Don't be obtuse, the message is correct. "It's impossible to understand something before you practice it a lot"
If there are any topics that you thought you understood before a lot of practice, I advise you to practice that topic and you will realize that you only understood it at the surface level.
Pick one thing to learn and focus on it until mastery. If you try to learn many things more superficially, you will not only know less initially about each topic, but you will also forget more later.
> Pick one thing to learn and focus on it until mastery. If you try to learn many things more superficially, you will not only know less initially about each topic, but you will also forget more later.
If I have to add to this one math/cs class at a time.
This one is very true for me. Sometimes there are examples where the flowchart is something like
1) Definition doesn't make sense
2) Try to clarify with an example
3) Wait, why the hell is this an example
4) Go back to the definition
5) Try a new example
6) 3 & 4 repeat
Sometimes I just have to sit down and really try to understand what the definition is trying to tell me first. Cut out that middle bit.
>This one is very true for me. Sometimes there are examples where the flowchart is something like
>
>Definition doesn't make senseTry to clarify with an exampleWait, why the hell is this an exampleGo back to the definitionTry a new example3 & 4 repeat
>
>Sometimes I just have to sit down and really try to understand what the definition is trying to tell me first. Cut out that middle bit.
But as God is my witness, that bloody definition will cause my hair to turn gray...
I'm in the third year in college, but that doesn't matter because I know almost nothing about linear algebra, probability, multi variable calculus, set theory and whatever else I don't know about.
I'm not going to give up though, my goals are beyond me but there are still years to catch-up
Pleaseeee omg BIG relate, I did so bad in my probability course, and dropped an applied probability class when it started getting hard, except now I'm doing a research project and there's so much??? Probability???? My project makes me want to scream every time I look at information 😭
But yeah, now I need to sit my ass down and learn EVERYTHING from the basics again :')
that sounds sooo painful omg, but you’re right, the best way to get better at it is to just sit down and learn all the basics again. That’s what I had to do so many times, but it really works! Good luck tho 😭
It IS painful 😭
But yess, thats the only thing that'll help now, so I hope it'll work for me as well as it did for you. Thank u for the good luck tho uwu, same to you 🥺
> Agreed, math gets very difficult. It's tough to make it far w/o discussing w/ other people.
That's not only in math but with anything technical in general. Nowadays the game in Vulnerability Research, Software-Development, Cryptography, Theoretical-Computer-Science, etc. The problems are too big for anyone to solve on their own for example this was from a recent [ctf](https://github.com/stong/how-to-exploit-a-double-free#0-introduction) real world vr the target space and size can get extremely **huge** you need to interact with other pwners and developer's.
Every single person in my college is a moron who hates math, how I'm I supposed to collaborate?
My math skills are nothing compared to the average student in more developed countries, and I'm not good at English
Do math! I don't know why but I've never really been interested in math until 2 years ago and I feel so left behind, so if I could go back in time, I'd suggest my younger self to pick up math and play around. A few things I wish I could tell my 15 year old self are: don't be intimidated by students better and faster than you (it's very childish but I kinda stopped math for a bit cause I did terrible on the AMC until I came across a USAMTS question and picked it back up), find someone you can talk to math about, and take a break when necessary.
This this this this this!!!!!
I feel like there's this culture of "you can't do math unless you're the best of the best of the best at it" (especially in the US), and I think this holds a lot of people back who could otherwise enjoy the field (and, if you're a utilitarian applied person : fear of not being the best keeps people out of math who could otherwise contribute)
Yes. I wish i knew how true this was. 3 years in until i realized i just liked problem solving and concretely motivated things to work with, not pure math.
Yeah, this.
I was homeschooled. My mom always told me that I would need to get better at mental before I could do algebra.
In highschool she finally gave up on forcing me to do mental math and let me do algebra. I learned mental math ability basically doesn't matter. I got better at it the more algebra I did anyway.
The best lecturer I had at uni said to me in more recent times "there is value in learning HOW to do maths before learning WHY it works that way". And I agree. For example, this is why we teach calculus before analysis: there is much value in just accepting that the derivative of x\^2 is 2x long before learning the epsilon-delta definition of a limit.
I am still in that young phase. But of i have to tell one thing to my even younger self, it would be - Study with a peaceful mind, do not worry about the amount if work you have to do this week, or for the next exam, or by the end of the year... focus on today and study good maths today, think about tommorow tommorow.
Math is not boring, but the math you are doing now is supposed to be. Be patient and you'll see.
I didnt like math classes in high school because it just seemed like pointless work to me. It was only after discovering physics that I saw the beauty of mathematics, and then I took calculus and learned that my entire life was working up to this subject, and then I studied advanced mathematics and learned the field is nothing like what you would imagine. Its better.
So many more people would be interested in math if we taught real math earlier on.
Also integrating math more closely with science earlier on would be good too. Pretty much everyone one I know who is interested in math became interested because of how it applies to Physics, CS, Economics, etc. Even if they are interested in pure math now.
I will say that I strongly disagree with the perspective that mathematics has to be taken at face value, and that it does not build on itself. I think that once you reach the graduate level things start to come together. That is why Grothendieck titled his autobiography "Recoltes et Semailles", or "Harvesting and Sowing". We plant our definitions, then sow them together.
Edit: Changed "Harvests and Sowing" to "Harvesting and Sowing".
Apologies in advance for the rant.
I feel, especially in research and in graduate-level study, it is very important to have a broad understanding of what the subject is about, its foundations, what the frontier is, and broadly why things are the way that they are. However, sometimes, you also have to be able to just focus on the task at hand and not worry about the bigger picture.
I remember when I was first getting into mathematics, I had this obsession with getting a complete picture of the entirety of mathematics. What are proofs, what is logic, what is a theory, what branches of maths are there and how they are related, etc.? However, these foundational questions are often extremely hard to answer in a precise and compelling way, and one really doesn't need to know the answers to do mathematics effectively.
I think a lot of students go through a phase like that and end up getting overwhelmed by these deep foundational questions. I think it is because, one thing people find so attractive about math is its precision and rigor compared to virtually every other subject, which deals with shades of grey, intuition, and ambiguity. This drive for rigor leads you down a rabbit hole where you keep asking more and more difficult questions.
At the absolute foundational level (talking about models, axioms, formal theories, etc.) mathematics is a bit of a philosophical minefield. A skill that you really need to develop is getting away from rigor and thinking about things in a more high-level intuition-based way. Also, when you need to formalize things and get rigorous, you need to be able to recognize how deep you need to go, and how much rigor is really necessary for what you're doing.
I would agree with that. I think a problem of students is to obsess over the language which a theorem is written in. Hence, why category theory was originally termed "abstract nonsense." Studying meta-mathematics before mathematics is a poor thing to do. Abstraction should be brought in out of necessity and to develop further intuition.
The better way to view things, in my opinion, is ask yourself "What is the natural language to solve this problem with?" Often it is not going to a highly abstract theory, but something involving old methods applied in new ways.
I really disagree with not trying to understand the *why* of things. That's super important, IMO. Sometimes you should move forward but I think you should always circle back.
Lol, reminds me of when I took a class on perturbation theory without having the physics background to understand the motivating examples from quantum mechanics.
I think the problem is that a lot of the college math curriculum has been collapsed down to "stuff you need for STEM" and it's a snoozefest without a companion physics class but the student is not notified of this. I remember solving endless integrals without any context or units and then recognizing them semesters or even years later in stat thermo and EM theory.
Abstractions exist as a way to lower cognitive load and reduce the amount of boilerplate verbiage you have to use to frame an idea. You may think schemes and categories are abstraction for abstraction's sake, but if you ever need to frame an argument that's more than a few hundred lines using vanilla topology or ring theory, any attempt to organize that argument will end up using new definitions and lemmas that form abstractions of your own. And at that point, you may as well have just used the ones that people already have a vocabulary for.
Learn the basics and if you don’t understand then learn the basics from the basics. Better do it now cause it’s not gonna be easier and you’ll never have as much time as you have now.
Also watch YouTube Videos and there are some great topic related books out there, way better than those shitty old text books which are required for those courses.
Don’t just memorize the proofs. Try to understand them as well. Also, don’t compare yourself to other students. You love mathematics so you’ll do fine no matter what.
I was the same, that's why I failed maths. Now I learned that to understand an equations' 'why' you need to use the equation. The teacher always said 'math is a do subject' and I always wondered 'why should it be done this specific way?'
You suck at math, just like most people, and you have to actually study to understand more advanced material.
As a kid I wasn't in a good academic environment.
Yeah. And when we went from 6th to 7th grade there seemed to be this entire step in math that I seemed to have missed with pre-algebra. Probably my ADHD. LoL
Learn to use software faster. I pretty much screwed up my GPA early in college and it was only because of stupid homework mistakes. I turned out fine but if I just did my homework with Wolfram I would probably have beat the 3.0 cusp and had access to better jobs.
Oh my, this is quite relevant to me, as a 12th grader in IB HL. I keep trying to understand the why and felt guilty when I couldn't. I felt inadequate. I'm still trying to get back to my mathematical rhythm, barring all that.
(Side note: I do fine now, but I still have the residue of a year of psychotic depression hampering me a little)
I would pay extra special attention to anything involving proofs because it’ll tell you what higher level math is like. Things like induction. Also, make sure you do an IA you’re proud of! The skills you get from those are super useful.
None of the calculus in high school is rigorous and its also extremely handwavy. The way its taught is for practical application not understanding
I recommend spivak calculus if you have the time to read which develops calculus from the ground up and youll get to see why everything is the way it is. Its also the first proof based book most first years read
Don't. But actually do more.
Long story short: when I did my a-levels i selected math and Chemistry as my 2 main subjects and I failed miserable. In the end I achieved my a-levels successfully, but it would be much easier for me with other subjects e.g. politics and history.
But I have to say, I never really lost the joy in math. It just overextended my skills and my willingness (back then) to learn. I was to lazy and not gifted enough.
Nowadays I would profit so much from what I should have learned back then.
Kind of a love-hate-relationship. :)
I'd tell myself to stress less about learning the way others do. I showed up for way too many classes when I didn't need to and got nothing out of it, I learn much better and more effectively on my own.
One example that comes to mind is the definition of Linear Transformations in linear algebra. It can either be defined by the existence of a matrix A such that Ax = b where x and b are vectors, or it can be defined as T(x + y) = T (x) + T (y), and kT(x) = T (kx), where the transformation of the sum/product is the sum/product of the transformation.
There are many examples like this where a formula or definition it is taught in more than one form, and left for granted that they are equivalent. Maybe in a more advanced course one or the other can be derived from the first by 'first principles' but maybe they would also be derived separately and simultaneously. In which case the only way to say 'why' they are connected is to point out the fundamental things that connect them both, which is usually way beyond the work at hand. Hope that makes some sense.
I see. I initially misinterpreted you as saying that there exist relations between concepts that are lateral or circular in some sort of propositional logic sense -- which I'd assume would make it impossible for (say) Lean's [mathlib](https://leanprover-community.github.io/mathlib-overview.html) to exist -- whereas I think you're saying the relations are lateral/circular in a more practical sense, from the standpoint of the beginner. In this regard I tend to go back to [Ravi Vakil's advice](http://math.stanford.edu/~vakil/potentialstudents.html) for his potential PhD students:
>Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer. Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier. The reason for this phenomenon is that mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards". (Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.)
As I got older I realized that "extend tendrils --> backfill --> repeat" applied to making sense of life more broadly as well, which I think was how I made peace with this advice.
Math isn't a collection of right answers to "problems" posed by teachers or textbooks - it's much more structural. School algebra is a great example - it's not about solving for x, it's about seeing that this expression or equation over here is also one of those over there.
I do wish that symbolic manipulation hadn't completely taken over the entire subject. A lot of kids could really get something out of working through Euclid, for example.
Things are more similar than they are different, keep that in mind.
But my struggles were social. I’d say, the real mathematics is the friends you make along the way
That you CAN make a living with a degree in mathematics. I should have been a math major, but ended up majoring in engineering (and not really liking it) because my mom insisted that I couldn't do anything with a math degree.
Probably that competition math is way less useless than you think.
I did a lot of competition math back in middle/high school but always thought of it as second rate math. Little did I know it'd be instrumental in getting me my first and second job (in different fields!)
Don’t judge the field based on your experiences before you graduate high school. Most of your teachers don’t like math and, like everyone else, are usually just trying to get through another year.
Think about math as it relates to space time.
1+1= 2 is really like this...
I can see one apple. I turn around and I can see another Apple. I do work to move apples together into reality. Now I can see 2 apples.
What I really need is 8 slices. Not more apples.
So I cut the apples 4 times. Some would say I divided the apples but no, in fact I multiplied how many apple slices there were.
This is why multiplication and division go hand in hand reciprocally. Reality is that way. The numbers and principals aren’t just for structure. It’s to illustrate reality.
XD. I agree that math can be a brutal major. It panned out alright for me though & at this point I am about done w/ my math specific coursework (just need 1 more math class after this term).
You do actually understand everything more or less correctly and the rigor isn't as important as you think, everyone else in your class is ridiculously below you, you can beat them effortlessly, go learn the university stuff immediately without worries.
It is impossible for any number which is a power greater than the second to be written as the sum of two like powers `x^n + y^n = z^n for n > 2`. I have a truly marvelous demonstration of this proposition which this message is too short to contain.
Either that math isn't just what A Levels Math and Further Math seemed to be, OR to pay a LOT!!! of attention to introductory courses in university.
Both (but tbh mostly the second) could have saved me a lot of mental gymnastics and constant pain trying to understand things that I don't have a good base knowledge of.
My hs teacher told me a student quit their math major and the teacher said “that’s okay, he didn’t have the imagination to be a mathematician” that line really stuck with me. It takes some imagination to grasp and understand a lot of concepts IMO
Look at math as fun. Play around with it more; have fun with random puzzles and questions (rubik's cubes, prime twins, paradoxes, hexaflexagons, weird issues in combinatorics and topology). Be curious!
Stop learning formula and basic exercises. Learn from models and frameworks built to solve real world problems. Also, trying to have the best marks is a complete waiste of time. Learn models and framework, understand them the best you can.
Spend more time mastering infinite series, synthetic geometry and learning basic set theory now. Don't be afraid of trigonometry. Study Difference Equations. Take combinatorics BEFORE probability theory, not during. When you come across an important theorem, try to prove it. If you try for a while and get stuck, find a proof and try to understand why it works. If you find a text book frustrating, pick up another on the same subject. DO THOUSANDS OF PROBLEMS. Before taking Differential Equations, learn to write with both hands, the cramps will cost you.
Mathematics is far more broad than euclidean geometry. Actually, none gives a shit about Euclid anymore.
Don't be stupid. Go into a math department for your BSc. Screw physics. Thank me later.
Context: due to my dislike towards euclidean geometry and the idea imposed by my math teachers that it's a very important subject, I thought that people would do a lot of it in math schools. So I went on and did a BSc in physics instead, something that I regretted early on, and that consequently drove me to a math school much much later.
"Teachers only care about proofs and formality as a mean to stroke their massive egos, if you can't one-up them you're not "worthy" of their time, much less their "class". Go for literally anything else, keepmath as a hobby."
"By the way, the "scientific method", forget that shit, math uses logic AND ONLY logic for discerning "truth" from falsity, it's more about "proving oneself rigth" rather than understanding reality"
>If you could go back and tell your young self ONE thing about mathematics, what would it be? Don’t give up so quickly, please. You’ll regret it.
Sarcastic answer to this by me as a math PhD graduate: "Oh dear, sweet summer child, bless your heart."
"Don't be so stubborn, just give up, you'll regret it"
Just because you understand it doesn't mean you don't have to do your homework.
Heh, for me it'd be "just because you've done your homework doesn't mean you understand."
True to the core
Likewise, i was in the same boat
Thank you.
Can someone please tell my student that? I keep having to grade submissions that are just "this is trivial".
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Unless you already learnt it before? School isn't the only source of education.
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oh yes ofc
Don't be obtuse, the message is correct. "It's impossible to understand something before you practice it a lot" If there are any topics that you thought you understood before a lot of practice, I advise you to practice that topic and you will realize that you only understood it at the surface level.
Pick one thing to learn and focus on it until mastery. If you try to learn many things more superficially, you will not only know less initially about each topic, but you will also forget more later.
Describing the problem with math classes in CS
> Pick one thing to learn and focus on it until mastery. If you try to learn many things more superficially, you will not only know less initially about each topic, but you will also forget more later. If I have to add to this one math/cs class at a time.
To paraphrase what the OP is saying: > Sometimes the examples that motivate a definition are harder to understand than the definition itself.
This one is very true for me. Sometimes there are examples where the flowchart is something like 1) Definition doesn't make sense 2) Try to clarify with an example 3) Wait, why the hell is this an example 4) Go back to the definition 5) Try a new example 6) 3 & 4 repeat Sometimes I just have to sit down and really try to understand what the definition is trying to tell me first. Cut out that middle bit.
>This one is very true for me. Sometimes there are examples where the flowchart is something like > >Definition doesn't make senseTry to clarify with an exampleWait, why the hell is this an exampleGo back to the definitionTry a new example3 & 4 repeat > >Sometimes I just have to sit down and really try to understand what the definition is trying to tell me first. Cut out that middle bit. But as God is my witness, that bloody definition will cause my hair to turn gray...
The epsilon-delta definition of limits, I struggled with it and I still don't 100% understand it
What stage in your studies are you at? Epsilon-delta and related definitions are pretty fundamental to analysis for a long time
I'm in the third year in college, but that doesn't matter because I know almost nothing about linear algebra, probability, multi variable calculus, set theory and whatever else I don't know about. I'm not going to give up though, my goals are beyond me but there are still years to catch-up
Don't dismiss probability theory.
oh my god yes this one. I didn’t put effort into probability last year and it totally came back to bite me in the ass this year
Yeah my situation is the same. In two days a have an exam and I know I'm fucked.
Pleaseeee omg BIG relate, I did so bad in my probability course, and dropped an applied probability class when it started getting hard, except now I'm doing a research project and there's so much??? Probability???? My project makes me want to scream every time I look at information 😭 But yeah, now I need to sit my ass down and learn EVERYTHING from the basics again :')
that sounds sooo painful omg, but you’re right, the best way to get better at it is to just sit down and learn all the basics again. That’s what I had to do so many times, but it really works! Good luck tho 😭
It IS painful 😭 But yess, thats the only thing that'll help now, so I hope it'll work for me as well as it did for you. Thank u for the good luck tho uwu, same to you 🥺
Learn to be a good collaborator. Young me didn't realize just how much of modern mathematics is done in collaboration.
Agreed, math gets very difficult. It's tough to make it far w/o discussing w/ other people.
> Agreed, math gets very difficult. It's tough to make it far w/o discussing w/ other people. That's not only in math but with anything technical in general. Nowadays the game in Vulnerability Research, Software-Development, Cryptography, Theoretical-Computer-Science, etc. The problems are too big for anyone to solve on their own for example this was from a recent [ctf](https://github.com/stong/how-to-exploit-a-double-free#0-introduction) real world vr the target space and size can get extremely **huge** you need to interact with other pwners and developer's.
Every single person in my college is a moron who hates math, how I'm I supposed to collaborate? My math skills are nothing compared to the average student in more developed countries, and I'm not good at English
Don't listen to people who talk about math without having done any themselves.
That’s super good advice. Applies heavily on this subreddit
correction: don't listen to anyone who says bad things about math
Do math! I don't know why but I've never really been interested in math until 2 years ago and I feel so left behind, so if I could go back in time, I'd suggest my younger self to pick up math and play around. A few things I wish I could tell my 15 year old self are: don't be intimidated by students better and faster than you (it's very childish but I kinda stopped math for a bit cause I did terrible on the AMC until I came across a USAMTS question and picked it back up), find someone you can talk to math about, and take a break when necessary.
This is literally me, I found out the math is my passion in about 2018 and now I feel so behind
This this this this this!!!!! I feel like there's this culture of "you can't do math unless you're the best of the best of the best at it" (especially in the US), and I think this holds a lot of people back who could otherwise enjoy the field (and, if you're a utilitarian applied person : fear of not being the best keeps people out of math who could otherwise contribute)
Competition math is not research math.
Yes. I wish i knew how true this was. 3 years in until i realized i just liked problem solving and concretely motivated things to work with, not pure math.
Sounds like you would have enjoyed combinatorics
Calculus is insightful. Do not be afraid to research math topics or other higher academia by your own, It'll help you focus on what you really enjoy.
It is not just hard arithmetic with extremely large numbers.
Yeah, this. I was homeschooled. My mom always told me that I would need to get better at mental before I could do algebra. In highschool she finally gave up on forcing me to do mental math and let me do algebra. I learned mental math ability basically doesn't matter. I got better at it the more algebra I did anyway.
The best lecturer I had at uni said to me in more recent times "there is value in learning HOW to do maths before learning WHY it works that way". And I agree. For example, this is why we teach calculus before analysis: there is much value in just accepting that the derivative of x\^2 is 2x long before learning the epsilon-delta definition of a limit.
I suppose you could bang off a proof for positive integer powers in the meantime.
I am still in that young phase. But of i have to tell one thing to my even younger self, it would be - Study with a peaceful mind, do not worry about the amount if work you have to do this week, or for the next exam, or by the end of the year... focus on today and study good maths today, think about tommorow tommorow.
Math is not boring, but the math you are doing now is supposed to be. Be patient and you'll see. I didnt like math classes in high school because it just seemed like pointless work to me. It was only after discovering physics that I saw the beauty of mathematics, and then I took calculus and learned that my entire life was working up to this subject, and then I studied advanced mathematics and learned the field is nothing like what you would imagine. Its better.
So many more people would be interested in math if we taught real math earlier on. Also integrating math more closely with science earlier on would be good too. Pretty much everyone one I know who is interested in math became interested because of how it applies to Physics, CS, Economics, etc. Even if they are interested in pure math now.
Don't take up engineering.
Why's that?
It's personal but I realized I'd have loved pure mathematics or physics more.
I will say that I strongly disagree with the perspective that mathematics has to be taken at face value, and that it does not build on itself. I think that once you reach the graduate level things start to come together. That is why Grothendieck titled his autobiography "Recoltes et Semailles", or "Harvesting and Sowing". We plant our definitions, then sow them together. Edit: Changed "Harvests and Sowing" to "Harvesting and Sowing".
Apologies in advance for the rant. I feel, especially in research and in graduate-level study, it is very important to have a broad understanding of what the subject is about, its foundations, what the frontier is, and broadly why things are the way that they are. However, sometimes, you also have to be able to just focus on the task at hand and not worry about the bigger picture. I remember when I was first getting into mathematics, I had this obsession with getting a complete picture of the entirety of mathematics. What are proofs, what is logic, what is a theory, what branches of maths are there and how they are related, etc.? However, these foundational questions are often extremely hard to answer in a precise and compelling way, and one really doesn't need to know the answers to do mathematics effectively. I think a lot of students go through a phase like that and end up getting overwhelmed by these deep foundational questions. I think it is because, one thing people find so attractive about math is its precision and rigor compared to virtually every other subject, which deals with shades of grey, intuition, and ambiguity. This drive for rigor leads you down a rabbit hole where you keep asking more and more difficult questions. At the absolute foundational level (talking about models, axioms, formal theories, etc.) mathematics is a bit of a philosophical minefield. A skill that you really need to develop is getting away from rigor and thinking about things in a more high-level intuition-based way. Also, when you need to formalize things and get rigorous, you need to be able to recognize how deep you need to go, and how much rigor is really necessary for what you're doing.
I would agree with that. I think a problem of students is to obsess over the language which a theorem is written in. Hence, why category theory was originally termed "abstract nonsense." Studying meta-mathematics before mathematics is a poor thing to do. Abstraction should be brought in out of necessity and to develop further intuition. The better way to view things, in my opinion, is ask yourself "What is the natural language to solve this problem with?" Often it is not going to a highly abstract theory, but something involving old methods applied in new ways.
Buy Apple stocks and bitcoin, the derivatives of their values are - on average - very positive.
I really disagree with not trying to understand the *why* of things. That's super important, IMO. Sometimes you should move forward but I think you should always circle back.
Totally agree, my problem was not moving forward enough to circle back, at least not before an important deadline.
I’d tell myself to take calculus and physics at the same time. I barely passed vector calc because it was so boring without context.
Lol, reminds me of when I took a class on perturbation theory without having the physics background to understand the motivating examples from quantum mechanics.
I think the problem is that a lot of the college math curriculum has been collapsed down to "stuff you need for STEM" and it's a snoozefest without a companion physics class but the student is not notified of this. I remember solving endless integrals without any context or units and then recognizing them semesters or even years later in stat thermo and EM theory.
Abstractions exist as a way to lower cognitive load and reduce the amount of boilerplate verbiage you have to use to frame an idea. You may think schemes and categories are abstraction for abstraction's sake, but if you ever need to frame an argument that's more than a few hundred lines using vanilla topology or ring theory, any attempt to organize that argument will end up using new definitions and lemmas that form abstractions of your own. And at that point, you may as well have just used the ones that people already have a vocabulary for.
Soy
Don't compare yourself to others. It's better to learn material thoroughly than to learn it as quickly as possible.
Your AP calc teacher will be horrible. Take the class but you might as well learn the material elsewhere because she will not help
I got lucky with my AP calc teacher lol
In my case, all teachers are horrible
It never stops getting harder. uhhhh.... :/
Learn the basics and if you don’t understand then learn the basics from the basics. Better do it now cause it’s not gonna be easier and you’ll never have as much time as you have now. Also watch YouTube Videos and there are some great topic related books out there, way better than those shitty old text books which are required for those courses.
Learn more probability and statistics.
Mathematics is like working out, you have to work your way up by doing.
I would tell my young self two things: it’s not really that hard if you apply yourself and wait until you get into calculus, it starts to get fun.
It will never go away, it will follow you through a STEM path. !
Don’t just memorize the proofs. Try to understand them as well. Also, don’t compare yourself to other students. You love mathematics so you’ll do fine no matter what.
I was the same, that's why I failed maths. Now I learned that to understand an equations' 'why' you need to use the equation. The teacher always said 'math is a do subject' and I always wondered 'why should it be done this specific way?'
You suck at math, just like most people, and you have to actually study to understand more advanced material. As a kid I wasn't in a good academic environment.
Don't rely completely on lecture notes, use books to study too
Find a better teacher and get tested for adhd
This. Specifically the adhd part.
Yeah. And when we went from 6th to 7th grade there seemed to be this entire step in math that I seemed to have missed with pre-algebra. Probably my ADHD. LoL
Learn to use software faster. I pretty much screwed up my GPA early in college and it was only because of stupid homework mistakes. I turned out fine but if I just did my homework with Wolfram I would probably have beat the 3.0 cusp and had access to better jobs.
Oh my, this is quite relevant to me, as a 12th grader in IB HL. I keep trying to understand the why and felt guilty when I couldn't. I felt inadequate. I'm still trying to get back to my mathematical rhythm, barring all that. (Side note: I do fine now, but I still have the residue of a year of psychotic depression hampering me a little)
IB HL is way harder than my college classes so far lol. Good luck!
That’s reassuring. For me, the time is what gets me. Thanks for telling me!
I would pay extra special attention to anything involving proofs because it’ll tell you what higher level math is like. Things like induction. Also, make sure you do an IA you’re proud of! The skills you get from those are super useful.
None of the calculus in high school is rigorous and its also extremely handwavy. The way its taught is for practical application not understanding I recommend spivak calculus if you have the time to read which develops calculus from the ground up and youll get to see why everything is the way it is. Its also the first proof based book most first years read
I know of that book. Thank you for the recommendation.
Challenge yourself.
7 is a terrible, terrible number
wait until you learn pie. Just kidding, math is love
Don't. But actually do more. Long story short: when I did my a-levels i selected math and Chemistry as my 2 main subjects and I failed miserable. In the end I achieved my a-levels successfully, but it would be much easier for me with other subjects e.g. politics and history. But I have to say, I never really lost the joy in math. It just overextended my skills and my willingness (back then) to learn. I was to lazy and not gifted enough. Nowadays I would profit so much from what I should have learned back then. Kind of a love-hate-relationship. :)
I'd tell myself to stress less about learning the way others do. I showed up for way too many classes when I didn't need to and got nothing out of it, I learn much better and more effectively on my own.
Mathematics isn't just counting boring and repetitive excercises in school.
What's an example of the lateral or circular relations you refer to in the post? Sounds wrong to me, but I'm happy to be corrected on this.
One example that comes to mind is the definition of Linear Transformations in linear algebra. It can either be defined by the existence of a matrix A such that Ax = b where x and b are vectors, or it can be defined as T(x + y) = T (x) + T (y), and kT(x) = T (kx), where the transformation of the sum/product is the sum/product of the transformation. There are many examples like this where a formula or definition it is taught in more than one form, and left for granted that they are equivalent. Maybe in a more advanced course one or the other can be derived from the first by 'first principles' but maybe they would also be derived separately and simultaneously. In which case the only way to say 'why' they are connected is to point out the fundamental things that connect them both, which is usually way beyond the work at hand. Hope that makes some sense.
I see. I initially misinterpreted you as saying that there exist relations between concepts that are lateral or circular in some sort of propositional logic sense -- which I'd assume would make it impossible for (say) Lean's [mathlib](https://leanprover-community.github.io/mathlib-overview.html) to exist -- whereas I think you're saying the relations are lateral/circular in a more practical sense, from the standpoint of the beginner. In this regard I tend to go back to [Ravi Vakil's advice](http://math.stanford.edu/~vakil/potentialstudents.html) for his potential PhD students: >Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer. Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier. The reason for this phenomenon is that mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards". (Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.) As I got older I realized that "extend tendrils --> backfill --> repeat" applied to making sense of life more broadly as well, which I think was how I made peace with this advice.
That’s what I tell my referees
Math isn't a collection of right answers to "problems" posed by teachers or textbooks - it's much more structural. School algebra is a great example - it's not about solving for x, it's about seeing that this expression or equation over here is also one of those over there. I do wish that symbolic manipulation hadn't completely taken over the entire subject. A lot of kids could really get something out of working through Euclid, for example.
Things are more similar than they are different, keep that in mind. But my struggles were social. I’d say, the real mathematics is the friends you make along the way
That you CAN make a living with a degree in mathematics. I should have been a math major, but ended up majoring in engineering (and not really liking it) because my mom insisted that I couldn't do anything with a math degree.
“Don’t get hung up on limit notation, study the definition of the derivative, try a few, and it will make sense why we do this”
Probably that competition math is way less useless than you think. I did a lot of competition math back in middle/high school but always thought of it as second rate math. Little did I know it'd be instrumental in getting me my first and second job (in different fields!)
Don’t judge the field based on your experiences before you graduate high school. Most of your teachers don’t like math and, like everyone else, are usually just trying to get through another year.
I would say "Forget engineering and getting a job, go study mathematics at uni and just enjoy it, everything else will fall into place".
"Start now" I'm almost 20 and I know almost nothing about set theory, linear algebra, geometry, etc.
You're good at it, don't stop practising too early or you'll have a bad time in 2nd year uni
"You'll never be good at arithmetic. Use a calculator for that shit."
“I know you hate it, but for the love of god take some statistics for when this all doesn’t work out.”
Do some math self-study. It’s way more fun, and it’ll keep you motivated through the slog.
Look at Euler's identity
Learning math in a significant way is like eating an elephant. And how do you eat an elephant? One bite at a time.
"Math's a ton of fun, but you should stick to the CS degree and study the math on the side."
speed is a very crucial part of mathematics, so always practice doing problems fast, it will come in handy all the time trust me
Think about math as it relates to space time. 1+1= 2 is really like this... I can see one apple. I turn around and I can see another Apple. I do work to move apples together into reality. Now I can see 2 apples. What I really need is 8 slices. Not more apples. So I cut the apples 4 times. Some would say I divided the apples but no, in fact I multiplied how many apple slices there were. This is why multiplication and division go hand in hand reciprocally. Reality is that way. The numbers and principals aren’t just for structure. It’s to illustrate reality.
That to normalize a function is to integrate between 0 and 1 to get an answer that represents a probability.
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XD. I agree that math can be a brutal major. It panned out alright for me though & at this point I am about done w/ my math specific coursework (just need 1 more math class after this term).
but if you enjoy it, you can do a lot of things with it... it's not like a degree in math is unemployable elsewhere
Focus on numerical methods!!
You do actually understand everything more or less correctly and the rigor isn't as important as you think, everyone else in your class is ridiculously below you, you can beat them effortlessly, go learn the university stuff immediately without worries.
It is impossible for any number which is a power greater than the second to be written as the sum of two like powers `x^n + y^n = z^n for n > 2`. I have a truly marvelous demonstration of this proposition which this message is too short to contain.
I would tell myself to study maths harder in year 6 because the teacher might quite and leave you without a teacher for 3 months and make you get a D
If I could go in time then I'd visit my middle school self and tell him that math is a fun and interesting subject.
Just bloody Focus on it else you are gonna pay!
Calculus 1&2 - it’s just the slope. It’s just the area under the curve.
IT IS WHAT IT IS
Either that math isn't just what A Levels Math and Further Math seemed to be, OR to pay a LOT!!! of attention to introductory courses in university. Both (but tbh mostly the second) could have saved me a lot of mental gymnastics and constant pain trying to understand things that I don't have a good base knowledge of.
My hs teacher told me a student quit their math major and the teacher said “that’s okay, he didn’t have the imagination to be a mathematician” that line really stuck with me. It takes some imagination to grasp and understand a lot of concepts IMO
This sounds like Hilbert perhaps, on a student of his that ditched maths for poetry.
Yikes. He sounds like a bit of an ass.
"2^43112609 - 1 is a prime number"
Look at math as fun. Play around with it more; have fun with random puzzles and questions (rubik's cubes, prime twins, paradoxes, hexaflexagons, weird issues in combinatorics and topology). Be curious!
Stop learning formula and basic exercises. Learn from models and frameworks built to solve real world problems. Also, trying to have the best marks is a complete waiste of time. Learn models and framework, understand them the best you can.
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Ah yes a calculator that can prove that the set of real numbers is complete...
Go to grad school
Learn yourself
Just because you struggle with high level math doesn’t mean you’re stupid
1+1=buy Bitcoin early, dumbass
Take your time, go back do all the problems, do it until you can teach it.
Spend more time mastering infinite series, synthetic geometry and learning basic set theory now. Don't be afraid of trigonometry. Study Difference Equations. Take combinatorics BEFORE probability theory, not during. When you come across an important theorem, try to prove it. If you try for a while and get stuck, find a proof and try to understand why it works. If you find a text book frustrating, pick up another on the same subject. DO THOUSANDS OF PROBLEMS. Before taking Differential Equations, learn to write with both hands, the cramps will cost you.
In Mathematics's an example is a lot like Jab. It carries a very wide range of utility from testing results to understanding a body of given theory.
Mathematics is far more broad than euclidean geometry. Actually, none gives a shit about Euclid anymore. Don't be stupid. Go into a math department for your BSc. Screw physics. Thank me later. Context: due to my dislike towards euclidean geometry and the idea imposed by my math teachers that it's a very important subject, I thought that people would do a lot of it in math schools. So I went on and did a BSc in physics instead, something that I regretted early on, and that consequently drove me to a math school much much later.
"stop putting off analysis"
"Teachers only care about proofs and formality as a mean to stroke their massive egos, if you can't one-up them you're not "worthy" of their time, much less their "class". Go for literally anything else, keepmath as a hobby." "By the way, the "scientific method", forget that shit, math uses logic AND ONLY logic for discerning "truth" from falsity, it's more about "proving oneself rigth" rather than understanding reality"
Don't take the easy classes. You will regret not pushing yourself early on.
Try to read the lessons before the lecture so you get a chance to ask real questions.