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Fantasy math

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Nothing ever changes...

...from personal experience.

I would like to come up with an outline of what undergraduate mathematics curriculum ought to be.

I have to ignore the reality of college education: prerequisites, "service courses", overlapping degree requirements, accreditation agencies, “assessment” (you don't want to know), transfer students, high school curriculum, SATs, GREs, etc.

Just as in a football fantasy league, the team will never play...

To get started, you are a victim of contemporary mathematics education if you can't make sense of the following statement:

  • The derivative of the composition is the composition of the derivatives.

(Of course, a typical freshmen does't understand compositions to begin with.)

Discrete functions

Consider the fact that nowadays data comes in digital format. This data is simply large tables of numbers. Just two examples:

  • experimental data: thousands of experiments each producing a list of hundreds of measurements,
  • digital images: 2d and 3d arrays of pixels each with a different vector value.

Sometimes there is a continuous function or process behind the numbers but often there isn't. The issues one has to deal with are the same though: max/min, increasing/decreasing behavior, rate of change, lengths, areas, volumes, etc.

What is commonly done is to go back to continuous functions via approximation, interpolation, curve fitting, etc. This approach is not the best solution as it introduces errors in your analysis (imagine approximate conservation of energy!). It also requires even more advanced math. Meanwhile, a typical graduate has never seen discrete functions and now at the workplace he doesn't connect what he does with Excel every day with what he learned in his calculus class.

The bottom line: the numerical/computational aspect should be built in!

How to do this? The answer is known: introduce discrete differential forms and discrete exterior calculus.

There is a lot of literature on discretization methods in modeling and simulation but it's quite complex. Waiting for these ideas to trickle down to undergraduate courses might take 20 years (more like 50).

These have always had their discrete parts integrated from the beginning:

Less emphasis on closed-form formulas

Once again, data comes in the digital format. Since there is no formula that produces this data, one has to deal with it as just functions. Then, surprisingly, taking the applied approach means more abstraction not less: more $y = f(x)$ than $y = x^2$. And here's a bad kind of abstraction. Students remain confused about compositions of functions well into calculus, why? Problems like this -- in every college algebra or calculus textbook I've seen -- "given $f(x)=\sin (x-1),g(x)=x^2$, find $f(g(x))$"; instead of: $f(y)=\sin(y-1)$ and $y=g(x)=x^2$. From the same bad place comes plotting a function and its derivative (or its inverse) on the same $xy$-plane, or writing $\partial f/\partial x=\partial f / \partial y =0$ etc. And if you want concrete, try Excel.

To make room for this new stuff, one has to cut something out. Fortunately, one can drop a lot of the following:

  • information about specific functions (polynomials, exponential, logarithm, etc),
  • trigonometry (If I see secant one more time! If you don't see it used in other math classes, why teach it?),
  • the algebraic tricks for solving equations, manipulation, and simplification: factoring, rationalization, partial fractions etc,
  • integration techniques:
    • the integration by substitution is important for a good reason -- it's about change of variables (another commonly missing part); but
    • figuring out which technique to use to evaluate a particular integral (trig substitution vs integration by parts vs algebra etc.) is so 19th century.
  • the polar, cylindrical, spherical coordinates, and conversion (a lot of that but nothing about Cartesian to Cartesian!), integration in these coordinates, etc.


Teaching proofs to students who aren't math majors is a lost cause.

Proofs still make sense when they are instructive.

Students still need more math. More math not more proofs. Consider calculus. So many new concepts but most proofs are beyond reach.

Also, fewer definitions and less new terminology. Don't introduce a new concept (or a word) unless you are to use it in other courses.

Breadth first and bottom up

Start calculus with sequences. Compared to the epsilon-delta, I find this approach much more intuitive. Another advantage is that you can deal with series later, after you've considered convergence of functions, based on sequences. Then, you can focus on the important stuff, such as Taylor and Fourier series etc.

I have never taught this way yet so it's hard to be specific yet. But I did learn calculus this way, back in Russia years ago. I even remember the professor, Vladimir Zorich. His textbook, Mathematical Analysis, is available on Amazon. It's fairly advanced and it teaches both sequences and epsilon-delta in parallel...

Introduce functions of several variables early. Do you see a lot of functions of single variable around? Waiting until calc 3 is unwise (some never get there) and unnecessary. Partial derivatives are manageable.

More bottom up. Removing structures from consideration is hard; adding is much easier. Compare how you start with point-set topology -- by removing the geometry from the Euclidean space -- and how you start with inner product spaces -- by adding this structure to vector spaces.

Discrete calculus as a subset of calculus

...on the pre-calculus level.

Quote: "Calculus Without Limits ... deliberately minimizes the use of limits, one of the major stumbling blocks initially standing in the way of calculus students." There are no limits in discrete calculus!

1. Derivatives.

Functions and forms. The velocity problem. Change and the exterior derivative. The product rule. The chain rule. Implicit differentiation. Rates of change in the natural and social sciences. Derivatives of exponential functions. Exponential growth and decay. Related rates. Maximum and minimum values. How derivatives affect the shape of a graph. Optimization. Anti-derivatives.

2. Integrals.

Areas and distances. The definite integral. The fundamental theorem of calculus. Indefinite integrals and the net change. Integration by substitution. Areas between curves. Volume. Work. Average value of a function. Applications to physics and engineering. Applications to economics and biology. Probability.

3. Parametric curves and differential equations.

Curves defined by parametric curves. Modeling with differential equations. Separable equations. Models for population growth. Linear equations.

4. Vector functions.

Three-dimensional coordinate systems. Vectors. The dot product. The cross product. Equations of lines and planes. Vector functions and space curves. Derivatives and integrals of vector functions. Curvature. Motion in space: velocity and acceleration.

5. Calculus of multiple variables.

Functions of several variables. Partial derivatives. The chain rule. Directional derivatives and the gradient. Maximum and minimum values. Double integrals. Iterated integrals. Applications of double integrals. Triple integrals.

6. Vector calculus.

Vector fields. Line integrals. The fundamental theorem of calculus for line integrals. Green's theorem. Curl and divergence. Surface integrals. Stokes' theorem. The divergence theorem.

7. What's next?

$\pi =4$ and what to do about it.