don't know, I think you're giving the US credit where it's not due. From what I understand, there are very few people that actually take advanced courses, whereas the vast majority gets an education that not many people in the world would envy. And I guess the regular versions of introductory maths courses really are just covering what high school should've covered, but a lot of universities in Canada now offer Honours courses, and no AP course that students in the US are taking covers what is covered in those. So by taking those I don't see how you'd be wasting university credit.

 

[Kindayr]

I may be overtly bitter about the situation. However, the unfortunate, cold, hard fact is that calculus is next to non-existent within Ontario high schools.

Its not that there isn't demand, because I know of at least 3 full classes (~25 students each) we held at my school, that has a lower population relative to other high schools in my area (~850 students). A large portion of my graduating classes went into business, biology, economics, and a relatively large group that includes math, physics, engineering, actuarial, and stats. So there should definitely be a reworking which includes calculus being taught earlier. I think math in general in elementary and high school in Ontario is bogus.

Grade 9 you're introduced to linear functions. Grade 10 is quadratics with an introduction to trigonometric. Grade 11 has a focus on quadratics including translations and transformations, etc. Grade 12 Advanced Functions goes beyond quadratics. And Grade 12 "Calculus and Vectors" works with lines, planes, and some projections (2.5 months), and then non-rigorous limits with introduction to derivatives.

I feel that is just not a good system, straight up.

I'm not necessarily at a disadvantage in my own university calculus and math courses (though, friends from BC already had some workings with integrals). But in comparison to other students at schools in the US where they are ahead of where I am. I just feel that some time was stolen from me. Yes its probably a failure of myself for not getting interested in these topics when I was younger. I just feel that a lot of this content can be taught at much, much younger ages.

Just look at the experiments the Khan Academy is doing in that one elementary school. Supposedly a larger portion of the students are learning more material and harder material, than what a normal math class would cover. Some of the students are even doing pre-Calculus and Calculus content in the 6th or 7th grade. Its not that this school is a school for genius children. I think it just goes to show how accessible this content is to younger children.

 

[Ryker]

From what you described, you really didn't cover that much. I can't remember my exact curriculum from back home, but I know we covered integration. However, I'm now studying in Alberta, and to me it seemed as if other students from the province knew that topic already, as well. I knew there were differences in provincial high school systems, and I've also read that Alberta's is one of, if not the best in Canada, but I assumed this is such a basic topic that everyone covers it.

What's funny is that I got rejected by an Ontario-based university due to my supposed lack of Chemistry prerequisites, even though I took three years of it in high school. The only thing I didn't do is take it in my last year, but you don't really introduce much new stuff then, you just consolidate knowledge for the final exams, the results of which are looked at by home universities when deciding upon admitting students. If Chemistry in Ontario is anything like maths you mentioned, then that's even more hilarious now. No sour grapes, either