The Washington Post recently published an op-ed by mathematician G. V. Ramanathan. The subject? Mathematics education. It is a mix of good points and bad points. Let's have a look.
Twenty-seven years have passed since the publication of the report “A Nation at Risk,” which warned of dire consequences if we did not reform our educational system. This report, not unlike the Sputnik scare of the 1950s, offered tremendous opportunities to universities and colleges to create and sell mathematics education programs.
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.
The calculus book I used in college had a problem that involved a farmer dropping a bale of hay from an airplane to some cows waiting below. Someone in class objected that it seemed a rather implausible scenario. The professor (actually a graduate student) suggested it was likely that the problem was originally conceived during WWII, and involved dropping a bomb onto some helpless civilians. But feeling that to be in poor taste, the authors changed it to dropping a bale of hay.
Anyway, there is a lot of truth in what Ramanathan is saying here. Certainly an over reliance on asinine, contrived applications is a weakness of many modern textbooks. The notion that you might just explore patterns and try to discover abstract relationships for the sheer satisfaction of it is mostly disallowed. On the other hand, I think he might also be overreacting. I very much doubt, for example, that the problem about the martini glass was intended as part of a practical treatise on fluid dynamics. I suspect the author was just trying to present a standard calculus problem in an amusing way.
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.
This I don't believe. That so many people seem to lack a basic number sense, not to mention a near total ignorance of basic probability and statistics, absolutely has practical consequences. Just to pick one of personal significance to me, I can't tell you how many creationists I've seen bamboozle their audiences with slick-sounding mathematical arguments. A closely related problem is the ease with which statistics are manipulated by unscrupulous pundits to justify whatever conclusion they like.
As for those studies to which Ramanathan refers, I think they mostly show that our kids perform poorly in mathematics relative to kids in other countries. Not that they're dumber. That is precisely the point, actually. Why do our kids perform poorly on math tests? If American kids were just dumber then we would have our answer. Since I assume no one is inclined to accept that answer, I'd say there's something worth looking into here.
We need to ask two questions. First, how effective are these educational creams and gels? With generous government grants over the past 25 years, countless courses and conferences have been invented and books written on how to teach teachers to teach. But where is the evidence that these efforts have helped students? A 2008 review by the Education Department found that the nation is at "greater risk now" than it was in 1983, and the National Assessment of Educational Progress math scores for 17-year-olds have remained stagnant since the 1980s.
Now he's back on track. Mathematicians tend to be unimpressed by specialists in mathematics education, for good reason. Mostly we are just lectured to stop lecturing, and also to use more group work (excuse me, “cooperative learning.”). Mostly my reaction is the same one my high school calculus teacher once gave to one of my classmates. After we had all done quite poorly on a test, my classmate asked, “Doesn't it reflect badly on you as a teacher that we all did so poorly?” My teacher replied, “Nope. I taught it fine. It reflects badly on you as a learner.”
The second question is more fundamental: How much math do you really need in everyday life? Ask yourself that -- and also the next 10 people you meet, say, your plumber, your lawyer, your grocer, your mechanic, your physician or even a math teacher.
Unlike literature, history, politics and music, math has little relevance to everyday life. That courses such as "Quantitative Reasoning" improve critical thinking is an unsubstantiated myth. All the mathematics one needs in real life can be learned in early years without much fuss. Most adults have no contact with math at work, nor do they curl up with an algebra book for relaxation.
Yikes! Now we're in crazyville. I might be willing to grant politics, and maybe even history (though it seems to me plenty of people manage to go their whole lives without thinking seriously about either one.) But how on earth are literature and music relevant to our daily lives? Who needs to read The Scarlet Letter, or The Grapes of Wrath, or The Great Gatsby, or any of the other novels we were forced to read in high school?
Ramanathan has far too narrow an understanding of what it means to use something in life. The fact is that virtually none of the topics you learn about in school is directly relevant to your daily activities. You learn about them in school precisely because you won't learn about them anywhere else. There is more to life than getting through the day without doing something stupid.
Those who do love math and science have been doing very well. Our graduate schools are the best in the world. This “nation at risk” has produced about 140 Nobel laureates since 1983 (about as many as before 1983).
Somehow I really don't think the authors of “A Nation at Risk” were concerned that America would no longer turn out ferociously talented people who will succeed in math and science regardless of what they learn in school.
As for the rest, there is no obligation to love math any more than grammar, composition, curfew or washing up after dinner. Why create a need to make it palatable to all and spend taxpayers' money on pointless endeavors without demonstrable results or accountability?
We survived the “New Math” of the 1960s. We will probably survive this math evangelism as well -- thanks to the irrelevance of pedagogical innovation.
The issue is not so much whether people love math, it is whether they reject it before developing any real sense of what it is. I also don't care if any given student likes reading literature, learning history, or eating spinach. We make kids do those things because it is good for them, whether they appreciate it or not.
There are certainly difficult problems to solve in mathematics education. For example, I think everyone, math major or not, ought to be able to give a coherent description of what calculus is, and what sorts of problems it was invented to solve. But the things I want everyone to know bear little resemblance to what typically gets taught in calculus classes.
Many schools, my own included, offers a course to non majors meant to introduce students to a swath of higher mathematics that they would otherwise never see. Such things, I believe, are great in principle, though I don't actually know how successful they are. Frankly, one of the biggest problems in math ed is simply large numbers of unmotivated students. People who are content to do minimal work and get by with C's. The best teaching in the world will not overcome a student who is unwilling to work hard at a subject.
There is much to fret about. But Ramanathan has not manage to put his finger on any of the real problems.
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I saw this article and was pretty appalled. But there are some valid concerns with math education. I teach college chemistry and every student I see (more or less) comes in taking calculus in high school, normally some AP course.
Yet very few can really handle more basic math like trig or even some more complicated algebra. They can do them in a classroom setting when it's the subject at hand, but don't full understand how to apply them outside of a math class. It's frustrating.
I'd prefer students to come in with a better understanding of mathematical concepts without feeling pressured to get through calculus.
And heaven knows that plumbers use Shakespeare daily, but never use math.
I spent some years on the Speakers' Bureau where I work. One of my favorite assignments was Career Day at elementary schools.
One of the things I would do is share "a secret" with the kids. The secret was, chances are they'd never actually use the algebra they were about to take.
So why study it?
For the same reason people on a football team do lots of sit-ups. Football teams never start games by seeing who can do the most sit-ups, but the game of football uses the muscles sit-ups strengthen. Similarly, algebra strengthens mental muscles you'll use throughout your life.
I don't know what effect it had on the kids, but the teachers seemed to approve.
I have long thought that maths education is under two conflicting pressures. One is to get a group of student proficient in reasonably advanced maths so they can progress further. This, naturally enough, is the group that most people prefer to teach. More people, however, need a familiarity with basic arithmetic and geometry, an understanding of basic statistical concepts and some finance. My experience has been that most students for whom this would be adequate never actually get a good feel for the concepts, but learn it all by rote.
I agree entirely, but I don't know what can be done about it. I've given adult learners problems involving regular deposits and accrued interest who, even after I pointed out that they had cheated themselves out of thousands of dollars and commented that they would be easy to swindle, were still not interested in learning how to do it. Perhaps if I'd offered them real money . . .
I went through high school in the UK, where I combined A level maths with biology. One thing that struck me was that I saw not one single calculus problem that involved biology although I now realize, of course, that many could have been used. As a result, I have always been conscious of the need to include relevant problems.
Another difficulty seems to be that many students have an extremely short attention span and are unwilling to attempt any slightly involved problem or one that they can't instantly solve. A couple of days ago a high school maths teacher was telling me that he had just looked over his notes for 15 years ago and considers that these days he could only cover about 40% of the material.
Andre3 @1
At the tail end of a test that was 50% Pythagoras' Theorem, I described how roads in the southern part of the province are on a one-mile grid and gave a diagram of a portion of old road with a one-mile offset in it and a new, diagonal road. I asked my adults how much shorter the new route was than the old, and not one student realized that it was just Pythag. It's as though everything is in its own little compartment.
I'm not sure that this is a new problem. I remember a statistics lecturer about 40 years ago saying that her students could handle the theoretical aspects well, but when she gave them some actual data for an analysis of variance they were completely floored.
Way back when I was in school the the problem in #4 was that people could not do what were then called story problems. I.E. you set a up a story that needs math to solve the issue raised, not just ask for a formula. Back then memorization of formulas was easier for most.
In addition for example to really understand Calculus you really need to go thru it twice. I really began to understand it in the senior level advanced calculus class I took. The freshman sophmore class taught some things but did not get to the base elements.
Really nice analysis. Now I don't have to blog about it and say pretty much the exact same thing. :)
The first time I studied maths at Uni, I bombed. I was the unmotivated student. I've since rectified that problem. I'm trying to teach myself higher maths so I can one day hope to grasp high end physics. The think I've noticed is I really don't have a good grasp of basic geometry and trigonometry. Algebra is fine, I can do single variable calculus good enough, but I didn't even know what vertically opposite angles were for example, so I didn't get some basic point in geometry. I wonder what crap I was getting up to in high school to miss that?
Oh well, I'm Australian, but it seems students are lazy the world over. :)
Well, to be honest, I'm not sure I'd meet the "basic description of what calculus is and what sort of problems it was designed to solve test". I wasn't a math major. So perhaps I'm a reasonable candidate.
So here goes:
I'd describe calculus as essentially two sorts of things. The first is to study how functions change - the most practical problem being acceleration, velocity, and displacement, and readily applicable to any quantity that changes over time; calculus is the tool you want for working out how fast it changes. The second is the inverse of this - given how fast something changes, extrapolate information about the nature of the function. It turns out that this second technique coincidentally has uses in determining area and volume.
I think that's probably an epic fail, but as you say - even in Australia, we're not talk directly what it's for.
I expect a lot of people value mathematics because they think it's hard and therefore character building. It's also apolitical. I've been told that the French decided to change the emphasis in their educational system from philosophy to mathematics because philosophy tended to encourage radicalism. I don't know if that's actually the case; but this explanation has, as they say, the ring of poetic truth.
Something I do know for sure: countless American community college students are flogged through a course called Intermediate Algebra even though the content of the course is pretty much entirely useless since most students are never even going to encounter an angry quadratic equation in the real world--the most practically useful part of algebra is already covered in basic algebra courses while the interesting parts of math lie beyond Intermediate Algebra. I don't deny the morale advantages of teaching people a tough course without any real use: the best algebra teachers I've met (and I've met hundreds of algebra teachers) are like good football coaches. I think of Intermediate Algebra as a sort of mass market version of the Kobayashi Maru exercise, but let's not pretend it or much else in math has a better rationale when retailed to nontechnical people.
1) The arguement that our educational system is doing OK because we produce X number of Nobel laureates, is hogwash equivalent to the arguement that our economy is doing OK because we produce Y number of billionaires. Nobel laureates and billionaires are meaningless measures in a culture where 60% of the people don't believe in the theory of evolution (AAAS survey 2010) and 20% of children go to bed hungry each night.
2) There are powerful interests that benefit from mass ignorance of science and math, and at some point they will have to be confronted directly, bluntly, and forcefully. The promoters of climate denialism and various forms of religious extremism (keyword search "dominionism") are the most obvious, but the problem extends to the everyday petty misleading of citizens by demagogues and of consumers by advertisers. Consider also the popularity of lotteries as a paradigm example.
3) Children should be taught probability & statistics starting at an early age so they become familiar with the concepts.
4) The unpopularity of math may have to do with teaching methods that emphasize memorization over the conceptual knowledge that is the glue that makes memory stick.
5) Don't underestimate undiagnosed dyslexia: visual inversion of character-strings. Speaking here from personal experience, and the difference it makes to know and be able to compensate: a D- in undergrad social science statistics, and an A- in graduate social science statistics. Kids who show any signs of trouble in math (or spelling) should be tested and then taught to compensate. Even something as simple as better (accessible) typographical design will help significantly.
6) Teaching people to become familiar with and comfortable with large numbers will have an inherently subversive effect against the agenda of religious extremists. Someone who has an intuitive grasp of what _billions_ mean, is less likely to fall for young-earth creationist BS that entraps people for whom six thousand years seem like a long time.
Who really needs math?
I polled 10 people and they said 5%, 10%, 25%, 33%, 50%, 66%, 95%, 95% 95%, and 100% of us. So I figure the real percent is within 3 standard deviations of the average of those.
Plus anyone who wants to really understand what that means.
There are certainly difficult problems to solve in mathematics education. For example, I think everyone, math major or not, ought to be able to give a coherent description of what calculus is, and what sorts of problems it was invented to solve. But the things I want everyone to know bear little resemblance to what typically gets taught in calculus classes.
Back when I was tutoring math to middle schoolers and high schoolers, my first session was always the same, no matter what class they were taking. We did a history of math concepts: the basic principles of how numbers came into being, how they represented information, why negative numbers and fractions were important, and why this all matters. Basically, I taught 17 year olds doing a second year of calculus what a number line is in 20 minutes or less. It was absolutely amazing how a little background information helped nearly everyone.
Unfortunately, I also had a large number of 1 time sessions and parents who would say: "I'm paying you to teach him Algebra/Geometry/Trig/Calculus, not history."
JR quotes the passage: "National Assessment of Educational Progress math scores for 17-year-olds have remained stagnant since the 1980."
But why wouldn't you expect them to remain stagnant? Americans' nutrition, their social structures, and, controversially, genetics, and many other factors that may affect test scores, have not much changed since 1980.
Maybe given constraints, there is only so much improvement in education that can now, reasonably, be done.
My students asked me what good is math in the real world, which caused me to realize that the tricks for calculation one learns in grade school and to a lesser extent in college as well are not particularly applicable. But the overall exposure to the structure of mathematics has been inestimably important to my intellectual life. Simply taking a logic and set theory or discrete math course is enough, I think, to make one a clearer, better thinker, and that's going to be reflected in every use of one's brain for the rest of one's life.
But I think the most important reason to learn mathematics at this point in history is actually a hard sell, because it has to do with circumstances that are unique in human history. For most of history, it was very hard to get information; now it's very easy to get information, but it's hard to sort out the useful stuff. As information technology becomes more tightly interwoven with the fabric of everyday life, it will become more and more of a handicap not to have ready experience with mathematics and logic that one can employ in trying to make sense of this new information-rich environment.
For most of history, mathematics was a highly specialized, derived study that had little bearing beyond the sorts of measurements one might need to make for surveying or other tradecraft. But as the world's economies and cultures become increasingly virtualized, mathematics will become more and more essential to success in life. Whereas one might have made the argument 40 years ago that mathematics is not an essential life skill, this has become and will become less and less true.
One catch is that while maybe only 20% of the students in an algebra class(*) will end up using what they were taught, we don't know which students will fall into that 20%. There's no way to know in advance whether a student is going to be involved in a math-heavy discipline like physics or engineering, or end up being an art student, factory worker, etc.
(*) And remember, 95.785326% of statistics are made up on the spot. :)
"Unlike literature, history, politics and music, math has little relevance to everyday life."
Hard to imagine a math professor making such a quasi-plebeian claim.
A lot of Americans took out mortgages they had no hope of ever being able to pay. And a lot of Americans are up to the eyes in credit card debt. Surely math ignorance isn't helping in these situations.
Students might not be motivated because they are taught from a very young age that it is OK to be bad at math because only especially talented people excel at math. This myth of excellence has allowed decades of math ed to be phoned in.
@g724 5) "Don't underestimate undiagnosed dyslexia"
I labored with undiagnosed dyslexia (mostly problems with larger than single digit numbers) until the 5th grade. This was back in the early 80's, so my teachers didn't know anything about dyslexia, they just thought I was one of those kids who was "dumb at math". It wasn't until I entered an early experimental gifted program in 5th grade that the specialty teacher noticed. By that time the well was poisoned. I hated math and I loathed the way it was taught. And a lack of good math skills pretty much ruined any chance I had to advance in sciences because I would hit the math and just freeze up.
Some anecdotes:
1. In elementary school (late '60s/early '70s), I was taught the New Math that everyone reflexively disparages. When, in fourth grade, my family moved to another state, I had about as good an intuitive grasp of set theory as any nine-year-old is likely to have. (I still think in terms of sets to this day.) My new school was working on rote memorization of "times tables." I was considered "slow," and put into a remedial class.
2. In my senior year of high school, I took an AP trig/pre-calc course even though I wasn't planning on a STEM career. All the other students were planning on majoring in STEM fields in school, and were full of self-confidence. I regretted taking the class, because after eight weeks or so I found myself earning the first C of my academic career. A week before the final, contra your HS experience, the teacher announced that after reviewing the class's scores, he had decided that he had completely failed to teach the subject, and to be fair to us, he would grade the class on a curve. I ended up with an A+. But I still felt like an idiot with no business doing math.
3. I went on to study arts and humanities in school, and after many twists and turns and false starts, became a technical writer working with CAE software. Today at work I got paid to teach myself about matrix decomposition methods and component mode synthesis. Not bad. Thanks, New Math!
We have a problem in our country that people do not understand numbers let alone math. We can never grow our Nuclear Energy base, no matter the bennefit, because people who do not understand what 20 Million to one really means demand a system the has "zero chance of failure".
Many of our poorest citizens dump what little reserves thy have into the local state Lotto because they fail to understand that just because the advertising says " someone will win" that it really truly is not going to be them. That imoral tax on their money is supported by the rest of us who either also do not understand the mat, or don't have the conviction to voice the moral outrage we should have.
OK - so I'm a little late to the party but I did want to recommend The Tiger That Isn't - a marvellous book which gently affirms the importance of maths in our daily lives, while providing guidance to those of us who are not experts how to avoid being bamboozled by dodgy stats, misleading numbers and meaningless targets.
Amazon Link
Great stuff. Written by a couple of Brisith people as I understand, so most of the examples given are UK-based. But no worse for that.
Although many people will not directly use math in the future unless their careers are directly correlated to math, but all those years studying math in school was of course necessary. Math isn't necessarily taught so one may use it in the future, but part of a process which teaches us how to think in a certain way that other subjects may not be able to teach us. Yes it may not seem pertinent to learn like other subjects but it is equally important as other subjects.
I believe and tell people that math is thinking (in the sense of reasoning) and thinking is math. Calculus and matrix algebra are just specialized forms for solving certain kinds of problems. If you have to do several errands in a certain amount of time, such as returning a book to the library, picking up clothes at the dry cleaner, and buying some food at the super market, as you figure out the best way to get them all done, you are doing math.
Studying math academically will teach you many methods that people have developed over the years to solve problems, and give you a lot of practice in reasoning your way through complicated situations. There is no guarantee it make your thinking better, but it can't hurt. Of course, people who go through life thinking poorly probably don't know what they are missing.
I never took many math courses in college or even high school, but I use math constantly - in cooking (changing quantities for different size pans, substituting ingredients with different qualities), in shopping, in creating project schedules at work, etc. Many people may never find occasion to sit down and work out a set of algebra or geometry problems as adults, but that doesn't mean they aren't using math.
Hi! You can create a fun (and even slightly intellectually provocative) calculus problem with a martini (or wine) glass!
Suppose you have a slightly exotically-shaped wineglass whose silhouette (without the stem) can be described by y=x^2 from x=-3 to x=3, with the glass being formed by revolving the curve around the y-axis. If the radius of the wineglass at the very top is 1 inch, and you fill the glass fully, and you drink the wine at the rate of one-half a cubic inch per second, answer these 2 questions: 1)How long will it take you to drink half of the wine? and 2)How high up (in terms of percentage) on the vertical axis will the wine reach when you've consumed half of it?
Bonjour,
Description : Mon Blog, présente le développement mathématique de la conscience c'est-à -dire la présentation de la théorie du Fermaton.La liste des questions mathématiques les plus importantes pour le siècle à venir, le No-18 sur la liste de Smale est; Quelles sont les limites de l'intelligence tant qu'humaine et artificielle.
(fermaton.over-blog.com)
Cordialement
Clovis Simard
There ONE HUGE difference between what we have been teaching since Sputnik, and what and how things were taught prior to that watershed moment. I spoke with a dear friend, recently, who worked as miller and cabinet maker for 34 years (he is now retired) about how he learned his craft. The schools during the 30 - 40's did not just have 'shop', they built entire houses. They had to level the foundation, measure and cut, weld and solder, use tools and create something real! Every house found a buyer and a family. Every year another house was built by each of the Cities high schools. They learned skills in math with several crafts which provided jobs for their own families after high school. Calculus? YES! There is more to building then wood! There are problems of the dynamics of water sources and pressures, and the ubiquitous issues of curves.
Did my friend learn math? "You bet I did! Or I would loose my job!I had a family to feed!"
Do you need math?
The short answer is "are you in a technical field?"
in an increasing technical world, if you want america to stay competitive, then you need to teach math. otherwise, you will become a consumer of tech instead of the creators of tech.
as to the teaching of math, I believe we are losing the talented math teachers to computers. Keep in mind that anyone with a good understanding of Math & Logic can easily be a computer programmer.
Average salary of a Math Teacher = $35,000
Average salary of a Computer Programmer = $72,010
I fully agree with this, especially the part about typesets:
In my case, it was a vision problem and the insistence of my parents that I'd have to go to a regular school. Seriously, you want to improve math education? Just don't use i and j as variables on a blackboard but, say, h and i. And no, m and n is no alternative! These things just look too similar! Not seeing something correctly and writing down things incorrectly are huge stumbling blocks.