I don't have full answers for all of your questions, but my colleagues doing Learning Science often mention this two-sigma improvement, and are working on computer-based cognitive tutors that can adapt and model what a student is learning. I vaguely recall that these cognitive tutors are at the 1-sigma level for certain subjects (primarily STEM subjects), but that was many years ago and I'm not sure of the details.

Here is one example abstract from one of my colleagues (sorry I don't have the full paper): https://link.springer.com/chapter/10.1007/3-540-44566-8_14

Cognitive Computer Tutors: Solving the Two-Sigma Problem

Individual human tutoring is the most effective and most expensive form of instruction. Students working with individual human tutors reach achievement levels as much as two standard deviations higher than students in conventional instruction (that is, 50% of tutored students score higher than 98% of the comparison group). Two early 20th-century innovations attempted to offer benefits of individualized instruction on a broader basis: (1) mechanized individualized feedback (via teaching machines and computers) and (2) mastery learning (individualized pacing of instruction). On average each of these innovations yields about a half standard deviation achievement effect. More recently, cognitive computer tutors have implemented these innovations in the context of a cognitive model of problem solving. This paper examines the achievement effect size of these two types of student-adapted instruction in a cognitive programming tutor. Results suggest that cognitive tutors have closed the gap with and arguably surpass human tutors.

Feels like the Khan Academy apps do something like this. You progress through a tree of knowledge and can’t move to any next node until you master the one you’re on, demonstrated by solving problems. It also gives you all the help you need to achieve that mastery.

> and can’t move to any next node until you master the one you’re on

Oh that's the absolute worse way of teaching people (or, ok, that's my pet peeve in education)

"but how do you expect to learn stuff without knowing the basics blah blah blah" well, because maybe actually knowing how it is used in the end helps with learning. Instead education seems to focus on wasting a lot of time with "basics" disconnected from reality then finally teaching things how it is.

Calculus was the worst for this: in my first class we spent literally two months dissecting the minutiae of limits and Lipschitz conditions and infinitesimals and blah blah etc, only to get the the punchline, "and you find the slope of a function by doing the obvious thing, which works in the obvious way every time you'll actually be using it in practice". I get it in a college level analysis class or something, but as an intro in high school that's just a great way to make students hate what is at its core a very simple and useful subject.

Not sure if I'd agree. Khan Academy is firmly rooted in the American tradition, it teaches facts, but it's like Feynman in Brazil, it does not integrate the concepts. For that you need problems tailored to your level. A good tutor can do that.

> it’s like Feynman in Brazil

I hadn’t heard that reference before. Here’s the story: http://calteches.library.caltech.edu/46/2/LatinAmerica.htm

Feynman's complaints will be familiar to anyone teaching university in the United States. Obviously Feynman makes a distinction between the science education that he knew and the travesty that he saw in Brazil, but today's education in the US is exactly like what he complained about. What happened, and when?

The Russian tradition of "math circles" (https://www.msri.org/people/staff/levy/files/MCL/Zvonkin.pdf) is altogether more productive than US-style kindergarden (no resemblance to the German original) where the 3-year olds sit in rows and have the ABC beaten into them. Of course the Russian approach relies on individual attention and problems, and class sizes are smaller.

Out of everything on the page, this grabbed my attention too. This seems to have immediate implications for how even experienced professionals train.

I'd like to become much better at math than I am now. Should I be putting a lot of effort into working out a one-on-one tutoring partnership?

That, and/or mastery learning. I haven’t read the research, but I hypothesize the focus on ‘mastery in each subtopic before moving to the next’ is what drives results, while 1-1 tutoring is what drives the speed of attaining them.

Why do you think that takes a lot of effort? I suspect it would not be hard to find a tutor at readonable rates. (Speaking as a former home-schooling dad. We used math toturs for 1:1 instruction at various times. )

I'd be looking for a tutor in graduate mathematics, and particularly the subjects that I can most readily apply to my work.

In Sydney I don't think there'd be that many available, and it would take time to find the right 'fit'. I don't think you can assume that once a reasonable rate is settled that the partnership would be a success.

A private tutor in graduate-level mathematics, with emphasis on subjects that can be applied to your work - what a great idea!

Like the sibling comment suggested, I imagine there must be an over-qualified and under-paid talent pool around universities, with enough candidates for you to select a suitable tutor.

No local uni with math PhD students?

I'm personally pretty confident that with the level of resources now available for learning almost anything online, you can replicate a lot of the benefits of tutoring by (1) getting students excited about a topic and then (2) teaching them the basic keywords and research skills for that topic.

(We've seen a lot of success with this model at CodeDay, n~50,000. It was very exciting when we first tried this as a legitimate educational strategy and saw it work, one of the reasons I decided to work for this nonprofit instead of another startup.)

Keeping and maintaining that excitement for self-driven learning is the most challenging part.

I assume the answer is... it depends. I've taken training/workshops that clearly depend on group dynamics. I've also had training presenting, skiing, individual projects in grad school, etc. that pretty much depend on 1:1 time and, to the degree you're in a group, it mostly means you just don't get as much attention (but it's cheaper).

Bloom's paper largely focuses on when/how students get feedback and less on the particulars of the assessments.

For example, maybe mastery learning works (in part) because it helps teachers pay attention in a group setting more like they would in a 1:1 setting. From Bloom's paper: https://share.getcloudapp.com/rRu7WJRz