Note: I just moved this post from Improveism, my old website. Some links and pictures might be broken.
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Do these sound familiar?
You can’t solve a problem when focusing, and only when the exam is over, you realize the solution.
You experienced difficulties in Algebra or Trigonometry before, but now, it’s “common sense” for you.
It turns out that these experiences are not unusual.
In fact, these results can be reproduced and even be used intentionally.
After reading A Mind for Numbers: How to Excel in Math and Science by Barbara Oakley, Ph.D, I was entirely convinced that getting better at math is yet another scientific process.
The author, by the way, flunked mathematics all the way to high school but would improve her math and learning skills to later on get a Bachelor’s degree in Electrical Engineering, a Master’s in Electrical & Computer Engineering, and a Doctorate in Systems Engineering.
When I read that I was like “Damn.”
In her book, she also stated that “the higher I went, the better I did. By the time I reached my doctoral studies, I was breezing by with perfect grades.”
In this bite-sized book summary, I’ll share with you the things that I’ve learned from this book as well as how to apply them to your own learning toolbox.
How to Get Better at Math: What Science Says
When I started implementing these strategies during my 4th year in College, I was not only able to spend less time studying problems, but also enjoy it more.
I’ve always considered myself a “Solutions” person, but I’ve never enjoyed practicing Math until I knew exactly how to make practice work.
Actually, I’m still just as lazy in practicing Math, because I only spent a maximum of 2 hours
Just using ONE idea from this list INSTANTLY improves your problem-solving skills.
It’s absolutely insane.
Anyway, let’s move on to the list of strategies.
Build a Library of Chunks (& How to Form a Chunk)
Chunks are like chains of memory that activate together.
If you’ve ever driven a manual transmission, you’ve already encountered the power of chunking.
At first, slowing down from 60 to 20mph goes like this:
You think of slowing down, you think of pressing the brakes…(but wait, brakes first before clutch) and then you shift down to a lower gear, time the clutch release appropriately before pressing on the gas pedal—done!
Once you’ve done these enough times during driving, your thought process simplifies to:
An immense amount of information, through chunking, can be accessed in a single retrieval ONLY.
All you need now is to form chunks yourself for your Mathematics subject.
To put it simply, this is done through understanding why the connection of steps work, not just how an individual step works.
Step #1. Intense Focus
Our brain has a limited attention resource called the working memory, and on average, humans have 4 slots of those.
When we’re building a chunk, we’re using up all of those slots at the same time, but as you may have guessed, when a chunk is formed, it will only take 1 slot at a time.
However, when we’re not focusing/we’re distracted, some of these slots are taken up, and we lose our ability to perform chunks effectively.
One technique that you can use is the Pomodoro Technique, as I’ve always preached on this blog.
Step #2. Understand, not Memorize
Understanding a piece of information allows you to see how it “fits” in the “big picture”.
That’s why memorizing solutions is the biggest mistake that a student can make when learning Math.
I can’t think of a better analogy than the one stated in A Mind for Numbers:
Understanding allows you to form a piece of a jigsaw puzzle that will fit perfectly in the big picture. Memorizing is like creating a circular puzzle piece. It will NOT be able to connect with anything in the big picture.
By understanding how each step in a solution is connected with one another, your mind essentially “chains” these ideas together and allows you to access them all at the same time. And easily, too.
Step #3. Interleaved Practice
Now that you’ve formed a chunk, it means you now understand the “how’s” of those pieces of information.
It’s now time to understand the “when’s” of these information.
Regardless of the chunk, you want to know when and when not to use the chunks in your library. Otherwise, they become useless.
To get better at discriminating problems, you can do Interleaving. (more on this later)
Master the Focused and Diffuse Mode of Thinking
Two modes of thinking are outlined in the book A Mind For Numbers, the attentive Focused Mode Thinking, and the relaxed Diffuse Mode Thinking.
You access the Focused Mode when you’re, well, focusing intently on a problem or a solution you’re learning.
As we’ve discussed earlier, focus is needed when building chunks. But sometimes, trying harder can become the problem.
It’s because when you’re focused, your mind can only access the nearby chunks.
It’s like looking for a ball in a barn filled with 5 haystacks. Except that you started in the wrong haystack, say Haystack #2.
You can focus your attention all day long in that haystack, but you’re probably better off just casually, rather than intently, searching each haystack.
That’s when the other mode comes in.
Have you ever got stuck on a problem for HOURS then figured out the solution on your way home?
That’s the Diffuse Mode in action. Your mind worked in the background, just like a computer service.
Your mind casually looked for your missing ball in ALL haystacks and found some clue that it may be in Haystack #3 rather than #2.
Because of that, you were able to find the right solution.
When our minds are relaxed, we access other areas of the brain that are not particularly related to anything. You can call it the Creative Mode or the Artist Mode, whatever.
In order to get better at Math, we should learn how to intentionally navigate between two modes of thinking.
By doing so, we reap both benefits of the focused, analytical thinking, and the scattered, creative thinking.
Avoid Illusions of Competence through Active Recall
How do you know if you’ve actually understood something? Well, when you’ve already retrieved them.
If you can get it out of your brain, it’s probably there, don’t you think?
Retrieval Practice, or simply Active Recall is a study technique that I talk about a LOT (I mean a LOT) in this blog.
It’s insanely effective because it not only strengthens a memory trace but also lets you avoid what we call Illusions of Competence.
Illusions of Competence are those moments that you “felt” like you learned something but actually can’t recall them.
And then you blame a thing called mental block for not learning it in the first place.
It’s not your fault, though. It does feel like you’ve actually learned something.
Sometimes even, textbooks or instructors teach SO well that it leaves you with an “aha” feeling.
Sadly, it doesn’t mean that we automatically understood them ourselves, let alone do the solutions ourselves.
It’s astoundingly common for students with great professors.
This is why we have to periodically test ourselves by doing problem sets. And that’s common sense.
You can easily find them online, and if you’re short on time, I recommend you check out Schaum’s Outlines.
It’s unusually available for ANY subject.
Just search “schaums outlines [subject]” in Google.
“Practice, Practice, Practice” is misunderstood
It’s misunderstood for the fact that some students do the SAME exact problem that requires the SAME solution again and again, hoping that this somehow increases their mastery.
Unfortunately, Problem-solving isn’t an exception to the Law of Diminishing Returns.
Einstellung is what happens when you get really stuck into a problem because of the first solution that comes into your mind.
And that’s related to overlearning itself–it makes you even more prone to interference and Einstellung.
Sure, when you’re just learning the solution, it’s kind of good to get the “hang of it” by doing 2 or 3 problems.
But once you’re able to get the basic idea, you should immediately hop on to Interleave your Practice.
Interleaving is a study technique that requires mixing up different questions from different topics that require different solutions.
It makes practice a little bit harder, but 10X more fruitful.
By doing Interleaving, you’re getting FIVE benefits:
- You get better at discriminating the questions’ topics.
- You learn Mathematics as a WHOLE instead of “algebra” or “trigonometry” independently.
- You get to learn WHEN and WHEN NOT to use a certain chunk, or solution
- You’re getting the MOST out of your time because you’re strengthening your chunk library; and
- Your exam anxiety will literally go to ZERO because you’re already adapted to solving different, mixed up problems
As I’ve mentioned first thing on this list, it is an important step in building a library of chunks. And chunks make you a hundred times more efficient when learning.
Deliberate Practice: Achieve Mastery
If you want to become a master at something, you’ll more or less have to practice deliberately.
In the book, Peak, Anders Ericsson (the human expertise psychologist behind the 10,000 hour “rule”) states that Doctors who have 20 years of experience aren’t always better than those who have 5 years.
It’s deliberate practice that matters.
There’s no such thing as improvement when you’re just solving problems but not learning about your mistakes.
One way you can improve is by practicing to solve faster.
Use less steps in your solutions.
Practice mentally doing Algebraic simplifications or Differentiations rather than doing them on paper. *
Ask yourself if there’s another faster, easier approach.
You can also try doing more difficult problems.
Instead of solving problems that use one concept, why not try those that use 3 or more in a single problem?
By doing deliberate practice:
- You create better mental representations of solutions in your toolbox
- You FORCE your mind to adapt, and lastly,
- You spark some creative ideas that make your previous hurdles easier to jump over
As Anders Ericsson says, “Mental representations explain the difference between an expert and a novice.”
*Note: Usually, thinking on paper leads to fewer errors, but we still can’t dismiss the benefits of chunking some steps when solving.
Easy Way to Learn Math Formulas
There are some math topics that require you to memorize formulas or just some mathematical facts that make things easier.
Especially in Geometry, you’re obviously better off memorizing formulas rather than deriving one spontaneously during an exam.
Of course, to make this easy and effective for you guys, I’ll always recommend doing Active Recall.
One excellent app that I swear by is a spaced-repetition App called Anki.
Anki combines Interleaving, Spaced Repetition, and Active Recall in one system.
A lot of people use it in the Language Learning and Medical School world.
These guys memorize INSANE amounts of information (10k+ flashcards), and to give you guys a perspective, I was able to memorize only 3k flashcards over a 5-month period containing ALL the information I need for my Engineering Board Exams.
Not only do these guys memorize easily, but they also encode it to their long-term memory.
So what’s there to lose? There’s everything to GAIN.
Check out my Beginner Anki Guide if you’re interested: (Awesome! It got pinned on a medical school Anki subreddit!)
Or, if you’re already using Anki (that’s great!) there’s also a more advanced guide that covers some effective Anki settings and life hacks that you MUST know.
Now, a lot of you may ask some questions related to getting better at Math, so here are some of my answers to those.
What does being good at math mean?
Mathematics is not just about how fast you can accurately calculate something in your head. It’s a common misconception that students have. Getting good at math also means that you can use previously learned concepts to make rational, sequential, and analytical approaches to gain new insights for solving a problem.
Can I become better at math?
People may think that they cannot get better at math because they’re too old or they may not be “born with it”. But problem-solving is just another skill that can be learned and improved. Everyone can become better at math by using better learning strategies and developing the right thought patterns.
Why do students fail mathematics?
Students fail mathematics because they have ineffective strategies and problem-solving approaches. It’s a common misconception that only “Practice, Practice, Practice” is the way to go. Sometimes they memorize solutions. Some of the better strategies are Chunking and Interleaving.
Bottom Line: How to Get Better at Math
Dr. Barbara Oakley gives us insanely effective strategies that she used herself to become successful at Math in her book, A Mind for Numbers: How to Excel in Math and Science (Even if You’ve Flunked Algebra). [afflink]
Trivia: She stated that she started doing this at age 27.
Now THAT’S what I call humbling.
Which tip will you try out first?
Which did you already have in your math toolbox?
Let me know in the comments and as always, thank you for reading!