I experienced exactly the same feeling you described concerning the quantum numbers. In high school and CEGEP (two years of college in Quebec before a 3-year university program), I memorized the rules for the quantum numbers etc. I was told that the magical numbers came from a magical formula, a formula which was represented by an intimidating greek letter psi, but that was it. Then, when I took a class in quantum mechanics at McGill, and a very good teacher decrypted the equations for me, I was very annoyed that this hadn’t been explained to me in earlier courses. It would have been an immense motivation for me to work hard on my math so that I could more fully appreciate what was going on.

One of the problems, as you point out, is that students don’t have calculus before they take chemistry and physics. I think most high school students these days would have a heart attack if I showed them what the Schrodinger equation looks like. Or maybe I’m making the mistake of underselling my future students. It isn’t their brains I doubt – but their stomachs. Right now, I tutor an extremely bright student in chemistry, and every time I try to show him the origins of the equations he’s using, I get an “ow, by brain hurts, stop!” It’s very frustrating to try to overcome that kind of resistance to intellectual challenge.

That being said, your approach to building formulas from fundamental principles and experiment would be an extremely powerful pedagogical approach. It teaches the scientific method by demonstrating it, rather than by beginning the course with a lecture on what the scientific method is (what a meaningless way to explain what scientists do!). It’s too bad that teachers these days aren’t able to teach this way because they have to drill their students to plug and chug numbers and grapple with those trivial unit conversions.

-K

Say in physics, where you learn a formula for the period of a pendulum. To actually derive this rigorously, you need to understand something about second-order differential equations; although it is not that difficult, most high school students won’t have seen this material. So instead, the “problems” we solve on this subject in high school just involve plugging in different numbers into this mysterious formula – simple calculations that don’t convey anything. They will give you two out of the three variables and ask you to solve for the third, ad nauseum. Maybe throw in some unit conversions to spice it up, which is just stupid. Not that bad, until you have to memorize a whole bunch of other formulas, too. Which is not bad either, but you are not actually learning anything about physics or how it is done.

There are two ways to fix this. One is to actually start teaching the necessary math earlier. My high school years were wasted away in subjects like “Algebra II” and “Pre-calculus.” I don’t need to be forced to see examples of every single combination of variables and fractions and etc. that might appear on a state-mandated exam to know how to do algebra. Nor do I need to memorize a bunch of trigonometric addition formulas, if you actually teach me how to derive them myself from complex numbers (which is not that complex to do at all, and leads to a whole bunch of other fascinating topics). So to solve this problem, we need to cut out all this extraneous material that becomes transparent later anyways. Starting earlier is definitely possible, too; people always think things are harder than they actually are, which is a crippling attitude, and one that I saw often in college.

The other is to actually teach those useful general principles. For example, you can find the period of the pendulum easily through dimensional analysis. Just ask, what could the period depend on? You could demonstrate, by simple experiments, that it doesn’t depend on the mass of the pendulum, just its length and, by intuition or thought experiments, the acceleration due to gravity. By finding the combination of these two quantities that gives a time, you’ll find the right period up to a numerical factor, which you can then estimate by experiment (plus you can check your functional dependence on length, at least, is correct). This may seem trivial, but it encapsulates how physics is actually done and teaches a critical and extremely useful tool. So instead of giving students a formula and asking them to calculate with it, why not ask them how they would find that formula in the first place? Once they have the basic intuition needed, then you can motivate the math behind the results more easily. So teach the general principles behind physics, which are always more interesting and beautiful than just ridiculous algebra calculations.

One thing that bugged me in chemistry was when I found out where the rules for the quantum numbers for atomic orbitals actually came from. I know that they don’t even explain this in the required chemistry course at my undergrad, and they certainly don’t explain it in high school AP chemistry, although you need to know the rules for the AP test. After I learned the quantum mechanics behind it years later in college, I wondered, why didn’t anybody tell me any of this earlier? It would have made much more sense to say, there is this equation that nature obeys, which only has these solutions, and numbers appear in this certain way in these solutions, and that is why we have quantum numbers. Maybe I wouldn’t understand the origin of the equation just yet, and you might argue that this would just be more formulas and funny rules. But in this case, the rules have some content; I could definitely plug in the solutions and verify that they work with just knowledge of calculus, and maybe this would interest me enough that I would go find out the rest myself. I would find more and more rules, each building on the last, until I found a set that no one has explained in terms of other rules, and to do so myself I’d have to do a bit of thinking. This is the basis of research. Instead, we are just given the very first level of these funny rules, with no hint of the structure and logic behind them or why they matter, and told to memorize them for the AP test or the final.

Feynman once said, “Know how to solve every problem that has been solved.” You can interpret this quote in two ways; unfortunately, the way often chosen by educators today is the more tedious and less enlightening of the two.

i am of the opinion, as i believe you are, that the content of the high school science curriculum is in need of reform. your comment about the awful “inclined planes” of high school physics rang very true for me (i hate inclined planes). in chemistry, i feel that there is too much physical chemistry in the AP course and not enough emphasis on biochemistry and organic chemistry (of course perhaps my bias comes from my own affinity for the biomedical arena). in the AP biology course, i feel these is too much breadth and not enough depth.

i am wondering if you could comment here on what kinds of changes you would suggest for the high school physics curriculum. what sort of topics would you like to see covered? and would you agree with me that topics like chemistry, phsyics and (real) mathematics should be introduced much earlier in a child’s education? it is my opinion that students think these topics are hard in part because they are avoided until grade 8 or 9. who not start with the basics in middle school?

-kristen