Ranty goodness: math in primary and secondary school

Caveat: I am not an educator. Most of what I'm saying is likely horseshit. I acknowledge this; I am merely posting my own, extremely biased opinion. I am merely talking about the US educational system, as I am not sufficiently well acquainted with any other to critique it. (I posted this also, more or less the same, at shadowmarch. Feel free to peruse the discussion there.)

I have a pet peeve about how primary and secondary education is structured for quantitative subjects (roughly, let's call that science and math; in reality, this is not a precise definition). I feel that primary school spends years wasting time with repetitive and useless information. This is justified by the argument that the "kids aren't ready for more advanced stuff yet". This "holding back" of students in their early years has disastrous reprecussions for curricula throughout primary and secondary school, culminating in students who are deficient in extremely important subjects and skills.

In specific, I feel that from grades 3-7, math and science curricula are hideously wasteful. The only things one learns after second grade (that's roughly 7 year olds, for those of you non-Americans) are the basics of fractions and decimals, some unit conversions, and a few random bits of unconnected knowledge (e.g. the very basics of statistics or probability). Only by seventh (or sometimes sixth) grade do schools again start to teach at something approaching a reasonable pace, with "pre-algebra" and algebra. Secondary schools generally teach algebra, Euclidean geometry, "Algebra 2" (which is essentially a made-up subject), trigonometry, "precalculus" (often a catch-all including Algebra 2 and trig), and very occasionally calculus (though this is most often seen as a "high level" class for smart seniors). Oh, you can toss in some linear algebra, matrix theory, etc. if you really want, too.

On the science end of things, students learn not only dumbed-down but incorrect versions of basic geology, biology and physics (chemistry is generally neglected entirely except for some nifty demos). They do not learn any quantitative ways of actually doing science; most science classes have an absurd idea of experimentation and the scientific method. Students are generally taught to memorize "facts" and vocabulary words about science, and to promptly forget them. What's worse is that math is intentionally left out of the curricula because the students simply don't know even the basics of manipulating equations, single-variable functions, or simple one-dimensional curves. Science teaching is handicapped by the lack of adequate math instruction.

This problem is compounded in secondary school. It would make sense for science to be taught from the basics to more complex systems - as such, physics should go first, followed by chemistry and then a life science. In most schools, this order is reversed for a simple reason: the students do not know enough math to do even non-calculus physics in the beginning of secondary school.

This is unacceptable. Science curricula are destroyed because of inadequate mathematics (and in general, quantitative applications) preparation; math curricula themselves waste at least four years in primary school on the grounds that students would not understand more complex ideas.

Obviously, I wouldn't be posting this if I didn't have a solution to propose.

The basics of algebra can be comprehended by a kindergardener. Hell, the basics of a derivative could be understood by a six year old, though not a rigorous application thereof. As such, I feel that grade school math programs should be completely overhauled. After teaching the basics of arithmetic - certainly by the end of third grade (eg, four basic functions, fraction operations, etc. - pretty much everything that sixth graders know nowadays), teachers can start teaching important stuff. Two dimensional plotting is a very simple concept to grasp. Jumping to the concept of "variables" and "functions" is not much harder. Kids don't need to be taught extremely rigorous algebraic applications in third grade - I think that may be beyond some eight year olds. Yet understanding the concept of a function is much easier. Here's a simple "function" game you can play: the kid gives you a "x" or independent value, and you spit them back a "y" value based on a function you came up with in your head (eg, you pick y = 2*x + 1, they give you a 3, you tell them 7). The kid tries to guess the function you've come up with in as few tries as possible. This could be easily integrated into a computer system, to apply to many students at once. The advantage to this sort of game is that you can ratchet up the difficulty as soon as they get the hang of it (also, it's best to let the kid come up with their own function often to keep them thinking of novel ways to combine numbers). You can introduce new types of operations and functions within the context of a very simple "function" game, without ever needing to utter the word "variable" or "equation".

I know this can work - it was tried on me (not by my school, alas), and I've since tried it on other young kids, with remarkable results.

The point being that without needing to teach extremely rigorous mathematical methods, you can already have the kids working out the basics of algebra in third grade. With a modest ratcheting up of difficulty, I can't see why kids shouldn't be learning and completing algebra by the end of fifth grade. With this foundation, science classes could already start introducting basic equations to describe natural phenomena - sure, the problems would likely be of the "plug and chug" variety at the beginning, but it's much better than waiting until 9th grade to do so. Even more importantly, science classes can emphasize real experiments with the focus on data capture and analysis - granted, with simpler words so you don't scare off a bunch of 10 year olds.

During the rest of primary school, students could easily learn Euclidean geometry, trigonometry, conic sections, and most importantly, basic statistics (for use in evaluating the validity of data). This is not an unrealistic dream, but merely one that assumes students can learn abstract subjects if given concrete ways in which to visualize them.

The benefits of this kind of curriculum are not merely limited to science classes. Students who can rigorously examine a proof or abstract concept can apply this skill to analyzing literature (another pet peeve of mine is how little emphasis is given to true analysis in primary and secondary school), in tracing historical origins, or in learning rhythmic patterns in music.

Anyways, this kind of primary school curriculum sets the stage for a very different sort of secondary school. Students will essentially know nearly everything they need to take calculus. I believe that the first two years of high school should be devoted to an in depth study of basic differential and integral calculus - roughly equivalent to two quarters of college calculus, or the AP Calc BC exam today. With a full two years, any understanding gaps from grade school can be carefully identified and dealt with, concrete and careful mathematical reasoning can be instilled into the students, and they can get a good understanding of what a derivative and integral means, exactly (again, I think in 5-8th grades there should be some non-concrete thought experiments about, essentially, derivatives and integrals, yet without the actual math backing it up).

I firmly believe that basic one-variable calculus is not only nice but essential for every person to know. A knowledge of the concepts put forth in calculus (though not necessarily knowing the exact way to take some hideously nasty integral) is invaluable in whatever field one enters. Calculus is a somewhat unique way of seeing how the world works, and I think that method of thinking can only be an asset in the real world. True, one may not have to take derivatives in a job writing newspaper articles, yet a knowledge of related rates, infinite limits, and infinite sums can be useful for conceptualizing... well, anything.

If students learn this in the first two years of secondary school, everyone will have these skills. Furthermore, science classes in secondary school with have a huge boon. Physics can come first (or even in 8th grade), replete with all the algebra and trig you want. Chemistry can be taught with real math (beyond silly dimensional analysis), and biology can actually address some of the more advanced maths involved (eg population growth, enzyme kinetics, etc.), as calculus will already be completed by then. The possibilities are boundless.

The rest of the secondary school mathematics curriculum could be filled with useful courses from which students could choose - multivariable calculus, advanced probability and statistics, number theory, linear algebra, ODEs, yadda yadda yadda. Plenty of useful things to learn there, depending on a student's interest.

The key is getting rid of the chaff in our current quantitative curricula. Once that is done, we can truly move our education ahead by years, and leave every high school graduate well equipped to take on the world (or college).

Feel free to post responses. Enjoy!