Should calculus be taught in high school?

[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

 

[Kindayr]

You're not alone in the stupidity of the Ontario system. My house mate is from BC and took his version of grade 12 Calculus where they covered some integration, and more differentiation than we ever did. However, his credit didn't count, so he had to take Calculus 1100A here at UWO, which is Ontario's grade 12 course +first year calculus all in one package.

Just doesn't make sense lol

But i still hold the fact that Calculus can and should be taught much earlier in a student's career. I do know for sure that any of my children will be learning very elementary logic (I'll be proud if they can understand implications) at a young age, and will be taught mathematics at a healthy rate to coincide with their mental progression. I won't be doing this to force them into math-oriented studies, but to just have access to the critical thinking and creativity that comes along with problem solving in mathematics and use its where ever they wish too.

But that'd be in a perfect world.

 

[kramer733]

I'm not going to lie but from grade 1-6 was completely useless and we only learned the 4 orders of operation.

Then in grade 7, we were finally introduced to the idea of integers.

grade 8, it was about accepting pythagorean's theorem and order of operations.

grade 9, we were introduced to cartesian plane.

grade 10, we were heavily doing up quadratic equations

grade 11, we were introduced to more functions.

grade 12, was a review of grade 11 and we got into a class called (calculus and vectors).

Basically we can learn all this stuff in 4 years. But the school program prolongs it. Along the way, we're introduced to geometry and accepting the truth of what geometry is without proof.

We can easily condense the material in 4 years too. Grade 1-6 is actually useless. We can definitely learn math at that age. We aren't stupid. In high school, we should be doing 2+ classes of mathematics with emphasis on proofs.

It's completely garbage.

I'm definitely not talking trash to the teachers though. It's not their fault. It's the school boards' fault. I'd like to one day change that though. I believe we can actually cover 3rd year university math classes when high school is done with.

Most people are not stupid at all. It's just that they are lazy.

 

[mathwonk]

this may not be comparable to a 1960's exam but this was an actual exam i gave at the university of georgia in 2004 in an honors, but not elite honors, calculus class.2310H final 2004 Exam, Smith,
I.(i) If f is a function defined on [a,b] and a = x0 ≤ x1 ≤ ...≤xn = b is a subdivision of [a,b], describe what an “Riemann sum” for f means, for this subdivision.

(ii) Define what it means for f to be “integrable” on [a,b] in terms of Riemann sums.

(iii) State two essentially different properties, each of which guarantees f is integrable on [a,b].

(iv) Give an example of a function defined, but not integrable, on [0,1].

II. (i) If f is defined by f(x) = 1/2 for 0 ≤ x < 1/2; f(x) = 1/4 for
1/2 ≤ x < 3/4; f(x) = 1/8 for 3/4 ≤ x < 7/8; ...; f(x) = 1/2n for
(2n-1 - 1)/2n-1 ≤ x < (2n -1)/2n; and f(1) = 0, explain why f is integrable on [0,1], and compute the integral. (The FTC is of no use.)

(ii) If f is defined on [0,1] by 1/sqrt(1+x4), explain why f is integrable, and estimate the integral from above and below. (The FTC is of no use.)

III. Compute the area between the x-axis and the graph of y = sin2(x), over the interval [0,π]. (At last the FTC is of use.)

IV. Compute the arclength of the curve y = (x2/4) - (ln(x)/2), over the interval [1,e2].

V. A solid has as base the ellipse (x2/25) + (y2/16) = 1. If every plane section perpendicular to the x-axis is an isosceles right triangle with one leg in the base, find the volume of the solid.

VI. Find the area of the surface generated by revolving the portion of the curve x2/3 + y2/3 = 1 lying in the first quadrant, around the y axis.

VII. Compute the following antiderivatives:
(i) =

(ii) =

(iii) =

(iv)

VIII. Determine whether the following series converge, and if possible, say explicitly what is the limit. Explain your conclusions.
(i)

(ii) 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 ±.....
(iii)

(iv)

IX. Compute the volume generated by revolving the plane region bounded by the x-axis and the curve y = 4 - x2, around the line x = 5.

X. Any function f: R+-->R which is
(i) continuous, (ii) not always zero, and (iii) satisfies f(ax) = f(a) + f(x) for all a, x >0 is a “log” function. Using this, prove that f(x) = is a log function, using appropriate theorems. [Hint: You will need to show f’ exists and then compare the derivatives of f(x) and f(ax).]XI. We know the only function f such that (i) f is differentiable, (ii) f(0) = 1, and (ii) f’ = f, is ex. Assuming an everywhere convergent power series is differentiable term by term, use the previous fact to prove that converges to ex. [Hint: First prove it converges everywhere.]

XII. Use the fact that y = tan(x) satisfies the diferential equation y’ = 1 + y^2, to find at least the first four terms of the power series for tan(x). Compare the coefficients to what Taylor’s formula a(n) = f^(n)(0)/n! gives you.

XIII.
a) If f is a continuous function on the reals, with f(1) = c > 0, what else must be checked to conclude that f(x) = c^x for all x?

b) If a,b are positive numbers, use the method above to prove that the function f(x) = (a^x)(b^x), equals (ab)^x.

 

[mathwonk]

it says i have attached a pdf file of this exam but i don't see it.?

 

[eumyang]

I'm definitely not talking trash to the teachers though. It's not their fault. It's the school boards' fault. I'd like to one day change that though. I believe we can actually cover 3rd year university math classes when high school is done with.

Thank god you're not in charge, then. No one would be able to graduate from high school.

On one hand, I agree with you that some of the material can be condensed, but on the other hand, some kids end up taking Algebra I too early because they can't handle the level of abstraction required.

I've also heard that some students are entering Calculus not prepared because of their weak Algebra skills. Is it because those students received a condensed treatment of their Algebra courses?

And yet... I've heard that in Asian countries like Japan and Korea, it's the norm to reach Calculus before finishing high school. My Korean is not that great, but from what I read, in Korea, a student in the liberal arts track can take an introduction to Calculus course. (It's not clear whether they HAVE to take this course, or it is an elective.)

 

[xdrgnh]

A lot of the posts on this thread disgust me, to those that suggest Calculus shouldn't be taught in high school are freaking pure math eletists. I have great respect for mathematicians and a lot of them are great people and good to know. However some them are math nazi's, if it doesn't involve rigor and proofs they will discredit it. Most people don't need pure calculus with the proofs and rigor they just need to know it's conceptual meaning and learn how to do problems that will arise in practical applications. Students who want to be physicist or engineering really need to the college credit for calculus so they can learn mechanics and E@m probably. Most good colleges require students to use multivariate calculus in E@M and it's good to have multi for mechanics to understand the line integral of work. So to you math elitists don't try to force everyone to learn pure and rigorous mathematics because it's unnecessary and in some cases harmful to those who want to just know how to apply to the real world.

 

[kramer733]

A lot of the posts on this thread disgust me, to those that suggest Calculus shouldn't be taught in high school are freaking pure math eletists. I have great respect for mathematicians and a lot of them are great people and good to know. However some them are math nazi's, if it doesn't involve rigor and proofs they will discredit it. Most people don't need pure calculus with the proofs and rigor they just need to know it's conceptual meaning and learn how to do problems that will arise in practical applications. Students who want to be physicist or engineering really need to the college credit for calculus so they can learn mechanics and E@m probably. Most good colleges require students to use multivariate calculus in E@M and it's good to have multi for mechanics to understand the line integral of work. So to you math elitists don't try to force everyone to learn pure and rigorous mathematics because it's unnecessary and in some cases harmful to those who want to just know how to apply to the real world.

How can understanding where something comes from be detrimental to somebody? If anything, they'll have a better understanding of calculus. Knowing where something comes from is useful. It's not just memorization of the formula but proofs are formal pieces of writing that make people understand where something comes from.

 

[xdrgnh]

How can understanding where something comes from be detrimental to somebody? If anything, they'll have a better understanding of calculus. Knowing where something comes from is useful. It's not just memorization of the formula but proofs are formal pieces of writing that make people understand where something comes from.

There's nothing wrong with understanding and talking about the origins of it. I'm referring to putting more emphasis on proofs then actual problem solving. Try to teach a 1st grader why 1+1=2, that would be detrimental to them learning addition. Try to teach limits using delta's and that would confuse someone in high school who doesn't intend to go into pure math and would intimidate him. I got nothing wrong with proofs being used in class to help to understand the concepts but they shouldn't be the center piece. Some people here say that the problem with teaching calculus in high school is that it's not rigorous enough, but most students don't need a rigorous class at the high school or even 1st year college level.

 

[mathwonk]

i may be a math nazi, as my knee jerk reaction to this question is always "no".

come to think of it though, it is based on a lifetime of experience having to deal with those students who think they learned calc in high school but didn't because the people they learned it from did not understand anything.

i am still probably a math nazi if that means i think i understand it and you don't.JUST KIDDING!

heil geometry!~ stop that! hey peter sellers, cut it out.

 

[WannabeNewton]

Try to teach limits using delta's and that would confuse someone in high school who doesn't intend to go into pure math and would intimidate him. I got nothing wrong with proofs being used in class to help to understand the concepts but they shouldn't be the center piece. Some people here say that the problem with teaching calculus in high school is that it's not rigorous enough, but most students don't need a rigorous class at the high school or even 1st year college level.

I finished Calc BC before the start of this summer and I can say from experience that I had to teach myself at home using Spivak because the Calc BC curriculum lacked in rigor left and right. I walked out satisfied only with what Spivak's text had given me in terms of preciseness; Calc BC just handed out assumptions to students and it was highly unsatisfying. I intend to go into physics but even I, an average high school kid, demand the same mathematical rigor that a pure maths individual would favor because intuition and thoroughness is much, much more important than knowing how to do calculations. You seem to assume that all high school students want the level of rigor that you keep saying should be maintained.

 

[xdrgnh]

I finished Calc BC before the start of this summer and I can say from experience that I had to teach myself at home using Spivak because the Calc BC curriculum lacked in rigor left and right. I walked out satisfied only with what Spivak's text had given me in terms of preciseness; Calc BC just handed out assumptions to students and it was highly unsatisfying. I intend to go into physics but even I, an average high school kid, demand the same mathematical rigor that a pure maths individual would favor because intuition and thoroughness is much, much more important than knowing how to do calculations. You seem to assume that all high school students want the level of rigor that you keep saying should be maintained.

If you like pure math a lot then you should consider minoring in math, I'm all about giving choices just like in college. In college you have the honor sequence which is more theory and then you have the standard which is more application. I went to Brooklyn Technical High school a elite math and science school and the kids who were in the math major loved all of that rigor while the kids in applied science major didn't. I'm just saying the emphasis of standard calculus class should be problem solving rather then theory. My calculus BC had proofs so we all can understand why the power rule works and ect, but it wasn't the main focus and it shouldn't be.

 

[mathwonk]

no serious argument should ever include the phrase "I got nothing wrong with..."

 

[xdrgnh]

no serious argument should ever include the phrase "I got nothing wrong with..."

No serious debater would resort to mud slinging like that.

 

[WannabeNewton]

I went to Brooklyn Technical High school a elite math and science school and the kids who were in the math major loved all of that rigor while the kids in applied science major didn't.

Wow that is freaky, I go to Bronx High School of Science at the moment o.0; we're pretty much neighbors. But I still think problem solving is not as important as the more rigorous conent. To give an example from the general relativity texts I learned from: both Carroll's "Spacetime and Geometry" and Wald's "General Relativity" were rigorous in differential geometry for a physics textbook and it made books like Schutz's "A First Course in General Relativity" much, much easier to work through and Schutz's book was more concerned with problem solving.

 

[xdrgnh]

Well that's GR a upper level physics class that uses abstract math like Differential geometry and tensors. We are talking about calc 1 and 2 which are the foundations of more rigorous math. If the emphasis is proofs and not applied problem solving then they will have a unnecessary harder class. Also again most science students aren't interested in proofs and shoving it down there throats is bad for everyone. If they are interested that is why they have the honors math sequence. Btw I live 5 minutes from Bronx sci but I choose to go to tech because I like being in Brooklyn and Manhattan more then Da Bronx.

 

[micromass]

Well that's GR a upper level physics class that uses abstract math like Differential geometry and tensors. We are talking about calc 1 and 2 which are the foundations of more rigorous math. If the emphasis is proofs and not applied problem solving then they will have a unnecessary harder class.

So we shouldn't challenge students??
And presenting reasons for formula's makes the class harder than just letting them memorize the formula's?

Also again most science students aren't interested in proofs and shoving it down there throats is bad for everyone.

Same analogy: most elementary school students aren't interested in learning. So shoving it down their throats is bad for everyone.

 

[xdrgnh]

So we shouldn't challenge students??
And presenting reasons for formula's makes the class harder than just letting them memorize the formula's?



Same analogy: most elementary school students aren't interested in learning. So shoving it down their throats is bad for everyone.

I'm for challenging students and making the overall math and science curriculum harder but math classes for math students and math classes for science and engineering students have to be different. No one size fits all, especially when we are talking about 1st year classes.

 

[micromass]

I'm for challenging students and making the overall math and science curriculum harder but math classes for math students and math classes for science and engineering students have to be different. No one size fits all, especially when we are talking about 1st year classes.

You keep on making statement like these. But could you actually present some evidence that your statement is valid.
In my country, engineering students and math students take the same rigorous math course. And both sides benifit from it. So I say that it IS possible. Now, can you present some evidence why such a thing is not possible?

 

[mathwonk]

no serious debater would have a picachu as an icon.

 

[BloodyFrozen]

no serious debater would have a picachu as an icon.

But anyways, I think students should be challenged more in high school and have atleast Calculus.

 

[n1person]

I am sure this view has been said before (somewhere in the 11 pages i didn't read), but the fact of the matter is calculus is extremely useful just as a plug and chug type of tool in most of science (biology, basic chemistry, basic physics, lots of engineering), and it is very important students interested in these fields learn it, even if it is just a cookbook sort of way.

And for those going into physics or math or some other similar field it is fine to learn it once the "easy" way and again in more rigor. You will have more experience and intuition the second time around.

I completely disagree with the statement that everyone should learn calculus. For the vast majority of people I think really Algebra (plus the most basic trigonometry) is enough, and I think the fact that we are shoving precalculus down high school students throats is very misguided.

 

[mathwonk]

i will modify my statement and say that i applaud anyone teaching anything that he/she actually understands, to anyone at any age who is prepared for it. I am afraid this may not include many high school AP calculus courses I am aware of in the US, although it does include some of course.

I have just spent the past 2 weeks teaching Euclidean geometry and the ideas of Archimedes to extremely gifted 8-10 year olds at a special camp for them. Nothing was crammed down their throats as these kids loved the subject and were excited to come to this type of camp.

During the process of discussing and analyzing these topics from Euclid I came to believe this is the best possible preparation for calculus.

Euclid discusses area and volume using finite decompositions as far as possible, and then transitions to using limits. Then Archimedes refines Euclid's technique of limits and obtains "Cavalieri's" principle for volumes.

(Euclid's theory of similarity also prepares a student for a careful analysis of the real line and rational approximations.)

Many basic facts about volumes and areas are got out beautifully by Euclid and Archimedes such as the volume formula for a cone, and a sphere, that still challenge many calculus students who think they have learned the subject.

E.g. Archimedes apparently knew not only that the volume of a sphere is 2/3 that of a circumscribing cylinder, but also that the same holds for the surface area, and even that the same facts hold for a bicylinder (intersection of two perpendicular cylinders of same radius) with respect to an inscribing cube.

I challenge any high school AP calc student , or any college calculus student, to prove all this using what he has learned about volume and surface area in his calculus class. These volume problems are among the hardest problems we assign calculus students, and I am not aware of anyone assigning the surface area of a bicylinder in college calculus.

(You AP calc graduates might try it and see. Maybe you'll get it and you can brag to your teacher.) The same ideas of Archimedes, such as the location of the center of gravity of a 3 dimensional cone, allow one to easily calculate the volume of a 4 dimensional ball, without calculus! How many of your AP classes do that (even with calculus)?

My advice to any good high school student is to study Euclid's Elements, then Euler's Elements of Algebra, and then Euler's Analysis of the infinities, as outstanding precalculus preparation. A little Archimedes is also useful but harder to read.

After this one could appreciate a good calculus book.

 

[mathwonk]

Even if the goal is to meet the needs of those scientists who need to use calculus to calculate things, this is not best served by traditional AP courses in my opinion. For those students much less theory should be presented, and questions as to the existence of the various limits which arise should be taken for granted.

The most important ideas should be emphasized with their geometrical meaning. Powerful and useful tools such as Pappus' theorems should always be presented, along with simplifying ideas like centers of mass. Both of these are often omitted even in college calculus classes.

Computation of tricky limits and tricky integrals has virtually no importance in my opinion.

 

[mathwonk]

I have just perused several AP calc syllabi available online and found as expected lengthy lists of tedious topics that make the subject seem hopelessly complicated and impenetrable.

The most important applications are treated briefly and without acknowledgment of the fact that hardly any of the painfully long theory is needed to understand them completely.

Important topics like Cavalieri's principle, the method of cylindrical shells, Pappus' theorems, are not visibly mentioned at all, although presumably Cavalieri's principle is hidden under the heading of "volumes by method of discs and washers".

Nowhere is it made clear for instance that Cavalieri's principle is already obvious just from the definition of volume as an integral, i.e. well before the fundamental theorem of calculus.

I have just read a sample AB AP calc test and found almost none of the questions to have any real interest. The only one that seemed useful to understand was the last question of part 1 on recognizing a slope field form a given o.d.e. most of the rest was just jumping through hoops.

 

[mathwonk]

heres an example of the sort of silliness i am talking about. I just looked up a calculus book by a professor at a major university, in which the problem of showing the surface area of a torus (result of revolving a circle of radius r, centered at (c,0) with c>r, around the y axis) equals 4π^2rc, is posed and a hint given about what complicated integral to use.

2,000 years before the invention of calculus, Pappus knew this problem has the trivial solution length of circle times distance traveled by center of mass of circle = (2πr)(2πc) = 4π^2rc.

Thus even an A student in this class struggles hard for a semester and comes out knowing less than someone knowledgeable from 2000 years ago who has never heard of calculus. The idea of applied math courses is to give people useful tools that make their problems easier, not harder.

Even books found online by famous professors at some of the best schools in the world, present ideas like center of mass and then omit to explain how this is useful in computing work. To give a calculus student a problem of computing work done pumping water from a conical tank and not mention that the center of mass is 1/4 the way up from the base and that this renders the problem trivial, is pretty useless I think.

By the way here (in an attachment) is a discussion of calculating the volume of a 4 dimensional sphere that uses only things Archimedes knew.