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Date: Fri, 02 Jul 1999 11:51:58 -0500
To: rdyarrow@elnet.com
Subject: Early use of Calculator Debate
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Status:
I have always closely followed the calculator debate on the K-12 Math Usenet
Group because I am always interested in teacher's attempts to defend the
ever-present calculator in the elementary grades. Since they have no research
support, I am not surprised that some of them use "boredom" for their defense.
Recently the debate has seemed more intense than two or three years ago, and I
thought that some of you might be interested in the postings of educators who
are opposed to early rampant calculator use. This first article is especially
interesting. I didn't include the pro-calculator posts which still make me
grimace. Mary
1. Article: 1 of 1
From: Domenico Rosa
Subject: FW: calculator article
Date: Tue, 15 Jun 1999 22:58:14 GMT
A slightly edited version of the following article has been published in the
May 1999 issue of Math ed Dialogues with the title "Do we need calculators?"
Dom Rosa
===========================
We all use calculators in daily life; why should we forbid them in school
mathematics classes, hobbling our children as we would never think to hobble
ourselves?
I believe, nonetheless, that calculators are often detrimental to the teaching
of mathematics. As the mathematician Ralph Raimi has written, "Education is not
imitation of life; it is an artificial process designed to put ideas into the
mind and not answers on paper." At the K-6 level, and
given the knowledge base of our average teacher, calculators only produce
answers.
From the TIMSS results it is clear that mathematical competence at the k-6
level does not require calculators. Two of the highest achieving countries at
the 4th and 8th grade levels, Singapore and Japan, use calculators sparingly in
elementary schools.
At the 7-16 level we might do well to heed the voices of those with the most
experience with calculator usage. John Duncan, a mathematics professor
at the University of Arkansas and an early advocate for technology in a college
setting, wrote these cautionary words in 1995, "Some of us who were very early
to use technology to alleviate drudgery, to
visualize graphs and surfaces, to conduct helpful experiments, etc. are now
alarmed at its use as a substitute for thinking. It even seems to deter
problem-solvers from producing general mathematical proofs by holding their
focus to computing a few numerical examples."[1] Overseas, Great Britain has
been at the forefront of using calculators in both elementary and secondary
settings since the 1982 Cockcroft Report. Anthony Gardiner, a mathematician
from the University of Birmingham, has graded thousands of competition papers
and has developed an excellent sense
of the shift in mathematical capabilities of students in his country over the
past 15 years. According to Gardiner the worst effects of calculator usage are:
(1) The loss of experience in SIMPLIFYING
and the consequent loss of student (and teacher/examiner) expectation that
expressions should have any MEANING.
(2) The DESTRUCTION WITHIN HALF A GENERATION
of a hard won, effective algebraic symbolism - developed and proven over
centuries, capable of being manipulated as a "calculus" for exact numerical and
symbolic calculations, and its replacement by slavish verbatim copies of what
appear in calculator displays. (3) The collapse within 10 years of arithmetical
fluency within the very best students - with the resulting loss of meaning for
symbolic generalizations of numerical expressions.
(4) The loss of all attempts to teach pupils to present solutions in forms that
others can make sense of - and the decline into mere personal jottings en route
to an answer.
(5) Which is related - I suspect - to the two most damning outcomes of
post-Cockcroftian innovations:
(i) The astonishing switch from solving simple problems (i.e. METHODS) to
caring only about ANSWERS (i.e. things which appear in the display of a
calculator) - exactly the opposite effect of what the innovators claimed they
wanted, and so horribly widespread that no one can pretend not to know what has
happened; (ii) The inability of students to solve two-step problems, because
teachers
and examiners have learned to accept mere answers since psychologically it is
almost impossible to train students who are expected to use a calculator to
write anything else down on paper.[2]
One might assume from what I have written and quoted that I am an anti-tech
Luddite who forces his students to do thousands of long division problems by
guttering candlelight. Not so. I have a dozen computers in my room and 10
ti-92's for student use. But I think too much of my students and the
mathematics they need to learn to condemn them to a black box paradise of
mindless button pushing merely for the sake of being on the cutting edge of
the math reform movement.
[1] American Mathematical Monthly, 102(1995), p.194. [2] personal email.
Bio: Kim Mackey has taught math and science in Alaska for 12 years. In 1994
he received a Distinguished Teacher award from the White House Commission on
Presidential Scholars, and in 1998 he was honored as the National Academic
Decathlon Coach of the Year.
Kim Mackey
Box 1996
Valdez, Alaska 99686
2.
Article: 3 of 3
From: Dan Vande Bunte
Subject: Re: Influence of Calculators
Date: Thu, 27 May 1999 13:15:06 GMT
I've made this argument before, and I will make it again. Because students are
allowed to use calculators too soon, high school students...HIGH SCHOOL
STUDENTS!...do not know that negative numbers squared are positive numbers.
Even worse, they do not know that ANY negative number times any OTHER negative
number is positive. I've warned them too many times that typing "-3^2" into
their calculators often yields incorrect responses due to the implicit order of
operations used in the programming of the calculator. Which reminds me, they
don't even know the order of operations!
THe fundamental issue at stake here is not calculators vs. no calculators. It
is philosophical. In order to appropriately answer the calculator question, we
must be able to clearly and coherently articulate a sound philosophy of
mathematics education.
Those who push calculator use tend to feel that mathematics education consists
of teaching students a pre-determined list of topics and subjects in the hopes
that they learn how to DO math. These are people who are generally considered
"applied" mathematicians. Those who urge against the use of calculators tend to
see mathematics education as consisting of numerous things: certain topics and
subjects, DOING math, LEARNING math, learning mathematical REASONING,
developing problem solving abilities, and generally giving students insights
into what mathematics is and is about (i.e. what is it that mathematicians do).
These people are generally considered "pure" mathematicians.
What's the real difference? Applied mathematicians tend not to care about how a
theorem or theory was developed, only how it is used. Hence the emphasis on
DOING math. Pure mathematicians care little about practical application and
more about the beauty that IS mathematics. They care very little about the
extra-curricular, and more about thinking mathematically. Hence the emphasis on
theory. I myself, if it isn't already obvious, consider myself a "pure"
mathematician, often "doing" mathematics just for fun. Hence my displeasure
with calculators. Don't get me wrong, they can be useful, but their overuse is
something I consider to be a great disjustice to mathematics education.
To complicate matters even further, applied mathematicians and pure
mathematicians seldom LIKE each other. Applied mathematicians often see
mathematical theory with no "real life" application or use as being, well,
useless. Obviously, such a philosophy of mathematics would be disconserting to
those who are good at, and generally enjoy mathematical theory. Pure
mathematicians often find "real life" application as irrelevant to the study of
mathematics. Mathematics is worthy of study in its own right, and not
contingent on its practical use. This, clearly, is upsetting to those who have
devoted their lives to applying mathematical theories to real life.
Personally, my own philosophy of mathematics education is a synthesis of both.
There are certain topics and subjects which must be taught. Certain abilities
must be mastered. Applications must be tied into theory, but theory does not
necessarily have to be tied into application. I.e. application must have basis,
but theory needs no ornaments. I also feel very strongly that mathematics
education must also teach students what mathematicians do. Notation, proofs,
reasoning, skill, memorization, and much more that, unfortunately, most
students do not see the importance of.
With this in mind, using calculators to teach multiplication facts is bad.
Using calculators to calculate standard deviations is good.
Daniel J. Vande Bunte dvbunt86@calvin.edu (stuff I can access on campus or at
home)* DanVandeBunte@msn.com (stuff I can only access from home)
danvandebunte@hotmail.com (stuff I can access on campus or at home)* * = your
best bet of having me see your message before 11pm.
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3. Article: 1 of 2
From: Sbehel1234
Subject: Calculators in the Classroom 4th grade
Date: Mon, 31 May 1999 16:27:43 GMT
I want to use both drill and calculators in the classroom for teaching math. I
love math and patterning(great book called Mathographics by Robert Dixon) and
the whole "game." What are the general knowledges of the professional readers
on this subject? Have there been research studies on this subject to your
knowledge? Please include experiences with the abacus. Enlighten me.
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REPLY:
Article: 2 of 2
From: Thomas W. Cowdery
Subject: Re: Calculators in the Classroom 4th grade
Date: Mon, 31 May 1999 17:33:40 GMT
My first reaction is YOU ARE GOING TO ROT THEIR BRAINS!
OK, now that the blood pressure is back to normal, an occasional project using
calculators in the 4th grade probably won't do any lasting damage, but PLEASE
actively discourage using them on a regular basis. Check out
www.coe.ilstu.edu/jabraun/students/cowdery/titlepage.htm
for a paper I once wrote for a grad class about using calculators in elementary
school (putting it out as a web page was also part of the class).
-- _.-'`'-._.-'`'-._.-'`'-._.-'`'-._.-'`'-._.-'`'-._.-'`'-._.-'`'-._.-'`'-._.-'
`'-.
"We Have Met The Enemy, And He Is Us" Walt Kelley's Pogo
tcowdery@usa.net (work) cowderyt@district87.org
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4. Article: 1 of 2
From: Doug Porter
Subject: Reply to "Influence of Calculators"
Date: Thu, 03 Jun 1999 13:30:39 GMT
I believe calculators should be limited in the high school classroom.
From geometry through calculus, calculators are necessary and important tools.
But those students who are taking algebra or pre-algebra, the use of
calculators should be limited. Many of these students cannot do basic fraction
operations, nor can they perform computations with positive and negative
integers. My pre-algebra students never use calculators and my algebra students
only use them with radicals and scientific notation. One big problem is the
fraction key. Many middle school teachers in our system let the students use
this key. Thus, the students do not learn the basic rules of fractions.
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5. Article: 2 of 2
From: Mark Bouwmeester
Subject: Re: Reply to "Influence of Calculators"
Date: Thu, 10 Jun 1999 01:07:03 GMT
IMHO, the problem here is that kids learn to do questions like 3/4 + 1/2 on a
calculator (which is what a calculator is for) but never learn the rules for
adding (or subtracting, or multiplying, or dividing) fractions. Then they get
to a question like (x+5)/(2x-3) + x/(x+1). Calculator can't help here. You need
to know how to do addition and subtraction of fractions to simplify this
correctly. The answer is certainly not (2x+5)/(3x-2), although I bet this is a
very common error.
(Of course some of the Newest, most advanced calculators may be able to do
this, but how many students have access to these calculators?)
A possible list of 4 most important things to know:
addition, subtraction, multiplication and division of fractions Pythagorean
Theorem
how to set up a proportion
addition, subtraction, multiplication and division of polynomials
Howard L. Hansen wrote in message
news:37597656.43223807@news.gte.net...
Doug,
Please tell me more about the "basic rules of fractions" and "fraction
operations." What do you believe are the 5 most important things a pre-algebera
student should learn in your class? Ditto for algebra.
6
Article: 4 of 4
From: William L. Bahn
Subject: Re: Use of calculators
Date: Tue, 08 Jun 1999 03:21:07 GMT
And, just in time to illustrate the points that many of us have been trying to
make, here is a post from one of the electronics newsgroups. It is quite
typical of far too many many posts.
RBanks wrote in message <7jg63s$knu$1@topsy.kiva.net>...
I am stuck on this problem in my electronics class. Can someone please help me?
_______1_______
fr = 6.28 * sqrt LC I need to solve for L. How do I get it by itself?
This person needs to be sent back to about 6th grade (or further) and when they
have actually learned the stuff that they should have learned the first time
through, then they should be allowed to pursue their education in electronics.
William L. Bahn wrote in message <375b4c1d.28187701@news.gte.net>...
And then, some day, this person gets to the point where they need to use a
microcontroller that can add and subtract single byte values and write a
program to multiply two bytes together. They will be clueless. I've seen it
happen. And when they need to divide two bytes they might as well drop out of
college and go get a job at Burger King so that they can rely on Burger King's
cash register to do their math for them (saw that happen once, too). I think
it's pretty sad when you have college seniors in electrical engineering and the
instructor has to spend several class periods teaching the basics of what a
positional numbering system is, how you perform multi-digit multiplication
using only single-digit multiplication and addition and how you perform
long-division - i.e., fourth and fifth grade math.
Your recommendation of only drilling them to the "fives" means that you are
expecting and actively encouraging that we make sure that people cannot
multiply ANYTHING without using a calculator. They buy lunch someplace and the
bill totals $6.00 and they can't figure out how much money to leave for a tip
because that involves multiples of 6 and we don't teach that anymore.
Sbehel1234 wrote in message <37589d83.53254152@news.gte.net>...
Objective: To get on with math applications and stretch the students'
quantitative skills.........
Why would one not teach the concept of multiplication to the "fives,"
drill
>and memorize them, and leave it at that, going on to the calculator? The long
.