The Co-Evolution of Calculators and

High School Mathematics

Dan KennedyBaylor 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 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.

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

1613Tan , so log tan

2871

log1613 log2871

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 Reason 1. b and c are real numbers Hypothesis 2. b + c is a real number Axiom of closure for addition 3. –c is a real number Axiom of additive inverses 4. (b + c) + (–c) = b + [c + (–c)] Associative axiom of addition 5. c + –c = 0 Axiom of additive inverses 6. b + [c + (–c)] = b + 0 Substitution principle 7. b + 0 = b Additive axiom of 0 8. b + [c + (–c)] = b Transitive property of equality 9. (b + c) + (–c) = b 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?

2. 24 6 3. Solve for : ( 2)(2 3) 49.x x x 4. Find cos( /3).5. Find the product: 874539 374958.

6. Find 239121.

7. What is ?ie

1. 5 2

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.

Y ear Raleigh Mesa 1980 150,255 152,404 1990 207,951 288,091 2000 282,956 397,776 2004 326,653 437,454

(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.

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?

5 3 5 19.40

3 2 2 11.05

6 4 3 21.25

h f d

h f d

h f d

After modeling the problem, there are two easy methods of solving it:

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.

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 Radicals

Geometry Trigonometry

FunctionsCalculus Statistics

Proofs

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 a e

f d c fx y

a b a b

c d 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.

2 4

2

b b ac

a

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

 

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

AP Calculus Calculator Survey ResultsWhich graphing calculator did you use?

(percent of students)

Participation and EligibilityBoth 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.

AMC 12 / AMC 10: American Mathematics Competitions

Meanwhile, the CAS conversations continue.

It just might be time for another change!

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.

dkennedy@baylorschool.org