Should calculus be taught in high school?

[Kindayr]

You're not alone in the stupidity of the Ontario system. My house mate is from BC and took his version of grade 12 Calculus where they covered some integration, and more differentiation than we ever did. However, his credit didn't count, so he had to take Calculus 1100A here at UWO, which is Ontario's grade 12 course +first year calculus all in one package.

Just doesn't make sense lol

But i still hold the fact that Calculus can and should be taught much earlier in a student's career. I do know for sure that any of my children will be learning very elementary logic (I'll be proud if they can understand implications) at a young age, and will be taught mathematics at a healthy rate to coincide with their mental progression. I won't be doing this to force them into math-oriented studies, but to just have access to the critical thinking and creativity that comes along with problem solving in mathematics and use its where ever they wish too.

But that'd be in a perfect world.

 

[kramer733]

I'm not going to lie but from grade 1-6 was completely useless and we only learned the 4 orders of operation.

Then in grade 7, we were finally introduced to the idea of integers.

grade 8, it was about accepting pythagorean's theorem and order of operations.

grade 9, we were introduced to cartesian plane.

grade 10, we were heavily doing up quadratic equations

grade 11, we were introduced to more functions.

grade 12, was a review of grade 11 and we got into a class called (calculus and vectors).

Basically we can learn all this stuff in 4 years. But the school program prolongs it. Along the way, we're introduced to geometry and accepting the truth of what geometry is without proof.

We can easily condense the material in 4 years too. Grade 1-6 is actually useless. We can definitely learn math at that age. We aren't stupid. In high school, we should be doing 2+ classes of mathematics with emphasis on proofs.

It's completely garbage.

I'm definitely not talking trash to the teachers though. It's not their fault. It's the school boards' fault. I'd like to one day change that though. I believe we can actually cover 3rd year university math classes when high school is done with.

Most people are not stupid at all. It's just that they are lazy.

 

[mathwonk]

this may not be comparable to a 1960's exam but this was an actual exam i gave at the university of georgia in 2004 in an honors, but not elite honors, calculus class.2310H final 2004 Exam, Smith,
I.(i) If f is a function defined on [a,b] and a = x0 ≤ x1 ≤ ...≤xn = b is a subdivision of [a,b], describe what an “Riemann sum” for f means, for this subdivision.

(ii) Define what it means for f to be “integrable” on [a,b] in terms of Riemann sums.

(iii) State two essentially different properties, each of which guarantees f is integrable on [a,b].

(iv) Give an example of a function defined, but not integrable, on [0,1].

II. (i) If f is defined by f(x) = 1/2 for 0 ≤ x < 1/2; f(x) = 1/4 for
1/2 ≤ x < 3/4; f(x) = 1/8 for 3/4 ≤ x < 7/8; ...; f(x) = 1/2n for
(2n-1 - 1)/2n-1 ≤ x < (2n -1)/2n; and f(1) = 0, explain why f is integrable on [0,1], and compute the integral. (The FTC is of no use.)

(ii) If f is defined on [0,1] by 1/sqrt(1+x4), explain why f is integrable, and estimate the integral from above and below. (The FTC is of no use.)

III. Compute the area between the x-axis and the graph of y = sin2(x), over the interval [0,π]. (At last the FTC is of use.)

IV. Compute the arclength of the curve y = (x2/4) - (ln(x)/2), over the interval [1,e2].

V. A solid has as base the ellipse (x2/25) + (y2/16) = 1. If every plane section perpendicular to the x-axis is an isosceles right triangle with one leg in the base, find the volume of the solid.

VI. Find the area of the surface generated by revolving the portion of the curve x2/3 + y2/3 = 1 lying in the first quadrant, around the y axis.

VII. Compute the following antiderivatives:
(i) =

(ii) =

(iii) =

(iv)

VIII. Determine whether the following series converge, and if possible, say explicitly what is the limit. Explain your conclusions.
(i)

(ii) 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 ±.....
(iii)

(iv)

IX. Compute the volume generated by revolving the plane region bounded by the x-axis and the curve y = 4 - x2, around the line x = 5.

X. Any function f: R+-->R which is
(i) continuous, (ii) not always zero, and (iii) satisfies f(ax) = f(a) + f(x) for all a, x >0 is a “log” function. Using this, prove that f(x) = is a log function, using appropriate theorems. [Hint: You will need to show f’ exists and then compare the derivatives of f(x) and f(ax).]XI. We know the only function f such that (i) f is differentiable, (ii) f(0) = 1, and (ii) f’ = f, is ex. Assuming an everywhere convergent power series is differentiable term by term, use the previous fact to prove that converges to ex. [Hint: First prove it converges everywhere.]

XII. Use the fact that y = tan(x) satisfies the diferential equation y’ = 1 + y^2, to find at least the first four terms of the power series for tan(x). Compare the coefficients to what Taylor’s formula a(n) = f^(n)(0)/n! gives you.

XIII.
a) If f is a continuous function on the reals, with f(1) = c > 0, what else must be checked to conclude that f(x) = c^x for all x?

b) If a,b are positive numbers, use the method above to prove that the function f(x) = (a^x)(b^x), equals (ab)^x.

 

[mathwonk]

it says i have attached a pdf file of this exam but i don't see it.?

 

[eumyang]

I'm definitely not talking trash to the teachers though. It's not their fault. It's the school boards' fault. I'd like to one day change that though. I believe we can actually cover 3rd year university math classes when high school is done with.

Thank god you're not in charge, then. No one would be able to graduate from high school.

On one hand, I agree with you that some of the material can be condensed, but on the other hand, some kids end up taking Algebra I too early because they can't handle the level of abstraction required.

I've also heard that some students are entering Calculus not prepared because of their weak Algebra skills. Is it because those students received a condensed treatment of their Algebra courses?

And yet... I've heard that in Asian countries like Japan and Korea, it's the norm to reach Calculus before finishing high school. My Korean is not that great, but from what I read, in Korea, a student in the liberal arts track can take an introduction to Calculus course. (It's not clear whether they HAVE to take this course, or it is an elective.)

 

[xdrgnh]

A lot of the posts on this thread disgust me, to those that suggest Calculus shouldn't be taught in high school are freaking pure math eletists. I have great respect for mathematicians and a lot of them are great people and good to know. However some them are math nazi's, if it doesn't involve rigor and proofs they will discredit it. Most people don't need pure calculus with the proofs and rigor they just need to know it's conceptual meaning and learn how to do problems that will arise in practical applications. Students who want to be physicist or engineering really need to the college credit for calculus so they can learn mechanics and E@m probably. Most good colleges require students to use multivariate calculus in E@M and it's good to have multi for mechanics to understand the line integral of work. So to you math elitists don't try to force everyone to learn pure and rigorous mathematics because it's unnecessary and in some cases harmful to those who want to just know how to apply to the real world.

 

[kramer733]

A lot of the posts on this thread disgust me, to those that suggest Calculus shouldn't be taught in high school are freaking pure math eletists. I have great respect for mathematicians and a lot of them are great people and good to know. However some them are math nazi's, if it doesn't involve rigor and proofs they will discredit it. Most people don't need pure calculus with the proofs and rigor they just need to know it's conceptual meaning and learn how to do problems that will arise in practical applications. Students who want to be physicist or engineering really need to the college credit for calculus so they can learn mechanics and E@m probably. Most good colleges require students to use multivariate calculus in E@M and it's good to have multi for mechanics to understand the line integral of work. So to you math elitists don't try to force everyone to learn pure and rigorous mathematics because it's unnecessary and in some cases harmful to those who want to just know how to apply to the real world.

How can understanding where something comes from be detrimental to somebody? If anything, they'll have a better understanding of calculus. Knowing where something comes from is useful. It's not just memorization of the formula but proofs are formal pieces of writing that make people understand where something comes from.

 

[xdrgnh]

How can understanding where something comes from be detrimental to somebody? If anything, they'll have a better understanding of calculus. Knowing where something comes from is useful. It's not just memorization of the formula but proofs are formal pieces of writing that make people understand where something comes from.

There's nothing wrong with understanding and talking about the origins of it. I'm referring to putting more emphasis on proofs then actual problem solving. Try to teach a 1st grader why 1+1=2, that would be detrimental to them learning addition. Try to teach limits using delta's and that would confuse someone in high school who doesn't intend to go into pure math and would intimidate him. I got nothing wrong with proofs being used in class to help to understand the concepts but they shouldn't be the center piece. Some people here say that the problem with teaching calculus in high school is that it's not rigorous enough, but most students don't need a rigorous class at the high school or even 1st year college level.

 

[mathwonk]

i may be a math nazi, as my knee jerk reaction to this question is always "no".

come to think of it though, it is based on a lifetime of experience having to deal with those students who think they learned calc in high school but didn't because the people they learned it from did not understand anything.

i am still probably a math nazi if that means i think i understand it and you don't.JUST KIDDING!

heil geometry!~ stop that! hey peter sellers, cut it out.

 

[WannabeNewton]

Try to teach limits using delta's and that would confuse someone in high school who doesn't intend to go into pure math and would intimidate him. I got nothing wrong with proofs being used in class to help to understand the concepts but they shouldn't be the center piece. Some people here say that the problem with teaching calculus in high school is that it's not rigorous enough, but most students don't need a rigorous class at the high school or even 1st year college level.

I finished Calc BC before the start of this summer and I can say from experience that I had to teach myself at home using Spivak because the Calc BC curriculum lacked in rigor left and right. I walked out satisfied only with what Spivak's text had given me in terms of preciseness; Calc BC just handed out assumptions to students and it was highly unsatisfying. I intend to go into physics but even I, an average high school kid, demand the same mathematical rigor that a pure maths individual would favor because intuition and thoroughness is much, much more important than knowing how to do calculations. You seem to assume that all high school students want the level of rigor that you keep saying should be maintained.

 

[xdrgnh]

I finished Calc BC before the start of this summer and I can say from experience that I had to teach myself at home using Spivak because the Calc BC curriculum lacked in rigor left and right. I walked out satisfied only with what Spivak's text had given me in terms of preciseness; Calc BC just handed out assumptions to students and it was highly unsatisfying. I intend to go into physics but even I, an average high school kid, demand the same mathematical rigor that a pure maths individual would favor because intuition and thoroughness is much, much more important than knowing how to do calculations. You seem to assume that all high school students want the level of rigor that you keep saying should be maintained.

If you like pure math a lot then you should consider minoring in math, I'm all about giving choices just like in college. In college you have the honor sequence which is more theory and then you have the standard which is more application. I went to Brooklyn Technical High school a elite math and science school and the kids who were in the math major loved all of that rigor while the kids in applied science major didn't. I'm just saying the emphasis of standard calculus class should be problem solving rather then theory. My calculus BC had proofs so we all can understand why the power rule works and ect, but it wasn't the main focus and it shouldn't be.

 

[mathwonk]

no serious argument should ever include the phrase "I got nothing wrong with..."

 

[xdrgnh]

no serious argument should ever include the phrase "I got nothing wrong with..."

No serious debater would resort to mud slinging like that.

 

[WannabeNewton]

I went to Brooklyn Technical High school a elite math and science school and the kids who were in the math major loved all of that rigor while the kids in applied science major didn't.

Wow that is freaky, I go to Bronx High School of Science at the moment o.0; we're pretty much neighbors. But I still think problem solving is not as important as the more rigorous conent. To give an example from the general relativity texts I learned from: both Carroll's "Spacetime and Geometry" and Wald's "General Relativity" were rigorous in differential geometry for a physics textbook and it made books like Schutz's "A First Course in General Relativity" much, much easier to work through and Schutz's book was more concerned with problem solving.

 

[xdrgnh]

Well that's GR a upper level physics class that uses abstract math like Differential geometry and tensors. We are talking about calc 1 and 2 which are the foundations of more rigorous math. If the emphasis is proofs and not applied problem solving then they will have a unnecessary harder class. Also again most science students aren't interested in proofs and shoving it down there throats is bad for everyone. If they are interested that is why they have the honors math sequence. Btw I live 5 minutes from Bronx sci but I choose to go to tech because I like being in Brooklyn and Manhattan more then Da Bronx.

 

[micromass]

Well that's GR a upper level physics class that uses abstract math like Differential geometry and tensors. We are talking about calc 1 and 2 which are the foundations of more rigorous math. If the emphasis is proofs and not applied problem solving then they will have a unnecessary harder class.

So we shouldn't challenge students??
And presenting reasons for formula's makes the class harder than just letting them memorize the formula's?

Also again most science students aren't interested in proofs and shoving it down there throats is bad for everyone.

Same analogy: most elementary school students aren't interested in learning. So shoving it down their throats is bad for everyone.

 

[xdrgnh]

So we shouldn't challenge students??
And presenting reasons for formula's makes the class harder than just letting them memorize the formula's?



Same analogy: most elementary school students aren't interested in learning. So shoving it down their throats is bad for everyone.

I'm for challenging students and making the overall math and science curriculum harder but math classes for math students and math classes for science and engineering students have to be different. No one size fits all, especially when we are talking about 1st year classes.

 

[micromass]

I'm for challenging students and making the overall math and science curriculum harder but math classes for math students and math classes for science and engineering students have to be different. No one size fits all, especially when we are talking about 1st year classes.

You keep on making statement like these. But could you actually present some evidence that your statement is valid.
In my country, engineering students and math students take the same rigorous math course. And both sides benifit from it. So I say that it IS possible. Now, can you present some evidence why such a thing is not possible?

 

[mathwonk]

no serious debater would have a picachu as an icon.

 

[BloodyFrozen]

no serious debater would have a picachu as an icon.

But anyways, I think students should be challenged more in high school and have atleast Calculus.

 

[n1person]

I am sure this view has been said before (somewhere in the 11 pages i didn't read), but the fact of the matter is calculus is extremely useful just as a plug and chug type of tool in most of science (biology, basic chemistry, basic physics, lots of engineering), and it is very important students interested in these fields learn it, even if it is just a cookbook sort of way.

And for those going into physics or math or some other similar field it is fine to learn it once the "easy" way and again in more rigor. You will have more experience and intuition the second time around.

I completely disagree with the statement that everyone should learn calculus. For the vast majority of people I think really Algebra (plus the most basic trigonometry) is enough, and I think the fact that we are shoving precalculus down high school students throats is very misguided.

 

[mathwonk]

i will modify my statement and say that i applaud anyone teaching anything that he/she actually understands, to anyone at any age who is prepared for it. I am afraid this may not include many high school AP calculus courses I am aware of in the US, although it does include some of course.

I have just spent the past 2 weeks teaching Euclidean geometry and the ideas of Archimedes to extremely gifted 8-10 year olds at a special camp for them. Nothing was crammed down their throats as these kids loved the subject and were excited to come to this type of camp.

During the process of discussing and analyzing these topics from Euclid I came to believe this is the best possible preparation for calculus.

Euclid discusses area and volume using finite decompositions as far as possible, and then transitions to using limits. Then Archimedes refines Euclid's technique of limits and obtains "Cavalieri's" principle for volumes.

(Euclid's theory of similarity also prepares a student for a careful analysis of the real line and rational approximations.)

Many basic facts about volumes and areas are got out beautifully by Euclid and Archimedes such as the volume formula for a cone, and a sphere, that still challenge many calculus students who think they have learned the subject.

E.g. Archimedes apparently knew not only that the volume of a sphere is 2/3 that of a circumscribing cylinder, but also that the same holds for the surface area, and even that the same facts hold for a bicylinder (intersection of two perpendicular cylinders of same radius) with respect to an inscribing cube.

I challenge any high school AP calc student , or any college calculus student, to prove all this using what he has learned about volume and surface area in his calculus class. These volume problems are among the hardest problems we assign calculus students, and I am not aware of anyone assigning the surface area of a bicylinder in college calculus.

(You AP calc graduates might try it and see. Maybe you'll get it and you can brag to your teacher.) The same ideas of Archimedes, such as the location of the center of gravity of a 3 dimensional cone, allow one to easily calculate the volume of a 4 dimensional ball, without calculus! How many of your AP classes do that (even with calculus)?

My advice to any good high school student is to study Euclid's Elements, then Euler's Elements of Algebra, and then Euler's Analysis of the infinities, as outstanding precalculus preparation. A little Archimedes is also useful but harder to read.

After this one could appreciate a good calculus book.

 

[mathwonk]

Even if the goal is to meet the needs of those scientists who need to use calculus to calculate things, this is not best served by traditional AP courses in my opinion. For those students much less theory should be presented, and questions as to the existence of the various limits which arise should be taken for granted.

The most important ideas should be emphasized with their geometrical meaning. Powerful and useful tools such as Pappus' theorems should always be presented, along with simplifying ideas like centers of mass. Both of these are often omitted even in college calculus classes.

Computation of tricky limits and tricky integrals has virtually no importance in my opinion.

 

[mathwonk]

I have just perused several AP calc syllabi available online and found as expected lengthy lists of tedious topics that make the subject seem hopelessly complicated and impenetrable.

The most important applications are treated briefly and without acknowledgment of the fact that hardly any of the painfully long theory is needed to understand them completely.

Important topics like Cavalieri's principle, the method of cylindrical shells, Pappus' theorems, are not visibly mentioned at all, although presumably Cavalieri's principle is hidden under the heading of "volumes by method of discs and washers".

Nowhere is it made clear for instance that Cavalieri's principle is already obvious just from the definition of volume as an integral, i.e. well before the fundamental theorem of calculus.

I have just read a sample AB AP calc test and found almost none of the questions to have any real interest. The only one that seemed useful to understand was the last question of part 1 on recognizing a slope field form a given o.d.e. most of the rest was just jumping through hoops.

 

[mathwonk]

heres an example of the sort of silliness i am talking about. I just looked up a calculus book by a professor at a major university, in which the problem of showing the surface area of a torus (result of revolving a circle of radius r, centered at (c,0) with c>r, around the y axis) equals 4π^2rc, is posed and a hint given about what complicated integral to use.

2,000 years before the invention of calculus, Pappus knew this problem has the trivial solution length of circle times distance traveled by center of mass of circle = (2πr)(2πc) = 4π^2rc.

Thus even an A student in this class struggles hard for a semester and comes out knowing less than someone knowledgeable from 2000 years ago who has never heard of calculus. The idea of applied math courses is to give people useful tools that make their problems easier, not harder.

Even books found online by famous professors at some of the best schools in the world, present ideas like center of mass and then omit to explain how this is useful in computing work. To give a calculus student a problem of computing work done pumping water from a conical tank and not mention that the center of mass is 1/4 the way up from the base and that this renders the problem trivial, is pretty useless I think.

By the way here (in an attachment) is a discussion of calculating the volume of a 4 dimensional sphere that uses only things Archimedes knew.

 

[xdrgnh]

Mathwonk everything you said proves my point. Unless someone wants to go into math they will have no need for 99% of the stuff you just stated. For 1st year students who want to go into engineering or science they have no need for this stuff. People like you want to force people who don't want this kind of math or need it into taking. It gets rid of choices and is not beneficial. Under your math plan I wouldn't be able to take multivariable calculus freshmen year in college. This would prevent me from taking the proper level of physics.

 

[mathwonk]

xd, it seems to me that you have neither read or at least not understood anything i have written in such detail for you. good luck to you.

by the way if you want to be taken seriously in your objection to "mudslinging" you might refrain from using the term "nazi's" (sic) to refer to your adversaries.

 

[mathwonk]

For the curious, in order to understand what 4 dimensional volume has to do with physics think about the idea of work, as the integral of the product of force acting in a given direction times mass. If density is assumed constant this amounts to multiplying distance times volume, a one dimensional concept times a three dimensional one, or a 4 dimensional quantity. This is why measuring work is essentially the same as measuring 4 dimensional volume, and this is the explanation of why Archimedes' arguments, which were based on physics, yield a nice computation of the volume of the 4 dimensional ball.

 

[Astronuc]

Has this been exhausted?

I think this question is relevant to the teaching of physics in high school. Advanced math and physics become rather abstract for most people.

How and/or when should math and physics be taught in primary school years?

 

[Andy Resnick]

Has this been exhausted?

<snip>

How and/or when should math and physics be taught in primary school years?

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'."

 

[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?