Calculating Against
Calculators

From "High Tech Heretic"

Clifford Stoll

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.

Well, no.

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*.