In a different thread I said
I estimate that the entirety of the average North American K-12 mathematical education can be compressed into a single book about 200 pages
long, while being entirely comprehensible to a 10-year-old.
Here I will attempt to devise a concise and complete curriculum for K-12 or pre-university mathematics. I will add new material to this thread
periodically, anything from lessons to exercises to bits and pieces of thoughts. It certainly won't be anything ready for publication. This thread is
only meant to be a proof of concept, or a proof that a concept will fail.
Without further ado, let's get to it.
Rough outline of topics. The names of the categories and the order of topics are rather arbitrary.
Elementary topics
- Natural (positive) numbers
- Addition
- Multiplication
- Geometry -- rectangles
- Non-positive numbers
- Subtraction
- Exponents
- Base systems (10, and perhaps 2?)
- Modulo operation (despite the scary name, this simply means finding remainders)
Intermediate topics
- Parentheses
- Prime numbers, the fundamental theorem of arithmetic
- Factorization
- Division of integers, fractions, rational numbers
- Operations involving rational numbers
- Radix points (usually only base 10)
- Geometry -- triangles, parallelograms, circles (what about compass-and-straightedge constructions?)
- Single-variable algebra
- Distributive property (perhaps introduced earlier, through diagrams)
- Graphing (plotting) equations of one variable
More intermediate topics
- Combinatorics
- Set theory
- Probability theory
- Statistics
- Logarithms
- Trigonometry
- Geometry -- basic identities and proofs
- Polynomials of second degree
- Proofs by induction, and contradiction
- Systems of linear equations
Advanced topics
- Linear algebra -- vectors and matrices
- Polynomials of higher degree
- Parametric equations
- Polar coordinates
- Conics, analytic 2D geometry
- 3D geometry
- Complex numbers (and de Moivre's theorem, anyone?)
Whew, that's rather a challenge. Best tackle that one piece at a time. For example, logarithms can be summarized in a few pages. Note: the estimate of
"200 pages" does not include exercises.
One way to motivate a student is to appeal to the inherent human characteristic of frugality. What number multiplied by itself equals 625? A naive but
sure-fire approach would be to try 1, 2, 3 ... all the way to 25. But there clearly exist ways which require less effort.
Formal logic and proof should be taught in parallel with elementary topics.
Nothing here is final, of course.
Feel free to contribute. I know I'll be back.
edit on 27-5-2012 by socialist because: (no reason given)