Should calculus be taught in high school?

[thrill3rnit3]

I believe students should discover such ideas themselves, but nowadays they are just given a list with all the "methods of differentiation and integration" that they MUST memorize if they want to get a 4 or a 5 in the AP test. Totally pathetic IMO.

 

[Tobias Funke]

I'm not definitely not against teaching applications of the mathematics, but it seems like teacher nowadays are too focused on the application that the theory behind it is lost.

I agree. One of the reasons is most likely that the teachers don't know the theory themselves. Especially considering how many people teaching high school math don't have a math degree (many have physics, chem, bio degrees and that's considered close enough I guess). Would they be able to explain how complex numbers came about? How to multiply and divide complex numbers geometrically, and how this illustrates (-1)(-1)=1? How they're no different than integers, rationals, and reals in the way they're formed from a smaller set? How you don't have to expand your set any further if your goal is that every nonconstant polynomial have a root, which is often the motivation for extending R?

Many probably can't, and so complex numbers remain some kind of mystery to students. Just some crazy thing those math people made up for no reason.

Practicality is fine, but nobody seems to ask for it when the kids are reading Huck Finn. How is that practical in 2009? It isn't, but they should read it because it's a great book. Why can't it be the same with math?

 

[thrill3rnit3]

They don't bother teaching the theory because they think it's "too hard for the kids". So instead, they just give the formula straight up, and tells them to plug-and-chug the numbers to get an answer.

But when the question is somewhat different from the sample exercises...they have no clue what to do, because all they've been told to do is "plug the numbers in the formula".

Anyways, this is getting off topic...we should be talking about if calculus should be taught in high school

 

[buffordboy23]

Practicality is fine, but nobody seems to ask for it when the kids are reading Huck Finn. How is that practical in 2009? It isn't, but they should read it because it's a great book. Why can't it be the same with math?

I'm not really familiar with the storyline and events. I was supposed to read it in school but just glossed over it, but it could be practical b/c of the experiences faced by the characters. How did they respond to these experiences and was their response appropriate? If you were in this situation or have been in this situation, what would you do or what did you do? Analysis and reflection are practical processes that we use constantly.

More importantly, Huck Finn is a book written by an author. Therefore it's an artwork, and according to many critics, it's so good that it's considered one of the Great American Novels. Expression through art is supposed to be pleasurable, not practical, so that is why Huck Finn is probably still read in schools today.

After reading Lockhart's article, I agree, math education should incorporate the artistic aspect.

 

[buffordboy23]

I believe students should discover such ideas themselves, but nowadays they are just given a list with all the "methods of differentiation and integration" that they MUST memorize if they want to get a 4 or a 5 in the AP test. Totally pathetic IMO.

Don't forget though that you have a biased perspective. You enjoy mathematics from what I see on your profile description. After being in college and reflecting back on things, you probably now feel that your high school math education ripped you off, and you are right. However, other students that have gone to pursue other majors not in the maths or sciences probably feel like they were tortured, and they are right as well. This is the result of poor structuring of the curriculum and unqualified teachers in the maths...nobody's needs are truly met.

 

[buffordboy23]

The AP calculus test should weed out students with inadequate understanding.

This would then suggest that mathematics is a special subject for only a small subset of the student population. Calculus can be for everyone. It's the different approaches (theoretical, practical, artistic, historical, etc.) that are for certain people but this is not the current focus of how teachers run such a class or how the students in calculus classes are organized.

 

[Astronuc]

Regarding the OP question - Should calculus be taught in high school?

I think it should be optional. I was ready to learn calculus, but many students were not.

Calculus should be available to students who are ready and willing.


Prior to that, I think there needs to be improvements in the way math is taught, so that students are ready for advanced math, but also that students are motivated to learn math.

I knew the utility of mathematics because I was interested in science: physics and astrophysics, so I knew that I needed calculus. I was also competitive in high school, and math and science came easy to me, while other kids struggled with those subjects. Some kids even struggled with trig, geometry and algebra.

 

[Sankaku]

This would then suggest that mathematics is a special subject for only a small subset of the student population. Calculus can be for everyone. (snip)

Yes, this idea of "weeding" people out is dangerous in our educational system. Essentially what you are weeding out is a group of people who may:
a) have a bad teacher and/or an early bad experience with math
b) have had a slightly slower start
c) have taken too many courses that semester
d) have no real interest

Really, you only want to have the last line go away. But if parents and Universities were not artificially pushing High-School calc, they wouldn't have gone into the course anyway. The first three lines are all people that could be good mathematicians. I am very concerned about this image that you have to be doing Differential Equations by age 16 or you will never make it in Math. If you "weed' talented people out of the field, they usually will never come back.

 

[RedX]

By weed out, I meant not being exempt from taking calculus in college. So if you are weeded out, you can still be a mathematician, but you have to take calculus again in college, because you didn't show you understood it well enough in high school.

In some countries like China, you are really weeded out if you don't show talent while in high school. That's not what I meant.

 

[snipez90]

While I have not read through this entire thread, I think some people are getting hung up on the AP Calculus exam. The point of the exam is clearly not to test whether you understand the theoretical underpinnings of calculus. This is the job of an introductory and more advanced analysis course. If a student finds regurgitating material for the AP exam is boring, there is a simple solution: read a more advanced textbook.

As for other students, I very much doubt that they are all memorizing a variety of formulas by rote and whatnot. I don't think any of my friends who are at engineering colleges (such as Cornell and MIT) learned anything more theoretical than what was taught in our calc BC class (which had no proofs), and most of them are just fine. If they really wanted to, they are intelligent enough to study more rigorous mathematics. For AP Calculus, having intuition is important, but knowing rigorous definitions and proofs is not particularly important. For many people, calculus is not even needed. I don't think this point can be emphasized enough. If you forget that you are on a math/physics forum for a moment, you will realize that this is a very reasonable point.

 

[ideasrule]

Using their critical thinking skills and knowledge of trigonometry, the former student realizes that they can accurately measure the baseline from some position to the tree and the angle from this position to the top of the tree and compute the height of the tree to good accuracy. Thus, the question is answered and learning trigonometry has proved useful to the student. There are many more examples that can convey the value of advanced mathematics, but it requires competency on the part of the teacher to show this to students, and unfortunately, this does not generally happen in our classrooms.

I don't agree that providing real-world examples are likely to improve students' interest. In your example, learning the trigonometry needed to calculate the tree's height is only interesting if the student is actually faced with the problem, takes out a measuring tape to measure the baseline, constructs a device to measure angles, and calculates the tree's height using the collected data. That would be interesting because the answer is a meaningful physical quantity of a real object, not a useless number that happens to be on the answer sheet.

I can't speak for other people, but I absolutely hated the "problem solving" questions in math class, many of which were similar to buffordboy's tree example. I considered them pathetic attempts at demonstrating the simplistic math we used was useful. At the same time, I often used math to calculate physical quantities, like the speed of a falling raindrop or the altitude of the Sun, because actually collecting the data was fun, not because the math was interesting. If a homework question asked me to calculate the speed of a raindrop based on somebody else's data, I would have considered that question as boring as the others.

 

[Count Iblis]

I am very concerned about this image that you have to be doing Differential Equations by age 16 or you will never make it in Math.

If you are taught math from an early age on, you can be sure that you will understand topics like differential equations well before you are 16. It is only because we hardly teach math at all in school does it sound impressive if someone has mastered differential equations before the age of 16.

Then given that our eductional system is severely flawed when it comes to math teaching, the typical math professor is almost always someone who was far ahead with math while in school.

The same is true for other subjects that are not taught in school, like music, sports, etc. etc.

 

[Count Iblis]

About real world problems: The fact is that we don't really give "real world problems" to students in school at all. What we do is we give artificially cooked up problems with no relevance at all to practical problems to children.

Real "real world problems" are usually very hard to solve if at all, and require advanced techiques you learn in theoretical physics courses or engineering courses. Giving such real world problems to children could actually make math very interesting. You can then motivate young children to learn calculus and other more advanced topics.

E.g., a high school project could be: "You are given a computer that can only do addition and subtraction. We want to program it so that it can compute all the special functions your calculator can do."

 

[snipez90]

I'm not sure what level of difficulty of problems you are referring to in the second paragraph. Your example in the last paragraph doesn't seem to coincide very well with the aim in the last sentence of the second paragraph. While I think introducing young students to great unsolved problems could certainly perk their interest, there is still the actual job of teaching these kids, and obviously you can't just give random unsolved problems to them. But again, what you described was kind of vague.

 

[snipez90]

If you are taught math from an early age on, you can be sure that you will understand topics like differential equations well before you are 16. It is only because we hardly teach math at all in school does it sound impressive if someone has mastered differential equations before the age of 16.

This is not necessarily true. If the students fail to actually pursue this knowledge on their own, then this idea of trying to teach them a bunch of things just so they could understand differential equations or whatever at 16 is probably not going to work out too well.

 

[Sankaku]

If you are taught math from an early age on, you can be sure that you will understand topics like differential equations well before you are 16. It is only because we hardly teach math at all in school does it sound impressive if someone has mastered differential equations before the age of 16.

Then given that our eductional system is severely flawed when it comes to math teaching, the typical math professor is almost always someone who was far ahead with math while in school.

The same is true for other subjects that are not taught in school, like music, sports, etc. etc.

I certainly understand that. However, just because you have talent doesn't mean you always get a head start. The problem is that we give all the attention to the lucky few who got good teachers, the right courses, and parental support.

There are plenty of stories of people who pick up a musical instrument as an adult and become very accomplished, why not math? I like the "Lament" where it says the worst thing we have done is to make it madatory!

More and more, to get into the "right" school, Calc is becoming "mandatory."

 

[thrill3rnit3]

I don't think calculus should be "mandatory", but I do think that IF they are offering the class, it should be taught by a well qualified teacher.

 

[buffordboy23]

In your example, learning the trigonometry needed to calculate the tree's height is only interesting if the student is actually faced with the problem, takes out a measuring tape to measure the baseline, constructs a device to measure angles, and calculates the tree's height using the collected data. That would be interesting because the answer is a meaningful physical quantity of a real object, not a useless number that happens to be on the answer sheet.

After thinking about, I agree with your point. The problem is illustrative but not of current consequence to the student, so it's not really motivational to learning.

I can't speak for other people, but I absolutely hated the "problem solving" questions in math class, many of which were similar to buffordboy's tree example. I considered them pathetic attempts at demonstrating the simplistic math we used was useful. At the same time, I often used math to calculate physical quantities, like the speed of a falling raindrop or the altitude of the Sun, because actually collecting the data was fun, not because the math was interesting. If a homework question asked me to calculate the speed of a raindrop based on somebody else's data, I would have considered that question as boring as the others.

I like what you said here. Basically, you like the freedom to choose your own problems. You choose these problems because they apply the content knowledge that you have learned. To ask such relevant questions is a skill. By Lockhart's perspective, we should consider it an art, along with answering the question. My tree example would be better suited as the spring-board to ignite the student's imagination and ask such questions like you have shared with us.

 

[Count Iblis]

I'm not sure what level of difficulty of problems you are referring to in the second paragraph. Your example in the last paragraph doesn't seem to coincide very well with the aim in the last sentence of the second paragraph. While I think introducing young students to great unsolved problems could certainly perk their interest, there is still the actual job of teaching these kids, and obviously you can't just give random unsolved problems to them. But again, what you described was kind of vague.

We don't need to focus in unsolved problems, simply on realistic problems, instead of artificially cooked up problems that have no relevance at all. Strangely the latter type of problems are often called "real world problems".

Being able to program a computer from scratch to do what you want it to do is certainly a real world problem. It does not have to be the way things are done in practice. What matters is that in the real world you don't have any artificial boundaries. The real world does not care whether or not a solution requires calculus. Since without calculus you can only evaluate rational functions, there are in practice almost no problems you can do without calculus.

Trigonometry without calculus is cheating, because you are then using your calculator to compute the trigonometric functions. I'm not saying that you cannot use your calculator. But I think students should know at least the basic principles about how calculators (can) compute trigonometric, exponential and logarithmic functions.

 

[Tobias Funke]

If a student finds regurgitating material for the AP exam is boring, there is a simple solution: read a more advanced textbook.

As for other students, I very much doubt that they are all memorizing a variety of formulas by rote and whatnot.

I'm not sure what you mean here. Other students meaning ones who don't find regurgitating material boring? Wouldn't they be the most likely to memorize a bunch of formulas by rote?

Anyway, memorizing by rote is still what most of my students try to do*. Why do they do that? Because they shouldn't be in AP Calculus, but schools make them think that if they don't take a lot of AP courses then they'll never get into the school they want. That's my problem with the AP program. If it was filled with kids who really liked math and were able to do it, it would be ten times better. This could all be solved if they didn't get college credit, and then when they entered the school they could try to test out of the class there.

*For example, if you know the quotient, product and chain rules, why memorize d/dx (tan x)? I ask them that, but they still try to memorize it.

 

[thrill3rnit3]

^^^

I agree. Most kids in my class were too worried about memorizing their differentiation and integration tables. As far as trig goes, all I knew by heart entering the test was the product rule, chain rule, and the derivatives of sin u and cos u and I did fine, even with the trig differentiation/integration.

I've asked a few of my friends personally, and they said that the only reason that they were taking AP Calculus was that they want to go to a good school (UCLA, Berkeley, Princeton, etc.).

 

[snipez90]

I'm not sure what you mean here. Other students meaning ones who don't find regurgitating material boring? Wouldn't they be the most likely to memorize a bunch of formulas by rote?

Anyway, memorizing by rote is still what most of my students try to do*. Why do they do that? Because they shouldn't be in AP Calculus, but schools make them think that if they don't take a lot of AP courses then they'll never get into the school they want. That's my problem with the AP program. If it was filled with kids who really liked math and were able to do it, it would be ten times better. This could all be solved if they didn't get college credit, and then when they entered the school they could try to test out of the class there.

*For example, if you know the quotient, product and chain rules, why memorize d/dx (tan x)? I ask them that, but they still try to memorize it.

Well the way I see it, you are regurgitating material either way. My calculus teacher was not particularly inspiring, but he still made sure many people got 4's and 5's. The easiest way of doing that is spending the couple of months before may assigning every Free Response packet from 1970 to the 2000's. Perhaps I spoke imprecisely, but what I meant was that if people are able to do the calculus problems assigned - well actually we never actually had to do our homework, but let's say the AP FRQ's - they probably don't have that much to be critical of. Many of the classmates I mentioned who went into engineering do not particularly care much for theoretical calculus, but they have the intuition and the computational fortitude. I guess I was responding to earlier posts that complained that the AP Calculus Exam is "not to be trusted" and those who had a theoretical leaning but do not understand how difficult it is to reform the current curriculum anyways.

As for your main point, shouldn't it be the job of the teacher and other administrators to try to persuade those who aren't doing well to reconsider taking the course in the first place? I still think that if one is able to do 80% of the AP Calc Exam correctly, then credit should be given. I don't think that college placement tests are really going to be much more precise in determining the right placement. I can give you two examples. The school that I attend has a very rigorous undergraduate math curriculum (very pure), but the computational portion of the exam was basically the AP Calc BC exam, perhaps easier. Although the free response portion was more theoretical (those who did particularly well on this portion placed into a very difficult analysis course), anyone who could do the computational part will get placement for calculus, or entry into our theoretical calculus course. My friend at MIT found their placement test to be of similar difficulty to the AP Exam as well. If people can do better than 80% on the AP Calc exam, they probably have a good intuitive and computational grasp on calculus, and there is no reason for them to have to do the same thing over again. But instead, you have people who essentially barely passed a math exam getting 5's and thinking they know calculus.

As for memorization, it would be terrible if someone approached everything in calc through memorization, but sometimes it's not a big deal. For instance, no one would really bother deriving the derivative of tan(x) all the time. I mean as long as know how to do it, I honestly don't see how hard it is to just memorize it. I mean if you use something like the derivative of tan(x) often, it really isn't something that's particularly hard to understand that you just all of a sudden forget that it's sec^2(x)?

 

[SonyAlmeida]

I've asked a few of my friends personally, and they said that the only reason that they were taking AP Calculus was that they want to go to a good school (UCLA, Berkeley, Princeton, etc.).

Why does this bother you? I don't see anything wrong with being competitive and demonstrating high achievement.

 

[Tobias Funke]

Just got back from an AP Calculus teacher's workshop. You'd think that we would talk about pedagogy, maybe whether or not to introduce the epsilon-delta definition of a limit, how to prove MVT, etc.

No, we spent almost all of the time doing standard AP problems because the teachers needed it. Think about that if you're entering AP Calc next year. Your teacher may very well have learned the material only a few months before (or possibly still not learned it). Think your teacher can do a straightforward, although tedious, derivative with 3 chains and an ln or tan thrown in? If you're lucky. Think they'll remember to change the limits of integration in a u substitution? Not many did. Think they can determine

\frac{d}{dx}\left(\int_0^{x^2}\sin(t^3)\,dt\right) ?

Don't be so sure. Not once did we discuss how to find a limit algebraically. We plugged points into the good old calculator and were encouraged to have our students do the same. When going over old tests, we noticed how lenient the grading is. A student who wrote "=V(x)" instead of the correct "=V(25)" was given full credit. Someone who defined a function O and then used O to mean two clearly different things in a formula was given full credit. None of the other teachers even noticed this either.

I liked the story about the official grader who started crying during a problem because she finally got it. And this was a simple problem about using the derivative curve to gain information about the function itself! Even the graders don't have to know what they're doing because they have everything laid out for them. If they see V=2,000, give one point, etc.

We discussed in class how to get more enrollment in the program. Well, dumbing down the math for the students is the only way*, and it's quite obvious that that's what's happening.

To summarize, if you or someone you care about enjoys math and wants to enter a career where you may use it, take AP calc at your own risk. DON'T assume your teacher knows what he or she is doing, and please don't skip calculus in college. Wait one year and you'll get a much better teacher. If you're a student who has to take every AP class and join every club to get into Harvard, then take AP calc. Nobody likes you anyway :). And if you respond with "well, my teacher was great!", then good for you. You got lucky. There were 3 or 4 other good teachers with me in the workshop and they were as shocked as I.

*Well, of course the only real way is to fix math education from the bottom up, but nobody, at least no teacher or education "expert", wants to talk about that issue because it's difficult and worthwhile.

Edit: Another scary thing is that courses like this count towards grad credit(in education, not math I hope) and are the basis for teachers to be called "highly qualified". What a joke.

 

[cristo]

No, we spent almost all of the time doing standard AP problems because the teachers needed it.

Wow! What qualifications does one need to become a maths teacher in the US?

 

[Tobias Funke]

Wow! What qualifications does one need to become a maths teacher in the US?

Well, a common complaint among teachers is that students keep getting passed along from grade to grade even though they don't know the math. When these people in turn become teachers, this is what happens. I'm apparently one of the minority who is crazy enough to believe that one should be pretty damn good at a subject before teaching it. I'm no PhD, but I majored in math. I don't know about some of these other people...

But you can't say anything. It's like a blind guy trying to be an art critic, but it's somehow rude of you to suggest that he should find another job. I got death stares in class for bluntly saying that we need more qualified elementary and middle school teachers.

 

[Count Iblis]

When I have to grade, I don't focus that much on whether the student got the correct answer. What matters is if the student understands the problem and understands the techniques needed to solve the problem. Then a student who makes a few errors can get an answer that it totally wrong, while a student who you can tell doesn't really understand much, can sometimes get a correct answer simply by using a correct formula by chance.

 

[Tobias Funke]

When I have to grade, I don't focus that much on whether the student got the correct answer. What matters is if the student understands the problem and understands the techniques needed to solve the problem. Then a student who makes a few errors can get an answer that it totally wrong, while a student who you can tell doesn't really understand much, can sometimes get a correct answer simply by using a correct formula by chance.

Makes sense, but it's not really the main complaint I have. It was just one more thing i didn't like. I would subtract a few points, but when there are 3 points to give for the subproblem, it's a choice between giving a 100 or a 67. There's no real freedom when grading, which isn't a good thing.

The main issue is teacher knowledge. It's scary.

 

[cristo]

Well, a common complaint among teachers is that students keep getting passed along from grade to grade even though they don't know the math.

Yes, but university is the place that rectifies this. I find it amazing that there are maths teachers teaching AP calculus who haven't got a degree in maths! Over here (in the UK) if you want to teach maths at the highest high school level, you need a degree in maths. Thus, I completely agree with you when you say that one should be good at something before teaching it!

But you can't say anything. It's like a blind guy trying to be an art critic, but it's somehow rude of you to suggest that he should find another job.

Not really: a blind man can't help being blind!


Out of interest, do you need teaching qualifications to teach in the US? Do people study education at university then go into teaching a subject, or do they study a subject at university then obtain a separate teaching qualification?

 

[Tobias Funke]

Out of interest, do you need teaching qualifications to teach in the US? Do people study education at university then go into teaching a subject, or do they study a subject at university then obtain a separate teaching qualification?

Yes, you need qualifications. They vary from state to state. I only needed a degree in anything and a passing grade on the (extremely easy) math test to get a preliminary license. But if a school needs a math teacher, even an AP teacher, and they're shorthanded, guess who gets asked? A Chemistry teacher, or a Biology teacher.

So while you need a certificate to teach in most schools, nobody is really checking. As to your remark about universities fixing the problems students have, maybe for math majors that's true. But from what I've seen, majoring in education is a complete joke. Just look at our education system and this makes sense. I think most math teachers have some kind of education with math degree and not an actual math degree, but I'm not too sure about this.

It's becoming more and more clear to me that AP is just a business like any other. How else can you explain the fact that underqualified students are let, and even encouraged, into the program? Our workshop leader was completely fine with saying that most of her students have trouble with precalculus topics like logs and exponentials. Why is this acceptable? Oh yeah, money.

And then when you say anything about the program, it's always your fault for "not seeing" the goals or somehow not understanding a great new way of teaching lol. People in education are wonderful because they're always right, even when nobody knows math!