Transcription
I've spent over 7500 hours learning math. And if I could start over, I'd study almost completely differently because nobody teaches us how to learn, especially when it comes to math.
But actually, if you just get these three things right, you will master math in half the time. If we haven't met, hi, my name's Han. I went to Columbia University. I majored in math and operations research where I graduated with straight A's.
So, even though I think I'm pretty good at math and enjoyed learning math now, when I was in high school, I struggled a lot. I think just at one point the material suddenly got more complex and the pace of teaching sped up so much that I felt completely behind. I remember sitting in the class and staring at the board and just basically have no idea what's going on. Nothing quite makes sense. I felt very overwhelmed and don't know where to start.
When I asked for help, I'd remember the frustration on my teacher's face because I was just so confused and I didn't even know what are the right questions to ask. I felt like I was working so hard and nothing clicked. It didn't feel fair to me that my teacher was relatively bad at teaching and very disorganized and the way they teach just doesn't work for me.
Eventually, at one point, I was just so sick of that and I decided, I'm no longer going to rely on anyone else. I'm going to figure out how to learn math myself because the biggest thing holding people back isn't talent. It's that they're learning math the wrong way.
And after lots of trials and errors, thousands of hours of learning math myself. And later on, I became a tutor and teaching other students, now I understand exactly the reasons why I wasn't getting it. What was holding me back and how I could actually improve. Along the way, I also discovered many of the strategies that I figured out myself are actually backed by research.
So, in this video, I'm going to share with you all the most important strategies that I learned so you can master math in half the time. So, here is the plan. Our goal is to become genuinely good at math. Not just barely passing the exam, but understand concepts deeply, solve problems easily, and learn new materials much faster.
To get there, we only need to overcome three obstacles. First, understand. You know, sometimes new concepts just doesn't click for you and you feel like you just don't get them.
Second, apply. Sometimes we listen to the lectures, we read the textbooks, and then we feel like we understand it, but then once we start doing the questions, we have no idea how to do that. We basically don't know how to use what we learned.
Third, optimize. So, sometimes you eventually figure out a math question and then you realize it's been two hours. Learning feels painfully slow. You're probably overwhelmed by the amount of materials that you need to get through and it feel like constantly feeling behind and you also forget a lot of things you learned the last month or last semester.
So, that's exactly how this video is organized and that's our road map. First, I will show you how to make math actually make sense. Then, we're going to turn that understanding into confidently and easily solving problems. And finally, optimize the way you learn so you will learn faster, remember more, and make them stick. And each section is built around the three simple evidence-based strategies. So, there are nine steps total together.
Before we jump in, I want to quickly mention something that is super exciting. If you enjoy this video and want a place to actually ask me questions directly, stay accountable, and learn aside other people that are also super serious about getting better at math. I created a community. If you'd like to check it out, there's the link in the description. I will also tell you more about it at the end of the video. For now, let's get into the first strategy.
Part one, understand.
1.1, read the definition. If you ever ask yourself this question, why do some people just seem to get it and I just don't seem to get it? I think there's a lots of mystery around how to understand math and I think lots of people, lots of teacher are actually teaching math the wrong way.
Let me introduce you to the three levels of mathematical thinking which is developed by math education researcher Tommy Drifus. And once I learned this everything like made sense. Essentially how we naturally learn math is through these three levels.
Level one is trial and error which is the most basic level like we just try things and see if it works. For example, you see 2 + 4 equals to 6 and 4 + 6 equals to 10. So you feel like evens plus evens are always even numbers or if you're in a test and you have a multiple choice question. So you just plugged in each one of those numbers from the options into the original question and then you just want to check which one is correct. So you're learning math through trial and error which means your conclusion is not always correct. It's just some random data that you kind of collected.
And level two is pattern recognization. So instead of just testing random cases, you notice a pattern and you generalize it. This is also a super useful skill to have. You know, lots of those IQ questions, they are pattern recognization questions. And continuing to the example of even numbers plus even numbers is always equal to even numbers. Instead of just trying two cases, now you've tried maybe hundreds of examples and then you feel like you just could not found a case that failed this statement. So you believed that's true or in exams instead of actually remember the formula you just found some patterns and plugged it in.
And level three is understanding and deductive reasoning. This is how you know exactly why something is true and how something theorems came from. The even plus even is always even example. You can actually explain it. Use a normal language term. When you're combining two even numbers, you're combining pairs with pairs and you still have no leftovers. So the total is still just bunch of little pairs and that's still even. That's kind of a intuitive explanation and that's how normal people understand things. However, that's actually deductive reasoning. If you wanted the formal version, you essentially had to understand the law of addition. You have to understand the definition of even number. It's not just as easy and as straightforward as, oh, you add the little pairs together, you still get pairs.
So when a teacher is explaining it to you in a way that you're supposed to understand it, it's the hardest level. They usually drop straight into level three before you get to play with level one and level two. For example, calculus. And when we were learning about integral, the first thing we usually learned is the definition. You know, we learn about the limit. They would learn about as n goes to infinity how the sum under the curve and all that sort of a thing and kind of expect us to understand it. However, that's not how normal brain works.
For mathematicians, the natural progression for them to develop new theorems, new rules, it's still through trials and error and then pattern recognization. They saw something and then they feel like that may be something interesting there and then they make a hypothesis and then they eventually try to prove it right.
So here is how we actually are supposed to learn. It's like completely backwards from what we usually are being taught. So usually the textbooks and the lectures give you the definition first and then examples. But our brain works in the opposite order. The examples are what actually makes the definitions click. So from now on we were learning new concepts. Yes, we probably still need to read the definition first but only to get oriented and get a rough idea and then just moving on to the examples and using the examples and actual questions to learn the definition. Don't sit there and try to force yourself to just get it cuz you're not going to perfectly understand everything.
1.2 understand it through the examples. Now moving on to the examples. There's a research actually conducted comparing two groups learning algebra. One group learned by worked examples. The other group learned by grinding through problems from scratch. And the worked examples group later solved the test problems in about half the time with about fifth of the errors. This means that when we're learning a new concept, learn from examples and understand it first before you jump into solving problems yourself.
Because our working memory which is a little mental workbench in our brain help us to actually think and solve problems is formed through existing things that we import in our brain in the past. And if you're learning something new for the first time and you're trying to solve a new problem from scratch, you don't have that working memory pathway formed yet. You basically don't know what you're doing and your brain is like panic and dead.
Of course, you don't want to rely on examples forever because once you already fully understand the examples, you want to actually apply those knowledge into independent problem solvings instead of worked through different examples over and over again. That can actually slow you down. So, here's what to do for a brand new topic. You want to read through the examples, learn step by step how it worked and then you want to do those questions yourself to prove that you actually understand the example and see those skills transferred. Once those problem fails easy, you want to move on to next type of questions like less similar problems.
1.3 found the gaps and fill them. The Feynman technique, which means that take a concepts or an example that you just learned, explain it in plain words, ideally feeling like you're explaining to someone that's like a young kid. That's how you can actually tell if you understand because if you can explain the whole thing very clearly, step by step out loud, you've actually got it instead of just thinking about, oh, I think I understand this in your brain. And the second you get stuck where you feel like there is a little bit wishy-washy feeling that you don't know how exactly that works, that's a fuzzy spot that you actually want to understand and there might be a knowledge gap.
There are also research behind this too. It's called the self-explanation effect. The studies found that the students who constantly explaining things to themselves why each step works learn way more from the same material and the better performing students ask themselves why and try to give an explanation to those wise more than three times than the students who perform not as good. So that's how you actually understand an example or a question which is by explaining each step and ask why behind each one.
And when you found a spot that you don't fully understand and you feel like kind of doesn't make sense to you and when you see one line jump to the next line and you just like super confused and feel like that makes no sense. That's because there's a knowledge gap. There is a missing piece that you forgot or you never learned. And those are usually the moment we feel like we're stupid, why we're not getting this, but it's solely because there's a knowledge that we missed. And that's how you can go look into your textbooks, Google it, watch YouTube videos, Khan Academy to figure out that very specific knowledge and very specific question. Instead of asking the entire example or the entire question, you just ask why this step become this step. So you're going to patch that little knowledge gap and then you're going to come back to your example and then continue to understand and explain the rest of the steps.
And if you're at school, you have someone else that able to help you, it's far more effective to ask a very specific question instead of just point at a question or point at a chapter and say, "I don't understand this entire thing." It's going to be really hard for them to help you and it's not going to do anything good for you either because you want to actively learn new things yourself. You want to have the ability to identify gaps and patch that gap. One of the most quoted lines in all of research and education, the most important single factor influencing learning is what the learner already knows. Math is cumulative. A missing prerequisite blocks everything that built on it. So if you feel like you're falling behind, you don't get things as quickly and you don't learn things as effectively as other people is most likely because of you don't know as much as other people.
Part two, apply.
The big idea behind this part is that math is something you do, not something you watch. I think lots of people trying to learn math and trying to read the math textbook the same way they would read a regular book. They just read it and try to understand it. That's not how math works. You really have to be hands-on and actively use your brain to engage in the learning process. You wouldn't say you can drive a car just by watching other people drive and watch tutorials online. You actually have to get in the car and practice a lot.
2.1 practice questions. So if we're learning a new concept and after we understood the examples, we should immediately start working and solving problems. And I think the ratio should be like 20% to 80% like 20% of time. Maybe read notes, read textbooks, work through examples. And 80% of the times should be practicing new problems.
Another common mistake lots of people do is when they're practicing questions, they just sit there trying to work through the problems all by themselves and they just try as hard as they can. And if they get stuck, they just feel stuck. And when they eventually can't, they feel so defeated. They feel like they're not smart enough. You want to solve the problems yourself at first. You would give it a genuine try. And once you got stuck, you just get the solution out and you study the solution and you fully understand it. Then you set the solution aside and then you redo the question yourself independently again completely on your own. And repeat this cycle until you can get this question right completely on your own.
So I would really against people sit there and just suffer for 45 minutes trying to solve a single problem. I think that's not the most effective way to learn especially you may be completely off the track. You don't even know what the question is asking about. And the non-negotiable part is that you should redo the whole question yourself independently again until you get it right. The way I like to think of it is say that you're not feeling very good and then you don't know why. So you go to the doctor and reading the solution is the diagnosis. You want to understand what went wrong. But the diagnosis is not the treatment or the cure. Redoing the problem and solve it independently is the follow-up appointment to see if you actually are treated and then confirms that you actually are better now. You're no longer sick. So, if you only ever read the solutions and just like nod and feel like you got it, that's like you only got the diagnosis, but you never actually follow through with the treatment and fix your disease. So, you probably still don't know how to do the question and you will still get it wrong on the test.
2.2 to active recall and spaced repetition. So, if you want to memorize things better, you probably heard of the terms active recall and spaced repetition. These are some of the most well researched strategies on effective learning. Active recall just means you're actively using your brain and retrieve the information from your brain instead of passively like reading or watching. Think about like quiz or answer questions. Spaced repetition basically just means revisiting the same information over over and then you spread it out over the days instead of drawing something 10 times on the same day. It's far more effective if you space it out like you visit the second day and then the fourth day and then one week later and then 30 days later. That's what transfer your short-term memory into long-term memory.
And I think a lot of people online talk about these strategies specifically in the setting of memorizing a lot of information like language learning or biology facts, anatomy or like history facts, but not really. Lots of people talk about how to use it to learn math. My favorite way to use these two powerful techniques is that a mistake notebook. Basically, you collect all the questions you got wrong, either something from your homework or something from your exams, and then you pull it all together and organize it in one single notebook. When I was in high school, I actually like to cut out all the questions from the papers and then stick it onto one single notebook. And then in college, I like to use iPad and then so I can just screenshot everything and then copy and paste in one single notebook on my iPad. And sometimes if I felt like that's too much work, I don't have a lot of time, I just like to put a little sticky note and mark where is it. I will mark which one I did wrong and I'll write the correct answer next to it as well.
So the idea of collecting all those mistakes and the questions you did wrong is not for you to feel bad. It's actually the opposite. You should feel really good about this notebook because this is like the golden diamonds like a gold mines for you because it's personalized to you on exactly what you don't understand very well. You identified those knowledge gaps and you have some logic flaws. It's not like common mistakes questions you found online which are I think are probably still very helpful and you want to periodically came back to revisit those questions and you want to redo that and just doing the questions yourself independently is active recall and if you're doing it periodically it's space repetition.
So here's what to do after each study session. Either you finished some homeworks or you just finished the exam and you got your answers. Log the questions you did wrong. Have the correct answer next to it. A few days later, redo those questions from a blank page. Do not peak the answers. Just really try to engage your brain and treat it as like a mini test again. And do the ones that you keep getting wrong sooner and the ones you feel like really comfortable and you got them easily done. Those ones you can come back later. That's the power of spaced repetition. I will talk about later about memorization. Right now I'm talking about notch memorization is more about engaging your brain and strengthen that memory of problem solving skills.
2.3 intuition and pattern recognization. Okay. Now let's build intuition. Sometimes you feel like there are people that are really good at math. They immediately see a question and they just understand what it's about. You see those people in the movies and shows and they're like just geniuses. But I would like to say that intuition is just mostly pattern recognization. It's level two thinking. Remember we talked about earlier is mostly made through practice.
There's a study that basically proves this. They looked at specifically chess masters versus beginners. So researchers showed both groups a real chess board from a real game just for a few seconds and then they took it away and asked them to rebuild it. The masters absolutely destroyed the beginners about three times better. Based just on that you may think oh the masters they must just have a better you know visual memories than the beginners that's why they're better. Except then the researchers showed both groups a random board. The pieces doesn't have any real game logic. Then on the random board, the masters had no advantage at all compared to the beginners. They were just as bad as the beginners. So the conclusion of that study is their seemed genius intuition wasn't just raw memory and ability. It was thousand of stored patterns from real games from all the practices they did.
So with that being said, anyone can build intuition, especially you. So think of math as a language. There are lots of theorems and there are lots of definitions that you kind of need to understand and you need to learn how to use them and you collect all of those information and you see them so many times that they're just tiny tools in your toolbox. So next time you see a question, you can just like get it out and then just throw it at the problem. And for a complex problem, you understand how to break it down into smaller little problems and then identify how to use your tools for each one of those.
And the way to build a better intuition is by mixing up your practice. The studies shows that mixing problem types feels like harder where you're learning it and lots of student feel like they are not as good or like as confident but they actually leads to better performance at tests. And the more type of questions you see and your practice, the more angle you understand how to solve one single problem, the better your working memory will function and the better you master the math as a language. Because all the knowledge is like a giant connected nets. You know how to exactly apply each one of those knowledge to each other. So what you want to do is once you did some basic questions and once you get really comfortable with a type of question you don't want to keep practicing it even though that may feels very good and then you feel like oh I finally understand this you really want to challenge yourself to different type of questions and the more mistakes the more challenging those questions become the better you're going to do.
Part three, optimize.
Now you understand the math concepts and you actually know how to do the questions. You can figure things out. However, you're just learning it so slowly. If you're in a school setting, you might feel like you're behind or you have a deadline that you want to meet. So, you don't get to all the topics that you actually need where you never had the time to reach different types of questions or harder questions. So here are some three powerful tools you can start using immediately and see a difference.
3.1 the 8020 rule. So the 8020 rule means that roughly 80% of your outcome come from only 20% of the input. And this is true in so many different ways. For example, if you're learning a new language, just getting the most common words and phrases can get you a long way. You can probably just speak in that language to express whatever you need. You don't actually need advanced vocab or structure phrases to just travel. And also it's true for exams. Roughly 80% of the exam scores are all about the basics. 20% of scores about something super hard and complex. Those actually have covered a lot of space and then you need to spend way much effort to get those 20% points.
Traditionally the way we're taught in school is we cover a topic and then we learn everything about this topic before we move on to the next chapter. And then the way I like to learn is for example we have 10 chapters. I will learn all the little basics about all these 10 chapters and then later I will get into the details and the harder examples and the harder questions of those each chapters. And then the way I like to do it is I sort things into three tiers. The first tier is the most important stuff, the core concepts that everything else is built on. The second tier is the secondary stuff that connect to the core and add depth. And the third tier is the nice to knows, you know, the extra interesting little details or like a super edge case. This is the stuff that you will only touch when you had extra time.
So, what I like to do is I want to learn and nail the most important tier and then the secondary to help me build a deeper understanding and then save the nice to knows last. And because of your studying in this order, once you get the core concepts down, you essentially can build a mental map and you connect all of like these 10 chapters of things together and you understand the bigger picture later. You want to dig deep into like a specific topic. Those details have things to attach on. So it'll also help you to learn faster.
3.2 memorize the basics. Memorization and math. It's kind of like a touchy subject. I think people get really weirded out when they're talking about memorizing the math. And I think we need to identify two different type of memorization. So number one, you don't have to memorize anything if you don't want to. Let me explain this. I think lots of people get bad experience from school is because the teacher doesn't really explain how things works and they will just tell you just memorize that you wouldn't understand it just memorize it and then you just apply that equation in your exams that's why you feel math is very dull and you don't get the engaging feelings and why I say you don't have to memorize anything is because if you truly understand math you can rebuild everything from scratch understanding meaning you understand the entire path. You can teach yourself from like the first step all the way to step five and you understand why step one becomes step two and why step two becomes step three and then like between each you understand the perfect logic. Memorize without understanding would mean that oh given step one here is step five. And then that can be frustrating and it's going to be really hard for you to apply those knowledge. If they like just twist something like if the questions asking about how to get to step four, then you wouldn't know how to do that.
But on the other side, if your goal is to become good at math and build that intuition and learn things faster, you should remember all the basic stuff. For example, if from like step one to step five, this whole thing is like a equation and that come up so many times given step one, how do you get to step five instead of every single time you have to work through that in like all the five steps logic, you should just memorize, oh, step one, then step five. Then it will save you so much time, especially if the questions are already complex enough. For example, you should absolutely remember what 5 * 6 equals to or in calculus, you should definitely remember what's the derivative of e x, right? Those type of thing because they will keep showing up and then they will be essential for you to sped up your problem solving time and those things are usually like the equations and the formulas.
So, what I like to do is have a piece of paper and then write down all of those basics. Essentially, it's like a cheat sheet and just every rule, every formulas, everything that you're learning and is relevant. I just like to have it while I'm doing questions. So, if I forget something, I just immediately look at the sheet instead of I have to go search it up in textbooks or online. That saves a lot of time. And then very so often like especially before exams I will just do active recall on this entire cheat sheet like the same way you would have for flashcards I would just try to write down everything that I can remember all the formulas and if I got it wrong I just do it again that way I just memorize those things in my heart so I don't have to spend the time and energy to worry about am I deriveting this right or wrong like we all talked about always ask why and understand things first and never just like wrote memorize something cuz I think that really took the fun out of math once you understand them then memorize some of the most basic things so it help you speed up in your learning and problem solving process.
3.3 focus. Focusing is probably one of the most helpful thing for you to learn faster and study faster and be more productive in general. So, I just want to give you a couple tips on how to focus better. So, tip number one is to put your phone in a different room. This is such a basic tip, I know, but I I think it would be so effective. One of the biggest distractors in my life and affecting my productivity is the damn phone. And there is actually research shows that even if that your phone is just sitting next to you and you're not using it, it's still spending your energy because you are fighting with the urge to touch it. So just by putting it out of the side, put it in a different room, leave it at home, that type of thing will give you a little bit more brain power back and just help you be more focused.
And tip number two is to work in sprints. I think some people would consider this is a Pomodoro technique which is the traditional one is set a timer for 25 minutes and then take a 5 minutes break and then set another 25 minutes timer for studying and then take another 5 minutes break instead of studying in a long period of time is like better to focus intensely and then take a break and then focus again. And in terms of math, the way I like to do is that I will set a timer and I'll say I will work on these questions like it can be four questions, five questions or like your homework problem sets and then I would take a break and then go back to more questions. So instead of just set a timer and just start working, I would like to have a goal and an amount of numbers of questions to get through. So that way I have that urgency and basically simulating on how to practice questions in like actual exam settings. It also doesn't interrupt you to the middle of questions. I really don't like getting interrupted. If I'm thinking in the middle of a a question and then the timer set off then I kind of distracted me a little bit.
And tip number three is one problem rule. So while you're solving a problem, only focusing on that problem. Don't think about what you're going to do next. Don't half read a question and then move on to the next question and then try to go back and forth again. In math, multitasking is really a destroyer of how productive you can be. So really focus on one problem at a time. Finish this entire thing and then move on to the next one.
Tip four is you can have a distraction pad. Ideally, you remove your phone and don't connect to the Wi-Fi on your computer if you're using your computer or iPad. And every time you have a a random thought that pop up, like think about like text mom back or replying to a specific email and then you really want to get to that thing or you feel like oh, you forget about it. Have a piece of paper or like a little sticky note on the side and whenever you have those thoughts come up write it down and just so you will know that you will get to it later and then right now you are just focusing on the problem you're working on. You know focus is something that you can also actively train to get better at. Maybe at the beginning it feels harder to focus a long period of time or focus on a math problem but the more you do this the easier it will become.
Finally, besides all the strategies I shared with you, I want to talk a little bit about mindset because mass anxiety is the highest form of academia anxiety among all the subjects. I think by the nature of it is very very frustrating if you don't understand a problem. Hopefully with all the strategies and the explanations on why we feel like stuck, why we feel like we couldn't solve a problem, you already feel better from now on. We really have to reframe our thinking. Next time when you stuck at a problem, instead of thinking, oh, I'm just bad at math, think about, oh, I'm just missing a specific step or knowledge gap. I will figure it out. I will learn it.
And I think a math is very tied to the idea of being smart. There are studies shows that your teachers, your math teachers, your parents all struggles with mass anxiety and that mask anxiety can subtly transfer and pass on to you. For example, if you ask your math teacher a question that they may not know the answer immediately and then they don't want to look bad. So they will start rambling and then they will say something like, "Oh, it just did this." And then they try to like brush you off instead of actually explaining it to you because they don't want to look bad. So just remember that everyone struggled with math anxiety. I still do. The first year when I was at Columbia University, the nights before my some biggest hardest exams, I was just like crying in the library because there was so much stuff that I need to get through and that I feel like I don't have enough time to get to them. If I don't understand something, I feel so defeated. So, it's not like it would ever go away. It potentially will still be there, but just you have to remind yourself and be patient with yourself when you're stuck at a question and give yourself full permission to look at the answers and studied in the way that we talked about earlier. Think about playing video games like Mario. You didn't pass a level. Instead of just crying and feel like you're so bad at this, you just learn something about that level and you do gonna do it again and you will eventually get through it. And those worries and anxieties can actually block you from performing at your full capacity cuz your brain is not going to be functioning at its best level if you're constantly worrying instead of in a relaxed state.
And one last thing is that this is quite personal. As a girl and I went to a high school that is very famous for its STEM subjects. I was told by so many people that I should have not pursued a STEM track because girls are bad at math or not as good. We have data and stats to show that's not true. So, don't let people tell you that you're bad math. Don't let people to convince you that you should have choose something else just because you're not meant for it. I think as we discussed, if you have the right strategies and you spend the time and you actually practice and you trained, you will get better at it and anyone will get better at it.
So, if you made it all the way to the end of the video, I know you're someone that are super committed and willing to spend the time and effort to get better at math. So, just give yourself a tap on the back. Good job. But I also know that watching a video is the easy part. The difficult part is actually applying all these and showing up next week and the week after that. Is staying motivated while life gets busy. Is knowing what to do when you get stuck. And having people who understand your goals. That's why I created Math Studio. Learning is just easier and more fun when you're not doing it alone. So, I want a place that we keep each other accountable like celebrate all the small wins, track your progress, hop on coaching calls, and build better study habits together. I will be there to give you feedback, help you think through challenges and support you as you build the skills to become a confident, independent math master. If that sounds like something you would like to be part of, I would love to have you join us. The link is in the description and I hope to see you inside. Otherwise, I will see you in the next video. I hope you have a wonderful day. Bye-bye.