People now use the Barbie quote as a general synonym for “I’m a bimbo.” I’ve seen people (both in person and online) after saying something stupid and realizing it, following it up with “Math is hard.”

I suppose I should include my own opinion as well as linking to someone else’s.

My opinion on the subject is that mathematics is not inherently any harder than any other intellectual area of human endeavour. There are hard parts to it (some of them are *very* hard), there are easy parts to it, and there’s an awful lot of stuff in between.

I think there are a couple main things which cause the common perception that maths is hard:

Firstly, mathematics as a whole is very badly taught (certainly at preuniversity level, as well as first year university).

Secondly, mathematics requires a higher degree of abstract thinking than most people are used to – ideas in mathematics don’t neccesarily correspond to real world concepts. Once you’re used to this such thinking isn’t really more difficult than other types, but until you’re used to it it is rather daunting.

Thirdly, there is an awful lot of jargon to take in, and until you see how all the concepts connect together it’s very hard to remember it all. This ties in with the previous point. For example, if you’re just told ‘A field is a set together with distinguished elements 0, 1, operations + and x satisfying the following axioms…’ from the word go you’re going to go “Huh??”, but once you have an idea of the sort of thing you’re talking about here, why one wants these particular operations and properties, etc. you already have the internal concept of a field, and the name and definition are just codifying it. You already know what a field is, you’re just tidying up your concept slightly and attaching a name to it.

Hmm. The length of my points seems to be growing exponentially, so I’ll stop here before I begin writing an essay for one point.

It is common for people to say you have to be good at abstract thought to be a good pure mathematician. I think the situation is more subtle than this statement suggests, and an ability to think abstractly is useful but not always essential in a mathematician. Ramanujan, Hardy and Nash (and maybe Erdos?) are examples of great mathematicians who were not so great at abstraction, IMHO.

Teaching undergraduate computer scientists has led me to the belief that if maths is hard, then computer science is harder. While it is true that mathematics CAN require abstract thought, not all mathematicians are good at the kind of abstract thought required to be a programmer.

Programming, by its very nature, requires an ability to abstract away from instances, since we almost never write a computer program to undertake a calculation or execute a process for only one input; the very idea of programing requires an ability to think beyond a single input case. I have found some good pure mathematics students are very poor at the type of abstraction needed to be good programmers, and conversely — some good programmers are poor at pure math.

I suspect that this is because most mathematics taught in school and at undergrad levels in University is aimed at problem-solving using abstraction rather than at undertaking abstraction for its own sake. So, someone can be good at solving math problems, even using abstract methods, without necessarily being good at creating and manipulating abstract structures themselves. This might also explain the general hostility to category theoretic approaches in many parts of mathematics (less so now than once).

I accept that math “…is very badly taught” at the preuniversity level, and I teach Algebra to eighth graders. I am slowed, even stopped dead from time to time, by gaps in either the knowledge or skill level of my students. The problem of ability progression is a constant source of meetings and posturing: teachers blame each other, university advisors (some occasionally have classroom experience) pontificate and the standards movement has created a de-facto triage system (who can I teach this year? Forget the rest…).

I reject the idea that “math is hard” because of its abstract nature. True abstraction in math is only introduced in the usual k-12 math curricula. I believe math would be much “easier” if the focus of education was on its vocabulary and its clear, exact basis on memorable concepts such as the number line. Mathematics is a language and should be taught as a language, but it is not. At least, not by most.

When I say “define hard”, what I mean is, there are obviously aspects of math that are hard by any definition, but on the whole, math is no harder than any thing that must be learned, even say learning how to use a hammer or a shovel properly, while some personalities may be more suited to certain types of skills than others, to say that one is harder than the other to my mind reveals a real lack of objectivity (if you doubt what I say about using a hammer or a shovel, try watching what a Master Carpenter can do with a hammer and try to imitate him, then you will know what I mean).

I guess it comes down to the notion of skill, learning the skills involved in an endeavor is learning the art of it, to be good with a hammer one must learn the art of using a hammer, this takes practice and _time_, this cannot be learned in a class room. The same is true with mathematics, there are skills involved that form the art of mathematics and those skills cannot be learned in a class room; they can be identified, even codified, but if someone wants to really learn (that is acquire) these skills they must spend the time. Unfortunately in most school settings it’s not the art of mathematics that is emphasized but the science (the knowledge), to have a real appreciation for math to my mind both are needed.

The other aspect is a matter of personality, just like there are some personalities more likely to become masterful at using a hammer those willing to suffer the smashed fingers and bent nails, and the other pains that are part of the learning process, in learning there is always pain, if there is no pain you’re not _actually_ learning, what spurs us on is the high we get after the lesson has been learned, that high is accomplishment. The same is true, again, with math, the pain is the frustration involved in problem solving, learning the usefulness of abstraction, and generalization, and I believe there are certain personalities that are more likely to enjoy, and therefore master these challenges.

I’ll stop now because as like Davis above I fear I’m about to create and essay.

Sorry for the delay in replying. I’ve been ill, so I didn’t really feel like starting an active debate on this.

First of all, my comment about the quality of mathematics teaching at pre-university level wasn’t really meant to impugn the abilities of the teachers. (I think you realised this, but I just want to make sure). There are some very good high school and lower mathematics teachers. The problem is that, as the general approach to mathematics that is required stands, they’re bound to either be not doing their jobs or not teaching mathematics well.

I wasn’t really claiming that it’s the abstraction that makes mathematics hard – as I wasn’t claiming that mathematics is hard. The fact of the matter is that concepts like numbers, formal manipulation of algebra, etc. *are* abstract. They’re abstractions of mostly concrete ideas, granted, but they don’t get presented as such very well. So even at it’s basic level mathematics becomes about little more than juggling abstractions.

I’m not going to get into a mathematics vs CS debate. For what it’s worth, I’m a pure mathematician attempting to move into a software developer job and I find the abstractions needed in mathematics and programming to be quite similar. There are differences of course, but I feel these are as more the result of different aims than different processes. Of course, this may just be an overapplication of the hammer principle ( “If all you have is a hammer then you’ll just have to solve your problem by hitting things.” ).

Posted in my classroom:
“It is impossible to learn and be perfect at the same time.”
One of the biggest hurtles in any math classroom is how unwilling students are to risk being wrong. Students would rather get an “F” or detention or whatever by not doing anything rather than take a chance that their answers to questions might be wrong.
That is why “math is hard.” There are correct answers and we live in a culture that is brutal when someone is not “right.”

I think there are genuinely hard concepts in math, although perhaps many of these don’t get taught at high school — eg, infinity, countable vs. uncountable infinities, complex numbers, probability, to name just a few. I think these concepts are hard because fall outside our general everyday life experience.

Probability is an interesting case, since most people who studied it these last 300 years thought they knew what it was. There have, however, always been dissidents (starting with Leibniz), who have argued against the standard view of probability (that formalized with the Kolmogorov axioms) as the only or best model of uncertainty. The most recent dissidents have been people in AI trying to build expert systems, who have since the 1970s developed several formal alternatives to the Kolmogorov axioms (although most statisticians and probabilists seem ignorant of this work). The concept is not an easy one at all, IMO.

What is easy is indulging ourself with our feelings and intuitions (what I’m doing right now !). As opposed to this, any form of rigourous thinking is hard. Being the most rigourous of all, math is probably the hardest in this respect. But there is another thing which is easy : that is repeating ceaselessly what we have learned to do. In this respect math can be very easy. To sum up I would say that the human mind is at ease when it is repeating or believing something, and has the greatest difficulties when it comes to rigour and invention.

Math is hard. I used to think it was ridiculous the way someone doing a PhD in chemistry, say, could get their qualification simply by carrying out the experiment their supervisor told them to. In mathematics you have to create new mathematics out of nowhere. In just about any other subject I have studied you can get away with simply applying yourself diligently – but in mathematics solving problems frequently needs a flash of insight that goes beyond anything you have been taught.

Math is easy. Like many of my colleagues I sailed through my degree hardly lifting a finger to work. Once you grok a theorem, say, it becomes obvious and more or less unforgettable. I used to feel sorry for people studying a subject like chemistry who had to memorise an immense body of empirically determined facts to get anywhere. There’s little to memorise in mathematics and no need to work late into the night cramming arbitrary facts.

Math is hard. Just look at how much effort it takes to explain even the most elementary mathematics to a non-specialist. Pick up a mathematics textbook you’ve never read before. It’s not at all unusual for anything just one page beyond where you have already read to be completely incomprehensible. Mathematics texts pile up layer after layer of definition and abstraction which you have to ‘get’ before you can proceed. I can pick up a textbook in a subject like genetics, say, and get the gist of just about any chapter. Conceptually, he subject is more or less trivial and mastering it is just about learning some details.

Math is easy. Time and time again when you get to what appears to be a difficult result you realise it’s fancy language for a trivial intuition or concept. Think of subjects like algebraic topology where much of the fancy machinery is just a way to allow formal notation to get a handle on things that a child already knows intuitively.

Math is hard. I can’t tell you how many times I’ve been stuck there looking at a theorem knowing that everything I need is spelled out there in front of me and yet not being able to put the pieces together. It’s not like I can blame it on a missing fact that I need from another book that requires the work of going down to the library to look something up. This is a completely different kind of work – just you battling against your own mind. I don’t know of anything harder, sometimes, than trying to put those pieces together so that they fit together as a logical whole. You can’t just try to memorise the theorem. If you don’t get it then you won’t be able to use it. There’s no guarantee that if you just spend 12 hours looking at it you’ll have done the required work. No other subject begins to compare with the difficulty of doing this for a challenging proof.

Math is easy. Of course it is, I’m a lazy ba*tard and couldn’t have studied anything else.

And just a general point about mathematics education following up on some earlier comments. I know quite a few people who lecture mathematics. They almost all have strong opinions on mathematics education and can hold forth for hours on how everyone else makes the subject harder than it is and if only everyone else used the methods that they themselves used people would find the subject easier. And they are mostly terrible lecturers and seem to have no clue that they are.

Another late reply. “Hard” by itself is a bit meaningless. Anything that you do that pays (deservedly) more than the average salary is hard by definition. However, math is so precise and hard to argue with that everybody can find a problem which they admit they cannot solve. You can’t get that kind of certainty in other fields. This is what gives the impression that math is hard.

All the mathematics that I have mastered is trivial, everything else is inescapably hard.

Math is hard, just like learning any OTHER language is hard.

Sigfpe Says: Think of subjects like algebraic topology where much of
the fancy machinery is just a way to allow formal notation
to get a handle on things that a child already knows
intuitively.

You know a lot children with an intuitive grasp of homological algebra?!?
I mean, the 5 lemma sure, but I am sure most of them have a little trouble with the snake lemma.

The infamous Barbie quotation. Nobody disputed the recording, just that Barbie- a girl!- said it and that would reinforce stereotypes.

I wonder if the reaction would have been as big if Barbie had said, “The book I have to read for English class is so boring!”

Not so! Mathematics is precisely that which is easy.

Define hard.

People now use the Barbie quote as a general synonym for “I’m a bimbo.” I’ve seen people (both in person and online) after saying something stupid and realizing it, following it up with “Math is hard.”

delon: NP hard.

I often say “math is hard,” but that happens when I’m actually doing math (usually abstract algebra) and am not getting where I want to.

I suppose I should include my own opinion as well as linking to someone else’s.

My opinion on the subject is that mathematics is not inherently any harder than any other intellectual area of human endeavour. There are hard parts to it (some of them are *very* hard), there are easy parts to it, and there’s an awful lot of stuff in between.

I think there are a couple main things which cause the common perception that maths is hard:

Firstly, mathematics as a whole is very badly taught (certainly at preuniversity level, as well as first year university).

Secondly, mathematics requires a higher degree of abstract thinking than most people are used to – ideas in mathematics don’t neccesarily correspond to real world concepts. Once you’re used to this such thinking isn’t really more difficult than other types, but until you’re used to it it is rather daunting.

Thirdly, there is an awful lot of jargon to take in, and until you see how all the concepts connect together it’s very hard to remember it all. This ties in with the previous point. For example, if you’re just told ‘A field is a set together with distinguished elements 0, 1, operations + and x satisfying the following axioms…’ from the word go you’re going to go “Huh??”, but once you have an idea of the sort of thing you’re talking about here, why one wants these particular operations and properties, etc. you already have the internal concept of a field, and the name and definition are just codifying it. You already know what a field is, you’re just tidying up your concept slightly and attaching a name to it.

Hmm. The length of my points seems to be growing exponentially, so I’ll stop here before I begin writing an essay for one point.

David

It is common for people to say you have to be good at abstract thought to be a good pure mathematician. I think the situation is more subtle than this statement suggests, and an ability to think abstractly is useful but not always essential in a mathematician. Ramanujan, Hardy and Nash (and maybe Erdos?) are examples of great mathematicians who were not so great at abstraction, IMHO.

Teaching undergraduate computer scientists has led me to the belief that if maths is hard, then computer science is harder. While it is true that mathematics CAN require abstract thought, not all mathematicians are good at the kind of abstract thought required to be a programmer.

Programming, by its very nature, requires an ability to abstract away from instances, since we almost never write a computer program to undertake a calculation or execute a process for only one input; the very idea of programing requires an ability to think beyond a single input case. I have found some good pure mathematics students are very poor at the type of abstraction needed to be good programmers, and conversely — some good programmers are poor at pure math.

I suspect that this is because most mathematics taught in school and at undergrad levels in University is aimed at problem-solving using abstraction rather than at undertaking abstraction for its own sake. So, someone can be good at solving math problems, even using abstract methods, without necessarily being good at creating and manipulating abstract structures themselves. This might also explain the general hostility to category theoretic approaches in many parts of mathematics (less so now than once).

LIfe is hard. Math is challenging and productive, thus worth doing.

I accept that math “…is very badly taught” at the preuniversity level, and I teach Algebra to eighth graders. I am slowed, even stopped dead from time to time, by gaps in either the knowledge or skill level of my students. The problem of ability progression is a constant source of meetings and posturing: teachers blame each other, university advisors (some occasionally have classroom experience) pontificate and the standards movement has created a de-facto triage system (who can I teach this year? Forget the rest…).

I reject the idea that “math is hard” because of its abstract nature. True abstraction in math is only introduced in the usual k-12 math curricula. I believe math would be much “easier” if the focus of education was on its vocabulary and its clear, exact basis on memorable concepts such as the number line. Mathematics is a language and should be taught as a language, but it is not. At least, not by most.

When I say “define hard”, what I mean is, there are obviously aspects of math that are hard by any definition, but on the whole, math is no harder than any thing that must be learned, even say learning how to use a hammer or a shovel properly, while some personalities may be more suited to certain types of skills than others, to say that one is harder than the other to my mind reveals a real lack of objectivity (if you doubt what I say about using a hammer or a shovel, try watching what a Master Carpenter can do with a hammer and try to imitate him, then you will know what I mean).

I guess it comes down to the notion of skill, learning the skills involved in an endeavor is learning the art of it, to be good with a hammer one must learn the art of using a hammer, this takes practice and _time_, this cannot be learned in a class room. The same is true with mathematics, there are skills involved that form the art of mathematics and those skills cannot be learned in a class room; they can be identified, even codified, but if someone wants to really learn (that is acquire) these skills they must spend the time. Unfortunately in most school settings it’s not the art of mathematics that is emphasized but the science (the knowledge), to have a real appreciation for math to my mind both are needed.

The other aspect is a matter of personality, just like there are some personalities more likely to become masterful at using a hammer those willing to suffer the smashed fingers and bent nails, and the other pains that are part of the learning process, in learning there is always pain, if there is no pain you’re not _actually_ learning, what spurs us on is the high we get after the lesson has been learned, that high is accomplishment. The same is true, again, with math, the pain is the frustration involved in problem solving, learning the usefulness of abstraction, and generalization, and I believe there are certain personalities that are more likely to enjoy, and therefore master these challenges.

I’ll stop now because as like Davis above I fear I’m about to create and essay.

Sorry for the delay in replying. I’ve been ill, so I didn’t really feel like starting an active debate on this.

First of all, my comment about the quality of mathematics teaching at pre-university level wasn’t really meant to impugn the abilities of the teachers. (I think you realised this, but I just want to make sure). There are some very good high school and lower mathematics teachers. The problem is that, as the general approach to mathematics that is required stands, they’re bound to either be not doing their jobs or not teaching mathematics well.

I wasn’t really claiming that it’s the abstraction that makes mathematics hard – as I wasn’t claiming that mathematics is hard. The fact of the matter is that concepts like numbers, formal manipulation of algebra, etc. *are* abstract. They’re abstractions of mostly concrete ideas, granted, but they don’t get presented as such very well. So even at it’s basic level mathematics becomes about little more than juggling abstractions.

I’m not going to get into a mathematics vs CS debate. For what it’s worth, I’m a pure mathematician attempting to move into a software developer job and I find the abstractions needed in mathematics and programming to be quite similar. There are differences of course, but I feel these are as more the result of different aims than different processes. Of course, this may just be an overapplication of the hammer principle ( “If all you have is a hammer then you’ll just have to solve your problem by hitting things.” ).

Posted in my classroom:

“It is impossible to learn and be perfect at the same time.”

One of the biggest hurtles in any math classroom is how unwilling students are to risk being wrong. Students would rather get an “F” or detention or whatever by not doing anything rather than take a chance that their answers to questions might be wrong.

That is why “math is hard.” There are correct answers and we live in a culture that is brutal when someone is not “right.”

I think there are genuinely hard concepts in math, although perhaps many of these don’t get taught at high school — eg, infinity, countable vs. uncountable infinities, complex numbers, probability, to name just a few. I think these concepts are hard because fall outside our general everyday life experience.

Probability is an interesting case, since most people who studied it these last 300 years thought they knew what it was. There have, however, always been dissidents (starting with Leibniz), who have argued against the standard view of probability (that formalized with the Kolmogorov axioms) as the only or best model of uncertainty. The most recent dissidents have been people in AI trying to build expert systems, who have since the 1970s developed several formal alternatives to the Kolmogorov axioms (although most statisticians and probabilists seem ignorant of this work). The concept is not an easy one at all, IMO.

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What is easy is indulging ourself with our feelings and intuitions (what I’m doing right now !). As opposed to this, any form of rigourous thinking is hard. Being the most rigourous of all, math is probably the hardest in this respect. But there is another thing which is easy : that is repeating ceaselessly what we have learned to do. In this respect math can be very easy. To sum up I would say that the human mind is at ease when it is repeating or believing something, and has the greatest difficulties when it comes to rigour and invention.

Math is hard. I used to think it was ridiculous the way someone doing a PhD in chemistry, say, could get their qualification simply by carrying out the experiment their supervisor told them to. In mathematics you have to create new mathematics out of nowhere. In just about any other subject I have studied you can get away with simply applying yourself diligently – but in mathematics solving problems frequently needs a flash of insight that goes beyond anything you have been taught.

Math is easy. Like many of my colleagues I sailed through my degree hardly lifting a finger to work. Once you grok a theorem, say, it becomes obvious and more or less unforgettable. I used to feel sorry for people studying a subject like chemistry who had to memorise an immense body of empirically determined facts to get anywhere. There’s little to memorise in mathematics and no need to work late into the night cramming arbitrary facts.

Math is hard. Just look at how much effort it takes to explain even the most elementary mathematics to a non-specialist. Pick up a mathematics textbook you’ve never read before. It’s not at all unusual for anything just one page beyond where you have already read to be completely incomprehensible. Mathematics texts pile up layer after layer of definition and abstraction which you have to ‘get’ before you can proceed. I can pick up a textbook in a subject like genetics, say, and get the gist of just about any chapter. Conceptually, he subject is more or less trivial and mastering it is just about learning some details.

Math is easy. Time and time again when you get to what appears to be a difficult result you realise it’s fancy language for a trivial intuition or concept. Think of subjects like algebraic topology where much of the fancy machinery is just a way to allow formal notation to get a handle on things that a child already knows intuitively.

Math is hard. I can’t tell you how many times I’ve been stuck there looking at a theorem knowing that everything I need is spelled out there in front of me and yet not being able to put the pieces together. It’s not like I can blame it on a missing fact that I need from another book that requires the work of going down to the library to look something up. This is a completely different kind of work – just you battling against your own mind. I don’t know of anything harder, sometimes, than trying to put those pieces together so that they fit together as a logical whole. You can’t just try to memorise the theorem. If you don’t get it then you won’t be able to use it. There’s no guarantee that if you just spend 12 hours looking at it you’ll have done the required work. No other subject begins to compare with the difficulty of doing this for a challenging proof.

Math is easy. Of course it is, I’m a lazy ba*tard and couldn’t have studied anything else.

Math is whatever you like.

And just a general point about mathematics education following up on some earlier comments. I know quite a few people who lecture mathematics. They almost all have strong opinions on mathematics education and can hold forth for hours on how everyone else makes the subject harder than it is and if only everyone else used the methods that they themselves used people would find the subject easier. And they are mostly terrible lecturers and seem to have no clue that they are.

Sigfpe (hopefully I don’t have to pronounce this…) I really like what you say, it’s really nice.

Another late reply. “Hard” by itself is a bit meaningless. Anything that you do that pays (deservedly) more than the average salary is hard by definition. However, math is so precise and hard to argue with that everybody can find a problem which they admit they cannot solve. You can’t get that kind of certainty in other fields. This is what gives the impression that math is hard.

All the mathematics that I have mastered is trivial, everything else is inescapably hard.

Math is hard, just like learning any OTHER language is hard.

Sigfpe Says: Think of subjects like algebraic topology where much of

the fancy machinery is just a way to allow formal notation

to get a handle on things that a child already knows

intuitively.

You know a lot children with an intuitive grasp of homological algebra?!?

I mean, the 5 lemma sure, but I am sure most of them have a little trouble with the snake lemma.

Of course it’s hard. It helps keep out the riffraff.