But even Von Neumann said that in mathematics, you don’t understand things, you just get used to them. A hyperbolic truth.

It isn’t mentioned but they made the same mistake with whole language, I think it was called, instead of phonics. They taught kids to read with hieroglyphics of the whole word instead of the basics. Phonetics are the main big advantage over graphic whole symbol based writing systems and they ignored it.

I think what they were trying to do was to mimic the people who read fast by seeing the whole word as one but that doesn’t work for everyone and probably doesn’t work as well if you skip the phonics part.

I wonder if the “new” math essentially did the same. Study accomplished math students and try to replicate their performance by skipping steps.

If your child is normal-ish, he needs repetitive drill, and he will not learn anything by being encouraged to “explore.”

90+% of the population wants to be told THE answer, and will not, cannot think successfully.

(By the way, I find most FAQs to be a waste of time.)

Ultimately the only way to learn something is to do it. The way to learn the foundations is to do the foundations. From which it follows that the way to teach the foundations is to drill the foundations. Not some stupid catechism of question-and-response, but actual application of the fundamentals to simple problems that approximate – in simplified form – real world situations.

The way to learn a language is to use the language. The way to learn a branch of math is to use the branch. The way to learn music is to play it.

Language: Start with a limited vocabulary and a fairly comprehensive grammar. Practice all the sentence structures this way. Learn pronunciation and hearing comprehension this way. Then add vocabulary to deal with ever more real world situations. The great bulk of vocabulary is specific to a domain. The vocab you need depends entirely on what you want to talk about.

Business math: Start by adding a column of numbers. From this progress to a ledger, and then to a balance sheet… your graduation exercise will be to start a business.

Engineering math: Start with exponential notation and algebraic notation. Then some canned algebraic equations to predict results of simple lab experiments. Check your math by doing the lab experiment. Then go on to more complicated machines that call for more complicated calculations. Here’s where you bring in algebra and calculus.

Number theory: Fiddle with cryptographic algorithms and data structures. Or else don’t bother.

Music: Start with an instrument that’s easy to play – a keyboard instrument. Learn the chords by playing the chords. Learn the keys and modes by playing the scales. Then learn improv by noodling. Not too much drill – that will slow you down. Skill will come in its own time, but learn the basics now by doing the basics now. And maybe switch to guitar at some point.

What of abstract knowledge, you ask? Screw abstract knowledge. It’s not actually about anything, is it?

Don’t talk to me of the evils of specialization. It’s a moot point. Kids come specialized. Teach according to the students’ aptitude. Some of them will sell widgets. Others will make moon rockets out of those widgets. It will all sort itself out.

In England they speak of maths in the plural. They may have the right idea.

In schools, the question is therefore not what is in the schools but what isnt.

I suggest that the liberal arts — in other words, the classics — are essential to producing a citizenry capable of governing itself. And naturally, a body politic capable, and not only capable but demanding, of self-governance is by far the most salient threat to any incipient occupational class. Nothing else comes close.

Ask, therefore, why Greek and Latin aren’t in the school. Ask why, if they are, they suck balls. Ask where the Great Books are in the school library. Ask where, when, and how the school teaches the students to organize themselves into a coherent body capable of making resolutions, and passing them, and keeping them.

Really, to ask the questions is to answer them.

Many students/people would have difficulties understanding the basic problem, let alone coming up with several ways of getting a solution.