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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Sign up to join this communityCan 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.
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.
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.
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.
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
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.
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.