• Support PF! Buy your school textbooks, materials and every day products Here!

Should calculus be taught in high school?

161
0
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.
 
324
0
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 highschool, 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

Science Advisor
Homework Helper
10,738
912
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

Science Advisor
Homework Helper
10,738
912
it says i have attached a pdf file of this exam but i don't see it.???
 

eumyang

Homework Helper
1,347
10
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.)
 
413
0
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.
 
324
0
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.
 
413
0
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

Science Advisor
Homework Helper
10,738
912
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.
 
Last edited:

WannabeNewton

Science Advisor
5,774
530
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.
 
413
0
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

Science Advisor
Homework Helper
10,738
912
no serious argument should ever include the phrase "I got nothing wrong with..."
 
413
0
no serious argument should ever include the phrase "I got nothing wrong with..."
No serious debater would resort to mud slinging like that.
 

WannabeNewton

Science Advisor
5,774
530
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 text book 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.
 
413
0
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.
 
21,992
3,274
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.
 
413
0
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.
 
21,992
3,274
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

Science Advisor
Homework Helper
10,738
912
no serious debater would have a picachu as an icon.
 
352
1
no serious debater would have a picachu as an icon.
:rofl:

But anyways, I think students should be challenged more in highschool and have atleast Calculus.
 
144
0
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

Science Advisor
Homework Helper
10,738
912
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.
 
Last edited:

mathwonk

Science Advisor
Homework Helper
10,738
912
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

Science Advisor
Homework Helper
10,738
912
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.
 
Last edited:

mathwonk

Science Advisor
Homework Helper
10,738
912
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.
 

Attachments

Last edited:

Related Threads for: Should calculus be taught in high school?

Replies
17
Views
12K
Replies
15
Views
2K
Replies
43
Views
33K
Replies
2
Views
9K
Replies
5
Views
7K
Top