“I don’t know how I even got up and took a crap in the morning before I knew linear algebra.”

I learned the basics of matrices and stuff in my Algebra 2 class, the year of math that came before trig/pre-calc calc. At the time, I did not see the motivation at ALL for learning them. I think it may have been mentioned that one can use them to classify and obtain solution sets of linear equations at the time, but the whole concept of determinant and all that just seemed so half-baked. It wasn’t until college when I took LA that I learned that basically this was a system concocted to reproduce the intuition we have of R^3, and extend that structure to arbitrary dimensions, different fields, etc.

I think LA is a perfect topic to spend far more time on in high school, since it doesn’t require calculus, and in my opinion, is a lot more inituitive and beautiful. One can

i) teach applications to computer graphics.

ii) go into complex vector spaces and do small toy exmaples (eg low dim spinors) in order to max out the visualization aspects of quantum systems. This one’s good because hopefully, quantum information will soon become a very important concept, and the math behind it is really all LA. The more early intuition we give kids about how quantum information flows, the better chance the bright ones will come up with some creative quantum algorithms that we’re too tunnel-visioned to improvise, ne?

iii) give students (somewhat) arbitary systems of information/objects/etc and challenge them to formulate it as a vector space concept, with a non-trivial degree of meaningfulness.

iv) tons of other shit that I can’t think of right now.

I think LA is critical to teach early, mostly because I think most kids who go through it in college and don’t do physics, don’t get a good intuitive feel for it. I can’t think of a more intuitively visual mathematical concept, and I hate to see someone who doens’t think LA is The Shit to the extent that I do. ]]>

Often, I’ve found that simply putting the problems in a familiar context is all that is needed, even if it’s just for color. Solving an algebra equation when the unknown is just ‘x’ is far less interesting that solving it if the unknown is ‘days off school’ or sometime (that’s more of an elementary school level, of course).

Statistics, probability, and combinatorics is a generally undertaught field that is full of real world applications. Error values on opinion polls and the false positive problem are some of the classics. ]]>

One thing I did like, though, was the idea of using trigonometry to measure heights from shadows. The joke among my friends in my math class in my freshman or sophomore year was that trigonometry was only useful if you were in the US Forest service (because you had to measure trees), or were dealing with large triangular buildings.

Investment, loans, and general financial stuff can be useful in showing some mathematics, but that is generally pretty simple stuff except for things like continually compounding interest (or whatever it is called involving e). ]]>