Maths to Learn
The previous post discussed the five categories of knowledge. These posts will go through different topics that are interesting, starting with topics in Mathematics.
Theoretical
Theory of computation - key ideas of computation. It's interesting how a mathematical idea about computation grew into physical computers.
Turing’s Halting Problem and Godel's Incompleteness theorems and how they relate to each other. (See Incompleteness Ex Machina)
Question: Is there a way one can exclude the halting problem and build a machine that can determine if almost everything will halt?
Intuitively understanding how high-level programming languages actually executes on a machine.
Review of basic calculus - Calculus is practical for scientists and certain engineers, but less practical for other people. The fundamental of it are interesting though:
Intuitive understanding of derivatives and integrals.
Optimization and related-rates problems
Basic applications to physics
Practical
Probability & statistics - Ultimately all knowledge comes down to probabilities. Statistics are useful for interpreting studies and experiments and everything else.
Fundamentals of probability
Bayesian probability and statistics
Pascal's triangle, the normal distribution, the central limit theorem
Applying statistics to real-world examples
Tools for statistics (e.g Spreadsheets, R, Python)
Statistics for machine learning
Using Mathematica for real world math problems - May as well use the most powerful tool one can.
Schools in general focus on more theoretical topics instead of generally practical topics today, such as applying statistics or using mathematical software. Some students could use more help understanding math basics, such as fractions and exponents. Other students can learn more advanced topics but it need not be limited to a narrow curriculum of trigonometry and geometry and specific parts of algebra. (See also my post from 2011.)