Sobre o uso de Calculadoras Gráficas
Date: Mon, 20 Feb 1995 15:05:16 -0500
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From: Robert Megginson (meggin@math.lsa.umich.edu)
To: Multiple recipients of list (calc-reform@e-math.ams.org)
Subject: [CALC-REFORM:2003] Darko's note on Murli Gupta's paper
X-Comment: From the CALC-REFORM discussion list.
Darko's comment:
>Don't let's forget that I'm still looking forward to hearing
one anecdote that supports the reformed axiom "GCs are good."
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
I must protest this drastic oversimplification of a complicated issue.
There is no such axiom, and I would challenge anyone making that
statement to find one individual who is willing to make the unqualified
statement that "graphing calculators are good." I believe that graphing
calculators can be a useful educational tool IF PROPERLY USED; I will
also be the first to assert that they can be badly misused by a teacher
who does not understand their proper role. For example, if the teacher
shows the students how to use the graphing calculator to locate local
extrema of functions, then devotes a large portion of an hour test to
seeing if the students have learned this skill, then the teacher has
missed the point of having the technology available. On the other hand,
the teacher can use the calculator to have the students explore the
graphs of the functions f_n defined by the formulas f_n(x)=x^n, where
n is a positive integer, then have the students make some conjectures
about the shape of such graphs for different values of n, then finally
justify their conjectures. It is my experience that this sort of
exercise is very useful in aiding the students' understanding.
If graphing calculators can be misused as a teaching tool, does this
make them dangerous? No more so than, for example, the standard
differentiation formulas. Many calculus teachers forget that these
are also just tools rather than the main mathematical content of a
course, and give hour tests whose main purpose is to see if the
student has learned these formulas. (How many times have you seen
calculus tests loaded with questions whose only purpose is to see
if students have memorized these rules and can apply them to
functions that require their repeated application in complicated
combinations? A colleague of mine here at Michigan calls such
exercises "feats of calculus," and he is not being complimentary
when he does so.) A teacher who emphasizes this is just as guilty
of teaching mechanical skills as is the one who emphasizes the
learning of sequences of buttons to be pushed on a calculator
to solve optimization problems. In short, graphing calculators
are not inherently good or bad, but are either useful or
counterproductive depending on the uses the teacher chooses
to make of them.
So perhaps the request made above should be for an anecdote that
shows how graphing calculators can aid in the students'
understanding, rather than the impossible request for an anecdote
that supports the notion that graphing calculators are an
unqualified good. I will stick my neck out by offering one.
During the summer, I teach a mathematics program for high school
students on the Turtle Mountain Indian Reservation in North Dakota.
One of the topics last summer involved trigonometry. I gave the
students a program for their TI-82 graphing calculators that did
the following. They would enter a number, then the TI-82 would
draw a unit circle on the screen and also draw the terminal side
of an angle in standard position having radian measure equal to
the entered number. The students could see where this terminal
side intersected the unit circle and then obtain the location of
that point (using either the TRACE or INTERSECT feature of the
calculator) and from that the sine and cosine of the entered number.
Of course, if the only point of this were to find these values,
then the students could do so much more easily with a few keystrokes
involving the trig keys on the calculator. The actual point, of
course, is to reinforce the students' understanding of the
unit-circle definitions of the trig functions. Having taught this
topic for years, and having watched past students cling to the
right-triangle definitions much longer than is appropriate, I can
confidently say that this exercise gave my Turtle Mountain students
a much better and more immediate understanding of the unit-circle
definitions than students I have previously had. (Of course,
textbooks perform much the same exercise with illustrations, but
having the students participate in the exercise rather than just
observe it happening on paper seems to enhance their understanding
immensely.)
Bob Megginson
University of Michigan at Ann Arbor
Date: Mon, 20 Feb 1995 23:12:08 -0500
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From: dkuhlman@IDEA.uml.edu (DougKuhlmann-PhillipsAcademy-Math)
To: Multiple recipients of list
Subject: [CALC-REFORM:2004] Megginson on GC's
X-Comment: From the CALC-REFORM discussion list.
Bob Megginson is on target about the proper use of GC's. I would
hope more or us would post how they have used the GC in class.
Using it as an exploratory tool, as Bob described with
f_n_(x)=x^n, or using a program to let students find the
intersection of the unit circle with the terminal side of an
angle are both good ideas.
I first realized how GC's changed things when in a precalculus class
we were studying the effects of transformations on graphs of
functions. Students usuually have an easy time with g(x)=2f(x),
g(x)=(1/2)f(x), g(x)=f(x)+2 and g(x)=f(x)-2 as they can easily
mentally compute what happens to the y-coordinate. However,
f(x+2) and f(x-2) always slows things down, as their (incorrect)
generalization doesn't work, i.e. +2 should shift in the positive
direction, hence right. In the past, before GC's (BGC) the students
would initially disbelieve me and I would have to spend some time
convincing them. Now, AGC, they aske me WHY the graphs shift the
way they do. They can graph several functions and see that f(x+2)
shifts to the left. Instead of being their adversary, I am their
assistant helping them understand what they can already see.
It was a pleasant change.
Thanks again, Bob, and I hope others post their favorite uses of the GC.
Doug
--
Doug Kuhlmann
Phillips Academy
Andover, MA 01810
(508) 749-4242 dkuhlman@idea.uml.eduDate: Tue, 21 Feb 1995 06:09:42 -0500
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From: Dick Beldin
To: Multiple recipients of list
Subject: [CALC-REFORM:2006] Re: Megginson on GC's
X-Comment: From the CALC-REFORM discussion list.
With reference to the "loss of authority" teachers have suffered,
I beg to differ. Teachers suffer a loss of credibility, just
like other authority figures who want to be accepted without
question. If my students have the temerity to challenge me,
that is just a sign that they are good students. All too often
they have been cowed into silence by some vindictive authoritarian
in junior high or high school. Education is a by-product of the
conflict between values and ideas with personal authority having
very little (at least in theory) to contribute to the resolution.
I teach my students that math is a language devoted to persuasion
that one's ideas are sound. Without vigorous challenges, there is
no need for such persuasive power.
The quest for "authority" is the antithesis of "empowerment".
If one knows his or her own capabilities, then challenges are
welcomed, not feared. First we must know ourselves before we
can teach others. Dick
Date: Tue, 21 Feb 1995 10:23:36 -0500
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From: Lou Talman
To: Multiple recipients of list
Subject: [CALC-REFORM:2009] Re: Megginson on GC's
X-Comment: From the CALC-REFORM discussion list.
On Mon, 20 Feb 1995, Doug Kuhlmann wrote:
>Students usuually have an easy time with g(x)=2f(x), g(x)=(1/2)f(x),
>g(x)=f(x)+2 and g(x)=f(x)-2 as they can easily mentally compute what
>happens to the y-coordinate. However, f(x+2) and f(x-2) always slows
>things down, as their (incorrect) generalization doesn't work, i.e. +2
>should shift in the positive direction, hence right.
Their generalization is less likely to be incorrect if one treats
the equations y = 2f[x], y = f[x]/2, y = f[x] + 2, and
y = f[x] - 2 as y/2 = f[x], 2y = f[x], y - 2 = f[x], and
y + 2 = f[x], respectively.
--Lou Talman
Metropolitan State College of Denver
Date: Tue, 21 Feb 1995 07:30:05 -0500
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From: dkuhlman@IDEA.uml.edu (DougKuhlmann-PhillipsAcademy-Math)
To: Multiple recipients of list
Subject: [CALC-REFORM:2007] Re: Megginson on GC's
X-Comment: From the CALC-REFORM discussion list.
>Doug,
>Two quick points:
>1) I believe that the graphing calculators should NEVER be used as
>an objective authority to "settle" student-teacher "disputes".
If I cannot convince my students that MY sketch of g(x) = f(x + 2)
is correct, then I really don't know how to teach the material, do I?
TTeachers have suffered tremendous losses of authority in the
classroom over the years, and to freely give up the role of a
higher academic authority is a grave mistake.
Dick Belden has responded to this better than I could. See his
recent posting. I was thinking of an exploratory exercise
where the sudtents graphed severel examples like f(x+2) and
then asked WHY.
>2) The graphing calculators do not answer the student questions WHY
>the functions shift in such a way---they only demonstrate THAT they
shift that way. If you have to answer their questions WHY (AGC), how
do the calculators really save much time? If the students understand
WHY, after you explain it (that's your job, not the calculator's),
then they will accept your sketches as authoritative. They will not
need (and, IMHO, neither do you) a higher classroom authority.
>John
I agree that the GC's do not answer WHY. I thought that was clear in
my first posting. I never expect the GC to explain why. How could
it? I would hope that some of my students would be moved to think
about the WHY after seeing some examples. I suspect that both BGC
and AGC many sudents merely memorized a rule and ignored our
explanations. (Finally, I'm up early this morning and a little
dense--what does IMHO stand for?)
>On Mon, 20 Feb 1995, DougKuhlmann-Phillips wrote:
>>I first realized how GC's changed things when in a precalculus
class we were studying the effects of transformations on graphs
of functions. Students usuually have an easy time with g(x)=2f(x),
g(x)=(1/2)f(x), g(x)=f(x)+2 and g(x)=f(x)-2 as they can easily
mentally compute what happens to the y-coordinate. However,
f(x+2) and f(x-2) always slows things down, as their (incorrect)
generalization doesn't work, i.e. +2 should shift in the positive
direction, hence right. In the past, before GC's (BGC) the students
would initially disbelieve me and I would have to spend some time
convincing them. Now, AGC, they aske me WHY the graphs shift the
way they do. They can graph several functions and see that f(x+2)
shifts to the left. Instead of being their adversary, I am their
assistant helping them understand what they can already see. It was
a pleasant change.
>>Thanks again, Bob, and I hope others post their favorite uses of the GC.
Doug
--
Doug Kuhlmann
Phillips Academy
Andover, MA 01810
(508) 749-4242 dkuhlman@idea.uml.edu
Date: Tue, 21 Feb 1995 20:11:12 -0500
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From: walter spunde
To: Multiple recipients of list
Subject: [CALC-REFORM:2017] Re: Megginson on GC's
X-Comment: From the CALC-REFORM discussion list.
Extracted from DougKuhlmann-PhillipsAcademy-Math
>... I was thinking of an exploratory exercise where the sudtents
graphed severel examples like f(x+2) and then asked WHY.
>>2) The graphing calculators do not answer the student questions
WHY the functions shift in such a way---they only demonstrate
THAT they shift that way. If you have to answer their questions
WHY (AGC), how do the calculators really save much time?
... and someone else wanted to classify calculators as either
good or bad. Of course, they are good when you know how to handle
them and bad when you do not. For students having problems with
functional notation, and they may be upper year engineering
students dealing with the Seconding Shifting Theorem, nearly
all graphing packages are bad because they hide what in truth
they are doing. They are not "graphing formulae" as students
may be forgiven for thinking, since the input to these packages
is a formula. They are in fact plotting a sample of points and
joining them with a sequence of dots roughly in a straight
line. When students are asked to obtain their own sample of
x values, increment them by 2, apply an executable
function f to these values to get y and then plot y vs x
with a package that accepts only points, there is no
confusion about f(x+2) or the direction of the shift.
To save time, one must have the right tools for the job.
% ______________________________________________________________________%
% W.G.Spunde, Mathematics Department, USQ, Toowoomba, Australia, 4350 %
% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%
Date: Wed, 22 Feb 1995 17:01:57 -0500
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From: Bradford R Findell
To: Multiple recipients of list
Subject: [CALC-REFORM:2031] Re: Megginson on GC's
X-Comment: From the CALC-REFORM discussion list.
On Tue, 21 Feb 1995, Forrest - John B. wrote:
>1) I believe that the graphing calculators should NEVER be used as
an objective authority to "settle" student-teacher "disputes". If I
cannot convince my students that MY sketch of g(x) = f(x + 2) is
correct, then I really don't know how to teach the material, do I?
TTeachers have suffered tremendous losses of authority in the
classroom over the years, and to freely give up the role of a
higher academic authority is a grave mistake.
By 'convincing your students' do you mean 'keep explaining it in
different ways until they agree'? If they then agree, can we
conclude that they understand? I would argue that NO amount of
explanation will promote understanding unless it somehow connects
with the student's experience, and aren't we interested in
understanding rather than just agreement? A GC exploration can
provide experience which doesn't depend on the deep algebraic
understanding that a purely algebraic explanation would require.
And perhaps the GC exploration can help them make more sense of
the algebra.
>2) The graphing calculators do not answer the student questions
WHY the functions shift in such a way---they only demonstrate THAT
they shift that way. If you have to answer their questions
WHY (AGC), how do the calculators really save much time? If
the students understand WHY, after you explain it (that's your
job, not the calculator's), then they will accept your sketches as
authoritative. They will not need (and, IMHO, neither do you) a
higher classroom authority.
I am concerned, too, about the 'authority' issue, but I am not
sure what you mean by authority. If you mean 'conviction' or
'source of belief' then I would argue that real understanding
is based on an internal authority, arising through connections
with other knowledge. Thus, I hope that each student is his or
her own final authority. If, on the other hand, you mean
'control of the classroom' and other discipline issues, then
I offer no argument.
It seems from your post that you believe that the teacher's
mission is to get students to accept the truth, and that he or
she should be the final authority on all truth, at least in
mathematics. Am I misinterpreting your view?
brad findell
Univeristy of New Hampshire
Date: Tue, 21 Feb 1995 05:41:50 -0500
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From: "Forrest - John B."
To: Multiple recipients of list
Subject: [CALC-REFORM:2005] Re: Megginson on GC's
X-Comment: From the CALC-REFORM discussion list.
Doug,
Two quick points:
1) I believe that the graphing calculators should NEVER be used as
an objective authority to "settle" student-teacher "disputes".
If I cannot convince my students that MY sketch of
g(x) = f(x + 2) is correct, then I really don't know
how to teach the material, do I?
Teachers have suffered tremendous losses of authority
in the classroom over the years, and to freely give up
the role of a higher academic authority is a grave mistake.
2) The graphing calculators do not answer the student questions WHY
the functions shift in such a way---they only demonstrate THAT
they shift that way. If you have to answer their questions
WHY (AGC), how do the calculators really save much time?
If the students understand WHY, after you explain it
(that's your job, not the calculator's), then they will
accept your sketches as authoritative. They will not need
(and, IMHO, neither do you) a higher classroom authority.
John
On Mon, 20 Feb 1995, DougKuhlmann-Phillips wrote:
>I first realized how GC's changed things when in a precalculus
class we were studying the effects of transformations on graphs
of functions. Students usuually have an easy time with
g(x)=2f(x), g(x)=(1/2)f(x), g(x)=f(x)+2 and g(x)=f(x)-2 as
they can easily mentally compute what happens to the
y-coordinate. However, f(x+2) and f(x-2) always slows
things down, as their (incorrect) generalization doesn't
work, i.e. +2 should shift in the positive direction,
hence right. In the past, before GC's (BGC) the students
would initially disbelieve me and I would have to spend
some time convincing them. Now, AGC, they aske me WHY the
graphs shift the way they do. They can graph several
functions and see that f(x+2) shifts to the left. Instead
of being their adversary, I am their assistant helping
them understand what they can already see. It was a pleasant change.
>Thanks again, Bob, and I hope others post their favorite uses of the GC.
>Doug
>--
>Doug Kuhlmann
>Phillips Academy
>Andover, MA 01810
>(508) 749-4242 dkuhlman@idea.uml.edu
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