Did you know that if you pick any three numbers in a single row, column, or diagonal on a dial pad, the sum of all the numbers is divisible by 3?
This is one of my favorite party tricks to make people think I’m a math genius.
You see, I was in an advanced math class through most of my elementary and high school years. Because everyone else in my advanced math class was keeping up and I wasn’t, I thought that they were all math gods… Except for me.
Through the years, though, I learned one important truth:
There’s no such thing as “math people,” because everyone has the potential to be good at math if they put in the work.
In this post, I’ll show you how to apply deliberate practice to build your math toolbox and get higher grades — even if you never thought you could.
(I’ll only cover how to study math in this post. If you want to learn how to take notes for math faster, check out this one too.)
There’s No Such Thing As Math People
In the book Peak: Secrets from the New Science of Expertise, psychologist Anders Ericsson wrote about how the ones who became the best had put in more work than anyone else, regardless of the field — whether it’s music, ballet, or chess.
Ericsson used chess grandmaster Bobby Fischer as an example:
Even Bobby Fischer, who at the time was the youngest person ever to become a grandmaster and whom many consider to have been the greatest chess player in history, studied chess for nine years before he reached grandmaster level. (96)
Despite all of the research that shows mathematicians are made and not born, I don’t blame you for thinking you’re simply bad at math, because math is friggin’ hard.
And if you slack on just one fundamental topic, your knowledge will collapse in the face of teeth-grinding all-nighters over problem sets and a palpitating heart on the big exam.
So the money question is this: How do you get better at math?
In engineering professor Barbara Oakley’s book A Mind for Numbers, she tells us the #1 way you won’t get better at math: rereading.
“When we read material over and over, the material becomes familiar and fluent, meaning it is easy for our minds to process. We then think that this easy processing is a sign that we have learned something well, even though we have not.” (xvii)
Instead, we need to master progressively harder math problems by employing a strategy Ericsson calls deliberate practice: “purposeful practice that knows where it is going and how to get there.”
In A Mind for Numbers, Oakley calls this information chunking: knowing not just how a formula or process progresses, but also when to apply it.
This is like having a set of tools in a toolbox and knowing when to use them. (For example, you won’t use a hammer to swat a fly off of someone’s head. Or use a sledgehammer to crack a nut.)
How to Get Better at Math: Chunking
“If you can’t solve a problem, then there is an easier problem you can solve: find it.”
— George Polva, How to Solve It
In her book, Oakley talks about the two modes of thinking you need to embed these chunks of information in your brain:
- Focused-mode — Approaching problems directly and analytically, like a laser pointer
- Diffused-mode — Relaxing and letting our subconscious work on the problem and find a big-picture solution, like a flashlight.
Contrary to what you may think, the diffused-mode — not the focused mode — is the one that causes lightbulb “Eureka!” moments.
Think of focused mode as laying the knowledge chunks to build your math house, and diffused-mode as letting the cement between the chunks dry so you can build on top of them.
To build strong math chunks in your brain by learning to which mode to use, you need to…
Step 1: Focus on the Information
“Diffused-mode insights often flow from preliminary thinking that’s been done in the focused mode.” – Barbara Oakley
There’s no getting around it — before your diffused-mode can help you get creative insights, you still have to put in the initial grind. This might mean locking yourself in your room, pencil in hand in order to:
- Isolate the concepts in the problem that you have the most trouble with and practicing again and again
- Work with similar problems that use smaller numbers to zero in on the concepts first
- Search for YouTube videos and explainer articles if your book and your notes aren’t enough
Going to office hours and asking your professor for help will get you on the right path.
Ericsson says, “A good math teacher, for instance, will look at more than the answer to a problem; he’ll look at exactly how the student got the answer as a way of understanding the mental representations the student was using. I needed, he’ll offer advice on how to think more effectively about the problem.”
Skilled teachers can often reword or re-frame concepts. If you’ve ever had an “OMG. That’s it?! I can’t believe I didn’t get it before,” moment after someone explains something to you, you know the feeling. Sometimes, all it takes is a slightly different word choice for a tricky concept to click into place.
Step 2: Understand the Main Idea of the Chunk
“Just understanding how a problem was solved does not necessarily create a chunk that you can easily call to mind later. Do not confuse the “aha!” of a breakthrough in understanding with solid expertise!”
– Barbara Oakley
To truly understand a concept, Oakley suggests using explanatory questioning and simple analogies as you go through problems, asking yourself, “How do I explain this to a 10-year-old?”
Asking questions like this allows you to deeply master a mathematical concept. To extend on my previous tool-toolbox analogy, this means that you’re not just picking up your new problem-solving tool and waving it around; you’re actually learning how to use it.
To fully understand a concept, though, you have to…
- Recall the solution from memory
- Apply spaced repetition
- Take breaks to let diffused-mode kick in whenever you get stuck
“Attempting to recall the material you are trying to learn—retrieval practice—is far more effective than simply rereading the material.”
– Barbara Oakley
Jeffrey Karpicke has written several papers on learning strategies for students. He says that when you read a paragraph multiple times, you feel like you understand because you can process it faster. In reality, though, this is an illusion of competence.
Instead, he found that attempting to retrieve information from memory is the most effective form of studying. It’s more effective, even, than multiple re-reads of the text or drawing concept maps. He advocated for methods like the Feynman Technique to help you recall and remember concepts deeply.
Focused-Mode and Spaced Repetition
Focusing on recalling information from memory for long periods of time can get exhausting.
Instead of spending long hours in the library, Oakley suggests starting your studying earlier in the semester so you can apply spaced repetition — shorter, more frequent study sessions that are spread out over weeks, instead of days.
Here are some specific tips that I’ve used myself…
- Make your math study session your first task of the day
- Make the session short — just one 25-minute Pomodoro session
- Have these sessions at least every other day for retention
And when you get stuck on a problem, it’s time to let your diffused-mode kick in.
Diffused-Mode and Taking Breaks
- The bed
- The bath
- The bus (not the beyond, unfortunately)
Oakley gives a bit more insight into how diffused-mode gets activated: “The key is to do something else until your brain is consciously free of any thought of the problem. Unless other tricks are brought into play, this generally takes several hours… Simply switch your focus to other things you need to do, and mix in a little relaxing break time.”
Step 3: Gain Context on When to Use the Chunk
To truly be a master of your tools, you need to learn how to adapt them to different situations.
To truly say that you’ve mastered a chunk of math, you need to see through beyond problems. You need to know which formula or process to use in a heartbeat.
To do that, Oakley recommends alternating different problem-solving techniques during your practice.
Don’t just solve problems of the same type that already know how to solve; challenge yourself with harder and harder problems.
Remember: Mastery is not about getting it right once. It’s about never getting it wrong!
Math Can Be Simple and Beautiful
“I don’t believe in the idea that there are a few peculiar people capable of understanding math, and the rest of the world is normal. Math is a human discovery, and it’s no more complicated than humans can understand.”
– Richard Feynman, Omni Magazine
Anyone can get better at math, but you need to put in the work to build a strong base of mathematical knowledge.
In order to do that you need to chunk your knowledge — know both how a formula works and when to use it — by employing both the focused and diffused-modes of your brain.
When you started putting in the right kind of work, you’ll enjoy feeling the gears in your brain click into place when you intuitively know how to solve the problem. That’s the simplicity of math: like the dial pad, if you get it right, everything works together perfectly.
Image Credits: Math on a blackboard