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.”
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