Maths to Learn

In the previous post I discussed the five categories of knowledge. These posts will go through different subjects that I'm interested in, starting with Mathematics.

• Theory of computation - key ideas of computation. It's interesting how a mathematical idea about computation grew into physical computers.
• Turing and Godel's theorems and how they relate to each other. Is there a way one can exclude the halting problem and build a machine that can determine if almost everything will halt?
• How high level code actually executes on a machine.
• Review of basic calculus
• Intuitive understanding of derivatives and integrals.
• Optimization and related-rates problems
• Applications to physics
• Probability & statistics - Ultimately all knowledge comes down to probabilities. Statistics are useful for interpreting studies and experiments and everything else.
• Review 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 stats (e.g Google sheets, R, Python)
• Stats for machine learning

• Using Mathematica for real world math problems

Besides for calculus, it's interesting how little these topics are taught in schools. Many students don't know basic topics like fractions well and schools should focus on teaching them better. Other students can learn more advanced topics but it does not need to be limited to a narrow curriculum of trigonometry and geometry and specific parts of algebra. (See also my post from 2011.)

Carthago delenda est 

Khan Academy and Math Education

Two professors just released a parody video critiquing Khan Academy.  Its pretty boring and too nitpicky, so you can instead read the article about it in the Chronicle of Higher Education.  Their basic critique is that Khan Academy focuses too much on procedural thinking and not enough on actual thinking.  This is an issue, but traditional education suffers from a similar flaw. Often, schools just teach for whatever test it is that students must take, and ignore actual understanding of the material. To properly explore a topic, one would have to hire an very good teacher to teach a small group of students who are all at a similar level. This would be very expensive. A more realistic plan would be for students to use quality interactive software to learn the material and be able to discuss it with their peers, and when needed, be able to consult with an expert or advanced student. This balanced option could probably produce the best result for many students at the lowest cost.

The other complaint in the parody is the quality of Khan's videos. It's true that Khan's videos are pretty simple, but he himself recognizes this. He stresses the main power of the site is the amount of data they have and their ability to add new features. For example, they added an intelligent quiz tool, and they will be adding additional interactivity in the future. This will improve the site, but there is also room for other players to contribute to online education. For example, if a site would make it easy for many people to create interactive educational content...

See also:
Math Education
The Future of Education 

Math Education

Intro - Education & Math
Mathematics is probably one of the most important subjects taught in school, and the school system does spend a large amount of time on it. But they do not teach it in the best manner. There are a few basic questions that need to be asked, such as Why, Who, What & How. 
Why teach math? 
Its useful. Its wise. But how well do are these reasons applied in practice? 
Who to teach math to? 
Many people barely use or appreciate any of the math they learned after division, so they don’t get much from it. There’s little point in causing them all that trouble. Maybe people who have no mathematical inclination, should not  go through such a system. Of course, an improved system might interest some of them… 
What they teach now
I never understood (at least since elementary school) why they have to teach so much by hand. What is the point when there are computers? People give many reasons to defend the practice, but they’re mostly just attempted justifications for keeping things as they are. For example, some people used to say ”What will you do if you don’t have a computer/calculator?” I’m not sure that rare occasion ever really justified spending so many years on it though. Anyways, that reason has become obsolete now that everyone has cellphones, and soon all will have smartphones.
A slightly stronger claim is that true understanding only comes when you do it on your own without a computer. Except there’s no real fundamental difference. People rarely actually understand what they’re doing, they just plug things into formulas they’ve memorized. In which case, they are just like a mindless computer, generating an answer. I bet many people don’t even know why the most basic formulas work, such as multiplication of 2-digit numbers.
Also, much of the math learned doesn’t even involve any true understanding, its just techniques to do things before computers. For example, most of calculus II is learning unnecessary techniques to solve integrals. There is no reason for so many people to learn such things when they can have silicon “formulas” to the work for them a billion times faster without errors. 
What they should teach
I’m not even sure how much of a value there is teaching the understanding of every formula. Maybe for mathematicians, or in certain areas. But the main thing they should be teaching is how to convert life into math so the computer can solve it. Computers cannot analyze life on its own, and wont be able to for quite some time. Students need to learn how to take questions in life and mathify them. People can focus on the higher-level interesting and useful questions, and let computers do their calculating thing. 

For more on this topic, see Conrad's Wolfram talk, which discusses similar issues.