Should calculus be taught in high school?

[symbolipoint]

The problem with answering this question is that it's too vague- perhaps the question shouldn't be 'when should [x] be taught', but rather 'how can topic [x] be taught better?'

For a specific example- my oldest, who has loved math until this year. The topic- (high school) geometry. I asked her what's the difference, and she replied: "Until this year, math was all about finding different ways to solve problems. With geometric proofs, there's only one way: starting with some 'obscure' rule that if you don't know, you can't solve the problem 'correctly'."

Andy Resnick,
Nobody can calculate for certain whether your daughter will be ready to learn Calculus before the end of high school. She has been learning about number properties and using them for number-problem-solving. NOW she is looking at shapes and directionality and several concepts described in horrible worded descriptions. Actually, you could be right, that Geometry could be taught differently to her, meaning also better for her. If your daughter is in ninth grade now, then there is some chance she may learn Calculus before graduating from high school, even if she does not get a C or better in Geometry. The reason is that, at least she is an "algebra" person, and she will rely on that when she studies Calculus. The amount of Geometry that she NEEDS to know for Calculus is much smaller than the amount of Geometry that students study in Geometry-the-course. See, in Calculus, you deal with functions, graphs, and numbers. In high school Geometry, you do not much deal with functions, and usually, the graphs are done -I say usually, not always- without cartesian coordinate systems. Actually, Geometry has a few topics requiring the cartesian coordinate system, and those particular sections of the course, she will probably find to be easier than most of the rest of her Geometry course.

Okay, this topic is supposed to be about learning Calculus in high school. Yes, it should be taught in high school but only to students who are ready for it. Students not being ready for it in high school is not bad. Learning Algebra 1 in high school before finishing grade 10 is more important than learning Calculus in high school. A student ready for Calculus in high school and wanting it but not learning it in high school is bad.

 

[Astronuc]

The problem with answering this question is that it's too vague- perhaps the question shouldn't be 'when should [x] be taught', but rather 'how can topic [x] be taught better?'

For a specific example- my oldest, who has loved math until this year. The topic- (high school) geometry. I asked her what's the difference, and she replied: "Until this year, math was all about finding different ways to solve problems. With geometric proofs, there's only one way: starting with some 'obscure' rule that if you don't know, you can't solve the problem 'correctly'."

I was thinking about this question in conjuction with the teaching of physics in high school, and the discussion of the thread about physics education in the US.
https://www.physicsforums.com/showthread.php?t=651649

Of course, calculus doesn't just happen; there are precursors: Algebra I, Geometry/Trigonometry, Algebra II, Analytical Geometry, all leading to Calculus

At my first high school, I would have been limited to Geometry, Trigonometry, Algebra II, and Analytical Geometry if I had taken a normal schedule. Fortunately, I was placed in Honors math program, so we did the Geometry in one semester instead of the normal year, followed by Trigonometry in the second semester in Grade 10. The high school did not offer Calculus. I then moved to a different high school (about 5 miles away in the same urban school district), which gave me the opportunity to take Calculus my senior year.

At the first high school, I would only be able to take one year of chemistry, and not allowed to do hands on chemistry in the lab. At the second high school, I did two years of chemistry from a teacher with an MS in Chemistry, and we did a lot of hands on chemistry in the lab, including analytical chemistry and synthesis of organic compounds. The second year included studying rate equations, so we received some practical applications of differential equations. That was in conjunction with Calculus program.

The physics course eventually included differential equations, but it was less coordinated with the math program, unfortunately.

I had started studying analytical geometry and calculus at home with the help of a summer program at a local university. IIRC, that was at the end of grade 10.

It would have been nice if the high school had a more coordinated math and science program for those students who were ready and willing to take on the math and science. I would have made a lot more progress early on had I had some guidance.

At a more advanced level are:
Multivariable/vector calculus
Linear Algebra
Ordinary Differential Equations
Group Theory
Abstract Algebra
Calculus of Variations
Partial Differential Equations
Differential Geometry and Topology

Could elements or precursors be taught in high school?

I had an exposure to matrices and determinants in junior high - 8th or 9th grade, but they were not tied to systems of equations, or vectors. Later, when I got to linear algebra, and systems of algebraic or differential equations (in university), I thought what a waste it had been not to have had some exposure years earlier.

 

[Andy Resnick]

Andy Resnick,
<snip>NOW she is looking at shapes and directionality and several concepts described in horrible worded descriptions. <snip>

No, high school geometry (at least hers) is all about proofs, e.g. 'prove segment AC is perpendicular to segment BD'. She *loved* algebraic geometry.

Okay, this topic is supposed to be about learning Calculus in high school. Yes, it should be taught in high school but only to students who are ready for it. Students not being ready for it in high school is not bad. Learning Algebra 1 in high school before finishing grade 10 is more important than learning Calculus in high school. A student ready for Calculus in high school and wanting it but not learning it in high school is bad.

I agree with this. My challenge question is "Can we get more kids interested in taking calculus (or taking the appropriate math track), and can we get more of those kids ready for calculus"?

I was thinking about this question in conjuction with the teaching of physics in high school, and the discussion of the thread about physics education in the US.
<snip>
It would have been nice if the high school had a more coordinated math and science program for those students who were ready and willing to take on the math and science. I would have made a lot more progress early on had I had some guidance.

I suspect my experience mirrors a lot of students- I learned most of my math in Physics class. This is a problem.

Ideally, a student would get exposed to a (mathematical) concept in math prior to it being applied in physics. Unfortunately, a student will likely be first exposed to a mathematical concept both in math and science class simultaneously, and at worst will be exposed to the math concept for the first time in Physics. For example, I handled diff. equations for the first time in physics (second semester freshman year) and didn't take the relevant math class until second semester sophomore year. Same thing for complex variables, linear algebra, and I've never taken a math class covering calculus on manifolds but instead learned the material in general relativity and continuum mechanics.

 

[symbolipoint]

Quote by symbolipoint

Andy Resnick,
<snip>NOW she is looking at shapes and directionality and several concepts described in horrible worded descriptions. <snip>

No, high school geometry (at least hers) is all about proofs, e.g. 'prove segment AC is perpendicular to segment BD'. She *loved* algebraic geometry.

I was being brief so left out some details. Of course Geometry is about proofs, but Geometry is different from Algebra and generalized Arithmetic in that Geometry now concentrates on points, lines, planes, directionality, and shapes; and certainly proving things about these.

 

[Sankaku]

No, high school geometry (at least hers) is all about proofs, e.g. 'prove segment AC is perpendicular to segment BD'. She *loved* algebraic geometry.

I was wondering if her high-school used Hartshorne for algebraic geometry, or something a little easier...?






[Sorry, I couldn't resist. You can now safely resume your on-topic discussion.]

 

[Andy Resnick]

I was wondering if her high-school used Hartshorne for algebraic geometry, or something a little easier...?

No clue, sorry.

 

[micromass]

I was wondering if her high-school used Hartshorne for algebraic geometry, or something a little easier...?

Imagine that...

 

[Andy Resnick]

I was being brief so left out some details. Of course Geometry is about proofs, but Geometry is different from Algebra and generalized Arithmetic in that Geometry now concentrates on points, lines, planes, directionality, and shapes; and certainly proving things about these.

At the risk of going off-topic again, this mirrors *exactly* my complaint about the way intro physics is taught, calculus or not. Teaching these classes using a pedagogical approach of "First, we define a whole bunch of obvious things (e.g slope, limits, velocity...) in terms of inscrutable symbols. We will also define a bunch of nonphysical abstractions (lines, points, vectors..) and claim they have physical relevance. Then, we manipulate these symbols to generate a fair number of formulas that obscure the underlying concepts. We then claim our computational results, even though based on nonphysical things, are an accurate description of the real world. For evidence in support of this claim, the class is often accompanied by a poorly-executed lab exercise, with mumbled excuses about 'errors'.

"On the exams, you (the student) will demonstrate that you 'understand' these symbolic scrawls by replicating the previously shown symbolic manipulation and sometimes plugging in arbitrary numbers to generate 'an answer'. Since there is only one correct sequence of manipulations and substitutions, your answer is either exactly right or exactly wrong. You have no freedom to think or explore because there are no alternative methods to 'solve this problem'. Welcome to science!"

Discussing whether or not calculus should be available in high school is moot- students can learn all about calculus on the interweb as soon as they can use a computer- the real question is 'how can teaching calculus in high school be improved?'

 

[mege]

I think I had a 'good' high school calculus experience. My instructor was amazing, and even though we were taught to the Calc AB exam - my first semester in college studying Calc II was a breeze (and mostly review). My high school instructor taught the class as symbolically as possible, but - he also knew what he was getting because he also taught the Pre-calculus class. (to be clear on his background: he was not a 'career mathematician' turned teacher but a local guy who wanted to be a math teacher)

Even in 1998 when I took high school calculus, there was a mindset of 'why do we need to know this when I can just use a computer?' The only CAS-like tool we had were TI-92s (which we did some projects on). Calculators were not required for most work in the course, and we focused on entirely manual approaches for the day-to-day work. I think the mind set of 'let me use the computer' is prevalent in college today too (I'm back at University for Physics now). Students look at some of the relatively 'complex' problems and jump to a computer. Now, there are problems with this as well since even with a computer some students still don't know how to answer the problem. This becomes a balancing act of: teach students the tools to solve complex problem at the same time as insisting they learn to solve the simple systems by hand.

Students need to be given motivation why they can't just jump to a computer. Computer/internet replacements are everywhere: typing has replaced cursive for papers, Wikipedia has replaced Encyclopedia Britannica, and Spell-check has replaced dictionaries. Should a computer replace learning fundamental math concepts? While I think we mostly agree that computers shouldn't be a replacement, it's an educator's job (IMO) to motivate students to enjoy and see the importance of doing math manually. (how to do this specifically, is the trick - maybe show that sometimes it can be just as time consuming and error prone to use a CAS as to just solve the problem by hand?)
Just musing, but has anywhere ever tried teaching Intro Physics and Calculus in the same class? It might involve rearranging topics, and obviously would make the whole sequence longer, but relating the two specifically and directly (for most pure scientists and most engineers at least) might be beneficial?

 

[Sankaku]

No clue, sorry.

If was a joke :-)

Even at our higher levels of competency, we are still subject to clashes of terminology similar to the ones that high-school students struggle with. Algebraic Geometry is a notoriously difficult branch of pure mathematics that students generally only meet in graduate school, if at all.

 

[Solid Snake]

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.

Well said. Not every child is going to want to grow up to be an engineer or a scientist or a mathematician, so worrying about calculus for high school students shouldn't be an issue. The students who want to be engineers and the such will learn calculus either in college or on their own. In other words they'll be fine.

BUT every child should know arithmetic, numbers (fractions!), some geometry, and math logic (something not taught in secondary schools). The reason these things should be more of a concern for the entire student body is because it is more likely that students will use these skills in their everyday lives no matter where they work or where life takes them (whether they realize it or not). The overwhelming majority of students in secondary schools today won't ever use the fact that we can measure the rate of change in a continuous function, or that we can find the area under curves, or that we can find upper bounds or lower bound, etc.

 

[A. Bahat]

Even at our higher levels of competency, we are still subject to clashes of terminology similar to the ones that high-school students struggle with. Algebraic Geometry is a notoriously difficult branch of pure mathematics that students generally only meet in graduate school, if at all.

When my classmates ask me what math class I'm taking, I tell them: algebra (sometimes I say abstract algebra, but it doesn't seem to make much difference). They usually think I'm joking, and ask, "Aren't you supposed to be taking, like, Calculus 10 or something?"

This brings to mind another important point: lots of people, at least those who haven't taken upper-division mathematics courses, get the impression that mathematics is just an endless progression of calculus courses, in which one just does increasingly complicated integrals...

Somehow we need to convey to our calculus students that there is more math out there, not to mention dispel the pervasive belief that math is synonymous with symbolic manipulation. Calculus is a very beautiful and deep subject (not to mention useful), and the way it is often taught does not do it justice. I believe this is in part due to the lack of preparation among students; you can't understand calculus well without understanding the idea of a function well. You must have a strong command of both basic algebra and Euclidean geometry, and all too often students are lacking in both. Nevertheless, high schools "have" to offer calculus so they can show that their students are being "challenged" and are "ready to do college-level work," when in fact:
(i) Their calculus class is anything but college-level. This has to do with both the teachers, students, and the AP system. (Don't even get me started about AP and the College Board. I will rant on and on, more so than I have already.)
(ii) There is no point, absolutely no point, in making high school students take calculus when they are not ready for it. Their time would be spent much more productively if they had stronger courses in algebra and geometry. Even other subjects like basic number theory, or probability and combinatorics, or an introduction to logic, might be more appropriate than calculus, because often students don't know how to reason logically i.e. prove things yet. This is a much more useful skill to acquire than knowing how to evaluate some tedious integral, especially when the student doesn't know what that integral means or why they should bother evaluating it, except that it counts for their grade.

Now, there are certainly high schools where it is a good idea to offer calculus. But it is silly to think that students' mathematical training is improved just by virtue of offering a "more advanced" course i.e. calculus. Who's to say calculus is "more advanced" than linear algebra? (Besides, linear algebra is, if you think about it, almost a prerequisite for really understanding differential calculus deeply; after all, derivatives are how we approximate nonlinear functions with linear ones.) What is the use of learning "more advanced" subjects shallowly if you don't know anything with any reasonable depth? What purpose is there in being able to recite the product rule if you don't know what a function is?

That's all I've got for now. Thoughts?

 

[lurflurf]

(ii) There is no point, absolutely no point, in making high school students take calculus when they are not ready for it.

A bigger problem is making high school students take elementary algebra when they are not ready for it.

Algebraic Geometry is a notoriously difficult branch of pure mathematics that students generally only meet in graduate school, if at all.

That is not an intrinsic property. Algebraic Geometry could be taught at every level. At the graduate level Algebraic Geometry seems unfamilar compared to calculus because the student has taken calculus 3-5 times and Algebraic Geometry 0.

 

[A. Bahat]

lurflurf makes a good point about algebraic geometry. Learning about, say, schemes and cohomology without any prior experience in the subject would be akin to learning about integration for the first time via abstract measure spaces. (This might seem like an exaggeration, but keep in mind that people learning schemes and cohomology have a lot more mathematical maturity than the average calculus student...so to be fair, let's assume that our hypothetical integration-learning student already had the mathematical maturity of a grad student, but somehow had never learned calculus.) Probably possible, but certainly not a desirable state of affairs.

 

[Sankaku]

That is not an intrinsic property. Algebraic Geometry could be taught at every level.

I certainly agree that any topic in mathematics can be made difficult or easy by a combination of good teaching and good preparation. However, some subjects take so much background that diving into them too early is inefficient. I have been thrown off the cliff a few times and, while I like a challenge, I probably didn't get as much out of the courses as if I had done things in the right order.

Really, this is the same problem as with much of calculus teaching in high-school. Cramming material that we are not prepared for is a bad idea. Is it good to offer calculus? Certainly for the small number of students poised to take advantage of the opportunity. For many, though, it seems like an arms race with admission standards.

I agree with A. Bahat that we could be emphasising different topics at high-school level, like linear algebra and discrete mathematics.

 

[chiro]

I certainly agree that any topic in mathematics can be made difficult or easy by a combination of good teaching and good preparation. However, some subjects take so much background that diving into them too early is inefficient. I have been thrown off the cliff a few times and, while I like a challenge, I probably didn't get as much out of the courses as if I had done things in the right order.

Really, this is the same problem as with much of calculus teaching in high-school. Cramming material that we are not prepared for is a bad idea. Is it good to offer calculus? Certainly for the small number of students poised to take advantage of the opportunity. For many, though, it seems like an arms race with admission standards.

I agree with A. Bahat that we could be emphasising different topics at high-school level, like linear algebra and discrete mathematics.

When I did a math practicum for the year 7 students I taught them what a two-dimensional convex hull was in the context of computational geometry.

I got them to draw the hull given a random spread of points and they all did it perfectly.

The concept was made clear and they all picked it up quickly.

But one thing I noticed is that math is taught horribly in high school and I would not be surprised if many students wanted to learn math but felt intimidated or inferior from prior experiences of being humiliated either publicly or privately (through test and exam scores).

Personally I think a lot of students could pick up university math quickly if it was taught in a certain way, but whether they would want this or need this is something that will be in debate long after I and many others are gone.

 

[Rono]

In my country, some schools are giving lessons of calculus to people who have shown a good level of math and makes Junior and Senior math during Junior year. I did it that way and, at least in my country, calculus shouldn't be taught at high school. If they can't understand basic algebra, it would be way too hard to teach calculus.

 

[Astronuc]

I certainly agree that any topic in mathematics can be made difficult or easy by a combination of good teaching and good preparation. However, some subjects take so much background that diving into them too early is inefficient. I have been thrown off the cliff a few times and, while I like a challenge, I probably didn't get as much out of the courses as if I had done things in the right order.

Really, this is the same problem as with much of calculus teaching in high-school. Cramming material that we are not prepared for is a bad idea. Is it good to offer calculus? Certainly for the small number of students poised to take advantage of the opportunity. For many, though, it seems like an arms race with admission standards.

I agree with A. Bahat that we could be emphasising different topics at high-school level, like linear algebra and discrete mathematics.

I don't believe anyone here has advocated 'diving into' the subject 'too early' or 'cramming material' for which students are not prepared. Clearly one's education is cumulative, and so if we advocate teaching calculus or other advanced math topics in high school, 11th or 12th grade, we must establish the appropriate prerequisite courses in earlier grades, perhaps starting in grade 6 or earlier. Word problems with two variables and two equations is an opporunity for linear algebra with 2x2 matrices.

Of course, the optimal situation requires capable students and teachers.

I found myself frustrated in primary school (in 4th grade my math workbook was confiscated because I went too far ahead) and otherwise held back because I wasn't supposed to able to solve certain problems when I could. So I pretty much had to go find resources by myself.

 

[Sankaku]

I don't believe anyone here has advocated 'diving into' the subject 'too early' or 'cramming material' for which students are not prepared.

Of course not.

However, the culture of admissions expectations may not match anything we would wish for. I am certainly not arguing against offering more challenging math classes in high-school. I am just wary of what that looks like when it is implemented in the real world.

Making a certain level of math performance mandatory at grade-school level ironically makes most kids dislike it more. At the next level up, weighting university admissions toward early performance in high-school calculus makes many kids take it for entirely the wrong reasons. Most people are not like yourself, where you obviously had an intrinsic motivation to learn math at an early age. They need to be inspired, and the heavy stick of compliance does not inspire anyone.

As well as Lockhart's Lament, I recommend reading Underwood Dudley's "What is Mathematics For?":
http://www.ams.org/notices/201005/rtx100500608p.pdf

It takes an extreme view, but one worth thinking about.

 

[mathwonk]

every subject can be taught at any level, but there are certain helpful prereqs. e.g. it is hard to teach much alg geom to someone who knows neither algebra nor geometry.

i might suggest as an early question in algebraic geom, how many times can a line meet the graph of a polynomial function of degree n?

 

[Dowland]

In my country, it's mandatory to learn some calculus in high school if you want to major in science or engineering. We also study calculus based physics parallel with the mathematics which I think enhance the intuition of the subject. All is done in a very intuitive and computational way. Then, in college, it's mandatory to take calculus in a more rigorous manner.

But we also have the problem with prerequisites. First of all, the mathematics professors often make complaints about the lacking algebra skills when students enter college. Further, we virtually never get exposed to any proofs in high school so one enters college without knowing what a proof is. I know that schools in the US tend to have a pretty proof-based euclidean geometry class but here we barely learn any euclidean geometry, and the learning of it consists of applying a bunch of rules/formulas (which we accept by faith) on geometric figures.

I'm currently self-studying euclidean geometry and algebra more in-depth, outside of class to be well prepared for college. It's fun and I think and hope that it will pay off.

 

[Astronuc]

Making a certain level of math performance mandatory at grade-school level ironically makes most kids dislike it more. At the next level up, weighting university admissions toward early performance in high-school calculus makes many kids take it for entirely the wrong reasons. Most people are not like yourself, where you obviously had an intrinsic motivation to learn math at an early age. They need to be inspired, and the heavy stick of compliance does not inspire anyone.

A certain level of proficiency at a given age is desirable, since there is a finite time to master any subject which generally grows in complexity with time. More advanced knowledge (and skills) is built upon a more basic foundation. Education is challenging because there is a spectrum/distribution of capability among the population of students. In the same classroom, one can find students who are beyond grade level, perhaps by years, and those who are struggling to keep up or who have fallen behind. Yet - all are being taught to a common schedule. Ideally, those struggling can be given extra help. Ideally, those who excel are given opportunity to continue to excel.

From where does "weighting university admissions toward early performance in high-school calculus" arise? The university? A university may wish to attract excellent students. Who makes many students (kids) take calculus for the wrong reason? Parents or the school system?

For me, calculus was an option only after I transferred high schools. At the new school, I sat down with a guidance counselor who gave me options, knowing that my interest was math and science. I was able to develop a schedule that included algebra with trigonometry, calculus (with analytical geometry), physics, and two years of chemistry. At the previous high school, I would not have had that opportunity. The two schools were 6 miles apart in the same urban school district, but they represented disparate opportunities.

At the end of 8th grade, I was required to develop a 4 year plan for 9th through 12th grade. I loaded up on math and science with a plan to take Algebra I (9th grade), Geometry (10th grade), Algebra II (11 grade), and Trigonometry/Analytical Geometry (12th Grade) - all at the honors level. In addition, I selected Biology (10th grade), Chemistry (11th grade) and Physics (12th grade) - again at the honors level. And I had to take the mandatory humanities, English, History, and Foreign Langauge (I could have done honors, but I didn't want to). The counselors weren't exactly encouraging, and my peers thought I was nuts. Nevertheless it was accepted. I was successful in achieving my goals in 9th and 10th grade, and even exceeded the math goal because the teacher gave as an intense program in which we did a year's worth of geometry in one semester, so we were then able to do a year's worth of trigonometry during the second semester. Had I stayed at the high school, I would have only contined with more advanced algebra and analytical geometry (pre-calculus). Instead, I was fortunate that my parents decided to move, and I was fortunate that we moved into the neighborhood of a really good high school.

The greater one's education, the greater the potential one has. There is no way to predict in the early grades which student will become a doctor, lawyer, scientist, mathematician, plumber, carpenter, welder, retailer, . . . , so the system attempts to provide a broad base of subjects in order to provide a wide opportunity to go in any direction.

I think the education system should provide a vehicles for those students who excel and those who are struggling - and everyone in between.

The pedagogical challenge is not only what to teach and when, but how to teach a subject in a way that is relevant and inspirational. I think many teachers understand that, and some educational administrators understand that, but it seems it is not universal, and in some cases, I've experienced individuals who seem hostile or obstructive to education.

As well as Lockhart's Lament, I recommend reading Underwood Dudley's "What is Mathematics For?":
http://www.ams.org/notices/201005/rtx100500608p.pdf

It takes an extreme view, but one worth thinking about.

I've read Lockhart's Lament, and Dudley's article. I'm one of those who uses algebra daily and often I use calculus or numerical analysis. Part of the work (computational physics) involves analysis of experimental data and numerical models of others, and then attempting to construct even better models. In reading the literature, one has to understand the mathematical principles in order to know how to apply the data or model to one's work, as well as whether or not the data or model are valid. What I've been finding (more recently) is that there is a certain level of error (and sloppiness) in the reporting of scientific/technical information (peer-reviewed journals are not exempt).

If it is determined that teaching calculus in high school is worthwhile, then it seems necessary to lay out the prerequisite courses and program in order to facilitate the teaching of calculus to students capable of learning calculus such that they are proficient in the understanding and application of calculus. Same goes for advanced mathematics (and science) in general.

 

[mathwonk]

i taught calculus to very bright 8,9, and 10 year olds this summer. I tended to skip over proofs of things that are visually obvious, like the fact that a polynomial graph that goes from below the x-axis to above it must cross it somewhere, and focus on other matters, such as the fact that the graph actually crosses from one side to the other only at a root of odd multiplicity. Rolle's theorem was taken as obvious as well and we used it to give an inductive proof of the rule of signs attributed (wrongly) to Descartes, and due rather to the Abbe' de Gua. We also analyzed different definitions of tangent line, from Euclid to Newton, and used differential calculus to solve max/min problems. then we studied questions of area and derived the area formulas for polynomials, noticing that they are antiderivatives of the height formulas. my notes are on my website at UGA math dept.

http://www.math.uga.edu/~roy/epsilon13.pdf