Math builds on itself, so trying to learn a math topic without first mastering its prerequisites is just asking for trouble. A good place to learn prerequisites is Khan Academy. Although it’s missing many advanced math topics, it covers the basics well. And the way it organizes practice problems ensures that you regularly get tested on the concepts you need the most practice with.
Practice Problems
Back in the day, Khan Academy was just a collection of YouTube videos. It still has plenty of those, and they’re useful for learning new concepts or refreshing your memory on old ones. It’s convenient that they’re integrated into the Khan Academy user interface, so it just takes a click to find a video related to the topic you’re studying. But Khan Academy is far from the only place to find free math videos.
Where Khan Academy distinguishes itself is in its practice problems. The way to learn math is to solve math problems, and Khan Academy provides unlimited problems on each topic it covers. If you’re having trouble with a topic, you can keep drilling practice problems until you have the concept down.
For students who are just learning math, Khan Academy provides a guided tour from $1 + 1 = 2$ through calculus and differential equations. But even for people who have already learned all the math topics Khan Academy covers, its practice problems can be a useful tool. They can pinpoint areas of weakness and let you strengthen them in preparation for future math studies or some other goal. It’s much more effective than leafing through a textbook chapter and saying, “I probably know that.”
The World of Math
Khan Academy uses the term mission to describe the user interface they created to guide students through math exercises and videos. The top-level mission, called The World of Math, contains all the other missions.
Missions contain skills, math topics you can practice and master. Skills are part of units, which are part of courses. For example, Limits of composite functions is a skill in the Limits and continuity unit, which is part of the AP®︎ Calculus AB course.
When you’re practicing a mission, you don’t have to worry about units and courses. You can just navigate your way through the skills. Last year, Khan Academy launched a new course and unit mastery system that may eventually replace the mission-based mastery system. For my purposes, I’m focusing on the mission approach. It’s simpler to navigate and I think it’s more appropriate for experienced students doing targeted learning: dipping into specific skills as needed, rather than progressing through courses in order.
The World of Math mission currently has 1498 skills. Khan Academy periodically adds more. For each skill, you get one of five ratings based on your performance on problems related to that skill: Not Started, Practiced, Level One, Level Two, and Mastered.
Using the World of Math mission, you can solve problems for these skills in two ways: 1) Practice, and 2) Mastery Challenge. Your can start a practice or challenge session from your World of Math dashboard.
Your dashboard will usually have a set of practice suggestions in the skills up next for you section. If you don’t like the suggestions, you can add to that section by clicking a square in the skill breakdown grid. You can then click the Practice button to practice a set of problems (five or so) for that skill. If you get enough problems correct, you’ll increase your skill level to Practiced.
In a browser console, you can also run JQuery code like $('.progress-cell')[658].click(), which lets you “click” a skill by number. Skill 658 is limits of composite functions, so running that code will add that skill to your practice list.
Also from your dashboard, you can click the Start button in the Mastery Challenge section. This is how you start a mastery challenge to access the higher skill levels. In a mastery challenge, as in a practice session, you get a few problems to solve. But in a mastery challenge, each problem is from a different skill. If you get a problem right, you increase your skill level in that skill. So with one mastery challenge, you can increase your level in several skills.
When I use Khan Academy, I usually use mastery challenges. Since a mastery challenge uses problems chosen by the system with no input from me, I can’t just pick familiar topics. It forces me to prove I still know the skills I previously practiced or mastered.
Here’s some trivia based on my experience with mastery challenges:
- If you answer fast enough, you can advance more than one level — e.g., you can go from Practiced to Level Two, or even Practiced to Mastered.
- If you get a challenge question wrong, your level goes down by one.
- A mastery challenge can give you a problem from a skill you never practiced (a skill at Needs Practice level). This allows you to master skills without using the Practice process, which saves you a few problems if you already know a skill.
- It can also give you a problem from a skill you already mastered, to make sure you haven’t forgotten it.
- The mastery challenge summary page doesn’t always accurately show the level changes that occurred during the challenge. For example, it may show that no level change happened, when you know it has (the new level is visible in the dashboard view). I assume this is a bug that no one has gotten around to fixing because the dev team is busy working on the new mastery system.
With some discipline, you can master all the World of Math skills, as some people have. (It’s a lot easier than finishing the Project Euler problems). Next week, I’ll discuss ideas for continuing to use the World of Math mission after reaching 100% mastery.
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.