When Math Gets Really HARD, This is What You Do

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In this episode of Office Hours with The Math Sorcerer I answer an email I received from a viewer. He feels like he is drowning because he lacks the necessary background in order to understand the material. The subject is Functional Analysis and he is learning with a professor via an independent study. Do you have any advice? If so, please leave a comment.
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Thank you:)

[@ronaldjorgensen6839]

rest the mind do not overcomplicate it, in retrospect the answer is simple any confusion was a illusion do not let it disrupt your minds focus

[@Itravelbackintime]

As a kid I use struggle with math to the point where I hated going to any math class because of being afraid of being called out by the teacher to solve a problem. I was struggling so badly that I was even a dyslexic and even had to go to see a speech therapist. That all changed when I first met my my 5th grade teacher who helped me go from that to having more mathmatical and reading literacy. Despite improving in my education over the years into high school I still had struggles with math. I recall my science teacher telling me I wasn't bright and was told I didn't amount to anything. As a kid that was soul crushing. That changed when I had another science teacher helping me discover my strentgh in visual learning. It was then after going from F's to being the 2nd highest in my class I realized that potential using it to my advantage. Even though I dropped out of high school I didn't give up by passing my GED and went onto graduating college. I never thought in a million years I could ever learn trigonometry and memorize formulas to design something. When I walk into a home I see numbers and dimensions. Math is not scary once you begin to understand the world around you better and how it's used. My goal is to eventually master Calculus and apply it for a project. Math can seem hard in the beginning but it was a few good teachers while growing up that can made the difference from the bad ones I had. I figure I could share this story that may help someone out there on the sidelines who are hesitant about math. My advice is to never give up no matter where you started. Math really can open a lot of doors of undestanding and how you can apply it to help make the world a better place. If I can go from not being able to count to 10 as a kid to graduating college after dropping out of high school I believe anyone can achieve better success.

[@pichirisu]

It would be really cool if you did a video on the philosophy of math, or had a discussion with a philosopher of math/philosopher that researches philosophy of math.

[@TheMathSorcerer]

Yes that would be cool. I like the idea of doing interviews.

[@devon9374]

I suggest you read books about mathematics and the history of math. Philosophy of math is in these. Books like Men of Math Mathematics, Love and Math and Paul Halmos autobiography are really good. Also the Princeton Companion to Mathematics is a great philosophical reference as well

[@cool_huip_]

You are literally a mathstar, big fan--thank you for all your content! From Ontario, Canada

[@SteveRuprecht]

+1 for extra textbooks on topics. Totally helps. Pro tip: go to your school library and check out books related to the topic you're studying before buying. Even your local public library has textbooks or can get them through interlibrary loans. Odds are some author has explained a topic in a way you understand so DO NOT rely on a single resource. Also join or form a math club. Good luck!

[@SteveRuprecht]

Sorry, All knowing Math Sorc if i cut into your commissions :(

[@nickadkins5679]

This is a very important topic honestly. No matter who you are, at some point, you will become hopelessly stuck on something that you can't wrap your head around. This happened to me my last semester in college when I did an independent study in graduate level differential geometry "Differential Geometry via moving frames and exterior differential systems". I really only had real analysis under my belt for a proofs class, and was in complex analysis at the same time as the independent study was going on. The kicker though, is that I had never studied differential geometry! My initial desire for an independent study was tensor calculus, but it was deemed too trivial by my professor(even I didn't think it was) and he picked this as a good option, and I went with it. The lesson I learned is, is that you will be overwhelmed by a topic if you do not have correct prerequisites and have not put in the time prior to solidifying your previous knowledge. It doesn't matter how smart you are, if you try to tackle complex analysis and you have only taken algebra 1 for instance, even though you have the capacity at some point to learn it, you will be hopelessly lost and demoralized. As you said, it is very important to isolate the things that you can understand, and build from there. For instance, take a couple of proofs or worked problems, and take a week or so and engrain the problems, solutions, and methodologies into your mind by repetition and analyzing the problems on a micro level. This mindset you develop each week of doing this will pay off, and definitely helps in not putting yourself in situations that are overwhelming. That's my two cents.

[@TheMathSorcerer]

excellent comment, thank you!

[@gunzafterwork1044]

Iikikkkiikkkkkkkkkkkkkkkkkkkkkkkkkkkkkk

[@johanponin8680]

This is something I'd love to see more. I believe this is a critical point.. how do smart people operate when they're in the dark, when everything is fuzzy, blurry, overwhelming. What kind of heuristics, process, pace are they using. Is it random .. is it aesthetic (aka following intuition bursts) driven, is it analytical/combinatorial ?

[@justgivemeanumber8215]

I like this transition into an influencer helping with mindset and everything. And I'm happy your channel has been growing well.

[@kapoioBCS]

My advice would be kinda obvious and maybe somewhat arrogant and maybe will also sound impossible time wise. In my experience if you feel this drowning feeling, it means that you have some very big gaps somewhere in your knowledge which must fill in order to be able to go forward with the specific subject. I would suggest, regardless if time allows, to go back and master every hole you have in your math background. For functional analysis is probably analysis and topology. So I would say that you should go back and master this two topics by solving at least a whole big book on both this subjects. In the meantime I would try to do the same with functional analysis, i.e. find solved problems from some other books or online solutions or solutions from online lectures and solve EVERYTHING, proof examples, exercises. If you cannot do it yourself, which is what will happened 75% of the time at least maybe even 90% then write down the solution and try to do it your self again until you don’t need to see the solution then mark the exercise with red. Then when you finish the particular chapter with all the proofs and exercises, go back and try as a test to solve all the marked exercises i.e. the exercises or proofs that you could not do yourself! And on and on and, this is the most safe way to master any subject imho!

[@Zendogg702]

This is what I’ve learned to do, more or less.. if I don’t understand something, I simply keep writing it over & over.. until the light of comprehension dawns. I will simply write as much mathematics as it takes to bring on comprehension or absorption of ideas. There’s always a certain amount of material I try to “know cold” so that when I write it, it almost writes itself automatically. I learned this from chess. Being a serious chess player requires knowing some amount positions, tactics, opening lines & whole games.. knowing them so well that you can recognize them & play them by hand, almost without thinking.

[@kapoioBCS]

@@Zendogg702 yes exactly, it is essentially pattern recognition

[@ivanrothstein8405]

Another book that is gentle: "A friendly approach to Functional Analysis" by Amol Sasane.

[@dkoxperiakoziukov3808]

I see no obstacles to contact a math skilled personal to get to know how to handle an issue or organize any embedded function

[@azibsubzar-bx2sr]

How to concentrate on math problems,how to solve that problems..

[@noorbhatia1200]

These thumbnails bruh i cant😂

[@juanpabloc.4002]

In my experience, learning set theory makes topics like topology, analysis and probability much more manageable (historically this was done backwards, but it is easier to learn the foundations first). I recommend you read A Book of Set Theory by Pinter. I think this is among the best introductions to the topic.

[@PowerK1]

You should do a QNA!

[@Sammy23806]

Love your videos ❤

[@explorerendeavour3009]

do what you can as a starting point!

[@soyokou.2810]

I recommend SIMMONS It's an actual undergraduate book and fun to read

[@orthodoxiechretienne847]

Hi ! Thanks for this great video Mister The Math Sorcerer. I was wondering what is an independant study ? Keep the good work. Corentin, from France.

[@oscardavidalarcon2673]

Gracias profe :)

[@tethyn]

3:05 has to stop video here because students are conditioned to think education is about content and not about thinking. Especially with the internet and ChatGPT, information is readily and easily accessible. It is what to do with it and how the information came to be understood is the beginning nor the end, of education. Don’t get me wrong that I am saying that the content is unimportant, because it is not; rather, how we use it. The difference between trade schools and technical schools are rather arbitrary. They are both concerned about learning specific content and development of critical thinking; however, the development of the human being in college undergrad is different than in a trade school. You do a great job with these videos stressing how we think about the subject because the value of that goes well beyond just math. Edit: finished video. Excellent advice. Baby Rudin was what I was using but it only helped me feel complete about where I was going. I hate to admit I have quite a large digital library of books (interpret that as you will) and I used many different books about the subject to fill gaps. Think of this way. Imagine the subject you are learning as the operator and the books is the vector and what you are trying to find is the right eigenvector (specific book, text, or KZbin video or whatever) that you can operate on in order to help you line up your understanding with subject matter so that you have an understanding of the subject (eigenvalue). Hope the analogy helps. Good luck to all.

[@patrickgambill9326]

I found the videos from Bright Side of Mathematics helpful when I was learning some of these concepts. Also, functional analysis is hard! It will take time to learn properly, and that drowning feeling is sometimes part of the process. Sometimes, it is good to take a few days away from the subject to give your brain a break and to reset. It is much easier to learn a subject when you do not feel stressed. I understand you do not always have that luxury when taking a class though

[@TheIllerX]

My biggest mistake when taking functional analysis, was to see it as a subject of formal symbol manipulations, a bit more like algebra than analysis. I could understand and read the proofs and get good grades, but I honestly did not really understand the subject anyway. The problem is that the presentation of functional analysis is quite abstract. You see a lot of operators T with different classifications, different abstract spaces satisfying some axioms and functionals and dual spaces. You can imagine it as an abstract game of symbol and axiom manipulation perfectly fine, but that is not the point of the subject. It took me years to fully realize that you are never going to find any operators named T hanging around in applications. What you will encounter is integral operators, differential operators, linear mappings in form of matricies and so on. You will se examples of what T could be. Therefore I really want to stress that you should never just look at the abstract names and definitions, but always try to come up with several examples of what would fit into that definition and theorems and what would not and what everything would look like in that settings. It is only then the theory comes to life and you will remember and understand it deeply. In functional analysis I would also really think about what the definitions and theorems would say if everything was finite dimensional, and we just had matricies and so on. Does the theorem still say something meaningful in that case? What does it say? Is it trivial in the finite dimensional case? If so, what is the exact reason the same conclusion is not trivial in a Banach or Hilbert space? The same goes for the difference between Banach and Hilbert spaces. Why is a theorem needed in a general Banach space, while it is trivial in a Hilbert space?

[@guidosalescalvano9862]

Fundamentally everyone in the exact sciences knows that it is hard. So it is ok to talk about it. In fact, that is the professional thing to do. And the better you can explain why you think something is hard, the more professional it is.

[@DonnieAllenCooper]

So you should write the intro material for future students

[@rjr230]

Here’s a good intro book for functional analysis: “Elementary Functional Analysis” by MacCluer. I found it to be really accessible and it’s relatively short. I really like this one

[@TheMathSorcerer]

Thank you!!!!

[@learninglad2985]

Daily dose of math sorcerer❤

[@emonymph6911]

What is a good book that teaches re-arranging formulas? So I can make a unknown the subject easily. In the real world you sometimes have 1 or 2 missing variables but if you can re-arrange the formula it won't matter.

[@bulbasaur_number1757]

Thank you, Lord of the Rings

[@fridayx02233]

Sir I am from Nepal, the education system and schools are awfully below than average USA schools I always wanted to learn math but the devoid of proficient mathematician masters, it remain the reason for me not to dive deep into math, I recently turned 18 I don't want to stay with the fact I didn't choose learn more of math cause of the lack of better teacher, I intuitively feel I can do better than majority. I tried to learn from KZbin however it is more difficult to grasp than it sounds via KZbin also because Counties lesson and problems are so different and mismatched it only gets baffling and the courses that are great to learn from are overly expensive for people like me who lives among the woods. what and how would you suggest me master, how do you want me to start? assist me sir.

[@Mr.Ali_Ryuji]

What your opinion on hard math exam or interview questions. Is it still relevant to measure someone understanding of mathematics?

[@TheMathSorcerer]

Well if you are hiring someone to teach math or use math on the job, then definitely yes, those questions matter a lot!

[@Mr.Ali_Ryuji]

@@TheMathSorcerer nice perspective.

[@courgette2572]

On which platform can we currently ask questions from you?

[@TheMathSorcerer]

Click contact on my website mathsorcerer.com

[@OrikiHotaro-rv4ku]

You look like Isaac Newton 😀, nice content keep it up

[@sp10sn]

Math doesn't get hard. We get greedy.

[@adventure2.028]

i like maths too

[@mannydossantos9603]

Thank you for the insightful advice, Math Sorcerer! You're the type of prof that anyone could have a coffee with, chat about any Math subject and the psychology of learning. You often refer about getting A's in various subjects... I think you like A's and I like that high standard too... everyone here should aim for the same goals. There's no place for mediocrity in Math, correct?

[@keirgordon8646]

kms? no? ok then

[@vania2931]

how I can send you an email?

[@TheMathSorcerer]

through my website freemathvids.com or click on about here on youtube

[@prostatecancergaming9531]

U

[@nsfeliz7825]

i dislike pure math. its a waste of time for engineers . yeah yeah math is useful. but pure math looking for absolute perfection diverts you from imperfect but realistic solutions.

[@vade248721]

Math teachers and schools ruin math. Will superfluous tests and assignments and projects and research. Bleh. Study math online solo.

[@jizert]

hey mathsorcerer, i sent u an email through ur website but it wasnt working so im gonna leave it here: "Dear TheMathSorcerer, I hope this email finds you well. My name is Micah Z, and I'm an avid viewer of your videos on math and math books. I am 14 years old and currently a freshman in high school, but I have a strong passion for higher mathematics beyond the standard curriculum. So far, I have completed reading "Book of Proof," "Linear Algebra Done Right," "Set Theory" (by Charles Pinter), a book on game theory, and now I'm exploring "Baby Rudin." Having learned the fundamentals of set theory and proofs, I am now looking for advice on what areas and books to explore next. Unfortunately, throughout grade school, I never developed good study habits because the material was not engaging or challenging enough. As a result, I now ace every test but struggle with doing homework consistently. I find my current geometry class to be under-stimulating and too easy, which further exacerbates this issue. I am seeking advice on how to overcome this under-stimulation and develop better study habits. In addition to improving my study habits, I genuinely desire to delve deeper into higher math. I feel a strong connection to the subject and find it to be incredibly stimulating and rewarding. However, I am unsure how to navigate this passion in the context of my current high school education. I would appreciate any advice you could offer on how to maintain my enthusiasm for higher mathematics while managing my school obligations. I am considering whether to tackle "Papa Rudin" next or explore other advanced math topics that you might suggest. I would greatly appreciate any guidance you could offer on this matter, both in terms of book recommendations, advice on developing study habits in the face of under-stimulation in school, and how to pursue my passion for higher mathematics. Your expertise in the field is truly inspiring, and I eagerly anticipate each of your insightful videos. In addition, I wanted to suggest a potential video topic that might benefit students like me. A discussion of the best math books for high school students who have a strong foundation in set theory and proofs and are interested in advanced topics in mathematics would be an incredibly valuable resource. Thank you for taking the time to read my email, and I look forward to hearing your recommendations and advice. Best regards, Micah Z"

[@penguins0392]

This is NSFW but I have a serious question. When I run across a head scratching math problem I get twitches in my penis. It starts to become unbearable that it makes moving onto harder problems an issue. Now that we had online school, i was able to masturbate to calm the twitches and it does help. BUT I fear that this would be a issue in the future. Idk how to even ask this question, but am I alone in this? Does anyone have any advice?

[@albertpiekarski4569]

Lmao. There were the days, holidays, when I decided that I want to do a lot of math problems during that time to get really good but I was getting so frustrated that I started masturbating more and more while doing them. And generally I'm often getting urges to do that as trying to solve some homework problem or sth like that makes me feel like failure and shit. So assuming that this question isn't just joke - I feel you bro.