Should calculus be taught in high school?

In summary, the conversation discusses the topic of teaching calculus in high school and whether it adequately prepares students for the rigor of college calculus courses. While some argue that it should be taught to develop mathematical maturity and better prepare students, others argue that the fail rates in college suggest otherwise. The conversation also touches on the idea of increasing standards in high school and the role of prerequisites in understanding calculus. Ultimately, the consensus is that while calculus should be taught in high school, it should not be counted for college credit and the curriculum should be reevaluated to better prepare students for higher level mathematics.
  • #1
brainy kevin
24
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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?
 
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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
snipez90 said:
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!
 
  • #6
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.
 
  • #7
I agree with Tobias!

Tobias Funke said:
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!
 
  • #8
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.
 
  • #9
Tobias Funke said:
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.
 
  • #10
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.
 
  • #11
Tobias Funke said:
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.


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  • #12
Count Iblis said:
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.
 
  • #13
yeongil said:
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.
 
  • #14
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.
 
  • #15
brainy kevin said:
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.
 
  • #16
Tobias Funke said:
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."
 
  • #17
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.
 
  • #18
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:
 
  • #19
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.
 
  • #20
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
 
  • #21
Count Iblis said:
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.
 
  • #22
Personally, I don't think there is any way out of this "education gap" between the United States and the rest of the world.
 
  • #23
Andy Resnick said:
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.
 
  • #24
qntty said:
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.
 
  • #25
Count Iblis said:
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.
 
  • #26
Count Iblis said:
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
 
  • #27
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...
 
  • #28
Count Iblis said:
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:
Count Iblis said:
...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?
 
  • #29
physicsnoob93 said:
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.
 
  • #30
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.
 
  • #31
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?
 
  • #32
mXSCNT said:
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.
 
  • #33
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).
 
  • #34
thrill3rnit3 said:
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.
 
  • #35
Astronuc said:
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.
 
<h2>1. Should calculus be taught in high school?</h2><p>This is a common question among educators and parents. The answer is not a simple yes or no, as it depends on various factors such as the curriculum, resources, and student population. However, many experts argue that calculus is an important subject that can benefit high school students in their academic and professional pursuits.</p><h2>2. What are the benefits of teaching calculus in high school?</h2><p>One of the main benefits of teaching calculus in high school is that it prepares students for college-level math courses. It also helps develop critical thinking, problem-solving, and analytical skills that are essential in various fields such as science, engineering, and economics. Additionally, calculus can open up career opportunities in these fields.</p><h2>3. Is calculus too difficult for high school students?</h2><p>Many people believe that calculus is a challenging subject, and some argue that it may be too difficult for high school students. However, with proper instruction and support, high school students can grasp the fundamental concepts of calculus and even excel in the subject. It is important to provide students with a strong foundation in algebra and geometry before introducing calculus.</p><h2>4. Are there any alternatives to teaching calculus in high school?</h2><p>Some educators propose alternative math courses, such as statistics or data analysis, instead of calculus in high school. While these courses may also be beneficial, they do not cover the same concepts and skills as calculus. Therefore, it is important to offer a variety of math courses to cater to the diverse interests and abilities of students.</p><h2>5. How can we make calculus more accessible to high school students?</h2><p>One way to make calculus more accessible is by incorporating real-world applications and examples in the curriculum. This can help students see the practical applications of calculus and make it more relatable. Additionally, providing extra support and resources, such as tutoring and online resources, can also help students who may struggle with the subject.</p>

1. Should calculus be taught in high school?

This is a common question among educators and parents. The answer is not a simple yes or no, as it depends on various factors such as the curriculum, resources, and student population. However, many experts argue that calculus is an important subject that can benefit high school students in their academic and professional pursuits.

2. What are the benefits of teaching calculus in high school?

One of the main benefits of teaching calculus in high school is that it prepares students for college-level math courses. It also helps develop critical thinking, problem-solving, and analytical skills that are essential in various fields such as science, engineering, and economics. Additionally, calculus can open up career opportunities in these fields.

3. Is calculus too difficult for high school students?

Many people believe that calculus is a challenging subject, and some argue that it may be too difficult for high school students. However, with proper instruction and support, high school students can grasp the fundamental concepts of calculus and even excel in the subject. It is important to provide students with a strong foundation in algebra and geometry before introducing calculus.

4. Are there any alternatives to teaching calculus in high school?

Some educators propose alternative math courses, such as statistics or data analysis, instead of calculus in high school. While these courses may also be beneficial, they do not cover the same concepts and skills as calculus. Therefore, it is important to offer a variety of math courses to cater to the diverse interests and abilities of students.

5. How can we make calculus more accessible to high school students?

One way to make calculus more accessible is by incorporating real-world applications and examples in the curriculum. This can help students see the practical applications of calculus and make it more relatable. Additionally, providing extra support and resources, such as tutoring and online resources, can also help students who may struggle with the subject.

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