Why you understand the math but CAN'T solve problems

Summary

In this video, Han, a Columbia University graduate, shares her journey from struggling with math to mastering it. She emphasizes the importance of active learning over passive learning, explaining that understanding concepts is not enough; practice is crucial. Han provides actionable tips to improve math skills, such as redoing questions until correct and using the Feynman technique to ensure deep understanding. She also addresses common issues like memorization, problem-solving tricks, and common mistakes, offering strategies to overcome them and enhance math performance.

Takeaways

• 📚 Han, the speaker, graduated from Columbia University and studied math and operations research, highlighting her credibility on the topic.
• 🔄 Han struggled with math in high school, receiving C's and D's, which shows that she overcame significant challenges in her math journey.
• 🤔 Han suggests that understanding math but not being able to solve questions correctly is a common issue among students.
• 📉 The motivation to practice math can fade with wrong answers, leading to frustration and the end of practice sessions.
• 🔍 Han emphasizes the difference between passive and active learning, stating that the latter is more effective for mastering math.
• 📈 Active learning involves doing something beyond just receiving information, such as discussions, practicing, and teaching others.
• 🚫 Passive learning, which includes listening to lectures and watching demonstrations, is less effective in math and science education.
• 🔄 Han recommends redoing questions independently when they are wrong, as understanding the solution is not enough.
• 💡 The 'Feynman Technique' is suggested as a method to improve understanding by explaining concepts as if to a child.
• 🧠 Active recall and spaced repetition are recommended for memorization, which is sometimes necessary in math.
• 🛠 Collecting and applying problem-solving tricks in a 'toolbox' can help with specific math tactics that are not commonly known.
• 🔍 Paying attention to common mistakes, such as calculation errors or misreading problems, is crucial for improvement.

Q & A

What is the main issue discussed in the video script?

The main issue discussed is the struggle of understanding math concepts but not being able to solve math problems correctly, and the importance of active learning over passive learning.

Who is the speaker in the video script?

The speaker is Han, a recent graduate from Columbia University who studied math and operations research.

What was Han's initial experience with math in high school?

Han struggled with math in high school, often receiving C's and D's despite investing a lot of time and effort.

What is the difference between passive learning and active learning as described in the script?

Passive learning involves receiving information from outside sources without internalizing it, such as listening to lectures or watching demonstrations. Active learning means being actively involved in the learning process, such as discussing, practicing, and teaching others.

Why is active learning more effective in math and science education according to the script?

Active learning is more effective because it involves doing something beyond just receiving information, which helps in better understanding and applying math concepts.

What is the 'fan technique' mentioned by Han?

The 'fan technique' is a learning method used by Richard Feynman, where one explains a concept as if to a child, which helps uncover areas of misunderstanding.

What should one do when they get a math question wrong according to the script?

One should redo the question independently until they get it right, which is a form of active learning and helps in understanding where the mistake was made.

What are the four types of problems one might face when studying math as outlined in the script?

The four types of problems are understanding problems, memorization issues, unfamiliarity with specific tricks or tactics, and making silly mistakes.

Why is memorization often unnecessary in math according to Han?

Memorization is often unnecessary because everything in math can be proven, and understanding the concepts is more important than memorizing specific equations or steps.

What is the recommended technique for memorization when it is necessary?

The recommended technique is active recall and spaced repetition, which involves actively trying to remember information and revisiting it over time to reinforce memory.

How can one prevent making silly mistakes when solving math problems?

One can pay close attention to common mistakes, understand why they occur, and adjust their approach or writing style to avoid such errors in the future.

Outlines

00:00

📚 The Struggle with Math and the Importance of Active Learning

In this paragraph, the speaker introduces their journey with math, highlighting their initial struggles in high school with poor grades and a lack of understanding despite their efforts. They mention their eventual success after learning from top teachers and realizing the importance of active learning over passive learning. The speaker emphasizes that merely listening to lectures or watching demonstrations is not enough; one must engage in discussions, practice, and teaching others to truly internalize mathematical concepts. They also mention the common issue of understanding math in theory but failing to apply it correctly in problems, suggesting that active learning is crucial for mastering math.

05:00

🔍 Identifying and Overcoming Math Challenges Through Active Learning

This paragraph delves deeper into the concept of active learning, explaining how it can help overcome various types of math problems. The speaker advises that when a question is answered incorrectly, it's essential to revisit it independently until it's correctly solved. They introduce the 'Fan Technique' for understanding concepts by explaining them as if to a child, which helps uncover gaps in understanding. The speaker also discusses the issue of memorization, suggesting that while it can be useful, understanding the underlying principles is more important. They mention the need for active recall and spaced repetition for memorization. Additionally, they address common mistakes, such as calculation errors or misreading problems, and the importance of being aware of these to avoid them in the future.

Keywords

💡Passive Learning

Passive learning refers to the process where students receive information from external sources without actively engaging in the learning process. In the video, it is contrasted with active learning, and the speaker argues that passive learning is less effective for mastering math. The script mentions that passive learning includes listening to lectures and watching demonstrations, which the speaker did extensively but found ineffective for truly understanding math concepts.

💡Active Learning

Active learning is a method where students are involved in the learning process by doing activities such as discussions, practicing, and teaching others. It is highlighted in the video as a more effective approach to learning math compared to passive learning. The script emphasizes that active learning helps in internalizing mathematical concepts and skills by engaging in problem-solving and teaching, which the speaker found transformative for their understanding and success in math.

💡Math Practice

Math practice involves working on mathematical problems to improve understanding and proficiency. The video script discusses the importance of practicing math problems actively rather than passively reviewing solutions. The speaker shares their personal experience of struggling with math and how active practice helped them overcome challenges and succeed in their math education.

💡Frustration

Frustration in the context of the video refers to the emotional response of feeling upset or discouraged when failing to solve math problems despite understanding the concepts. The speaker describes how initial enthusiasm for learning math can fade into frustration due to repeated failures in problem-solving, which can hinder the learning process.

💡Understanding vs. Application

Understanding versus application is a key distinction made in the video. Understanding refers to knowing the concepts, while application is the ability to use those concepts to solve problems. The script illustrates that one can understand math concepts but still struggle with applying them in problem-solving, which is a common issue the speaker addresses.

💡Diagnostic Process

In the video, the diagnostic process is likened to identifying a problem and then finding a solution, similar to a medical diagnosis. The speaker uses this analogy to emphasize the importance of revisiting and understanding mistakes in math practice. The diagnostic process involves comparing one's incorrect solution with the correct one to pinpoint errors and learn from them.

💡Mistake Analysis

Mistake analysis involves reviewing incorrect solutions to identify specific errors made during problem-solving. The video script encourages viewers to analyze their mistakes to understand why they went wrong and to learn how to avoid repeating those errors in the future, which is a critical part of the learning process.

💡Active Recall

Active recall is a learning technique where students actively attempt to remember information without looking at the material. In the context of the video, active recall is recommended as a method for memorizing mathematical equations or steps, which can be particularly useful when memorization is necessary for solving problems efficiently.

💡Spaced Repetition

Spaced repetition is a learning method where material is reviewed at increasing intervals over time to enhance long-term retention. The video mentions spaced repetition as an effective technique for memorizing mathematical concepts or steps, suggesting that it can be particularly helpful for students who struggle with memorization.

💡Problem-Solving Toolbox

The problem-solving toolbox is a metaphor used in the video to describe a collection of strategies, tricks, or tactics that a student can use to solve different types of math problems. The speaker recommends that students pay attention to new tricks they encounter, understand how they are applied, and then store them in their 'toolbox' for future use, making problem-solving more efficient.

💡Silly Mistakes

Silly mistakes refer to simple errors made during problem-solving that are not due to a lack of understanding but rather oversight or carelessness. The video script mentions these types of mistakes, such as misreading a number or making a calculation error, and advises students to be mindful of common mistakes they make to avoid them in the future.

Highlights

The speaker has a background in math and operations research from Columbia University.

Initially struggled with math in high school, receiving C's and D's despite effort.

Realized the importance of active learning over passive learning for effective math comprehension.

Passive learning involves receiving information without internalizing it, while active learning is engaging in the learning process.

Research shows active learning is more effective in math and science education.

Math is a skill that requires practice to be mastered, similar to driving a car.

The speaker suggests that understanding math without being able to apply it is a form of passive learning.

When a math question is answered incorrectly, it should be revisited independently until solved.

Reviewing wrong answers is a diagnostic process to identify gaps in understanding.

The speaker emphasizes the importance of 'treatment' after diagnosis, which involves understanding and correcting mistakes.

Four types of problems are identified: understanding, memorization, tactics, and simple mistakes.

The 'Feynman technique' is recommended for better understanding of concepts.

Active recall and spaced repetition are effective for memorization.

Cultivating a 'math problem-solving toolbox' helps in applying tactics to various problems.

Common mistakes should be noted and avoided in future problem-solving.

The video provides actionable tips that transformed the speaker's experience with studying math.

The speaker's journey from struggling with math to succeeding in hard classes at Columbia University.

Transcripts

you've just learned about the new math

concept everything makes sense to you

you feel energetic and motivated to

tackle the questions until you get a

wrong answer the motivation slowly Fades

away you feel the frustration building

bringing your math practice to an end

you can't help but wonder why you

understand the math but just can't get

the questions right hi welcome to my

channel my name is Han I just graduated

from Columbia this may I studied math

and operations research in college as

someone who has been battling with math

for over 17 years now I have not always

been a math person when I was in high

school I really struggled with math I

often received C's and D's in my math

classes and no matter how much time and

effort I put in it I just cannot get the

questions right and receive the scores

that I wanted on exams I thought I might

not be smart enough and couldn't pursue

a career in stem especially there were

students in my class put in way less

time than I did but Reed really good

grades so after thousands of hours

practicing math and learning from some

of the best teacher in the world I

finally realized that I was just doing

it wrong so in this video I'm going to

explain to you why you understand the

math but you just couldn't do the

questions right I will also show you

that how you can fix it with some

actionable tips that completely

transformed my experience with studying

math and helped me succeed in some of

the hardest classes Columbia University

let's just get into

it for a context let me explain to you

how I was studying before I realized

something was wrong I would really pay

attention in class and carefully take

notes I spent hours looking at how to

solve different math problems going over

each problem carefully to see how it's

done you may stop me and be like hold on

that doesn't sound bad it sounds like

you were a really hard worker but if you

pay close attention the problem here is

it's solely passive learning and no

Active Learning at all passive learning

is a method of learning where students

receive information from outside sources

and try to internalize it which include

listening to lectures reading and

watching

demonstrations and active learning means

you're actively involved in the learning

process basically means you're doing

something besides just receiving the

information and which includes

discussions practicing questions and

teaching others in our education system

there are lots of passive learning and

very little Active Learning many

research Studies have shown that passive

learning is not as effective as active

learning in math and science education

so if you feel like you're spending lots

of time in studying math but still not

get good results it's probably because

you are doing too much passive learning

and not enough active learning math is a

skill to help you solve problems you

have to know how to use it by doing it

you wouldn't say that you can drive a

car just by watching someone else drive

and memorizing all the traffic rules you

have to get into the car and practice

this way you can know how it feels which

part you have problems or any

complications you may run into so I hate

to break this to you but if you only

read and listen how the math is done

sometimes you may think oh I understand

the math but actually you

don't so how can we actually do more

active learning in our math studying if

there's one thing I wish you can take

away from this video is if you get a

question wrong always redo it

independently until you get it right

lots of time people get really upset

when they get a question wrong and start

thinking oh what is wrong with me I just

cannot understand why I cannot do math

you know what the question in front of

you is literally telling you what's

wrong what I often did was if I get a

question wrong I just read through the

answers and try to understand it and

just move on but again the problem here

is if I only read the solutions and try

to understand how it's done it's still a

form of passive learning you wouldn't go

to the doctor spend two days doing all

the lab work for a physical exam and

receiving a result that says you have a

heart issue and you just go oh crap they

say I have a heart issue and just move

on with life you would actually follow

up with the doctor take pills every day

or pursue whatever the necessary

treatment it is until the disease is

cured or under control every time you

practice a question it's a diagnostic

process to see whether you understand

the math concept and your ability to

solve the questions and you need to

realize that the diagnosis is not the

most important part and what truly

matters is the treatment you've already

invested all your time and efforts in

reading the question trying to

understand the question thinking about

the question write the question if you

don't follow up with what's wrong it's

really just a waste of time so follow up

with the question that's wrong figure

out precisely which Step you made a

mistake really trying to understand why

you made that mistake and how you can

prevent it from happening again and

doing the question again is the followup

doctor appointment that they will tell

you whether the disease is truly cured

or not cuz the doctor May tell you yeah

you know what you're good to go or they

may tell you it's still there you still

haven't fix it or they may even say that

ah your heart disease is cured but now

you have a liver disease so make sure

you test yourself by doing the questions

again until you get it right so how do

you actually follow up with your math

disease and then found the treatment for

it AKA how to fix the problem so ENT

entally you compare your wrong answer to

the correct answer and figure out which

exact step that you had the problem in

my experience there are four type of

problems let's just dive into each one

of those firstly an understanding

problem might arise when you feel to

grasp a math concept or the equation you

recently learned or perhaps you don't

understand the question itself to help

you understand math better consider use

the fan technique it is a learning

method that used by the no prize winning

physicist Richard fman basically the way

you understand a concept is by

explaining it as if you're explaining it

to a child while doing this notice which

part did you struggle to explain what

details did you miss what part is really

hard for you to put into simple words

answering those questions will show you

which part are you missing the act of

explaining to yourself or teaching

others is a form of active learning

sometimes when we thinking about things

in our own mind we think we understand

it well but we may miss some important

stuff so the best way to uncovering

those details is to explaining it out

loud verbalize it and teaching someone

else who doesn't know about the topic

and the more you can put it into simple

terms the better understanding you have

about the topic secondly you might face

issues with memorization when you can't

remember an equation function or the

specific step GS required to solve a

particular type of problem in my opinion

in math memorization is really often

unnecessary because everything can be

proven if you're really motivated not to

memorize anything you don't even need to

memorize 1 + 1 = to 2 because there are

162 pages of proofs in abstract algebra

to prove y 1 + 1 equal to 2 therefore I

highly recommend you try to understand

everything before you just straight jump

into memorization but in our daily math

studying memorization does have an

important role because it really can

save us time in situations where

memorization is necessary the best of

techniques is active recall and space

repetition s some questions may need a

specific trick or tactic that you're not

just familiar with and my recommendation

will be just practice more questions but

whenever you encounter a new trick pay

attention how it applied in what type of

question and then collect that trick in

your math problem solving toolbox next

time you see a similar question you can

just pull up your toolbox and use the

tricks that you've collected like a

Pokemon situation the fourth you do

actually know how to answer questions

and you could have done it correctly if

you hadn't just made a single silly

mistake believe me I understand the

frustration I experience it all the time

especially with more challenging

problems the answer just become really

lengthy and requires so many steps

making it really hard not to make

mistakes so pay close attention to the

common mistakes that you make and keep

these in mind the next time you work on

the question is a calculation error or

did you misread the problem for example

I used to make silly mistakes like in

the last line it was 45 but the next

line I just randomly magically write 6 5

so I noticed that and trying to figure

out why turns out the way I wrote four

was really close to Six so I just

changed the way I write for and then

just pay more close attention so that's

today's video thank you so much for

[Music]

watching