School Mathematics VS University Mathematics

Subset Association 2021

A recording of a discussion by students in the group Subset Association, reflecting on their Theory Building Assignment for Seminar 4 in Foundations of Abstract Mathematics I (2021, Stellenbosch University).

 

Student impressions in 2021, towards the end of the first term

Transcripts from group work, where students reflected on what they have learned in the first term of 2021:

• Knowing how to approach proving something.
• Reading and understanding equations. We knew about logical operators, but now we know how to use them more effectively to get actual results. This equips one with a toolbox to use in other math modules. Reading and interpreting equations in other modules specifically.
  
• The idea of assumption and conclusion: how every statement in a mathematical argument is either an assumption or a conclusion.
  
• How the course was presented: great to have emphasis on effort and passion in the beginning of the term, rather than your skill. Intuitive example and then connecting with math - great method of teaching.
  
• Philosophical look at mathematics: instead of being given a problem and asked to solve it, now we look at the mechanics of how we can solve the problem and what really encompasses mathematical activity. Comparison with language is fascinating. It is a good life skill to understand logic, which this term contributed to.
• Originally, I thought of this like every other math course: numbers and calculations. Now I view this course more as a course in logic which teaches you how to think. This was very cool, very unlike to what I have done before. Excellent pacing: it was important not to go fast to get a good understanding of what we are working on.
  
• This term gave me a deeper understanding of mathematics - it was not just about learning a method and solving problems. It was nice that in the beginning more emphasis was placed on effort rather than accuracy. Instead of trying to get it right, one had opportunity to engage deeper and learn more about the subject, than in other modules where the emphasis is to learn something to get it right. In this module, you learn to understand. The focus was more on understanding concepts rather than grasping the language used to interpret the concept.
  
• Usually the student is on the receiving end - now it is the student who was expected to produce a precise mathematical statement that others would be able to interpret correctly.
  
• It was interesting how assumptions that we do not know if they are true or not can still be used to draw conclusions from. It was also interesting how false implies anything. It is interesting to see how proofs work in a step-by-step logic, even if your assumption is false. For example, it is interesting to see how you prove that the empty set is a subset of every set.
  
• The concept of breaking things down and unpacking in proofs. A cool skill to learn. Mathematics is neither invented or discovered. Mathematics is rather something that is within every human being.
• How anything can be turned into math. Mathematics can be made from a normal conversation. How to write down logical reasoning through mathematical steps.
• How mathematics is really so broad around us. I kind of new this, but I did not realise the actual broad extent if this.
  
• Nice to start out the seminar as a normal conversation where we can learn from each other, instead of starting through some sort of rigid system. Nice to have the summarising videos from the lecturer at the end.
  
• The seminar does not force you to parrot learn - it is much more understanding based. It is a nice thing that the focus is on understanding the work.
  
• Instead of repetitive information, the lecturer gives us information and lets us build on it while learning from each other. I wish other modules were like that too.
  
• It was easy to communicate in chat and express opinions as the lecturer would listen and take them into account. We were not afraid of being wrong. This usually does not happen. It was nice to have a proper conversation. The demis were helpful in terms of unpacking and explaining things.
  
• It is mind boggling how false implies everything. It is interesting how false can break entire proof. It is interesting how mathematical concepts can be explained without necessarily using numbers. This is what makes mathematics a universal language. Mathematics is not a numbers game.
  
• This is probably the only course that brings thought into it. After the lecture, instead of being happy that the lecture is done, you are still thinking about the lecture. Assessments reflect this too. Putting in extra thought and creativity give you marks. So assessments allow thought input. You also have freedom to interpret things in your own way.
  
• Assignments - doing the work, was really enjoyable. It did not feel like I was hating what I was doing.
  
• Coming up with your own problems and solving them.
• Figuring out what is being asked (which may take time) instead of immediately starting to solve the problem. Without right/wrong answer memos it was difficult at start, but eventually one gets the hang of it.
  
• The examples of formulas made me realise that math is not limited to e.g. science and engineering but applicable everywhere -- a universal language.
  
• This module teaches you how to formulate your thoughts and structure them in terms of assumptions and conclusions. You must think carefully and understand the process, rather than go through everything step by step or parrot learning, as is often the case in other math modules.
  
• This module has given me a greater appreciation for math and broadened my understanding from numbers and variables to something that's fundamental to everything and interlinks everything. It has changed my view and challenged my understanding of how to approach math, because there is no simple formula, and you have to think logically and then apply. I found out I enjoy some parts of math that I never realised existed.
  
• The group work really helped for the individual work. If I don't understand a concept, talking about it as a group helps. It's nice to see other people's perspectives, even if you already understood something in your own way.
  
• In the business world they want people who think outside the box. This module would be beneficial to that kind of thinking even if the content is not relevant.
  
• I like that the module is not tested purely on accuracy but more on the effort you put in. Someone can know a subject really well and still make a small mistake, which would usually make you lose marks even though you understand the concept. Other subjects end up testing how good you are at writing tests. This module actually tests your understanding, and it is graded in a way that could suit every student who tries hard enough.
  
• This is almost like having the bonus question at the end of math papers as a module. It's not just going through a routine. They keep saying in school that they encourage "out of the box" thinking, but they never actually teach you how. This module does this.
  
• What I like the most about this module is how hands-on the lecturer is. I love the fact that he's on the WhatsApp group -- some lecturers don't even answer their emails. The fact that he says he likes constructive criticism makes me feel like he cares. He's also very passionate about math and teaching, which rubs off on you, and it's easy to learn from him.
  
• This module would have been impossible without group work. Last year I did not talk to anyone about math, and just speaking through it and understanding how other people's thought processes work helped me. The group work keeps you accountable for staying up to date with the work. We all found a new appreciation for math because we had to dig deeper than before. I often did the individual work first and after the group work I would go back and rethink what I did. The back-and-forth helped me understand the content faster than in other modules.
  
• I've never enjoyed group work before, but it has been very helpful in this module because we were not given a textbook. After watching the lecture and seeing the assignment I'd be confused and overwhelmed, so it was nice to have the group sessions to confirm my understanding. The way the groups were chosen worked really well. I would not have survived in a more "advanced" group -- everything would go over my head.
  
• I like the way the proofs are broken down -- I used to hate proofs and never knew what was going on, but I got used to the bracket notation and breaking it down made it simple for me. I like how the statements we had to prove were very simple. I found them hard to prove in the beginning, but it was helpful to learn the process. It's nice to know that you can prove something very basic.
  
• This module is a lot more flexible than others. It was new and refreshing to me to be told that the final answer is not as important as the process (our personal journey).
  
• I wish they'd introduce something of this nature (basic logical thinking) in high school or earlier, rather than the mechanical way to work everything out.
  
• I appreciate the effort that goes into feedback to our work.
  
• I have used the bracket notation to help me figure out how to do the proof in a linear algebra exam. This module has helped me in computer science in breaking down how things should be structured.
  
• This seminar was interesting to me. I enjoyed it, but it was confusing. I was speaking to a friend about how proofs are "too easy" for me to get them. You have to be explicit about the steps you take, and I tend to skip over those steps. E.g. when I "take something over" to the other side, I don't add to both sides to get it to the other side. I do it all in my head in one go, which messes my proofs up. Initially it was quite a difference, but I have enjoyed the proofs and it has been interesting to think about math so abstractly.
  
• This seminar was fun for me because I love the logical thinking behind proving something. We started out with the essence of mathematics and the way other people see it. I enjoyed the group work, because without it I couldn't see my own faults -- in the discussions, the others' point of view changed mine and helped me understand things better. The group work is a real bonus in this module. The fact that everyone is so excited and passionate about the module makes me more passionate about it and it is easier to delve into the content and try to figure it out.
  
• At the start it was hard to come to terms with the fact that it doesn't have the normal structure. You really have to know your work to get those 2 marks. Usually, when something is out of 50 marks you know more or less how much effort you will have to put in. I like the mindset shift in this module that you have to put in the same amount of hard work for everything you do, and it is for the benefit of your own understanding and mental development. It is a very different thought process. The marks are few, but so valuable.
  
• Choosing this module, I wasn't sure what it was about. It changed my perspective of mathematics.
  
• A fun part was writing the English formulas in mathematics, which makes me feel like we understand them in ways that average people don't. Now I'm confident about translating language to mathematics, and this module helped me with that. The fact that we actually deal with examples like "Peter is friends with Jane" is so casual, whereas I used to think of math in terms of "x+y=z". Relating it to everyday life, and seeing that you can change something small and mean something different has been interesting to me. This module has helped to think of math as something that is everywhere, more than something you just do in a classroom. It is weird for me to think of math as something you can discuss (like philosophy or psychology) rather than just being told facts.
  
• I really enjoyed this course. I was actually learning something that wasn't as stressful as the other courses. The most interesting thing I've learnt is that there is no absolute truth in math -- there are axioms that can't be proved, therefore it is also based on a little bit of belief. The other thing I never knew is that false implies everything, which is interesting to think about.
  
• This module has made me very interested in mathematics and given me a broader idea of what mathematics is and what makes it interesting. I had no idea that you could take an ordinary statement and make it into something mathematical. I didn't like the abstract math module in first year, but I took a chance on this one and I don't regret it at all. I am getting fond of abstract math because of the way Prof. J makes it interesting. I actually want to figure things out. I used to struggle with proofs, but now I feel like I can prove anything. Having a lecturer like Prof. J is really important in getting you to be interested in the field of study. What keeps me interested in this module and why I'm keen on watching all the lectures is because Prof. J really wants you to understand -- you can see he himself grasps it and he's very passionate.
  
• I enjoyed the fact that we can prove things and write out logical statements, because I always dreaded the proofs in my first year -- I wasn't able to do it properly, and this course made it clear and satisfying for me.
  
• I haven't learnt anything different about how I view mathematics, but I do like the fact that we're told that there is no absolute certainty and we're always working in a certain framework. I love how this course was structured -- freedom is given to the student. We could pick the problems we wanted to solve. As long as you demonstrated careful thought about it, your answers would be taken seriously. Compared to most other modules, I find it quite attractive. I can't remember much from the other modules I take from one week to the next.
  
• This is the second time I'm doing this module. The first time, I didn't enjoy it -- I wanted to, but I struggled to grasp the content and fully understand it. It's only now that I started over with the knowledge I really had that I enjoy it and find it satisfying and manageable. Last year it was completely new, which made it difficult and frustrating. But if you put in the time and effort, it's manageable. I really like the individual assignments when I take the time to sit down and focus. Once you really think about the work and put in the effort, it is not so difficult.
  
• Hopefully seminar 2 will be somewhat similar to this, because I have no complaints. I feel like I'm being scammed in some modules. They're getting some other person not even to teach the students, but to read verbatim from the slides. In one module, what was once a one hour in person discussion has turned into a ten minute video. It was such a horror show, how they phrased the questions. They're forcing you to think in this stereotypical box -- learn this and that's it. The answers are just straight from the text -- either you get the answers right in their context or you get 0. This module is totally in the opposite direction. I cannot explain how much I love the freedom. I'm not shy to give an argument and talk about it.
  
• Apart from learning mathematics, the biggest learning curve was the change of mindset when it comes to mathematics -- thinking about it from different angles and not just accepting something because it has been said, delving into what is logical, what is not logical, and what may seem logical but isn't. I definitely feel like it questioned logic as we know it -- things that usually make sense. It is strange that if we're told 1+1=4 we can work with that mathematically. There were weird examples that mess with your head and you actually had to think about it. Also real-life examples that bring math to your doorstep. It was pretty abstract but felt so much more real.
  
• I really liked Prof. J's approach to teaching -- it's like we discover it with him. It feels like learning a new language, and not just math. It's interesting that there is something called a sentence in logic, which makes you think about it like a language.
  
• I liked how Prof. J loves how he's doing and he really made the lectures interesting. I like how he's on the WhatsApp group and sends extra information. It's very interactive and he's constantly working towards giving you knowledge. He expects effort from us because he puts in so much effort. You have no excuse not to work if the lecturer is right there with you. It really is amazing that he puts so much time in, even after hours on WhatsApp. I probably would have used the time better if I weren't so crushed by my other modules. I'm sad that I didn't have enough time to put in.
  
• It really excites me to see that math is essentially logic and the part of our language that gets translated is the logical meaning of it. I love logic independent of math, and now I can see how they're actually integrated.
  
• I never know what was meant by "abstract" math. I don't feel like what we did was that abstract, because there were always practical examples -- maybe logic is abstract in the sense that you can't touch it. I think the purpose of abstraction is to take something practical and abstract it enough that you can solve it without the original example. From a software perspective, when you abstract something you separate it from the specific example -- you try to build models and objects to describe what you're working with, because they're easier to work with than the entire problem. It's roughly the same in mathematics.
  
• I've learnt many things -- one of them is that math can be used as a more efficient form of communication. I used to see it as a tool to solve problems, but my view of it changed. Instead of using words to express something, you can use mathematical symbols. The same mathematical symbols are used around the globe, so someone who speaks a different language can understand your mathematical statement even if you don't speak the same natural language. A proof written in bracket notation, for instance, can be understood regardless of home language. Furthermore, the mathematical representation can be shorter than writing it out in words, which adds to the efficiency.
  
• The course helped me a lot with coding, because there, basically any statement can be set to true or false. Programming languages are mathematical languages, so learning to translate English into mathematical language in this course has been useful for programming.
  
• Before starting this module, I never thought of math as something you can do in a group. I really enjoyed the group work and discussing the math and working on it together. It helped me look at it from different perspectives.
  
• You have to think hard about what you want to put on paper, because in the end it is not about getting the work done but about understanding what you are writing about. You have to think about the logical steps and the reasoning behind them. You are free to interject if you do not understand it. If you rush through the work just to get it down, the chances that you miss steps are high, and hence you lose marks.
  
• If you feel overwhelmed by a problem you really don't know how to approach because you're not used to e.g. quantifiers or bracket notation, the best thing to do is to start writing things down and refine them later on. In the group work, we wrote a bunch of things down and drew pictures -- they weren't necessarily correct, but we could narrow it down and come to a conclusion. We didn't just decide on one thing and submit it. Everyone contributed and we thought about it together.
  
• I learnt that in mathematics, you're required to imagine things creatively. You have to imagine things you haven't seen and make sense of it in order to understand the concepts.
  
• One of the biggest things I learnt was how to write a proof using mathematical symbols. It shortens the proof so much once you understand the symbols. I learnt to understand implication and quantifiers better -- I knew the implication symbol, but I never understood what it meant the way I do now.
  
• I learnt how to simplify and work efficiently in order to structure proofs well. This seminar taught me to think deeply, even outside this module.
  
• In other math modules you know you will be calculating and solving, but this one turned out to be different, and I started seeing math differently. It is all about understanding the foundations and background of what you learn, and about team work, and it's not only about getting marks at the end.
  
• An interesting thing I learnt is that the process of calculation is not just studying and remembering the steps -- there is actually an underlying logic to it. This module has helped me think more critically about the methods we use for calculation.
  
• Open-mindedness to problem-solving and thinking outside the box. Not about solving problems by applying methods without understanding what is really going on.
  
• Assignments were enjoyable - like playing a sudoku (which I love).
  
• I went through high school being really good at maths. Abstract maths has really made me go back and ask: how much of what I know I actually understand? I realized that by breaking things down into basic logical components makes me understand the topic better. It gives nice foundation for learning mathematics that would I have never thought to be possible.
  
• At first I thought mathematics was discovered, since two people far from each other could discover the same thing. Afterwards, stepping back, I realized that the question whether mathematics is invented or discovered is not the right question to ask. Now I realize that mathematics exists because we have a sense of counting. More generally, mathematics arises as an elaboration of our basic senses.
  
• This course differs highly from any other mathematics I have been taught: instead of just telling me how to apply mathematics, it is trying to make me understand the basis of it. This is very useful since to apply something you need to understand it well first. With normal math you always have a clear direction what you need to do. Here you had to figure out things yourself and be more creative. Towards the end the direction was given as well and that was useful too. Logic taught in this course is applicable to real life.
  
• This course has taught me a lot how to be precise in mathematics, beyond the level usually presented in mathematics courses. It has given me satisfaction in the abstractness and precision which in other courses might have been left unattainable.
  
• A breath of fresh air - usually mathematics is a subject where you only care about numerical answers. The seminar was a lot more about understanding and exploration than calculation, which is what my view is about the essence of mathematics. It has forced me to understand things that I have been using before more precisely. The proofs have never been explained before concretely and in terms of what is actually going on in them. I think this is why most students struggle with proofs too. Knowing what you are doing and why you are doing something in contrast to knowing how to do something, sums up for me what this seminar has delivered.
  
• Deeper understanding of mathematics and proof structures. We are usually given proofs to memorize them instead of us learning how to compose proofs ourselves, which is what this seminar delivered. Seeing the difference between the spoken language and the mathematical language was quite interesting for me. Specifically, seeing how the English language can be ambiguous as opposed to precise mathematical language. It was remarkable to see how some English statements translated to mathematical statements looked longer than the original. The seminar was focused on hard work rather than performing, which was important as it helped to enjoy mathematics.
  
• Coming from high school where I enjoyed mathematics, to the university where I very much did not like the mathematics modules, this seminar allowed me to enjoy mathematics again. It mainly comes down not necessarily the material but the presentation style, which is why I genuinely enjoyed this seminar, where presentation is oriented towards understanding. Looking back in general I never really liked proofs since it always felt like reciting a poem as opposed to showing why we are allowed to use a certain property. In this seminar we got to nitty-gritty of the proof design and as someone studying computer science and enjoying designing algorithms, I really enjoyed that. I was fairly familiar with mathematics up to this point, but not enjoying it to the point that I was thinking of dropping mathematics, but getting to see the side of mathematics that this module showed, I really started enjoying mathematics.
  
• This module taught me to think slow rather than fast. I usually like thinking fast and just live off from my intuition. But I have come to realize that while a lot of intuition is involved in setting up a proof, when it comes to finer detail you have to start being precise and start thinking about it. The slower you work the more precise you become. This module has really taught me how to be more precise. This module made me feel that if I work systematically I will be able to understand any piece of mathematics eventually. You can break down every concept! It was hard for me in the beginning to switch to the new style. In the beginning I did not spend that much time on this module because of difficulty to transition to the new free style.
  
• What I really enjoyed in this seminar is to see how mathematics is a combination of slow and precise thinking, but also intuition. To formulate the proof you need to know where you are going (intuition), but you also need to know how to take each step (precision).
  
• That is a perfect mix of structure and rules with creativity and intuition, which really showed in this seminar. What I also really enjoyed is translation from normal language to mathematical language, as well as how proofs on the mathematical side can be applied back (maybe to different) real-life situations. This shows how abstract on the one hand mathematics can be, but at the same time being completely true, intuitive and logical. It made me realize how easy it is to have flawed logic. A perfect seminar to become a lawyer. It was also nice to develop the skill of distinguishing between assumption and conclusion.
  
• Linear algebra was a lot more enjoyable because of the relation to this module.