Best advice I have so far (still a student myself): As you read or study, if you don’t know why something is true, make it a lemma and prove it, anytime a question pops into your head, make it a lemma and prove it, any time the author says something is clear and it isn’t, make it a lemma and prove it. When taking proof based math classes, you should have a notebook that ends up getting cover to cover front and back covevered in lemmas, examples and counter examples, separate from hw and exams study. That’s why, in my opinion, grad school is so hard for people. There were a lot of things you probably took as true, but you never actually figured out why, you just believed it. Believing without seeing was a very hard habit to break. One line at a time friends, and thank you for your videos sir!
Another tip: when you get stuck, before you look at the answer, ask yourself if there's some assumption or intermediate result that, if true, would enable you to solve the problem. That will help you improve your analytical skills. Also, when you look at the answer, you may discover that you were almost there; you were just missing the one piece. That will boost your confidence.
100% agree. The more general version of this advice is to modify the problem by tweaking some aspect of it (e.g., adding an intermediate result, removing a restriction) and then trying to solve that.
One thing I try to do is when I've exhausted all of my resources and I decide to look at the solution: only look at a tiny part of the solution. Cover up the page or scroll very slowly. I want to see just enough to get me unstuck so that I can try to finish the rest of the proof on my own. I learn a lot more that way than if I look at the entire solution and see every detail laid out for me.
have a question: If every time I go to solve a math exercise and I don't get it because it is too difficult, I look at the answers/ask for help or solve it with some "cheat" method, am I really learning anything? And if I don't learn anything, how do I solve the exercise without help or without having to spend 1 day trying to solve it on my own?
There's a lot of great advice in this video and this comment section. One thing that I have struggled with and still do when faced with hard problems is a feeling that I'm stupid or other people in the class are understanding the material so much easier than I am. This often leads to wanting to give up. I think that it's really important to realize that this negative self talk is not based on reality and is mostly made up by your imagination.
Wow what an espectacular pair of comments you both made. Thanks a lot. That it is what I feel about myself - story of my life - in a few lines. Hats off
That negative self-talk about how everyone is more equipped than you are is called Imposter syndrome. When I started grad school I struggled with the same thing, but I eventually convinced myself with help from advisors/friends that I am good enough to be there. Trust the process and trust the admission committee's decision to admit you to grad school. Good luck!
It also helps to remember that everyone has different experiences. I loved maths as a kid, spent my summer holidays doing math camps for fun, and I gained expiences with proofs and the basic concepts of advanced maths there. I spent a lot of time struggling and feeling stuck and not understanding "basic" ideas. Many of my coursemates weren't like that, so they struggle with the same concepts now. Often, if someone seems to be "quicker" than you they just know the material already. Chances are they spent just as much time struggling as you do now, and in a year you'll find todays material much easier too.
Totally agree with MS on using the material help solve the problem. Also, rather than spend endless hours on one problem, come back to it after a break of several hours or even a day. I often filter out key information or make a wrong assumption at first glance. Many times I can return with a fresh perspective and solve the same problem easily. Best of luck
As someone who's repeating their first year in their math major (and someone who's slowly getting the hang of coming up with proofs) I would say that most of the tips I would've given to myself two years ago are things that you have already said: understand the definitions (this was something important to me, since I thought I understood them at first glance when it wasn't the case), look up previous things that have been mentioned before, such as theorems, propositions and definitions and apply them to your problem and reflect on the solution that are already given (or the one you just came up with.) Another thing I would have told myself (similar as looking up the solutions to proofs and summarizing them in your own words) is to look up and also summarize the proofs given in the books you're reading. This will give you a feeling on how proofs should be written, when can certain proof techniques be used and also keep your thoughts organized when it comes to a specific problem. Finally, sometimes the issue I had in not being able to solve a problem is that I wouldn't even understand what the problem was (as in, I didn't understand what exactly it was that I wanted to prove.) What I had to do in order to get over this hurdle was to relearn the habit of writing down what I already knew about the topic of the problem and the goal I wanted to achieve solving this problem (regarding definitions, propositions, etc. Sometimes I write these using the logical quantifiers and symbols to get an overview of the problem.) This habit now helps me get hints on how to solve a specific problem and it's also something my high school physics teacher was very adamant on so we could solve physics problems easily (these would mostly involve equations and numbers instead of definitions and concepts, but I have to say that in the end the results are similar in regards to developing problem-solving abilities.) I hope these tips are helpful!!
There is a type of math course for majors to show how to do proofs. Often it is taught in college sophomore year after calculus and before abstract algebra. The textbooks for those kinds of classes often have a variety of proof ideas and techniques. Search for proof methods, mathematical logic, and discrete math to get a variety of answers. Look for well-reviewed and highly rated books, then look for used copies of these texts. Most of these ideas are not brand new, so cheaper, older books may be OK for logic and proof methods.
Having a strong background is always vital. Without decent algebra skills you're gonna suffer in calculus. That applies to number theory too. Also, make sure you don't spend 3 hours on one problem. Take a 5-10 minute break whenever you get stuck for too long (30-60 minutes of struggle). Ponder on the problem on a high level. Let your mind free to find the out-of-the-box solutions for you.
I was known as the roomate that "walks a lot" Every time I would get agitated I'd go on a small walk, "There she goes again" my flatmates would say haha
The part about thinking how you could come to the answer on your own is really important. It applies to other things too: I often use it when solving chess puzzles or watching someone solve sudoku/pencil puzzles. It's also important to realize the path for solving is often different from the path of understanding the problem and finding the proof. A lot of times, it's completely backwards. You may start with "ok, this is what I want to prove. What would have to be true for my conclusion to follow? And what would have to be true for _that_? What are some _consequences_ of this conclusion? Maybe that consequence is an "if and only if", so it's also a prerequisite and j could just prove that the consequence is true". Completely backwards or even wrong from a logic standpoint, but very useful in practice.
My obvious but not obvious advice would be to be methodical. I always start with a literal written list of things I know (givens, assumptions, definitions) and then write a mathematical expression of what I'm trying to prove (eg: proving a number n is even means showing that n = 2k for some integer k). From there, I often let the definitions lead me, constantly referring to the "end goal" as a lodestone. Also, working backwards can help (starting with your conclusion and then see if you can create a backward path from that statement) - I often use the analogy that mazes are often easier to solve backwards.
I am so glad I found this video. Also grateful to that person who asked this wonderful question. I am going through almost a similar kind of problem. I am doing a lot of problem solving but I feel like I haven't reached that level where answers or hints would naturally come to your head. Solving lots of problems or seeing the answers won't necessarily allow you to reach that level cause it's not about solving similar problems, it's something else. I couldn't figure out what that "something else" is till now. Thank you so much. Both of you.
I suggest in addition to everything the Math Sorcerer has said (particularly understanding the provided solution - it may leave out some key points) - is after writing out the solution from memory is (i) make a note of the part of the solution that you didn't find yourself - if you have really tried the problem the key step will leap off the page and (ii) go back to the problem a few days later - certainly no longer than one week - and try the problem again and (iii) once you have mastered the original problem find a very similar but different problem then try that. The part you got stuck on will eventually become clearer in your mind and become a key technique when you see a similar problem in the future. Such an approach takes time and effort.
Teach someone, teach, teach and more! I can't tell you how much trying to teach someone has helped me figure out things. Concepts that I thought I knew, really gave me interesting twists when I would teach, and slowly but steadily the CONCEPTS would solidify. Just giving free lessons to folks has helped me immensely in the learning journey!
One thing you can always do while studying (not only doing exercises and problems but also when reading the theory) is to try to play spontaneously with the theory before attacking the problem set. This freedom will allow you to discover small but interesting facts, which might be key to the solution of some subsequent problems.
true. But the really valuable thing is WHY it works. This is often not even in the textbooks. The really cool thing is coming up with the definitions yourself. Try it. This gives you real intuition.
Had a professor recommend learning a theorem prover for proofs, using it now for solving a good chunk of proofs as a PMATH major, it will definitely be worth it.
You mean you are using artifical intelligence to give u hints ? Theorem prover is lacking fantasy though, so be careful in case you also want to write aesthetic proofs
You grow not at the moment you solve the problem, but during the hours you are trying to solve it. It's completely okay to be trying to solve a hard problem for a week sometimes. Upd: and never read the entire solution. Read it until the "ahh, that's the next step I haven't tried", and then try to finish it yourself.
thank you for this video! i’m in geometry and often get hung up and even skip proofs; i know i’ll have to face them eventually so this motivates me! continue making videos!! God bless you and have a wonderful day ! 😄😄
number theory is a surprisingly difficult start to higher level math. Don’t feel bad if you struggle with it, I think Real Analysis is actually easier at first.
The beauty about proofs is that it’s basically like writing a musical symphony concerning the level we are at , especially when we write aesthetical proofs 😊
Hey, can you make a video on the benefits of mathematics and knowing how to do mathematics? A video which will give us more insight on how math helps you to think logically and analytically in any situation.
When you are stuck on a problem, sometimes you just need to sleep on it. There have been many, many times when I was befuddled with a topic or problem, and I just had to set it aside and go to bed. The next day I would get up, look at the problem, and it would seem almost trivial. I've even woken up in the middle of the night with the solution to a problem, and I would get up, sit at my desk and write it out. Some time away allows information to be integrated in your brain and open new pathways to understanding.
The camera isn't quite focused enough for me to read the titles of the wonderful books that you have in the background, although I know from other videos what to expect since you went through them. Still, it's fun to see them.
I googled what I wanted a proof of, got this video (slap in the face from CZcams lmao), and the best part is the video helped. It was induction. I looked at similar shit and it all was induction.
I recently did my first proof from Bartle Intro to Analysis, starting with writing the definition helped, then something just clicked and I understood what was going and why the case was true. It took me a 2 years to come back to it after being too intimidated
Your first tip is exactly what I would have said - looking at the solutions is not a problem. I remember following this strategy a lot in my abstract maths courses (abstract algebra, analysis): (1) Spend a few minutes trying to solve a problem (2) If stuck, take a peek at the solution (just a peek), looking for that one step to get unstuck (3) Analyse the techniques employed (usually, some kind of simplification or reduction because of a result I didn't know well enough) in that one step (4) Continue the rest of the solution yourself
Indeed, understanding the definitions, coming up with examples & counter-examples develops one's intuition about the subject. "Ya gotta know what we're talkin' about!"
I've been waiting to hear that in doing proofs or ANY math problem, one should take into consideration ALL of the math that they remember and understand and consider applying it to problems where a part might seem difficult but then recall something simple or not to difficult that one can apply like " since a < b and b < c, then a < c " so keep in mind that one shouldn't assume the opposite according to what's needed in the given problem whatever. I know this is very simple and corny, but , . . .
In my experience the questions at the end of the chapter do not just include the thing that was taught in that chapter, rather they include things that were perhaps taught in high school like algebra, trigonometric identities etc. Since they are not explicitly mentioned in the question so no one thinks about them.
I’m taking my first real analysis class and loving it, but I need to pull out an absolutely stellar grade on my next exam. I went looking for some secret ingredient to my proofs on CZcams and seem to have found this video at the perfect time. I wonder if trying out this sort of mindset is what I need to do what feels impossible right now!
you inspired me to launch myself into the world of real analysis! thank you so much, although I know this is hard, I know there are people like you, someone who I can really feel is a human being, even through the screen, who have found ways to get through these tough subjects! thank you so much :D
Great advice! Last night I was working on a set theory proof (symmetric difference property of sets) by proving it directly ..and was really stuck....till I realized to change my approach and switch to a contrapositive method of proof....and it was so easy!
Para mi Read How to do and read proofs de Solow es el mejor libro para aprender una metodología para hacer mucho tipos de demostraciones. Aunque en español solo existe hasta la segunda edición. Con ese libro aprendí a demostrar. Que como ya sabemos, demostrar es otro tipo de problemas en matemáticas. Otro libro bastante útil es: Para Pensar Mejor de Miguel de Guzmán Ozamiz. No sé si exista una versión en inglés pero reúne una gran cantidad de técnicas y maneras de proceder para ejercitarse en la solución de problemas (polya, Masson, Burton yStacey, schoenfeld) mediante la implementación de un protocolo del proceso para el auto análisis y auto corrección de nuestro modo de pensar al querer solucionar problemas. Lo recomiendo bastante para quienes puedan acceder a este libro.
one of the trick I use is to go backwards, start from the conclusion you want to prove, to see if there is any equivalent statement: to prove… it suffice to prove… often the equivalent statement provides some directions
The textbook I'm reading (Elementary Topology Problem Textbook, Viro, Ivanov, Kharlamov, Netsvetaev) assumes that you know a lot of stuff beforehand, and since I've only got rudimentary knowledge on Real Analysis at best. Though the point your late Indian maths professor made on how to solve induction questions sounds like a really good way to build general intuition for the subject. I'm currently on a chapter on Metric Spaces, and with questions here, it's really helpful to know why the restrictions provided by the axioms of a metric space force a particular result or theorem to be true. Though, the thing that's helped a ton with me, is just to walk around while thinking of the problem, or forget the difficult problem, and come back with new eyes.
Do you ever find that proofs often have multiple ways to be proved? I don't know if I'm doing the proof right if my proof doesn't closely mirror the answer.
Having the solution is helpful for me. Just last night I did a radical equation and I didn't know if I was supposed to multiply the square root of 81 or 9 by the square root of 4 or just 4. The book has most of the answers in the back, but some answers it doesn't. It did not have the answer to this problem. Had it had the answer, I might have been able to go back over the problem and understand how someone gets that answer. So I have been on CZcams watching videos about how to do radical equations. Today there was another radical equation. This time the book had the answer and I was able to figure how to get that answer. This is all my attempt to teach myself math. I was watching another video on this channel about a precalculus book and I found it on eBay today. It was the book by James Stewart. I had misconceptions about precalculus until I watched the video here. Precalculus seems to have a little bit of everything-trigonometry, algebra. I think I might like precalculus. Not sure about calculus though. Insert laughing emoji here if I could. I think calculus might to be too advanced. I also ordered Modern Algebra, a Dover book after seeing it on one of these videos, and 2 Blitzer books: College Algebra, and Precalculus. I went a bit overboard today ordering math books. Need a laughing emoji here too.
Study some basic Set Theory first: the use of Sets and Logic are two fundamental tools used in proofs and problems in advanced mathematics. I learned this many years after dropping the second semester of an Abstract Algebra course. (I needed to learn Group Theory at the time, but fell off the Ring.)
Hey Douglas, I really appreciate this advice, and thank your for sharing. I would also like to share, that I also failed my physics: E&M 3 times, and Thermodynamics 2 times. I still hope I can do some astronomy some day, I'm good enough to apply for maths, so every little bit helps get their. Don't give up on your dreams, we can do this!
thank you... i just want to pick up on your point that trying to memorise a solution (after understading it) and being able to rewrite the proofs from memory is actually quite a powerful technique. It means you the techniques are more accessible to you when you need to use them in finding some other proof.
Another GREAT video, I really mean it. You actually managed to *predict* with, mere seconds of advance, my little concern that hit me about memorizing the solution after you look it up in the middle of the video (and resolve it as well). That caught me off guard ahaha! You really love math and helping others loving it as much as you do, if not even more! Keep up your great work :D
Do problems as many as you can; read the book from A to Z, and then from Z to A repeatedly, and try to solve some more problems. Looking different examples from different number theory books will help. That is how I studied mathematics in high school and college.
In my experience those proofs often skip or gloss over steps that are not alway clear or obvious. There is almost always an intuitive leap and conclusion that doesn't makes sense and leaves the student frustrated because there is often a small part missing that is crucial to understanding the proof.
they love to assume you remember some random property of a certain class of functions from a course you muled through a year or two ago. I need refreshers, as an ADHDr. The level of selfishness of many proof writers is audacious in my opinion. because this experience I always explain every step clearly so my reader isn't left feeling the way I did! I'm with you Jon. its like they oversimplify proofs, and overcomplicate the lessons. this has the benefit many people allude to (more time becoming intimated with the material and methods), but thats if you actually have some information (a solution, etc.) or are adept enough at that point to not get stuck. Math education in Uni is a joke. I always learn more at home.
I think this is a problem with Math books in general. The authors are accomplished Mathematicians with decades of experience, I think they forget that some things that are just trivially easy to them (and their colleagues) and so are just skipped over as givens in proofs fly right over the heads of the vast majority of undergrads!
One good thing to do is memorize the problem, and memorize some of the main topics in the section of text that the problem appears in. Get this totally memorized in your head. AND THEN start thinking it through when you are out for a walk or travelling by bus or train or other activity where your mind is free to ponder. Look around the problem from all angles. Try different things, and then mull them over, push hard but retreat back when you get in a muddle, don't completely exhaust yourself. Think creatively, then push hard again when the pieces start coming together and you are starting to see something. Keep asking yourself WHY is this, WHY is that, WHY do these things CONNECT, ask question after question after question after question. When you can answer your own questions you are BECOMING a mathematician. Its like detective work - NOTICING DETAILS - like Lieutenant Columbo or Inspector Morse, deep in thought. THEN the lightbulb comes on - that's a great moment ! One thing I would say is NEVER follow other people's thinking - always ask YOUR OWN QUESTIONS - no matter how foolish a question it might seem - often these questions lead to critical steps forward.
I taught my self how to read and do mathproofs by reading and working through "How to Prove it: A Structured Approach" by Daniel Velleman. Proofs are a different skill set than the cook book methods taught in elementary math classes.
A great proof book, I actually had a hard time starting with this book for my first self-study proofs book, Many people love this book as their first one though! I'm jealous of all of you xD I can't remember the book that finally got it for me, it was a lightish green book I want to say intro to mathematical analysis, If I remember I'll try to look in my bookshelf. The book that helped me would almost test you with exercises after like every paragraph or so, This helped me get in the habit of writing questions in other books like the "How to Prove it!" 🙂
I have actually taught Number Theory using Burton ..Math Majors found the exercises difficult ..Our friend needs to go with a more student oriented book ...The number theory book by Silverman comes to mind ...There are also many great Indian books with lots of solved examples ...Math in general is not a good subject for independent study .... I also want to add that the book "Number Theory Step by Step " by Kuldeep Singh is a very recent book , which is a World Class Masterpiece in Number Theory....
Thanks, what a beautiful resource , and it's free (pdf) and videos to go along with it....much more approachable ( readable ) ... I am gathering acorns for my return to school ...
I'm going to say something similar, but word it differently. 1. Do you have the teachers solutions manual or are you going by the odd answers in the back of the book? True story, bought an out of print Calc book I didn't recognize at a yardsale and the very first answer in the first chapter (review of algebra) was WRONG. Pity the student who beats their head trying to come up with that answer! 2.In most math books every chapter starts with "What you will learn: 1,2,3,4. At the end of this chapter you should know or demonstrate knowledge of these things: 1,2,3,4. Go back and read them! As you are looking at each section it may have some things highlighted, in sidebars or whatever. This is what the questions at the end will be about. They assume you learned the previous chapter, so those skills could be needed too, but the excercise will be to show that you grasp THIS section or chapter. Most math books are organized the same and the chapters even have the same titles. Your bookstore may have previous edition. It will be the same (usuay) as the newest edition, only with different problems and errors that haven't been fixed. If there's 25 problems to solve in one, there will be another 25 in the other, different. Now you have double the problems to practice with. One more thing. Unplug your TV and ps5, Nintendo, Xbox, cancel your Netflix, etc. You don't have time for all that nonsense if you want to study math! :)
"Unplug your TV and ps5, Nintendo, Xbox, cancel your Netflix, etc. You don't have time for all that nonsense if you want to study math! :)" Good advice. I am lucky, I don't have most of those things anyway... Fortunately when I do math I can shut everything else out. For instance, when I do math I usually have my computer right next to me and turned on. And my phone is nearby as well. Nonetheless, they don't distract me. When I am in the midst of maths it's as nothing but my mind, a pencil and some paper is relevent to me. My mind locks out everything else.
where to start is my problem. i go in circles for hours at a time and 7 months has passed. my house is in forclosure and i dont think I'll ever have the confidence again to get another job. I'm so embarassed and ashamed of my actions all around when it comes to math
@@romansynovle990 I'm not there with you so it's hard for me to say. I think you should start with the properties of our numbers and in every math excercise or 'problem',, identify the relevant property or identity and include it in your answer. When you apply the associative property spell it out, commutative, distributive, identity, whatever. I think many teachers would disagree with me, but I can't imagine trying to do math if I didn't know my multiplication tables. How would I factor? You learn to factor polynomials by doing it until your brain smokes. You will start to see the connections in groups of numbers like 3,5,8,15,-2, etc and not waste test time thinking about it If you are past this freshman problems, and not checking your phone every few minutes, I don't know what to say. Start at the beginning and just think about the chapter, what it's about, what you should know or be able to do at the end of it, and try to see the big picture before wading into the exercises. A TIG welder doing fieldwork, plant shutdowns and such makes over $200K a year, and doesn't need much math beyond the simple arithmetic they use to count the bundles of money they rake in weekly.
From your videos "Learn Mathematics from Start to Finish" i could see that the organization of the math knowledge it's a little bit different in each country. From my perspective (i'm brazillian), the best way to learn proofs (i'm sill learning, as a physicist, it's not my core area) it's trought books of "Discrete Mathematics" or "Arithmetic", or even, "Number Theory"... That being said, in my opinion, i think that learning how to write proofs, including logical and set theory thinking, are the best way to start doing more advanced math... About logic, i'm not familiar with english references, but in set theory i would recommend Paul Halmos book "Naive Set Theory"...
Took a Formal Logic class in college as an elective and today I wish I had taken more. I took a lot of math in college. Anyway, in med school I used what logic I could recall in medical problem solving and my classmates believed I just thought differently from everyone else. Science uses many of the same processes in problem solving. First, I'd ask myself am I understanding the question correctly, next, I ask myself to identify which principle is the professor testing my comprehension? Finally, with review of the chapter no answer presents itself I label the missing data as "Blackbox," where the missing Blackbox information falls on a timeline of past data, current data or future data. If past data I would review past math that I may have forgotten as in using calculus to find the volume of a complex three-dimensional structure whether I forgotten an algebraic, geometry or trigonometric function was used. Or in medicine in my current practice a patient problem occurs, and the textbook does not solve I go back to every level of a structure and physiology and see if I can discover the ""Blackbox' issue if still no luck I will read other disciplines texts that may encounter the problem. In math maybe and engineering book, in medicine radiology or surgery or Rheumatology. Cross reading has solved many problems for me, it however is laborious and time consuming. Current Blackbox issues I will consult other texts written by different authors, so on and so forth. I hope this helps young men and women.
Thank you professor! Really important advice in the video, useful to a first time learner as well as to a math teacher or a mentor. And loved the ending: "I feel like doing some problems right now" :)
The techniques learned from solved problems should enhance the learner's ability to come up with techniques of with his/her own in future problem solving bids.
Measure Theory and Lebesgue integration is very difficult . However the fact that very few good books are available on these subjects makes it even more difficult
My best advice is not to feel with low self esteem when you can not figure something out, because math and everything you learn in life is about progressive failure. Everyone talks about the great mathematicians achievement but not about the amount of times they failed to get to that achievement. So do not feel bad if you get stuck and can not produce the result that you want or get frustrated like what happens to me sometimes when doing all types of math proofs and I am 41 years old and still learning about doing math proofs.
I can give an advice that the way solutions are written is not equal to the thought process. To illustrate this, in math olympiad geometry, solutions frequently use some special observations, but thats not how the THOUGHT process a person who solve qn undergoes. So blindly reading the solution might give new math student the wrong idea that math solutions are super different from their approach. I suggest discussing with friends for math problems so that y ou learn with your friends
Me personally the hardest part is understanding the question, usually a solution presents itself once the problem becomes clear. Knowing if a statement is true or false is also difficult for me.
Got 2 and a half weeks til my optimisation exam which requires a surprising amount of proof. Quite anxious and trying to run through a lot of questions, very difficult and pressed for time and quite anxious.
have a question: If every time I go to solve a math exercise and I don't get it because it is too difficult, I look at the answers/ask for help or solve it with some "cheat" method, am I really learning anything? And if I don't learn anything, how do I solve the exercise without help or without having to spend 1 day trying to solve it on my own?
Dear math sorcerer. I am currently taking a calc 3 course in school. Can you please make a video giving advice specific to calc 3? Thank you. From Marcel.
@@marcelgunadi2429 for Calc 3 I honestly think doing as many problems as possible is the best way to study. For each concept do multiple types of problems so you can learn their “tricks”
Sir, in your opinion, what would be the best time for a person (i.e. me) having passed 12th grade to begin with mathematical logic? The field is quite interesting as it looks at maths from a very different way... Thankyou!!
immediatly, do not wait. It is super important not just for math or stem but for just mental growth. What is amazing is if you can get just the basics down and store it long recall, when you take courses that require it you can recall some of the principle and build a great mental map. I beseech you personally also, to do it ASAP. I found logic very scary, and still do, and that is a handicap I set for myself as to not approach it, I'm not doing a good job explaining, but I believe your future self will thank you tremendously! Also there is a lot of notation/symbols that are SOOO NEW, bizarre, AND WEIRRRRD!!! So when you open a book andyou haven't seen logic symbols before you're like OMG!!!! And in pure panic
Hi I’m 17 taking differential equations, multivariable calculus and linear algebra and I wanted to ask how long does it take you to read the average math book and how do you get started with reading math books?
You should also look at the information you were given and ask yourself, "what can I prove from this information?" Also, look at the conclusion and ask yourself, "which results from this section have the same conclusion as the result?" And then try connecting these new endpoints.
Really interesting advice in the video and comments. He touched on it, but commenters, say you have quite a heavy chapter / section, just theorem proof theorem proof, little exposition for many pages. There is a set of questions at the end. Before you do the questions, is there a way you can articulate how you filter for importance? Sometimes I find it hard to know out of all these theorems and proofs which are the most important, would a good heuristic be the earlier ones are usually the most general / important because usually other proofs flow from them? It's just hard to study so much content in one section only to arrive at the questions with a vast library of stuff to categorise. I preferred in other books where the questions were at least after every couple of pages, so I would have a nice bow around the things I had covered, the scope was manageable. Maybe something like, study the first n theorems in a section, then jump to the questions relevant to that immediately? Would you guys often look at the questions first, then study the material with them in mind to see where the emphasis is?
I haven't had anyone explain to my satisfaction why I need to solve problems instead of just looking at the solutions , if they are available. If I look at enough solutions to problems, then solving another similar problem should more or less be a routine thing, I would think. If I can't find solutions in one book, then I just move on to another book that has solutions.
"You'll look at it and you'll say there's no way I would be able to figure that out on my own"...I've said this so many times and I'm only starting my math journey...guess it's time to re-open Book of Proof.
I feel like "solve more problems" is a loose advice because it's notn solely about solving more problems but more like "what can i learn by solving more problems?" "why do they do it that way?"
Best advice I have so far (still a student myself): As you read or study, if you don’t know why something is true, make it a lemma and prove it, anytime a question pops into your head, make it a lemma and prove it, any time the author says something is clear and it isn’t, make it a lemma and prove it. When taking proof based math classes, you should have a notebook that ends up getting cover to cover front and back covevered in lemmas, examples and counter examples, separate from hw and exams study. That’s why, in my opinion, grad school is so hard for people. There were a lot of things you probably took as true, but you never actually figured out why, you just believed it. Believing without seeing was a very hard habit to break. One line at a time friends, and thank you for your videos sir!
What's the time complexity of that solution in big(O) notation? Asking for a friend with not a lot of time...
@@philippezevenberg1332 good joke 🤣 At least the space complexity is clear i.e. one notebook 😁
Mind blown
What’s the difference between a hypothesis and a lemma?
Hypothesis comes out of genuine curiosity
Another tip: when you get stuck, before you look at the answer, ask yourself if there's some assumption or intermediate result that, if true, would enable you to solve the problem. That will help you improve your analytical skills. Also, when you look at the answer, you may discover that you were almost there; you were just missing the one piece. That will boost your confidence.
Yeah!
100% agree. The more general version of this advice is to modify the problem by tweaking some aspect of it (e.g., adding an intermediate result, removing a restriction) and then trying to solve that.
One thing I try to do is when I've exhausted all of my resources and I decide to look at the solution: only look at a tiny part of the solution. Cover up the page or scroll very slowly. I want to see just enough to get me unstuck so that I can try to finish the rest of the proof on my own. I learn a lot more that way than if I look at the entire solution and see every detail laid out for me.
have a question: If every time I go to solve a math exercise and I don't get it because it is too difficult, I look at the answers/ask for help or solve it with some "cheat" method, am I really learning anything? And if I don't learn anything, how do I solve the exercise without help or without having to spend 1 day trying to solve it on my own?
There's a lot of great advice in this video and this comment section. One thing that I have struggled with and still do when faced with hard problems is a feeling that I'm stupid or other people in the class are understanding the material so much easier than I am. This often leads to wanting to give up. I think that it's really important to realize that this negative self talk is not based on reality and is mostly made up by your imagination.
Wow what an espectacular pair of comments you both made. Thanks a lot. That it is what I feel about myself - story of my life - in a few lines. Hats off
@@JohnDoe-kh3hy keep pushing you’re able to get through it! God bless you! have a wonderful day 😄
That negative self-talk about how everyone is more equipped than you are is called Imposter syndrome. When I started grad school I struggled with the same thing, but I eventually convinced myself with help from advisors/friends that I am good enough to be there. Trust the process and trust the admission committee's decision to admit you to grad school. Good luck!
It also helps to remember that everyone has different experiences. I loved maths as a kid, spent my summer holidays doing math camps for fun, and I gained expiences with proofs and the basic concepts of advanced maths there. I spent a lot of time struggling and feeling stuck and not understanding "basic" ideas. Many of my coursemates weren't like that, so they struggle with the same concepts now.
Often, if someone seems to be "quicker" than you they just know the material already. Chances are they spent just as much time struggling as you do now, and in a year you'll find todays material much easier too.
Totally agree with MS on using the material help solve the problem. Also, rather than spend endless hours on one problem, come back to it after a break of several hours or even a day. I often filter out key information or make a wrong assumption at first glance. Many times I can return with a fresh perspective and solve the same problem easily. Best of luck
Super Good Advice!!!!!
@D hickey
Gotta agree there was number theory problem I couldn’t solve at 6am and I did it next day in like the11am lol it’s super effective
@@rssl5500 so you started it at 11am? or you finished/solved at 11?
As someone who's repeating their first year in their math major (and someone who's slowly getting the hang of coming up with proofs) I would say that most of the tips I would've given to myself two years ago are things that you have already said: understand the definitions (this was something important to me, since I thought I understood them at first glance when it wasn't the case), look up previous things that have been mentioned before, such as theorems, propositions and definitions and apply them to your problem and reflect on the solution that are already given (or the one you just came up with.)
Another thing I would have told myself (similar as looking up the solutions to proofs and summarizing them in your own words) is to look up and also summarize the proofs given in the books you're reading. This will give you a feeling on how proofs should be written, when can certain proof techniques be used and also keep your thoughts organized when it comes to a specific problem.
Finally, sometimes the issue I had in not being able to solve a problem is that I wouldn't even understand what the problem was (as in, I didn't understand what exactly it was that I wanted to prove.) What I had to do in order to get over this hurdle was to relearn the habit of writing down what I already knew about the topic of the problem and the goal I wanted to achieve solving this problem (regarding definitions, propositions, etc. Sometimes I write these using the logical quantifiers and symbols to get an overview of the problem.)
This habit now helps me get hints on how to solve a specific problem and it's also something my high school physics teacher was very adamant on so we could solve physics problems easily (these would mostly involve equations and numbers instead of definitions and concepts, but I have to say that in the end the results are similar in regards to developing problem-solving abilities.)
I hope these tips are helpful!!
Excellent thank you!
There is a type of math course for majors to show how to do proofs. Often it is taught in college sophomore year after calculus and before abstract algebra. The textbooks for those kinds of classes often have a variety of proof ideas and techniques. Search for proof methods, mathematical logic, and discrete math to get a variety of answers. Look for well-reviewed and highly rated books, then look for used copies of these texts. Most of these ideas are not brand new, so cheaper, older books may be OK for logic and proof methods.
Having a strong background is always vital. Without decent algebra skills you're gonna suffer in calculus. That applies to number theory too. Also, make sure you don't spend 3 hours on one problem. Take a 5-10 minute break whenever you get stuck for too long (30-60 minutes of struggle). Ponder on the problem on a high level. Let your mind free to find the out-of-the-box solutions for you.
I was known as the roomate that "walks a lot"
Every time I would get agitated I'd go on a small walk,
"There she goes again" my flatmates would say haha
The part about thinking how you could come to the answer on your own is really important. It applies to other things too: I often use it when solving chess puzzles or watching someone solve sudoku/pencil puzzles.
It's also important to realize the path for solving is often different from the path of understanding the problem and finding the proof. A lot of times, it's completely backwards. You may start with "ok, this is what I want to prove. What would have to be true for my conclusion to follow? And what would have to be true for _that_? What are some _consequences_ of this conclusion? Maybe that consequence is an "if and only if", so it's also a prerequisite and j could just prove that the consequence is true". Completely backwards or even wrong from a logic standpoint, but very useful in practice.
My obvious but not obvious advice would be to be methodical. I always start with a literal written list of things I know (givens, assumptions, definitions) and then write a mathematical expression of what I'm trying to prove (eg: proving a number n is even means showing that n = 2k for some integer k). From there, I often let the definitions lead me, constantly referring to the "end goal" as a lodestone. Also, working backwards can help (starting with your conclusion and then see if you can create a backward path from that statement) - I often use the analogy that mazes are often easier to solve backwards.
I am so glad I found this video. Also grateful to that person who asked this wonderful question. I am going through almost a similar kind of problem. I am doing a lot of problem solving but I feel like I haven't reached that level where answers or hints would naturally come to your head. Solving lots of problems or seeing the answers won't necessarily allow you to reach that level cause it's not about solving similar problems, it's something else. I couldn't figure out what that "something else" is till now.
Thank you so much. Both of you.
I suggest in addition to everything the Math Sorcerer has said (particularly understanding the provided solution - it may leave out some key points) - is after writing out the solution from memory is (i) make a note of the part of the solution that you didn't find yourself - if you have really tried the problem the key step will leap off the page and (ii) go back to the problem a few days later - certainly no longer than one week - and try the problem again and (iii) once you have mastered the original problem find a very similar but different problem then try that. The part you got stuck on will eventually become clearer in your mind and become a key technique when you see a similar problem in the future. Such an approach takes time and effort.
Teach someone, teach, teach and more! I can't tell you how much trying to teach someone has helped me figure out things. Concepts that I thought I knew, really gave me interesting twists when I would teach, and slowly but steadily the CONCEPTS would solidify. Just giving free lessons to folks has helped me immensely in the learning journey!
There is a deeper truth behind all this -- by helping others you in fact help yourself.
One thing you can always do while studying (not only doing exercises and problems but also when reading the theory) is to try to play spontaneously with the theory before attacking the problem set. This freedom will allow you to discover small but interesting facts, which might be key to the solution of some subsequent problems.
This is a key skill if you wanna become a researcher as well.
Understanding how it works is more important than memorizing it
YEAH
true. But the really valuable thing is WHY it works. This is often not even in the textbooks. The really cool thing is coming up with the definitions yourself. Try it. This gives you real intuition.
Had a professor recommend learning a theorem prover for proofs, using it now for solving a good chunk of proofs as a PMATH major, it will definitely be worth it.
You mean you are using artifical intelligence to give u hints ? Theorem prover is lacking fantasy though, so be careful in case you also want to write aesthetic proofs
Math proof advice: ignore math proof rules and just use philosophy proof rules. Same destination, different road.
You grow not at the moment you solve the problem, but during the hours you are trying to solve it. It's completely okay to be trying to solve a hard problem for a week sometimes.
Upd: and never read the entire solution. Read it until the "ahh, that's the next step I haven't tried", and then try to finish it yourself.
solid advice! I especially like the last part about reading it until the "ahh that's the next step I haven't tried" moment. Thank you!
thank you for this video! i’m in geometry and often get hung up and even skip proofs; i know i’ll have to face them eventually so this motivates me! continue making videos!! God bless you and have a wonderful day ! 😄😄
number theory is a surprisingly difficult start to higher level math. Don’t feel bad if you struggle with it, I think Real Analysis is actually easier at first.
The beauty about proofs is that it’s basically like writing a musical symphony concerning the level we are at , especially when we write aesthetical proofs 😊
Hey, can you make a video on the benefits of mathematics and knowing how to do mathematics? A video which will give us more insight on how math helps you to think logically and analytically in any situation.
When you are stuck on a problem, sometimes you just need to sleep on it. There have been many, many times when I was befuddled with a topic or problem, and I just had to set it aside and go to bed. The next day I would get up, look at the problem, and it would seem almost trivial. I've even woken up in the middle of the night with the solution to a problem, and I would get up, sit at my desk and write it out. Some time away allows information to be integrated in your brain and open new pathways to understanding.
The camera isn't quite focused enough for me to read the titles of the wonderful books that you have in the background, although I know from other videos what to expect since you went through them. Still, it's fun to see them.
I googled what I wanted a proof of, got this video (slap in the face from CZcams lmao), and the best part is the video helped. It was induction. I looked at similar shit and it all was induction.
I recently did my first proof from Bartle Intro to Analysis, starting with writing the definition helped,
then something just clicked and I understood what was going and why the case was true.
It took me a 2 years to come back to it after being too intimidated
Not the big brother we deserved but the big brother we needed.
Your first tip is exactly what I would have said - looking at the solutions is not a problem. I remember following this strategy a lot in my abstract maths courses (abstract algebra, analysis):
(1) Spend a few minutes trying to solve a problem
(2) If stuck, take a peek at the solution (just a peek), looking for that one step to get unstuck
(3) Analyse the techniques employed (usually, some kind of simplification or reduction because of a result I didn't know well enough) in that one step
(4) Continue the rest of the solution yourself
Indeed, understanding the definitions, coming up with examples & counter-examples develops one's intuition about the subject.
"Ya gotta know what we're talkin' about!"
I've been waiting to hear that in doing proofs or ANY math problem, one should take into consideration ALL of the math that they remember and understand and consider applying it to problems where a part might seem difficult but then recall something simple or not to difficult
that one can apply like " since a < b and b < c, then a < c " so keep in mind that one shouldn't assume the opposite according to what's needed in the given problem whatever. I know this is very simple and corny, but , . . .
In my experience the questions at the end of the chapter do not just include the thing that was taught in that chapter, rather they include things that were perhaps taught in high school like algebra, trigonometric identities etc. Since they are not explicitly mentioned in the question so no one thinks about them.
The same once struck me... Thanks for dealing with this, Math Sorcerer!
I’m taking my first real analysis class and loving it, but I need to pull out an absolutely stellar grade on my next exam. I went looking for some secret ingredient to my proofs on CZcams and seem to have found this video at the perfect time. I wonder if trying out this sort of mindset is what I need to do what feels impossible right now!
Best wishes on your real analysis course Maddy!
you inspired me to launch myself into the world of real analysis! thank you so much, although I know this is hard, I know there are people like you, someone who I can really feel is a human being, even through the screen, who have found ways to get through these tough subjects! thank you so much :D
Great advice! Last night I was working on a set theory proof (symmetric difference property of sets) by proving it directly ..and was really stuck....till I realized to change my approach and switch to a contrapositive method of proof....and it was so easy!
Para mi Read How to do and read proofs de Solow es el mejor libro para aprender una metodología para hacer mucho tipos de demostraciones. Aunque en español solo existe hasta la segunda edición. Con ese libro aprendí a demostrar. Que como ya sabemos, demostrar es otro tipo de problemas en matemáticas.
Otro libro bastante útil es: Para Pensar Mejor de Miguel de Guzmán Ozamiz. No sé si exista una versión en inglés pero reúne una gran cantidad de técnicas y maneras de proceder para ejercitarse en la solución de problemas (polya, Masson, Burton yStacey, schoenfeld) mediante la implementación de un protocolo del proceso para el auto análisis y auto corrección de nuestro modo de pensar al querer solucionar problemas. Lo recomiendo bastante para quienes puedan acceder a este libro.
one of the trick I use is to go backwards, start from the conclusion you want to prove, to see if there is any equivalent statement: to prove… it suffice to prove… often the equivalent statement provides some directions
This is something i needed badly💝
I think it will help, A LOT!! Good luck:)
The textbook I'm reading (Elementary Topology Problem Textbook, Viro, Ivanov, Kharlamov, Netsvetaev) assumes that you know a lot of stuff beforehand, and since I've only got rudimentary knowledge on Real Analysis at best. Though the point your late Indian maths professor made on how to solve induction questions sounds like a really good way to build general intuition for the subject. I'm currently on a chapter on Metric Spaces, and with questions here, it's really helpful to know why the restrictions provided by the axioms of a metric space force a particular result or theorem to be true. Though, the thing that's helped a ton with me, is just to walk around while thinking of the problem, or forget the difficult problem, and come back with new eyes.
Do you ever find that proofs often have multiple ways to be proved? I don't know if I'm doing the proof right if my proof doesn't closely mirror the answer.
Having the solution is helpful for me. Just last night I did a radical equation and I didn't know if I was supposed to multiply the square root of 81 or 9 by the square root of 4 or just 4. The book has most of the answers in the back, but some answers it doesn't. It did not have the answer to this problem. Had it had the answer, I might have been able to go back over the problem and understand how someone gets that answer. So I have been on CZcams watching videos about how to do radical equations. Today there was another radical equation. This time the book had the answer and I was able to figure how to get that answer. This is all my attempt to teach myself math. I was watching another video on this channel about a precalculus book and I found it on eBay today. It was the book by James Stewart. I had misconceptions about precalculus until I watched the video here. Precalculus seems to have a little bit of everything-trigonometry, algebra. I think I might like precalculus. Not sure about calculus though. Insert laughing emoji here if I could. I think calculus might to be too advanced. I also ordered Modern Algebra, a Dover book after seeing it on one of these videos, and 2 Blitzer books: College Algebra, and Precalculus. I went a bit overboard today ordering math books. Need a laughing emoji here too.
Study some basic Set Theory first: the use of Sets and Logic are two fundamental tools used in proofs and problems in advanced mathematics.
I learned this many years after dropping the second semester of an Abstract Algebra course. (I needed to learn Group Theory at the time, but fell off the Ring.)
Hey Douglas, I really appreciate this advice, and thank your for sharing.
I would also like to share, that I also failed my physics: E&M 3 times, and Thermodynamics 2 times.
I still hope I can do some astronomy some day, I'm good enough to apply for maths, so every little bit helps get their.
Don't give up on your dreams, we can do this!
@@leeming1317 I learned the difference between Mathematicians' Math and Physicists' Math.
thank you... i just want to pick up on your point that trying to memorise a solution (after understading it) and being able to rewrite the proofs from memory is actually quite a powerful technique. It means you the techniques are more accessible to you when you need to use them in finding some other proof.
Another GREAT video, I really mean it. You actually managed to *predict* with, mere seconds of advance, my little concern that hit me about memorizing the solution after you look it up in the middle of the video (and resolve it as well). That caught me off guard ahaha! You really love math and helping others loving it as much as you do, if not even more! Keep up your great work :D
AYYYYY TRIG COURSE IS OUT WOOOO
Do problems as many as you can; read the book from A to Z, and then from Z to A repeatedly, and try to solve some more problems. Looking different examples from different number theory books will help. That is how I studied mathematics in high school and college.
great advice, thank you!
In my experience those proofs often skip or gloss over steps that are not alway clear or obvious. There is almost always an intuitive leap and conclusion that doesn't makes sense and leaves the student frustrated because there is often a small part missing that is crucial to understanding the proof.
they love to assume you remember some random property of a certain class of functions from a course you muled through a year or two ago. I need refreshers, as an ADHDr. The level of selfishness of many proof writers is audacious in my opinion. because this experience I always explain every step clearly so my reader isn't left feeling the way I did! I'm with you Jon.
its like they oversimplify proofs, and overcomplicate the lessons. this has the benefit many people allude to (more time becoming intimated with the material and methods), but thats if you actually have some information (a solution, etc.) or are adept enough at that point to not get stuck. Math education in Uni is a joke. I always learn more at home.
I think this is a problem with Math books in general. The authors are accomplished Mathematicians with decades of experience, I think they forget that some things that are just trivially easy to them (and their colleagues) and so are just skipped over as givens in proofs fly right over the heads of the vast majority of undergrads!
One good thing to do is memorize the problem, and memorize some of the main topics in the section of text that the problem appears in. Get this totally memorized in your head. AND THEN start thinking it through when you are out for a walk or travelling by bus or train or other activity where your mind is free to ponder. Look around the problem from all angles. Try different things, and then mull them over, push hard but retreat back when you get in a muddle, don't completely exhaust yourself. Think creatively, then push hard again when the pieces start coming together and you are starting to see something. Keep asking yourself WHY is this, WHY is that, WHY do these things CONNECT, ask question after question after question after question. When you can answer your own questions you are BECOMING a mathematician. Its like detective work - NOTICING DETAILS - like Lieutenant Columbo or Inspector Morse, deep in thought. THEN the lightbulb comes on - that's a great moment ! One thing I would say is NEVER follow other people's thinking - always ask YOUR OWN QUESTIONS - no matter how foolish a question it might seem - often these questions lead to critical steps forward.
I truly don’t know what made me go down this rabbit hole *but it’s a long time coming & I owe it to myself to do it right* #Subscribed
Fantastic video, thank you!
you are very welcome!
I taught my self how to read and do mathproofs by reading and working through "How to Prove it: A Structured Approach" by Daniel Velleman. Proofs are a different skill set than the cook book methods taught in elementary math classes.
A great proof book,
I actually had a hard time starting with this book for my first self-study proofs book,
Many people love this book as their first one though! I'm jealous of all of you xD
I can't remember the book that finally got it for me, it was a lightish green book I want to say intro to mathematical analysis,
If I remember I'll try to look in my bookshelf.
The book that helped me would almost test you with exercises after like every paragraph or so,
This helped me get in the habit of writing questions in other books like the "How to Prove it!" 🙂
I have actually taught Number Theory using Burton ..Math Majors found the exercises difficult ..Our friend needs to go with a more student oriented book ...The number theory book by Silverman comes to mind ...There are also many great Indian books with lots of solved examples ...Math in general is not a good subject for independent study .... I also want to add that the book "Number Theory Step by Step " by Kuldeep Singh is a very recent book , which is a World Class Masterpiece in Number Theory....
Thanks, what a beautiful resource , and it's free (pdf) and videos to go along with it....much more approachable ( readable ) ... I am gathering acorns for my return to school ...
I'm going to say something similar, but word it differently. 1. Do you have the teachers solutions manual or are you going by the odd answers in the back of the book? True story, bought an out of print Calc book I didn't recognize at a yardsale and the very first answer in the first chapter (review of algebra) was WRONG. Pity the student who beats their head trying to come up with that answer! 2.In most math books every chapter starts with "What you will learn: 1,2,3,4. At the end of this chapter you should know or demonstrate knowledge of these things: 1,2,3,4. Go back and read them! As you are looking at each section it may have some things highlighted, in sidebars or whatever. This is what the questions at the end will be about. They assume you learned the previous chapter, so those skills could be needed too, but the excercise will be to show that you grasp THIS section or chapter.
Most math books are organized the same and the chapters even have the same titles. Your bookstore may have previous edition. It will be the same (usuay) as the newest edition, only with different problems and errors that haven't been fixed. If there's 25 problems to solve in one, there will be another 25 in the other, different.
Now you have double the problems to practice with.
One more thing. Unplug your TV and ps5, Nintendo, Xbox, cancel your Netflix, etc. You don't have time for all that nonsense if you want to study math! :)
Amazing advice, thank you for posting this!
"Unplug your TV and ps5, Nintendo, Xbox, cancel your Netflix, etc. You don't have time for all that nonsense if you want to study math! :)"
Good advice. I am lucky, I don't have most of those things anyway... Fortunately when I do math I can shut everything else out. For instance, when I do math I usually have my computer right next to me and turned on. And my phone is nearby as well. Nonetheless, they don't distract me. When I am in the midst of maths it's as nothing but my mind, a pencil and some paper is relevent to me. My mind locks out everything else.
where to start is my problem. i go in circles for hours at a time and 7 months has passed. my house is in forclosure and i dont think I'll ever have the confidence again to get another job. I'm so embarassed and ashamed of my actions all around when it comes to math
@@romansynovle990 I'm not there with you so it's hard for me to say. I think you should start with the properties of our numbers and in every math excercise or 'problem',, identify the relevant property or identity and include it in your answer. When you apply the associative property spell it out, commutative, distributive, identity, whatever. I think many teachers would disagree with me, but I can't imagine trying to do math if I didn't know my multiplication tables. How would I factor? You learn to factor polynomials by doing it until your brain smokes. You will start to see the connections in groups of numbers like 3,5,8,15,-2, etc and not waste test time thinking about it
If you are past this freshman problems, and not checking your phone every few minutes, I don't know what to say. Start at the beginning and just think about the chapter, what it's about, what you should know or be able to do at the end of it, and try to see the big picture before wading into the exercises. A TIG welder doing fieldwork, plant shutdowns and such makes over $200K a year, and doesn't need much math beyond the simple arithmetic they use to count the bundles of money they rake in weekly.
Very good tips!
From your videos "Learn Mathematics from Start to Finish" i could see that the organization of the math knowledge it's a little bit different in each country. From my perspective (i'm brazillian), the best way to learn proofs (i'm sill learning, as a physicist, it's not my core area) it's trought books of "Discrete Mathematics" or "Arithmetic", or even, "Number Theory"... That being said, in my opinion, i think that learning how to write proofs, including logical and set theory thinking, are the best way to start doing more advanced math... About logic, i'm not familiar with english references, but in set theory i would recommend Paul Halmos book "Naive Set Theory"...
Took a Formal Logic class in college as an elective and today I wish I had taken more. I took a lot of math in college. Anyway, in med school I used what logic I could recall in medical problem solving and my classmates believed I just thought differently from everyone else.
Science uses many of the same processes in problem solving. First, I'd ask myself am I understanding the question correctly, next, I ask myself to identify which principle is the professor testing my comprehension? Finally, with review of the chapter no answer presents itself I label the missing data as "Blackbox," where the missing Blackbox information falls on a timeline of past data, current data or future data.
If past data I would review past math that I may have forgotten as in using calculus to find the volume of a complex three-dimensional structure whether I forgotten an algebraic, geometry or trigonometric function was used. Or in medicine in my current practice a patient problem occurs, and the textbook does not solve I go back to every level of a structure and physiology and see if I can discover the ""Blackbox' issue if still no luck I will read other disciplines texts that may encounter the problem. In math maybe and engineering book, in medicine radiology or surgery or Rheumatology. Cross reading has solved many problems for me, it however is laborious and time consuming. Current Blackbox issues I will consult other texts written by different authors, so on and so forth. I hope this helps young men and women.
Thank you professor! Really important advice in the video, useful to a first time learner as well as to a math teacher or a mentor. And loved the ending: "I feel like doing some problems right now" :)
Trying to solve a problem is never a waste of time. Never.
I still have the unsatisfied feeling of having failed, when I am forced to peek at a solution. Is it possible to overcome that?
Excellent advice in the video.😀
The techniques learned from solved problems should enhance the learner's ability to come up with techniques of with his/her own in future problem solving bids.
I share this opinion and relate to what you've said, it really does work
Very good points of advice. Thank you
I certainly need this video for Real Analysis. Measure Theory and Lebesgue Integration are very challenging for me.
Measure Theory and Lebesgue integration is very difficult . However the fact that very few good books are available on these subjects makes it even more difficult
Measure Theory by Malik , Gupta , and Mittal (stay away from American Authors)
Simple cliche that is true: Math is hard, but there’s value in that.
I use this same strategy when I rewrite my homework exercises. I rework the problems without looking at the answer.
My best advice is not to feel with low self esteem when you can not figure something out, because math and everything you learn in life is about progressive failure. Everyone talks about the great mathematicians achievement but not about the amount of times they failed to get to that achievement. So do not feel bad if you get stuck and can not produce the result that you want or get frustrated like what happens to me sometimes when doing all types of math proofs and I am 41 years old and still learning about doing math proofs.
I can give an advice that the way solutions are written is not equal to the thought process. To illustrate this, in math olympiad geometry, solutions frequently use some special observations, but thats not how the THOUGHT process a person who solve qn undergoes. So blindly reading the solution might give new math student the wrong idea that math solutions are super different from their approach. I suggest discussing with friends for math problems so that y
ou learn with your friends
Great channel, very encouraging.
And go step by step yes
Beautiful word.
Mathematician George Polya has written two classics on creativity and problem solving: "How to Solve It" and "Induction and Analogy in Mathematics".
Me personally the hardest part is understanding the question, usually a solution presents itself once the problem becomes clear. Knowing if a statement is true or false is also difficult for me.
Really good teacher
Got 2 and a half weeks til my optimisation exam which requires a surprising amount of proof. Quite anxious and trying to run through a lot of questions, very difficult and pressed for time and quite anxious.
have a question: If every time I go to solve a math exercise and I don't get it because it is too difficult, I look at the answers/ask for help or solve it with some "cheat" method, am I really learning anything? And if I don't learn anything, how do I solve the exercise without help or without having to spend 1 day trying to solve it on my own?
Geez, this one made a lightbulb turn on in my head!
do you have more tips on this? I struggle really bad with this and can't even solve a single simple problem!
yes tons! After making this video I realized I have much more that can help! I will make another video:)
Dear math sorcerer. I am currently taking a calc 3 course in school. Can you please make a video giving advice specific to calc 3? Thank you. From Marcel.
P.S I have a final in about a week, so I would like your thoughts on how to go about studying
@@marcelgunadi2429 for Calc 3 I honestly think doing as many problems as possible is the best way to study. For each concept do multiple types of problems so you can learn their “tricks”
Sir,
in your opinion, what would be the best time for a person (i.e. me) having passed 12th grade to begin with mathematical logic?
The field is quite interesting as it looks at maths from a very different way...
Thankyou!!
immediatly, do not wait. It is super important not just for math or stem but for just mental growth.
What is amazing is if you can get just the basics down and store it long recall,
when you take courses that require it you can recall some of the principle and build a great mental map.
I beseech you personally also, to do it ASAP.
I found logic very scary, and still do, and that is a handicap I set for myself as to not approach it,
I'm not doing a good job explaining, but I believe your future self will thank you tremendously!
Also there is a lot of notation/symbols that are SOOO NEW, bizarre, AND WEIRRRRD!!!
So when you open a book andyou haven't seen logic symbols before you're like OMG!!!! And in pure panic
Really helped. Sir.
Hi I’m 17 taking differential equations, multivariable calculus and linear algebra and I wanted to ask how long does it take you to read the average math book and how do you get started with reading math books?
He has a video titled "How to read a Math Book".
You should also look at the information you were given and ask yourself, "what can I prove from this information?" Also, look at the conclusion and ask yourself, "which results from this section have the same conclusion as the result?" And then try connecting these new endpoints.
Really interesting advice in the video and comments. He touched on it, but commenters, say you have quite a heavy chapter / section, just theorem proof theorem proof, little exposition for many pages. There is a set of questions at the end. Before you do the questions, is there a way you can articulate how you filter for importance? Sometimes I find it hard to know out of all these theorems and proofs which are the most important, would a good heuristic be the earlier ones are usually the most general / important because usually other proofs flow from them?
It's just hard to study so much content in one section only to arrive at the questions with a vast library of stuff to categorise. I preferred in other books where the questions were at least after every couple of pages, so I would have a nice bow around the things I had covered, the scope was manageable.
Maybe something like, study the first n theorems in a section, then jump to the questions relevant to that immediately? Would you guys often look at the questions first, then study the material with them in mind to see where the emphasis is?
❤️❤️❤️
Correct me if I’m wrong, isn’t the base for mathematics propositional logic??
very helpful
I have the worst situation when I solve in my way, I can not figure out if my solutions are correct or not, do you have any advice for that?
I feel guilty and defeated if I look at the solutions.
I have a question which I'm not satisfied. A professor said 0 divides 0. Is it true?
It starts with a question
What type of math notebook should I get for pre calc throughout calc? What do you recommend as a good cheap option besides a 70 sheet page notebook?
Put some good problems on the math sorcerers forums so I have something to do!
Odd question... but would you ever consider doing videos on implementing numerical algorithms in Python (or whatever language of your choice)?
Hey review number theory by Burton . I am too doing that book lately :()
I haven't had anyone explain to my satisfaction why I need to solve problems instead of just looking at the solutions , if they are available. If I look at enough solutions to problems, then solving another similar problem should more or less be a routine thing, I would think. If I can't find solutions in one book, then I just move on to another book that has solutions.
Studying/Mastering Math, Physics, Music, etc. are all learn-by-doing activities.
What ideas
I feel like Issac Newton is teaching me math lol
Hi, i am 10th grade what should i learn at mathematics?
"You'll look at it and you'll say there's no way I would be able to figure that out on my own"...I've said this so many times and I'm only starting my math journey...guess it's time to re-open Book of Proof.
I feel like "solve more problems" is a loose advice because it's notn solely about solving more problems but more like "what can i learn by solving more problems?" "why do they do it that way?"