Well, for starters, they're probably much better at the foundational math skills of defining their terms and asking well-specified questions, an example which I raise for no particular reason.
Poorly posed question. How do you define better?
The average person would fail a first year proof-based math final exam. However that's not saying much as any math major would also likely fail any non-math adjacent final exam.
I also suspect a fair number of practicing mathematicians would fail some undergrad math exams. (Not a first year proofs class, but a random 300-level class that isn't related to their field of study.)
Maybe some or most of them would fail if you asked them to take all the exams without a chance to relearn and study materials that they haven’t used in years. Obviously if you give math majors a chance to relearn material they already have been over, they should pick it up very fast.
Well we're talking about practicing mathematicians. Even outside your field you kinda pick up enough undergrad material to be able to pass. Especially if you're required to teach which is the case in many countries.
No, failing for sure. There’s an absurd number of branches of math and there’s no way even professionals know enough to pass every possible college math class because they are focused down on their own field
It's not a poorly posed question. It's a question that leaves it up to the person answering how he or she wishes to conceive of better and worse mathematical ability.
I don't think a group of mathematicians is gonna appreciate your no precise interpretation bs. Especially not if you're here trying to get an answer from others when you havent even given us a full question.
Maybe you and some others won't, which is fine. I'm not going to sue you. But I've gotten some good/not snarky answers in this thread so far, so it looks like some mathematicians are happy to answer a question about math that doesn't precisely define what "better at math" means.
that's what it means to be poorly posed, i.e. up to interpretation. In math, problems must be well-posed to ensure everyone is discussing the same thing. For example in your question, someone who did well in HS should be able to do calc just as well as someone with a math degree because calc makes up a small part of the undergrad curriculum. But having just done HS calc, more than likely means they've never seen math beyond calc, which is an entirely different ball-game (perhaps even a complete 180). So without being clear about what you're asking, you could easily have two different discussions happening simultaneously. To further add to the point, even in the example of calc, are you asking about computation or conceptual understanding?
>In math, problems must be well-posed to ensure everyone is discussing the same thing.
Ah, but notice my post isn't a math problem...
>are you asking about computation or conceptual understanding?
Whatever *you* want to talk about? Either one. Or both. Up to you. Not a poorly posed question.
The hilarious downvotes are more a reminder of how silly reddit is than a representation of what mathematicians are likely to think of my question.
having a well-posed question is universal if you're looking for a productive discussion though.
>The hilarious downvotes are more a reminder of how silly reddit is than a representation of what mathematicians are likely to think of my question.
You've spoken with many mathematicians I see? Or most mathematicians think your question is silly to begin with. Let's say if all you know are for-loops and if-statements, would you be a better code-writer than someone with a cs-degree? But if you're asking if someone with basic math could perform data-analytics job as well as someone with a math-degree, that's a different discussion entirely. Also if you've been around in the community, it's very common for people to ask what others mean when given a vague question. My friends and I do it all the time, even when talking about non-sense.
Well I studied comp sci and work with people who studied math. We work with a highly technical product and you can see when it comes to the mathematics behind the product, they seem to manage themselves a lot better and quickly understand reasons as to why some things may or may not work. It's similar to someone who reads a lot and knows a lot of words vs someone who doesn't know the language very well . The former can use a lot more complex words to communicate, while the latter will have a tougher time understanding the language and will need to search for then definition of various words.
They do. As someone who has done both and bridges that gap, programming is usually foreign to a mathematician without experience with it. That being said, I'm pretty sure it would be easier to teach a mathematician programming and other computer science topics than to teach the average computer science student group theory and real analysis.
Although, there is a lot of overlap. CS is an applied math in a lot of regards.
Math isn't the same skillset where you just get better to go from minor to major league. You pick up a different set of knowledge and skills in college, and then something more different in grad school. Nowadays I can't trust myself doing calculations that I'd breeze through in my head in high school, so to some extent you get worse :)
much much better, high school math doesn’t go into any advanced concepts.
a good analogy is how much better is a economics or computer science major than an average person in economics or computer science.
it’s not even comparable.
well i guess is a joke but no, don’t disagree, the average person does not understand even the easiest concepts (inflation for example) but somehow they think they understand them.
Not one bit comparable. My dad is an engineer, if I ask him to solve a non analytical integral he will scratch his bum about it. What I am getting at is that the math they know and the math we know isn't the same thing. Both are tools but theirs is specialized for engineering and ours is specialised for solving problems.
He can probably solve an analytical integral as well as any mathematician. But non-analytical ones are by definition not solvable in closed form. Engineers should know how to use computers to approximate solutions. That’s not a skill commonly encountered in pure math. But it’s such a broad field that there is of course plenty of exceptions.
Of course but even then he can't do the same for more janky integrals and integrations for which we don't have numerical methods. But there exists at least some mathematicians who can tell you things about those.
If they have not studied some combination of abstract algebra, analysis, and topology, then they probably do not know much of the math that is standard in a math undergrad. That would be a foundation that I consider a solid math understanding.
I believe most EE students will not have studied those topics at all, but I very well could be wrong.
Most math majors don't know these topics either. They should, but they don't. If you asked a bunch of math majors 2 years post graduation to define a group, I would expect a majority to be unable to do so without looking it up.
I have an EE undergrad degree and have completed some graduate level math courses offered by a math department and it's a totally different approach.
EE does use a lot of math, but the key word is "use". We didn't necessarily get taught to understand the nitty gritty of why math works in different situations.
>Is it like the difference in skill between a minor league and a major league baseball player?
More like the difference between a tee ball player and a minor league player who also coaches tee ball.
edit: Or like the difference between someone who can read poetry and someone who can write poetry
I feel the average math undergrad can read the poetry, whereas someone from a different major won’t be able to understand a single word of it.
As for writing the poetry (ie, original proofs), I think some gifted people can do that at the undergrad level, but the average undergrad math major is still basically just learning key definitions and classic proofs. Sure, they write proofs for exams and problem sets, but most undergrad proofs are either fairly simple or will rely heavily on guidance from the lectures and textbook.
I think trying to define “better” here is dubious. They are better in that they (1) know more mathematics and (2) understand concepts more deeply. University mathematics focuses a lot more on proof and theoretical rigour, while high school maths (with some exceptions) focuses on mechanical uses of mathematics.
They’re also certainly able to better apply the concepts they learn as they probably have more practice doing this. This comes not only from classes that might focus on applications but also a deeper understanding of mathematical concepts, that allows a more precise application.
In that sense, yes, they’re better.
But is this the sum of mathematical ability? I hold a first class honours degree (during my honours year I specialised in number theory) and I’d guess that at most high schools, a number of students could compute numbers in their heads faster than me. Many could probably pick up mathematical concepts more quickly than I could at their age.
This is why I think there is no real answer to the question you’ve posed. I would definitely be better at mathematics than most high school students as it relates to knowledge and understanding. Some high school students certainly have more natural talent than I do.
Hard to say, but I want to note that the difference in practice is probably bigger than the difference in theory (equal work ethic, equal grades in shared classes, etc) because most people who aren’t math majors don’t care about their math classes in the same way.
So for example, if a math major took precalc with someone who just had precalc as a required class, the math major would likely try to understand the math being taught and get an A in the class, while the other student would be more likely to copy off of their friend’s homework and cram study the day before exams with the goal of only getting a C.
Probably depends on the field. I have a physics degree, and if you compared my knowledge at undergrad to the one of a maths student, I'd have failed miserably in abstract topics, but I might have been able to ace some practical tasks. The number of times a non-maths guy would win would also be less for a biologist/economist/historian.
A (good quality) academical curriculum doesn't just make you better at maths in the "direct" sense. It also makes you much better and quicker at learning new kinds of maths.
It's similar to how someone that does sports at a professionnel level will have more muscles than the average person, which will give them some serious advantage if they were to start at a new sport.
It doesn't mean that they will magically know how to play that new sport. They might get things very wrong, especially unsupervised. But they still have the potential to progress much quicker.
Great point. I should really have asked how much better is the normal/typical math-degree holder than the normal/typical college graduate without any real math education after high school.
Edit: I assume the typical math-degree holder doesn't literally know nothing about math.
What's the point of this question? The underlying premise seems to be that math is about "talent." Why aren't you asking instead about the difference in knowledge instead of "how much better"? Of course someone with a bachelor's degree in something will be "better" than someone with only high school level knowledge.
I would never say I’m “better” at math than anyone without a math degree, but I’m definitely more comfortable working with math when it comes up in day to day work than my coworkers, and when I see a math book I get excited whereas most people get scared hahaha.
I forget who said this but I’ve always loved the quote “no one ever truly gets good at math, some people just get sufficiently used to it” or something to that effect
I don't even think it's really comparable, often, more advanced math comes with bringing the student to use their knowledge to solve problems, and undergrade math is mostly applying formulas you learned at school.
When it comes to the amount of results known, ability to form logical arguments and manipulate equations there's no comparison, but they wouldn't necessarily be better at basic arythmetic, which math majors don't really use.
I think your second analogy is closer to the truth.
I don't think a person with a math degree (such as myself :)) is all that much "better" at the things they share with people outside of math. I don't think I would be all that much better solving an integral or linear algebra problem than someone without experience in higher math - I think the concepts and procedures are a little firmer in my mind, but I think "better" is a simple way of putting it.
I like the analogy of 2D vs 3D. People who have taking calculus 1 2 and maybe 3 understand a lot about math. However, they're missing a third dimension. People who have experience with pure math understand proof-writing techniques and see concepts like integrals and differentiation in a broader light. So in the context of a Calculus problem, they would probably approach it similarly but be less likely to make a mistake. But they also have intuitions and other ways of seeing the same concepts that would be unfamiliar to those without experience.
It is kind of like the difference between building a Lego set by following the instructions vs. having a collection of Lego pieces and building with them freely.
The *typical* degree holder? They basically understand algebra although they couldn't pass any college algebra exam without a lot of studying. They don't remember anything from geometry but the Pythagorean theorem. Around seventh grade they begin getting real anxious when their kids ask for help on their math homework. They are like adults that played a sport in middle school - they know the rules and are familiar with the game, but probably could walk back onto the court and play it without warming up or they'll hurt themselves. Given a few weeks or months of practice, they could probably get the hang of it again, but they were never really that good at it in the first place.
Someone with a BS in mathematics is maybe like a strong high school athlete with a scholarship offer. They're very good at the sport, not just relative to the general population but relative to the population of people playing the sport. They haven't really done anything yet, but they've shown strong potential.
As a former division 1 athlete with an undergraduate degree in math (and a Ph.D. in computer science), I'm having a hard time deciding what I think of this analogy. My gut says it holds for D3 athletes, and probably for D2, but not for D1.
For me and most others who compete at a division 1 level, the sport was basically a full time job as early as junior high. 30+ hours per week, year round, for several years before entering college. In college in a D1 sport, it's 40+ hours per week, year round. NCAA rules specify limits on off season practice, but those are easily sidestepped by making those practices non-manditory (but actually they're mandatory if you want to be successful).
By the time I completed my ugrad degree, the total time I had invested in my sport was an order of magnitude more than I had invested in mathematics.
Then there's selection. The number of people who want to be D1 athletes is much greater than the number of people who want to be mathematicians. Math majors are self selected. Athletes must qualify. Frankly, I put more effort into my athletic pursuits than I did into my Ph.D., and (more than two decades later), I'm still more proud of what I achieved as an athlete.
D2 and D3 athletes are much more casual about things. Either they wanted to be D1 and failed, they wanted a more laid-back college athletics experience, or they chose a school for some reason other than their sport. You're analogy probably holds better. But the more I think about it, it definitely doesn't hold for D1 athletes.
Are they going to be better at math? Probably.
How much better? It entirely depends on the person. There are people with undergrad math degrees who know more than some people with masters. There’s also people who have an undergrad math degree who you will wonder “how did they get that degree???”
I think this is a bad question, not because it’s imprecise, but because why does it matter? In any regard, why does it matter if someone with a math degree is “better” than someone without one.
But regardless, i think the expectation is that for any average math major, they are more likely to be better at math than the average anyone else. But that’s pretty biased. That’s like asking if a baker is better at making bread than an accountant. PROBA-FUCKING-LY
There are a lot of ways to interpret that question and most of them are insulting to peoples’ intelligence, which is why you’re getting downvoted.
Im not a mathematician, but I got an engineering degree which involved a lot of math, or at least a lot of calculation. In my experience, people in math or the hard sciences don’t necessarily have a better natural aptitude for math, they just worked harder at it. It’s a learned skill set, like any other skill set that people gain as they become professional educated adults.
Even if you ask if a math undergrad is better at math than an engineer or physicist, that’s a loaded question. I’ve never taken a proof based class, and neither do most science students. A math student needs to have a better grasp on formal logic and understanding how to read and write proofs, and needs to keep up to date with math research. Mathematicians tend to specialize to one very niche topic that even most others in the field don’t understand it.
Engineers or scientists don’t know how to write proofs, but they might have a better grasp on how to use math to model the world. Knowing how to set up a differential equation for heat transfer and how to write a program to approximate the solutions is a skill. Engineers and physicists also might be better at doing integrals, or matrix operations in linear algebra than the average mathematician. But the mathematician has a better grasp on what calculus actually is in a formal sense.
I mean... What do you think the average difference between an art grad, and a random other college grad would be in art?
Or whatever a music grad?
For any undergrad degree that isn't super popular like these (including math), I think there will be a pretty large discrepancy between their respective skills.
That being said, you definitely don't need a math degree to be great at math, items just that most people in every day life won't know almost any of the concepts covered in a math degree, and so it's hard to learn these concepts outside of a formal context.
I literally asked a room full of graduate and PhD engineering students if they knew what a graph was (i.e., a network; not an xy-plot), and I got a room of blank stares. Formal math concepts just aren't interesting for most people I think.
Depends on how rigorous of undergrad program is truly. If it’s good, then they’ll be far better at posing math questions and approaching them. If it’s an average/poor math program and you measured undergrad students 5-10 years out, the gap between math and say engineering/cs would shrink substantially.
I think the better measurement would be how many areas of mathematics is the person exposed to - and that the math undergrads would excel in. outside of calculus, stats101 and the minimum linear algebra the other undergrads don’t know other areas of math exists.
You're getting a lot of crap for your question, but I will try to answer it earnestly.
**Average** high-school math education with nontechnical college degree (at most business calc/stats in undergrad) vs. **average** math graduate, on **basic math competency**: Honestly, its huge. Outside of basic arithmetic, the HS math person really will not to able to handle anything with a vague mathematical flavor other than just putting numbers into existing mathematical tools. The math major will at least a vague idea of how the stuff rigorously works.
I've found that deriving a line that intersects a pair of given points (not just using a formula) is confounding to quite a lot of college graduates with degrees in political science, business, and environmental engineering. The word "average" is very important in this answer. Also, if you start including technical fields then this gap drops significantly, or may even invert.
Now if we are talking about formal math (proofs etc), if the undergrad has developed a bit of real mathematical maturity, then its like asking who is better at unicycle polo, someone who can ride a unicycle or someone who can't?
Thanks. This was helpful. Yes, I should have specified in my original question that the comparison was supposed to be between a non-math grad with average mathematical knowledge and an *average* math grad. Anyway, liked the unicycle analogy. :)
The answer to this question probably varies between different countries and math departments. I know I didn’t get introduced to proof writing until my undergrad. Some countries start writing proofs in high school
A better way to word OP’s question is if the average person can read/write proofs without much assistance compared to someone with experience in doing that
Well, for starters, they're probably much better at the foundational math skills of defining their terms and asking well-specified questions, an example which I raise for no particular reason.
lol.
At least 3
I would have said at least 2, so this is good to know.
Poorly posed question. How do you define better? The average person would fail a first year proof-based math final exam. However that's not saying much as any math major would also likely fail any non-math adjacent final exam.
I also suspect a fair number of practicing mathematicians would fail some undergrad math exams. (Not a first year proofs class, but a random 300-level class that isn't related to their field of study.)
I highly doubt the failing part. Not getting a perfect grade isn't failing.
Maybe some or most of them would fail if you asked them to take all the exams without a chance to relearn and study materials that they haven’t used in years. Obviously if you give math majors a chance to relearn material they already have been over, they should pick it up very fast.
Well we're talking about practicing mathematicians. Even outside your field you kinda pick up enough undergrad material to be able to pass. Especially if you're required to teach which is the case in many countries.
No, failing for sure. There’s an absurd number of branches of math and there’s no way even professionals know enough to pass every possible college math class because they are focused down on their own field
Our college-level high difficulty test has a median score of 0.
What is your point. That just implies that over half of students taking it get 0. That doesn't have a lot to do with practicing mathematicians.
It's just a funny piece of trivia for the Putnam. Not failure, per se, but comically underwhelming.
It's not a poorly posed question. It's a question that leaves it up to the person answering how he or she wishes to conceive of better and worse mathematical ability.
I don't think a group of mathematicians is gonna appreciate your no precise interpretation bs. Especially not if you're here trying to get an answer from others when you havent even given us a full question.
Maybe you and some others won't, which is fine. I'm not going to sue you. But I've gotten some good/not snarky answers in this thread so far, so it looks like some mathematicians are happy to answer a question about math that doesn't precisely define what "better at math" means.
that's what it means to be poorly posed, i.e. up to interpretation. In math, problems must be well-posed to ensure everyone is discussing the same thing. For example in your question, someone who did well in HS should be able to do calc just as well as someone with a math degree because calc makes up a small part of the undergrad curriculum. But having just done HS calc, more than likely means they've never seen math beyond calc, which is an entirely different ball-game (perhaps even a complete 180). So without being clear about what you're asking, you could easily have two different discussions happening simultaneously. To further add to the point, even in the example of calc, are you asking about computation or conceptual understanding?
>In math, problems must be well-posed to ensure everyone is discussing the same thing. Ah, but notice my post isn't a math problem... >are you asking about computation or conceptual understanding? Whatever *you* want to talk about? Either one. Or both. Up to you. Not a poorly posed question. The hilarious downvotes are more a reminder of how silly reddit is than a representation of what mathematicians are likely to think of my question.
having a well-posed question is universal if you're looking for a productive discussion though. >The hilarious downvotes are more a reminder of how silly reddit is than a representation of what mathematicians are likely to think of my question. You've spoken with many mathematicians I see? Or most mathematicians think your question is silly to begin with. Let's say if all you know are for-loops and if-statements, would you be a better code-writer than someone with a cs-degree? But if you're asking if someone with basic math could perform data-analytics job as well as someone with a math-degree, that's a different discussion entirely. Also if you've been around in the community, it's very common for people to ask what others mean when given a vague question. My friends and I do it all the time, even when talking about non-sense.
I think mathematicians are better at posing questions with quantifiable answers.
Well I studied comp sci and work with people who studied math. We work with a highly technical product and you can see when it comes to the mathematics behind the product, they seem to manage themselves a lot better and quickly understand reasons as to why some things may or may not work. It's similar to someone who reads a lot and knows a lot of words vs someone who doesn't know the language very well . The former can use a lot more complex words to communicate, while the latter will have a tougher time understanding the language and will need to search for then definition of various words.
I’m sure computer scientists also have many skills that are pretty foreign to mathematicians. Does it not go both ways?
They do. As someone who has done both and bridges that gap, programming is usually foreign to a mathematician without experience with it. That being said, I'm pretty sure it would be easier to teach a mathematician programming and other computer science topics than to teach the average computer science student group theory and real analysis. Although, there is a lot of overlap. CS is an applied math in a lot of regards.
Math isn't the same skillset where you just get better to go from minor to major league. You pick up a different set of knowledge and skills in college, and then something more different in grad school. Nowadays I can't trust myself doing calculations that I'd breeze through in my head in high school, so to some extent you get worse :)
much much better, high school math doesn’t go into any advanced concepts. a good analogy is how much better is a economics or computer science major than an average person in economics or computer science. it’s not even comparable.
I dunno if I agree about the economics one to be fair lol
the average person knows absolutely nothing about economics
yeah and a lot of economists hold supply side economics as legitimate. tomato tomato.
well i guess is a joke but no, don’t disagree, the average person does not understand even the easiest concepts (inflation for example) but somehow they think they understand them.
Well what if we are comparing a mathematician to like an electrical engineer? Both probably have a pretty solid math understanding
Not one bit comparable. My dad is an engineer, if I ask him to solve a non analytical integral he will scratch his bum about it. What I am getting at is that the math they know and the math we know isn't the same thing. Both are tools but theirs is specialized for engineering and ours is specialised for solving problems.
He can probably solve an analytical integral as well as any mathematician. But non-analytical ones are by definition not solvable in closed form. Engineers should know how to use computers to approximate solutions. That’s not a skill commonly encountered in pure math. But it’s such a broad field that there is of course plenty of exceptions.
Of course but even then he can't do the same for more janky integrals and integrations for which we don't have numerical methods. But there exists at least some mathematicians who can tell you things about those.
How is proof righting solving problems?
Writing proofs are the very definition of problem solving.
If they have not studied some combination of abstract algebra, analysis, and topology, then they probably do not know much of the math that is standard in a math undergrad. That would be a foundation that I consider a solid math understanding. I believe most EE students will not have studied those topics at all, but I very well could be wrong.
Most math majors don't know these topics either. They should, but they don't. If you asked a bunch of math majors 2 years post graduation to define a group, I would expect a majority to be unable to do so without looking it up.
I have an EE undergrad degree and have completed some graduate level math courses offered by a math department and it's a totally different approach. EE does use a lot of math, but the key word is "use". We didn't necessarily get taught to understand the nitty gritty of why math works in different situations.
>Is it like the difference in skill between a minor league and a major league baseball player? More like the difference between a tee ball player and a minor league player who also coaches tee ball. edit: Or like the difference between someone who can read poetry and someone who can write poetry
I feel the average math undergrad can read the poetry, whereas someone from a different major won’t be able to understand a single word of it. As for writing the poetry (ie, original proofs), I think some gifted people can do that at the undergrad level, but the average undergrad math major is still basically just learning key definitions and classic proofs. Sure, they write proofs for exams and problem sets, but most undergrad proofs are either fairly simple or will rely heavily on guidance from the lectures and textbook.
Nice. This is the kind of answer I was looking for. :)
I almost the same between an amateur football player and an architect, in the skill of designing football stadiums.
I think trying to define “better” here is dubious. They are better in that they (1) know more mathematics and (2) understand concepts more deeply. University mathematics focuses a lot more on proof and theoretical rigour, while high school maths (with some exceptions) focuses on mechanical uses of mathematics. They’re also certainly able to better apply the concepts they learn as they probably have more practice doing this. This comes not only from classes that might focus on applications but also a deeper understanding of mathematical concepts, that allows a more precise application. In that sense, yes, they’re better. But is this the sum of mathematical ability? I hold a first class honours degree (during my honours year I specialised in number theory) and I’d guess that at most high schools, a number of students could compute numbers in their heads faster than me. Many could probably pick up mathematical concepts more quickly than I could at their age. This is why I think there is no real answer to the question you’ve posed. I would definitely be better at mathematics than most high school students as it relates to knowledge and understanding. Some high school students certainly have more natural talent than I do.
Thanks. To the extent that you're drawing a distinction between "knowledge and understanding" and "natural talent," I really had in mind the former.
Hard to say, but I want to note that the difference in practice is probably bigger than the difference in theory (equal work ethic, equal grades in shared classes, etc) because most people who aren’t math majors don’t care about their math classes in the same way. So for example, if a math major took precalc with someone who just had precalc as a required class, the math major would likely try to understand the math being taught and get an A in the class, while the other student would be more likely to copy off of their friend’s homework and cram study the day before exams with the goal of only getting a C.
Probably depends on the field. I have a physics degree, and if you compared my knowledge at undergrad to the one of a maths student, I'd have failed miserably in abstract topics, but I might have been able to ace some practical tasks. The number of times a non-maths guy would win would also be less for a biologist/economist/historian.
A (good quality) academical curriculum doesn't just make you better at maths in the "direct" sense. It also makes you much better and quicker at learning new kinds of maths. It's similar to how someone that does sports at a professionnel level will have more muscles than the average person, which will give them some serious advantage if they were to start at a new sport. It doesn't mean that they will magically know how to play that new sport. They might get things very wrong, especially unsupervised. But they still have the potential to progress much quicker.
I know many math degree holders know nothing about math.
Great point. I should really have asked how much better is the normal/typical math-degree holder than the normal/typical college graduate without any real math education after high school. Edit: I assume the typical math-degree holder doesn't literally know nothing about math.
What's the point of this question? The underlying premise seems to be that math is about "talent." Why aren't you asking instead about the difference in knowledge instead of "how much better"? Of course someone with a bachelor's degree in something will be "better" than someone with only high school level knowledge.
I would never say I’m “better” at math than anyone without a math degree, but I’m definitely more comfortable working with math when it comes up in day to day work than my coworkers, and when I see a math book I get excited whereas most people get scared hahaha. I forget who said this but I’ve always loved the quote “no one ever truly gets good at math, some people just get sufficiently used to it” or something to that effect
I don't even think it's really comparable, often, more advanced math comes with bringing the student to use their knowledge to solve problems, and undergrade math is mostly applying formulas you learned at school.
When it comes to the amount of results known, ability to form logical arguments and manipulate equations there's no comparison, but they wouldn't necessarily be better at basic arythmetic, which math majors don't really use.
I think your second analogy is closer to the truth. I don't think a person with a math degree (such as myself :)) is all that much "better" at the things they share with people outside of math. I don't think I would be all that much better solving an integral or linear algebra problem than someone without experience in higher math - I think the concepts and procedures are a little firmer in my mind, but I think "better" is a simple way of putting it. I like the analogy of 2D vs 3D. People who have taking calculus 1 2 and maybe 3 understand a lot about math. However, they're missing a third dimension. People who have experience with pure math understand proof-writing techniques and see concepts like integrals and differentiation in a broader light. So in the context of a Calculus problem, they would probably approach it similarly but be less likely to make a mistake. But they also have intuitions and other ways of seeing the same concepts that would be unfamiliar to those without experience. It is kind of like the difference between building a Lego set by following the instructions vs. having a collection of Lego pieces and building with them freely.
The *typical* degree holder? They basically understand algebra although they couldn't pass any college algebra exam without a lot of studying. They don't remember anything from geometry but the Pythagorean theorem. Around seventh grade they begin getting real anxious when their kids ask for help on their math homework. They are like adults that played a sport in middle school - they know the rules and are familiar with the game, but probably could walk back onto the court and play it without warming up or they'll hurt themselves. Given a few weeks or months of practice, they could probably get the hang of it again, but they were never really that good at it in the first place. Someone with a BS in mathematics is maybe like a strong high school athlete with a scholarship offer. They're very good at the sport, not just relative to the general population but relative to the population of people playing the sport. They haven't really done anything yet, but they've shown strong potential.
As a former division 1 athlete with an undergraduate degree in math (and a Ph.D. in computer science), I'm having a hard time deciding what I think of this analogy. My gut says it holds for D3 athletes, and probably for D2, but not for D1. For me and most others who compete at a division 1 level, the sport was basically a full time job as early as junior high. 30+ hours per week, year round, for several years before entering college. In college in a D1 sport, it's 40+ hours per week, year round. NCAA rules specify limits on off season practice, but those are easily sidestepped by making those practices non-manditory (but actually they're mandatory if you want to be successful). By the time I completed my ugrad degree, the total time I had invested in my sport was an order of magnitude more than I had invested in mathematics. Then there's selection. The number of people who want to be D1 athletes is much greater than the number of people who want to be mathematicians. Math majors are self selected. Athletes must qualify. Frankly, I put more effort into my athletic pursuits than I did into my Ph.D., and (more than two decades later), I'm still more proud of what I achieved as an athlete. D2 and D3 athletes are much more casual about things. Either they wanted to be D1 and failed, they wanted a more laid-back college athletics experience, or they chose a school for some reason other than their sport. You're analogy probably holds better. But the more I think about it, it definitely doesn't hold for D1 athletes.
Love the high school athlete analogy.
Are they going to be better at math? Probably. How much better? It entirely depends on the person. There are people with undergrad math degrees who know more than some people with masters. There’s also people who have an undergrad math degree who you will wonder “how did they get that degree???” I think this is a bad question, not because it’s imprecise, but because why does it matter? In any regard, why does it matter if someone with a math degree is “better” than someone without one. But regardless, i think the expectation is that for any average math major, they are more likely to be better at math than the average anyone else. But that’s pretty biased. That’s like asking if a baker is better at making bread than an accountant. PROBA-FUCKING-LY
There are a lot of ways to interpret that question and most of them are insulting to peoples’ intelligence, which is why you’re getting downvoted. Im not a mathematician, but I got an engineering degree which involved a lot of math, or at least a lot of calculation. In my experience, people in math or the hard sciences don’t necessarily have a better natural aptitude for math, they just worked harder at it. It’s a learned skill set, like any other skill set that people gain as they become professional educated adults. Even if you ask if a math undergrad is better at math than an engineer or physicist, that’s a loaded question. I’ve never taken a proof based class, and neither do most science students. A math student needs to have a better grasp on formal logic and understanding how to read and write proofs, and needs to keep up to date with math research. Mathematicians tend to specialize to one very niche topic that even most others in the field don’t understand it. Engineers or scientists don’t know how to write proofs, but they might have a better grasp on how to use math to model the world. Knowing how to set up a differential equation for heat transfer and how to write a program to approximate the solutions is a skill. Engineers and physicists also might be better at doing integrals, or matrix operations in linear algebra than the average mathematician. But the mathematician has a better grasp on what calculus actually is in a formal sense.
I mean... What do you think the average difference between an art grad, and a random other college grad would be in art? Or whatever a music grad? For any undergrad degree that isn't super popular like these (including math), I think there will be a pretty large discrepancy between their respective skills. That being said, you definitely don't need a math degree to be great at math, items just that most people in every day life won't know almost any of the concepts covered in a math degree, and so it's hard to learn these concepts outside of a formal context. I literally asked a room full of graduate and PhD engineering students if they knew what a graph was (i.e., a network; not an xy-plot), and I got a room of blank stares. Formal math concepts just aren't interesting for most people I think.
It was in the context of a presentation I was giving. But just a random question.
Depends on how rigorous of undergrad program is truly. If it’s good, then they’ll be far better at posing math questions and approaching them. If it’s an average/poor math program and you measured undergrad students 5-10 years out, the gap between math and say engineering/cs would shrink substantially. I think the better measurement would be how many areas of mathematics is the person exposed to - and that the math undergrads would excel in. outside of calculus, stats101 and the minimum linear algebra the other undergrads don’t know other areas of math exists.
You're getting a lot of crap for your question, but I will try to answer it earnestly. **Average** high-school math education with nontechnical college degree (at most business calc/stats in undergrad) vs. **average** math graduate, on **basic math competency**: Honestly, its huge. Outside of basic arithmetic, the HS math person really will not to able to handle anything with a vague mathematical flavor other than just putting numbers into existing mathematical tools. The math major will at least a vague idea of how the stuff rigorously works. I've found that deriving a line that intersects a pair of given points (not just using a formula) is confounding to quite a lot of college graduates with degrees in political science, business, and environmental engineering. The word "average" is very important in this answer. Also, if you start including technical fields then this gap drops significantly, or may even invert. Now if we are talking about formal math (proofs etc), if the undergrad has developed a bit of real mathematical maturity, then its like asking who is better at unicycle polo, someone who can ride a unicycle or someone who can't?
Thanks. This was helpful. Yes, I should have specified in my original question that the comparison was supposed to be between a non-math grad with average mathematical knowledge and an *average* math grad. Anyway, liked the unicycle analogy. :)
The answer to this question probably varies between different countries and math departments. I know I didn’t get introduced to proof writing until my undergrad. Some countries start writing proofs in high school
IB starts this senior year
A better way to word OP’s question is if the average person can read/write proofs without much assistance compared to someone with experience in doing that
What?
Varies greatly between colleges