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Background: I’m completing a BSc Mathematics, currently in year 4.

A lot of people who have almost no mathematical background ask me what I study in university. I understand that this is a very common way (at least in my country) to start a conversation to an undergraduate, but I think that this is inappropriate.

I am not going to blame them because I believe that they truly don’t understand what mathematicians are doing (perhaps I don’t either). Also, explaining what to study in other majors appears to be much easier (at least for me). For example, a CS student may explain with an applied course (such as mobile app development). This is even easier for job-oriented subjects (law, social work etc).

I usually start by saying that MATH students study rigorous maths like theorems and proofs. There are several domains like PDE and applied maths (I’m not going to talk about analysis and algebra because the discussion will soon go into chaos, and I believe they know nothing about elementary calculus and linear algebra). My most “real-life” example is image processing (and perhaps, no other examples). However, I think emphasising such an example does not give a very clear picture of what a math student studies. They may even have difficulty understanding cooperative strategy in game theory.

Worse still, they

  1. start feeling that studying maths is useless (and I believe that a lot of people think so).

  2. start by teasing at people who study math. That’s fine because it is OK for me to simply ignore them after a detailed explanation (e.g. studying math trains one’s reasoning skills), but this is in no way a good situation.

  3. start by saying something irrelevant, especially for those who entirely don’t know advanced maths. For example, they think that a 10-year-old child who won in a competition by correctly doing a large number of mental arithmetic calculation in a short time is very good at maths.

I usually try to explain in detail, slowly and patiently, but in vain. Any hint?

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    What's your goal? Are you just trying to come up with a socially acceptable response? – apaul Sep 5 '18 at 11:32
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    Very similar question, if not a duplicate: interpersonal.stackexchange.com/questions/3213/… – Em C Sep 5 '18 at 11:53
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    “A lot of people who have almost no mathematical background ask me what I study in university. … I think that this is inappropriate.” — This sounds very odd to me. Are you really saying that you think it’s inappropriate to ask someone what their occupation is? Obviously there are situations where any kind of personal question is inappropriate, but in normal, everyday conversation, asking someone what they study or do for a living is not considered inappropriate by most people. – Janus Bahs Jacquet Sep 5 '18 at 14:25
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    What are you planning to do with your Math degree? Are you looking to continue in academia? Are you planning to consult in industry? Use your skills in a tech field? People making small talk probably dont want to actually know about the intricate details of your field. – Tal Sep 5 '18 at 15:15
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I personally think that the right answer for you will depend strongly on your mathematical philosophy. If you are a platonist, then mathematics is completely about discovering truths (not about evaluating mathematical expressions as per your point 3). If you are a formalist, then mathematics is just about proving theorems within your chosen foundational system, and whether they are 'nice' or not is a matter of mathematical aesthetics. If you are a finitist, then almost all higher abstract mathematics taught in university are meaningless to begin with, and you may not have a good answer. If you are unconcerned with foundations and philosophy of mathematics, then I think you should at least figure out what you personally believe in and wish to gain from mathematics, because that ought to drive your mathematical exploration, and hence how you explain that exploration to others.

To specifically address your points:

I usually start by saying that MATH students study rigorous maths like theorems and proofs.

As long as you are not anti-platonist, phrase it not as "theorems and proofs" but as "facts and justifications".

There are several domains like PDE and applied maths (I’m not going to talk about analysis and algebra because the discussion will soon go into chaos, and I believe they know nothing about elementary calculus and linear algebra).

You can if you wish mention very basic results like the intermediate value theorem, but phrased in a layman way or even weakened. For example:

If you draw a curve from one corner of a square to the opposite corner through the interior of the square, without lifting your pen, you must at some point cross the diagonal between the other two corners.

Yes, I know that this is not accurate; it is only about piecewise smooth curves, and does not convey the complexity of continuous curves. But it is just to get the point across that it is a fact that can actually be proven and not just left to intuition or guesswork.

My most “real-life” example is image processing (and perhaps, no other examples). However, I think emphasising such an example does not give a very clear picture of what a math student studies.

Well image processing is not on the pure mathematics side, so if you are a pure mathematics student then clearly using such examples might not sit well with explaining what you study. So instead try to find examples in your specializations.

They may even have difficulty understanding cooperative strategy in game theory.

If you like game theory, you can try to use simpler examples. There is no need to explain the most complicated knowledge that you have gained from university.

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Generally when people ask these kinds of questions, they're just trying to get to know what sort of person you are. Your answer gives them a sense of your interests, and how you spend your time. Beyond that, whether consciously, or unconsciously, they're feeling out your soft skills. Small talk gives people a sense of how approachable you are, your sense of humor, your personality.

The way you choose to answer a simple question about what you study ends up telling the person a lot more than your field of study. So, you'll have to ask yourself what sort of impression you want to leave on your conversation partner. Are you intellectually smug? Are you intelligent, yet grounded? Are you nerdy, but quick with a joke? Would you rather not be bothered?

What you actually study has little to do with that impression, unless the person has a similar background or interests. Granted, high level math has a rather small target audience, and most people won't understand what you're studying, but you can choose to think less of them, or you can navigate the conversation with a little more skill and find some common ground. Once you've found common ground, you'll probably leave a much better impression.

Rather than focusing on your field, you could segue into campus life, student loan debt, the cost of textbooks, or pretty much anything that's study/university adjacent. These are topics that a much wider audience will be familiar with, and will likely lead to a more natural conversation.

For instance, my brother is an electrical engineer. Most people, in the small town we grew up in, would have a hard time understanding what he was studying, but he went to a university with a rather well known football team. This made for an easy place to segue to. He could tell a story about going to see a game, or the school's hyper focus on its sports program and shift the conversation towards something that people could relate to and have a light conversation about.

Think of small talk like tennis or ping-pong, but rather than trying to score points, you're looking for a volley. Nice gentle serves that people can return. If you're the guy who spikes the ball at every opportunity, and complains that people can't return the serve, you'll eventually be limited to a very small set of people who are interested in playing.

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I agree with the answers saying this is basically small talk, but have a different approach on how to respond.

In my experience (biomedical electrical engineering), nobody cares what exactly you do (programming an algorithm to extract stuff from signals, calculating electrical fields, simulating a circuit), they simply want to be entertained.

In my case, I say we do stuff like pace makers, mri, ct, etc. That's enough for 90% of people, because they don't even care how exactly you do that. It's enough to visualize what you do though.

I'd recommend the same to you. Don't say “bayesian summarizing of unexplicable vectorized vodoo data“, make it so the other person can “touch“ what you do. Say for example

I could calculate credit scores based on how your friends behave on social media.

Or

I can consult with firms that don't know why they produce so much waste instead of product by analyzing how much waste is created at the different stations and predicting how changes will improve that.

Or whatever floats your boat. Just don't make it about math. Frame it differently. Understandable.

If someone insists (but doesn't have the knowledge to understand it), you can still throw the 7-dimensional integration of qubit-quarks at them. Then they can say “ohhh wow, that sounds hard“ and you're over it. I've never come across someone who was interested in more details.

Details are boring. A conversation is entertainment. Think about a few standard lines that sound cool and you're fine. They don't want a clear picture. If you're just into theoretical stuff without application, you can still say

I invent the stuff the engineers can use in a 100 years to make flying cars. Just like Einstein invented the theory of relativity which helped us get to space.

(Accuracy = details = boring). If you want something more accurate, spend some time finding something equally catchy.

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So is the question really about what you study? Because you answer What do you study with Math.

Or is this question on how to explain what you are studying? So after you respond with math they follow up with question and what do you do exactly?

In the latter, from personal experience, people don't want to know what you do exactly. It is, as you stated, just a conversation starter. So you can just say "theorems and proofs". It may not be true but any other answer will give the same impression. Also from personal experience, you can just say that the whole field is big as universe and your section is closer in size to solar system.
I don't go into detail about my work, unless my interlocutor shows me they knows something we can talk about.
Also I picked up few "tricks" that look amazing but are actually totally useless. In math department I think it may be those "Asian" way of multiplying large numbers.

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Another way to answer this question is what do you plan to do with your math degree.

I study computer science which often leads to the joking (or sometimes serious) question "Can you fix my computer/printer/wifi?"

I usually use this stereotype to segue into what I hope to do with my degree.

"Oh, I'm not a IT/hardware/networking expert. I'm studying so that I can later ..." Then finish up with what you plan to do with you degree

This addresses why studying math is practical (you are planning a career), any stereotypes they might have about math careers (you reorient the conversation around your goals and interests, not ten-year old protégées), and usually stops teasing (because you give a serious reply).

For example, if someone asks "What do you study", you would reply, "I'm studying math, so that I can later..."

  • Teach math
  • Consult on X/Y or Z
  • Become an actuary
  • Become a day trader
  • Whatever it is that you hope to do in math

Talking about what your future plans are is another way to make your degree accessible. Like many of the other answers, people have a mental picture of day traders, actuaries and consultants, while they might not understand the theoretical basic of their jobs, so they can continue the conversation in a relevant direction.

The one disclaimer for this answer is if you hope to become a theorist. No one knows what theorists do (joke). Then I would suggest using DonQuiKong's answer to flesh out what area you hope to contribute to in theory.

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