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Should calculus be taught in high school?

  • #201
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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.
 
  • #202
Astronuc
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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 Language (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.
 
  • #203
mathwonk
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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 [Broken]
 
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