I love this channel and it has kinda become my comfort channel. The way you make your videos is relatable and inspiring. I mean this in the sense that I see you and am like “oh, these math wizards are people too. I can be like them if I try.” I’m not the best at writing so I hope this makes sense.
There seem to be two competing styles of mathematics pedagogy. One of them is what I think you and I would both lean towards, which is patiently teaching young math students how math really works by starting with the simpler concepts, building up to more complex ideas, and being rigorous about the math along the way. The other is to teach some rather advanced math to students in such a way as to skip the proofs and explanations, but get them to memorize and have some intuition for the methods. I really began to love math when I was a student at a community college where two of the math professors really taught their courses in a rigorous, logically clear way. The first of these courses I took was a linear algebra course. Instead of a matrix-oriented, memorization-based class for engineering majors, which evidently is the norm at some colleges, this was a rigorous class about the math itself. The professor was really enthusiastic and loved math. This was an entirely new experience for me, and it was the first time I really enjoyed math.
I agree that we should start teaching proofs earlier. I think more people would be interested in math if they had a taste of the philosophical aspect of it that you only really study in math major courses
To imply a depravity of (one aspect of) applied logic to a student for the sake of “other interests” is the reason for the death of all thought and interests. Outside of reason man can have no interest. Eason belongs in all Aspects of man’s life from his ethics (from metaphysics and epistemology), to everything.
math proof would have definitely got me into maths more cuz all I wanted to know was how things work, without any practicality... i ditched it before getting into limits because I was so bad at using the calculator... now that I have went through history & philosophy UG to understand better how things work and a librarianship PG for me to organise knowledge, I find myself having to learn physics and neuropsychology to further my understanding in how things *actually* works, both are so interesting but just say I hope I am more familiar with the mathematical language...
Most of the time I don't comment on videos, but I just need to say that your channel is great! One of your videos appeared on my home channel randomly, I believe it was the first video you made? I would love if you went into more detail about the problems! Edit: I'm from europe, and they make us use proofs in middle school, and they really do help!
im not taking a math major in college but i can safe to say for me ill prob. stop at bachelors and maybe masters but going for PhD is kinda a waste of money/time at least for me.
I’d love to go into research and get a phd but I'm struggling to get my bachelor's and my grades are mediocre at best but I can’t bring myself to quit bc I really love my subject even if my grades don’t reflect it
Because when you multiply the 2 through all the terms the 1 becomes a 2 we are taking the 2 from 2n+2m+2, not 2n+2m+4 which would be what you get if you distribute 2(n+m+2)
I agree we should introduce proof writing at a younger age. Proof writing is The Skill you get from studying mathematics formally, and there is no reason children in middle school or high school couldn’t be introduced to it at an elementary level. I used Hammack for math proofs class. It’s very accessible, I think, but what the reader gets is proportional to what they give. I wish I had tried harder in that class. Thanks for sharing
In India, we had proof writing as a separate chapter called 'Mathematical Induction' since sixth grade. I bought Hammack and was surprised to find a lot of problems in common with my 7th and 8th grade textbooks. Although I didn't pursue pure math, instead pursuing engineering, proof writing still helps me understand many core concepts in different fields. It's a fundamental concept that should be taught in all education systems because of the critical thinking capabilities that it nurtures in young students, which is important for any STEM field.
Nice to hear you reminisce about some things! My introductory proof writing course at university was centered on metric spaces and I learned a lot from that. It definitely helped a lot with real analysis, abstract algebra and topology later. Some middle school students can handle elementary proofs for sure. I didn't see any until high school geometry though. Nice video !
This channel is very interesting, I have pretty limited knowledge of math (B.S. in astronomy + one failed group theory class + binge-watching math channels on youtube) but I enjoy your content anyway. It's cool to see how my life could have looked like if I decided not to drop from M.S. and actually go deeper into academia.
I appreciate your openness and your thoughts on being an academic in general. Although I am not a PhD student in Mathematics but in Finance, I can absolutely relate. Thanks a lot!
As a young student who wants to go into math research, i'm so glad i discovered this channel because i really had no idea what the atmosphere of math research was like, i just love the subject. it's funny, i also started loving math by studying geometry, at the same age as you
I was not too much into art but I like to draw and paint, but also asked myself what is this reality and this planet, what is the point of life. How is it that I did not exist a few years back, but now I do.But I won't forever exist. Then I read somewhere this quote from Michelangelo "The true work of art is but a shadow of the divine perfection. Only God creates. The rest of us just copy." The obvious conclusion from this is that God is the greatest artist. I always liked math, but I never had a good background for competitions, but I really liked to go to the core of each topic learnt in class, obviously never got to that point because for example a simple set can be as complex for a grad student as for a kid, but in this manner I always liked to think of math results as some type of divine brush that God once put his effort on. This kept me on the subject and still is, it's the purest form of art. I'm still an undergraduate, don't think I'm able to do research but I discovered trivial things on my own before actually learning, and made conclusions most of the times incorrect during lecture without previously reading before it, it's an amazing feeling to even have conclusions, doesn't matter if it's right or wrong. I have an ok background on Algebra, but the analysis part I need to improve, also number theory, really obscure results such as reciprocity law, Artin and Grothendieck were truly remarkable mathematicians I look up to, because they worked on the areas I want to work on. I have a personal goal which is to get kids to study mathematics and do research, while my "career" goal is to learn about the langlands program, at least from the algebraic number theory point of view. Having said all of that I was never a book person, only went to books once I learnt the subjects, I find it more enjoyable this way. This is why I would really enjoy a mesure theory series. I can do math for a whole day but I really need the base knowledge either from a video or a person in order to touch a book. This has been my weakness since school, with this given information I would like to know your second opinion about it. Thanks for all your videos, it really motivates me without even you trying to do so.
Quick note: Having a master's degree or a PhD. will NOT guarantee you a job opportunity at the level you're expecting. Unless you're planning on staying in academia, companies couldn't care less if you did your best during your graduate studies; they only care about EXPERIENCE. So the focus of your career should be getting at least 2 internships during your studies and maximizing your networking connections, making sure that you will be guaranteed a job right after graduation. Tldr: dont get too lost on proofs. Focus on opportunities to apply what you've learned to the real world, unless you wanna stay in academia
US yes/maybe (at least that is what I heard). In Germany for example, a master's is the norm for many careers still. Furthermore a Masters is not intertwined with a PhD but a completely separate degree after which you can apply for ~3 year PhD positions. And of course a master's is cheaper there than in the US. I am just assuming that you are from the US, because it is the only country I ever heard about that does not seem to appreciate a math major. Here in Germany a math major means easy and rich employment opportunities, even without internships (even though that is still a smart thing to do IMO). But then there are these INSANELY high paying jobs in the US that simply seem to not exist in Europe... But these are almost all IT huh? Would love to hear, if you could shed some light on the issue...
Mostly true, except that getting 2 internships and maximizing your networking connections also doesn't guarantee you a job. (It helps a lot, not saying it doesn't, but it's no guarantee.) In rare instances, companies care about your research ability. Specifically, if you're applying for a position in a corporate research lab, such as Microsoft Research, NTT Research, IBM Research, etc., then your graduate work (as well as post-graduate experience, if any) is relevant. However, these jobs are corporate jobs in name only; spiritually they're much closer to academic jobs.
I think maths should become a global subject, especially mathematical logic; in the broad view, you are not just teaching math but the principles of human thinking.
I think starting with proofs asap makes any math topic and book way easier. I don't think learning proof writing and reading is the hardest. Not many symbols you got to learn and you can get started after learning some basic algebra as you showed. I think because it's held off till the very last point till you get to real analysis, it's feels like a gigantic mountain to climb. But I blame real analysis and the assumption of where you are mathematically at that point is more at fault.
Hey, I was wondering if you had any textbook you'd recommend for self studying analysis? Or if you could do a video on different textbooks you'd recommend for different subjects, I'd very much like to see that, and I think I've seen other commenters echo the sentiment. Love your videos, keep it up!
I have a video showing some of the books I use to self study, but in short my favorite analysis book is by Royden and Fitzpatrick. A free pdf copy exists on the internet :)
To anyone still on this video, I recommend the book "Mathematical Proofs - A Transition To Advanced Mathematics", I worked through it last summer holidays and it was a very very well written book. And I need ehat I've learned there all the time in contest math And to your comment about middle school: I am in highschool, so yeah this topic is very accessible for anyone interested
If you want to be a researcher, go to the grad school. If you want to earn money, apply for jobs rather than go to grad school. I wanted to study the elliptic curve with my professor when I was studying in America(he is a well-known professor in his field), but I didn’t have enough money to make a living. Thus, I started to search for available job positions. Now, my goal is to make money for the math grad school I like and study more there, even though I will be in my 40s or 50s. I don’t want to wait just to die. I want to study math more.
I’m pursuing my applied mathematics bachelors! Just about finished my 2nd proof course. My 1st professor in proofs HARPED on how we wrote ours in terms of it being clear etc. that really helped me out seriously, but still proofs are a little hard
I know it's trivial but for the proofs you presented, how would you prove that the sum of two integers n and m is also itself an integer? Is that an axiom?
Idk if it's a wall or because I was homeless but this trig stuff is so confusing I was homeless in geometry so precalc or advanced algebra is really difficult I find algebra itself very easy no test less than 90% but that and logs are little tricky
I took a proofs class in my uni a few years ago and barely passed it. Took a few more undergrad maths and fortuantely passed those but my marks were mid. I only understood proofs and proofwriting when i took analysis
I have started a channel doing proofs. Because I’m seeing a subject, this semester that is called advanced calculus. We have been through all of these, so I feel like motivated to upload some proofs to the channel because I think that some people will find them useful and will help them in their writing.
When I teach proofs, I place heavy emphasis on providing specific justification for each step in the proof. Having a hard requirement for justification helps to prevent students from making up their own nonsense when they are doing proofs on their own. Here is how I might present the proof to my students: Definition: n ∈ ℤ is even if ∃ k ∈ ℤ, n = 2k. Theorem: ∀ n m ∈ ℤ, n even ∧ m even ⇒ (n+m) even. Proof: Let n ∈ ℤ and m ∈ ℤ with n even and m even. By definition of even, ∃ k ∈ ℤ, n = 2k. By definition of even, ∃ l ∈ ℤ, m = 2l. Then n+m = 2k+2l. By the distributive law, 2k+2l = 2(k+l). By transitivity of equality, n+m = 2(k+l). Hence there exists an integer x, namely x=k+l, such that (n+m) = 2x. By definition of even, (n+m) is even. QED Different variations are possible. I might leave out transitivity of equality (in fact transitivity of equality is already low-key needed to get n+m = 2k+2l). But I think I would not leave out the distributive law under any circumstances, since that to me is the crux of this proof.
@@bestgun9994 kind of relevant info. Huge difference between PhDs. The video is asking a question directly tied to someones experience. It like a video: should you invest in crypto? Kind of matters if it is someone from blackrock or some random influencer
Btw if you don’t mind, you should really upload an asmr math video of you solving questions or working on problems whatever? You sound rly cool and it might motivate the few of us at least to maybe kick start our study session too. Great vid tho!
Hi, I love your videos. I have a question, though. I’m currently discerning what major to switch to (I’m in engineering but I’m not really keen on it). I love math and I love physics especially when I get to prove something, so I think I’m interested in maybe doing a double major or a major in math and a minor in physics. My biggest qualm with engineering is that it takes up so much of my time and really forces me to sacrifice the things I want to do (like learn more about math, take art classes, exercise regularly). Basically my question is does math require you to make more sacrifices than necessary? I’m not against working really hard for something. Especially, if it allows my some freedom, but before I jump into math I want to make sure I can be both a person who majors in math and a person who has a life. Would you say that’s possible?
If you want to do college right, and work hard, then no matter what subject you pick, it will take up as much time as you let it. That may not be a satisfying answer but it is the most honest one I can give. Personally, I spend a lot of time doing mathematics, because I am in a PhD program and it requires me to do so.
Math and physics will also eat up a ton of time. They're arguably the 2 hardest majors out there so unless you're naturally great at it you'll have to study a ton to keep up. If you want to have a lot of free time during university you have 2 options, pick a major that would be easy like say, business, or take less classes at a time which will reduce your work load but also mean you'll take longer to graduate. Depending on your level and if you have elective classes at your university you might not need to study all day every day, but if you want to do well you need to be prepared to put a lot into it.
I really disagree with the other commenters who replied to you here. I’m an American, so I can comment on how engineering and engineering majors are treated at American universities. Engineering is seen as being superior in some sense to other majors, and it’s treated in a special way in American universities. It’s seen as a major that prepares the students for a specific profession which is assumed to require high working hours. Engineering students are assigned much more time-consuming projects and homework than college students in any other major. Although mathematics and physics might be more intellectually demanding than engineering, the math and physics majors in college are given a much lighter workload than engineering majors are. If you are a math or physics major, you will do many difficult homework and exam problems. However, as long as you are following the lectures, doing the reading, and have some mathematical talent and passion for the subject, you can solve the problems. A typical weekly assignment load in an upper-level math class is one homework assignment of several (roughly 3 to 10) problems. The assignment will likely take you between 3 and 10 hours to complete (some problems will go quickly for you, others will require more time and thought). It’s just this special treatment of engineering in American universities that creates this distinct gap between the workloads of engineering majors and all other college students. So, in my opinion, a college student who has the talent and interest to major in math or physics is able to enjoy studying and obtaining a degree in a subject that is hard for most people but doesn’t require the grueling hours of engineering.
hi new subscriber! I'm a woman currently in my second year of my co-terminal program (bachelors and masters) in applied math and statistics. currently there just aren't very many other women in my major, how does graduate school compare? What is the diversity like? Thank you!
@@sbnwnc true! I'm still going to study math regardless I love the subject! but it can definitely feel isolating at times and there definitely needs to be more resources for women in stem so we can better the field
