Justifications for: Why learn mathematics?

Question

I wonder how you teachers walk the line between justifying mathematics because of its many—and sometimes surprising—applications, and justifying it as the study of one of the great intellectual and creative achievements of humankind?

I have quoted to my students G.H. Hardy's famous line,

The Theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics.

and then contrasted this with the role number theory plays in contemporary cryptography. But I feel slightly guilty in doing so, because I believe that even without the applications to cryptography that Hardy could not foresee—if in fact number theory were completely "useless"—it would nevertheless be well-worth studying for anyone.

One provocation is Andrew Hacker's influential article in the NYTimes, Is Algebra Necessary? I believe your cultural education is not complete unless you understand something of the achievements of mathematics, even if pure and useless. But this is a difficult argument to make when you are teaching students how to factor quadratic polynomials, e.g., The sum - product problem.

So, to repeat, how do you walk this line?

Answer 1

Most of what you learn in school isn't directly useful. When I was in primary school I was taught the difference between warm and cold-blooded animals. I've never used that information; should I not have been taught it? Another example is that in any English-speaking country you still have to take English courses even after you speak the language fluently. In day-to-day life you'll never need to know about Shakespeare or whatever literature the curriculum exposes you to. But it's still important to develop your communication and reasoning skills, your ability to think critically and creatively, to expose you to things that are influential on our culture, to show you a field or a way of looking at life that might be interesting to you or that might influence your career choice, and so on. All of this is applicable to mathematics.

Answer 2

My answer is mostly about higher education, and mostly suited for undergrads (which make the largest part of my students).

The main purpose of their studies is to become more intelligent.

This is what I say my undergrad student a lot. It applies to math as well as to any other fields; as was mentioned in other answers, many aspects what we teach, from reproducing algorithmic tasks to handle abstract reasoning, makes one mind more flexible, more powerful.

A couple of years ago, I came up with a comparison I like, but the effectiveness of which I cannot yet evaluate: I tell my students that they are like high level sport competitors, except they work their brain rather than their body. This has many interesting consequences (among which the most important to me is not related to your question: it makes it clear that the student have to work by themselves to learn, and the teacher, like a coach, can only help them, but cannot work for them), and it partially answers the "why learn math" question.

Knowing how to use mathematical tools is useful.

Knowing how to use mathematical tools is indeed useful for sciences as well as in any quantitative field. Let me digress to partially answer "Lockhart's lament" mentioned in a comment. that mathematicians should not conflate our research activity "doing maths", and what we mainly teach (this is why I wrote about mathematical tools). We also should help students not to conflate the two, and know a little bit about what "doing math" really is, but for most of them this is not what we are asked to teach them. That said, it would be good to make what we teach interesting, of course.

Back to usefulness. Knowing about probabilities is useful in every day's life, just as is basic arithmetic. Let me give an example I heard about recently: a biology professor gave her students yes/no tests where each correct answer was rewarded 1 point, and each incorrect answer was penalized 0.25 point. The professor was not really aware that this grading system gives a not good, but not too bad grade on average to a monkey answering randomly. A students knowing one fourth of the answers and giving random answers to the other questions would often get a passing grade. This is a classical example of mathematical illiteracy (here, the lack of understanding of expectation and the law of large numbers), and shows that a basic understanding of a variety of basic mathematics comes right next to reading and writing in terms of usefulness.

I also try to give examples of the usefulness of mathematical tools as often as possible, in order to give an idea of why we chose to teach them their precise curriculum. Let me name a couple: to justify the differential equations to freshmen, I gave a dozen examples from biology (various population growth models), physics (from mechanics, radioactivity, etc.), economy (I derived the "second law of economics" that I recently read in Thomas Picketty's book from a differential equation, and mentioned the context to show that the example was not made up for them), chemistry (cinetics); to justify the differential geometry course on curves and surfaces, I had a session devoted to students measuring distances on a globe and a map and comparing them, before I went on to prove that no map can be faithful.

Answer 3

Some reasons (random order):

• Math is fun and beatiful. Not all marvel at the beauty of mathematics and not all enjoy working with math. In fact, daily mathematical work can be tedious, boring, uninspiring even for those in love with it. But every now and then you discover something great, and the feeling you get can be matched by very few other things (it does help if you have someone to share it with).

  Moreover, math can help you appreciate the world even more deeply than before. For example, during a nice clear sky night, try to imagine the size of Andromeda galaxy from how far it is (about $2.5\times 10^6$ light years$) and the fact that its apparent size (angular diameter) is about six times that of full moon.

• Math makes you more intelligent. I have no citation for this, but it seems to work that way. There are some papers linking happiness to intelligence, but to the best of my knowledge they are inconclusive, so this may be a double-edged sword.

• Math rewires your brain. I don't know how it happens, but it is true. To give an example, usually people don't consider $-5$ (negative $5$) a valid number of apples, but that would be perfectly normal to a mathematician (unless assumptions imply otherwise). In most cases such a rewiring gives a superior perspective (of course, it does not help much if it is your only one; insight is not wisdom), although it may also cause friction or trouble with communication.

• Math can teach you how the world works. There are many non-intuitive real-life phenomena, for example, adding new roads may increase traffic jams (see Braess's paradox). Still, there is math that models it in an intuitive way (here it is game theory). Knowledge of such math allows you to better predict the outcome of actions.

• Math is useful. Almost everybody and everything uses math now, whether it is a small MP3 player or big spaceship. We wouldn't be where we are now without probability, eigenvalues, Fourier transform or differential equations to name a few.

• Math is considered important. Even if math was not important by itself, people thinking it is important make it important.

• Math is gatekeeper. As @Benjamin Dickman pointed out, math is already thought of as a gatekeeper, but this might be even more true in the future, see a nice article by Alexandre Borovik: Calling a spade a spade: mathematics in the new pattern of division of labour.

• Finally:

Also, this post is somewhat related.

I hope this helps $\ddot\smile$

Answer 4

I'm not a teacher/professor but I'd like to give my input as a person who hated math for a big part of his life, especially at high school.

I'd like first to explain why I hated math that much and for so long. At high school we were basically receiving some input and had to spit out some output. I'm referring to the countless resolution of derivatives, integrals well design to apply some seen methods. At the end I didn't need my brain at all. Fortunately, It didn't stop me to go in a technical field (chemical engineering) but during all my years at the university I never reconsidered my feeling for the mathematics and I simply considered it as a necessary evil, some tool only needed to do some other cool stuffs. It's only lately that I've started to enjoy math by among other things reading about its history and doing some math puzzles.

Now this is what I'd have loved my teacher at high school to tell us. I'd have loved her to be bluntly honest. That there are some problems with the program, that it won't be necessary useful. That what is called math at their level is actually a tiny bit of what it is reality. Someone mentioned Lockhart's Lament, why not make them read it.

Of course, just that is not enough. I think it should go along with that other thought: humans don't do something because it's useful but because they enjoy it. Unfortunately we also have to do things we didn't choose to and this is they case with their curriculum. But people that are happy in their life are those who manage to find some enjoyment even in the things they have to do (are forced?). So encourage them to find a way to enjoy it (some might enjoy it because it's useful - but not much). Each of them will find something different. After all if you tell them that math is useful in chemistry you'll get the attention of only those interested in chemistry.

Of course you should give them some pointers. For instance numerical analysis was a big revelation for me,especially the methods for solving differential equations. For years I wonder how scientists who really had to solve these kind of equations in real life were doing that. I don't say to explain it, just mention it (sometime, just knowing something exist is an eye-opener).

Others things that reconciled me with math:

• Coursera's course Introduction to mathematical thinking
• The puzzles of Martin Gardner
• Reading about the history of mathematics
• Some blogs written by fun mathematicians like this one
• Some resources explaining basic concept using intuition like this one
• Some Math/Computer programming challenges like the project Euler

tl;dr : I wouldn't try to show them it's useful, I'd show them it's in their best interesting to try to find enjoyment in it while being honest about the shortcoming of the educational system.

Answer 5

Edit (Feb 2016): Since the OP mentioned Hacker's Algebra opinion piece in the NYTimes, perhaps this is a good place to point out his most recent follow-up in a similar direction (I exclude here my own assessment of either): The Wrong Way to Teach Math by Andrew Hacker (Retrieved: 2016 Feb 28).

---

Edit (July 2015): In a similar vein, here is a link to Is Math Important? David Leonhardt of the New York Times acts as host for the panel discussion; to quote directly from this blog post:

From the mathematical world there are Steven Strogatz of Cornell University and Jordan Ellenberg of the University of Wisconsin, and from mathematics education research there is Jo Boaler of Stanford University. They are joined by David Coleman, President of the College Board, education writer Elizabeth Green, author of the recent book Building a Better Teacher, Pamela Fox, a computer scientist working with Khan Academy, and financier Steve Rattner.

(The two hyperlinks were added by me: one to an MESE answer about Ellenberg's book; the other to an MESE answer about Boaler's comments on timed tests and math anxiety.)

---

In providing a justification for learning mathematics, I would like to split this question into two pieces (even though there are certainly more) and comment briefly about one of them (even though the other may be closer to the intended question).

A first interpretation of the italicized text above: Why does a subject, mathematics, that covers so much "abstract" material, occupy such an important place in our schools?

This is the question that I believe is being asked, and it is the sort of consideration that underlies the linked piece on the necessity of algebra, from which I quote:

The toll mathematics takes begins early. To our nation’s shame, one in four ninth graders fail to finish high school. In South Carolina, 34 percent fell away in 2008-9, according to national data released last year; for Nevada, it was 45 percent. Most of the educators I’ve talked with cite algebra as the major academic reason.

You can find other comments in this direction in an earlier opinion piece in the New York Times, Garfunkel and Mumford's (2011) How to Fix Our Math Education. Again, quoting directly:

Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers. Science and math were originally discovered together, and they are best learned together now.

Traditionalists will object that the standard curriculum teaches valuable abstract reasoning, even if the specific skills acquired are not immediately useful in later life. A generation ago, traditionalists were also arguing that studying Latin, though it had no practical application, helped students develop unique linguistic skills. We believe that studying applied math, like learning living languages, provides both useable knowledge and abstract skills.

In math, what we need is “quantitative literacy,” the ability to make quantitative connections whenever life requires (as when we are confronted with conflicting medical test results but need to decide whether to undergo a further procedure) and “mathematical modeling,” the ability to move practically between everyday problems and mathematical formulations (as when we decide whether it is better to buy or lease a new car).

A second interpretation of providing a justification for learning mathematics: Why should we encourage students to study school mathematics now?

This is the question that I would like to respond to, briefly.

I do not disagree with studying mathematics for its aesthetic value; I do not disagree with studying mathematics for the opportunities it provides to express ourselves and be creative; I do not disagree that pure mathematics may turn out to have important applications. But I think the strongest argument right now for studying mathematics is its role as a societal gatekeeper (google scholar).

There are normative and utilitarian meta-questions about where mathematics' place should be in school and academic endeavors, but the current reality is that "learning mathematics" is essential to moving forward (or up) in the world; such a competence seems, to me, necessary but not sufficient for working towards a "successful" life.

At present, I have been teaching mathematics to elementary school teachers. Do I try to get them excited about mathematics? Yes. Do I try to get them to think about mathematics creatively? Yes. Do they sometimes latch on to applications of their own in our discussion of pure mathematical concepts? Yes: If only you could see the dawning of epiphanies (!) as many soon-to-be-married teachers in my Spring semester course began to brainstorm, collectively, about applications of LCMs and GCFs to the construction of flower arrangements and seating charts at upcoming weddings.

But I also realize that many of them are teaching students at high-needs schools, and that their students' futures (in our current set-up - speaking specifically about the United States) can be derailed by problems that start with an inability to factor quadratic expressions - or, in many cases, even earlier.

Answer 6

I'm no teacher, but my position is if you're only studying what has some useful application now, how are you ever going to discover the new applications that we don't know about? I think your Hardy example sums that up perfectly.

I know, that still probably somewhat misses the point of your question because it still implies that mathematics (or specific branches of it) are only worth learning for practical applications. I certainly don't think that's the case, but there are some people you are going to have a very difficult time convincing of that (much as there are some people that don't see the value in whatever human endeavor, from art to manned spaceflight).

I think at the very least you might be able to win them over somewhat with the prospect of discovery. The whole point of studying these things is because we don't know what they can be used for. And the only way to find out is to get them into the heads of as many people as possible so that when each of them goes out into the world, one of them stumbles onto something.

Answer 7

Quoting the question: "the study of one of the great intellectual and creative achievements of humankind"

The conventional high school course except for statistics and perhaps some aspects of geometry exists only for the purpose of preparing students for calculus. 99% of the students who take calculus will not use it after the final exam. Therefore the calculus course they take should be about "one of the great intellectual and creative achievements of humankind"; otherwise it's worthless.

Unfortunately, what is actually done is that students learn a bunch of algorithms for evaluating limits, derivatives, and integrals without actually understanding the subject. The only way to fix that is to drop the requirement that students must learn all the elementary ways of finding limits, derivatives, integrals, etc., to leave room to explain why the subject is one of the great intellectual, cultural, and aesthetic achievements.

Answer 8

You are not learning math.

You are learning the very useful, highly transferable skills of abstraction and problem solving. It's not that math can solve many problems, it's the skill you learn in order to do math are useful in so many other problems.

Answer 9

I am not a formal teacher or a mathematician, but a mechanical engineer who loves to learn and derives great satisfaction from mentoring other up-and-coming “S.T.E.M.” professionals/students. Due to my lack of educator credentials or particular knowledge on the subject, my response reflects personal views and experiences.

I was terribly bored by math during my primary and secondary education. I was much more prone to want to tinker with things to see how they worked. I used the knowledge that I gained to build and create things that I enjoyed or found useful. When the time came to select a major while filling out my university application I chose mechanical engineering because it would extend my understanding of the mechanical objects that had captured my fancy. I made this choice in spite of my mathematical insecurities because of my passion for the mechanical and my desire to understand it.

My first math and math based courses were extremely difficult for me, but I persevered and received good grades. However, my grades were more a result of my ability to regurgitate information effectively than a result of a sound understanding. Here and there a light would come on and things would make sense, but by far and large I was being led on a blindfolded path, copying the recipes that I was instructed to follow.

Through it all, I did feel very empowered as I learned that I could make, design, or mathematically model nearly anything that I wanted to if I could just find an equation in some textbook for the problem. This spurred me to develop a can-do attitude toward anything that I wanted to do. This new mental paradigm and tenacity was soon to be called to good use in awakening my understanding of what math really could be.

While studying heat transfer I was introduced to PDE’s. I had actually previously had a formal introduction in a class on differential equations (focusing of course on ODE’s), but the regurgitation education model had left me with virtually no recollection of what a PDE even was so it was like discovering it for the first time. I was intrigued, so I enrolled in a PDE’s math class during the last semester of my undergrad. My professor had a doctorate of art in mathematics, which contrasted greatly with my purely applied engineering background.

At first I scoffed at the professor’s insistence that math was a creative form, but as the class went on I was forced to dig up and actually learn a great deal of the topics that had been presented in earlier math classes. As I did, I began to really understand a few concepts and realize that they actually made sense in and of themselves. Math began to take on a meaning of its own beyond just its applications.

One of the biggest revelations was the ideas of vector spaces and that of variable transformations to map from one vector space to another. After asking my professor for help with just such a problem, I asked how he had known that that particular variable transformation would work. His answer changed the way that I have looked at math ever since. He said “I defined it that way because it was convenient”. The idea was so foreign to me that I had to think about it for some time before it really made sense. I had always thought that there was only one correct variable transformation, one correct proof of every mathematical fact, one best way to solve every problem. The idea that I could pick something, define it, and work out the implications on my own was amazing to me.

My interest in PDE’s led me to begin a study of them on my own. This study has led to a multitude of other topics that are interrelated with PDE’s including higher dimensional vector spaces and other wondrous ideas that lead the imagination to ask a lot of “what if” type questions. One of my most recent reads was a book on the history of mathematics. It was such an eye opener. Math and science progressed hand in hand in most eras (to me, those seemed to be the most fruitful). Many mathematicians were also classified as scientists or experts in other disciplines. Their processes for discovery varied, some liked rigorous proofs, others relied on inspiration/intuition and worked out the details later, some even published erroneous solutions to problems, notations changed and evolved, and creativity flourished. No longer was math a cold, exact, and deterministic subject. It had come to life for me.

I apologize for taking so long to get to my point, but I felt the background was necessary to justify my position since I do not know any educational theories. My point is that I think the best way to balance the applications and the art (the art part is a newly discovered part to me, but I am so glad that I have come to be able to view math in such a way) is to follow the historical development itself. Don’t ask students to solve totally stupid and uninteresting problems about Sally and Rob’s ages, travel distances, etc. Why not use the real questions of the giants that paved the way for us? There are a great deal of artistic and applied problems that spurred the development (and may I add understanding) of mankind. Why rob students of the richness of that history? I think their minds will tend to evolve in understanding over time much the way the ancients did. They can then see our wealth of knowledge as accessible to them. They can ponder on things and question about how the ancients determined the mass of the earth, the percent of gold in a crown, the value of pi or even the fact the ratio of the circumference of a circle to its diameter is constant.

In short, I think the best way to discover math is to relive the discovery of it with a little help from a skilled mentor so as to avoid the intellectual pitfalls of the ancients and of course so it will not take thousands of years to get an education. My opinion is that this would turn math from a dry subject into an interesting narration. It would also give a balance of art and application since neither deserves full credit for the present state of our knowledge.

What if a student will never use it again? My answer to that is that from an art and creativity perspective it will open their mind to consider all the possibilities to any given scenario they may meet and from an applied perspective that it will enhance their appreciation for the world around them just like a study of art an poetry turns a mundane artwork into something with meaning.

Answer 10

Sometimes learning in general is not about the actual usefulness of the subject matter in question, but how it changes and expands your thinking. Inspiration can also stem from many different places.

The main issue is that in many High school classes, students are taught to look at a problem and work it like a machine. For example, taking the quadratic equation and putting in all the inputs. I believe the best way to teach is be a role model and inspire an interest in learning. Unless there is an interest, teaching is an uphill battle against all the distractions that occur. A great teacher can often teach students ideas and concepts that can stay with them a lifetime.

Most people forget things quite quickly when they aren't put to use. However if they can remember what the math was about, with some time and google, its pretty easy to solve any high school level problem. For people that really don't care about math, I would say its not that hard to live in society without it -- however scary that thought might be.

Even if you are interested in a certain subject, perhaps the first thing to consider is if you can get your students interested as well (as well as if they can actually understand it). For the gifted teachers, this eventually becomes less of an obstacle as the level of their own interest, charisma, or teaching method is enough to spark an interest in any subject matter.

Answer 11

My simplest answer to any student asking "Why study math?" is this

Math gives you power to get what you want.

This is not the most "politically correct" way of "selling math", but it's both true and often persuasive to students and adults who want more power (which ends up being a majority of students and adults).

The real question being asked here is this: "How do we sell math?" We are not just interested in selling math to students, but also in selling it to people in general. Some parts of math are easier to sell because the sales pitch is simple: you need to know how money works so that you can buy and sell things without feeling like you've made a "bad deal" or feeling like you've been "cheated". Most of the economic reasons for learning math are based on our inherent fear of loosing economic security or our inherent want for economic power.

Outside of this economic sphere there are deep social reasons for studying mathematics. Perhaps the most valuable is that of forethought. Mathematics, unlike history, demonstrates that by systematically applying a basic problem solving process like Polya's you can put together facts which show clearly and exactly the future consequences of present actions.

So far I've discussed the economic motivation, and the general problem solving motivation, now I move into the motivation from love. When someone asks a question like "Why study math?" we might hear, in the back of our curious mind, another question "Why do anything?" and this is sometimes what a pesky philosopher is apt to ask. I do not know how to answer the question "Why study x?" or "Why do x instead of y?", but I do know of at least one statement which provides a popularly accepted way of approaching these general problems:

The good life is one inspired by love and guided by knowledge. - Bertrand Russell

From this simple statement I say that we study math because some of us are inspired to love it: some people study math because they love it. Ask them why they love it and I'm sure they can tell you all sorts of silly facts that really get their blood pumping, but the truth is that they do just love it. Why do we love anything? Why do we do anything for love? It seems that there is often no use in questioning the love that inspires us, but only in using the knowledge that we have to guide that inspiration towards the good life.

Answer 12

Mathematics is often thought of as a bunch of clever people doing weird things with symbols for reasons that are difficult to relate to reality. Whilst this is not entirely false, it is for the most part (almost humorously so in my opinion), the antithesis of the actual role of mathematics in today's world.

I challenge anyone to seriously tackle the world's major problems without first being an expert in mathematics.

Our planet is on the verge of human extinction due to climate change. There will be wars over water. There is currently a global pandemic. There are billions of people living in poverty. There is famine, disease, etc...

Has any of the progress made (towards solving these problems) in the past been done without rigorous problem-solving? Without collecting and analysing data (statistics)?

Ah, rigorous problem solving. This is, in my opinion, the most accurate definition of mathematics.

Does anyone seriously think we can tackle any of these problems without greatly advancing technology for example machine learning or computer simulation, in which being good at mathematics is a prerequisite to even understanding to what is going on, let alone being able to work towards solutions?

Why would / how can you not learn mathematics?

Even if you don't care about the world's serious problems, but just want to further your own career, you may want to advertise your product or service to make it arrive at the top of a google search. In which case, it may help to understand how google's pagerank algorithm works and how you can exploit it (although there are already people who are specialist in this type of work- but clearly they need to be good at maths.)

Then there are Scientists who must use maths to both gather and analyse data and also to help solve problems e.g. Mathematical modelling.

What about trying to come up to solutions to economic problems. If you take a look at economics stackexchange. Take a look at some of the questions on there and notice the high proportion of questions with some reasonably heavy mathematical machinery in them.

I could easily give another thousand examples, but you can find lists of the applications of mathematics if you do a quick google search.

Again, in response to the original question of, "Why do I need to learn mathematics?". I don't see how you can do without mathematics.

Not to the extent of air and water, I'm not delusional. For example, you can avoid mathematics and pursue a career of art or music or entertainment or "social media influencer" and whilst I don't think these people are entirely useless to society, the fact remains that they are not the problem-solvers.

Answer 13

I wish I could find an excerpt to link you to, but the book Made to Stick had a great reason as one of its examples. You learn math in order to make you better at thinking. It's like athletes lifting weights; they don't lift barbells because they are going to need to

Answer 14

I don't know that there is such a line to walk. Justification usually requires some specific context, and an answer that is out of context assumed by the asker usually does not convince the asker.

I would not dare to convince someone that study in mathematics should have higher priority than study in physics, or biochemistry, or psychology, or architecture, or art. Nor would I ask a funding organization that mathematics education should deserve a greater share of funds than educational efforts in other disciplines.

I would note that if they attempted a variety of studies without mathematics or a mathematical viewpoint, they would be missing out on an opportunity to help better understand much of their world. I would point out that trying to understand music without ratios, to understand realism in painting without some knowledge of geometry, to understand social and economic forces without some notion of function and rate of change, to arrange and prepare parties and supplies without basic notions of recipe/algorithm and algebra, much of it would have to be invented, arranged, and taught as some class called "Ratiogeomfunctionrecipalgebrology". Fortunately, older and wiser heads already chose to group such things under "Mathematics".

Gerhard "Head Getting Older At Least" Paseman, 2015.05.11

Answer 15

Maybe teachers should have some of these posters (available free from the AMS) hanging around the place.