The Co-Evolution of Calculators and High School Mathematics

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The Co-Evolution of Calculators and High School
Mathematics
Dan Kennedy Baylor SchoolChattanooga, TN
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Change makes everyone less comfortable ..but we
change because we must. Calculators have changed
quite a bit in the last 20 years.
And so has high school mathematics.
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Some people seem to think that pre-college
mathematics is timeless. If it was important for
our parents, how can it be unimportant today? But
technology has been rendering our parents
mathematics obsolete for decades. For example,
consider log tables.
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Here is a 1928 College Board mathematics
achievement exam. It looks a lot like todays
college placement tests. But that is another
talk.
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Notice that problem 7 is from the 1928 version
of the Real World. You must find the angle of
elevation of a balloon by using logarithms.
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In the old days (e.g. 1970), any good algebra
book had a table of 5-place logarithms to solve
problems like 7 which was posed in 1928.
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log 1613 log (1.613 103) 3.20763
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log 2871 log (2.781 103) 3.45803
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9.74959 10
So log (tan ?) 9.74959 10.
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Now we go to a log trig table and look for
9.74959 in the L Tan column. We find some
success on the 29 page.
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Since 9.74959 is two-thirds of the way between
9.74939 and 9.74969, we conclude that
? 29 19 ' 40 "
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But that was then. This is now
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And speaking of logarithms
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And do any surviving Algebra I teachers remember
these?
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A sobering thought There are people walking the
streets of your town right now who became
convinced years ago that they could not do math
-- because they could not do some things that
we no longer teach today!
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And who defines what it means to do math? MATH
TEACHERS! This a big difference between the
ability to do mathematics and the ability to read!
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Someone who can read this sentence knows how to
read. How about this sentence Ontogeny
recapitulates phylogeny.
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What does it mean to do mathematics?
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The fact is that calculators keep changing the
definition of what it means to do mathematics.
They have done so before and they will surely do
so again.
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The main catalyst for change in high school
mathematics in recent years has been technology.
The passing of log tables and slide rules are
obvious consequences.
Other changes have been more subtle.
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Graphing calculators have brought the
power of visualization to young students of
mathematics.
Bert Waits and Frank Demana
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The AP Calculus Test Development Committee
realized in 1989 that graphing calculators would
change the way that students learn mathematics.
In 1990 they set a goal to require graphing
calculators on the AP Calculus exams by 1995.
This was eventually to become the AP programs
finest hour.
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1990 The College Board Calculator Impact Study
Nearly 8000 students from more than 400 schools
field-tested new test items. 300 college
mathematics departments were surveyed. A diverse
panel of mathematical experts was assembled to
advise the AP committee.
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1991 The Decision was Announced. AP teachers
would have four years to make the transition to
Calculus for the New Century.
Incredibly, they actually did.
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Technology Intensive Calculus for Advanced
Placement (TICAP) was the launching pad.
John Kenelly
Clemson University
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Soon TICAP graduates were conducting AP workshops
across the country, exposing more and more
teachers to the power of visualization for
teaching AP Calculus. And many of these teachers
taught other math courses.
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Graphing calculators have liberated students,
teachers, and real-world textbook problems from
the tyranny of computation.
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Graphing calculators have made more meaningful
data analysis accessible to young students of
mathematics
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Graphing calculators have made word problems more
accessible to students. The emphasis has shifted
much more toward modeling.
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An example of a problem that used to be hard for
students but that now is easy
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After modeling the problem, there are two easy
methods of solving it
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The former paradigm Learn the mathematics in a
context-free setting, then apply it to a section
of word problems at the end of the chapter.
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In 2000, the BC Calculus exam had two lengthy
modeling problems about an amusement park. They
appeared consecutively. Nobody complained much.
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For teachers, these changes have not come easily.
We have made changes, hopefully for the better.
You might think we could pause, reflect, and
enjoy what we have accomplished. But that is not
how technology works! Here are a few changes we
have yet to make
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We need to stop thinking of a students
mathematics education as a linear progression of
skills that must be mastered.
Arithmetic
Fractions
Factoring
Equations
Inequalities
Radicals
Geometry
Trigonometry
Proofs
Functions
Calculus
Statistics
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If students who have not mastered our traditional
mathematics skills can solve problems with
technology, should it be our role as mathematics
teachers to prevent them, or even discourage
them, from doing so?
That does not count, Miss Nouveau. Put that thing
away.
Dr. Retro, Ive got it!
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We ALL must teach fundamental mathemics skills to
our students, who probably will not have mastered
them. Patiently. Casually. As a matter of course.
Mr. Oiler, if there are twice as many dogs as
cats, doesnt that mean that 2d c?
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Mr. Jones, if that is all you learned last year,
you had better drop this course before it drops
you.
Good question, Mr. Jones. Lets see what would
happen if there were 4 cats
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We must honestly confront the goals of our
current mathematics curricula. Just because it
is good mathematics does not mean that we have to
keep teaching it.
Nor is it necessary, advisable, or perhaps even
possible to teach everything that is in your
textbook.
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Example AZ, OK and MA still have Cramers Rule
in their state standards. The purpose of
Cramers Rule is to solve systems of linear
equations using determinants.
Recall
How can we possibly still mandate the teaching of
Cramers Rule?
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Example AL, OK, and CT want students to know how
to compute a 3-by-3 determinant.






0
2
1
(4)
(4)
0
11
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Compare this to
So how do we justify teaching a meaningless
computational trick that is ONLY good for
computing 3-by-3 determinants? It does not
generalize to higher orders. It does not even
suggest anything important about how determinants
work!
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We should treat every mathematics course as a
history course at least in part. We will
probably always teach some topics for their
historical value.
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In fact, if you love Cramers Rule, go ahead and
teach Cramers Rule. Just admit to your students
that you are teaching it for its historical
value. Do not make them use it to solve
simultaneous linear equations!
Cramer Himself
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We must honestly assess every advance in
technology for its appropriate uses in the
classroom. As noted before, we must also
determine what is meant by important mathematics.
Important? Expendable?
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The Skandu 2020 It has the potential to scan any
standard algebra textbook problem directly into
its memory for an analysis of key instructional
words, solve it with CAS, and display all
possible solutions. It will do the same for
standard geometry textbook proofs.
The Skandu 2020 (Not its real name)
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HA HA! Im only kidding. At least for now.
If there is no Skandu 2020 in our classrooms in
five years, I doubt it will be because the design
is impossible. It will be because teachers do not
feel that it would improve the teaching and
learning of important mathematics. This is still
a co-evolution!
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AP Calculus Calculator History 1983 Calculators
allowed, not required 1985 Calculators
disallowed again 1990 Calculator Impact
Study 1993 Scientific calculators required 1995
Graphing calculators required 1997 Reformed
course description 2000 Free-response split
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AP Calculus Calculator Survey ResultsWhich
graphing calculator did you use?(percent of
students)
  Calculus AB Calculus AB Calculus AB Calculus AB Calculus AB Calculus BC Calculus BC Calculus BC Calculus BC Calculus BC
  2002 2003 2004 2005 2006 2002 2003 2004 2005 2006
Casio 6300, 7300, 7400, 7700 TI 73, 80, 81 1.0 1.1 0.9 0.6 0.5 0.6 0.7 0.7 0.4 0.5
Casio 9700, 9800 Sharp 9200, 9300 TI 82, 85 6.6 3.8 2.4 1.4 1.0 4.5 2.5 1.4 0.8 0.5
Casio 9750, 9850, 9860, FX 1.0 Sharp 9600, 9900 TI 83, 83 Plus, 83 Plus Silver, 84 Plus, 84 Plus Silver, 86 74.1 75.7 76.9 79.5 79.9 66.1 67.4 68.2 70.5 70.8
Casio 9970, Algebra FX 2.0 HP 38G, 39, 40G, 48, 49 TI 89, 89 Titanium 17.2 18.2 18.3 17.9 18.2 28.1 28.7 28.7 27.9 27.9
Other 1.1 1.2 1.4 0.6 0.5 0.8 0.7 1.0 0.3 0.3
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Participation and Eligibility Both AMC 10 and AMC
12 are 25-question, 75-minute multiple-choice
contests administered in your school by you or a
designated teacher. The AMC 12 covers the high
school mathematics curriculum, excluding
calculus. The AMC 10 covers subject matter
normally associated with grades 9 and 10. To
challenge students at all grade levels, and with
varying mathematical skills, the problems range
from fairly easy to extremely difficult.
Approximately 12 questions are common to both
contests. Students may not use calculators on the
contests.
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Meanwhile, the CAS conversations continue.
They are not just about technology, nor should
they be. They are about the teaching and learning
of mathematics. Stay tuned. Be informed. Join the
conversation.
It just might be time for another change!
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dkennedy_at_baylorschool.org