How to Excel at Math and Science

Summary

This video explores the book 'A Mind for Numbers,' which offers learning techniques applicable across various subjects, including math, science, and even chess. It emphasizes the importance of alternating between focused and diffuse modes of thinking for deep learning and overcoming the Einstellung effect. The script suggests daily learning, active recall, and tackling problems methodically. It also addresses common issues like procrastination with strategies like the Pomodoro Technique and mental contrasting. The key takeaway is the power of patient persistence and a positive mindset in mastering math and science.

Takeaways

• 📚 The book 'A Mind for Numbers' offers learning techniques applicable to various subjects, not just math and science.
• 🧠 Two essential modes of thinking for deep learning are 'focused mode' and 'diffuse mode', each serving different cognitive functions.
• 🔍 The 'Einstellung effect' is a cognitive bias where having a preconceived idea can hinder finding better solutions to problems.
• 🤔 Diffuse mode thinking allows the brain to relax, enabling it to make broader connections and potentially discover new solutions.
• 🎓 Alternating between focused and diffuse mode learning is crucial for improving skills and understanding complex subjects.
• 📈 Daily learning and attempting to recall material are more effective than cramming for deep learning and retention.
• 📝 When tackling problems, understanding the fundamentals is key before attempting to solve them independently.
• 🚫 Avoid the illusion of learning by not just reading solutions but ensuring comprehension and independent problem-solving.
• 🔄 Mixing up problem sections can enhance learning by simulating the unpredictability of real-world problem-solving.
• ⏱ The Pomodoro Technique can help with procrastination by setting focused work intervals followed by short breaks.
• 🌟 Persistence and a positive mindset are more important for success in math and science than natural talent.

Q & A

What is the main theme of the book 'A Mind for Numbers'?

The main theme of 'A Mind for Numbers' is to teach learning techniques that can be applied to various subjects, including math, science, and even games like chess, to improve efficiency and performance at any level.

What are the two modes of thinking mentioned in the book?

The two modes of thinking mentioned are focused mode and diffuse mode. Focused mode is when you are highly concentrated on a task, while diffuse mode is when your mind is relaxed and wandering, allowing for subconscious problem-solving.

What is the Einstellung effect and how does it relate to learning?

The Einstellung effect refers to the phenomenon where having an already formed idea in your head can prevent you from finding a better solution to a problem. It's related to learning as it can hinder the ability to see alternative approaches or methods.

How does diffuse mode thinking help in problem-solving?

Diffuse mode thinking helps in problem-solving by allowing the brain to relax and make connections between different parts of the brain, enabling a broader perspective and potentially leading to the discovery of better solutions.

What is an example of how a chess grandmaster uses diffuse mode thinking?

An example given is of a chess grandmaster who gets up from his seat and looks away from the game during a match. This short break allows him to distract himself and let his mind wander, preventing a narrow way of thinking and possibly leading to new insights.

Can you explain the story of Henri Poincaré and how it illustrates the concept of diffuse mode thinking?

Henri Poincaré was a French mathematician who struggled with a difficult problem. He decided to take a vacation, and while boarding a bus, the solution suddenly came to him. This story illustrates how diffuse mode thinking can allow the brain to continue working on a problem in the background, leading to insights that might not come from focused attention alone.

What is the recommended study method for learning new material according to the book?

The recommended study method is to attempt to recall the material you have just learned instead of rereading it repeatedly. This involves reading new information, then closing the book and trying to recall what you've read, repeating this process for better retention.

Why is it not effective to look at the solution to a problem and then immediately move on?

Looking at the solution and moving on without fully understanding it creates an illusion of learning. To truly learn, you should work through a problem, understand each step, and then repeat the process on a different problem to reinforce the concept.

What is the Pomodoro Technique and how can it help with procrastination?

The Pomodoro Technique involves setting a timer for a short period, such as 25 minutes, to focus solely on work, followed by a short break. This technique can help with procrastination by creating a structured work routine and building the habit of focused work sessions.

Why is it suggested to mix up the sections of the book when doing problems?

Mixing up the sections ensures that you can apply the concepts from different parts of the book independently, preparing you for situations where you may not know the specific section a problem is from, thus enhancing your problem-solving skills.

What is the significance of persistence in learning math or science according to the book?

Persistence is highlighted as more important than intelligence in learning math or science. It suggests that with time and consistent effort, one can reshape their brain and optimize their learning and problem-solving abilities.

How does the author's personal story illustrate the power of mindset in learning?

The author's story shows that despite hating math and failing in high school, a shift in mindset and determination led her to excel in math and achieve advanced degrees in engineering. This demonstrates that mindset can significantly influence learning outcomes.

What is the significance of the four-minute mile story in the context of the book?

The four-minute mile story illustrates how a shift in collective mindset, from believing something is impossible to possible, can lead to breakthroughs and achievements that were previously thought unattainable.

Outlines

00:00

📚 Learning Techniques for Math and Science

The video discusses the book 'A Mind for Numbers' and its applicability to learning techniques across various subjects, including math, science, and even chess. It introduces two modes of thinking: focused mode for intense concentration and diffuse mode for subconscious problem-solving. The Einstellung effect is highlighted as a common barrier to finding better solutions. The video emphasizes the importance of alternating between focused and diffuse modes for deep learning and understanding, suggesting that taking breaks can lead to fresh insights and improved problem-solving.

05:02

🧠 Study Techniques and Overcoming Procrastination

This section delves into specific study techniques for math and science, advocating for spaced repetition and active recall rather than passive rereading. It advises against the illusion of learning by merely looking at solutions, urging viewers to tackle different problems to solidify understanding. The importance of mixing up problem sections to simulate test conditions is also discussed. To combat procrastination, the Pomodoro Technique and focusing on the process rather than the product are recommended. The video also suggests doing the most disliked tasks early in the day to build momentum and using mental contrasting to visualize goals as a motivation to overcome procrastination.

10:03

🚀 Mindset and Persistence in Learning

The final paragraph emphasizes the power of mindset and persistence in learning, especially in subjects like math and science. It shares the story of a woman who overcame her dislike for math to achieve advanced degrees in engineering through a change in mindset and learning strategies. The video concludes with the inspirational story of the four-minute mile, illustrating how a shift in collective belief can lead to significant breakthroughs. The message is clear: with patience, persistence, and the right approach, anyone can excel in areas they initially struggle with.

Keywords

💡Focused mode

Focused mode refers to a state of intense concentration where one is deeply engaged in a task, such as solving a complex math problem, with minimal distractions. It is a key concept in the video as it discusses the importance of this mode for learning and problem-solving in subjects like math and science. The script mentions that while in focused mode, one can become highly efficient at tackling specific problems but may also become too tunnel-visioned to see alternative solutions.

💡Diffuse mode

Diffuse mode is a more relaxed state of mind where the brain is allowed to wander and make connections between different ideas. This concept is central to the video's theme, as it explains how taking breaks and allowing the mind to relax can lead to innovative solutions and a deeper understanding of complex subjects. The script provides examples such as a chess grandmaster looking away from the board to allow his mind to work on the problem in the background, leading to a fresh perspective when returning to the task.

💡Einstellung effect

The Einstellung effect is a psychological phenomenon where having an established idea or method for solving a problem can hinder the discovery of a better or more efficient solution. The video uses this concept to illustrate why it's important to alternate between focused and diffuse modes of thinking. An example from the script is students using the quadratic formula when a simpler method is available, demonstrating how being stuck on one idea can impede learning.

💡Deep learning

Deep learning in the context of the video refers to the process of truly understanding and internalizing knowledge, rather than just memorizing information. The script emphasizes that both focused and diffuse modes of thinking are essential for deep learning, as they allow for the exploration of ideas and the formation of connections between concepts, which is particularly important in fields like math and science.

💡Recalling material

Recalling material is a study technique where one attempts to remember information without looking at the source material. The video suggests this as an effective method for learning and retaining information. It contrasts this with the less effective strategy of rereading, which the script argues does not promote deep learning. The technique is recommended for reinforcing memory and understanding of newly learned material.

💡Conceptual understanding

Conceptual understanding is the ability to grasp the underlying principles and ideas of a subject, rather than just memorizing facts or procedures. The video stresses the importance of this type of understanding in learning math and science, advising viewers to ensure they understand the fundamentals before attempting to solve problems. The script uses the example of working through a problem and then repeating it on one's own to solidify this understanding.

💡Practice

Practice, as discussed in the video, is the act of repeatedly working on tasks or problems to improve skill and understanding. It is a key component of the learning process, especially in subjects like math and science. The script advises that while practice is essential, it should be balanced with the understanding that learning deeply and efficiently is more important than simply doing a large volume of problems.

💡Pomodoro Technique

The Pomodoro Technique is a time management method designed to enhance productivity by breaking work into intervals, traditionally 25 minutes in length, separated by short breaks. The video mentions this technique as a strategy to combat procrastination and improve focus. It suggests setting a timer and working with concentration for the set period, then rewarding oneself with a short break, which can help manage time effectively and reduce the feeling of being overwhelmed.

💡Mental contrasting

Mental contrasting is a motivational strategy where one visualizes their desired future state and contrasts it with their current reality, along with the reasons for pursuing their goals. The video describes this as a tool to help with procrastination, by reminding oneself of the long-term benefits of completing tasks. An example from the script is a student who framed their expected starting salary to motivate themselves to work hard.

💡Persistence

Persistence in the video is portrayed as a critical factor in learning and achieving success in subjects like math and science. It is defined as the continuous effort and determination to keep working towards a goal, even in the face of challenges. The script shares a personal story of the author who, despite initially struggling with math, persisted and eventually excelled, highlighting the power of persistence over natural talent.

Highlights

The book 'A Mind for Numbers' offers learning techniques applicable to various subjects including math, science, and even games like chess.

Learning is a science with practices to improve efficiency and performance at any level.

Two essential modes of thinking for deep learning are focused mode and diffuse mode.

Diffuse mode allows the brain to connect different parts and solve problems in the background.

The Einstellung effect is a cognitive bias where preconceived ideas hinder finding better solutions.

Diffuse mode thinking can help overcome the Einstellung effect by providing a fresh perspective.

Alternating between focused and diffuse mode is crucial for improving skills in math and science.

Learning should be done in small increments daily for better retention.

Recalling material after learning is more effective than rereading for memorization.

Understanding the fundamentals is key before attempting to solve problems.

Mixing up problem sections can improve the ability to apply knowledge in various contexts.

The Pomodoro Technique can help manage time and reduce procrastination.

Focusing on the process rather than the product can lead to deeper understanding.

Doing the most important and disliked tasks in the morning can build momentum for the day.

Mental contrasting can motivate by reminding of the end goal.

Persistence is often more important than intelligence in learning math and science.

A shift in mindset can significantly improve learning outcomes.

The story of the four-minute mile illustrates the power of mindset in achieving the impossible.

Transcripts

this video is going to cover the book a mind for numbers how to excel at math

and science and although the title is very specific the learning techniques in

this book can apply to many different subjects all the way to games like chess

now whether you're in high school struggling with math and science or

you're in college studying math science or engineering and doing well the

techniques taught here can apply to anyone because learning itself is a

science and there are practices to improve efficiency and performance no

matter what level you're at so let's get started. Now there are two modes of

thinking you need to be aware, of focused mode and diffuse mode. Focus mode is

obvious it's when you are highly focused on something like attempting a difficult

math problem with no distractions. Diffuse mode is what happens when you're

more relaxed and let your mind wander basically just like taking a break but

what this does is allows different parts of your brain to connect and solve

problems kind of in the background and I will elaborate on this soon but both

these are actually essential for deep learning and understanding when it comes

to especially math and science. Now to explain the significance of diffuse

learning I want to introduce something called the Einstellung effect. This is where

where you have an already formed idea in your head of how to solve a problem

which prevents you from finding a better solution, you're kind of zoomed into one

idea. I've tutored personally a lot in math and I've seen this all the time for

high schoolers here's a common one we'll be learning the quadratic formula and

I'll give them multiple problems that require it then I'll give them something

like this which they have done before and they go straight for the quadratic

formula forgetting that they can just take out an X and it's already factored

they were zoomed into one idea and it prevented them from finding a better way

to do the problem. This has also been shown in chess players sometimes players

have an idea in their head of what move they should do they try to continue to

scan the board for better moves however through studies of their eyes they've

shown that their focus stays on the original move they thought of, making it

more difficult to come up with a better one and this is where diffuse mode

thinking comes in. It allows your brain to relax and the neurons to connect and

kind of work in the background which allows you to see more of the big picture.

In the book the author shows a picture of a chess grand master during a game who

gets up from his seat and looks away from his game. He takes a short amount of

time to distract himself and let his mind wander so he doesn't become stuck

in a narrow way of thinking and this is what diffuse mode thinking really does

you just take a break from that intense focus then when you come back to the

problem you have a fresh mindset and often the solution or a better solution

comes to you much faster. A very famous french mathematician henri poincaré

struggled to crack a difficult problem so he decided to take a vacation and when

he was getting on a bus the answer suddenly came to him from a part of his

brain that continued to work on the problem in the background and you'll

hear of many stories where this kind of thing happens. So this is the foundation

for improving your skills, you need to alternate between focused mode learning

and diffuse mode. No you cannot just let your mind wander and wait for the

answers to come. You need to focus intently on a problem then when you are

stuck take a break for maybe two minutes maybe an hour and then come back and

focus again and the solution may come to you much quicker as you have a fresh

look at the problem. It may not just happen like you did for this

mathematician but resume the problem and it may surface faster than you think

ways to get into diffuse mode include going to the gym, going for a walk, going

for a drive, taking a shower, sleeping, and much more. And remember you can't just

look at a problem you have no clue what to do on and then take a break and think

you'll get to the solution you have to be logical you need to actually

understand the basics then focus on the problem and then alternate between that

focused and diffuse mode thinking. Now let's get into some more specific study

techniques and tips. When it comes to learning math and science learning fast

is like trying to gain muscle fast it'll just burn you out. Although you can maybe

memorize things very quickly you can't learn on a deep level in one or

two nights so that's why you need to learn a little every day or at least as

much as possible. Now the best study method they found for any topic is

attempting to recall the material that you have just learned. What you don't

want to do is reread material until you get it, this is one of the worst ways to

memorize or learn new material. What you want to do is read something new and

turn away or close the book and try to recall what you just read as much as

possible then read it over again and repeat that

one more time. They found just with two repetitions this worked better than

nearly any other method done in the same amount of time. But okay what about when

it comes to doing problems like in math, physics, engineering, etc which aren't

just memorizing things. Again here's what you don't want to do and this I thought

was very insightful because I know it will apply to a lot of people watching.

What you don't want to do is attempt a problem then when you don't understand

it you look at the solution and say "oh I see how they got that" and then move on.

This is one of the biggest illusions of learning math and science out there so

you should not fall for it you haven't learned the concept even though you

think you may have. If you need to look at the solution that's no problem but

then you need to do a totally different problem on that topic and solve it

completely on your own. So if you're struggling to learn a new concept what

you want to do is first read about the topic and make sure you understand the

fundamentals, ask your professor, your friends, or whatever it takes so you can

just get the basics, you don't just want to dive into problems without learning

the fundamentals. Then what you want to do is work a problem all the way through

try to have the solution available but don't look at it unless you really need

to, then if you didn't fully understand the problem you need to repeat it all on

your own and make sure as you're going through each step you understand why

it's being done not just how it's done, this allows your brain to make

connections rather than just reading the solution and thinking you have it. Then

you want to take a small break after a few problems, go into that diffused mode

thinking so you can come back with a fresh mind and if you don't get

something you need to practice it a lot, that's basically what this all comes

down to. It will take time but your brain is going to keep making

connections especially as you accomplish problems on your own. Now many students

on the other hand will do a certain math problem correctly but then right after

that they'll do more of those same problems, now this can be a good thing

and improve test scores for sure, but this doesn't maximize your time in the

most efficient way. If you want to learn the most in the shortest amount of time

then once you successfully learn a concept and do maybe couple problems

correctly, move on. If you had all the time in the world

then of course practice as much as you can but often that's not the case so

this is something to keep in mind. And another tip which I really thought was

interesting was to mix up the sections of the book you're doing problems in. Like

let's say your homework is in section 5.4 of the book, well do

some of those problems then do some from 5.3, then go back to

5.4, then maybe go back to 5.2. Especially if you're

struggling you'll need this practice. Now why do you do this? Because you want to

make sure that you could do all these problems on a test if you didn't know

what section things were on. If section 5.4 is all about synthetic

division, even if you get an abstract problem you still can assume it requires

synthetic division, but just given that problem randomly on a test would you

know to do that? This is something to ask yourself and a reason to kind of mix up

the sections you work on so that you can fit the pieces together. Now in another

video I talked about procrastination but this book had a few takes on it as well

and I wanted to offer some more advice for those who have issues with

procrastination. One thing to try it is the Pomodoro Technique which there are

many videos on but essentially to sum it up, set a timer for something like 25

minutes, maybe a little less or more but then solely focus on work for that time.

This is short enough where anyone can really focus for that long, then

afterwards reward yourself with some internet surfing or texting or whatever

but not for too long, maybe just a few minutes, and then repeat the process. Some

people argue that they feel rushed with the timer in the background when doing

something like math, but studies have shown that this actually helps because

it will make you become accustomed to being on the clock and feeling a little

stressed so that when you're taking a test and that stress occurs you're able

to perform better. Also you want to make sure you focus on the process not the

product. As in you should care more about the time you're putting in not how

much you're getting done. It doesn't matter if it takes two or five hours to

complete your homework (assuming you have the time) you really

want to make sure you understand the material on a deeper level and learn how

to apply it. Another very effective tool is to do the most important and disliked

jobs in the morning when you get up or maybe immediately when you get back from

school. This then do that momentum to keep going. Another

thing to try is mental contrasting, where you basically picture where you want to

be compared to now and why you are doing this. One student from the book was

quoted as saying that he put what his starting salary would most likely be on

his wall and framed it to remind himself why he's doing all this work and it

helped him with procrastination. Now I'm going to stop there with procrastination.

Now when it comes to getting better at math or science it basically all comes

down to patient persistence. After so much time of doing all these techniques

it will reshape your brain and optimize the way you learn and solve problems.

Persistence is often more important than intelligence. This author said that she

absolutely hated math all throughout high school and failed her way through.

She started studying just basic trigonometry when she was 26 years old

after being in the military, but she had a shift in her mindset and decided that

she was going to get good at math. She ended up getting a bachelor's in

electrical engineering, a master's in Electrical and Computer Engineering, then

a doctorate in systems engineering with a background in various subjects and she

said that it got easier for her over time because of how she was learning to

learn. And remember things typically get more fun when you're better at them.

People often are bad at something then because of that they get bored which

makes them not try and it's just a downward spiral from there. And

throughout the book she talks about many success stories or people who weren't

naturally gifted at math who then excelled later in life.

Just a simple shift in your mindset can do more than you may think. I'll end this

video one of my favorite stories of the four-minute mile. The four-minute mile

was assumed by experts to be impossible for the human body. In over a hundred

years of the timed mile no one had done it until one day in 1954 one person

finally beat it by less than a second. But then less than two months later

another person beat it, and since then thousands of people have beat it

including a few high school kids. And people say it was that switch in the

mindset of others that the impossible was actually possible that then allowed

the other people to beat the time more and more. So yes your mindset is more

powerful than you may think. Math is something I personally like a

lot but I still learned plenty and found lots of helpful techniques and tips from

this book which I've linked below if you want to buy it for yourself and support

the channel. I hope this summary helped you, if you liked the video don't forget

to comment, like, and subscribe and I'll see you all next time