Should calculus be taught in high school? [Archive]

snipez90

May9-09, 05:10 AM

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: http://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.

cristo

May9-09, 05:24 AM

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.

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!

snipez90

May10-09, 02:59 AM

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.

physics girl phd

May11-09, 04:01 PM

I agree with Tobias!

Well, I teach AP calc, so I'll say a few words. I think calculus should be taught, but no college credit given...

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!

snipez90

May11-09, 04:56 PM

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.

cristo

May11-09, 05:19 PM

Well, I teach AP calc, so I'll say a few words. I think calculus should be taught, but no college credit given.

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.

Count Iblis

May11-09, 05:36 PM

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.

yeongil

May11-09, 06:55 PM

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.
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.
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.
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

thrill3rnit3

May11-09, 07:19 PM

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 agree. Good fundamentals are a necessity in any field, not just math.

Tobias Funke

May11-09, 07:25 PM

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.)
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.

thrill3rnit3

May12-09, 10:14 AM

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.

buffordboy23

May12-09, 10:44 AM

Anyone have any arguments for or against teaching calculus in high school?

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.

Moonbear

Jun5-09, 08:31 PM

Well, I teach AP calc, so I'll say a few words. I think calculus should be taught, but no college credit given.

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."

Andy Resnick

Jun8-09, 08:35 AM

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.

Count Iblis

Jun8-09, 08:56 AM

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.:approve:

physicsnoob93

Jun8-09, 09:41 AM

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.

zetafunction

Jun8-09, 09:43 AM

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

Andy Resnick

Jun8-09, 01:28 PM

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.:approve:

Walk into any bookstore (or big-box store with a 'books' section) and you will find scads of already-existing workbooks specifically with this aim. A cursory interweb search will likewise net you a nearly uncountable set of similar materials.

The problem is not availability; the problem is lack of interest.

thrill3rnit3

Jun18-09, 01:31 PM

Personally, I don't think there is any way out of this "education gap" between the United States and the rest of the world.

qntty

Jun18-09, 01:35 PM

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?

I don't think the issue is whether advanced coursework should be offered, but rather whether that coursework should be calculus. If the college fail rate of calculus is high then that means that kids don't know the fundamentals well enough. Maybe, rather than introducing calculus sooner, we should make sure kids understand everything up to the point of calculus better.

Count Iblis

Jun18-09, 02:10 PM

I don't think the issue is whether advanced coursework should be offered, but rather whether that coursework should be calculus. If the college fail rate of calculus is high then that means that kids don't know the fundamentals well enough. Maybe, rather than introducing calculus sooner, we should make sure kids understand everything up to the point of calculus better.

The reason why students are bad a math is precisely because we don't teach enough of it early enough. The age at which most children could start to learn math is somewhere around the age of 8. But we start to teach very elementary math at the age of 12, so that's four years lost, which is the same amount of time students spend at the undergraduate level at university.

Also, if we were to start teaching math at the age of 8 then more of what the children learn will be hard wired in their brains. Things like manipulating algebraic expressons etc. will be as natural as speaking English. While if you learn these things at a later age, it is like learning to speak Chinese at a very late age. It is more difficult to get fluent at it.

buffordboy23

Jun18-09, 11:41 PM

yeongil

Jun20-09, 10:27 PM

thrill3rnit3

Jun20-09, 11:23 PM

Math is not emphasized enough at those levels. For heaven's sake kids don't fully understand how to add/subtract "unlike" fractions until the 6th grade...

physicsnoob93

Jun20-09, 11:45 PM

The reason why students are bad a math is precisely because we don't teach enough of it early enough. The age at which most children could start to learn math is somewhere around the age of 8. But we start to teach very elementary math at the age of 12, so that's four years lost, which is the same amount of time students spend at the undergraduate level at university.

Also, if we were to start teaching math at the age of 8 then more of what the children learn will be hard wired in their brains. Things like manipulating algebraic expressons etc. will be as natural as speaking English. While if you learn these things at a later age, it is like learning to speak Chinese at a very late age. It is more difficult to get fluent at it.

People start learning math when they are 6 in Primary School over here in Singapore. I thought they would do the same in the US too? And are you sure about:
...But we start to teach very elementary math at the age of 12...?

We have an International called Kyle from North Carolina, he is probably the most advanced math student in our level, and hes an year younger than us. He learned math through calculus when he was in Elementary school. I think its the difference between private and public schools?

thrill3rnit3

Jun20-09, 11:59 PM

People start learning math when they are 6 in Primary School over here in Singapore. I thought they would do the same in the US too? And are you sure about:
?

We have an International called Kyle from North Carolina, he is probably the most advanced math student in our level, and hes an year younger than us. He learned math through calculus when he was in Elementary school. I think its the difference between private and public schools?

Well Kyle most likely fits in the category of "outlier".

No elementary school here teaches calculus. In fact, only a small number teaches algebra in 6th grade.

Elementary, middle school, and high school education here in the U.S. is crap.

And Count Iblis is right. Most kids don't have their "basic" maths straightened out until age 12, at the least.

thrill3rnit3

Jun21-09, 12:00 AM

Plus, most private schools are worse because of lack of funding. Of course there are exceptions like the Philips Exeter Academy.

Most of the good high schools are public high schools.

mXSCNT

Jun21-09, 01:39 AM

My personal opinion on math education in the US is that our problems stem from the anti-intellectual culture that many youth get drawn into. The culture glorifies soldiers, musicians, actors, athletes, anything but scientists, who are derided as stuffy and useless. There isn't much emphasis on a work ethic, either. It's all about quick gratification. The result is, most students don't value math much, and if they do value it they are less inclined to work at it. The best students, who both value achievement and are willing to work, are ostracized as geeks. With that kind of peer pressure who would want to be smart?

j93

Jun21-09, 02:17 AM

thrill3rnit3

Jun21-09, 02:26 AM

I believe that if reform is to be done to the curriculum it should start with the bottom (preschool - elementary education), working its way to the top (high school curriculum).

Astronuc

Jun21-09, 08:54 PM

Math is not emphasized enough at those levels. For heaven's sake kids don't fully understand how to add/subtract "unlike" fractions until the 6th grade... I learned that in 4th grade in the US. But I've found US schools uneven. Some are great and many are poor. I probably had the best teachers in the schools I attended, but that's because I got shuffled into Major Works (MW) or Honors courses.

thrill3rnit3

Jun21-09, 09:33 PM

I learned that in 4th grade in the US. But I've found US schools uneven. Some are great and many are poor. I probably had the best teachers in the schools I attended, but that's because I got shuffled into Major Works (MW) or Honors courses.

It's supposed to be "taught" at that stage. But because of the lack of emphasis by the teachers, and thus the lack of interest by the students (I'm talking about the middle tier-lower tier students), they don't fully understand the concept until middle school.

Which is pretty pathetic IMO.

PhysicalAnomaly

Jun21-09, 11:09 PM

thrill3rnit3

Jun22-09, 01:45 AM

Maybe you could adopt the asian method and just make the students do more and hope it works. XD

The australian maths syllabus is a year behind malaysian and singaporean syllabi and their students are no more competent at what they learn either. The students in the asian countries do more questions a day and by the time they graduate from high school, they are expected to have done thousands of calculus questions. There's also the massive peer and parent pressure. They go for tuition classes and spend a lot of time just doing problems. We also learn so many different methods of doing things that it's quite shocking to find that the australian students only know a single method.

I personally don't think much of mindlessly doing hundreds of questions. But if it works, it works.

Well I don't think doing a lot more problems would solve the issue either. What I'm talking about the way it is being taught to the students. Here in the U.S. the "plug and chug" method is the prevalent method in use by most of the students AND teachers alike.

So when the kids are given a problem a little bit different from the sample exercises, they are lost and have no clue where to even begin.

Andy Resnick

Jun22-09, 09:27 AM

Feldoh

Jun22-09, 10:24 AM

Eh, maybe my experience is unique but I took AP Calculus, got a 5 and passed out of Calc I and II at college. I've gone on and passed Calc 3, and differential equations easily with a's. Next semester I'll be taking real analysis, and I've been going over the book over the summer and although it's difficult doesn't really seem over-the-top. On top of that the only way I make money is tutoring students in math (Calc I-III).

Basically my point is is that I've done just fine without ever having to retake the first few intro calculus courses in college, so to be honest I really don't think it's a solution that really makes sense.

On top of that people I know that have taken the AP test and opted to skip on college credit now find that they (two of my close friends) dislike math just because they've had the same old information for two year in a row, which starts to get stale.

symbolipoint

Jun22-09, 01:38 PM

Andy Resnick:
Some people, even I, share your opinion that parents need to care and encourage. On the other hand, some parents mishandle this, destroying the childrens' motivations for Math and are unable to give or find sensible help. A few children are lucky that their Math instruction in their school may actually be good; better than just "plug & chug" Algebra.

Feloh:
Some students NEED to study material or courses more than once. They also need opportunities to use the Mathematics which they study. Part of this is just having good variety of Algebra and Calculus exercises with derivations and analytical thinking; and some of this is having science lab exercises or real-life work situations which can be understood or managed with Mathematical topics.

Astronuc

Jun24-09, 09:40 PM

I've said this many times before, and will say it again-

Looking over this thread, there is not one single comment (excepting mine) that admits the role of *parents* in their child's education. Part of the problem with the US public education system is that many parents have completely ceded their role in the education of their children to the schools. As long as parents consider their children's mathematical education (or any other part- history, composition, etc) not worth discussing over dinner, and parents make no effort to show their children that the material they learn in school has value outside of the classroom, no amount of time and effort spent in the classroom will compensate.

Teachers have an incredibly difficult job and get paid very little money. Is it any wonder that high-caliber educators are not created and nurtured? The US curriculum is now results-based: school funding hinges on how well the students perform on idiotic standardized tests. Is it any wonder that increasing amounts of classroom time are spent teaching to the test rather than providing an educational environment?

To you folks who claim to be so concerned about how poorly students are being educated, I challenge you to do something about it- offer to teach a 'science day' in an elementary school classroom. Volunteer for "Teach for America". Stop whining about how the larger public doesn't give a rat's a** for the subjects you hold so dear. Engage the public and get their attention. Yep - parents' lack of involvement in their childrens' education is a big problem - has been for 3+ decades since I left high school, and years I went to primary and secondary school.

After the students leave the school, teachers cannot make the students do homework or study. That is when the parents need to enforce the discpline and ensure their children do their homework and class assignments. That should the be the priority - not watching TV, or playing video games, or running the streets, or playing sports or some other extracurricular activity in place of studying. But this is OT.

Analysis and calculus should be taught as early as possible, and certainly by 12th grade, but that requires the pre-requisites be taught in earlier classes. One difficulty is disparity in the ability of students and also in the capability of teachers - not only from state to state or from school to school, but even within schools.

By the time I was in 11th grade, I was well ahead of my parents ability, so I pretty much took responsibility for my studies and academic program. I was one the fortunate students who got the best teachers in the school who were also the heads of the mathematics and science departments, and I had the best academic counselor who was well aware of university programs around the country, and each year directed students to NSF and university summer programs in the academic subjects of interest. Many of us in the honors/major works/AP programs did a summer program between 11th and 12th grade. One of my classmates went to MIT for a summer program in math and science, and he attended MIT out of high school. I did a summer program at Colorado School of Mines in EE and NucE. One of the kids from that same summer as CSM is now a professor of astrophysics at Caltech.

The high school I attended in 11th and 12th grade was on the trimester system and one typically took 4 courses per trimester period. I took 5 courses in order to add an extra course. I had taken geometry and trigonometry in 10th grade, so 11th grade was a second year of algebra, with some linear algebra and more trigonometry. The 12th math program consisted of analytical geometry (one trimester) and two trimesters of differential and integral calculus. Only about 30 students out of more than 700 did that math program. All the rest did up through analytical geometry, if that, in their senior year.

buffordboy23

Jun24-09, 11:25 PM

I definitely agree that parents are a large part of the problem, but we shouldn't place all the blame concerning the poor state of our education system on them. If we allow ourselves to believe this folly, then the only solution toward a better education system is to change the mindset of a nation…highly unlikely. Educating our youth does not require us to educate their parents. Therefore, we should focus on educating qualified and competent mathematics/science teachers, modifying instruction to actually connect with students lives, and modify the K-12 curriculum to the student’s educational goals. This “educational” population is smaller than the parent population and values education more as well, so reform in this area should be more realizable.

First, let’s look at the number of qualified math/science teachers. I don’t have recent statistics in front of me, but I bet that many teachers out there teaching these subjects are not qualified to teach them but do so with a temporary certificate or something similar. So students of these teachers get the shaft.

Now, the remaining teachers are qualified to teach these subjects. What exactly does it mean to be a qualified teacher? Usually, it means that the teacher has a degree in the area they are teaching and passed a general and subject-specific certification exam. Do you really think that graduating students with teaching degrees really know their subjects? From what I have seen, the methodologies often employed in college instruction only require memorization to pass a multiple-choice test, so there is no real understanding to be had unless the student takes their own initiative. What about the general and subject-specific certification exams? It’s a net to catch the dumbest of the dummies and keep them from actually entering the classroom. Don’t believe me? I knew an elementary education major that had her degree for two years and still couldn’t pass the mathematics portion of the general exam, and so she couldn’t teach in the classroom. So while a student may have a “qualified” teacher, they are actually getting the shaft.

This is a difficult problem to overcome, since our educational system is cyclical through time and is affected by numerous factors.

In the future, I plan to discuss some simple solutions to this dilemma and discuss the other two points when I get the time. This post may appear somewhat to wander from the OP (teach calculus in school?), but I assure you that I am working my way there.

buffordboy23

Jun25-09, 12:47 AM

The simple solutions to the first point of my last post are to modify college instruction to ensure real understanding of the subject, to learn appropriate and creative methodologies to transmit this knowledge effectively to the student, and to provide incentives to aspiring teachers entering these fields. All of these are done in some college-level settings, yet it hasn’t become widespread. Even if it were widespread, we would have to wait years to see real results.

Now, let’s suppose that your one of the few to get such training in college or that you acquired it on your own through initiative and hard work. Would you actually employ these skills in the classroom? With so many cookie-cutter lesson plans available on the internet and resources offered by textbook manufacturers, the demanding work load that our current teachers face makes it so tempting to sacrifice the time needed to employ the skills learned in college in favor of these time-saving crutches. This leads to my second point, modifying instruction to actually connect with students lives.

In regard to mathematics, the typical and most simplistic form of instruction is rote memorization. While I do agree with this at the elementary level, since this is the foundation of all advanced mathematical subjects, like algebra and so on, I disagree with this method of instruction during math education in grades 7-12, yet this method still persists.

The connection of elementary mathematics education is easy to connect to students lives (they see examples everyday), but advanced subjects are more of a challenge to convey. It requires a large sacrifice of time on the part of the teacher to develop such lessons, since the resources readily available don’t usually have the necessary focus—check the research studies done on mathematics textbooks and their associated resources and you will find that they are rated poorly in most instances. Furthermore, real-life scenarios/problems for math subjects offered in grades 7-12 require more critical and creative thinking on the part of the student…something they are not used to and is a skill in and of itself. Research shows that, in general, students value the learning of a subject if it appears useful or important to them, so we must not neglect this fact and target it in our instruction.

Here’s a simple example to show why knowledge of trigonometry is important to the student. In the future, the student will likely buy a house. They may eventually decide that they want to cut down a tree that resides on their property, and that they want to do this task themselves to save money, yet the layout of their property and the general appearance of the height of the tree makes this appear like a risky endeavor. If the tree has only one cut at its base, will it fall on the house, or will two cuts and the extra work be necessary? 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.

The last point, modifying the K-12 curriculum (mostly 7-12), is connected to the discussion of the second point. What mathematical knowledge is really necessary for students who choose vocational studies vs. college prep studies in the sciences or liberal arts? Usually, students in the vocational studies don’t take calculus, while students in college prep do and for many of them it will never be of any use except for a well-rounded educational background. Instead of requiring calculus for these particular students, it should be offered as an elective vs. another class that explores familiar mathematical subjects and their connections to real-life scenarios in order to build problem solving skills. This aspect of mathematics education should be specifically tailored to the student's chosen path of study and should provide the student with freedom of choice rather than required restraint. So, if a student is planning to pursue the sciences then calculus should definitely be taught in high school.

Astronuc

Jun25-09, 10:00 AM

I took introductory mathematics and science courses through a local university program during junior high and high school. I had to go out a buy my own analytical geometry and calculus textbooks, and I think that was during the summer before I started 10th grade. My dad took me the main technical bookstore in the city, so I could browse the aisles for math and science books. I was able to learn bits and pieces, but I had no formal direction from a mentor.

buffordboy23 raises several good points, which are all aimed at improvements in primary and secondary education, which in theory would lead to having schools in which calculus is taught in high school to those students are prepared to learn it.

Teachers need support and appropriate training.

Parental involvement is essential.

mgb_phys

Jun25-09, 10:32 AM

Probably everyone here is familiar with this article (Lockhart's lament), but it is a good read
http://www.maa.org/devlin/devlin_03_08.html

thrill3rnit3

Jun25-09, 11:32 AM

buffordboy23

Jun25-09, 12:22 PM

Why are we so concerned about connecting math to "everyday lives"?? I think the teachers try to hard to make that connection and application, in which they are losing the theoretical side of their lectures.

There are only two reasons why an individual would choose to learn something. They are practicality and pleasure. Neither is significantly evident in our mathematics curriculum but they should be.

Practice with real-world scenarios develops experience and foresight, which then enables one to solve problems when actually confronted by them. Here's an example a former colleague of mine gave to her middle school students. The student got to choose any car that they wanted to have in the future and that they think they could someday afford. Many students picked really expensive vehicles. They calculated their monthly loan payments as part of the project. Most of them crapped themselves when they saw the final figures, and some noted that their parents salary wasn't even sufficient. Yet we still see people placing themselves in bad financial situations due to lack of critical reasoning or just plain temptation. If you are fortunate to have tools, but have no experience or knowledge of using them, then they are useless.

physics girl phd

Jun25-09, 01:12 PM

There are only two reasons why an individual would choose to learn something. They are practicality and pleasure. Neither is significantly evident in our mathematics curriculum but they should be.

Practice with real-world scenarios develops experience and foresight, which then enables one to solve problems when actually confronted by them. Here's an example...

And here's another:

Our rising fifth grader has been picking at dinner then sneaking peanut butter in the middle of the night (after having some for breakfast and lunch)... so to motivate him to do otherwise, I prepared a worksheet for him where he estimated his daily consumption of peanut butter (8 servings in his case) and he then would look at the nutritional value of peanut butter on the jar and see how much of various dietary needs were being neglected, met or exceeded by his daily consumption. After seeing he gets 144% of an adult's daily fat needs, and none of certain nutrients, we're hoping that now reconsiders his decisions on things.

Unfortunately he didn't get to the part where he looked at the few other things he eats and how they might fill the gaps... mostly what little he does eat at dinner, a packet of raisins, a snack bag of crackers, and a bottle of apple juice (and boy did he protest when I insisted he get a 100% no sugar added juice when we were at the store!). However, I noticed this morning that his preferred bread contains 5% of saturated fat per slice and no vitamin A or C (two of the biggies that were missing from the PB). Ouch!

While yeah, it was practice with math... but he was genuinely interested (probably was hoping things would turn out better for him and his peanut butter diet would be justified). And then for us it had the desired result -- at least he ate his regular dinner and didn't sneak peanut butter last night!

Andy Resnick

Jun25-09, 01:20 PM

Probably everyone here is familiar with this article (Lockhart's lament), but it is a good read
http://www.maa.org/devlin/devlin_03_08.html

I have not seen this before- thanks for posting it!

Astronuc

Jun25-09, 02:05 PM

Astronuc

Jun25-09, 02:17 PM

Probably everyone here is familiar with this article (Lockhart's lament), but it is a good read
http://www.maa.org/devlin/devlin_03_08.html I too have not heard of Lockhart.

I like Lockhart's idea "that there is a playground in their [students] minds and that that is where mathematics happens." That's brilliant.

How do the educators tap into the imagination of students' minds?

buffordboy23

Jun25-09, 02:19 PM

And here's another:

Our rising fifth grader has been picking at dinner then sneaking peanut butter in the middle of the night (after having some for breakfast and lunch)... so to motivate him to do otherwise, I prepared a worksheet for him where he estimated his daily consumption of peanut butter (8 servings in his case) and he then would look at the nutritional value of peanut butter on the jar and see how much of various dietary needs were being neglected, met or exceeded by his daily consumption.

Your right. This is just mild form of coercion though. In my post, I was specifically referring to the freedom of choice concerning the individual.

The educational standards require that coercion be the epitome of our education system and coercion usually doesn't work by itself. It also needs an offering of practicality (how will this benefit me now or in the future?) or pleasure (will learning this be fun? will I have large freedom with my approach?). It appears that your son saw the practicality of your proposed lesson and complied. I can't confidently comment on the pleasure though.:smile:

buffordboy23

Jun25-09, 02:53 PM

I like Lockhart's idea "that there is a playground in their [students] minds and that that is where mathematics happens." That's brilliant.

I agree. Also, Lockhart suggests that math can be perceived as an art form, and this is true.

How do the educators tap into the imagination of students' minds?

Initially, it probably requires carefully planned use of scaffolding. For example, it's easy to provide scenarios that can lead elementary-level students to the concept of infinity without initially telling them what infinity is. How does the student then make sense of the results of this scenario and analogous ones? They can create a definition that characterizes it. Another example that borrows from Lockhart is to let students choose the geometric figure inscribed in a square and to determine how much area of the square that the figure consumes--the scaffolding is in place, but with a sense of individual freedom for exploration in this case.

By the teacher modeling this type of behavior and by seeking original and slight variations on the problems from the students, we expect that they eventually will ask their own questions. This is analogous to how a scientist operates. By studying the questions asked by other scientists (mainly the teacher and sometimes the students) and learning about their results, they (mainly the students) learn to ask new questions that are relevant to the current body of knowledge that exists.

This is usually absent from our education system.

Sankaku

Jun25-09, 08:30 PM

Why are we so concerned about connecting math to "everyday lives"?? I think the teachers try to hard to make that connection and application, in which they are losing the theoretical side of their lectures.
I think there is an important distinction here. Kids need to be INSPIRED to want to do math. One way to do it is to make it something that they see as useful. When I was in high-school, half the boys wanted to know mechanics because they liked cars. A means to an end.

Although the "useful" angle is sometimes good, I think the "Lockhart's Lament" article points out that you get more interest if something is "beautiful" rather than just "useful."

You can read "beautiful" as "cool" if it fits modern semantics better. For me, I have a rebellious love of things that are beautiful but of no (obvious) practical importance whatsoever. Whatever the motivation is, you can't expect 13 year old kids to be interested in math just because some grownup thinks that "the theoretical side" is important. The "theoretical side" has to have some kind of relevance, no?

As far as teaching Calc in high-school. Yes, if kids are ready for it and enthusiastic about it. Otherwise, No. The question seems to be devolving into "should we force smart kids to study Calc in high-school?" I don't like this. Why are we in such an awful blinding rush?

SonyAlmeida

Jun25-09, 10:15 PM

Taught but not forcibly so, which is... the status quo, pretty much.

Andy Resnick

Jun25-09, 10:38 PM

I too have not heard of Lockhart.

I like Lockhart's idea "that there is a playground in their [students] minds and that that is where mathematics happens." That's brilliant.

How do the educators tap into the imagination of students' minds?

That's a good question, indeed! Unfortunately, I have to agree with Lockhart's conclusion- in order to treat Math like the other Arts, one must give up standardized tests. Personally, I don't think that is ever going to happen as long as I am alive- it's too hard to argue against "minimum competency requirements", because there are some perfectly valid reasons for having minimum standards.

Perhaps this means moving toward a more European (i.e. German) model, with separate 'vocational' tracks established early on- but OTOH that is exactly the system that drives away talent (to the US, currently).

thrill3rnit3

Jun25-09, 10:42 PM

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.

Same reason why calculus books nowadays are considered "watered down", for example they are relying too much on the calculator which I think is counterproductive.

buffordboy23

Jun25-09, 11:37 PM

I think there is an important distinction here. Kids need to be INSPIRED to want to do math. One way to do it is to make it something that they see as useful. When I was in high-school, half the boys wanted to know mechanics because they liked cars. A means to an end.

Exactly. There is practicality here. This is exactly why I believe that parental involvement is not the most important factor in regard to the education of our youth. Whether the parents are neglectful or not to their child's education, their sons/daughters know how to use many technologies, such as ipods, cell-phones, computers, etc., better than their parents and most of us older adults. Why? Because one major reason is that it keeps them socially connected to their peers...social Darwinism in effect. They see value in learning how to use the technology and take initiative to teach themselves.

thrill3rnit3

Jun26-09, 02:13 AM

Simplicio and Salviati's conversation about teaching the "practicality" of math is definitely what I was talking about.

RedX

Jun26-09, 02:44 AM

I didn't read everything so apologies if my opinion is repeated.

Maybe you can have a compromise: only teach differential calculus in high school, and spend the other half of the year strengthening algebra. Integrals you can learn in college.

A lot of calculus is, in my opinion, memorization, until you take an analysis course. But teaching students d/dx(x^n)=nx^(n-1) is not a waste because they'll have to learn it eventually, and eventually everyone has that down by rote. It's the age old question - do you learn by rote first, and then teach why? Or do you teach why first, and then the rote? High school can lay down the rote, and college can provide the aha that's why. Maybe reform should be on the college side. Each major has a definition of what amount of math is acceptable, each college has applicants of differing abilities, so maybe the colleges should cater to the students and not the high schools to the college?

The AP calculus test should weed out students with inadequate understanding. In theory at least. But I don't trust them. Perhaps it's better to ask for AP-test reforms than refining what's taught in high schools.

thrill3rnit3

Jun26-09, 03:00 AM

Tobias Funke

Jun26-09, 10:59 AM

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

Jun26-09, 11:33 AM

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 :smile:

buffordboy23

Jun26-09, 12:04 PM

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

Jun26-09, 12:09 PM

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

Jun26-09, 12:21 PM

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

Jun26-09, 01:57 PM

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

Jun26-09, 04:48 PM

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

Jun26-09, 05:12 PM

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

Jun26-09, 05:56 PM

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

Jun26-09, 06:55 PM

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

Jun26-09, 07:00 PM

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

Jun26-09, 07:17 PM

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

Jun26-09, 07:28 PM

snipez90

Jun26-09, 07:33 PM

Sankaku

Jun26-09, 08:11 PM

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

Jun26-09, 08:22 PM

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

Jun26-09, 09:04 PM

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

Jun26-09, 09:49 PM

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

Jun27-09, 10:54 AM

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

Jun27-09, 12:03 PM

^^^

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

Jun27-09, 09:09 PM

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

Jun30-09, 02:05 PM

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

Jul11-09, 09:58 AM

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 in to 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

Jul11-09, 10:01 AM

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

Jul11-09, 10:09 AM

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

Jul11-09, 10:14 AM

Tobias Funke

Jul11-09, 10:19 AM

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

Jul11-09, 10:30 AM

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

Jul11-09, 10:41 AM

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!

cristo

Jul11-09, 11:09 AM

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

Jul11-09, 11:11 AM

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

Jul11-09, 11:12 AM

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

Jul11-09, 11:18 AM

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

Jul11-09, 02:30 PM

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

Jul11-09, 11:09 PM

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

Jul12-09, 12:41 AM

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

Jul12-09, 01:07 AM

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 knowledgable 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

Jul12-09, 01:55 AM

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

Jul15-09, 09:55 AM

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

Jul15-09, 11:55 AM

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

Jul16-09, 02:46 PM

snipez90

Jul20-09, 04:42 AM

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

Aug25-09, 10:57 AM

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

Oct6-09, 12:43 AM

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

Oct6-09, 01:35 AM

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

Dec14-09, 05:23 PM

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.

CheckMate

Dec15-09, 09:19 PM

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

Dec16-09, 12:44 AM

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

Dec27-09, 09:32 PM

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.

iratern

Mar17-10, 09:45 PM

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

May25-10, 12:48 PM

Feldoh

May25-10, 03:26 PM

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

May25-10, 04:19 PM

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

May25-10, 07:30 PM

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

May25-10, 07:40 PM

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

May25-10, 09:18 PM

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

Jun10-10, 02:18 AM

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

Oct28-10, 06:36 PM

Astronuc

Oct29-10, 10:52 AM

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, . . . .

Angry Citizen

Oct29-10, 01:08 PM

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.

G037H3

Oct29-10, 01:14 PM

High school kids shouldn't take calculus, as they don't have the right prerequisites. I agree with mathwonk.

mathwonk

Oct29-10, 01:33 PM

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.

Klockan3

Oct29-10, 02:43 PM

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.

DR13

Nov2-10, 11:23 PM

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).

mathwonk

Nov5-10, 12:12 AM

Well i don't know you and it is of course quite possible that everything you say is correct, DR13. If so however, then you are very different from almost all the students I have met in my professional career lasting over 40 years. In all that time i have seldom met any students at all, who understand even a modicum of calculus, no matter what score they obtained on BC/AP tests and classes.

They were often deceived into thinking they understood college level calculus however because most colleges have had to dumb down their courses to accommodate these AP students. Hence although high school AP students do not understand calculus at what used to be a college level, colleges have lowered the level of their classes so as to prevent all these students from failing.

In the present day curriculum, we now offer three or four different college classes in calculus, at different levels. For the most gifted students, the best advice I can give them is to take calculus again from the beginning in college, but take it in a high level honors class, so as to get the deepest experience of it. I.e. to take a "Spivak style" class.

The reason a student should not take college credit for AP calculus and then begin in sophomore calculus is that he will have moved himself from an honors level high school course to a non honors level college course. I.e. there are almost no (you could of course be another of the one or two exceptions I have met in 30 years) graduating high school AP students who are qualified to begin college in a second honors level calculus class, i.e. a course from say Apostol volume 2, or from Loomis and Sternberg.

The few exceptions tend to wind up at Harvard or MIT, and have prepared by taking genuine college level classes in high school from real colleges, or from super high schools like Exeter and Andover, or the Bronx high school of science.

Hence NOT starting in a first year honors level college calculus class in college is usually doing yourself a disfavor, and lowering the level of your education. I.e. if you take the regular second year course you are likely qualified for, you will never again be able to enter the honors level work and you will never learn it, and you will likely never achieve the level of preparation needed to become a professional mathematician, if that is one of your possible goals.

Here is a little test for you: did you learn to prove that a continuous function on a closed bounded interval has a maximum in your high school AP class? This was covered in the first semester of my college class when I was a student, and I taught it in my first semester honors class at an average state university, not the higher level first semester Spivak class, just the class for people who had done well in AP courses.

Easier: can you state and prove the fundamental theorem of calculus? I teach this even in my non honors classes in college. Of course if you can really do these things, then indeed you have learned a lot in your high school classes and your preparation is unusual. But very few students at my university have this preparation from high school. It is certainly not included in the usual AP syllabus or tests I have seen.

Sankaku

Nov5-10, 03:32 AM

...you will never again be able to enter the honors level work and you will never learn it, and you will likely never achieve the level of preparation needed to become a professional mathematician...
Wow - that is a pretty severe statement. Just because someone doesn't enter the course-stream where YOU think they should curses them forever to not understand math? Give me a break!

There are many routes to the same goal. Just because you have one sanctioned path for the blessed does not mean it is the only one.

mathwonk

Nov6-10, 12:04 AM

i am apparently not being clear. There is not just one stream, there are several streams. At Harvard when I was a student there, one could take the 3 non honors calculus courses math 1,20, 105, or one could take the honors courses math 11, 55, and then go right into graduate courses.

In the non honors courses the subject was taught in the traditional way, old fashioned approach more common in physics, more numerical, less conceptual. This is paralleled at UGA today by our non honors sequence math 2250,2260,2500, and perhaps 4100, or our honors sequence 2400, 2410, 3500, 3510, and then maybe 4200, 4210, or also 4100.

In the honors sequences the material is taught in a more modern way, with more use of linear algebra, more topology, the way a practicing mathematician uses it.

But a student only takes one sequence, not both. Thus the students who take the non honors sequence never see the modern approach at all, and usually by the time they finish, they do not have time to go back and do it all over again, or maybe not even the mental flexibility, having already learned to think in the old way.

Thus I noticed at Harvard that students who knew more than I did, and seemed smarter than me, were nonetheless not learning the more powerful approach to the material that I was, and ironically it was precisely because they had not started at the beginning at Harvard, but had gotten their start in prep school.

Thus not only did they get a lower level version of the math, they also had contact with less strong students, non honors students, and they also did not get the most stimulating professors who tended to teach the honors courses.

So they never got the same perspective on the material. Moreover this lack of stimulation caused some of them to begin to find the subject uninteresting, and eventually to drop out. They were not getting the stimulating viewpoint, the stimulating professors, nor contact with the most stimulating peer group.

This eliminates them from consideration for admission to top grad schools, although it is true there are other schools where they can perhaps slowly come up to speed.

You are of course correct it is possible to take many different paths to ones goal, but it is harder, and fewer people find it, especially people who do not have the wisdom to listen to their elders.

DR13

Nov6-10, 12:14 AM

To mathwonk:

Trust me, I am no genius. I am in the non-honors calc sequence (by choice). And even in that class I am no genius. I would say that I am in about the 75th percentile (pretty solid but nowhere near genius). There were kids at my high school that were much smarter than me. If I do not have a problem starting with calc III then these kids definitely will not. Also, I am in engineering so to be honest I do not want to take an ultra-rigorous calc class based on proofs and what-not. I do like math but I do not love it.

One note on what Astronuc said: I definitely agree that there is too much redundancy in pre-high school math

Fragment

Nov6-10, 05:47 AM

I think a fundamental issue here is determining how rigorous of an understanding one wants. Surely engineering students do not need a Spivak-style course, where some math students do. However, this is not a rule, having taken a few courses in higher mathematics myself, and having seen the average student in those classes, I can attest that you need not know the basic of anything in order to pass. Regurgitation and memorization is still a wide-spread means of passing courses, especially in mathematics.

mathwonk

Nov6-10, 03:17 PM

Well I could be wrong, I am just reacting to a lifetime of trying to teach students who went from high school AP to one of my own college classes. They never had the same background from high school that I would have given them in college. Now many of my friends did lower the level of their classes to accommodate this high school preparation but I found it hard to do that.

My attitude was that this was bad because the reason they took AP classes in high school was because they were smart honors students. Thus it made no sense to me that the best honors level high school students should be funneled into the non honors college classes.

You may be the best judge of what course you should take to meet your own goals provided you understand the options, but if you are undervaluing yourself, and not giving yourself the most challenging and useful background you deserve, then it is the job of your college counselor to suggest you try something else.

But for someone who does want to be a mathematician, and who thinks that a top student proves it by skipping calc 1, and going into non honors calc 2 or 3, I am just trying to explain to those people what they are really doing.

One of the saddest group of people I see is the first semester incoming class who have signed up for my calc 2 or calc 3 class, thinking they are prepared from an AP class in high school.

It puts me in a bind because I have to choose between losing a lot of them, or else dumbing down the course below what a college class should be. Both choices harm my students. I just wish the weaker ones would listen to me and retake calc 1 but on an honors level, or in some cases the non honors calc 1. These students are seduced by the offer of free college credit. Colleges know that AP classes are not really worth college credit but they feel pressured to offer it because the students will go elsewhere where they can get the credit. So in some cases AP credit is a sort of dishonest bribe to bring in strong students.

Unfortunately i admit the choice is even harder because not all professors are the same and many have chosen to dumb down their college classes. Taking one of those calc 1 classes would be mistake. This is why I emphasize interviewing the professor first to be sure you are in the right course.

There is no one size fits all program, but non honors college classes in calculus are often not intended for future mathematicians, and it seems unfortunate to me for the profession, if most honors high school AP students wind up in those without realizing this fact.

thrill3rnit3

Nov6-10, 11:49 PM

I feel you with your concern mathwonk, but if the CollegeBoard tried to increase the level of difficulty of the AP Calculus courses, then there wouldn't be enough teachers qualified to teach it. Then the schools would just drop AP Calculus altogether and stick to the "PreCalculus" course that is nowhere to what it should really be.

I think if they are going to make a change, it should start early on because changing it in the high school level, I feel, would be too abrupt and there wouldn't be a continuity from the math courses they were used to taking into this all-new "rigorous" style of mathematics.

Again, I feel that the issue lies with the lack of proper instruction due to the lack of qualified teachers, and with the course already hard as it is (to a regular student), increasing the difficulty would need a simultaneous increase in motivation.

I really envy the students that have/had the chance to go to schools that have a really good mathematics programs. I'm a senior right now, and my school would probably fit with the 98% of the schools in the country in terms of mathematics curriculum.

mathwonk

Nov7-10, 03:46 PM

Maybe I am giving the wrong advice here (although I think most of my colleagues agree with it.) But maybe I should just warn people that although AP courses and tests are a fait accompli in high schools, and that many college professors teach the same course as in high school, nonetheless there are some old dinosaurs like me, who still teach the way they were taught in the 1960's, believing that challenging courses are more useful than ones in which an average decent student is guaranteed an A.

Since these courses are much harder than most AP courses (based on the questions on AP tests I have seen) a student needs to be careful to understand what type of course he is getting into by either interviewing the professor or some of his previous students, and perhaps looking at some materials from the course in the recent past.

Along those lines, although this may be irrelevant since I am now retired, and maybe few other people teach like me, here are some of the tests I gave in an honors level calc 2 course a while back, that was taken by first semester students which had only high school AP preparation. As I recall, they largely felt it was the hardest course they had ever had, and at least one student dropped out after test 1 because she only got an A-.

mathwonk

Nov7-10, 03:48 PM

heres another one.

DR13

Nov7-10, 04:56 PM

So I looked through the tests (didnt actually do them but I was honest with my self). The first one was super easy. The second one would trip me up a bit as of this moment but if I actually studied for it then it would not be a problem. The third one would be the only one to give me real trouble as we did not cover series that well in my AP calc BC class.

thrill3rnit3

Nov7-10, 07:55 PM

Lipschitz continuity in a Calc 2 class? I like it :smile:

edit: It's an honors class so I think it's cool. I thought it was a regular Calc 2 class.

And I take it that you never use LaTex typeset mathwonk?

mathwonk

Nov7-10, 10:10 PM

this was an ordinary honors class, not a high level spivak honors class. this class is taken by anyone in our general "honors" program.

mathwonk

Nov7-10, 10:13 PM

DR13. I am curious as to your answer to III i in test one. (Hint: It is a theorem of Newton.)

mathwonk

Nov7-10, 10:29 PM

Well I thought it interesting to generalize the fundamental theorem of calculus. I.e. if f is continuous then an indefinite integral G of f is characterized by being differentiable everywhere with derivative equal to f. So I thought it should be interesting to describe an indefinite integral G of any riemann integrable function f. It turned out to be any G which is differentiable wherever f is continuous with derivative equal to f at those points, but also G is lipschitz continuous.

I.e. I was fully aware that every indefinite integral of any integrable function was continuous but i did not realize they were also lipschitz continuous and that without this one cannot nail down a G which computes the integral of f. I.e. if f is only integrable, there can be continuous G, which are differentiable with derivative equal to f wherever f is continuous, and yet G(b)-G(a) does not equal the integral of f. Whenever I teach a course I rethink all the material and try to introduce something new, so it is not the same every year. I am trying to pass on the tradition of discovery whenever I teach anything. This is something I like to think a researcher should bring to a course.

DR13

Nov7-10, 11:36 PM

DR13. I am curious as to your answer to III i in test one. (Hint: It is a theorem of Newton.)

I would say that even if the function does not exist at 1 the integral of f over [0,1] is the same as the integral of f over [0,1) because the integral of any f over [1,1] equals 0.

(Hopefully this is right and I didnt make myself look like a fool. Its been a couple of years since I went over the rules that make a function differentiable)

mathwonk

Nov8-10, 01:26 PM

Good. That is correct. You can ignore the value at a single point in determining integrability. But it still requires a hypothesis to show it is integrable. In this case the function is not continuous nor even piece wise continuous, so the usual theorems in most books do not apply.

Here the function is integrable because it is "monotone" decreasing. Newton already proved before Riemann that all monotone functions have a definite integral. It is certainly not a sign of foolishness to forget or not have seen this. I am just indicating the level of subtlety that a college course may contain that is not usual in high school.

In most books the theorem that all continuous functions are integrable is stated but not proved. It seemed to me that since most functions people actually encounter are piecewise monotone (e.g. all polynomials, rational functions, exponential, log, and trig functions), it would be nice to give the actual proof that such functions are integrable, since the proof is very easy. I then can also give the proof of the Fundamental theorem of calculus (FTC) for monotone continuous functions, since that is also easy and conveys the full idea behind the FTC.

I.e. to me it is a sign of the carelessness with which many books are written that they give the wrong impression that what they are doing is hard. They just haven't thought about it enough to realize that what they are doing is easy if done right. They just seem to copy the same stuff from one book to another year after year.

How can the student be expected to understand the material if the author does not even think about it deeply? This theorem on integrability of all monotone and hence all piecewise monotone functions does appear in the excellent college level honors book by Apostol. It also appears, with credit to Newton, in the excellent book by Michael Comenetz. That latter book also conveys very carefully the physical intuition behind the concepts of derivative and integral.

It is usual by the way for my students to do fairly well on my test one, which covers mostly material they have seen in high school, and then to bomb on test 2, which requires actually learning something new that has been presented in my course, and learning it rather more quickly than in high school. I.e. the one or two years of high school usually lasts about 3 weeks into the college course.

In this second semester honors course I also presented the L1, L2, and sup norms on the metric space of bounded continuous functions, and proved that sup norm convergence is preserved by taking indefinite Riemann integrals over a bounded interval. This was used to deduce convergence of the derived series of a power series by the usual trick of integrating back and using the FTC.

This sort of thing is sometimes not seen until a senior analysis course in most non honors programs, and essentially never in most high schools.

The concept of lipschitz continuity was presented in order to answer the question: suppose f is Riemann integrable but not continuous on [a,b]. Then we can still define a function H = definite integral of f from a to x, and we will have the integral of f over [a,b] equalling H(b)-H(a). But how do we recognize such an H? I.e. how d we a recognize an "antiderivative" function G for f in this case such that the integral of f over [a,b] must equal G(b)-G(a)?

The answer is that G should be any lipschitz continuous function which has a derivative equal to f at those points where f is continuous. (Since f is integrable, it must be continuous at most points, as Riemann himself showed.)

This sort of thing is probably not done in any high school course anywhere.

In the first semester of the honors course I proved that all locally bounded functions on [a,b] are globally bounded there, in particular all continuous functions on [a,b] are bounded, the main result usually not proved in first semester calculus. Then one derives the mean value theorem and hence the main corollary that a differentiable function is determined on an interval up to a constant by its derivative.

I thought through the usual proofs and remade them into more elementary arguments using infinite decimals instead of abstract axiomatic arguments, to render them easier and more concrete. I have not seen such arguments in any books.

My experience is that even strong high school AP calc students are challenged by my first semester honors course, and that is where I advise most of them to begin. The rare student who is beyond that level is advised to take the first semester spivak style "super honors" course. Hardly anyone is recommended to take a later (second semester or higher) course. The honors level ones are too hard, and the non honors level ones may be as well, but they also run the risk of falling below the honors level of challenge that an AP student deserves.

However it could be reasonable for a student who does not want to be a mathematician, but is interested in engineering, or another application of calculus, and who has the desired level of computational skill in calculus, to begin in a later non honors course. This is provided they are not interested in learning calculus at a theoretical level and are happy in a non honors class as a means to a practical end, and may not be as intellectually challenging.

deluks917

Nov8-10, 10:42 PM

Do your students find the idea of metric confusing? How much of your honors class is spent doing the standard epsilon delta proofs? L1, L2 is somewhat advanced for calc2. Everything in your post seems like it would make for a great class but it does seem pretty hard.

mathwonk

Nov9-10, 08:17 PM

I think they found the class hard. One of them said it was the most challenging math course he had ever taken. Then idea that courses should be easy is not one I adopt.

DR13

Nov9-10, 08:48 PM

I think they found the class hard. One of them said it was the most challenging math course he had ever taken. Then idea that courses should be easy is not one I adopt.

Just curious. How do you curve your course? Do you curve the class average to some grade or alter your grading scale or go curveless?

mathwonk

Nov9-10, 11:04 PM

kramer733

Aug9-11, 09:10 PM

well when the class is hard I try to give higher grades. I am trying to challenge but also encourage people. If they do poorly I adjust, and feel somewhat responsible for their poor preparation. In a really hard honors class like the one I described I try to give mostly A's and B's. In that class a C means they did not really get much of it. I think all those who stayed got A or B.

Is it true that colleges have exams backing up to their earlier days in the library? Can you give us an example of 1960s level course load and questions from the exam?

Kindayr

Aug12-11, 06:17 PM

I think its incredible that Ontario schools hardly even begin to touch the concept of Calculus. The content is shared in a 1 semester course in grade 12 with an introduction to vectors. So you have grade 12 students going into university with 2.5 months of elementary Calculus (half of which is taken up with limits) and 2.5 months on elementary linear algebra. (we barely learn derivatives, let alone begin to discuss anything past or including integration).

Going into university, students have only briefly been introduced with 2.5 months of "calculus" before a 2 month break. This forces university calculus to have to waste time on simpler concepts due to the failures of our education system.

The worst of it is you have math majors, like myself, put a complete disadvantage without even knowing it until the damage has been done. That is, we waste university credit on learning what would be considered high school concepts to our US counterparts.

I'm going into my third year of university and I legitimately feel that I've wasted time and money in my first two years of school.

Ryker

Aug12-11, 08:07 PM

The worst of it is you have math majors, like myself, put a complete disadvantage without even knowing it until the damage has been done. That is, we waste university credit on learning what would be considered high school concepts to our US counterparts.I though the Canadian high school system in general is better than the one in US. Is that not the case then? Or is it just Ontario that's so bad comparatively? I know many countries have a better high school system than Canada, but I didn't think it was even worse than in the US. I don't know, I think you're giving the US credit where it's not due. From what I understand, there are very few people that actually take advanced courses, whereas the vast majority gets an education that not many people in the world would envy. And I guess the regular versions of introductory maths courses really are just covering what high school should've covered, but a lot of universities in Canada now offer Honours courses, and no AP course that students in the US are taking covers what is covered in those. So by taking those I don't see how you'd be wasting university credit.

Kindayr

Aug13-11, 01:16 AM

I may be overtly bitter about the situation. However, the unfortunate, cold, hard fact is that calculus is next to non-existent within Ontario high schools.

Its not that there isn't demand, because I know of at least 3 full classes (~25 students each) we held at my school, that has a lower population relative to other high schools in my area (~850 students). A large portion of my graduating classes went into business, biology, economics, and a relatively large group that includes math, physics, engineering, actuarial, and stats. So there should definitely be a reworking which includes calculus being taught earlier. I think math in general in elementary and high school in Ontario is bogus.

Grade 9 you're introduced to linear functions. Grade 10 is quadratics with an introduction to trigonometric. Grade 11 has a focus on quadratics including translations and transformations, etc. Grade 12 Advanced Functions goes beyond quadratics. And Grade 12 "Calculus and Vectors" works with lines, planes, and some projections (2.5 months), and then non-rigorous limits with introduction to derivatives.

I feel that is just not a good system, straight up.

I'm not necessarily at a disadvantage in my own university calculus and math courses (though, friends from BC already had some workings with integrals). But in comparison to other students at schools in the US where they are ahead of where I am. I just feel that some time was stolen from me. Yes its probably a failure of myself for not getting interested in these topics when I was younger. I just feel that a lot of this content can be taught at much, much younger ages.

Just look at the experiments the Khan Academy is doing in that one elementary school. Supposedly a larger portion of the students are learning more material and harder material, than what a normal math class would cover. Some of the students are even doing pre-Calculus and Calculus content in the 6th or 7th grade. Its not that this school is a school for genius children. I think it just goes to show how accessible this content is to younger children.

Ryker

Aug13-11, 01:58 AM

From what you described, you really didn't cover that much. I can't remember my exact curriculum from back home, but I know we covered integration. However, I'm now studying in Alberta, and to me it seemed as if other students from the province knew that topic already, as well. I knew there were differences in provincial high school systems, and I've also read that Alberta's is one of, if not the best in Canada, but I assumed this is such a basic topic that everyone covers it.

What's funny is that I got rejected by an Ontario-based university due to my supposed lack of Chemistry prerequisites, even though I took three years of it in high school. The only thing I didn't do is take it in my last year, but you don't really introduce much new stuff then, you just consolidate knowledge for the final exams, the results of which are looked at by home universities when deciding upon admitting students. If Chemistry in Ontario is anything like maths you mentioned, then that's even more hilarious now. No sour grapes, either :biggrin:

Kindayr

Aug13-11, 02:05 AM

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.

kramer733

Aug13-11, 02:50 PM

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

Aug15-11, 12:18 AM

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

Aug15-11, 12:22 AM

it says i have attached a pdf file of this exam but i don't see it.???

eumyang

Aug15-11, 04:15 AM

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.)

xdrgnh

Aug16-11, 12:05 AM

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.

kramer733

Aug16-11, 12:34 AM

xdrgnh

Aug16-11, 12:39 AM

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

Aug16-11, 05:25 PM

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.

WannabeNewton

Aug16-11, 05:43 PM

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.

xdrgnh

Aug16-11, 05:53 PM

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

Aug16-11, 05:54 PM

no serious argument should ever include the phrase "I got nothing wrong with..."

xdrgnh

Aug16-11, 05:58 PM

no serious argument should ever include the phrase "I got nothing wrong with..."

No serious debater would resort to mud slinging like that.

WannabeNewton

Aug16-11, 06:09 PM

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.

xdrgnh

Aug16-11, 06:26 PM

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.

micromass

Aug16-11, 06:33 PM

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.

xdrgnh

Aug16-11, 08:14 PM

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.

micromass

Aug16-11, 08:29 PM

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

Aug17-11, 11:34 AM

no serious debater would have a picachu as an icon.

BloodyFrozen

Aug17-11, 11:44 AM

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.

n1person

Aug17-11, 12:06 PM

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

Aug17-11, 01:48 PM

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.

mathwonk

Aug17-11, 02:23 PM

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

Aug17-11, 02:31 PM

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.

mathwonk

Aug17-11, 03:56 PM

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.