23 thoughts on “The Peacock-Tail Theory of AP Calculus”

This is the first time I strongly disagree with one of your blogpost. Calculus is nearly the most usefull discipline in Secondary Education. Its concepts are of utterly importance to Mathematics itself. And its by far the most applicable discipline in Economics, Physics, …. Third argument: it is beautifull, just like a peacock!

maybe his point is that the way calc is taught in high school is pretty useless. i teach at a school where most students test right into calc 3, but I find they have very little understanding of calc 1 and 2. i would much rather everyone just start learning calculus when they got to college because so many high schools just teach students some rules to memorize instead of any conceptual understanding.

I’m with you, Hans, that a well-taught calculus class is a great thing! Intellectually enriching, full of connections to other disciplines, planting the seeds for a whole garden of ideas in later mathematics…

But that doesn’t seem to be what we do in AP Calculus. We spend a lot of time on computational and technical issues. We linger in the halfway house of “limit notation,” somewhere between hand-waving intuition and actual rigor, often achieving the benefits of neither.

I find I can see high school calculus in one of two ways: (1) as a very unsuccessful pair of wings, or (2) as a pretty successful peacock tail.

I’m inclined to believe the latter. Most (though not all!) students take calculus because of the effect it will have on their college applications. And colleges like to see it on applications because they know that only academically capable and ambitious kids will take it. AP Calculus is largely a signal between student and admissions officer; the teacher is only a middleman.

This matters to me because I want the classes I teach to do more of the things you describe – to provide an enriching intellectual experience. But first, I find it useful to understand the role these classes play in the educational institutions. That helps me know which battles to pick.

I, being Dutch, am an outsider regarding the UK Education System, so I don’t know what exactly AP Calculus implies or Calc 1, 2, 3 etc. In The Netherlands the way Calculus is taught is not about rules (like the chain rule), but about concepts, linking the discipline with other subjects, like Economics and Physics. Being trained myself as a (theoretical) physicist, Calculus was the far most important thing in Mathematics I learned in Secondary Education (velocity, acceleration, …). Even concept in Mathematics itself like limits and asymptots become so much clearer if there is a sound base in Calculus.

Daniel: Absolutely not. I knew I was going to study theoretical physics and mathematics in college and graduate school. I took AP calculus my junior year so I could take calculus II through dual enrollment during senior year. Waiting until college to have any calculus would have made me want to kill myself. Maybe it is rules at first (certainly wasn’t for me), but for any serious math student that doesn’t matter because the good ones will take the time to understand things more deeply. And once you take analysis, you understand where everything comes from anyway. Conversely, if you are the type who’ll never even contemplate touching analysis, then you probably are fine off taking just the surface level material in calculus. For example, someone I know who took AP calculus in my high school is now an animal vet; I doubt that they’ll need to know how to prove L’Hospital’s rule, but they were the valedictorian senior year. So having plenty of calculus in high school is critically important in education for a plethora of different types of students.

Daniel, as an aside, if students are testing into calc 3 with little understanding of calc 1 and 2, is it possible to revisit the specs on the placement test?

Or perhaps the point is that Calculus isn’t for everyone… because it’s not! If you’re not good at math but through a bad system and/or bad parenting you were in Precalculus as a junior, why would you take Calculus, both a class AND a topic that are going to be of essentially no use to you? You would be MUCH better off taking a good Statistics class (and likely not AP Stats, either) than you would be taking Calculus. When *I* read the comic, I read it from the perspective of “you don’t need Calculus on your transcript for you to look good to colleges… or at least you shouldn’t need it.” I LOVE Calculus (second only to its perhaps-opposite topic, Discrete Math), but as a high school teacher of all levels of Calculus, I see FAR to many students trying to stuff calculus onto their report cards instead of taking something they will actually a) understand, and b) use later in life.

I think a lot of it is going to depend on the teacher, both in high school and in college. I didn’t take calc in high school. Though I probably should have, I also heard that the teacher was very much about busy work and focused little on concepts. No one in his classes scored above a 3 on the AP test. That being said, when I got to college, I had a TA from eastern europe whom I couldn’t understand and the book was so theoretical that I didn’t learn any practical computational skills. I ended up taking a class at another college that had that right balance, partially because I had a very good and experienced teacher. Because of that teacher, I really loved calculus, and math in general, while prior to that, the whole thing was drudgery.

Calculus is certainly very useful, to those who will be using it. This includes most engineers many scientists, and many applied mathematicians. It doesn’t include most computer scientists.

I don’t know the statistics on this, but my guess is that calculus is useful in this manner to between 15% and 30% of those who actually take calculus. That’s a lot of people. Perhaps it is a reason to expect it of students who aspire to get into a good college. Perhaps not.

Discrete math is much more useful for those who want to be computer scientists, or who want to understand computer programming.

Linear algebra is very useful and widely used in many fields, especially fields that deal with numerical computing.

Probability, statistics and data analysis are useful for anyone who wants to understand interpretations of data. It’s very important to understanding the world in which we live.

For many people (myself included), calculus is the least important of the four fields of math that I mentioned, unless you count the fact that it makes it easier to get into a good college. In that domain, (as was the point of this post), calculus is important to aspiring students just as the tail is important to an aspiring (male) peacock.

Are students being given ‘peacock tails’ because they are taught facts instead of trained to think in a mathematical way?

In some subjects recalling facts, dates and names may suffice but in a mathematical context if you’re not learning how to reason and manipulate ideas for yourself then the whole rotten edifice collapses. You might as well waste your time somewhere else.

Also, I think that given the wide and wonderful world of mathematics we could probably find something more interesting and relevant to teach than calculus. It’s pretty dry material for the most part. Maybe other topics would help student’s engage with mathematics better? It’s unfortunate that so much time is dedicated to calculus when it isn’t interesting or useful to many people.

Ha ha – very fair! I don’t intend to use it for emphasis, but rather for color (I find it makes a block of handwritten text more pleasing to the eye). But if it’s distracting, then that’s counterproductive!

I think you have properly tagged the role Calculus plays for many, many students.
I think it’s a pretty effective response to the people suggesting that “most people don’t really need higher math so we shouldn’t make them take it,” too.

My third year high school algebra class was considerably harder than first year calculus. I ended up reading a calculus book at the local community college near the end of high school as I was interested in physics, and it’s really hard to do much in physics without calculus. That’s why Newton invented it.

The most useless thing taught in high school math is complex numbers. Unless you do Laplace transforms (mechanics or circuit theory) or quantum mechanics, you are unlikely to ever use them again in your life.

“For most students, taking calcuclus in high school serves the same purpose as a peacock growing a tail.” Really? Most?

I suggest that the order of common reasons for H.S. students to take AP Calculus is:
1) They like math.
2) Why not take a course that gives you college credit if you can handle the material?
3) Less coursework in college.

I think “to impress”, would be farther down the list.

Interesting – my own thoughts, I have to admit, are grounded very much in anecdote, personal impressions, conversations with students and with adults, not any kind of systematic study. But I would put “they like math” quite far down the list, and “to strengthen college applications” at the very top.

I imagine our different perspectives come from different backgrounds – can I ask what experiences have shaped your impressions?

I defer to your experience as a teacher over mine as a dad.

I love math and see beauty in it in many unexpected places (e.g. Basel Problem, i^i is a real number, non-existence of a quintic formula, etc.). I’ve shared my enthusiasms with my son, who (brag alert) just scored 5 on his AP Calc AB. When he signed up for AP classes, neither he nor I even considered that colleges cared about them as regards to admissions, just as credits. He just loves math. Sample size n = 1, I know.

I guess it’s just wishful thinking that kids wouldn’t be so cynical that they’d take an AP class just to impress someone besides themselves. Sigh.

Yeah, I wrestle with my own cynicism on this stuff – I’d like to think that the function of our education system is to enlighten and enrich, rather than to sort and label, but I think schools play a lot of functions in society, not all of them particularly pretty.

Anyway, I’m glad that your son is such a fan of mathematics – it’s great that you’ve got two generations of math appreciation in your family!

Calculus is a difficult subject to learn. If a student has some familiarity with the subject, however broken it may be, it is very likely that the student will develop much better understanding during college coursework. In all honesty, most students do not fully understand all of the material that the Calc courses offer by the end of the semester.

Gaming the college admission system is nothing new. Example, there was an article recently in NYTimes on how rich parents are booking exotic “charity” trips to countries specifically for college admissions (“http://www.nytimes.com/2016/08/14/opinion/sunday/to-get-to-harvard-go-to-haiti.html?_r=0”). Some students who cannot afford that exotic trip are going to resort to advanced calculus course to impress colleges. The problem is not with AP courses, but, lies within the admission process.

This is the first time I strongly disagree with one of your blogpost. Calculus is nearly the most usefull discipline in Secondary Education. Its concepts are of utterly importance to Mathematics itself. And its by far the most applicable discipline in Economics, Physics, …. Third argument: it is beautifull, just like a peacock!

maybe his point is that the way calc is taught in high school is pretty useless. i teach at a school where most students test right into calc 3, but I find they have very little understanding of calc 1 and 2. i would much rather everyone just start learning calculus when they got to college because so many high schools just teach students some rules to memorize instead of any conceptual understanding.

Yeah, Daniel, I think that’s a large part of it.

I’m with you, Hans, that a well-taught calculus class is a great thing! Intellectually enriching, full of connections to other disciplines, planting the seeds for a whole garden of ideas in later mathematics…

But that doesn’t seem to be what we do in AP Calculus. We spend a lot of time on computational and technical issues. We linger in the halfway house of “limit notation,” somewhere between hand-waving intuition and actual rigor, often achieving the benefits of neither.

I find I can see high school calculus in one of two ways: (1) as a very unsuccessful pair of wings, or (2) as a pretty successful peacock tail.

I’m inclined to believe the latter. Most (though not all!) students take calculus because of the effect it will have on their college applications. And colleges like to see it on applications because they know that only academically capable and ambitious kids will take it. AP Calculus is largely a signal between student and admissions officer; the teacher is only a middleman.

This matters to me because I want the classes I teach to do more of the things you describe – to provide an enriching intellectual experience. But first, I find it useful to understand the role these classes play in the educational institutions. That helps me know which battles to pick.

I, being Dutch, am an outsider regarding the UK Education System, so I don’t know what exactly AP Calculus implies or Calc 1, 2, 3 etc. In The Netherlands the way Calculus is taught is not about rules (like the chain rule), but about concepts, linking the discipline with other subjects, like Economics and Physics. Being trained myself as a (theoretical) physicist, Calculus was the far most important thing in Mathematics I learned in Secondary Education (velocity, acceleration, …). Even concept in Mathematics itself like limits and asymptots become so much clearer if there is a sound base in Calculus.

Daniel: Absolutely not. I knew I was going to study theoretical physics and mathematics in college and graduate school. I took AP calculus my junior year so I could take calculus II through dual enrollment during senior year. Waiting until college to have any calculus would have made me want to kill myself. Maybe it is rules at first (certainly wasn’t for me), but for any serious math student that doesn’t matter because the good ones will take the time to understand things more deeply. And once you take analysis, you understand where everything comes from anyway. Conversely, if you are the type who’ll never even contemplate touching analysis, then you probably are fine off taking just the surface level material in calculus. For example, someone I know who took AP calculus in my high school is now an animal vet; I doubt that they’ll need to know how to prove L’Hospital’s rule, but they were the valedictorian senior year. So having plenty of calculus in high school is critically important in education for a plethora of different types of students.

Daniel, as an aside, if students are testing into calc 3 with little understanding of calc 1 and 2, is it possible to revisit the specs on the placement test?

Or perhaps the point is that Calculus isn’t for everyone… because it’s not! If you’re not good at math but through a bad system and/or bad parenting you were in Precalculus as a junior, why would you take Calculus, both a class AND a topic that are going to be of essentially no use to you? You would be MUCH better off taking a good Statistics class (and likely not AP Stats, either) than you would be taking Calculus. When *I* read the comic, I read it from the perspective of “you don’t need Calculus on your transcript for you to look good to colleges… or at least you shouldn’t need it.” I LOVE Calculus (second only to its perhaps-opposite topic, Discrete Math), but as a high school teacher of all levels of Calculus, I see FAR to many students trying to stuff calculus onto their report cards instead of taking something they will actually a) understand, and b) use later in life.

What happened?

I think a lot of it is going to depend on the teacher, both in high school and in college. I didn’t take calc in high school. Though I probably should have, I also heard that the teacher was very much about busy work and focused little on concepts. No one in his classes scored above a 3 on the AP test. That being said, when I got to college, I had a TA from eastern europe whom I couldn’t understand and the book was so theoretical that I didn’t learn any practical computational skills. I ended up taking a class at another college that had that right balance, partially because I had a very good and experienced teacher. Because of that teacher, I really loved calculus, and math in general, while prior to that, the whole thing was drudgery.

Calculus is certainly very useful, to those who will be using it. This includes most engineers many scientists, and many applied mathematicians. It doesn’t include most computer scientists.

I don’t know the statistics on this, but my guess is that calculus is useful in this manner to between 15% and 30% of those who actually take calculus. That’s a lot of people. Perhaps it is a reason to expect it of students who aspire to get into a good college. Perhaps not.

Discrete math is much more useful for those who want to be computer scientists, or who want to understand computer programming.

Linear algebra is very useful and widely used in many fields, especially fields that deal with numerical computing.

Probability, statistics and data analysis are useful for anyone who wants to understand interpretations of data. It’s very important to understanding the world in which we live.

For many people (myself included), calculus is the least important of the four fields of math that I mentioned, unless you count the fact that it makes it easier to get into a good college. In that domain, (as was the point of this post), calculus is important to aspiring students just as the tail is important to an aspiring (male) peacock.

Are students being given ‘peacock tails’ because they are taught facts instead of trained to think in a mathematical way?

In some subjects recalling facts, dates and names may suffice but in a mathematical context if you’re not learning how to reason and manipulate ideas for yourself then the whole rotten edifice collapses. You might as well waste your time somewhere else.

Also, I think that given the wide and wonderful world of mathematics we could probably find something more interesting and relevant to teach than calculus. It’s pretty dry material for the most part. Maybe other topics would help student’s engage with mathematics better? It’s unfortunate that so much time is dedicated to calculus when it isn’t interesting or useful to many people.

I love your blog, but can I just suggest that underlining becomes a lot less effective when you underline nearly every word?

Ha ha – very fair! I don’t intend to use it for emphasis, but rather for color (I find it makes a block of handwritten text more pleasing to the eye). But if it’s distracting, then that’s counterproductive!

Ditto! To my eye, too, it does look like over-eager underlining.

I think you have properly tagged the role Calculus plays for many, many students.

I think it’s a pretty effective response to the people suggesting that “most people don’t really need higher math so we shouldn’t make them take it,” too.

My third year high school algebra class was considerably harder than first year calculus. I ended up reading a calculus book at the local community college near the end of high school as I was interested in physics, and it’s really hard to do much in physics without calculus. That’s why Newton invented it.

The most useless thing taught in high school math is complex numbers. Unless you do Laplace transforms (mechanics or circuit theory) or quantum mechanics, you are unlikely to ever use them again in your life.

I was lucky. I had a great teacher for AP Calculus. It was the first time math was interesting.

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“For most students, taking calcuclus in high school serves the same purpose as a peacock growing a tail.” Really? Most?

I suggest that the order of common reasons for H.S. students to take AP Calculus is:

1) They like math.

2) Why not take a course that gives you college credit if you can handle the material?

3) Less coursework in college.

I think “to impress”, would be farther down the list.

Interesting – my own thoughts, I have to admit, are grounded very much in anecdote, personal impressions, conversations with students and with adults, not any kind of systematic study. But I would put “they like math” quite far down the list, and “to strengthen college applications” at the very top.

I imagine our different perspectives come from different backgrounds – can I ask what experiences have shaped your impressions?

Thanks for your reply.

I defer to your experience as a teacher over mine as a dad.

I love math and see beauty in it in many unexpected places (e.g. Basel Problem, i^i is a real number, non-existence of a quintic formula, etc.). I’ve shared my enthusiasms with my son, who (brag alert) just scored 5 on his AP Calc AB. When he signed up for AP classes, neither he nor I even considered that colleges cared about them as regards to admissions, just as credits. He just loves math. Sample size n = 1, I know.

I guess it’s just wishful thinking that kids wouldn’t be so cynical that they’d take an AP class just to impress someone besides themselves. Sigh.

Keep up the good work on your blog.

Yeah, I wrestle with my own cynicism on this stuff – I’d like to think that the function of our education system is to enlighten and enrich, rather than to sort and label, but I think schools play a lot of functions in society, not all of them particularly pretty.

Anyway, I’m glad that your son is such a fan of mathematics – it’s great that you’ve got two generations of math appreciation in your family!

Calculus is a difficult subject to learn. If a student has some familiarity with the subject, however broken it may be, it is very likely that the student will develop much better understanding during college coursework. In all honesty, most students do not fully understand all of the material that the Calc courses offer by the end of the semester.

Gaming the college admission system is nothing new. Example, there was an article recently in NYTimes on how rich parents are booking exotic “charity” trips to countries specifically for college admissions (“http://www.nytimes.com/2016/08/14/opinion/sunday/to-get-to-harvard-go-to-haiti.html?_r=0”). Some students who cannot afford that exotic trip are going to resort to advanced calculus course to impress colleges. The problem is not with AP courses, but, lies within the admission process.