Should calculus be taught in high school?

[brainy kevin]

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]

Well, I teach AP calc, so I'll say a few words. I think calculus should be taught, but no college credit given. That way, the serious and mathematically gifted students can take it and the students who are only there because it's another AP class to pad their applications will be mostly weeded out.

If the system functioned ideally and only students who mastered the previous material passed I'd reconsider, but there are too many students who don't know basic trig or logarithm properties (nor have any clue how they may go about rediscovering them) that somehow make it to my class.

As for the students who could handle the course but wouldn't take it because they see no reason too, that's fine. Let them wait until college.

 

[Howers]

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]

Hmm, the solution you outlined sounds nice, but it's a lot to ask of the current education system in America. But I think I'm more concerned about your use of the term "exhaustive". The prerequisites for understanding calculus are actually very finite. A strong understanding of the very basics is required of trigonometry is required (a good calculus book will give a more rigorous treatment anyways). For algebra, the ability to solve equations, not necessarily very difficult ones, is required, but this is fundamental.

This should be enough to tackle a book such as Stewarts. In turn, a good computational background in calculus and an overall perspective on the various topics can prepare one to tackle a book such as Spivak. I had the very good computational background, but not much knowledge of proofs, which is needed for a more theoretical treatment of calculus. It turns out by going through some of the links here: https://www.physicsforums.com/showthread.php?t=166996 (the first one is especially good imo), that was enough to understand Spivak.

I think an honest attempt to go through Stewart while giving the explanations and proofs provided in the book is a lot more instructive than what you'll find in many high school calculus courses. Indeed, this is one reason why I don't think it's harmful for someone to read Stewart before a more rigorous introduction (of course, the person should judge for themselves by comparing to a more theoretical book) because if you really read and understand everything in Stewart and perhaps do the problems in the problems plus section, you can learn a lot. The route I outlined above is of course subject to many contingencies and is certainly not exhaustive, but it is practical.

 

[cristo]

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 learned 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]

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]

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]

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]

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 let's 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]

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]

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]

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]

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]

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]

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]

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]

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]

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.

 

[physicsnoob93]

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]

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]

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.

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]

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]

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]

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]

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.

Interesting statement...I can't agree or disagree at the moment, since it is a generalized statement. Do you have any sources that support your remark? What about links to the national mathematics curriculum for foreign countries? We can compare their standards by grade to those of the U.S.

 

[yeongil]

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.

I am confused by this statement. Are you saying that what students are learning in Math class in grades K-2 isn't "elementary math" at all? What are they learning, then?


01

 

[thrill3rnit3]

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]

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 he's 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]

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 he's 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]

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]

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]

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?

This still doesn't account for the fact that in the US kids spend 7 years learning how to add and subtract due to the curriculum.

 

[thrill3rnit3]

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]

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]

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]

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.

 

[thrill3rnit3]

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]

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.

 

[Feldoh]

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]

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]

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]

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]

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]

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]

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]

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.

 

[buffordboy23]

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]

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]

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]

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.

Some students need tangible examples, while others have no problem with abstractions like n-tuples.