How to get better at painting without painting anything (2015)

At a low level, you can point out all the patterns that are in multiplication tables, but they won’t be remembered until the student has internalized the facts.

At a higher level, teaching epsilon-delta proofs isn’t a good way to learn calculus. Memorizing the building blocks and the chain rule is.

Understanding requires time to play and explore something. It enables slow but powerful problem solving into new territory.

But fluency requires practice. And it’s benefit is quick and accurate automatic responses, freeing up the brain to tackle novel things much faster.

Together they work really well. and the repetition in obtaining fluency can also push understanding deep into stable memory where it will remain accessible years later.

I have seen a lot of education where only one side was emphasized. When kids are pushed for time, neither side gets done well.

I don’t think anyone learning anything should move on to more difficult tasks until they both really understand and become fluent at the prerequisites.

But that would take education off it’s cohort centered time schedules. That system is so ingrained in most of our non-online education systems.

3 things an artist make:

1. Finishing and finishing often. You should not be making the same thing all the time, but you should be consciously iterating in any direction you like as long as you are still finishing. Great painters, writers, musicians, are constantly making stuff, you only see the polished stuff that ends up through the filter.

2. Studying the great works in whatever your chosen field/style/genre. Know the "rules", the normal directions on the map, before you break them and decide on a shortcut or to buck convention all together. All great artists, even the enfant-terrible avant-garde artists, are extremely knowledgeable in art-history in their chosen field and can explain extremely precise opinions on the merits of one artist or movement. Most postures of naiveté are just that: posturing. Great artists know their stuff.

3. Surrounding yourself with other artists. You need people who will look at whatever you finish and tell you about it. Hopefully, these people should be as interested in good art as you, and as well versed or more well versed in art history as you. Engaging in communities of artists will make your imagination and creativity soar.

You absolutely need 1 and 2. You can get by without 3, but you will likely never achieve true greatness without it.

Everything else after that is luck.

You see, the students who failed at the interleaved problems initially, were rocking it when I had them work through like 3 of these 20-similar problems worksheets before moving on to the 'pedagogically designed' problems.

And epistemically I think it makes a lot of sense to train basics and build upon that.

I think Math education could benefit a lot if we split the subject in two courses, 3h per week on drill (Arithmetics), 2h per week on the beautiful math (can also expose the student to axioms there, functions, mappings, etc, more complex problems and solving that with math, potentially with CAS support). Best separated with different teachers.

Fact is, most high school graduates will find it challenging in their lives to apply the 'rule of three'. During the covid pandemic we have seen that members of the executive branch have no understanding of exponential growth (bad during the pandemic, but I wonder how the fiscal policy is affected by that??).

Maybe we need to rethink mathematical education once again.

Same with understanding anything else in life - you don't really understand anything until you get to the point you can, in your head, predict a specific outcome, test it, and be proven correct, repeatedly.

(At some point you may learn to gain robust understanding purely from simulating things in your head, deriving insight from lower-level principles. But this itself is a skill, a hard one, which few people master. It's not something a random kid, or adult, can do.)