Should calculus be taught in high school? [Archive]

brainy kevin

May8-09, 09:10 PM

While on the surface, this appears to be a no brainer, (Of course it should, if the students are ready) I actually seriously question the practice of letting high schoolers, usually seniors, take calculus. You see, the college calculus fail rate is about 50%, which is not good at all. It is a complex problem, but it has a great deal to do with the fact that incoming college students have minimal mathematical maturity, and have only a tenuous grasp of trig and advanced algebra. Most high school textbooks teach by working out a few problems, and having a grossly oversimplified explanation. Classics like Jacobs, Sullivan, and the like are rarely used. Why not, then, take a slower pace with some of the great textbooks throughout high school, have an exhaustive understanding of the subjects, develop mathematical maturity and thereby adequately prepare students for truly rigorous calculus in college. (Like Apostol's Spivak's or similar calculus texts?)

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

Tobias Funke

May8-09, 10:47 PM

Howers

May9-09, 12:18 AM

Teaching students deep mathematics in high school was tried and tested in the 60s... the failure rates were even more alarming. Simply put, there is no point in designing the curriculum to meet the needs of less than 1% of the students. Very few students will need that kind of depth, and most are served better by a skimpy version of calculus which is used in engineering and science - by far the most popular majors that require any math. Also, most people lack the ability and interest to pursue mathematics at that kind of level.

Having said that, I think the standards should be increased for students in high school. You can pull an A off without having a clue what you are doing.

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

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

Thoughts?

snipez90

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.