To recap: 1. Don’t be scared! 2. Embrace the variety of mathematics on offer 3. Review lecture material after the lecture 4. Make recipe cards, practice questions and figure out how you complete example sheets best 5. It’s okay if if gets hard at times 6. Enjoy!! Any questions, just comment below! 😌
Math PhD here. Nice video! Long post ahead.. :) I like that you bring up the fear most, if not all math students feel. When I started I assumed everyone else would be some much more clever and better at math than me. And I constantly thought that the next course or exam would be way beyond my capabilities and I would eventually make a fool of myself. Nothing of this was true. But the constant struggle against your fears and impostor syndrome was very real. What you eventually realize though, is that you are not alone. Almost everyone feels and thinks the same way to some extent. You are not supposed to be perfect. You are there to learn, to question, not to give an impression to be some perfect all knowing student. I would like to add what is my most important point if I could go back in time and start over: * Try to really understand what the intuitive meaning behind everything you learn is and why and when what you learn is important. Especially how it is connected to other stuff you already know, and also how it is new and different. Don't skip examples and applications. The professors/teachers will be very happy if you ask them about such things and you can benefit a lot from their experience. When I started studying I did not care about applications at all. I had read that one should see math as some kind of art where the beauty of the subject itself should be enough. I did not question or reflect much about what I learned and why the contents of the definitions and theorems where chosen as they were. This was a mistake. You should do the opposite as much as possible. Never rely on some authority argument that what you learn is good for you. You need to internalize it for yourself. When you see a new definition, think about all parts of it and try to figure out why the definition is as it is. Why is it useful? Can you find examples satisfying it, examples which do not? What would happen if you skipped or added some part of the definition, would it be meaningful? How does the definition connect to your previous knowledge? What new stuff does it bring? Do not just read and think "Ok, I understand what it says, but it seems pretty random and strange to me." Then you don't really understand it. The same goes with reading and understanding proofs. You can read a proof, understand every symbol and every logical step from start to finish, but still totally miss the intuition and deeper understanding of why the proof works. It would be comparable to reading a chapter of a book word for word, understanding every word but not paying attention to the story told. Almost all proofs have deeper meaning which you typically miss at a first reading. Understanding and uncovering those deeper insights is an important aspect of really learning math and also gives it much more meaning. Otherwise everything will just feel like facts on top of each other without structure.
Thank you so much for this comment! I am just starting my maths course and I really struggle with understanding the deeper meaning of a proof, I always feel as if I should have more information to get how it is useful and gives insight. Although this is sometimes true, and I will understand the proof a few lectures later, I will absolutely take your advice and try to look for the insight I can access immediately and reflect more deeply on definitions and proofs. I just wanted to thank you for the advice!
@@emmam.1864 I am glad if I could help. Finding that deeper or more intuitive meaning is pretty hard in general, especially since you are learning the stuff for the first time. But it is worth the effort. Everything you learn will feel more meaningful and this also helps remembering it. Also, ask the teacher if he/she can help you by providing some insight, for example describing in words what is really happening and what is the core ideas behind the scenes. It is very easy to get lost in a mess of symbols when learning math.
Hey, hope not to bother you. If you were self-studying, how do you think you could get the "intuition" and "deeper understanding" of how the proof works? I feel the words are vague and I haven't really experienced that distinction, often I just learn by doing and playing the material. Does it only become important once you complete undergrad?
@@owlheracross Hi! This deeper understanding becomes important when you really want to use the math you have learn and make it a part of your problem solving skills. If you only learn stuff on a more surface level you will have a much harder time to see when and why your knowledge can be applied. When you take some math course at university, you know that the exercises and exam questions will be adapted to fit perfectly to what you have just learned. In the "real world" at work or if you are doing research in math to come up with something new, no one is telling you what math could be useful for the problem at hand. You need to rely on your intuition and deeper knowledge to guide you. Learning by doing and playing the material is excellent. That is how you quickly discover gaps in your knowledge. If you only read without doing, it is easy to start to believe you know the material better than you actually do. What I mean with deeper understanding and intuition here are things like being able to explain why the proof works in words, using visual geometrical arguments and so on where you don't just rely on the exact mathematical symbols in the proofs. I don't know exactly what level you study, but in the first year or two, math is mostly about calculating, solving and doing. Later on almost everything is about proofs. And in research it is about coming up with new proofs. So if you haven't really started to read and understand proofs yet, then I understand if what I wrote does not yet makes much sense. Often a proof can be very long and consists of lots of smaller arguments. You will never remember the proof or have any real use of it either if you just memorize the logical steps. Maybe there are 50 smaller logical steps for example. Then you have to sit down and think what the main greater steps are. Maybe there are just two or three main ideas and all the other logical steps are just arguments to why these main ideas work. For those main ideas try to form some picture in your head why everything fits together without going into details. Think of a long proof as your vacation plans. You will not write down you vacation plans from start to finish like "Get up from bed, take first step, go to kitchen, ..., call taxi, take your bags to taxi.." and so on. You will, of course, plan the big things first. Where do you want to go? How will you get there? For how long will you be away? What kind of accommodation do you prefer? When you have done so, you can start to plan the details of your trip around these steps. "I want to go to the US, lets see what state. I will travel by plane, lets search for the time table and look for prices and available flights. I think two weeks are good. Exactly what day will I leave?" And so on. You try to make a top down plan for your vacation. This is how you ultimately want to look at your mathematical proofs as well. Unfortunately, most math books are not written with that top-down view, but rather like my first example "Get up from bed.." and so on. This is not how a human thinks intuitively. You can always ask your professor for the main point in the proofs in the beginning. They will be happy to explain. In time you should try to get better by yourself doing this. It is often not easy, but you will get better in time. Regarding forming pictures in your head. Lets take a very simple example to illustrate the point. Let us say you have two linear equations in three variables. If you go ahead and solve them you will notice you either get zero solutions or infinitely many solutions. If you had formed a geometrical picture in your head you would have known this without doing any calculations: Each of the equations can be seen as the equation of a plane in three dimensions. And two planes intersect either on a line (infinitely many points) or else the planes are parallel in which case they don't intersect at all (or are equal). Finding a geometrical picture or some other intuitive way of looking at it, is often hard work though. But it will be worth it. You will remember the proofs, be able to use them and not the least, the math you learn will feel much more meaningful.
@@TheIllerXThank you. I am not exactly a Mathematics student at any University however I am using Apostol's two introductory volumes on Calculus for self-study. Sometimes, I feel that a proof is like a journey through various streets in our mind to the final destination, i.e. the theorem. Anyways, proofs are hard to understand in one go. I will apply your tips the next time I study.
I'm MSc maths like you, and your advice is spot on. I do hobby research, and are working on two papers at the moment with new stuff. The best advice I will give on top of yours is, that that is possible with a math degree. You can do new math. Build up contexts where you use tools from the uni-toolbox, and prove properties of objects in the context. Evening after evening with hard problems and where the outcome is unknown. The great satisfaction it gives when you succeed, and can type the whole thing up in Latex. In the projects I've used number theory, group theory, vector spaces, metric spaces, measure theoy, topology and real analysis. Without my master degree, the whole thing would be impossible!
Great advice, Ellie. Couldn't agree more. 👍 I'd only add that there is now a great wealth of help on-line, with videos of worked examples and lectures here on UA-cam, and elsewhere -- which wasn't the case when I did my degree, but is the case now.
I’ve just started my computer science degree, and I’m already diving into some of the math-heavy content. One of my first modules is on set theory, which is discrete math, which I’ve learned is a whole branch of mathematics. It hasn’t been too difficult yet, but the lecturers say the pace will pick up gradually. I’m looking forward to the challenge as the content becomes more advanced. Thanks for the advice.
Something I'd add from my studies of mathematics at Cambridge, if you're not from a private school, is don't be intimidated by seemingly confident/cocky privately educated posh sounding students around you asking or pointing out clever sounding things in mathematics lectures (some are lovely, some are not..), if you follow Ellie's advice you'll likely do way better than them - most did crap in their exams compared to us non-privately educated students at Cambridge mathematics certainly in my year cohort, despite the tens of thousands that had been spent on them prepping for and giving them a leg up into university :)
@@Kristianres Correct, not all are; several friends both from the Cambridge mathematics degree and my professional colleagues (and some of my relatives) went to private school and like I said were/are lovely - but that said a fair proportion on the Cambridge mathematics degree and noticeably a bigger proportion on other degrees were cocky/quite arrogant, yet didn't do so well in their exams (I suspect their abilities had 'peaked' just to get in having had the extra benefit of private school which as an aside makes the whole "what school did you go to?" thing a total nonsense - not a criticism of their ability, the Cambridge mathematics degree is hard even for smart people). This shouldn't put anyone off doing the maths degree or Cambridge in general, I was also pleasantly surprised by how many down to earth and lovely people both private and state school background there were.
Just because all cocky students are privately educated doesn't mean all privately educated students are cocky. Why are you looking at a maths channel if you don't understand basic set theory?
@@theartisticactuary In both my original comment and my subsequent reply to someone who commented, I stated that some privately educated students were lovely. Furthermore I literally stated that I did mathematics at Cambridge, which is why the channel is interesting to me.
I’m a first year studying mathematics. I took the advance version of math courses that everyone is required to take for their math degree. It’s a little hard and I’m so nervous about midterms and exams because most of the times I can’t solve the assignments completely on my own and it makes me nervous about exams. Would be great if you can give an advice about this situation :’) by the way I love your videos and I always watch them :)
hey Ellie a quick question: what are the exams like? Are they similar to example sheets? Or they would throw you some questions that require "wit" and "inspiration" to solve?
Did you ever do any self studying? I'm currently a 2nd year at a Russel group university in the UK studying maths but I often find the course having a smaller scope and having a much slower pace covering less content. I want to Study part 3 at Cambridge for pure maths; do you have an advice on trying to bridge the gap? (currently going through Dexter's lecture notes at cambridge).
Taught math>won't use it for anything useful in the world>get a degree in math>teach it to others who won't use it for anything useful>they get degrees and then teach it to others who won't use it for anything useful
math is a way of thinking which helps us to make good decisions and it is not just a subject, it is the first thing to use it for living, after food. Math is the ground that we built the other science on.
My best advice for math students is always try to find applications for your math study and apply more step by step , otherwise math science will be just for nonsense ....
To recap:
1. Don’t be scared!
2. Embrace the variety of mathematics on offer
3. Review lecture material after the lecture
4. Make recipe cards, practice questions and figure out how you complete example sheets best
5. It’s okay if if gets hard at times
6. Enjoy!!
Any questions, just comment below! 😌
I love you deeply! 😍
My question: Do you love me?
How to be really good at Topology and real Analysis. Kindly Make videos on it
How to be really good at Topology and real Analysis. Kindly Make some videos on it
@Andrew-TsegayeNice question😂
My advice: Don't think you're the only one who doesn't understand a certain problem or concept. You're not.
right
Math PhD here. Nice video!
Long post ahead.. :)
I like that you bring up the fear most, if not all math students feel. When I started I assumed everyone else would be some much more clever and better at math than me.
And I constantly thought that the next course or exam would be way beyond my capabilities and I would eventually make a fool of myself.
Nothing of this was true. But the constant struggle against your fears and impostor syndrome was very real.
What you eventually realize though, is that you are not alone. Almost everyone feels and thinks the same way to some extent.
You are not supposed to be perfect. You are there to learn, to question, not to give an impression to be some perfect all knowing student.
I would like to add what is my most important point if I could go back in time and start over:
* Try to really understand what the intuitive meaning behind everything you learn is and why and when what you learn is important. Especially how it is connected to other stuff you already know, and also how it is new and different. Don't skip examples and applications.
The professors/teachers will be very happy if you ask them about such things and you can benefit a lot from their experience.
When I started studying I did not care about applications at all. I had read that one should see math as some kind of art where the beauty of the subject itself should be enough.
I did not question or reflect much about what I learned and why the contents of the definitions and theorems where chosen as they were.
This was a mistake. You should do the opposite as much as possible. Never rely on some authority argument that what you learn is good for you. You need to internalize it for yourself.
When you see a new definition, think about all parts of it and try to figure out why the definition is as it is. Why is it useful? Can you find examples satisfying it, examples which do not?
What would happen if you skipped or added some part of the definition, would it be meaningful? How does the definition connect to your previous knowledge? What new stuff does it bring?
Do not just read and think "Ok, I understand what it says, but it seems pretty random and strange to me." Then you don't really understand it.
The same goes with reading and understanding proofs. You can read a proof, understand every symbol and every logical step from start to finish, but still totally miss the intuition and deeper understanding of why the proof works. It would be comparable to reading a chapter of a book word for word, understanding every word but not paying attention to the story told.
Almost all proofs have deeper meaning which you typically miss at a first reading. Understanding and uncovering those deeper insights is an important aspect of really learning math and also gives it much more meaning. Otherwise everything will just feel like facts on top of each other without structure.
Thank you so much for this comment! I am just starting my maths course and I really struggle with understanding the deeper meaning of a proof, I always feel as if I should have more information to get how it is useful and gives insight. Although this is sometimes true, and I will understand the proof a few lectures later, I will absolutely take your advice and try to look for the insight I can access immediately and reflect more deeply on definitions and proofs. I just wanted to thank you for the advice!
@@emmam.1864 I am glad if I could help.
Finding that deeper or more intuitive meaning is pretty hard in general, especially since you are learning the stuff for the first time.
But it is worth the effort. Everything you learn will feel more meaningful and this also helps remembering it.
Also, ask the teacher if he/she can help you by providing some insight, for example describing in words what is really happening and what is the core ideas behind the scenes.
It is very easy to get lost in a mess of symbols when learning math.
Hey, hope not to bother you.
If you were self-studying, how do you think you could get the "intuition" and "deeper understanding" of how the proof works? I feel the words are vague and I haven't really experienced that distinction, often I just learn by doing and playing the material. Does it only become important once you complete undergrad?
@@owlheracross Hi!
This deeper understanding becomes important when you really want to use the math you have learn and make it a part of your problem solving skills.
If you only learn stuff on a more surface level you will have a much harder time to see when and why your knowledge can be applied.
When you take some math course at university, you know that the exercises and exam questions will be adapted to fit perfectly to what you have just learned. In the "real world" at work or if you are doing research in math to come up with something new, no one is telling you what math could be useful for the problem at hand. You need to rely on your intuition and deeper knowledge to guide you.
Learning by doing and playing the material is excellent. That is how you quickly discover gaps in your knowledge. If you only read without doing, it is easy to start to believe you know the material better than you actually do.
What I mean with deeper understanding and intuition here are things like being able to explain why the proof works in words, using visual geometrical arguments and so on where you don't just rely on the exact mathematical symbols in the proofs.
I don't know exactly what level you study, but in the first year or two, math is mostly about calculating, solving and doing. Later on almost everything is about proofs.
And in research it is about coming up with new proofs. So if you haven't really started to read and understand proofs yet, then I understand if what I wrote does not yet makes much sense.
Often a proof can be very long and consists of lots of smaller arguments. You will never remember the proof or have any real use of it either if you just memorize the logical steps.
Maybe there are 50 smaller logical steps for example. Then you have to sit down and think what the main greater steps are. Maybe there are just two or three main ideas and all the other logical steps are just arguments to why these main ideas work.
For those main ideas try to form some picture in your head why everything fits together without going into details.
Think of a long proof as your vacation plans. You will not write down you vacation plans from start to finish like "Get up from bed, take first step, go to kitchen, ..., call taxi, take your bags to taxi.." and so on. You will, of course, plan the big things first. Where do you want to go? How will you get there? For how long will you be away? What kind of accommodation do you prefer?
When you have done so, you can start to plan the details of your trip around these steps. "I want to go to the US, lets see what state. I will travel by plane, lets search for the time table and look for prices and available flights. I think two weeks are good. Exactly what day will I leave?"
And so on. You try to make a top down plan for your vacation. This is how you ultimately want to look at your mathematical proofs as well.
Unfortunately, most math books are not written with that top-down view, but rather like my first example "Get up from bed.." and so on. This is not how a human thinks intuitively.
You can always ask your professor for the main point in the proofs in the beginning. They will be happy to explain. In time you should try to get better by yourself doing this.
It is often not easy, but you will get better in time.
Regarding forming pictures in your head. Lets take a very simple example to illustrate the point. Let us say you have two linear equations in three variables.
If you go ahead and solve them you will notice you either get zero solutions or infinitely many solutions.
If you had formed a geometrical picture in your head you would have known this without doing any calculations: Each of the equations can be seen as the equation of a plane in three dimensions. And two planes intersect either on a line (infinitely many points) or else the planes are parallel in which case they don't intersect at all (or are equal).
Finding a geometrical picture or some other intuitive way of looking at it, is often hard work though. But it will be worth it. You will remember the proofs, be able to use them and not the least, the math you learn will feel much more meaningful.
@@TheIllerXThank you. I am not exactly a Mathematics student at any University however I am using Apostol's two introductory volumes on Calculus for self-study. Sometimes, I feel that a proof is like a journey through various streets in our mind to the final destination, i.e. the theorem. Anyways, proofs are hard to understand in one go. I will apply your tips the next time I study.
I'm MSc maths like you, and your advice is spot on. I do hobby research, and are working on two papers at the moment with new stuff. The best advice I will give on top of yours is, that that is possible with a math degree. You can do new math. Build up contexts where you use tools from the uni-toolbox, and prove properties of objects in the context. Evening after evening with hard problems and where the outcome is unknown. The great satisfaction it gives when you succeed, and can type the whole thing up in Latex. In the projects I've used number theory, group theory, vector spaces, metric spaces, measure theoy, topology and real analysis. Without my master degree, the whole thing would be impossible!
Great advice, Ellie. Couldn't agree more. 👍
I'd only add that there is now a great wealth of help on-line, with videos of worked examples and lectures here on UA-cam, and elsewhere -- which wasn't the case when I did my degree, but is the case now.
Doing Mathematics is fun.
I’ve just started my computer science degree, and I’m already diving into some of the math-heavy content. One of my first modules is on set theory, which is discrete math, which I’ve learned is a whole branch of mathematics. It hasn’t been too difficult yet, but the lecturers say the pace will pick up gradually. I’m looking forward to the challenge as the content becomes more advanced. Thanks for the advice.
im so happy that im early!! :)
Could you please make a whole video on your journey?
Something I'd add from my studies of mathematics at Cambridge, if you're not from a private school, is don't be intimidated by seemingly confident/cocky privately educated posh sounding students around you asking or pointing out clever sounding things in mathematics lectures (some are lovely, some are not..), if you follow Ellie's advice you'll likely do way better than them - most did crap in their exams compared to us non-privately educated students at Cambridge mathematics certainly in my year cohort, despite the tens of thousands that had been spent on them prepping for and giving them a leg up into university :)
I don’t think all private students are cocky and overconfident like that.
@@Kristianres Correct, not all are; several friends both from the Cambridge mathematics degree and my professional colleagues (and some of my relatives) went to private school and like I said were/are lovely - but that said a fair proportion on the Cambridge mathematics degree and noticeably a bigger proportion on other degrees were cocky/quite arrogant, yet didn't do so well in their exams (I suspect their abilities had 'peaked' just to get in having had the extra benefit of private school which as an aside makes the whole "what school did you go to?" thing a total nonsense - not a criticism of their ability, the Cambridge mathematics degree is hard even for smart people). This shouldn't put anyone off doing the maths degree or Cambridge in general, I was also pleasantly surprised by how many down to earth and lovely people both private and state school background there were.
Just because all cocky students are privately educated doesn't mean all privately educated students are cocky.
Why are you looking at a maths channel if you don't understand basic set theory?
@@theartisticactuary In both my original comment and my subsequent reply to someone who commented, I stated that some privately educated students were lovely. Furthermore I literally stated that I did mathematics at Cambridge, which is why the channel is interesting to me.
@@electrocademyofficial893Wasn't aimed at you, mate. It's Kristianres who doesn't seem to understand set theory.
The BEST MATH UA-cam CHANNEL.
Mind Your Decisions! I am Presh Talwalkar.
I’m a first year studying mathematics. I took the advance version of math courses that everyone is required to take for their math degree. It’s a little hard and I’m so nervous about midterms and exams because most of the times I can’t solve the assignments completely on my own and it makes me nervous about exams. Would be great if you can give an advice about this situation :’) by the way I love your videos and I always watch them :)
realmente os videos dela sao bons.
List of math books you studied plz - abhihiro
Why didn't you go for a PhD? Will you do one in the future?
A dream PROFESSION☺
I can do it 💪
hey Ellie a quick question: what are the exams like? Are they similar to example sheets? Or they would throw you some questions that require "wit" and "inspiration" to solve?
I did not know I have to calculate for R
hello, i need advice on proofs, how to mature mathematically and write better proofs?
Pls explain Dirac delta function
Seus videos sao bem didaticos.
You are amazing Ellie, In this time where most of people are making time wasting and sexual video you make videos on mathematics, ❤ Amazing...
Did you ever do any self studying? I'm currently a 2nd year at a Russel group university in the UK studying maths but I often find the course having a smaller scope and having a much slower pace covering less content. I want to Study part 3 at Cambridge for pure maths; do you have an advice on trying to bridge the gap? (currently going through Dexter's lecture notes at cambridge).
Estou estudando Matematica no brasil mais precisamente na Universidade Estadual de Goias na UEG, acredito que nao conheça.
Why so many formulas
Do you recommend a double major in both math and CS?
this is a very broad question that the depends on your specific life circumstances
Taught math>won't use it for anything useful in the world>get a degree in math>teach it to others who won't use it for anything useful>they get degrees and then teach it to others who won't use it for anything useful
math is a way of thinking which helps us to make good decisions and it is not just a subject, it is the first thing to use it for living, after food. Math is the ground that we built the other science on.
NICE
😊😊😊😊🧠🧠🚀🚀you are a precious!!!
My best advice for math students is always try to find applications for your math study and apply more step by step , otherwise math science will be just for nonsense ....
You are good girl.
Wth
log off bro
Ok man.
My advice, which is not a really serious one but sort of important one is, learn LaTeX asap
voce acha que deve apreender isso? primeiramente?
You are looking gorgeous 😊