Mathematics can be frustrating. Sometimes we need a mental break. I know I've been to the point of quitting math a couple of times. One of the undergrad advisors has said that when you are working on your Ph.D. in math, your love for math needs to be just a little greater than your tolerance for the frustration of the work. That's a paraphrase. It was an important idea. Perspective is important in anything. Why do we do the things we do? Psychology suggests: we don't always know. We need to seek to understand what is really important to us in life.
Thanks for the video!!! 🙏🙏🙏 hope the new year is going great 🎉🎉🎉 By the way I have a suggestion for you: I'm reading a book on differential forms by Fortney (Springer edition), and it's wonderful!! Should you ever wanna learn df, it's the best introductory book, with a lot of pictures and intuition! PS: i found a used copy of the Oprea book on differential geometry and will couple it with the book by Tapp when the time comes
Thanks! I hope the year is going great for you as well. Yeah, that's a great suggestion! I had started to read it when I started getting back into math in ~2021. The one by Collier is also very good. It's much more informal. Maybe I'll give DF another try, but I just so happened to find myself another good book on functional analysis! Thanks for that Tapp recommendation, by the way. I've been considering getting it, but I already have quite a few books on the subject. What are you currently working on, besides the Fortney book? Also, thoughts on DF as a subject?
@MathematicalToolbox Well, since I work full time, I can't really afford to study two subjects at once. If I get tired of working on a single subject, I pause it (like i put on hold Complex Analysis to work on df). Differential forms is a very cool subject, conceptually not so easy (they are multilinear alternating forms on the tangent space of a manifold), and feels more advanced than the usual stuff, probably because they are less known. they're also a good way to get into differential geometry since integration on manifolds is done essentially through differential forms. My interest in differential forms and differential geometry is mainly to get to study general relativity and cosmology. Functional analysis looks always a bit intimidating 😁 what new reference did you found?
Today I show my most read books of 2024. If you want to buy any of these books be sure to check out our affiliate links! Also check out our Patreon or our TH-cam Memberships! As of the upload of this video, I am still in the process of importing content from Patreon. I ask for your patience. Thank you for supporting us! Shoutout to @CrazyShores for encouraging me to do this video! I also apologize for the echo present in the video.
at 2.19 what do you mean by the example of R as an R-vector space being too much for someone who has learned Linear Algebra? Isn't that the whole point of a first course in Linear Algebra?
Yeah, you're exactly right with that particular example and the purpose of a first course in linear algebra! What I'm trying to say is that the amount of abstract examples (e.g., "X has this property" or "X satisfies this condition" without first showing the reader how to do it) in this book might be overwhelming for an audience equipped only with linear algebra and calculus. A student with analysis (and linear algebra) should be better prepared for this book in general. Whether we agree or not, I hope I've made my point clear. Thanks for your comment!
I think I would also enjoy the book by Malik. But I just finished Calc I so I don't think I'll be picking it up any time soon. I think I'm ready to dive into Real Analysis, do you recommend the book by Tao? Its hilarious that you hate discrete probability lol.
I would recommend an intro to math proof first, but feel free to give Tao a shot. Tao is good if you want a hard book that starts from first principles. You can also try Ash's book on Real Variables and Basic Metric Space Topology. It has full solutions.
@@MathematicalToolbox Thanks for the additional Recco's! I think I'm pretty good on proof, I've been doing all the Geometry and Calc proofs. I even did some basic Topology proofs. Haven't gotten to the Metric Spaces yet, I will check that out.
As far as books that I dedicated more than around five hours to, that's about it. There are some other books that I worked on for the Patreon articles, but the list is probably greater than 20 books. Mostly analysis and differential equations type stuff. What did you read last year?
@@MathematicalToolbox Just the ones that were in my syllabus. Here's the list - 1. Discrete Math with Graph Theory - Goodaire/Parmenter 2. Abstract Algebra - Gallian 3. Operations Research - Taha 4. Elements of Real Analysis - Denlinger
Great reviews. A very nice and friendly book on functional analysis is "An Introduction to Functional Analysis" by James C. Robinson (2020) - solutions to all exercises in back! ...
I could've sworn I had checked that one out before, but I think I'm confusing it with another! That one looks very well written and easy to read. Great suggestion, sir!
Depends on the book and the layout of the exercises. If they are dispersed throughout the text then yes, I will. If they are at the end of a section then I'll do 5-10 exercises. What about you?
Did you know that a meta study from 2023 that looked at tens of others studies on worked examples in the field of math education has been determined to be MORE beneficial than hard core problem solving itself for students learning mathematics? The reason is simple. Any problem that is truly fully solved in detail will identify your gaps in understanding. Pure problem solving will take longer and there will exist WAY more frustration and you will never truly have insight if there are small nuances you have not mastered on some math building blocks that are prerequisite to your topic of study. You can then LATER add problem solving but only after you have basic mastery of the concepts that are new to you. Books on any undergraduate math and graduate math that will be structured in the style of fully worked examples will be a goldmine. Sadly I doubt many will take the lead to make those kind of books even though there is right now actual hard core evidence that for students learning new math topics or new math areas, worked examples done right will ALWAYS be significantly better than you having to be subjected to enormous cognitive load that is typical of standard problem solving, aka MATH PHOBIA. Math phobia exists only when cognitive load is high(typical problem solving) and specially in middle school and high school when you never get proper explanations of the concepts and mechanisms of WHY every formula works as it does, that you are forced to just accept. If you find some pioneer educators out there who tried really hard to make formal logic and proofs accessible(without over simplification) to students in high school and even maybe at the end of middle school please let me know. I'll probably buy every one of those books because I'm interested in the pedagogical tools and examples they use to make proofs and logic digestible to way younger people than what's currently done today where mostly only once you get to college/university you start learning how to read , understand and write proofs.
@@MathematicalToolbox To find the article paste into google GUEST POST: Worked Examples: An Effective Tool for Math Learning I suggest you also read the meta study itself. But the conclusion in the post is why I wrote to you just so you're aware that in proper math pedagogy research academia this fact is well known. Sadly this info did not pass down to actual amateur and pro mathematician "normies" common knowledge. Conclusion of the article: "As we learned in this post, studying worked examples leads to better performance than problem solving. But studying worked examples has another advantage: it can help you gauge (or monitor) your knowledge. We tend to overestimate what we know, which can lead us astray when studying - by spending too little time on material we need to learn. Fortunately, a few studies have found that studying worked examples reduces this overconfidence (5, 6), which can help you direct your study efforts more appropriately."
Amazing review.
@@achunaryan3418 thank you! Which was your favorite from here?
Also, what books did you read in 2024?
Thank you!
@@MathematicalToolboxo Reilly programming quantum computers, understanding deep learning j.d. prince, and applied akka patterns by michael nash
Mathematics can be frustrating. Sometimes we need a mental break. I know I've been to the point of quitting math a couple of times. One of the undergrad advisors has said that when you are working on your Ph.D. in math, your love for math needs to be just a little greater than your tolerance for the frustration of the work. That's a paraphrase. It was an important idea. Perspective is important in anything. Why do we do the things we do? Psychology suggests: we don't always know. We need to seek to understand what is really important to us in life.
This is really good philosophy/advice. The next time I talk about a related topic, I'll share your input and shout you out!
Thanks for the video!!! 🙏🙏🙏 hope the new year is going great 🎉🎉🎉
By the way I have a suggestion for you: I'm reading a book on differential forms by Fortney (Springer edition), and it's wonderful!! Should you ever wanna learn df, it's the best introductory book, with a lot of pictures and intuition!
PS: i found a used copy of the Oprea book on differential geometry and will couple it with the book by Tapp when the time comes
Thanks! I hope the year is going great for you as well.
Yeah, that's a great suggestion! I had started to read it when I started getting back into math in ~2021. The one by Collier is also very good. It's much more informal. Maybe I'll give DF another try, but I just so happened to find myself another good book on functional analysis!
Thanks for that Tapp recommendation, by the way. I've been considering getting it, but I already have quite a few books on the subject.
What are you currently working on, besides the Fortney book? Also, thoughts on DF as a subject?
@MathematicalToolbox Well, since I work full time, I can't really afford to study two subjects at once. If I get tired of working on a single subject, I pause it (like i put on hold Complex Analysis to work on df). Differential forms is a very cool subject, conceptually not so easy (they are multilinear alternating forms on the tangent space of a manifold), and feels more advanced than the usual stuff, probably because they are less known. they're also a good way to get into differential geometry since integration on manifolds is done essentially through differential forms. My interest in differential forms and differential geometry is mainly to get to study general relativity and cosmology. Functional analysis looks always a bit intimidating 😁 what new reference did you found?
@MathematicalToolbox oh yes Collier ... I forgot about that one! I will probably check it out, it should complement well the book by Fortney...
A note on the review of the first book: In Europe, as I understand it, calculus is synonymous with what is called real analysis in the US
Today I show my most read books of 2024. If you want to buy any of these books be sure to check out our affiliate links!
Also check out our Patreon or our TH-cam Memberships! As of the upload of this video, I am still in the process of importing content from Patreon. I ask for your patience.
Thank you for supporting us!
Shoutout to @CrazyShores for encouraging me to do this video! I also apologize for the echo present in the video.
at 2.19 what do you mean by the example of R as an R-vector space being too much for someone who has learned Linear Algebra? Isn't that the whole point of a first course in Linear Algebra?
Yeah, you're exactly right with that particular example and the purpose of a first course in linear algebra!
What I'm trying to say is that the amount of abstract examples (e.g., "X has this property" or "X satisfies this condition" without first showing the reader how to do it) in this book might be overwhelming for an audience equipped only with linear algebra and calculus. A student with analysis (and linear algebra) should be better prepared for this book in general.
Whether we agree or not, I hope I've made my point clear. Thanks for your comment!
I think I would also enjoy the book by Malik. But I just finished Calc I so I don't think I'll be picking it up any time soon. I think I'm ready to dive into Real Analysis, do you recommend the book by Tao? Its hilarious that you hate discrete probability lol.
I would recommend an intro to math proof first, but feel free to give Tao a shot. Tao is good if you want a hard book that starts from first principles. You can also try Ash's book on Real Variables and Basic Metric Space Topology. It has full solutions.
@@MathematicalToolbox Thanks for the additional Recco's! I think I'm pretty good on proof, I've been doing all the Geometry and Calc proofs. I even did some basic Topology proofs. Haven't gotten to the Metric Spaces yet, I will check that out.
@jammasound I did the first chapter of the book by Pugh and it's great. It's also suggested by another channel (mathematical adventures)...
Awesome MT! interested to know ALL the books you read in 2024..
As far as books that I dedicated more than around five hours to, that's about it. There are some other books that I worked on for the Patreon articles, but the list is probably greater than 20 books. Mostly analysis and differential equations type stuff.
What did you read last year?
@@MathematicalToolbox Just the ones that were in my syllabus. Here's the list -
1. Discrete Math with Graph Theory - Goodaire/Parmenter
2. Abstract Algebra - Gallian
3. Operations Research - Taha
4. Elements of Real Analysis - Denlinger
Great reviews. A very nice and friendly book on functional analysis is "An Introduction to Functional Analysis" by James C. Robinson (2020) - solutions to all exercises in back! ...
I could've sworn I had checked that one out before, but I think I'm confusing it with another! That one looks very well written and easy to read. Great suggestion, sir!
Do u solve all the exercises?
Depends on the book and the layout of the exercises. If they are dispersed throughout the text then yes, I will. If they are at the end of a section then I'll do 5-10 exercises. What about you?
@MathematicalToolbox
I usually solve the easy ones first, and then pick 2/3 problems that seem 'interesting'.
Did you know that a meta study from 2023 that looked at tens of others studies on worked examples in the field of math education has been determined to be MORE beneficial than hard core problem solving itself for students learning mathematics?
The reason is simple. Any problem that is truly fully solved in detail will identify your gaps in understanding.
Pure problem solving will take longer and there will exist WAY more frustration and you will never truly have insight if there are small nuances you have not mastered on some math building blocks that are prerequisite to your topic of study.
You can then LATER add problem solving but only after you have basic mastery of the concepts that are new to you.
Books on any undergraduate math and graduate math that will be structured in the style of fully worked examples will be a goldmine.
Sadly I doubt many will take the lead to make those kind of books even though there is right now actual hard core evidence that for students learning new math topics or new math areas, worked examples done right will ALWAYS be significantly better than you having to be subjected to enormous cognitive load that is typical of standard problem solving, aka MATH PHOBIA.
Math phobia exists only when cognitive load is high(typical problem solving) and specially in middle school and high school when you never get proper explanations of the concepts and mechanisms of WHY every formula works as it does, that you are forced to just accept.
If you find some pioneer educators out there who tried really hard to make formal logic and proofs accessible(without over simplification) to students in high school and even maybe at the end of middle school please let me know. I'll probably buy every one of those books because I'm interested in the pedagogical tools and examples they use to make proofs and logic digestible to way younger people than what's currently done today where mostly only once you get to college/university you start learning how to read , understand and write proofs.
Wow, I have not heard of that. I'll try looking for it. Do you have a link? Thanks for sharing this
@@MathematicalToolbox To find the article paste into google GUEST POST: Worked Examples: An Effective Tool for Math Learning
I suggest you also read the meta study itself. But the conclusion in the post is why I wrote to you just so you're aware that in proper math pedagogy research academia this fact is well known. Sadly this info did not pass down to actual amateur and pro mathematician "normies" common knowledge.
Conclusion of the article:
"As we learned in this post, studying worked examples leads to better performance than problem solving. But studying worked examples has another advantage: it can help you gauge (or monitor) your knowledge. We tend to overestimate what we know, which can lead us astray when studying - by spending too little time on material we need to learn. Fortunately, a few studies have found that studying worked examples reduces this overconfidence (5, 6), which can help you direct your study efforts more appropriately."
I aspire to be an educator,do you mind sharing some sources/the meta study?
@@monishrules6580 I already shared what you need to find the article and the study.