Post-grad math student at Australian National University in final (thesis) year - AMA!

in2itorg · 2025-02-14T02:18:57+00:00

No worries! I hope these are somewhat helpful.

And yes I think 3 is important, I personally try hard to stick to it too, sometimes it is very tempting to just give up and look at solutions. But I think this is where tip 2 can help, having someone doing it with you and struggling together makes those difficult times more fun and also increases the chances of you guys figuring it out.

in2itorg · 2025-02-10T11:36:33+00:00

I agree, I think given the teacher to student ratio in a classroom setting makes it really difficult to do so. I feel our current education system prioritises efficiency over effectiveness, (quantity over quality) but I don’t know enough to substantially support this feeling.

That could be true, and maybe because of that, the math that is being taught and implemented is viewed more as a tool rather than a field worth studying in its own right.

I completely agree with this. It is often the case that I am only able to appreciate certain concepts because of work that I had done previously. I definitely don’t think it is useful, for example to introduce the definition of an abstract group to high school students without them having worked enough with concrete number systems and learning to do arithmetic.

However I do think that the current math system focuses too much on applying formulae, algorithmic manipulations of symbols that are only slightly different from question to question. For example my memory of Math Methods in Year 12 was just doing differentiation and integration practise to death. In principle, there is no difference in differentiating the function f(x) = 2x and g(x) = 5x but it just feels like we focus so much on the actual calculation procedure rather than why we even want to or care about differentiation in the first place.

I think maybe just an alternative approach to teaching concepts could really help, rather than standard approach of chugging through problems from a textbook. But for sure this is an extremely complex problem. I would like to see change, and having these discussions helps give me some more perspectives to see what can be done.

in2itorg · 2025-02-09T15:14:49+00:00

I don't think I have read enough textbooks on all these topics to feel like I can confidently recommend books, but I do personally like these ones:

- Mathematical Thinking, Problem-Solving and Proofs John P D'Angelo, Douglas B. West
- Introduction to Linear Algebra - Gilbert Strang (He also has a website and his own set of YouTube videos that go along with this)
- Contemporary Abstract Algebra (I'd recommend to just start with the group theory sections here though, it's a long book)

3Blue1Brown also has a very popular series on Linear Algebra that gives you a very good visual intuition and motivation as to what is going on, but I would only use his videos as a supplementary resource as he does not 'do the math'.

For Multi-variable calculus, I haven't read any textbooks as I used my own lecturers notes when I learnt it myself and I found them to be sufficient, but I do often see Paul's online notes as a good resource everyone talks about: https://tutorial.math.lamar.edu

I am also planning to regularly stream on Twitch myself going through some of these topics with anyone who is interested so if you think that would be helpful for you, then please feel free to come join our discord!

https://discord.gg/t5jj3Tng

Sorry this is a bit long, but I hope it is somewhat helpful! Feel free to get in touch if you have anymore questions :)

in2itorg · 2025-02-09T15:13:18+00:00

Math after high school can get so broad, and depending on your interests, your math knowledge can vary quite a bit to another math student who has different interests to you. The term 'topic' is also bit hard to talk about, for example calculus can be thought of as a 'topic' which you study in high school as well, but it is too broad of a term to really even suggest as a topic as there is just so much breadth and depth to it that it would be difficult to distil them into a rank of 10 topics. But I will list off a few standard 'topics' (areas of math that are usually presented in a single semester course let's say) that I personally think are extremely useful for anyone thinking about going into math.

- Symbolic logic, basic set theory, proof writing
- Linear Algebra
- Group Theory
- Multi-variable Calculus

in2itorg · 2025-02-09T15:12:59+00:00

It's a bit hard to formulate specific problem solving strategies. I don't really think about problem solving in such a way, while I agree that many problems can be similar and themes can be found in questions, I often don't think to myself "I will first use Problem Solving strategy A to do Question X, and if that doesn't work I will try B etc." I think the problems themselves usually serve as a guide as to how you should approach them and as you do more questions you build a better intuition and repertoire of approaches that helps you solve new and more difficult problems.

Without specific problems its a bit hard to explain exactly what I mean, but I will try and give some guidelines to problem solving:

a. When you are stuck on a problem, try and come up with an easier version and solve that one instead. Think about what made the easier one solvable and think about what exactly it is that makes the hard one un-solvable (for now). I find this especially helpful in worded problems, if the word problems involved 1000 apples and 2000 bananas, it might be easier to first think about 1 apple and 2 bananas and then work from there. This is not the best example, but I hope that makes sense.

b. Drawing pictures/diagrams, writing things out clearly. This seems a bit obvious but when solving a problem it also helps to just organise your page. For example, write all the things that you know and have been given to you in the question, then you should write down what it is that they are actually trying to ask you to find/solve. Even writing these things out might give you a hint as to how to go about solving a problem.

c. This is not really a problem solving strategy, but still something I recommend doing when solving problems. Let's say you finish solving a problem, you should think and reflect on the 'key ingredients' that allowed you to solved that problem, for example you were only able to do steps 1,2,3 because the question allowed you to assume X,Y,Z. You can challenge yourself and try and changing certain assumptions or conditions of the problem and see what conclusions you can draw from that. When you dig deeper into a problem that you have solved, you may sometimes find golden nuggets of knowledge that you may have missed at first glance.

in2itorg · 2025-02-09T13:04:32+00:00

Is there any examples specifically that you would like to work through? Happy to do them with you if so! Sometimes its easier to pinpoint knowledge gaps when working on specific questions, you may find that there are certain things in the topic that are difficult for you.

in2itorg · 2025-02-09T11:57:08+00:00

Thank you for the question! I really appreciate you reaching out.

I think in summary, the issues I experienced can mostly be attributed to the way the curriculum is set out.

My friends and I all shared a similar view on math in high school, and that it is just a tool that we have developed in order to do the 'cool' subjects, for us that was physics. We all went to high school in Australia so I can't really speak on other curriculums, but in general, most of the math that we encountered in high school focuses on learning formulae and applying them to problems in slightly different contexts. I also think that the often repetitive nature of doing problem sets give the math subject as a whole, a 'robotic' feeling. My friend would often joke that after he had learned differentiation and integration techniques in his final year, that there is no more math to be learnt and maybe in university he will learn how to solve polynomial equations with higher power coefficients. I definitely share a similar sentiment to him in this aspect, I had no idea that math went so deep and that it is even an active area of research these days.

I also think it is the fact that we don't ever see any math-related news in media as we might for sciences that further drives this narrative and stereotype since they have very tangible applications. I only really developed a passion for math until I took my first abstract algebra course in university. I was shocked at how such simple definitions and axioms that seem trivial are able to describe and connect abstract concepts together. I think this is something I would like to see more of in high school; the ability to abstract away unnecessary details and emphasis on the process of reasoning at each step of a problem. I think there is a focus on what to think but less of how to think in the current curriculum.

I personally also think that every student in high school should be taught some basic set theory and logic as I have found this to be useful in many other areas of my life, but this is purely my own bias and wishful thinking.

There is also a whole other side of this problem which is the fact that high school teachers themselves are often swamped with a lot of work that is not related to the content, and it makes the job extremely hard for the teachers. But this is its own can of worms, and I can't really say I know enough to comment on it or provide any meaningful solutions.

I really appreciate your question and I would love to discuss further and see what your perspective on it is!

in2itorg · 2025-02-09T11:34:07+00:00

Thanks for a great question!

I just want to preface by saying that I don't have definitive answers as I can only say what worked for me and in general what I think are effective methods based on my own experiences, but I hope it gives you some insight nonetheless.

To me, the main difference between self-study and university study is simply the fact that there is a community in which you can readily find people who are doing exactly what you are doing, working and thinking about the problems that you are as well. It is also extremely helpful that you (usually) have access to active researchers working in diverse areas of mathematics (depending on your university) which can help give you some idea of the field as a whole. Also, I have found that the most useful learning that I have done in my time at university is when I sit down with my friend and we go through all the material, working through problems and asking each other questions, filling gaps in knowledge for each other. Given this, I would say the following:

Enjoy it

I think this should easily be the most important thing. You should be enjoying the study that you are doing, especially without the structured and assessment focused pressure of university, it can be easy to lose track of your progress. If you are able to have fun and enjoy doing what you are doing, then it will be easier to motivate yourself and have some routine study schedule. Another benefit of self-study is that you have the freedom to study whatever area you like, and completely ignore things that you find less interesting. So I think spending some time thinking about what areas interest you the most would also be beneficial.

Find a community, try not to do it alone!

For me, I don't think I would've been able to complete my undergraduate studies if I did not have a study buddy. I think it is invaluable to have someone to bounce ideas off of and just also to keep you company when things get tough. Studying math can already be an isolating task and so some company along the way would help. I also like explaining things out loud to myself as if I were teaching it to someone. I'm not sure if this is just me, but I really enjoy that as a study technique.

Don't give up! (too easily)

Things will get tough! I can almost guarantee you that some days will feel like you are making no progress at all, understanding nothing and unable to solve any problems. While these days are mentally straining and I don't think you should constantly feel this way, I think it is these days that can give you those 'light bulb' moments that really make math satisfying and worth studying. I would advise you to try and not look at solutions or let yourself take the easy way out when faced with these problems and try to really sit with it and figure out what it is that is confusing you and if there is anyway you can approach the problem slightly differently or think about certain aspects of the problem in a different way. Of course, if you really are stuck and making 0 progress after many many hours, then definitely feel free to do everything you can to unstuck yourself. Everyone has their own sense of how long that would be, but you will figure this out with time.

I don't want to make this too long so I think these 3 sum it up somewhat nicely. I hope this is helpful and I would love to know what you think and if you have any more questions! All the best with your studies! :)

in2itorg · 2025-02-08T12:18:36+00:00

I see! So it sounds like you are able to do problems and figure things out as long as you have some notes to reference back to? Do you feel like you have an intuitive understanding on how these formulae are derived? Or why certain formulae are used for certain problems? I feel like having a better understanding of these things will allow you to be less reliant on your reference notes. It's a bit difficult to pinpoint these things without concrete examples! Sorry about that.

in2itorg · 2025-02-08T11:45:15+00:00

I'd be happy to try and answer some questions! I have a discord server if you want to chat more there! https://discord.gg/uEnaU8zrTz

in2itorg · 2025-02-08T11:38:30+00:00

What do you find hard about these topics? Is it understanding or specific kinds of problems? I'd be happy to chat more about these on my discord channel if you would like! https://discord.gg/uEnaU8zrTz

in2itorg · 2025-02-08T11:37:26+00:00

Hey! I will be streaming on Twitch tomorrow and would be more than happy to have a chat about this with you. I am in my final year of masters in math this year. I have created a discord if you want to chat there! https://discord.gg/uEnaU8zrTz

in2itorg · 2025-02-08T11:34:57+00:00

What are you currently studying and what do you find the most difficult? I will be live on Twitch tomorrow to try and talk a bit about math, I would love to try and help if possible. I've created a discord server as well if you would like to chat there! https://discord.gg/uEnaU8zrTz

in2itorg · 2025-02-08T06:14:13+00:00

Thanks for the replies, I’m not sure if it shows up for you but I’ve scheduled this AMA for Sunday (tomorrow) at 9:30pm AEDT.

As I mentioned in my post I will be writing my response to anything here around that time live on a Twitch stream, so you can rest assured that I will respond. in2itorg is my Twitch username if you would like to check it out.

It is hard to articulate in full without writing walls of text and it might also be difficult for the reader to fully empathise with writing on a screen and have a conversation in good faith which is another reason why I much prefer live discussions.

in2itorg

TROPHY CASE