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

Should calculus be taught in high school?

  • #101
1,101
3
Well the simple solution is to cut AP Calculus. If not that many students demonstrate interest or commitment, then many probably won't pursue math/science as a major. In this case, they should take stats instead of calculus.

On the other hand, the syllabus of a precalculus course should be fairly flexible, so maybe varying the emphasis of the topics covered may help. I felt that in my precalculus class, we covered a few topics that were not particularly helpful for the subsequent AP Calc course. For instance, there was no need to cover trigonometry in great depth. The basic identities and reasoning with the unit circle should suffice. We also covered vectors, conic sections, and applications of complex arithmetic (up to DeMoivre). Although these topics may be of interest, they should not take the place of more direct ways of building algebraic manipulation skills if the students need it. I think a good precalculus curriculum should emphasize on reinforcing algebra skills, introducing basic trig, and then move straight into limits and derivatives.

Or yet another way is to teach geometry before algebra 2. My high school did not have an honors algebra 2 course, and that might be why algebra 2 was taught first. The more motivated students took algebra 2 in 8th grade, so when I moved to my new high school, I took honors geometry with them freshman year. Then I took algebra 2 and then precalc. I think it makes a lot more sense to teach precalculus right after algebra 2. The algebraic manipulation skills in typically encountered in algebra 2 are crucial.
 
  • #102
thrill3rnit3
Gold Member
713
1
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?
 
  • #103
1,101
3
If I had to pick one of the most utility to high school students in general, I would pick linear algebra, despite that fact that I have more affinity for calculus. I mean just on the surface, vectors and matrices and their underlying theory seem far more applicable in a general scope than derivatives and integrals. I don't think specific examples would be that hard to find.
 
  • #104
866
0
I just wanted to add my personal experience to the mix here. I took BC calc in high school as well as some other AP courses. The courses were challenging but most of my class did well. My first semester in college I started in calc3 (multivariate) and the second semester of the introductory calculus based physics series.

My GPA would have been higher if I hadn't skipped those initial courses, but I ended up with Bs anyways. I'm very glad that I took the AP courses and got a jump on college. I was able to double major with honors in both mechanical engineering and philosophy. Without my AP credits that would have been impossible. I feel that I learned a lot more in college and am much better off now because of the jump I was able to get. It opened up a lot of doors that would have been closed otherwise. The only downside was that I bit off slightly more than I could chew early on, but I would much rather see students have the opportunity to be challenged and face their limits than be held back.
 
Last edited:
  • #105
Well Calculus was a compulsary part of your Math courses in the last two years of my high school. I think most people found it easy and had more problems with co-ordinate geometry where a lot of algebraic manipulaion was involved.
 
  • #106
1,341
3
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.
 
  • #107
157
0
I think that this issue will always come up and that there is no simple answer. I believe what happens alot 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.
 
  • #108
58
0
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.
 
  • #109
thrill3rnit3
Gold Member
713
1
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.
 
  • #110
Astronuc
Staff Emeritus
Science Advisor
18,704
1,717
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 2x = 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.
 
Last edited:
  • #111
82
0
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?
 
  • #112
422
74
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.
 
  • #113
1,341
3
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.
 
  • #114
274
3
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.
 
  • #115
hotvette
Homework Helper
988
3
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.
 
  • #116
422
74
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.
 
  • #117
274
3
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!
 
  • #118
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.
 
  • #119
mathwonk
Science Advisor
Homework Helper
10,765
937
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.
 
  • #120
Astronuc
Staff Emeritus
Science Advisor
18,704
1,717
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, . . . .
 
  • #121
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.
 
  • #122
277
0
High school kids shouldn't take calculus, as they don't have the right prerequisites. I agree with mathwonk.
 
  • #123
mathwonk
Science Advisor
Homework Helper
10,765
937
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.
 
  • #124
601
0
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 wont 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.
 
  • #125
150
0
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 dont 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? Im 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 dont 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).
 

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

Replies
17
Views
12K
Replies
15
Views
2K
Replies
43
Views
33K
Replies
5
Views
7K
  • Last Post
Replies
8
Views
6K
Replies
37
Views
5K
Top