Should calculus be taught in high school?

[Feldoh]

Why stop teaching classes just because some people don't do that well. I'm sure people have passed all the college math classes with high grades after skipping out of intro calculus classes. It's not really fair to them to be denied taking the classes because of the competencies of others.

 

[kjohnson]

I think that this issue will always come up and that there is no simple answer. I believe what happens a lot is you get kids who took calculus in high school who think they have no need to revisit it in college. It's like they think that the 1yr or semester they took in high school is all there is to it. I have also seen many students who took calculus in high school who can only do basic algebra. They don't really even understand what a function is, yet they are finding its deravitives and integrals. Its the sad truth that many students don't actually understand what is going on, its just rules and formulas to them. For example if I have a problem similar to 'A' i solve it with this formula. It takes the thinking and learning out of the puzzle. For this reason I believe that a deeper understanding of algebra is a better option for most students. There are exceptions such as if the school has a teacher that can connect with students better and make them want to learn. Otherwise I would have to say it is more beneficial in high school to double down on algebra and leave the calculus to college. On a last note, in high school there is rarely if ever any science courses offered that are calculus based. To understand the math better it often helps to see it applied in areas such as physics. With most AP physics courses being algebra based (physics B) I believe it to be more benificial to the students to be mastering algebra at that time.

 

[CheckMate]

In my high school (French HS in Ontario Canada), calculus is taught like the calculus taught in first year universities. Normally, we'd have calculus and vectors but the teachers decided to teach vectors with precalculus and do more "calculus" in the Calculus course. Therefore, the students are generally more prepared for university.

 

[thrill3rnit3]

Well, elementary vector operations are usually taught in precalculus anyways. Stuff like dot and cross product and all that.

Unless you're talking about vector calculus, which is a different beast altogether.

 

[Astronuc]

I was reading the thread "Who wants to be a mathematician" by the good ol' mathwonk (I wish he'd come back), and he asserts that the focus of the high school math program (and AP) should be linear algebra instead of calculus.

Thoughts?

I wish they had done linear algebra in junior high school. I did a program on matrices sometime about 9th or 10th grade, but there wasn't really any tie to linear algebra or systems of equations or rotations.

In fact I found the pure disjoint between mathematics and physics during junior high and high school, as well as at the university.


Richard Feynman apparently kept notebooks as far back as 9th grade.

Having learned the meaning of an exponent as a high school freshman, it was intuitively clear to him that the solution of 2^x = 32 was x=5. As a sophomore, in 1933, he worked hard on the problem of the trisection of an angle with only compass and ruler and had fantasies about the acclaim he would receive upon solving the problem. During that same year, Feynman taught himself trigonometry, advanced algebra, infinite series, analytical geometry, and differential and integral calculus. . . . What is noteworthy about their [his notebooks] content is the thoroughness and the practical bent they display.

Ref: Silvan Schweber, QED and the Men Who Made it: Dyson, Feynman, Schwinger and Tomonaga, Chapter 8, p. 374 Princeton University Press, 1994.

 

[iratern]

Well I don't know much but I went to school in Turkey for 3 years (I am actually Canadian), and they fly through math. I they teach much more topics in high school math than in North America. For example algebra is mostly done by mid 10th grade, and trig is done soon after, as well as probability etc. In 11th grade exponential functions, complex numbers, and introductory linear algebra are taught, as well as series/sequences. 12th grade is devoted purely to calculus which I know is more rigorous than AP calc BC. Also calculators are not allowed, so we all had to have good skills in calculation. Bu the way this is a normal public school I'm talking about. Maybe here in North America students have it too easy?

 

[romsofia]

Well, at my high school they have a lot of options. Currently, we have a freshman in AP calculus BC, then he'll go onto calc 3 and DEQ's and the highest math course is Abstract Math and Linear algebra. I'll be taking Calculus BC next year, which should be fun.

 

[Feldoh]

I went to a public high school and took Calculus BC. Honestly I feel well prepared for skipping a college course on calculus. I'm currently a physics major with a 3.98 GPA entering my junior year currently I've gone through QM and E&M and the level of Griffiths and Stat Mech w/ Kittel (I placed out of the intro physics classes with AP credit so I'm a little ahead).

I think a lot of people are putting calculus into too theoretical of a background. You wish a deeper foundation was taught however what is the point? People tend to learn the math that they need, which is not necessary analysis/abstract algebra. I've taken those classes and I'm glad I had calculus at an introductory level first to be honest.

I'm a TA for calc 1 & 2 and I can tell you that people are learning the exact same things in both settings. The whole point o f doing it in high school is for those students who have demonstrated proper knowledge of the pre-requisites and from what I've experienced the system works under this context.

 

[Wellesley]

Well, at my high school they have a lot of options. Currently, we have a freshman in AP calculus BC, then he'll go onto calc 3 and DEQ's and the highest math course is Abstract Math and Linear algebra. I'll be taking Calculus BC next year, which should be fun.

Who teaches the Calc III and DE? Is it part of a dual enrollment program with a local university? If it isn't and it is at your high school, I would be very leery of a high school teacher teaching DE.

I took BC Calculus as a Sophomore. I really do think Calculus in high school is more beneficial than detrimental. If nothing else, it is a base for Calculus I and II in college.

 

[hotvette]

I'm in favor of going a bit slow. My daughter (HS Junior) is getting calc A during the last few weeks of a 'Math Analysis' class and will have B/C in the fall as a senior. I'm shocked by how fast the material is being introduced. She started just a few weeks ago and has blasted by limits and derivatives (including chain rule, product/quotient rules) and is now doing relatively complicated optimization problems. For the most part she's able to do the homework (because it is so procedural), but I can't believe she really has much of a true grasp of basics. On the other hand, I think she will have a much easier time in College Calculus having been introduced in HS. I'm certainly in favor of no college credit.

My HS (rural America) had no AP or calculus classes at all. I took pre-calc during my first college semester and calc I 2nd semester (Calc II & III were during Soph year). In retrospect I'm glad things were slow and believe I got a great fundamental introduction (having a fantastic professor really helped). I don't feel I was disadvantaged at all by waiting until 2nd semester freshman to take Calc I. On the other hand, that was a long time ago and I need to admit that what was OK a gazillion years ago may not necessarily work today. I still think slow is better. I guess we all have different perspectives based on our own experience.

 

[romsofia]

Who teaches the Calc III and DE? Is it part of a dual enrollment program with a local university? If it isn't and it is at your high school, I would be very leery of a high school teacher teaching DE.

I took BC Calculus as a Sophomore. I really do think Calculus in high school is more beneficial than detrimental. If nothing else, it is a base for Calculus I and II in college.

It's not a dual enrollment, but I've heard that she teaches it pretty well.

 

[Wellesley]

It's not a dual enrollment, but I've heard that she teaches it pretty well.

I wish my school had a teacher willing to teach Calculus III and Differential Equations. It would have made my life a lot easier...

Good luck next year!

 

[sudarshanabcd]

yeah
i think it is very logical that calculus is taught in high school.
i started learning it in 7 th grade. i found it was very logical,just took some time.
it is good for students if they are exposed to the(0/0) concept very early.

 

[mathwonk]

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. Well how does that compare to a course from a professional mathematician with 5-30 years of research experience? It doesn't.

 

[Astronuc]

Does it go without saying (i.e., is it intuitively and blatantly obvious) that students need proper preparation to learn and understand calculus?

Learning calculus in high school necessarily means learning and mastering the pre-requisite mathematics and analyses, and analytical skills.

So then - what is the ideal program starting as early as 3rd/4th/5th grade?

Before 9th grade, I felt there was a lot of redundancy in mathematics. It would also have helped if the math one learned was more explicitly applied (discussed) in science classes. I'm not sure it was obvious to many students that science used tools like simultaneous or systems of equations, or algebra.

At what stage should students learn algebra, analysis, geometry, trigonometry, linear algebra, . . . .

 

[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)