The Co-Evolution of
Calculators and
High School Mathematics
Dan Kennedy
Baylor School
Chattanooga, TN
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.
Some people seem to think that precollege 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.
Here is a 1928
College Board
mathematics
achievement
exam.
It looks a lot
like today’s
college
placement
tests.
But that is
another talk.
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.”
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.
log 1613 = log (1.613 × 10^3) = 3.20763
log 2871 = log (2.781 × 10^3) = 3.45803
1613
Tan  
, so log  tan   
2871
log1613  log 2871
3.20763
 3.45803
 0.25040
= 9.74959 – 10
So log (tan θ) = 9.74959 – 10.
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.
Since
9.74959 is
two-thirds
of the
way
between
9.74939
and
9.74969,
we
conclude
that
θ = 29° 19 ' 40 "
But that was then.
This is now:
And speaking of logarithms…
And do any surviving Algebra I teachers
remember these?
Theorem: (b + c) + (–c) = b
Statement
1.
2.
3.
4.
5.
6.
7.
8.
9.
b and c are real numbers
b + c is a real number
–c is a real number
(b + c) + (–c) = b + [c + (–c)]
c + –c = 0
b + [c + (–c)] = b + 0
b+0=b
b + [c + (–c)] = b
 (b + c) + (–c) = b
Reason
Hypothesis
Axiom of closure for addition
Axiom of additive inverses
Associative axiom of addition
Axiom of additive inverses
Substitution principle
Additive axiom of 0
Transitive property of equality
Transitive property of equality
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!
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!
Someone who can read this sentence
knows how to read.
How about this sentence:
Ontogeny recapitulates phylogeny.
What does it mean to do mathematics?
1. 5  2 
2. 24  6 
3. Solve for x : ( x  2)(2 x  3)  49.
4. Find cos( / 3).
5. Find the product: 874539  374958.
6. Find 239121.
7. What is e i ?
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.
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.
Graphing calculators have brought the
power of visualization to young students
of mathematics.
Bert Waits and Frank Demana
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 program’s
finest hour.
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.
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.
Technology Intensive Calculus for
Advanced Placement (TICAP) was the
launching pad.
John Kenelly
Clemson University
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.
Graphing calculators have liberated
students, teachers, and real-world
textbook problems from the tyranny of
computation.
Graphing calculators have made more
meaningful data analysis accessible to
young students of mathematics
Data Shown in the table below is the population growth for the cities of Raleigh, NC and Mesa,
AZ, using census numbers for 1980, 1990, and 2000, and estimates for 2004.
Year
1980
1990
2000
2004
Raleigh
150,255
207,951
282,956
326,653
Mesa
152,404
288,091
397,776
437,454
(a) Using a graphing calculator, find quadratic models for both populations as functions of time.
(Use t = 0 for 1980, t = 10 for 1990, and so on.) Notice that quadratic models fit the data very
well in both cases.
(b) Graph the quadratic functions. The graphs suggest that the two cities will eventually have the
same population. In approximately what year will this occur?
(c) Discuss some reasons why this prediction is probably not very reliable.
(c) The main reason is that 2023 is so far in the future. The conditions in
Raleigh and Mesa might change dramatically from the conditions that gave
rise to the data, for any number of sociological, political, meteorological, or
historical reasons.
Graphing calculators have made word
problems more accessible to students.
The emphasis has shifted much more
toward modeling.
An example of a problem that used to
be hard for students but that now is
easy:
Three families order lunch at a fast food restaurant. The Jacksons
pay $19.40 for 5 hamburgers, 3 small fries, and 5 soft drinks. The
Garcias pay $11.05 for 3 hamburgers, 2 small fries, and 2 soft
drinks. The Lorenzos pay $21.25 for 6 hamburgers, 4 small fries,
and 3 soft drinks. How much would a person pay at this restaurant
for one burger, one small order of fries, and one soft drink?
5h  3 f  5d  19.40
3h  2 f  2d  11.05
6h  4 f  3d  21.25
After modeling the problem, there are
two easy methods of solving it:
The former
paradigm:
Learn the
mathematics
in a contextfree setting,
then apply it
to a section
of “word
problems” at
the end of the
chapter.
In 2000, the BC
Calculus exam
had two lengthy
modeling
problems about an
amusement park.
They appeared
consecutively.
Nobody
complained
…much.
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…
We need to stop thinking of a
student’s mathematics education as a
linear progression of skills that must
be mastered.
Arithmetic
Fractions
Factoring
Equations
Inequalities
Geometry
Trigonometry
Calculus
Statistics
Radicals
Proofs
Functions
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?
Dr. Retro, I’ve got it!
That does
not count,
Miss
Nouveau.
Put that
thing away.
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, doesn’t that
mean that 2d = c?
Mr. Jones, if that
is all you learned
last year, you had
better drop this
course before it
drops you.
Good question,
Mr. Jones. Let’s
see what would
happen if there
were 4 cats…
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.
Example:
AZ, OK and MA still have Cramer’s Rule
in their state standards.
The purpose of Cramer’s Rule is to solve
systems of linear equations using
determinants.
Recall:
How can we possibly still mandate the
teaching of Cramer’s Rule?
Example:
AL, OK, and CT want students to know
how to compute a 3-by-3 determinant.
2 1 1
1 4 2
1 1 0
–
–
–
2 1
1 4
1 1
+
+
+
0 + 2 + 1 – (–4) – (–4) – 0 = 11
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!
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.
In fact, if you love Cramer’s Rule, go
ahead and teach Cramer’s 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!
e b
f d
x
; y
a b
c d
a e
c f
a b
c d
Cramer Himself
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.
b  b  4ac
2a
2
Important?
Expendable?
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)
HA HA! I’m 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!
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
AP Calculus Calculator Survey Results
Which graphing calculator did you use?
(percent of students)
Calculus AB
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
AMC 12 / AMC 10: American
Mathematics Competitions
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.
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!
dkennedy@baylorschool.org