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Can these be taught at a high school level?

  • multivariable calculus

  • linear algebra

  • abstract algebra

  • real analysis

  • complex analysis

  • (non-measure theoretic) probability theory

Maybe even other more math courses.

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    $\begingroup$ Yes. In fact, these should be required for graduation :) $\endgroup$ – James S. Cook Aug 29 '18 at 0:59
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    $\begingroup$ What's the situation and goal here? Do you actually have a class's worth of students who are interested? A single student who wants a tutor? Idle speculation? $\endgroup$ – Adam Aug 29 '18 at 1:08
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    $\begingroup$ Virtually anything can be taught at school level if you simplify the material enough. What constitutes knowing a particular subject is a more important question to ask. Also, note that teaching a basic version could negatively impact students if they take a fuller course later, as students have been shown to do worse on material they think they already know. $\endgroup$ – Jessica B Aug 29 '18 at 5:48
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    $\begingroup$ In high school I took AP Calculus AB/BC, which covered single-variable calculus and the like, and a higher-level IB course that touched on some early aspects of linear algebra. Those were equivalent to first-year undergraduate math courses at the university I did my undergrad. $\endgroup$ – JAB Aug 29 '18 at 6:15
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    $\begingroup$ In India the ncert (national council of educational research and training) has provided a syllabus in which they suggest topics of upto differential equations for highschool, but keep in mind that India is a country where sciences and math are highly stressed as important and thus would have different standards to other countries which may have a more holistic approach to education. $\endgroup$ – Karan Shishoo Aug 29 '18 at 7:35
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In Spain, some linear algebra is taught in high school and is examined on the exams required to enter the university. Topics taught include matrices, determinants, solvability of and solving linear systems of equations, and Euclidean geometry of lines and planes in the plane and three space. Basic probability is also taught, focusing on the binomial and normal distributions. One variable calculus is taught in more detail, and notions such as continuity and differentiability are included.

One might ask how much students really learn. The answer is that most of this material has to be retaught, albeit rapidly, in the first few weeks of the university.

In the US, with the great flexibility of its otherwise generally undemanding school system, advanced students often find a way to learn multivariable calculus, linear algebra, and more. The limiting factor is often the availability of an instructor competent to teach these things.

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  • $\begingroup$ This is very similar to france. The problem is students just learn the techniques without understanding any theory. $\endgroup$ – Paracosmiste Aug 30 '18 at 15:07
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Why not? As long as the students have the appropriate prerequisites and aptitude, go for it. Our local private/boarding school regularly offers one of multivariate, linear algebra, or discrete math to the seniors. A lot of these kids take precalculus freshman year and are looking for more once they complete second semester calculus.

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This paper discusses Linear Algebra as an elective ("Mathematics III") for science high-school students in South Korea.

Choe, Young Han. "Teaching Linear Algebra to High School Students." Research in Mathematical Education 8 (2004). PDF download.

So certainly Linear Algebra can be and has been taught at the high-school level.

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Well, in India some of the topics you mentioned above are already being taught at high school level, infact they are necessary for students who want to become future engineers. You can check out the JEE-MAINS syllabus India(earlier known as IIT-JEE).

Edit: link to syllabus - look at page 23 for required topics for mathematics

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    $\begingroup$ Welcome to matheducators.se. The answer would be improved by adding a link to the syllabus. $\endgroup$ – Tommi Aug 29 '18 at 6:58
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In the UK, some students take A-Level Further Mathematics, which covers bits of linear algebra (determinants, solving and interpreting systems of 3 linear equations, possibly also eigenvalues and eigenvectors and diagonalizing matrices) and some simple first and second order ODEs.

It also gives students the option to study some elementary group theory (the definition of a group, basic examples, and Lagrange's Theorem) and elementary number theory (Euclidean algorithm, modular arithmetic, solving congruence equations).

One syllabus can be found here.

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If you want to teach more advanced courses and have sufficient students, I would advise 3rd semester calc, or ODEs. The reason is these are all "no regrets" moves from the student point of view. Pretty much any hard science, math or engineering will require them.

Abstract algebra is only required for math students which is a tiny tiny percentage of your class population (with undeclared major high school students). Just because they are good or advanced in math do not assume they will become math majors. Common sense and personal experience and statistic should show that most will go into engineering or chemistry or physics or the like.

Linear algebra is not a bad choice either, especially if it is one with emphasis on matrices and calculation (not proofs). For one thing, it is an easier course than 3rd sem calc or ODEs. (Perhaps important for kids who already feel they are essentially doing an elective.) It won't be as central for engineers or physicists (most don't have a semester requirement, just pick up a little in midst of engine math class). But it will be more central and required for comp sci students.

Maybe do the LA because of the easiness and serving the future comp sci students. It won't hurt the engineers either. Plus it will appeal to you more. At least it is algebra even if not abstract.

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  • $\begingroup$ Elementary differential geometry seems very good. $\endgroup$ – Paracosmiste Aug 30 '18 at 15:10
  • $\begingroup$ I took linear algebra, it was easily my favorite course. However, I was sad that when it was over and we didn't get to cover the 'interesting' subjects such as normal forms and orthogonal basis as much. $\endgroup$ – Lenny Aug 30 '18 at 17:41

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