From "High Tech Heretic"
How could anyone argue against bringing computers into a mathematics
class? After all, computers are natural number crunchers. Wherever
you find algebra, equations, and logic, there's a computer. Of
all places that computers belong, shouldn't it be math class.
The computer and its kid brother, the hand calculator, work against learning the basic arithmetic. They work against appreciating the nature of math. Against familiarity with numbers. Against acquiring an understanding of algebra.
Thanks to digital electronics, students get answers without manipulating concepts: Problem solving becomes button-pushing. It's not necessary to understand how to formulate abstract quantities. Rather, you go straight from numbers to answers. Calculators deliver answers with minimum thought. Confronted by a numeric problem, students naturally reach for electronics, rather than experience. The device first adopted to strengthen mathematical understanding has become a crutch which hobbles the development of numeric literacy.
You no longer need to memorize the times tables to multiply. No surprise that students weaned on calculators can't multiply in their heads. They can't divide. They're minimally cognizant of basic arithmetic.
Watch kids solve math problems with a calculator. They punch the buttons, see the result, and accept what the machine tells 'em. A geography student hands in an assignment where she calculates the height of a Toronto broadcast tower as 0.0034117 millimeters. Numerical literacy tells us to question each result, but the authority of the calculator dulls our critical sense.
Today's students receive only passing exposure to arithmetic tables*. Burned into every calculator, these tables are fundamental to all numeric understanding. Without knowing how to add and multiply, you can't tell when you're being rooked.
Sad to say, learning to multiply isn't a feel-good project, so beloved of the self-esteem movement. Rather, learning the times tables requires rote drill work. It's not fun, like shooting down Martians on a computer screen. But it's one of those must-learn-or-else lessons, without which you're eliminated from many fields of human endeavor.
Kids once had to learn basic arithmetic in second grade. this was back when you often confronted numbers: making change, tipping a waitress, recognizing bargains, balancing checkbooks, understanding public expenditures.
Hey -- those demands haven't disappeared. Today there are more numbers in our life than fifty years ago. We're confronted with sales tax, toll roads, discounts, balloon mortgages, and state lotteries. Indeed, our economic and scientific worlds demand more -- not less -- familiarity with math.
"Thanks to calculators, we get through arithmetic fast," educators tell me. "Children can then advance quickly to more advanced topics. There's so much to learn in math that we shouldn't waste time on arithmetic tables."
This, of course, is backwards. You gotta learn the basics before progressing to advance topics. Without a firm grounding in arithmetic, you'll drown when you hit the high-power stuff. Calculators and computers short-circuit the stepwise progression in learning. Shall we teach reading without requiring children to tediously memorize the alphabet?
Most students won't continue on to higher math. For them, the most important thing they can learn in math is basic arithmetic ... the kind of math that lets you make change and balance your checkbook. These students get cheated when they're taught to solve the problem with a calculator.
Yep, lots of kids trip over arithmetic and algebra. That is the very reason why we should emphasize these areas and spend more time on them. It's why we shouldn't hand out calculators.
Mathematician Neal Koblitz tells of playing math games with sixth graders at Seattle's Washington Middle School. To explore rounding off, they'd divide numbers by seven and give the answer to the nearest integer. What's sixty divided by seven? Good answers include "About nine" or "A little more than eight." But the calculator-equipped kid got 8.5714285 ... and couldn't interpret the answer. They read out numbers, but didn't understand the decimal point.
Moreover, those blinking digits obscure deeper truths. The circumference of a circle isn't 3.14159 times its diameter, but rather Pi times the diameter. And how do you grade a student who says that two thirds times three is 1.9999999?
getting around in life depends on ballpark estimates, approximations, and rounding off. These become second nature after you know manual arithmetic ... after all, they're mental shortcuts. But if you've always depended on your calculator, well, good luck figuring a 20 tip. Don't ask that teenager at the checkout counter -- she probably can't make change without the cash register.
Math educators, discouraged by centuries of bad feelings directed at their profession, have searched for new ways to teach math. I was the victim of one of these evangelical movements: the New Math.
Back in the 1960s, when smart foreigners threatened America, modern educators invented the New Math. And Buffalo's Millard Fillmore Junior High School was modern. We study some old-fashioned algebra. We learned the latest: set theory, Venn diagrams, and number bases.
After three years of New Math, I knew all about the communicative law of addition and the union sets. I could multiply in base seven. And I could write the symbol for the null set -- the collection of nothingness.
So I start high school without knowing algebra. My chemistry teacher writes an equation on the board. A's and B's and C's ... and they all add up to zero. Here I am, budding science jock, and I can't balance a chemistry equation. I'm sweating blood, looking for something to hang a solution on. But there's nothing about set theory or number bases. Just letters dancing on the chalkboard. And a most unwelcome equal sign.
Thanks to the New Math, I knew lots of squiggly symbols but hardly ever saw an equal sign. Algebra -- so essential to science -- had to be learned outside a math class. I wasn't taught how to solve equations, but I sure knew that the null set was a collection of nothingness.
The null set pretty much sums up the New Math. Within two decades, it was recognized as an utter flop. But not before a generation of college-bound students -- smart and stupid alike -- dutifully learned all about theoretical laws of multiplication. By 1975, the New Math collapse like a punctured whoopee cushion.
But if you think the New Math was idiotic, you haven't heard of the New New Math. It goes under the name of Complete Math, Connected math, Reform Math, Constructive Math, or Fuzzy Math. It all adds up to Mickey Mouse Math.
Instead of teaching arithmetic and algebra, students get a half-baked casserole of "real-world problems" to be taught through "discover learning" and solved by group work." Lucille Renwick, who covers education for the Los Angeles Times, reports how New New Math handles a problem of in two-digit arithmetic: Children "would join together in groups and discuss how many beans or sticks it takes to make 10 ... They'd collect the tens and discuss the ones. The students would work out the problems until they understood tens and ones. Reaching an answer would not be paramount."
This isn't math. At best, it's math appreciation, And the result of group work isn't understanding. Perhaps you wind up with a bunch of kids who are good at splitting up work and copying answers. Most likely, a few kids will do most of the work and several confused onlookers try to follow along. But everyone gets an A.
The New New Math is promoted by the National Council of Teachers of Mathematics, joined by the National Science Foundation. They recommend "the integration of the calculators into the school mathematics program at all grade levels in class work, homework and evaluation" and that "every mathematics teacher at every level promote the use of calculators." They justify this with feel-good mumbo jumbo: "The cognitive gain in number sense, conceptual development, and visualization can empower and motivate students to engage in true mathematical problem solving."
Learning algebra in eighth grade is a key to success in college. Yet school boards hate to inflict rigorous courses on unwilling students. They advertise rigorous algebra, but deliver a vapid curriculum, with a heavy emphasis on computing. The natural effect is a continual dumbing down of both curriculum and graduates.
In the heart of Silicon Valley, Palo Alto's school board has provided a decade of New New math, along with a heavy dose of computers. In 1996 they polled parents only to discover a deep dissatisfaction over the number of math facts and techniques the children were learning. Depending on grade, between half and two thirds of families reported hiring outside tutors for their children.
Palo Alto parents, among the nation's wealthiest and best educated, can often afford after-school programs and private math tutors. Other parents have a rougher time when their schools latch onto these touchy-feely math programs.
Arizona State University Professor Marianne Jennings noticed that her daughter Sarah had received an A in high school algebra, yet couldn't solve an equation. Sara's textbook, Secondary Math: An Integrated Approach, Focus on Algebra, includes photos of Mali wood carvings, Maya Angelou's poetry, and praise for the wife of Pythagoras. Questions posed to students include "What role should zoos play in our society?"
It's Rain-Forest Math according to Dr. Jennings. despite the book's title, algebra isn't actually being taught. Sarah's class would measure their wingspans one day; the next they'd toss coins all period. Showing their work on homework or tests was optional. Rather than learning math, children were taught mathematical concepts.
Mathematical concepts? Yale's Professor David Gelernter writes, "Most people have no use for 'mathematical concepts' anyway -- arithmetic yes, group theory no. For the others, the theory that 'real math' has nothing to do with arithmetic is wrong -- engineers and hard scientists are invariably intimate with numbers. They have to be. So if you don't go into math, basic arithmetic is crucial. Whereas if you do go into math, basic arithmetic is crucial."
Dr. Jennings naturally spoke with Sarah's math teacher, only to hear that "We don't plug and chug anymore. We're teaching them to think." Outraged by this justification of academic pblum, she approached the school administrators. Ultimately, a school board member told her, "You may have to face the fact that your daughter won't get algebra." Whee!
Professor Gelernter writes, "The yawning chasm between ed-school doctrine and common sense has already swallowed up (to our national shame) a whole generation of American kids. Big reformers are needed, but the electronic calculator perfectly captures what the struggle is about. When you hand children an automatic, know-it-all crib sheet, you undermine learning -- obviously. So let's get rid of the damned things. Professional educators are leading us full-speed toward a world of smart machines and stupid people."
Practice -- repeated drill -- is out of favor today ... witness that Arizona math instructor's scorn for "plug and chug." Yes it's the core of math competency. If we want students that can handle math, they've got to memorize the times tables. Do plenty of problem sets. Master algebra ... don't just talk about it in vague, fuzzy terms.
"don't forget that computer programming teaches students to think," says a friend of mine who's a computer jock in Silicon Valley. He's deeply invested in technology and has no kids. "programming is a logical system that rewards clear reasoning."
Uh, sure. Nineteenth-century schoolmasters used the same reasoning to justify teaching ancient languages. According to computer scientist Joseph Weizenbaum, "There is, as far as I know, no more evidence that programming is good for the mind than that Latin is."
Anyway, programming teaches us to think of problems as solvable through the sterile Boolean logic of AND, OR, NOT, and IF. Snaggly real-world problems require common sense, buttressed by familiarity with numbers. The ability to construct an argument and spot logical fallacies has little to do with creating computer codes.
Then too, calculators and computers trivialize mistakes. When he gets a wrong answer, a student will typically dismiss the mistake with "Oh, I just pressed the wrong button!" rather than recognizing that he went about solving a problem the wrong way.
Good algebra teachers demand that students show their work to see whether the mistake's due to calculation or misunderstanding. You got a good grade if you took a good path to the answer but made a mistake in calculation. Students were graded on method, not just the final answer. But when a calculator's in use, there's no trail of reasoning and telling why a mistake occurred, so the instructor can't tell the difference.
The Educational Testing Service, purveyors of standardized tests like the SAT and GRE, compared fourth- and eighth-grade students who used computers to learn math. Analyzing a 1996 survey, Harold Wenglinsky found that when eighth graders used computers mainly for math drills like dividing fractions, test scores averaged half a grade lower than for other students.
Mr. Wenglinsky reports that for both fourth and eighth graders, "The frequency of school computer use was negatively related to academic achievement."
Those who used computer programs that encouraged "higher-level cognitive practices" scored slightly higher than those who did not. Similarly, fourth graders who used interactive math games scored slightly higher than those who used the computers mainly for math drills.
His survey showed that a teacher's computing competence has the most effect on whether technology helped or hindered the students. Mr. Wenglinsky concludes that schools should insist on teacher training in technology. But his survey shows that over three fourths of all teachers have already taken specialized professional class in in how to use technology -- that's pretty close to saturation. Teaching more computing to teachers isn't going to make better students.
Look: Teachers should teach ... not become computer jocks. When educational films were the rage, plenty of teachers had to learn how to thread movie projectors. We didn't send them to summer schools for projectionists. Who'll make the better math teacher: the guy that loves math and wants to tell you his latest topology adventures, or the propellerhead who can surf the Web and is an expert at downloading files.
Samual Sava, executive director of the National Association of Elementary School Principals, writes that "37 percent of students used computers in at least some lessons. yet this increased use seems to make no difference to our math results. In sum, if computers make a difference, it has yet to show up in achievement."
Even in college, computers work against mathematical competence. Calculus professors have long been frustrated by high dropout rates. Sad to say, calculus, like much of math, is tough. But it's essential throughout engineering and science ... it's the common language of the physical sciences.
So the University of Illinois developed a calculus course centered on the computer program Mathematica. The software neatly solves problems of arithmetic, algebra, trigonometry, and calculus. Instead of teaching how to integrate functions, instructors teach how to integrate using Mathematica.
Predictably, college students learn how to run the Mathematica program, but they don't learn calculus. Adam Eyring, studying environmental sciences and engineering at the University of North Carolina at Chapel Hill, felt his course was a disaster because "it was hard to go from a program that does your computations for you to doing it by hand."
Professor Jonathan Reichert has taught college physics for some thirty years. His advanced undergraduates learn about nuclear magnetic resonance by flipping spinning electrons in magnetic fields. "The data collection part isn't challenging stuff," Dr. Reichert says. "After the kids set up the experiment, they read off points on an oscilloscope, take the log of the number, then plot the data on the graph. You've got to plot the data as you take it, so that you can uncover your mistakes, root out systematic errors, and keep the apparatus working right. By plotting the answers by hand, crazy answers pop out right away."
But for the past three or four years, students refuse to plot data as they collect it. Instead they write down the data, take it home, and draw the graphs on their computers. It wastes time, since their first graphs inevitably show a problem that they could have quickly uncovered in the lab.
"They want automatic answers," observes Dr. Reichert, shaking his head. "Students can't stand the manual labor of drawing simple graphs. Or maybe they no longer know how. Either way, they miss out on what it means to do physics."
Now I know Professor Reichert -- twenty-five years ago, he taught me the physics of electricity and magnetism. Right after his section on circuit analysis, he stops me in the hallway and hands me six electronic resistors. "What happens if you connect twelve 100 ohm resistors into a cube, with a single resistor on each edge?" he challenges me. "What'll be the resistance across opposite corners?"
I roll the problem around in my mind, knowing that there's go to be a way to answer this question. But it's hairy -- when I draw the schematic, some resistors are in series, others in parallel. I can't see a simple way to analyze this circuit. It's not obvious whether the answer will be more or less than a hundred ohms.
What to do? Well, I get out the soldering gun, clip leads, and ohmmeter. Damn, but it's about 83 ohms. How come? I don't know why, but that's what the meter says. Next day, without blinking I tell Dr. Reichert the answer. He won't let me off the hook. Wants to know how I got it. I show him my carefully constructed cube of resistors, and he rolls his eyes.
I'd cheated -- or at least taken a shortcut. Instead of recognizing a chance to apply what I should have learned in physics, I reached for the soldering gun. Today, students would fire up the SPICE electronic circuit analysis software to get the answer to six digits of precision, yet still not understand Ohm's law, Kirchoff's law, and how to deal with resistors in series and parallel.
Ironically, Professor Reichert notes that today, the one area of computing hat's hardly ever taught is data collection. "Students don't mind fooling with pretty software, but they're lost if you ask them to wire up a computer to control an experiment.
"Hey, I love teaching physics and I've paid my dues on computers," he says. "If the computers such a wonderful learning tool, show me the evidence that our students are better prepared. Even though they're constantly in contact with electronic devices, they're certainly don't know much electronics. They don't know how to assemble and manage even simple experiments. the best students that come through here are the immigrants whose parents can't afford computers."
Mathematician Neal Koblitz believes that computers are unnecessary in the study f mathematics from kindergarten through college calculus. He writes that computers in schools drain resources, corrupt educators, work for bad pedagogy, and hold a broad anti-intellectual appeal.
The drain on resources is obvious, as is the corruption of educators (who do you think sponsors all the Computers in Education conferences?). The bad pedagogy shows up in television shows about math and science, such as the public television shows show "Square One." It's heavy on gimmerickry and pays little attention to content. The ambiance carries into computer and software as well: Program developers work hard on graphics and sounds, and only rarely consult with teachers.
The anti-intellectual appeal of the Internet? Look at the many popular World Wide Web pages which ridicule learning, knowledge, tradition, authority, and scholarship. Dr. Miles Everette wrote about television but his words could equally apply to much of what crosses the Internet: "its methods, pace, and style constantly denigrate the values essential to schooling, concentration, disciplined analysis, wrestling with complexity, and pursuit of understanding."
Larry Braden has received the presidential aware for excellence in teaching. You'll find him in front of a chalkboard at St. Paul's School in Concord, New Hampshire. It's one of those old private schools, charging tens of thousands a year. No shortage of good students.
"I've been teaching math thirty years," he says. "But much as I love computers and enjoy working through algorithms, I sense that computers are ruining math education. Math is a deductive endeavor and computers substitute punching numbers for understanding concepts.
"On the computer, anyone can program the quadratic formula," Larry tells me. "You hit the coefficients a, b, and c, then voila, there are the roots. With the answer on the computer's screen, you think that you know math. But you don't."
I'm chatting with Larry -- he looks a bit like Mr. Chips -- when he looks up and asks point-blank: "What's the average speed of the earth around the sun?"
"I ask every entering student this question," he continues. "Twenty years ago, three quarters of 'em could solve it. Now, it's maybe one in three."
I scramble for a second and suddenly realize that Larry's not asking an astronomy question so much as a simple math problem. High school kids know that the earth goes around the sun once a year. At least they ought to know. They know the distance from the sun to the earth -- again, they ought to know -- so apply the formula for a circle.
Plenty of answers: The earth travels some 67,000 miles per hour. Or about 2.5 million kilometers each day. Or 2Pi astronomical units per year. The main thing isn't getting an exact number. Rather, it's recognizing one more place where the circumference equals Pi times the diameter.
What's the effect of using computers in math education? Over the past fifteen years, colleges have seen an astounding growth in remedial math classes. Pre-algebra, once the mainstay of seventh and eighth graders, has become a common college class. Indeed, two thirds of college math enrollment is in courses which are ordinarily high school classes.
math, of course, also means recognizing those problems which simply aren't numerical: if you need two bananas for each loaf of banana bread,** what'll you do with the overripe bunch in your kitchen? Your calculator might tell you to bake three loaves, but common sense might suggest tossing 'em out.
Much of mathematics means translating problems into abstract representations and converting numerical solutions into understanding. It's something that neither calculator nor computer program can do. It's what each of us struggles with whenever we enter the world of numbers. It's why Larry Braden teaches algebra. It's why we'll forever need arithmetic, algebra, and calculus. And it's why computers don't belong in math class.
* Also true for handwriting. Penmanship, spelling, and
grammar aren't considered worthy of instruction, so they're pushed
aside for word processing. result: Surprisingly few high school
students can write clearly.
** mash 2 ripe bananas and mix with 2 eggs, 2/3 cup sugar, 1/3 cup warm butter, 2 cups flour, and 2 teaspoons baking powder. 1 1/4 hours at 350 F in a 9x5x3 loaf pan.
Clifford Stoll, an MSNBC commentator, lecturer, and a Berkeley astronomer, is the author of the New York Times bestseller The Cuckoo's Egg and Silicon Snake Oil: Second Thoughts on the Information Highway.