Some readers have called to my attention a pair of recent stories from the New York Times that you may find interesting.
First, Audrey noted another dispatch on the eternal struggle over how math ought to be taught:
For the second time in a generation, education officials are rethinking the teaching of math in American schools.The changes are being driven by students' lagging performance on international tests and mathematicians' warnings that more than a decade of so-called reform math -- critics call it fuzzy math -- has crippled students with its de-emphasizing of basic drills and memorization in favor of allowing children to find their own ways to solve problems.
You should read the whole thing. My take: basic skills are really important, and understanding is too. Veering too far toward either extreme is probably a mistake. Messing around with numbers (and geometrical objects, and sets, etc.) improves one's facility at messing around with such things, but it can also be used to help students think about problem solving more broadly. And to the extent that rote drills bore the heck out of a lot of kids (especially very bright kids), they should not be overused.
And, here was Audrey's quick reaction to the piece:
I feel a little bit like playing spot-the-false-dichotomy when I read it, but my favourite is the way the article seems to suggest that either you teach math rigourously or you use "fuzzy math", which involves "language" to "explain things".
Sure makes me ashamed of having used all those silly English words proving the Completeness Theorem...
RMD flagged the other piece, a story suggesting (and a mere month behind the curve) that Mythbusters is the best science show on TV:
As the name implies, the program tests what the creators call myths, hypotheses taken from folklore, history, movies, the Internet and urban legends. Can a skunk's smell can be neutralized with tomato juice? Did the Confederacy come up with a two-stage rocket that could strike Washington from Richmond, Va.? Can a sunken ship be raised with Ping-Pong balls? Could a car stereo be so loud that it would blow out the windows?
Mr. Hyneman and Mr. Savage, who produce Hollywood special effects and gadgets for a living, come up with ways to challenge each thesis and build experiments with a small crew. If fire and explosions or, say, rotting pig carcasses happen to be involved, well, that's entertainment.
Though not a regular watcher of the show, I've enjoyed it the times I tuned in. It does have a jaunty This-Old-Exploding-Car feel to it, and models hypothesis testing very nicely. However, I do wonder what to make of the fact that even science programs seem to need to appeal to the viewer's lizard brain in order to achieve commercial success.
Then again, the Mythbusters team claims not to be aimed at teaching us science:
Mr. Hyneman ... insists that he and the "Mythbusters" team "don't have any pretense of teaching science." His wife, he noted, is a science teacher, and he knows how difficult that profession is. "If we tried to teach science," he said, "the shows probably wouldn't be successful."
"If people take away science from it," Mr. Hyneman said, "it's not our fault." But if the antics inspire people to dig deeper into learning, he said, "that's great."
So, do science teachers nowadays have much opportunity to blow things up, set them afire, launch them into the air, and so forth? (You don't have to name names here, I'm just curious as to whether flash TV attention grabbers are useful -- or necessary -- in classrooms filled with TV-watchers.)
My kids are being taught using the "Investigations" program, which is one of the ones criticized by the NYT. It does lack some of the basic drill and the school has begun supplementing the program. Another lack is simple facility -- the traditional algorithms, once learned, are much faster. The problem is that the algorithms (carry the '1', etc.) don't do anything for understanding. In the past, the kids who have come out of the "Investigations" program have been very good at problem solving, especially dealing with new kinds of problems, but have lagged in basic computation.
I agree that there's a baby-and-bathwater approach to math -- it's just like the phonics vs whole-language debate for reading. You need both the "fuzzy" approach and drill/rote learning in math, just as you need both phonics (for decoding words) and whole-language (for understanding what they mean.)
I had a childhood love of math. I used to read math books and really loved "math mysteries" sort of things (like the books Colin Bruce writes). My little sister struggled with math so my mom enrolled her in "Kumon." In the interest of not having her throw tantrums I was enrolled too. It was page after page of math practice. Think of hours of time tests. It was awful and I stopped liking math. I completely pulled away during middle school and didn't start coming back around until after college when I realized that I really missed out on a lot of interesting math. I've tried to learn on my own with mixed results.
This is a long way of saying that "drill and kill" killed math for this student. On the plus side though I am really good at basic computations in my head, so it depends on what you're looking for.
I watch Mythbusters regularly with my children (currently ages 9 and 11). It's definitely not teaching science per se, in the sense of teaching the particulars of what any scientific discipline has discovered. But it certainly does exemplify some important general concepts, about experimental design, and about perserverance in the face of inconclusive results. And that ain't nothin' ...
Still, that 'lizard brain' comment is certainly on point. One of my fondest memories was the conversation between my wife and me, across the house, during one episode:
"Scott, can you come in here for a minute."
"Hold on. Adam and Jamie are about to blow up a cement truck."
.
.
.
"Um, why?"
"WHO CARES?!"
Fill a room with forty kids, then it doesn't matter what syllabus or style you teach. They will do poorly.
Flashy things in science? At one point, we lived in northern BC, a pretty isolated area (120 miles from the school board offices so the teachers had a lot of freedom), and I remember demonstrations using a pistol to investigate echos, crushing a jerrican with atmospheric pressure, and innumerable movies of weird and wonderful things.
As for math, maybe kids would benefit from separating the rote work (times tables, mad minutes etc.), and the thinking aspect, putting them on slightly different tracks. Once you reach criterion on rote work, switch and focus more on the reasons and extensions.
The debate over fuzzy math vs "drill & kill," or phonics vs whole language, is like that over nature vs nurture -- near zero proponents of "drill & kill," phonics, or nature have suggested that the situation is/ought to be at one extreme; whereas many (most?) proponents of fuzzy math, whole language, and nature *have* suggested that the situation is/ought to be at their extreme.
Back when algorithms & rote math were taught, we still learned how to translate words into math symbols, how to set up & solve word problems, etc. -- so the argument that opponents of fuzzy math want *only* or *mostly* drill & kill is a non-starter. It was always a mix of the two. Ditto for opponents of whole languge -- we always learned how to summarize the main idea, infer the author's intent, etc., alongside basic phonics.
The proponents of fuzzy math & whole language, on the contrary, do suggest extreme things like junking algorithms (they "stifle creativity") and rote memorization, or junking basic phonics drills. Now that they've been caught with their pants down, they're trying to spin it as if it were a battle between two extremist points, when the reality is that they were advocating for one extreme while their opponents were arguing for a centrist approach.
Note the parallel to emphasizing the role of genes plus environment (nature / hereditarian) vs the role of environment only (nurturist), and the lack of a genes-only position.
Sorry, but an addendum about getting kids interested in math -- most of them abhor all mathematics. The math nerds should be tracked and put into special classes where they can go beyond rote memorization & basic word problems, learn how to apply math to all aspects of the world, etc.
Everyone else -- and I know this from tutoring kids who aren't math nerds -- finds the homework problems on "applications to real life" to be more painfully boring than simple computation. For them, math is like a toxin that they want to be exposed to as little as possible: they accept contact with it in math class and the part of their private life that involves completing HW assignments. Aside from that, they don't want math invading any other aspect of their lives. It would be like bringing your grandparents to a school dance.
And of course, these kids aren't ever going to be using most high school math in the real world -- they're not going to be engineers or what-have-you -- so as long as they get down the basics, they'll be set. They *will* need to know how many taxis to call if there are 20 people and each cab fits 4 people, how to estimate the total cost of 48 widgets that each cost $51, etc. Brainy people could figure out strategies to solve these on their own w/o much math instruction, but for the bottom 84% of the IQ distribution, they need reliable tricks taught to them so they don't have to figure it out themselves (and fail in the process).
"It would be like bringing your grandparents to a school dance."
Thank you for this comment. The kids are correct in this case. Mathematics and calculus are the programming languages of the 17th century still taught because academic scholasticism cannot adjust itself to changing times. The kids are not the problem, the math is the problem. Calculus must be dumped in favor of modern computer languages. Thanks to Dr. Free Ride for choosing such interesting subjects.
WTF?? What modern computer language replaces calculus?
Who doesn't love the science teacher who throws a magnesium strip in some water?
"What modern computer language replaces calculus?"
Any modern language would do. It will also depend on why you are teaching calculus. What is the benefit students are getting by going through calculus? From the original post and comments it appears that math and calculus are taught to give students problem solving abilities that they can use in later life. Calculus fails in this respect. Every student forgets whatever calculus she was forced to learn as soon as the school is over. And rightly so. I have not the occasion to use calculus to solve a real life problem last 20 years. Similarly, I have had no occasion to use Latin to communicate in order to conduct my daily business. Human beings learn problem solving abilities by practice in real life, not by cramming theoretical calculus concepts.
Students who are judged by teachers to be unable to solve real life problems because they fail to grasp *theoretical* problems of calculus do pretty well in real life and solve truly complex problems such as relationships, finding a job, making money and just going through life. You don't need calculus in the 21st century to live your life successfully.
The one single reason why calculus should no longer be taught is that students do not remember calculus after they pass the last exam. Instead, calculus makes students hate mathematics and science. Is this the goal teachers wish to achieve?
On the other hand, in our society computer languages are useful at many levels. Today everyone at some point in their daily life interacts with some kind of programming by using computers. The twenty first century citizen interacts with software (programming languages) not with calculus. Calculus does not exist in real life.
Calculus was invented as an algorithm to compute tangents. In today's world the computation of tangents is not anyone's priority. In fact, I don't know anyone, including any professional mathematician, who would consider computation of tangents to be a priority. Calculus is a 17th century solution but a 21st century problem.
Then why teach a dead and useless language which is calculus? The only reason is that once a scholastic subject enters the curriculum it will not be removed except by force. Why do you think Latin is not taught today as the language of science? Calculus replaced Latin as the professional communications language of scholasticism. Teachers can spare the kids the torture of learning this dead language which exists only in academic scholasticism.
Give a student the assignment of writing a "hello world" program in any computer language and this simple exercise will be more beneficial to the student than the entire calculus torture.
The most valued concept of mathematical proof was invented because there was no computer to do the evaluation of the algorithm. The mathematician was also the evaluator and there was a need to establish strict and standard rules to evaluate algorithms. This is called proof. When you run a comptuer program it is evaluated either properly (proved) or you get an error message.
As the language to be taught I would recommend LISP because it is mathematics and it is fun and there is nothing students will miss if they studied LISP instead of calculus. As an added benefit they could program similuations they see in Mythbuster. They cannot do this with calculus. Gregory Chaitin describes LISP as "a beautiful and highly mathematical formalism for expressing algorithms." Similarly, calculus can be described as an ugly and bloated formalism for expressing *an* algorithm.
Calculus is a dead language. When Newton entered Cambridge he had to learn Latin as the language of science. It is not a coincidence that Newton's Principia signaled the end of Latin and established calculus as the new Latin. (Why do you think this is the case?) Calculus is the programming language of the seventeenth century while computer languages are the mathematics of our day.
Thanks again and I apologize for writing such a long comment. I think both students and teachers will benefit from a reevaluation of the place of calculus in modern curriculum.
Hunh. That's interesting, GP1. I *despise* math, in general. I had a number of appallingly bad experiences in math education, and have been assessed as borderline learning impaired with respect to 'basic' functions.
But I thought calculus was pretty cool. It made *sense.* How things change, and how they're changing in relation to other things that are (or are not, I suppose) changing, was very useful. I can't do it, of course, because I have an immense amount of trouble with algebra, but the way it made me think stuck with me. Given that my field is sociology, the ability to think about the way variables are changing in relation to each other is one that is pretty important.
(I have the same difficulty I have with algebra when I'm working with computer languages. Symbol systems make my brain melt, while things that I can see as pictures (calculus) or words (geometry) are easy. From a pure 'learning styles' standpoint, I would say that your suggestion to use computer programming languages is one that would leave me, personally, in just as poor a position.)
People learn to problem solve differently. And yes, much of math education is awful and turns kids (and adults) off math, science, and anything vaguely related for good. I think a solid, reasoned approach integrating both drill (because calculating a tip is a lot easier if you know your multiplication tables) and intuition (because sometimes you need to set up a lot of things in relation to each other, and knowing how to think about your problem is going to help) is key.
In that sense, I agree with Agnostic. I have, however, definitely come across advocates of an 'algorithm only' math setup. Current standardized testing, especially under No Child Left Behind, emphasizes this sort of teaching style (because time limits and multiple choice/short problem setups don't lend themselves well to creativity, even of the fairly constrained 'how do I solve this equation' sort). It's a lot easier to evaluate large groups for 'learning' when the criteria are easily quantified (multiplication tables, geometric theorems, etc.), and actual problem solving ability doesn't fall anywhere in this category (and thus is dubbed 'useless' by a fairly significant percent of the population. Besides, as is frequently mentioned, how often are you going to need to know how to think like a mathematician?)
Very few people advocate a true mix, but quite a few people are stuck on one side or another (even if they'll concede that the other side might have one or two little things going for it, like computational speed or slight increases in student retention). Fewer still people advocate what I advocate, which is a situational mix. Not every child needs to have their time tables shoved down their throats, and not every child will thrive being told to 'just think about it.' If the goal is learning and skill acquisition, then the program must be tailored to the children in it.
Periphrasis wrote "Given that my field is sociology, the ability to think about the way variables are changing in relation to each other is one that is pretty important."
Thanks for your post. I wonder if there is a clearly expressed definite goal why calculus and mathematics are taught at all and why they are taught the way they are taught? Do you know if there is a universally accepted justification for teaching math?
I agree that the most interesting part of calculus for me too was its premise that it is a system to study variables and how they change. But this is not an innovation of calculus and you don't need calculus to study it. The fundamental concept humans used for millennia before calculus was the concept of proportionality. If you look at Principia Newton uses only ratios and proportionalities. Calculus is another layer of notation on the concept of proportions and ratios. It is the algebraic expression of rules of proportionality. So if you study Euclid and Appolonius and Archimedes and Galileo you would not need calculus at all. In a proportionality you keep one term constant while you imagine the other terms to vary. There is nothing more to variables and how they change. In any case it would be nice to know why mathematics is taught the way it is taught. I have a feeling that teachers do not have the answer since they have no choice in what to teach or how to teach it.