Should calculus be taught in high school?

[cristo]

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.

I agree that it is not ideal for a teacher holding a degree in education should be teaching higher maths! From what I gather of the system over here, degrees in education are incredibly useful for people wanting to teach primary, or lower secondary school (elementary or middle school, in your terminology), since for the former, one needs to teach most subjects, and for the latter, one teaches at least a few subjects. But.. the more complicated stuff should be taught by mathematicians.

 

[thrill3rnit3]

Most of the teachers just obtain a degree in general education and test for a specific subject credential (say math). But that test is like a joke, really.

 

[thrill3rnit3]

I agree that it is not ideal for a teacher holding a degree in education should be teaching higher maths! From what I gather of the system over here, degrees in education are incredibly useful for people wanting to teach primary, or lower secondary school (elementary or middle school, in your terminology), since for the former, one needs to teach most subjects, and for the latter, one teaches at least a few subjects. But.. the more complicated stuff should be taught by mathematicians.

Unfortunately, "mathematicians" here won't go to teaching secondary schools because the pay is low. Of course they would rather work in a university or in a private sector because the pay is much, much higher.

 

[cristo]

Unfortunately, "mathematicians" here won't go to teaching secondary schools because the pay is low. Of course they would rather work in a university or in a private sector because the pay is much, much higher.

By "mathematician" I meant someone with a degree in maths (or maths major, as you lot would say). Some people don't just judge their job on income! Anyway, how much is "pretty low"?

 

[Tobias Funke]

The pay isn't great, but teachers do tend to exaggerate how incredibly poor they are. If you factor in all the vacation time, most veteran teachers make quite a bit of money, at least in MA. The main problem for some is huge, unruly classes and just a general lack of respect from society. Teachers do have to put up with a lot of crap that just gets in the way of actual teaching, and our system doesn't place them correctly or asks too much of them, especially elementary school teachers. So many of them are bad at math and I'd imagine they really want to change this, so we need to do a better job at helping them.

So even though I was shocked that we spent an hour doing a left Riemann sum, I guess that's what teachers need. I'd just feel more comfortable if every AP teacher had to pass the test with a 5. For some reason, I bet that would be fiercely resisted by a lot of teachers.

 

[symbolipoint]

Snipez90, what you describe in your post #82 is ridiculous. What happened to Praxis? What about CSET? What about the meaning of "highly qualified teacher" including possession of minimum of 32 nonremedial units of Mathematics? Saying that teachers do not get misassigned is not for me to say, since I really do know better than that; but the way you described misassignments for teaching of Calculus - ... If that is true, then it is really very disappointing.

 

[snipez90]

Um, exactly what part of my post are you responding to? Since when did I imply anything about teacher misassignments? Much of this thread has been on the focus of the student and I was mainly addressing issues brought up in that regard. The only statement I made about teachers in general was that they should share the responsibility in deciding who should stay in the course. Even if I am wrong on this, I still don't see how this is an extremely pertinent point. In light of Tobias Funke's description, I could see how I could have had a worse teacher. But still, my teacher knew the material, he was just not very good at teaching. Asking students to work through every FRQ and MC test in existence and telling them to discuss the solutions among themselves without further guidance is pretty terrible, but if you learned enough to pass a calculus test, you could probably get a 5, or at least a 4. Of course, Tobias Funke's description of the graders is rather troubling.

 

[symbolipoint]

In clearer wording, misassignment of a teacher is putting a teacher into a situation to teach something which that teacher is technically not authorized to teach; mainly because that teacher does not have the fitting subject knowledge to teach a particular course. Check back again to post #82. Teachers do actually get assigned to jobs which they should not be, but we seem to understand that this is a bad thing. Students hoping to learn Calculus need both a good textbook and a very knowledgeable Calculus teacher. A teacher without sufficient knowledge of Calculus can simply not give effective guidance on the Calculus topics. Even some highly motivated students need guidance from well qualified teachers.

I really can not say how frequent is the misassigning of teachers to courses. At the very least, I know that it happens. Misassigning teachers for Calculus seems to be worse than other courses for misassignment.

 

[snipez90]

All right, I am still not sure what gave you the impression that I made any general comment about the misassignment of teachers. Yes, I know what it means, but you expressed great incredulity at my earlier post for some reason. Highly motivated students will know when they require resources that the teacher won't provide. One of my best friends despised the calc teacher, claiming that the only reason I did well in calculus was because I studied it on my own. Students hoping to learn calculus can do it themselves, but many are lazy. Now any reasonably intelligent student probably knows what the current education system in America is like. But there are students in this category who still refuse to do the work or resort to whining. I've seen this attitude even in my honors calculus course this past year as a freshman in college.

I agree that highly motivated students need guidance, but if the teacher is not up to the task, it is the responsibility of the student to find outside resources. Motivated students will make their efforts worthwhile. Many students have access to an internet connection and a library, but if they continue to rely on an incompetent teacher, then I would not call them motivated at all. Now obviously, I am not proud of the fact teachers are missassgined, nor am I refusing to believe this is often the case. I simply hold the view that one solution to such failures largely depends on exactly how motivated a student is. Most people will never use calculus, but if some student wants to learn it, then he or she had better utilize other resources.

 

[yeongil]

Should calculus be taught in MY high school?

Should calculus be taught in MY high school?

In the school where I teach (private, Catholic high school for girls) we do offer an AP Calculus AB course. (We also offer an AP Calculus BC course, but it is offered only sporadically and students usually take it as an independent study.) Despite having taken both tests myself when I was in high school, I have been lately becoming anti-AP, and I wonder if they do more harm than good to our students. The students who take our AP Calculus AB class come in with not-so-strong algebra skills. I teach Pre-calculus (designated an honors course), and I've b**ched-and moaned more than once here about the incredible Algebra mistakes my Pre-calculus students make.

I should mention that not all of juniors who take PreCalc proceed to AP Calc. Those who really struggle in PreCalc are placed into Stats (non-AP) their senior year. Those with A's and B's in PreCalc junior year go to AP Calc as a senior -- and many of these students still have not-so-strong algebra (and now trig) skills.

And it's not just the Precalculus students. Pretty much most of the school body enter our school with weak math skills. We give an entrance exam to 8th graders that tests English and Math, but Administration/Admissions admits students they shouldn't, because many of the ones we initially offer admission to will end up going to one of our competing schools for whatever reason. As we are a small school (and especially with the financial troubles that all of our area private schools are facing), we have no choice but to admit students who may not have done well on the math portion of the entrance exam. And for the foreseeable future, I don't think this is going to change.

With all of this as a preface, if you were in my shoes would you make a proposal to Admin that we drop AP Calc? Maybe in its place, we can make Pre-calculus a two-year course. A number of schools offer Pre-calculus as two year-long courses -- in fact, the public school system in the district where I live offers separate "College Algebra" and "Trigonometry/Analytic Geometry" courses. Or, would you just suck it up and keep the honors track in place (Algebra 2 - Geometry - PreCalc - AP Calc), because we don't have many graduates who will major in math/science anyway? Or do you have any other ideas?


01

 

[snipez90]

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.

 

[thrill3rnit3]

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?

 

[snipez90]

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.

 

[kote]

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.

 

[Chairman Lmao]

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.

 

[Feldoh]

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

 

[kjohnson]

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

 

[CheckMate]

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

 

[thrill3rnit3]

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

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

 

[Astronuc]

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

Thoughts?

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

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


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

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

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

 

[iratern]

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

 

[romsofia]

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

 

[Feldoh]

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

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

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

 

[Wellesley]

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

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

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

 

[hotvette]

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

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

 

[romsofia]

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

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

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

 

[Wellesley]

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

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

Good luck next year!

 

[sudarshanabcd]

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

 

[mathwonk]

Nothing should be taught to students who are not ready for it. Calculus requires prerequisite understanding of polynomial algebra, trig, geometry, and preferably logic. Hence most high school students should not be offered it. But since it is considered a political coup for a high school to offer calculus, most of them have made room for it by deleting their previous Euclidean geometry courses, replacing those by phony precalculus courses. This is ludicrous. To make room for a calculus class by deleting its proper prerequisite? Duhhh. I agree with post 2, teach it if you will, but do not deceive students by offering college credit for it. I guarantee you if you take my college class it will not be the same as your high school class, unless you went to Bronx high school of science or maybe Andover, and maybe not then. I have been told by the high school coordinator of one of the top private schools in the state that anyone who has had a college class in calculus is qualified to teach it in his high school. Well how does that compare to a course from a professional mathematician with 5-30 years of research experience? It doesn't.

 

[Astronuc]

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

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

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

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

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