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: 
> sin(38.0); 
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 
answer down. 
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 
> evalf(sin(38/180*Pi)); 
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. 
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