25 Aug 00
Here is my tirade on calculators. Something a little more constructive shall follow
In 1988, when I took trigonometry in high school, graphing calculators
were an expensive new tool and calculators hadn't really been integrated
into the mathematics curriculum. We mainly used calculators to add,
subtract, multiply, divide, and sometimes even take a square
root. However, even these most rudimentary calculators were forbidden my
first quarter in trig.
How could we function, one might ask. How did we know the sines, cosines,
and tangents of arbitrary angles? Well, first of all, we memorized some
sin, cos, and tan values for standard angles: 30 degrees, 45 degrees, 60
degrees, etc. We also knew the half-angle, difference of angles, sum of
angles, and double angle formulas; we also had trig tables.
Indeed, we were forced to do linear interpolation =by hand= for a full two
months of trigonometry. This tedious procedure, involving long division,
made one plan very carefully before doing anything. One made sure that it
was a cosine, and not a sine, that needed to be calculated (mainly because
we had not learned how to take square roots by hand). One set up one's
equations well in advance, using as many identities and geometrical
theorems as possible to minimize the amount of numbers that needed to be
calculated. Even so, many of us did not complete our tests in the given
The next quarter, we were allowed to use calculators. Unfortunately for
me, my sisters kept stealing my calculators. So I was stuck with trig
tables or my dad's slide rule, which was indeed much faster than
interpolation but still involved some estimation and calculation by
hand. Interestingly, I would finish tests before the other people in
class. Why? I had learned the lesson of efficiency in
problem-solving. I still set up my equations carefully and determined
well in advance of any calculations which functions would need to be
calculated. Many of my fellow students hadn't learned this lesson, and
so, when faced with a few named angles, they would apply every
trigonometric function we knew to the angles and do various things with
those results and the side lengths of given triangles. I have no idea as
to how they determined which of the four or more numbers was the answer
they actually wanted. And woe to them if they didn't pay attention to
whether the calculator was in degree or radian mode.
Fast forward to 1996, in which very sophisticated calculators and computer
programs have been incorporated into the math curriculum, from pre-algebra
to trigonometry to calculus and beyond. I spent four years as a
computer consultant for the math department at North Carolina State
University, which had fully incorporated the symbolic math program Maple
into its Calculus curriculum. I saw many of the students doing the same
thing as my fellow students from so many years before: taking the
functions they had and applying all sorts of things from example Maple
worksheets to it, hoping they would recognize the answer when they saw
it. If they were lucky, the homework problem exactly paralleled the
Usually, they were not lucky, and they, like Cinderella's step-sisters
trying on the dainty slipper, would hack at the problem given trying to
make it fit one of the examples that had previously been done.
This, obviously, is a stupid way to apply technology to math problems.
Much has been made of the use of calculators and computers in math, and
they are indeed very useful, powerful, and even necessary tools in modern
math research. However, I feel that the focus of the use of these tools
has been misplaced. Too often they are seen as something that can remove
the tedium from math, as opposed to tools that remove tedious calculations
that one understands very well and can do by hand one's self.
People claim that many students are turned off by math early on due to
excessive rote memorization of addition tables, multiplication tables, and
the like. Math is about recognizing patterns, not simply arithmetic, they
enthusiastically proclaim, and let us use calculators to cut through the
tedium of practicing long division and graphing lines.
I would agree with them -- mathematics has very little to do with
arithmetic and has everything to do with finding patterns and relations
and using these things to solve problems. Indeed, I rarely do long
division by hand, or even integrate by hand anymore. However, I do not
agree with the reasons as to why students are getting turned off from
They get turned off because they do not understand it.
They do not understand it because they don't have enough practice with
basic problems and are rushed onto harder problems that are grounded in
one's knowledge of what multiplication or division means. They haven't
dirtied their hands in the earth of numbers, so when they're asked to tend
a 10-acre field of word problems, they become flummoxed.
Let us consider other subjects which children are taught. In music, one
is usually forced to practice scales. Indeed, my guitar teacher makes me
do them rigorously. This is very boring and very tedious. However, if I
want to do smooth jazz improvisations or agile sight-reading, I need to
practice these basics. In piano, I had to do finger exercises that were
not only tedious but sometimes painful to build up individual finger
strength. In crew, I had to practice on rowing machines (which involved a
great deal of pain and tedium) so that my team could fly across the
water. Many other disciplines involve practice of the basics, usually a
very boring chore, so that one can be proficient at higher levels. If one
tries to skip the practice phase, one finds that not only can one not do
the task one wants to do, one becomes very beat up in the process.
No, I didn't really enjoy memorizing my times tables or solving very
similar linear equations or taking countless derivatives or proving
continuity through delta-epsilon proofs. But I did them, knowing that I
was developing my mathematical intuition and making my life in math that
much more smooth for years to come.
Why am I allowed now to use the computer programs that I disallow my
students? Because I already know how to do these and through my
experience, know what answers to expect. Technology nowadays allows the
taking of a derivative, minimization of a function, numerical and symbolic
integration, plotting of a complicated function, and much more difficult
mathematical tasks with little effort on the part of the user of the
technology. However, students often get incorrect answers, mainly because
they are asking the wrong questions.
For example, say I want to know the sine of 38 degrees. I'll type that
into my handy-dandy Maple input line:
Looks good, right? I've got a number between -1 and 1, 38 degrees is in
the first quadrant, so the sine must be positive. Let me write that
However, if I had been thinking, even before I typed in this I would've
known that this number couldn't possibly be correct. Why? Because sine
of 30 degrees is .5, and sine of 45 degrees is about .707. If I looked at
a graph of sine I would know that sin(38 degrees) has to be somewhere in
the middle, say about .6 (indeed, linear interpolation tells me it should
be around .6104). What happened?
Oh, silly me. Maple sees angles as =radians=, not degrees. Let me fix
Now, that's more like it. You see, I was thinking before I calculated any
numbers as to what kind of answer I should get. That way, when I make a
mistake in what I'm calculating, I catch it right away. Students make
this degrees/radians mistake all the time, and one might say, well they
just have to be careful. However, this was just an illustration.
Mistakes are being made like this all the time when students use
calculators and computers (not a problem), and they're not being spotted
(a big problem).
It is true that one can learn to think critically about problem-solving in
math while one always has a calculator in one's hands, and one need not
ever memorize anything in math anymore, for the procedures are all
programmed into the computers. However, I feel that actually doing
calculations by hand =forces= students to be careful, because making
mistakes mean one wastes a great deal of time. If students must do things
by hand, they start being able to see where they've gone wrong and become
more aware as to all the different ways they can go wrong -- misplaced
parentheses, changed signs, dropped digits, etc.
Some people complain that calculator use in math classes is a crutch. I
disagree - they can be very helpful. However, when I see current freshmen
in first semester calculus classes, I realize that often calculators are
not only crutches given to people who can't use their legs but crutches
given to those who also can't use their arms. Students come into
Calculus, with perfectly fine SAT scores, unable to solve algebraic
equations and unable to graph lines. Giving these students graphing
calculators which also can do symbolic math helps noone. For when they
come to a related rates problem which states "You have a circle whose
radius is expanding at the rate of 2 centimeters per second. How fast is
its area growing when the radius is a meter?" they will complain to the TA
"But we didn't know what the area of a circle is!" (this happened to me in
the first class I TAed.)
These are students, having had Calculus the previous year in high school,
classes which went all the way through integrals, who have the slightest
idea of a derivative. They claim to understand Calculus and are in the
class for an easy A. So I ask them to differentiate x^x. This is how
most of them answer:
Okay, so I bring the exponent down and then subtract one: x* x^(x-1)
Then I ask them to simplify that.
Um... (after much banging of head)... x^x.
Then I tell them that the only function whose derivative is itself is
C*e^x. Sometimes they have calculators that can take symbolic
derivatives, which will give them the answer: x^x * (ln x + 1).
Still, looking at that answer, they have no idea why their first solution
was incorrect and even how one would get that second answer. They've not
had enough practice taking derivatives by hand to recognize that (ln x +
1) is the derivative of x ln x, which would clue them into the fact that
some kind of chain rule thing is going on, involving x ln x somehow.
By all means, students should be using calculators to do the tasks they
understand very well. Calculus students should not have to do long
division by hand, or even solving linear equations by hand. Students
taking differential equations shouldn't have to do integrals by
hand. However, students should not be pushed into deep, complex math
before they can master the essential basics. We do not need more
university students who have difficulty adding fractions.
At certain levels of math, one finds that technology doesn't help at
all. I was recently looking at functions that increase only on the Cantor
set, basically a bunch of fractal dust. A computer could approximate the
graphs of these functions for me, but wouldn't give me any real insight as
to what is going on. I recently looked at a bunch of false proofs, many
of which come from plausible graphs and algebraic steps that would be
verified by a computer every step of the way. A computer sometimes does
not warn you not to divide by zero, or take the square root of a negative
number, when one is doing symbolic calculations. Even when I'm doing some
mundane things, such as integrals of seemingly innocent functions, Maple
explodes with a mess that would be cleared away if one applies the proper
trig identities (which I end up doing by hand).
A computer or calculator will give error messages if you misspell a
command, don't give it enough inputs, or ask it to calculate something
impossible (like log(-3)... a calculator will probably spit at you, but
Maple will give you an answer... try it out, and think about
it). However, a calculator or computer will not yell at you: "Those are
the wrong limits for that integral!", "No, you're solving for the wrong
variable!", "You want to multiply, not divide!". These are things only
your brain will tell you, and you must train it accordingly.
The "tedious" math you practice now will give you mathematical intuition
and flexibility. The math problems you will be doing in 40 years are not
the math problems you are doing now; you will probably not have answers in
the back of the book or examples you can copy. You may do no math
problems whatsoever, you think. How about planning for retirement? How
about estimating the surface area of walls you need to paint? What about
comparing two possible car buying options? What about trying to
understand newspaper articles about statistical results of medical
research? Or demographic change? These are not necessarily about using a
calculator, but knowing what you need to calculate.
Anyway, at the very least, before you use a calculator or a computer for
any math problem, think about what kind of answer you expect. This will
get you one step farther in intelligent use of technology.