Beyond Calculus, an Invitation to Dream Higher for High School

Question

From what I see in the curriculum we use for my children if we stay on track with the current trajectory they'll finish by grade 8 what is usually called Precalculus (USA terminology, includes solutions of algebraic equations, factoring techniques, graphing rational functions, basic matrix arithmetic, solutions of inequalities, basic linear programming, trigonometry, logs, exponentials, basic theory of functions). Hypothetically, this leaves 4 years of high school to learn mathematics beyond this foundation.

Question: which course sequence would you suggest for a high school student who is ready to take calculus in their first year of high school ?

Assume the children are home-taught with a parent who has a graduate education in mathematics for the convenience of your answer if you wish. More to the point, don't worry much about the plausibility of your answer to "real-world" school situations. In particular, set-aside worries about keeping all students on the same track and/or a lack of expertise in the instructors in your ideal school. The goal here is not to create mathematicians, but, to give mathematics a proper scholarly introduction. To borrow Frenkel's terminology, to show them artistry, not just fence painting.

In Why do we teach calculus in high school rather than a different math course? I made a comment which prompted the creation of this question.

Answer 1

You can explore combinatorics, like through Bogart's "Combinatorics through Guided Discovery". Or you could take a peek at Chen's lecture notes, they cover a hefty part of college math curriculum in detail, and are well written.

Answer 2

My current opinion is that the ideal curriculum is one which returns to the material learned in the past with an eye towards establishing the relevance of formal methods in addressing such troubling questions as "What are numbers?" or "What is a function?" or, gasp, "What is proof?"

B.F. Skinner demonstrated that a pigeon is capable of performing ever more complex sequences of behaviors, and that, in this way, a completely "mindless" bird could behave as if it "knew" what it was doing, or perhaps even "understood" what it was doing. But, I would find it hard for anyone to admit into science the statement "That pigeon knows it's playing ping pong like a boss" without some trepidation or clarification.

My point is that it is one thing to rehearse the mathematical methods of the past, but it is something completely different to see, within yourself, why humans perform the mathematical behaviors that they do.

An alternate way of saying this is that what we must strive to develop is all that would be lost to the world if everyone over 50 suddenly disappeared.

Answer 3

Where I live, many fences are works of art. If someone around here bothers to paint a fence, it will be a pretty fence.

Here are some practical skills for scientists and engineers:

Tolerances and estimation

• Significant figures, and the difference between accuracy and precision.
• How to estimate the number of bits in a tolerance.
• Linear interpolation along non-linear functions (such as Production Possibility Frontiers in Economics)
• How to estimate the roots of functions
• Error analysis, and how to choose algorithms that mitigate errors
• Big-O notation for the amount of time or space required to perform an algorithm

Calculus through the Reynolds Transport Theorem

• Integration and differentiation of polynomials in one-dimension
• Integration and differentiation of exponentials and sinusoids in one-dimension
• Area integrals
• Volume integrals
• Flux (Surface integrals, with time derivatives)
• The Reynolds' Transport Theorem

Convolutions

• Macaulay's notation for integrals and derivatives of singlets and doublets (e.g., Dirac delta functions, step functions, and ramp functions)
• A few especially useful sequences and series: Taylor series, polynomial decay, exponential decay
• Laplace transforms (and inverse Laplace transforms) or Fourier transforms (and inverse Fourier transforms)

Statistics and Data Analysis

• How to Lie with Statistics
• Basic statistics (R², variance, slope, meaning of standard deviation, normal distribution, binomial distribution, central limit theorem, Poisson distribution, Weibull distribution, cumulative density function, probability density function, control charts as used in statistical process control, how to use a "significance test" in a misleading way)
• Relational Algebra or Relational Calculus or SQL coding, through being able to write SQL code to perform a left anti semi join.

Answer 4

I would keep it down the middle (accelerated, not enriched). Think that gives you more option value.

Frosh: Calc 1 and 2 (standard version) Soph: Calc 3 (multivariable), ODEs

ODEs is the first place where you may have issues as almost any textbook covers more material than is comfortable to do in a semester (unless you treat it by doing double hours). Whereas calc, you can just work through a text. My recc is to either double up hours or take a little longer. But run all the way through Tenenbaum. Most colleges handle this by just selecting topics. But it is different from calc where you can work straight through most texts. Kreyszig has a very clean, short diffyqs approach, but it is inside a fat book of other stuff.

Junior: PDEs, Complex. (You will have some similar issues on PDEs but not as bad as ODEs since it is mostly about the standard three problems. I would lean towards using the easier Farlow text that Dover has for PDEs. Can come back for harder later. For complex, there are some good simple applied semester long texts.

Senior: Lot of choices, really. You are kind of done with the standard engineer/science math route. I would say linear algebra. (Actually now that I think of it, do it earlier. Little break from calculus track. Choose an easy text.) maybe a math methods course to hit the other semester. There are some texts that get recommended a lot for self studiers. But you could pick something else too. (simple probes and shafts course for instance.)