Fortnightly Issue #9
Math education, working memory, and more
Hello, dear readers!
Today’s text will be densely packed with topics, ideas, and recommendations so I will allow myself the liberty to omit a lengthy introduction. Let’s cut to the chase.
Mathematics education
A couple of years ago, a friend of mine said that he wanted to design a curriculum that would make students able to attack the Millennium Prize Problems upon graduation. That sounded ridiculous then — still sounds now — but so much is wrong with the current education system that we don’t know what students are really capable of. There is plenty of room for improvement.
Last Friday, I participated in a roundtable discussion on how to teach mathematics to kids. Well, primarily to kids but not exclusively. The prompt was to build a bottom-up approach to learning mathematics: problems should be atomic units clustering to form topics, topics would form semantically meaningful blocks.
The tricky question is — how to link the problems together to suggest a near-optimal path that would inspire deep understanding? This is not an easy task even with a couple of hundreds of problems but there are no less than ten thousand of those that pave the way to advanced mathematics. It’s near impossible to tackle this manually.
My current proposal is for the experimental groups of scholars to massively use graph note-taking apps. If they all cling to a certain markup notation, it will be possible later to merge their graphs and measure linking patterns statistically. Or to apply machine learning algorithms. Or whatever. There are issues with this too. We need to organize many people to act following the notation. Most of them will be schoolkids which only makes our situation direr.
Levels of ascension
Once the problems and topics are tied together to form a roadmap, they need to be split again — now into several tiers. These are the strategic parts of our curriculum: what people are expected to know after completing a given module. Each tier features different subfields of mathematics but at a different depth and level of understanding.
Unlike most current programs that see entire subfields as more or less complex, our program will feature parts of many domains in each tier. Even the first one, designed for primary school, will have basic ideas from probability theory and calculus.
Art and technology
Many people (myself included) have complained about their math classes for showing the lack of usefulness. I think I know how to address this problem.
Two classes involve handcrafting: arts and technology. The former yields no tangible utility, all that is done there is done to please the senses and to teach kids to see beauty. The latter is all about applications in real life — you learn how to dismantle, fix, and build things.
Math classes can be split into these two categories as well. We may have a class for hands-on skills: probability to assess risks, economics math to learn what a loan will cost us, and so on. The exquisite math pearls can be taught separately: Pascal’s triangle, Pythagorean theorem, Euler's formula, you name it. Surely some of these topics are indispensable in certain scientific domains, but not in one’s life if that life is devoid of academic endeavors. Maybe it’s better to teach those pearls as arts and not force people to remember strict proofs?
Working memory
Recently, we had a long walk with Nikolai and spoke of many things. The capacity of our working memory was one of the most frequently recurring topics.
There will be some numbers in the next paragraph that are often spread on the Internet but I cannot vouch for their validity, and I haven’t yet converged on the list of papers to read so if you have something to recommend I’m all ears. For now, let’s take these numbers at face value.
Modern humans have very limited mental space to retain and handle objects. We can keep track of about 7 objects at a time, with no spare attention left. Trained people can sometimes operate more but there is a strong guess that this depends on the way they encode those objects within their brain, not on some task-specific additional space flickering into existence. So, let’s raise the bar a little and say that we can focus on up to 15 (a huge overestimate) distinct objects or primitive data points.
But what the world would look like if we could handle 100 objects? 500? 1000?
It is pretty easy to imagine our eyes being enhanced with ultraviolet receptors, or our ears upgraded with ultrasound detectors. But what lies in the lands of greater working memory capacity — I find it impossible to picture. I’m sure this will tremendously improve our ability to find links and associations between existing concepts but to imagine it is what I find so hard.
Multipliers
Sleep, physical exercises, psychotherapy — all these demonstrate a multiplier effect on our productivity and the quality of our life. Working memory is a multiplier too. The good news is that it can be trained.
Stray insights
One of the topics has swollen to a volume that requires me to move it to the next issue and make it one of the two major subjects. It’s interesting to follow the undercurrents of how the structure of these issues is being crystallized — on most occasions, there are two main ideas, some insights, and several recommendations. I think I like it.
Dropping the first paragraph
This is one of the most underrated techniques in writing I've encountered so far. It makes a text much livelier and rarely impacts clarity. I applied it when working on Love and Death, and this method alone roughly doubled the quality of the essay.
Though don’t think it moved the essay anywhere close to perfect! Avery Bedows wrote a whole doc of comments on what is wrong or dubious about it.
The Sun and the Moon
Avery and I had a call later whence a metaphor sprung into existence:
Please, take it with a grain of salt as there is no thorough sophistication behind it, I only want to share our minute desire to create something poetic.
Recommendations
This thread by Paul Graham. Great startups emerge at the intersection of basic forces.
Logarithmic time planning: make plans at hand as detailed as possible, and simply outline the events that are farther in the future. From this post by Ed Boyden.
Top-10 research papers in AI in the last 5 years, more or less. Exciting read!
A beautiful article on how to build a really expensive machine learning rig. But think twice before buying your own hardware: sometimes it’s wiser to rent some GPUs.
It’s 2 AM in my timezone, I’m feeling tired as hell but also happy as much. The toil of a periodic newsletter writer is demanding but rewarding. Thank you for reading!