Ultralearning, the new book by Scott Young, comes out in August. Last week, I briefly covered some key ideas from the book, including the ultralearning philosophy and the nine principles of ultralearning. But ultralearning is about projects, so this week I’d like to explore how you could use the ideas in the book to optimize a mathematics learning project.
Chapter 13 of Ultralearning contains a guide to planning, executing, and reviewing an ultralearning project. I’ll focus on the first three steps: research, scheduling, and execution.
Recall the philosophy of ultralearning: You’re responsible for your learning outcomes. One way to increase the odds of a good outcome is to start with a good plan. There are no guarantees, and the plan usually changes once you get started, but planning forces you to think through the details of your project before you’re occupied with project execution.
In the book, Scott recommends spending 5%-10% of your total project time (with a lower percentage for longer projects) on research. So for a year-long project, you might spend the first few weeks researching. While that might seem like a long time to postpone the start of project execution, there’s plenty to do in the research phase.
Two key aspects of planning are a clear definition of the topic and scope, and a list of learning resources. In my 2019 project definition, I defined my project scope as learn discrete math for competitive programming. That is more specific than “learn math,” but according to ultralearning best practices, I should probably narrow it down more. Maybe “learn these discrete math topics used in programming contests,” followed by a specific list of topics gathered during the research phase.
A narrower project definition would also require a more specific list of learning resources. I mentioned a few resources in my project definition, but I didn’t reference specific chapters, page numbers, or web pages. One goal of the research phase is to work out these details so during the execution phase you can focus on learning.
Since ultralearning is an intense process, you can’t rely on getting it done in your spare time. The only way it works is if you commit to a schedule and stick to it. Since I started blogging a few years ago, I have been using a fixed morning and evening schedule on weekdays. On days when I’m not writing, I use that same schedule for project work. Everyone’s schedule is different, but in my experience, working the same specific times each day as much as possible makes a schedule easier to stick to.
Once an ultralearning project is planned, scheduled, and underway, there are nine principles (defined last week) to keep it on track. Here’s how they might apply to a math project:
Metalearning (studying the learning process) continues while a project is underway, using some percentage of your project time. For math, it’s important to learn not just how to solve each specific type of problem, but also how to approach problem-solving in general. There are some books on this topic, and math textbooks may address it, but there’s no substitute for observing how more experienced problem-solvers approach problems. If none are available in person, you can find them virtually on Quora and Stack Exchange.
As Ultralearning suggests, different kinds of focus work best for different kinds of problems. For math, intense focus is helpful for absorbing the details of a problem and maybe starting on a first approach. If you get stuck, a more diffuse focus (including stepping away from the problem entirely, and maybe taking a nap) can jostle some ideas from the subconscious. Then intense focus can return as you work out the details and write up your solution.
The directness principle means practicing a skill in the same context where you’ll be using it. If you’re learning math for the purpose of learning more math, then the standard textbook problem approach is fairly direct. But if you have an application in mind, it’s better to find problems in that application area. For example, if you’re learning math for competitive programming or coding interviews, then the right approach would be to study a math topic and then find programming problems that require it. uHunt has a section that you could use as a source of math programming problems. And of course there’s always Project Euler.
This principle is perfectly suited to learning math, which requires drilling on problems to practice procedures. But you have to practice the right problems for your level of expertise. At a high level, that means practicing problems at the edge of your ability (not too hard and not too easy). But Ultralearning also introduces the concept of the rate-limiting step, the skill that contributes most to how long a process takes. To use an example from the book, if you’re studying calculus but your algebra skills are weak, algebra might be your rate-limiting step in each problem. By focusing for a while on algebra, you might improve your calculus ability more than if you focused directly on calculus.
Math relies less on memory than other subjects do. You need to be able to derive results, not just remember them. But it takes too much time to derive every formula you need whenever you need it. To practice retrieval, one approach is to make a cheat sheet of formulas, but try to get by without it. When you need a formula that you don’t remember, look it up and put it on your list. But the next time you need it, try to recall it without consulting the list. Even if you eventually check the list to refresh your memory, you’ll gradually learn the most useful formulas.
One area of math where feedback is especially useful is proof writing. You can check your proof in the answer section at the back of a textbook, in a solution guide, or by attempting an example proof in the body of the text before looking at how the textbook author proved it. These options will all give you some feedback. But to get better at proofs, you eventually need a human to read your proofs and tell you what they think. They don’t necessarily need to be an expert. As long as they know enough math to follow your argument, they can tell you if they think your proof is complete, convincing, and understandable.
One ultralearning technique for improving retention is to turn factual information into procedures, which the brain can remember more easily. Going back to the algebra example: you may know the theory of factoring polynomials, and you may even have studied the steps required to do so. But the way to remember how to factor a polynomial is to get familiar with factoring procedures, so that you can carry them out without thinking about each step. That’s why practice problems are so important for mathematical subjects.
While automatic knowledge of procedures is necessary for efficiently working out the details of a problem, strong mathematical intuition requires the opposite. The first step to building intuition is understanding the fundamentals, which means thinking about why each step is necessary. Mathematical proofs are a good way to verify that your intuition is correct. Another way is to explain a concept to someone else, especially someone who is a beginner in that concept.
While there are common math learning techniques that work for most people, it’s still up to each individual to test what works best for them. So if you try any of the suggestions above, take notes on how well each one works. That will help you develop the unique learning techniques that are best optimized for you.
This week’s and last week’s post are just a selection of the advice from Ultralearning, with an emphasis on math. While Scott wrote the book as a guide to learning any subject, the ideas have been well-tested in technical subjects, as shown in Scott’s year-long MIT computer science learning project, and his recent month-long exploration of quantum mechanics. So if that’s the type of thing you want to learn, I recommend taking a look when the book comes out in August.
(Image credit: Revise_D)
I based this article on a review copy of Ultralearning, which was provided by the author.
I’m writing about discrete math and competitive programming this year. For an introduction, see A Project for 2019. To read the whole series, see my Discrete Math category page.