Monday, July 14, 2014

On the Importance of Mathematics

I came to mathematics relatively late in life, as such things go.  Generally, when someone is passionate about something, they get “bit by the bug” early.  In that, I’m rather unconventional in that many of my passions came to me much later than they did to my colleagues.  I’m a magician who, though I did have a magic kit as a child, never became serious about the art until I began to realize its deep connections with psychology.  While I was always interested in science, I didn’t actually choose to go to university to pursue it until much later than most people make those decisions (indeed, I didn’t even go to university at all right out of high school, instead choosing to take several years off to pursue other things).  And when it comes to mathematics, most people likewise get bit by the bug early.  They learn a little trick for computing some arithmetic operation more quickly, and they develop a lifelong passion.  Not so for me.  I left high school barely knowing any algebra at all.  But now I’m pursuing a degree in mathematics.

Why do I point this out?  It’s to set the stage for a discussion that, while nothing new, is perhaps more relevant than ever.  That’s because what we’re talking about today is math education.  What I’m talking about today is not about how to improve math education, though I will touch on some of those issues toward the end.  It’s simply a defense of math education as something worth the effort to improve.  I’m afraid this is the discussion we need to have first, because it has become common for people not only to unfairly dislike mathematics, but to actually believe they have no use for it.

Every mathematics teacher in the world is familiar with the dreaded question: “When am I ever going to use this?”  This is annoying enough, I’m sure, because it means the students are questioning the very validity of the subject the teacher is spending her time and energy trying to bring to their attention.  But while it may be annoying coming from inexperienced students, it’s hard to blame those students because it is indicative of a much larger societal devaluation of mathematics, particularly in the United States (though I’m sure such attitudes are common to varying degrees throughout much of the world).

Evidence of this trend against mathematics can be found in that great new repository of society’s attitudes: the Facebook meme.  I’ve seen all of these come across my newsfeed at one time or another, and have found independent sources of the images so I can share them here.

Here is an entire blog post devoted to the idea that as an adult, its author has never used long division, fractions, or algebra.  The page is broken up with embedded images, many of which I’ve seen before, like this one:

Similar sentiments are commonplace.  Consider this one from an author who claims that “As a marketer and communicator, I’ve never had much use for sophisticated math,” but that he found value in taking algebra only in that it taught him the “life lesson” of “being forced to do something uninteresting simply because someone in authority told me I had to do it.” (Nevermind the fact that algebra hardly qualifies as “sophisticated math,” as evidenced by the fact that universities consider everything up to and including Calculus to be lower-level mathematics courses.)

There are other examples:

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Let me make something absolutely clear at this point.  It’s probably true that everyone is good at something, and everyone is bad at something.  This is not about some people being better at mathematics than other people.  Indeed, it’s not at all necessary that everyone should be a mathematician.  But it’s curious, isn’t it, that mathematics is (as far as I’m aware) the only field that is so commonly hated that people actually take pride in their ignorance of it.  No one would take such pride in illiteracy as they do in innumeracy.

There are many things of which I am ignorant.  For instance, I speak only one language.  Though I hope soon to rectify this situation, it is the truth at the moment, but you don’t see me going around pridefully remarking that I have never needed to know Spanish or French.  I also don’t know how to knit, but I don’t spend any time trying to convince people that being a non-knitter is such a good thing.  Only mathematics is widely regarded as such a hated subject in school and such a difficult field of study that people actually make snide remarks about their inability to perform mathematical operations, while trying to convince people that their ignorance is justified, because the entire field is useless (unless you happen to be a scientist or engineer--this being the limited concession offered by most of the anti-mathematical crowd).

Now that I’ve set the stage by showing you the attitude mathematics educators are up against, I want to spend some time talking about why these people are wrong, and perhaps to help you to understand why mathematics is not only useful (yes, even to you), but fun, rewarding, and profitable.

Let’s begin by looking at why it’s patently false that all of these people don’t use mathematics in their day to day lives.  We should first clarify that I am, in fact, talking about mathematics as distinguished from arithmetic (a small subset of mathematics).  For the duration of this paper, arithmetic refers to basic operations: addition, subtraction, multiplication, division, exponents, roots.  While it is absolutely important as part of a complete education that someone should learn how these arithmetic operations work and how to do them by hand (yes, that does include long division), this is, I would argue, the least important application of any sort of mathematics in daily life.  It’s also the only application most people think of, which is why we hear arguments like “I don’t need to learn math because I can balance my checkbook with a calculator.”  And that’s very true.  Because calculators are now ubiquitous, it’s significantly less important to be adept at arithmetic than it once was.  Arithmetic education’s importance these days has more to do with understanding how the operations work and knowing when to apply them than being able to recite multiplication tables from memory.  Mathematics, however, which we can broadly (and not quite accurately, but it’s good enough for our purposes here) describe as anything you learn in a math course beginning with algebra, is another matter entirely.  This is the sort of mathematics that people think they do not use in day to day life, and yet it is precisely the sort of thinking we all must do every day.

To be sure, most people probably never see a problem like this: 3x^2+2x-3=0 (the solutions, by the way, are x=-1 and x=2/3).  That’s very true, and that’s what people are thinking of when they say they never use algebra.  But that isn’t all algebra is.  It is algebra, yes, but it’s the very formalized algebra you learn in school.  It’s important to learn how to do algebra formally because when you’ve developed that knowledge and ability, you’re able to incorporate algebraic concepts into your life.

Here’s a very brief (and certainly partial) list of some day-to-day applications in which one must use mathematics:

Cooking
Mortgages
Accountancy
Tax returns
Calculating gratuities
Managing your car (fuel economy, distance traveled, etc.)
Making purchases

Depending on one’s profession, there are plenty of others, ranging from the very basic to the very advanced.

Another application recently came up when a family member was shopping for a television.  They knew the diagonal measurement and aspect ratio of the television but not its vertical and horizontal dimensions and needed to determine whether it would fit where they wanted to put it before making a purchase.  It’s a matter of simple geometry or trigonometry (trig makes it easier if you happen to know trig, but the Pythagorean Theorem and a bit of algebra will do the job, too) to figure that problem out.  I managed to provide the correct answer in a couple of minutes, ensuring that the purchase would work out for this person.

Mathematics is everywhere, and everyone can use it in direct applications.

But that’s not why it’s important.  That’s just a quick couple of examples to prove the lie that people don’t regularly use algebra in day to day life.  The real importance of mathematics is that it’s a formalized way of analyzing information about the world around us.  As in my example with the television, we can use numbers to solve problems.  Mathematics is abstract, but its value is that we use that abstraction to tell us something about the concrete.

Algebra is an Arabic word which, roughly translated, means “to reassemble from broken parts.”  Algebra is about manipulating variables to learn something about some unknown.  It’s not necessarily about “solving for x,” though that’s an easy symbolic way to solve problems.  Consider a simple example.

Story problem: Jim and Jon have five apples.  Jon has three.  How many does Jim have?

Algebra: 3+x=5

In either case, we use some basic algebra, move a couple of numbers around, and we can determine that x=2, so Jim has two apples.

If you’ve ever done a problem like that (and yes you have--we all do problems like that every day), you’ve done algebra.  You may not have written it symbolically, and that’s fine.  You’ve still done the math.  Of course, the value of doing things symbolically is that problems become very complex, even in “real world” examples, so it’s useful to understand a shorthand so that you can write everything out and make sure you solve it correctly.  Our ancestors did not have such systems, and their problem-solving abilities were limited by it.  We do have such systems, so it’s pointless to ignore the tools we have.  Solving the problem symbolically is certainly no more difficult than doing it using a more “intuitive” method, as anyone would do in one’s head.  It’s exactly the same operation!  The symbolic method just gives us a tool to solve more complicated problems than those we can do mentally.

Mathematics is also not just a set of operations.  There are plenty of people who will seldom (if ever) need to solve a quadratic equation.  Does that mean it has no utility for those people? Certainly not!  Mathematics is a method of formalized problem solving.  Learning mathematics, even if you never use it (which you do, though I will actually argue here that the direct applications are less important than the mental training), trains your mind to be able to tackle more and more difficult problems.  Even problems that have nothing to do with mathematics directly can benefit from mathematical thinking.  For instance, when I play chess, I tend not to mathematically analyze each move in the game (though I would be remiss if I didn’t point out that I do calculate some types of positions by “brute force” mathematical operations).  I’m not counting pawns and performing complex operations in order to determine the best move.  But I am problem-solving, and those analytical skills are not something we’re born with.  They are developed over a lifetime of education, and mathematics is the discipline which most directly builds our analytical abilities.

Suggesting that mathematics is an unimportant part of a standard education is clearly wrong, and for similar reasons to the suggestion that art is an unimportant part of a standard education.  I would argue that mathematics is of higher value in the limited sense that it is of greater utility in every (and yes, I do mean EVERY) profession, but let’s not worry about ranking.  Let’s just consider that the very people who seem to take pride in their failure to understand mathematics are often the first in line to defend the teaching of art and music in the public schools.  They are right to do so, but if they object to mathematics because of a misguided view that they never use it, I think it’s a fair question to ask them when was the last time they used their knowledge of Leonardo da Vinci in their day to day life?  I would guess that they probably have never used that knowledge.  And yet, no one would dare to suggest that a study of Leonardo is anything less than essential for a complete education.

For the more economically-minded readers, it’s also worth pointing out that mathematics is a great opportunity for profit. The most obvious way is simply that those with a degree in mathematics are among the most employed (and highest paid) of college graduates.  A teacher once told me that a greater proportion of math majors are admitted to medical school, for instance, than pre-med majors.  The same professor also told me of an acquaintance he once knew who specialized in consulting for government and industry on the mathematical optimization of their systems, a service for which he was easily able to command fees in excess of \$10,000 per hour.  While these types of financial incentives may apply more to those who actually spend their lives studying mathematics (you don’t get \$10,000 per hour just by knowing how to do basic calculus), there is also a more general argument that any skill one possesses adds a certain value to one’s marketability to employers, and I would suggest that mathematics, with its universal applicability in any field or profession, is arguably one of the greatest (if not the greatest) adder of value to one’s resume.  The more math you know, in other words, the better your job prospects will be.

Just as much as our culture is defined by our art, our cinema, our literature, it is also defined by mathematics.  This has always been true, but it is perhaps truer today than ever before.  In The Demon-Haunted World, Carl Sagan famously wrote, “We’ve arranged a global civilization in which the most crucial elements--transportation, communications, and all other industries; agriculture, medicine, education, entertainment, protecting the environment; and even the key democratic institution of voting, profoundly depend on science and technology.  We have also arranged things so that almost no one understands science and technology. This is a prescription for disaster. We might get away with it for a while, but sooner or later this combustible mixture of ignorance and power is going to blow up in our faces.”  He might as well have been writing about mathematics as well.  In many ways, he was, because I have long contended that a significant (perhaps not the most significant, but surely a significant) reason for this cultural ignorance of science and technology is the widespread understanding that science and technology are dependent upon mathematics, and people are afraid of mathematics.

Let us imagine, then, that you have no desire to extract yourself from the well of ignorance, that you don’t mind falling behind as mathematically literate people change the world, that you still don’t believe you use mathematics in your daily life (though you bloody-well DO), and that you’re perfectly happy to divorce your consciousness from an important part of your culture.  Why should the person I have just described still care about mathematics?  Let us move away from the daily toil and look at some of the other applications of mathematics which arise, not every day, but still in every lifetime.

We can begin this discussion by thinking about the legal profession.  Mathematical literacy amongst lawyers (and hence, judges) is famously low.  Many of my colleagues share my hypothesis that a large part of the reason the market is beyond saturated with more lawyers than we as a society know what to do with is that there is a perception that a law degree is the most prestigious and highest-yielding graduate program one can undertake which does not require significant amounts of mathematics.  People, desiring a post-graduate education but still afraid of mathematics or insecure because of their ignorance thereof, flock to the law schools in an attempt to do the best they can do without ever having to crack open a calculus book.  And yet, this is precisely the opposite of the way things should be.  The legal profession is failing to properly utilize one of its greatest tools, with the result that trials are presented before juries in which statistical analyses play a determining factor, and none of the players involved (the judge, the jury, or the lawyers on either side) realize that the mathematics has been perverted, misunderstood, and gotten wrong.

I have been reading a remarkable book lately entitled Math on Trial, which is a compendium of cases in which the lawyers presenting criminal trials have gotten their statistics wrong.  It includes such novice mistakes as multiplying non-independent probabilities to create the illusion of guilt where happenstance may be a more likely explanation.  Of course we need mathematics in the courtroom, but mathematics only works if the players involved know enough mathematics to actually do it correctly.  Otherwise, they might as well just bring in a kindergartener to write numbers on the whiteboard at random for all the good it will do in the pursuit of justice.

Why do I spend precious paragraphs talking about the legal profession when I know very well that a small minority of my readers are lawyers?  Because it isn’t just the lawyers who get it wrong.  So do the juries.  Any one of you can--and probably will--be called to serve on a jury (why you shouldn’t try to get out of that duty is a topic for another essay).  I think it’s safe to say that, if impaneled on a jury, every one of you would want to get things right.  None of you want to let the guilty go free and you certainly don’t want to wrongfully imprison the innocent.  There is also a fair chance, if the trial is of any importance at all, that statistical analysis will play some role in the evidence you’re presented when you’re sitting in that jury box.  And if not statistical evidence, then perhaps some other branch of mathematics.  The case in question may hinge completely on the geometry of the crime scene or the physics of the objects involved.  However it might manifest itself, there will likely be some mathematics.  So when you’re sitting in that jury box, do you believe the prosecutor when he tells you the odds that the defendant is guilty?  He could be mistaken; he could be lying.  There’s strong precedent for both.  Lawyers often get their mathematics wrong, and certainly prosecutorial misconduct is, unfortunately, not unheard of in the pursuit of winning the case.  But if you can’t trust him, do you trust the defense attorney to be able to adequately explain to you why he’s wrong?  Even if they both make mathematical arguments, how are you to determine whose mathematics is correct?  In order to serve justice in such a situation, you must be mathematically literate enough, if not to check their calculations, at least to ask the right questions.

Okay, so maybe you’re not worried about jury duty.  Perhaps you’re content to assume that the other jurors know enough mathematics that you can rely on them.  (This is obviously not the case, as most of the jurors are likely to be thinking exactly the same thoughts, but let’s imagine for a moment.)  How can you be a fully functional citizen and protect your own interests without at least a rudimentary understanding of mathematics.  Consider one of my blog posts of the past, in which I explain how pyramid schemes work.  The mathematics there is relatively simple (especially in the simplified form in which I presented it).  Most people can understand that math, but it serves as an example of which I speak.  If you did not understand the math, you might be inclined to think the scheme was a good idea.  Similarly, if you don’t understand the math, you might be inclined to think more subtle schemes are good ideas.

The mathematically illiterate are ill-equipped to handle the challenges of the world, particularly in the 21st Century.  They may be the victims of fraud, they may misunderstand science, they may fall victim to predatory practices within the economic system.  They are not prepared to conduct business or vote as educated actors, but instead must rely on the often biased (whether intentionally or otherwise) information provided to them by outside sources.  No one knows everything, so of course it’s not reasonable to expect all people to be able to do all types of mathematics, but I think it is very reasonable to expect people to have a sufficient understanding of mathematics to a) solve the mathematical problems that occur in their lives, b) recognize mathematical problems with sufficient sophistication that they can seek their answers from the correct sources, and c) know well enough when basic errors in mathematics have resulted in erroneous conclusions in that information they’ve been provided.

Journalists certainly have failed in their duties to remain mathematically literate.  Examples abound of newspapers publishing shoddy statistics or even simple arithmetic errors.  Perhaps you’ve seen political polling data.  Do you believe it?  Well, do you know enough math to understand how those numbers work?

And it’s not just about practical uses either.  Mathematics is the language of the universe.  It is the language of nature.  It amazes me how many people can claim to be lovers of nature, but fail to see the mathematical beauty it holds.  Consider the spiral of a nautilus shell.  There is a deep mathematical principle at work in that simple geometric pattern.  The motion of every object can be expressed mathematically, from the gentle floating of a dandelion seed drifting on a gust of wind to the motion of the entire Earth around the Sun or the entire solar system about the center of the Milky Way Galaxy.  Mathematics is everything, and it can be used to reveal the hidden patterns of everything we ever see or do.  Even the relatively mundane is the result of mathematics.  The advertisements you see online are determined by algorithms designed to match advertisements specifically to you based on data profiles of people who visit websites similar to the ones you visit.  Whether you like that or not, it’s mathematics.  And it is understanding of mathematics that allows one to form an informed opinion about such matters.  I’m certainly not saying that everyone should be able to write an algorithm capable of effectively targeting advertising.  But I am saying that everyone ought at least to have some idea of the mathematics behind that process.

No less a person than Abraham Lincoln refused to continue his legal education until he had read, understood, and mastered Euclid's Elements (this text on geometry is arguably one of the greatest products the human mind has ever produced).  Lincoln had no intention to become a mathematician, but he recognized within mathematical thinking the foundation of an agile mind.  The sixteenth President of the United States of America would not complete law school until he mastered Euclid.  Think about that when you consider whether mathematics has relevance in non-mathematical fields.

So mathematics is important.  Why, then, are so many people afraid of it?  Why do people think they’re bad at it?  Why have they allowed themselves to persist in the delusion that they have never had need of it in the “real world,” despite mathematics’ deep connections to literally everything in the real world?

I think it comes down to a number of relatively simple factors.  The first is arguably the most insidious.  Issac Asmiov knew a thing or two about a thing or two.  He wrote over 500 books and is a very rare person indeed, having been published in nine of the ten major categories of the Dewey Decimal System.  So it is with deference to his great mind that we note his famous observation: “There is a cult of ignorance in the United States, and there always has been. The strain of anti-intellectualism has been a constant thread winding its way through our political and cultural life, nurtured by the false notion that democracy means that ‘my ignorance is just as good as your knowledge.’”  I think he nailed it.  I think people feel justified in their ignorance--and indeed, even proud of it to some extent--precisely because of the bizarre idea Asimov described.

My personal feeling is that this dangerous attitude stems from a fundamental misreading of the Declaration of Independence.  Thomas Jefferson did write that “all men are created equal,” but surely he could not have meant that everyone should achieve equal results in life despite varying degrees of effort, but that is the view that we collectively seem to have taken.  We live in a culture where children’s sporting events no longer have winners and losers but all children are given a ribbon for participating.  We’re to value self-esteem above introspection and hard work.  Similarly, we have come to believe that “all men are created equal” means, beyond equal protection of the law, that all men’s ideas are created equal.  This is clearly an unjustifiable position to take, but evidence of its pervasiveness in our culture is overwhelming.  A simple look at commentary on Facebook (authored in reply to my postings and those of my acquaintances) provides clear examples: “it’s just my opinion, and it’s as valid as yours,” or “we’re all entitled to our opinions,” or “let’s just agree to disagree.”  These phrases signal an end of the intellectual debate and a surrender to anti-intellectualism because they confuse facts for opinions and assume that all of these “opinions” are of equal merit.

Once we as a people decided that we were collectively going to buy into this deadly lie, it was not a great stretch for many of us to come to the conclusion that mathematical illiteracy is just as good as mathematical literacy.  Similarly, I have recently seen it argued that belief in astrology is just as valid as belief in astronomy and that creationism and evolution should be given equal time in public school science classes.  We seem to have decided that mathematics is only for some people, and that the pride due to those who have spent their time developing a true mastery of advanced mathematics is also due to those who have no mathematical ability at all.

The second and, I think, far more important factor is the way mathematics is taught.  I don’t believe there is a country in the world whose educational system quite gets mathematics right, but I think our educational system in the United States gets it more wrong than most in the industrialized world.  While the above is, I think, the reason people feel pride rather than shame in their ignorance, I think educational failure is the reason for that ignorance in the first place, and I think that failure is a composition of several factors.

1) Peer influences poison the well.  It has been allowed to become common knowledge that mathematics is an intellectually difficult field of study, and should be limited to the “nerds.”  I will write elsewhere about why nerdiness is indeed something to be proud of, but for many students, this culture stereotype creates a major problem.  It convinces the student that, unless he or she is one of the “nerdy” crowd, he or she will struggle with mathematics.  Self-fulfilling prophecies are common in the study of psychology, and this is no exception.  As Guinan once said in Star Trek: The Next Generation (4.1; The Best of Both Worlds, Part 2), “When a man is convinced he’s going to die tomorrow, he’ll probably find a way to make it happen.”  Similarly, if you’re convinced that you’re going to struggle or fail at mathematics, you probably will unconsciously sabotage yourself to the point that you do so.

2) Mathematical opportunities are limited.  American primary and secondary schools are relatively limited in their mathematical options.  Students in elementary and middle school learn arithmetic and pre-algebra.  In high school, they take Algebra 1, Geometry, Algebra 2, and then (depending on the school) some other math course(s), probably in the following sequence: Trigonometry, maybe Pre-Calculus, then Calculus.  Never are they exposed to any other branch of mathematics such as probability and statistics, number theory, or game theory.  Even at university, many of those fields of mathematics remain entirely unknown to students who do not major in mathematics.

Now, I’m not saying that everyone should master all of these. Not at all.  I do think that statistics should be added to the mandatory sequence, at least to the extent that students learn the basics of interpreting statistical information so they can be informed consumers of information.  But do I think they need to study game theory? No.  On the other hand, do I think students should hear about game theory and number theory and many others? Absolutely.  I think part of the reason people think they dislike mathematics is because by the time they get through all the arithmetic (which is, quite frankly, the most boring part of mathematics), they’ve decided that mathematics isn’t for them, and they never realize the beauty of the more advanced disciplines.

In a short but powerful TED talk, Arthur Benjamin argues (go and watch that video now--it’s worth it) that calculus is the wrong summit of mathematical education for most students, and suggests teaching statistics before calculus.  Given the limited time and resources of the high school math department, I’m not sure if I would go quite as far as he does, because those students who do wish to study science or engineering have a significant leg up if they learn calculus while in high school (certainly, I would have saved a fair amount of time and money if I had entered university with a more complete math education).  But on the other hand, Benjamin is absolutely correct that statistics is a more important discipline in mathematics for the majority of students (every single person would benefit greatly from an understanding of probabilities and statistics), and it is beyond doubt that our educational system does a great disservice to our children by not at least offering statistics as an alternative for students who prefer not to take the “calculus track.”  And even those who do want or need to take calculus (which really is a lot of fun) would benefit greatly from some greater exposure to statistics.

While discussing the issue with my girlfriend, she suggested that Trigonometry is a longer course than it needs to be.  Our schools devote an entire year to it.  She suggested as an alternative that students should take a semester of Statistics and a semester of Trigonometry.  I would amend that slightly to say that every student should, after completing Algebra 2, take a semester of Statistics.  Then, those students who wish to move on to Calculus should take Trigonometry for the second semester of that year, while the others should continue with a course devoted to just exploring the big picture of mathematics.  In such a course, they needn’t develop the skills to do advanced mathematics, but would be exposed to both the history of mathematics and the beauty underlying some of the current areas of mathematical research.  This leaves even the non-scientist and non-mathematician with an education in mathematics comparable to the education in the arts that non-artists receive.

Because the educational opportunities in mathematics are narrowly tailored to a particular type of student with particular needs and expectations, mathematical education has failed to reach the rest of the students who might very well have been more interested in mathematics if they knew there were other disciplines.  Yes, you do need calculus to actually do a lot of those other types of math, but at least students would feel less ignorant and less proud of that ignorance if they knew what work was being done.

3) Many people had bad teachers.  This is not to put teachers down by any stretch.  Unfortunately, though, the bad teachers who are currently working have more opportunity to do harm in mathematics than in many other fields, because our culture has already begun to turn students off to mathematics.  When such a student encounters a bad teacher, they often just give up, instead of pushing through and waiting for a good teacher, as they might be inclined to do in a subject of which they’re more forgiving because they already have a passion for it.

What do I mean by bad teachers in this context?  I mean those who, instead of working to help their students understand, make their students feel inadequate for not understanding.  While I do think there is shame in life-long mathematical illiteracy, there is never shame in ignorance when the ignorant person has a legitimate desire to learn.  I use the word “ignorance” here in is true sense, referring to a lack of knowledge, not to belittle the people in question. All of us are ignorant of something, so there’s no shame in ignorance if we are willing to correct those deficiencies.

I have worked with students who have had teachers bluntly call them “stupid” for not understanding something.  That’s enough to make anyone not go back to their lectures.  So those bad teachers truly are to blame for at least some of the mathematical illiteracy in the world.

4) Teacher selection and training is inadequate.  Separate from the bad teachers mentioned above, there are many who are simply unqualified to teach mathematics.  Teaching mathematics requires knowledge of two fields: mathematics, and education.  That’s true of any field.  You need to know your subject, and you also need to know how to teach it.  Many teachers in the high schools and lower did not study mathematics.  They studied education.  What’s more--many of them were the very same students who disliked and feared mathematics a generation earlier.  They get through their education degree with as little math as possible, and end up teaching elementary mathematics simply because that’s where there was a teaching job available.  How can anyone teach a subject about which they are not passionate and knowledgeable?  And yet, that is exactly what is expected of far too many teachers.

On the other extreme, there are those few who did study mathematics, but many of them don’t understand how to teach mathematics.  They never struggled with mathematics (because most people with math degrees never had a hard time with the subject), and so they are ill-equipped to understand the struggles their students have with the subject.  We need a community of mathematics teachers who understand both mathematics and education.  There are some of them out there, but we need a concentrated effort to create more of them.

5) Mathematics is cumulative but math education is age-based.  Mathematics, as much as any and more than most fields, builds upon itself.  You cannot master fractions until you master multiplication and division; you cannot master algebra until you master fractions; you cannot master logarithms until you understand exponents; you cannot master calculus until you master algebra.  Each new course in mathematics assumes the previous courses as prerequisite knowledge.  There’s nothing wrong with that--indeed, that’s the way it must be (though I have some crazy ideas about tinkering with the order things are taught, this is not the essay to go into that).   So where’s the problem?

The problem is that, with the rare exceptions of students who are exceptionally good or bad, most students will progress through their education based on age rather than ability.  Let us imagine a typical young student.  In his first math course, he gets 90% on his final exam.  He’s labeled an “A” student and passed along to the next course.  But there’s still 10% of the information he’s missing.  That might not seem like much, but that next course will build upon everything he’s previously learned.  He struggles to catch up with that extra 10%, and in this course, he gets 80% on his final because he lost time catching up.  In the next course, he gets 70%.  Then 60%.  Sooner or later, he convinces himself that he’s just bad at mathematics and doesn’t like it.  But it’s not really that case.  If education were based on mastery instead of based on pushing children through according to their age, he would have eventually gotten that initial 10% he missed, and the entire problem would have been averted.  Quality teachers would be free to deviate from the standard curriculum and teach students at their own pace, passing them along to the next course when they demonstrate mastery rather than at the end of each year.

This requires a compete rethinking of how our educational system is structured, so I hold out limited hope for the immediate future.  However, it remains true that these cumulative failures are part of what convinces people that they are bad at mathematics.  I would contend that, putting aside those who have legitimate mental disabilities, no one is just “bad at math.”  Some will take to it more quickly than others for a variety of mysterious reasons psychologists might struggle to understand, but anyone can learn it.  It is only through failures in education that people come to believe that math is just not for them.

6) Informal education is limited.  We can bemoan the rates of illiteracy and scientific illiteracy right along with that of mathematical illiteracy.  However, I think the problem is worse in mathematics, and I think that part of the reason is because there are not many opportunities to gain informal education in mathematics.  Just as with any field, you could actively seek out that education at a library or bookstore (though even then, you’ll struggle to find books that simultaneously provide a depth of understanding and an ease of comprehension befitting the autodidactic student.

Public television (and some network television) is full of programs for children trying to help them understand science or teach them to read.  LeVar Burton recently raised millions of dollars to bring back Reading Rainbow.  For adults, Neil deGrasse Tyson’s reboot of Carl Sagan’s Cosmos was a great television series teaching about science.  Where are the equivalents for mathematics?  The closest I have been able to find are the little segments about counting in Sesame Street.  While admirable, that just isn’t enough.

People often use the word “infotainment” as a derogatory phrase.  It’s meant to convey displeasure at the state of media, often news media, placing more emphasis on keeping people entertained than keeping them informed.  However, there is no reason education should not be entertaining.  We desperately need more high-quality entertaining books, websites, television programs, etc., capable of keeping audiences engaged long enough that they will learn some good information while reading or watching.  Though we need more of them in all disciplines, in mathematics, these programs are virtually nonexistent.

7) There is a fundamental disconnect between the student and the mathematician.  A video on the YouTube channel Numberphile compares this disconnect to art in this way: the way we teach mathematics is akin to teaching someone how to paint a fence and calling it art education.  I hadn’t thought of it in those terms before, but of course that’s absolutely correct!  The average mathematics student learns a series of operations to perform in order to solve the type of math problem that shows up in textbooks.  While they’re doing so (even moreso if they’re engaged and paying attention), they’re developing the analytical tools I mentioned earlier, but on the surface, they’re solving “cookie-cutter” problems which do bear little resemblance to the real world applications of what they’re learning.  Never do they gain an understanding of the history of mathematics as art students would learn about the great masters.  This leaves students not only exhausted by the work they’ve been frustrated about (for the reasons I’ve discussed), but absolutely unaware of the kind of work that real mathematicians are doing.  Students should be taught mathematics in a way that gets them excited about mathematics, rather than in a way that leaves them afraid of it.

8) Cultural stereotypes limit student performance.  As I mentioned earlier, someone who feels doomed to fail at mathematics will probably do so, for purely psychological reasons that have nothing whatever to do with their actual ability to perform mathematical operations or to reason mathematically.  There is an added cultural stereotype which I think adds to this problem, and which is worth pointing out.  Namely, it seems to be a popular belief in the United States that natural ability determines success in mathematics; that some people “naturally” are better analytical thinkers while others are better at, say, emotions.  Psychologists have not fully unlocked the secrets of why some people perform better than others.  While there may be some genetic elements, it is almost certain that environmental influences and hard work make a greater difference.  In many Asian cultures which routinely outperform the United States in tests of students’ mathematical abilities, the success of those students is attributed to hard work and discipline.  In the United States, success is often attributed to “talent.”

I’m not going to sit here and tell you there is no such thing as natural talent.  What I will say is that this cultural view that only a relative few who are “mathematically gifted” can succeed in mathematics is false.  I firmly believe that anyone, with hard work, can become an expert mathematician if they so desire.  It means putting aside our culture of instant gratification and putting in the time it takes to master these skills, but it is achievable for everyone except those with the severest of mental disabilities.

So why do I care?  Why do I take the time to write an essay of this length to tell people a) that mathematics is important and b) that mathematics is within the reach of everyone?  Well, there are a number of reasons.  Of course, the reasons I mentioned above hold true, that mathematics is important in day-to-day life for more people than seem willing to admit it.  But that may be viewed as personal for those people.  However, when mathematical illiteracy affects the way juries rule, the way the news is reported, and the way people vote, it affects all of us.  I don’t think I hyperbolize when I suggest that our survival depends on greater levels of mathematical literacy.

But there’s also a more personal reason.  I find mathematics beautiful, interesting, and fun.  It’s the language of the universe.  With mathematics, we can understand our world, we can understand each other.  We can unlock the secrets of nature if only we speak their language, and that language is mathematics.  Speaking of science, which is also a passion of mine, Carl Sagan said “When you’re in love, you want to tell the world.”