While on the surface, this appears to be a no brainer, (Of course it should, if the students are ready) I actually seriously question the practice of letting high schoolers, usually seniors, take calculus. You see, the college calculus fail rate is about 50%, which is not good at all. It is a complex problem, but it has a great deal to do with the fact that incoming college students have minimal mathematical maturity, and have only a tenuous grasp of trig and advanced algebra. Most high school textbooks teach by working out a few problems, and having a grossly oversimplified explanation. Classics like Jacobs, Sullivan, and the like are rarely used. Why not, then, take a slower pace with some of the great textbooks throughout high school, have an exhaustive understanding of the subjects, develop mathematical maturity and thereby adequately prepare students for truly rigorous calculus in college. (Like Apostol's Spivak's or similar calculus texts?) Anyone have any arguments for or against teaching calculus in high school?
Well, I teach AP calc, so I'll say a few words. I think calculus should be taught, but no college credit given. That way, the serious and mathematically gifted students can take it and the students who are only there because it's another AP class to pad their applications will be mostly weeded out. If the system functioned ideally and only students who mastered the previous material passed I'd reconsider, but there are too many students who don't know basic trig or logarithm properties (nor have any clue how they may go about rediscovering them) that somehow make it to my class. As for the students who could handle the course but wouldn't take it because they see no reason too, that's fine. Let them wait until college.
Teaching students deep mathematics in high school was tried and tested in the 60s... the failure rates were even more alarming. Simply put, there is no point in designing the curriculum to meet the needs of less than 1% of the students. Very few students will need that kind of depth, and most are served better by a skimpy version of calculus which is used in engineering and science - by far the most popular majors that require any math. Also, most people lack the ability and interest to pursue mathematics at that kind of level. Having said that, I think the standards should be increased for students in high school. You can pull an A off without having a clue what you are doing.
Hmm, the solution you outlined sounds nice, but it's a lot to ask of the current education system in America. But I think I'm more concerned about your use of the term "exhaustive". The prerequisites for understanding calculus are actually very finite. A strong understanding of the very basics is required of trigonometry is required (a good calculus book will give a more rigorous treatment anyways). For algebra, the ability to solve equations, not necessarily very difficult ones, is required, but this is fundamental. This should be enough to tackle a book such as Stewarts. In turn, a good computational background in calculus and an overall perspective on the various topics can prepare one to tackle a book such as Spivak. I had the very good computational background, but not much knowledge of proofs, which is needed for a more theoretical treatment of calculus. It turns out by going through some of the links here: https://www.physicsforums.com/showthread.php?t=166996 (the first one is especially good imo), that was enough to understand Spivak. I think an honest attempt to go through Stewart while giving the explanations and proofs provided in the book is a lot more instructive than what you'll find in many high school calculus courses. Indeed, this is one reason why I don't think it's harmful for someone to read Stewart before a more rigorous introduction (of course, the person should judge for themselves by comparing to a more theoretical book) because if you really read and understand everything in Stewart and perhaps do the problems in the problems plus section, you can learn a lot. The route I outlined above is of course subject to many contingencies and is certainly not exhaustive, but it is practical.
If a student is planning on going to university to study maths/science, then these are the sorts of things he should have learnt by about 16. As to whether calculus should be taught before university: of course it should, as is the case in most of the education systems around the world!
Right, I was just trying to emphasize the fact that calculus isn't something one needs to make completely thorough preparations for. I'm not saying that one should blow past the basics, but there's no need to confine oneself to just the basics. Of course, the solution to learning the prerequisites deeply is to pick up a book and read it on your own.
I agree with Tobias! I was taught most of Calc I and II material in high school under the mysterious course title of "Math Five" (implying a fifth year of high school level math given that Algebra I was taken in eighth grade). We (or at least I) therefore thought this math was fun filler for math credit... as some of the other course material (in the last few weeks) included probability theory and symbolic logic. I got to college and was surprised I'd already had the material in Calc... but sitting through the college course and doing the homework to be SURE I had the proper math background at the proper level was probably a good idea. I'm personally rather glad my teacher never even called it "calculus" (although we did use the terms "differentiation" and "integration" etc.). It still makes me think Calc is fun!
Hmm, the AP Calculus exam, which many schools will require their students to take (which seems reasonable), is the most popular way of gaining credit for college calculus. Most, if not all schools that offer college credit for calculus will give credit for a 5 on the Calc BC exam (many will give some credit for a 4, some for a 3). But to get a 5 on the calc BC exam, you effectively have to pass the exam to get a 5 in recent years, i.e., a 5 is given if you can get about 60% of the points on the exam. Now I would in most circumstances give the credit to someone who can do about 80% of the exam correctly and let them decide he or she wants to use it. But unfortunately, I doubt this would ever happen. Of course, college calculus placement exams are a reasonably good way to gauge performance and the merit of credit, but this is not always true.
I've never really understood this part of the American system that lets you basically skip fundamental classes. I don't think 'college credit' should be given for any course taught in high school! The way it worked for me was that in the last two years of high school, calculus is introduced. Then, in the first term of university, a core course is given to all taking mathematics which basically skips through the same material, at a much quicker pace. Not only does this help students get to grips with independent studying at university with a subject they basically know, it also ensures that everyone is on a level playing field by the second term of university.
One should focus on primary school not high school. From the age of 6 to 12 children learn almost nothing about math. It seems to me that a great deal of math could be taught in this stage.
I remember something the AP Calculus teacher at my school told me. She has this rule where if you take the class and take the AP exam, you're exempt from her final exam. There was one student who, when taking the AP exam, wrote her name on it and put her head down for the entire exam. (!!!!) I don't remember if the AP Calculus teacher did anything when she found out. I agree that calculus should be taught with no college credit given. This AP Calculus teacher is actually retiring after this year, and I was offered to teach this class next year. I first said yes, but I changed my mind and said no. I became anti-AP and anti-College Board in the meantime. I know many people don't agree, but now I wish that the AP exams be abolished. I mentioned in mathwonk's "Teaching Calculus Today in College" thread about some of the incredible errors that my precalculus students make, and these errors were in algebra. (I'm wondering if it's because our high school math books these days are so packed with material that in the teachers' attempts to cover as much as possible students aren't getting enough practice in many concepts.) Half of my precalculus class are juniors, and many of them will be taking the AP Calculus AB course next year, with less than solid algebra skills. Oh, boy. 01
Yep, I know all to well what you mean. I suppose I'm part of the problem in a sense. My school refuses my (and others') requests for a much needed prealgebra class and throws all freshmen into algebra 1. Count Ibis is right. These kids are not ready at all and it's just unreasonable to expect them to learn much algebra. The result is a dumbed down class- prealgebra with the name algebra 1. Unfortunately, most of them never really do catch up. Even the honors students seem weak, and it's not just me forgetting how it was back then. I remember listening to my classmates' conversations in honors trig and wondering what the hell was so hard. I think worrying about calculus in high school, at least in the US, is less important than just making sure they learn up to algebra 2.
I think what matters most is the WAY IT IS BEING TAUGHT to students, especially to the younger ones. Even if you put all sorts of Calculus and AP classes in there, if it isn't taught very well, serves no purpose. Unfortunately, the plug and chug approach has taken over the US education system, and that doesn't work as well once you hit college.
If your talking about the U.S. education system, then to me, it is a no-brainer and it should be taught. My thoughts are that if we cut-back on the math curriculum then we would become even less competitive in the international arena. Your right about the poor-performance of students. Two large reasons for these results are (1) the unmotivated study habits and respect for one's education by the students and (2) the inadequate number of competent and qualified teachers to teach the subject. Competent and qualified are two different characteristics, and in my opinion, being certified (qualified) to teach math does not mean one is competent. I would focus my efforts more towards the latter (2) than the former (1) as means for improving math education.
As someone who took AP calculus in high school and was given credit for the first semester of calculus in college, I absolutely, completely, unequivocally agree with this statement. It was good to learn calculus in high school, mostly because I then understood physics in college better. But, by skipping a semester at the college level, I had just enough time to forget what I had learned in high school and fell behind when I took second semester calculus. I never really caught up and struggled through multivariable calc too. Actually, my own experiences with AP credits leads me to this argument regarding all AP courses now...they are good to make college courses a little easier, but should not count for credit, especially if they are in any way remotely related to your major. You can pass the AP exam while still having substantial knowledge gaps that would be filled in during your freshman courses, and it's more hindrance than help to miss those freshman courses. Edit: Regarding the OP, where do you get the statistic that the failure rate is 50% for college calc? That certainly is far from consistent with my own experience, so I'd like to see some evidence supporting that "statistic."
I guess I'm a little confused about everyone's posts- I took AP calc in high school, took the AP test (Calc BC? I can't recall) and passed out of math I, for reference. First, taking AP math is not required in high school, and second, my understanding is that it is up to the university if any AP credit is granted. I see nothing wrong with offering advanced coursework in high school as an option- remedial coursework is offered, why not the converse? As to Moonbear's post, I kinda-sorta agree that there are pitfalls in passing out of freshman courses. However, because I did have a reasonable amount of credit, I was able to take a lot of elective courses that I would not otherwise have had the opportunity to take (and still graduate in 4 years). And, while I agree that in a perfect world math and science concepts would be introduced earlier, even unto elementary school, in the real world (US public school) parents have, by and large, ceded all responsibility for all facets of their child's education to the whims of the school system. So, given elementary school teachers with inadequate math and science knowledge on top of disinterested parents, also with substandard math and science knowledge, it's not realistic to simply introduce the concepts earlier and expect any real increase in ability.
It should be possible for universities to make downloadable lecture notes for primary school children. Many parents are interested but they are incomptent to help their children. They do want to get their children to the best universities. So, if the universities themselves where to say: "To make sure your child doesn't drop out in the first year, we recommend that your child studies from our specially prepared lecture notes", the problem would be solved.
I think it should be offered as an elective to students who do give a damn. There are many who dont, honestly. And a lot have interest in other subjects.
As i scientist i must say Calculus is fundamental and almost needed as breeze to breathe or as the food to live the problem is those people involved in 'Social Science' , or take a career about Art, History, Filology,... so they will NEVER need it , or in case they need could be taught at University however the cultural impact of calculus is so high that any person considered 'intructed' or 'wise' should know