Math is hard
December 26th, 2005 by Walt
Math is hard. Discuss.
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December 27th, 2005 at 1:26 am
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!”
December 27th, 2005 at 4:13 am
Not so! Mathematics is precisely that which is easy.
December 27th, 2005 at 7:05 pm
Define hard.
December 27th, 2005 at 7:38 pm
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.
December 28th, 2005 at 12:58 am
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.
December 28th, 2005 at 4:50 am
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
December 28th, 2005 at 6:13 am
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).
December 28th, 2005 at 9:43 am
LIfe is hard. Math is challenging and productive, thus worth doing.
December 28th, 2005 at 11:16 am
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.
December 30th, 2005 at 8:18 am
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.
December 30th, 2005 at 1:26 pm
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.” ).
December 31st, 2005 at 5:00 am
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.”
December 31st, 2005 at 12:30 pm
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.
January 1st, 2006 at 9:09 pm
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January 2nd, 2006 at 2:53 am
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.
January 3rd, 2006 at 9:57 am
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.
January 3rd, 2006 at 10:37 am
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.
January 3rd, 2006 at 12:10 pm
Sigfpe (hopefully I don’t have to pronounce this…) I really like what you say, it’s really nice.
January 4th, 2006 at 12:18 am
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.
January 9th, 2006 at 12:06 pm
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.
January 11th, 2006 at 8:19 am
Of course it’s hard. It helps keep out the riffraff.
December 11th, 2007 at 12:59 pm
I’m merely an undergraduate, but here is my opinion of math.
Math is the most difficult subject I have ever taken. When I study for it, I actually don’t do well in my other courses in addition to not doing well in what I studied for. I often fall behind before I know it. It’s the only thing I can spend hours studying and still not understanding.
Math is abstract thinking? I consider myself good at abstract thinking. I can write poetry and analyze pictures and figures. Math is a foreign language? If math WERE greek to me, I would understand it better. I excell in learning foreign words and grammer. But math? It’s kind of like trying to analyze a picture that hasn’t been drawn or write and speak a language I don’t know.
It’s challenging, maybe even fun, but does it have to be like this? I would like to be able to take a math course for once in my life and remember the concepts, formulas, etc. for the duration I remember the other pertenant material I study.
How do people learn math when all they have is a serious of interrelated abstractions? For me, studying mathematics is mostly just memorization. Manipulating the pieces of the puzzle is difficult if one does not even know what the picture should look like, after all. I easily forget the concepts, and I find taking math courses very frustrating, because its a vicious circle that depleats my energy harms my other subjects, and usually leaves me feeling defeated, lazy, tired, and disappointed.
If there is any way to tame the study of mathematics, I would love to know, because I don’t want to be another semi-mathematically-illiterate professional in the American workforce. I want to understand and remember these concepts, but why do I have to learn how to speak a foreign language before I know what I’m saying?
I view this as kind of intellectually insulting. Why can’t I know what I’m learning? Don’t the teachers understand it? I wish I didn’t feel like I was in some kind of foreign country where no one was willing to teach me the language.
If someone would teach me the language, I too could become proficient in mathematics, but, as of now, I only know how to say a few words, and maybe ask for directions, and order food.
Another topic
Math should be taught as a complete foreign language with its own words, grammer rules, and pronunciation. Speaking a formula would certainly help one remember it better, at least it would me.
Why can’t someone wrap the concepts of math in the packages other curricula come in? For some people, learning math using the current methods is like trying to load a DVD into a CD ROM drive.
Well, that’s all I have to say.
December 11th, 2007 at 1:05 pm
Whew, I was pretty long winded. Please read the entire comment before responding. I really would like to know what to do.
December 11th, 2007 at 3:50 pm
Dear undergrad,
I thought that was very eloquently put, as far as it went. But it would help to know also: (1) what is your background in mathematics, and (2) why would you like to understand it better (besides doing well in courses). Do you expect you will need the mathematics you are studying now in your future career, or do you feel that some understanding of mathematics should be part of one’s general culture? What motivates you?
I’m guessing from what you’ve written that you may have had a sequence in calculus and perhaps a bit beyond that. I think your analogy to learning a foreign language is apt in many respects: mathematics is a precise, specialized language, with a complete grammar of its own and a rich, vibrant literature. The foreign language aspect becomes especially apparent when one first begins taking mathematics courses which emphasize an axiomatic development and mathematical proofs (and I think a good instructor in such an introductory course would be very sensitive to that foreign language aspect). But that’s not all: the cultural referents of the language (the structures of modern mathematics) may be something quite alien as well, e.g., the concept of a vector space.
There’s no royal road that I know of: learning mathematics requires a lot of patience and asking yourself or your peers or your teacher a lot of questions and thinking a lot on your own about what the hell some piece of mathematics is talking about (not just memorizing, but really getting underneath it). It’s pretty hard learning it by yourself; one piece of generic advice is to form a study group with similarly interested peers, or perhaps better, seek out someone who is a bit further along than you to talk to (it need not be a professor). This is no different from people who are learning Spanish getting together in informal discussion groups.
But it’s hard giving more specific advice without knowing a bit more about you.