Albert Einstein, thinker

← I'm not sure exactly what Albert Einstein meant when he said that. He was, of course, a great mathematician, and (contrary to myth) he was known to have been a good math student. But I think he was acknowledging that whatever your level of understanding, learning new math—pushing your limits—can be difficult.

Humans are innately mathematical beings; we notice things like symmetry and we're good at estimating and comparing numbers. But we're really not that good at concentrating for long periods of time on challenging problems. That may be why we revere thinkers like Einstein, who was capable of focusing deeply on the most challenging problems of his time.

Don't compare yourself to others (especially Albert Einstein) when learning math. Intelligence, including your knowledge of math, can be increased by sustained effort. Keep at it and remind yourself once in a while how far you've come. Even Einstein had a lot to learn.

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There are many ways to order the subjects when teaching math at various levels like Algebra, Geometry, and so on. This is roughly how I do it, but I've tried to write the sections so that they pretty much stand alone. There is some repetition of some sections between major disciplines. I'm always happy to hear suggestions about what else to include or to remix.

**Algebra**- Fractions– Everything you need to know to be an expert with fractions
**VIDEO**– Fraction arithmetic: Adding, subtracting, multiplying and dividing fractions

- Basics of algebra
- Rational & negative exponents
**VIDEO**– Examples of how to use rational and negative exponents in problems

- Solving for x
- Pitfalls – common mistakes and how to avoid them
- Lines
**Special topic**: Scientific notation**Special topic**: The metric system

- Fractions– Everything you need to know to be an expert with fractions
**Geometry**- Slope
- Distance between two points
- Midpoint between two points on a line segment
- Lines – equations of lines
- Analytic geometry
- Triangles
- Parallel lines
- Circles
- Polygons
- Area & volume
- Conic sections
- Parabola
- Circle
- Ellipse
- Hyperbola

- Fractal geometry

**Functions**- Linear–lines as functions
- Quadratic
**VIDEO**– Video examples- Roots of all functions: See also,
*Calculus*→ Newton's method

- Polynomial
- Graphing polynomial functions
- Polynomial long division
**VIDEO**– Video examples– factoring, graphing, long division

- Rational
**VIDEO**– Sketching graphs of rational functions

- Exponential
- Rational and negative exponents
**VIDEO**– Video examples of problems involving rational and/or negative exponents

**VIDEO**– Video examples of exponential and log functions problems- Scientific notation

- Rational and negative exponents
- Logarithmic
- Trigonometric
- Non-right triangle trigonometry
- Inverse trigonometric functions
- Analytical trigonometry
- Trigonometric equations
- Polar coordinates
- Complex plane
- Special topic: Trigonometry of rock climbing
**VIDEO**– Video examples

- Parametric functions
- Domain and range
- The Factorial function and its properties

**Probability & Statistics**- Probability-discrete – discrete random variables
- Probability-Bayesian
- Probability-continuous – continuous random variables
- Combinatorics – counting: permutations & combinations
- Statistics
- Average / Mean
- Running average
- Geometric mean

- Median & Mode
- Correlation
- Law of large numbers
- Probability distributions
- Gaussian
- Binomial
- Geometric

- Average / Mean

**Calculus –***the poetry of mathematics*- Limits
- Existence theorems
- Continuity
- Mean value theorem
- Rolles' theorem
- Intermediate value theorem

**Differential calculus**- The derivative
- Product rule
- Quotient rule
- Chain rule
- Derivatives of trigonometric functions
- Derivatives of exponential and log functions
**VIDEO**- The derivative

- Derivatives of inverse functions
- Implicit differentiation
- Related rates
- Linear approximation
- Curve sketching
- Newton's method of finding roots

**Integral calculus**- Antiderivatives / indefinite integrals
**VIDEO**- u-substitution examples

- Definite integrals & integration by substitution (u-substitution)
- Integration by substitution
- Area between curves
- Area of polar functions
- Fundamental theorem of calculus (FTOC)
- Integration by parts
- Rational decomposition
- Trigonometric integrals
- Integration: Trig. subst.
- Integration by rational decomposition of functions
- Tough integrals– some examples of complicated integration
- Volume integrals
- Arc length
- (
*see also*: Parametric functions)

- (
- Surface area

- Antiderivatives / indefinite integrals
- Differential equations
- Separable differential equations
- The logistic differential equation

**Infinite series**- Infinite series –Introduction
- Polynomial long division (insight into infinite series)

- Alternating series
- p-series (see the integral test first)
- Power series
- Taylor / MacLaurin series
- Tests for convergence
- Fourier series

- Infinite series –Introduction
**Linear algebra / Matrix algebra****Multivariable mathematics****Game theory**

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