How Much Math Do We Really Need?

That title was lifted from a Washington Post essay by G.V. Ramanathan, a professor emeritus of mathematics, statistics and computer science at the University of Illinois at Chicago.

For my Twitter- and text-obsessed readers, I’ll answer the question immediately: for me, a lot. For ordinary citizens, not that much. Ramanathan’s take is not much different:

Unfortunately, the marketing of math has become similar to the marketing of creams to whiten teeth, gels to grow hair and regimens to build a beautiful body.

There are three steps to this kind of aggressive marketing. The first is to convince people that white teeth, a full head of hair and a sculpted physique are essential to a good life. The second is to embarrass those who do not possess them. The third is to make people think that, since a good life is their right, they must buy these products.

So it is with math education. A lot of effort and money has been spent to make mathematics seem essential to everybody’s daily life. There are even calculus textbooks showing how to calculate — I am not making this up and in fact I taught from such a book — the rate at which the fluid level in a martini glass will go down, assuming, of course, that one sips differentiably. Elementary math books have to be stuffed with such contrived applications; otherwise they won’t be published.

Many of the citizens I meet, after hearing that I’m teaching calculus, all tell me something along the lines of, “Oh, I wish I knew calculus. I even bought a book that explained it, but I haven’t been able to read it.” It’s even worse when people hear that I’m a statistician: that’s when they notice that their glasses are empty and need immediate refilling.

I can attest to use of asinine examples in math books, too. The textbook I use for my Algebra Sans Algebra class assumes the reader is a child. There are pictures galore: of candy bars, popsicles, soda pop, of skiing, sledding, snowboarding, or dollars bills, ten dollar bills, hundred dollar bills, all tied to goofy “mathematical” examples. The publisher must surmise that students will not assimilate the math unless it is turned into a Saturday morning cartoon. The writer of that book, incidentally, uses more exclamation marks than even I do. Sorry: that should read than even I do!

Ramanathan continues:

You can see attempts at embarrassing the public in popular books written by mathematicians bemoaning the innumeracy of common folk and how it is supposed to be costing billions; books about how mathematicians have a more clever way of reading the newspaper than the masses; and studies purportedly showing how much dumber our kids are than those in Europe and Asia.

Buried in there is a good dig at There-Is-No-God bandwagon book-writer mathematician and newspaper reader John Allen Paulos (I liked Paulos’s first book; but then he turned literary).

Ramanathan asks the important question: “how effective are [the] educational creams and gels?” More government money taken from citizens and applied liberally to the educational bureaucracy has not made the situation better. In fact, statistically speaking, there is a robust negative correlation with money and mathematical education. This implies that we should spend less to learn more!

In a true college education, all should know at least calculus, and then perhaps some analysis or linear algebra, perhaps number theory, certainly a smattering of probability. But for the ordinary citizen, addition, multiplication, and some geometry are all they need and—be honest—all they can master.

So what if most people can’t integrate? Not everybody can master dunking a basketball through a crowd of 250-pound, elbow-throwing, impolite sweaty behemoths either. Nor can everybody read through a ream of 350-years of common law to distill a point of legal philosophy. And so on.

There are plenty of areas of expertise that are unattainable for most of us. Having a firm grip on the basics for these areas is enough. There will always be enough people who have the smarts and the desire to go into math: we needn’t browbeat the rest of their society for their inability or lack of desire. Being good at math is no different than being tall.

As Ramanathan says, “Those who do love math and science have been doing very well. Our graduate schools are the best in the world [and] produced about 140 Nobel laureates since 1983…” I don’t love the idea of measuring success by the number of trophies given by an organization that called Yasser Arafat a peacemaker, but we get the idea. The States are doing fine at the top.

The real problem are the number of people we unleash on the world thinking they know math because they have a piece of paper which says they do.

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Thanks to reader James Erlandson for bringing this article to my attention.

34 Comments

  1. Calculus teacher in high school got our attention the first day by giving us a problem unsolvable using Algebra and letting us contend with it for 20 minutes – most of us could see we couldn’t do it on sight – then showing how you could get there with calculus.

    The joy of being able to solve the seemingly impossible carried us through the first semester without any “cute” problems although I do like the martini problem – look into adsorption rates of vermouth by olives or onions, or a twist, evaporation rate of the alcohol with differing surface area as drink is consumed – a lot to work with there.

    And then the experimental confirmation.

  2. RE: Yasser Arafat.
    You made the mistake of confusing the three Academies in Sweden which confer the real Nobel Prizes with the subset of the Norwegian Parliament which awards the fake one.
    It is a very common mistake, though.

  3. The only math that’s important is that which increases my income and reverses the sign on my taxes. Despite the WaPo slogan (“If you don’t get it; you don’t get it”), most of its content is of questionable utility. Even this essay was likely filler created by an advert drop.

    I’ve been wondering what happened to all the exclamation marks.

  4. @DAV

    Reverses the sign on your taxes? You mean you qualify for the EITC? Or have you fallen for the “tax refund” nonsense, that really just means you’ve overpaid your taxes?

    If it’s the former, you need more maths so you can earn a better income. If it’s the latter, you could use a good logic course.

  5. Always liked the ladder sliding off the wall problem myself, although it contains as much physics as math. How about the ladder maneuvering around the hallway corridor? Math examples should not necessarily be practical, but focus on the math which they illustrate. This is why Lewis Carroll used nonsense syllogisms; to focus on the logic! Thus the educrats have it backwards, as usual. The applications come later.

  6. Amen to this. It applies not just at the college level, though – because it’s perfectly clear to every K-12 teacher that many students are either not able or not willing to learn all that is offered in the standard K-12 curriculum. And they are certainly not all able or willing to learn it well enough to make use of it down the road.

    So why do we insist that they all spend large chunks of their day attempting to learn higher math, or advanced composition, when they are still struggling with carrying the one and basic literacy? It makes no sense. It’s a pure result of the rhetorically pleasing but false idea that everyone can reach high standards in all academic areas if only we (a) keep them coming to school, and (b) do “the things” that makes them motivated to learn and makes the lesson make sense, which are presumed to be always known (it’s just that the stupid teachers won’t implement what the academics say will work this week!)

    Which I guess is why the college textbooks tie calculus to alcohol consumption, as alcohol consumption is widely considered to be a prime motivator of college students. Psychologically, I’m not sure this is a valid approach – seems to me you’d have to provide the alcohol as a reward of some kind for mastering the material – but perhaps this is the best they can do from a practical perspective.

    Still, we keep wanting to “bundle” capabilities in the form of a diploma or degree, which requires certain standard certifications of learning. And we don’t want people to fail to get the degree they paid for or the diploma that is “necessary for success”. Never mind that we can’t help but cheapen the signal sent by the diploma or degree as a result of lowering standards.

  7. But for the ordinary citizen, addition, multiplication, and some geometry are all they need …

    And most who have received a genuine High School diploma have mastered them. What is too often missing is the ability to apply those skills to solve real-world problems.

  8. Yes, I agree. I’ve only used Calculus once in “real life,” i.e. outside my job. The USPS limits parcel post to parcels with length+girth less than or equal to 130 inches. What is the maximum volume of a parcel you can ship by parcel post? I would say that Algebra has more general application. For example, spreadsheet math uses some algebra, e.g. A23=(A12+B24)/C9. So I’d add a little algebra in with addition, multiplication and some geometry.

    This line of thought is a slippery slope though. If we start basing education on what will be “needed” beyond school, you’d gut a lot of high school studies. Chemistry? Completely useless. Foreign language? Almost surely useless unless you learn Spanish. History? Much of it is useless. Etc.

  9. The problem with what to learn is that you don’t find out what you should have learned until later on. The line that the child is father to the man seems borne out by experience. Some of us are the children of idiots.
    I chose a career based on its imagined reduced dependency on the things I found hard to learn when I was in high school – a big mistake as a I later discovered when job circumstances required me to quickly come up to speed on thermodynamics, on my own. My advice if you’re confronted by anything like this is to get three different textbooks and hopefully what wasn’t clear in one will be clear in one of the others.

  10. To be fair most of America’s math gap comes from the way educational statistics are gathered. Like with infant mortality, American education stats include all the kids, unlike abroad where the pool is usually the students who scored above a certain level on a test, hence their results will always be nominally higher.

    Then again, every time America tries to implement that kind of testing the liberals howl to the moon and scream RAAAAAACISM and so forth…

  11. “There are pictures galore: of candy bars, popsicles, soda pop, of skiing, sledding, snowboarding, or dollars bills, ten dollar bills, hundred dollar bills, all tied to goofy “mathematical” examples. :

    My son’s high school science textbook was a picture book. I think it devoted more area to pictures than text and the organization was terrible. Next thing you know they will be making them coloring books.

  12. Why fight it? Just go with the flow. For instance, my next money-making scheme is to open a school where I teach the students how to become tall.

  13. This post reminded me that every Stats department in the planet has repeated Hal Varian’s ‘I keep saying that the sexy job in the next 10 years will be statisticians’. Now the departments are getting all these people that 1. have no real interest in stats, 2. want to be millionaires before they are 30 and 3. would normally struggle with stats101 using excel.

    Learn linear regression and you’ll work for Google!!!!!

  14. MikeD.

    “Why fight it? Just go with the flow. For instance, my next money-making scheme is to open a school where I teach the students how to become tall.”

    If I can judge from the spam I get, this dimension isn’t the one of greatest current interest.

  15. What I think we should be considering what kind of mathematics (sorry, calling it math just grates) most people need for everyday life and make sure that is what is taught in school.

    In addition to William’s suggestion of addition, multiplication and some geometry (I’d personally question the geometry), a basic understanding of percentages as they apply to interest rates because pretty much everyone will get a loan of one kind or another at some stage in their life. I would also put in a plug for some basic probability understanding as well – since we are bombarded with this in the media all the time. How many people really think that one in a million actually means around 240 people in the US?

    I have to use a lot more statistics now than I ever thought I would, but that is a specialization that I am much happier to learn for myself and – having had a university education – I have the confidence that I can learn it for myself.

    That is what I feel is now lacking in college-level students – the confidence that they can learn something (for themselves) as opposed to being told what they need. This is at the bottom of William’s problems with his current class(es) and I for one don’t envy him his job!

  16. But for the ordinary citizen, addition, multiplication, and some geometry are all they need and—be honest—all they can master.

    In other words, all they need is a calculator. ^_^

    The question is: what does it mean to know mathematics?

    Sadly, “knowing math” is never in the list of tips for happiness. Perhaps it’s time to add it into the list. It’s true that knowing math will give you more career options, which probably would make you a bit happier.

  17. Our education system fails us in many ways not the least of which is requiring subjects that are useless to most students. They do this for the obvious reason that educators are in charge and to them their particular field is the most important. Every high scholl student must study a language but 10 years after graduation 99% of them couldn’t speak that language well enough to order a meal. So much time is wasted in or K-12 system and so little of real value comes from it. How many graduating high school seniors can balance a checkbook or prepare a household budget.
    Robert Heinlein said ” human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly.” In general a high school graduate today has no knowledge of these things but does know how to make a baby.
    I have a stepson who has a natural mechanical ability but cannot master algebra. Because of this he cannot enroll at our local community college to become a mechanic because algebra is required. Why? I could see why it might be encouraged but why require it? The obvious reason is one of the people who sets the curriculum is a math teacher.
    Our schools are no longer designed to prepare our youth for their future life they are designed to provide a living for teachers.

  18. Very true, I’d say that’s also applies for most science classes as well. Do we really need our lawyers or mechanics or etc etc to know biology or physics? At most I say only require an intro class just so students can get a taste of math, science, foreign languages, etc so if they find they enjoy it, the student can continue on in those classes in the future.

    No, the only classes I would require are basic math, some stats & economics (just because it seems like learning the basics would cure a lot of people of liberalism), history of the country they are in, and these 10 classes.

  19. Briggs In a true college education, all should know at least calculus, and then perhaps some analysis or linear algebra, perhaps number theory, certainly a smattering of probability. But for the ordinary citizen, addition, multiplication, and some geometry are all they need and—be honest—all they can master.

    There are plenty of areas of expertise that are unattainable for most of us. Having a firm grip on the basics for these areas is enough.

    You can’t have a firm grip on anything unless you’ve studied things beyond it. That’s because education is inherently inefficient; you retain less than what was taught to you. However, there’s no boundary between what you retain and what you don’t. Instead, your grip is weaker as the thing to be gripped is further from the core of what you’ve retained.

    This is a good match to what you need to know. Your need to know any given thing is smaller as the thing is further from the core of what you use in your “practical” life, but it does exist. I think it exists in many different forms and ways, but they’re hard to identify and describe, so I resort to an obvious example:

    Shouldn’t a person have any idea of whether it makes sense to say that the top (say) 1 percent of earners should pay the entire income tax? Sure, he should, but some don’t, and not all of them are morons – not by a long shot. The reason they have no idea is that they don’t know anything about statistical distributions. In some great majority of cases, this ignorance cannot be eliminated by a 10-minute explanation. (I’ve tried, and it hasn’t worked.) The person needs some prior “steeping” in the concept, and this could only be the result of exposing him to more than he “actually needs” to criticize that taxation idea. If he had that, then he will have retained what he “actually needs.”

    I don’t mean to confine this to statistics, or to mathematics, or even to what is needed for intelligent voting, but it would take too much effort to extend it to other things. I can only say that a person who only knows what he “actually needs” to know must be an idiot.

    The idea of only teaching people what they “actually need” to know is pernicious.

  20. Nate Winchester:

    “Do we really need our lawyers or mechanics or etc etc to know biology or physics?”

    Are you sure about the lawyers? Isn’t that the population that congress comes from? I’d like them to have more math, statistics, physics, and bio. It’s bad enough to be ignorant but to be ruled by the ignorant?

  21. @View from the Solent: you are wrong. The Nobel Peace Prize may be fatuous but it’s genuine. (Ditto the Literature Prize, in my opinion.) It’s the Economics Prize that’s pretendy. It ain’t from the old boy’s will but was faked up by a bank decades later.

  22. Smoking Frog (incidentally, given your choice of drug, you should read today’s story),

    You say: “You can’t have a firm grip on anything unless you’ve studied things beyond it.”

    My dear sir, just think about that, will you?

  23. Smoking Frog,

    I agree with you in the sense that we can gain a broader and deeper understanding of a subject by learning more about it. (Of course, the more we learn, the more we realize that how little we know.) For example, what is an area? Calculus tells us that it’s more than the formulas we learn in algebra.

    Anyway, just being curious, is there a correct answer to the question? Yes or No? ^_^

    …whether it makes sense to say that the top (say) 1 percent of earners should pay the entire income tax?

  24. Are you sure about the lawyers? Isn’t that the population that congress comes from? I’d like them to have more math, statistics, physics, and bio. It’s bad enough to be ignorant but to be ruled by the ignorant?

    I think that proves my point. XD

    If anything, we need less lawyers and more people in congress who are statisticians or physicists or biologists or mechanics or… well random citizens. You get too many in there from a single class/profession/strata/whatever and you’re going to get what we have now.

  25. Nate Winchester,
    I couldn’t agree more. we need more engineers and fewer lawyers in congress, but what do you think the chances of that are? A term in congress is a career waypoint for the legal profession, and seemingly not for anyone else.

    It’s interesting that the group running China consists of 6 or 7 engineers and one PhD in Psychology (mix and credentials of the last may be wrong). Our current president who seems remarkably bright is an attorney and clewless (not having any grip at all) on scientific and technical issues at least to judge from his pronouncements, CO2 the poison gas etc.

    How do you recommend we rebalance our congress?

  26. Maybe I am misunderstanding, but isnt the whole point of the high school education to give students enough background so that they can succeed in college? If schools just taught people the minimum in an area to get by, how will students manage if they go on to college and realize that they now have to spend extra time and resources to get the pre-requisites?

    If we just taught students “enough”, then we would have to have multiple tracts of studies in schools – one for people who want to be an artist (and thus doesnt need to know any science/math), one for people who want to be scientists, one for engineers and so on?

    I agree that dumbing down math textbooks with coloring pictures is not correct, but maybe they are just catering to the perceived understanding level of students – and to keep them interested in the subject? Note, I dont agree with this approach, when I was in high school back in India, I had to wade through textbooks with theorems and problems etc. and it was painful sometimes. Maybe this is symptomatic of the culture here, where people want everything to be spoonfed to them, and dont expect to make an effort to understand the subject. Probably the educators who are approving such textbooks are contributing to the problem.

  27. I am not opposed to offerring courses like Spanish or calculus to students, which is what you seem to imply. What I am opposed to is mandating that unless you take Spanish or calculus and pass them with a C or better you cannot take your major classes and graduate. Should an Einstein have been denied the chance to go to school because he didn’t study poetry or Classical music??? Then why should a mechanic not be allowed to go to school because he cannot pass algebra? Should everyone who cannot pass algebra be denied future work or only allowed to work in fast food? What I am talking about is the requirement that a specific subject be mastered. When you consider the major failure that is our public school system were less then 75% of students graduate and less then 50% of those who graduate have mastered any of the subject they took why would you want to prevent someone from attaining mastery of a subject they were good at simply because they could not attain mastery of a subject YOU are good at?

  28. Briggs Smoking Frog (incidentally, given your choice of drug, you should read today’s story),

    As I’ve told you, Smoking Frog was a Mayan warlord of the 4th century. His name has nothing to do with smoking cigarettes or the like, and I’ve never thought of it that way. I think of him as smoking the way a smoldering thing smokes, without thinking about what exactly the metaphor might represent. As I’ve told you, his real name was “Born of Fire.” “Smoking Frog” is from some combination of joking by scholars and misconceiving by them in the past, when they didn’t understand the hieroglyphics very well. I wish his name were “Smoking Frog,” because “smoking” and “frog” are incongruous, so it would make the Mayans more interesting.

    You say: “You can’t have a firm grip on anything unless you’ve studied things beyond it.”

    My dear sir, just think about that, will you?

    I think it’s true.

    JH I agree with you in the sense that we can gain a broader and deeper understanding of a subject by learning more about it. (Of course, the more we learn, the more we realize that how little we know.) For example, what is an area? Calculus tells us that it’s more than the formulas we learn in algebra.

    That’s the sense in which I meant it, but I’m not suggesting that people who would otherwise only learn arithmetic should be made to plow through all the way to calculus.

    <i?Anyway, just being curious, is there a correct answer to the question? Yes or No? ^_^

    …whether it makes sense to say that the top (say) 1 percent of earners should pay the entire income tax?

    You only want yes or no? Ah, but to which question? Whether there’s a correct answer, or whether the statement makes sense? I say yes to the 1st, no to the 2nd. If you want more, I wish you’d said so, since these comments threads go obsolete in a hurry.

  29. I always thought of maths and other ‘technical’ subjects as meant to encourage thinking and reasoning, help you develop a thought process that gets you to a conclusion that you can rationally explain to others.

    As for the cartoons and other visuals in textbooks, I am currently reading a book called ‘Head First Design Patterns’ which was recommended as a good source of information and which is advertised as written in ‘a visually rich format designed for the way your brain works. Using the latest research in neurobiology, cognitive science, and learning theory, Head First Design Patterns will load patterns into your brain in a way that sticks.’ It has drawings, pictures and speech bubbles next to all sorts of characters.

  30. Having taught mathematics in middle school before pursuing advanced degrees in physics and materials science, I taught in college for 27 years before retiring to a normal existence at a ranch in New Mexico. Dr. Ramanathan’s thesis is that the education proponents are pushing the idea that mathematics is essential to life. If the proponents were right, a lot of people including teachers, college and university graduates would be endangered species.
    I have also seen the growth of coffee table picture books purporting to make mathematics interesting and therefore motivate the student to want to learn a lesson. I had the pleasure of teaching mathematics in middle school when the Cleveland Ohio schools systems decided to implement the “new math” which apparently just confused teachers, parents, and students.
    What I see is a population of people who may or may not have studied mathematics in college who are innumerate. By innumerate I mean, they do have any inkling about the information contained in a number. Like the aborigines in Australia who had only three words for a numbers, one, two, three, and many. The meaning of numbers has been obliterated by the calculator. I know this to be true because when my students in college would obtain a number that was the answer to a calculation, they had no idea whether it was the correct number. They believed that if they pushed the right buttons, they would get the correct numeric answer. What I am saying is that people today do not understand order of magnitude in a number. Just look at the growth of the national debt over the last two years. People in California do not see a problem with the state owing 160 billion dollars because they cannot fathom the implications of the size of the number.
    Some years later I entered in the ranks of academe in a CSU School of Engineering. In the time period between when I started teaching in a secondary school until the time I taught in the university mathematics education had changed dramatically. The students had access to powerful computers and had calculators which I would have died for in graduate school. The calculator made it possible are our intricate mathematical operations which I had learned to do the hard way such as square roots, exponents, trigonometric functions, logarithms. While the advent of the hand held computer may have been a blessing for people that hate mathematics, it has also shaped their minds in a negative way. The calculator has made mental arithmetic obsolete. In fact operations, like division of large numbers and using exponents in calculations have lost their meaning to an identifiable button. The order of mathematical operations is unimportant. We have made in possible for someone to accurately calculate the size of an atom of copper from the crystal structure to 10 decimal places with an exponent of a power of 10 and the person making the calculation would not know if the answer was close to being correct.
    I have heard teachers of mathematics in public schools say there are no reasons to make students memorize addition, subtraction, multiplication, and division tables; there is no need to make students do mental arithmetic. Just teach them to push the right buttons. If they sometimes get the correct answer, well give them credit.
    Apparently Dr. Ramanathan is correct. Mathematical knowledge is not essential for a good life in America. Information in the form of numbers just gets in the way.

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