Graphing Calculator Lending New Insight Into Intricacies of Math, Advocates Say
Yet, for several hundred mathematics teachers who gathered here for a weekend conference recently, encounters with the device seemed the stuff of a near-religious experience. Their testimonials rang out like those of a tent-revival meeting.
"It's changed my life," one beaming participant confessed to the knowing chuckles of others.
Others, novices and veterans alike, crowded into the large auditorium to exchange their personal frustrations and share their latest success stories.
At the heart of all the commotion was a device--frequently described as a cross between a traditional electronic calculator and a microcomputer--that proponents say helps give students new insight into the power and intricacy of mathematics.
"This is a damned good way to teach math," one participant declared. "I'm addicted to it."
The machine allows users to graph equations, at the touch of a button, that otherwise take hours of tedious pencil-and-paper work to understand, if they can be attempted at all.
Classroom applications are obvious, teachers and researchers say.
The device, they assert, appeals strongly to visual learners, who often struggle with the symbolic nature of math, and opens the way to teach high-school students not just the rudiments of computation and repetition, but the beauty and precision of such subdisciplines as probability and statistics and matrices.
"They're going to make big changes in content," said Frank Eccles, who recently conducted a seminar on the new calculators at the prestigious Phillips Academy in Andover, Mass. "We can solve whole classes of equations that we've previously avoided in high school."
While there are no hard figures on the numbers of districts or students using the devices, participants at the weekend conference came from across the country, and at least one large public-school district and one private school--Pittsburgh and Phillips Andover, respectively--are requiring the calculators in some classes.
Graphing calculators are a relatively new addition to the math teacher's tool kit.
Similar in design to the basic electronic scientific calculator, they also incorporate a small screen where the graphic representation of a complex equation can be made to appear at the touch of a few keys.
"They're really pocket computers," said Franklin D. Demana, an Ohio State University math professor who is co-author of a pre-calculus curriculum based on use of the graphing calculator.
And while the devices are only slightly larger than a conventional calculator, they are capable of storing as many as 37 individual software programs, he explained.
In fact, one of the more popular graphing calculators now available, he noted, "is more powerful than the [early] Apple II" personal computers.
The power and capabilities of the devices have made them the centerpiece of a small but growing grassroots curriculum-reform effort led by a cadre of enthusiastic math educators.
Given the fervor of the several hundred such math teachers who gathered here for the conference, the uninitiated could be forgiven for confusing a new pedagogy with a new belief system.
"You would think it's a religious experience, too, if you were in the situations that some of these people, particularly those in the urban areas, were in," Mr. Demana said.
While primarily intended for such higher-math courses as trigonometry and calculus, the devices have applications in algebra and pre-calculus classes as well.
No one is contending that the machines themselves will spark a complete overhaul of high-school math.
And proponents of the new approach recognize that many teachers, especially those who argue that a "back to basics" approach to arithmetic is a more effective pedagogy than one that focuses on technological aids, will not readily embrace their world view.
But they do assert that field tests among precollegiate math students indicate the devices--some of which sell for around $80--are, in many schools, already making math more meaningful for a broad range of students.
"For the price of a trendy pair of tennis shoes, we can put this very powerful piece of technology into our students' hands," said Ellen S. Hook, a 20-year veteran teacher from Norfolk, Va., who has used graphing calculators for three years. "Mathematics is changing so much just because of this."
Out of the Lab
Mr. Demana, a leading proponent of the graphical approach to teaching higher math, developed the technique together with Bert K. Waits, a colleague at Ohio State.
He traces the development of the approach to the late 1970's, when the two began using traditional calculators as teaching aids to assist remedial students.
"We were able to do good problems and discover that these kids were not stupid, they were just unable to work symbolically," he said.
They subsequently incorporated those insights into software programs to help ease the transition for college students from pre-calculus to calculus courses.
And while the computer was an effective tool, the strategy was not without its drawbacks, Mr. Demana said.
There were never enough computers in the classroom, those that existed were not portable and often isolated in computer labs, and students rarely had them at home, he noted.
Then, in the mid-1980's, Mr. Demana discovered an early graphing calculator produced by Casio Inc., a New Jersey-based electronics firm. The device, he argues, has a "legitimate claim to the first graphing calculator."
Finding such a machine and learning how to use it effectively, he added, "was a real revolution because it removed us from the lab."
A major step in the development of the new approach came in the late 1980's when the researchers invited calculator manufacturers to attend a session of the "Calculator and Computer Pre-Calculus," or CPC Project, a series of technology-based workshops that they had been holding for math teachers.
Although several companies, including Casio, Hewlett-Packard, and Sharp Electronics, were marketing graphing calculators at the time, only one--Texas Instruments, Inc.--accepted the invitation.
Engineers from the firm watched the teachers as they worked out problems and asked them how to improve the existing machines to better meet their needs.
The joint research-and-development effort by the teachers and engineers culminated in the production of the TI-81, the company's popular graphing calculator.
"It's been a very good merger," Mr. Demana said. "The teacher excitement about this product has been unbelievable."
Acceptance of the new method has been fostered by publication of a series of curriculum standards two years ago by the National Council of Teachers of Mathematics that call for increased use of technology in teaching math.
The standards, to which Mr. Waits and other Ohio State researchers contributed, specifically cite graphing calculators as a valuable tool for pre-calculus students.
Candy for Calculators
But the excitement generated by the TI-81, perhaps the most popular graphing calculator now on the market, surprised even officials at Texas Instruments.
"There were some people who were pessimistic all along," said Ann Phipps of Texas Instruments. "But there were others who thought all along it would be a good product and a big seller. And it is."
And the grassroots interest among math teachers is fueling renewed competition among manufacturers to improve their machines.
Casio announced earlier this year that a new version of its graphing calculator would be available this spring. And Hewlett-Packard representatives, along with those from Texas Instruments, attended the most recent CPC conference, and lent their products to teachers overnight.
Yet, because the TI-81 has been on the market less than a year, "we feel like we're still at the beginning of a big wave of potential purchases," Ms. Phipps said.
Such is the excitement the machines are causing at the local level that Ms. Hook and her students in Norfolk have taken to selling candy bars to raise money to buy the machines for their own school.
Although the technology is still new to many teachers, at least one public-school district has adopted the graphing approach on a wide scale.
The Pittsburgh district this year supplied each student in its upper-level math courses with a TI-81 and the appropriate textbook, said Diane J. Briars, director of the district's division of math.
She said the district, which has previously required universal use of scientific calculators in lower-level math classes, already was relying strongly on microcomputers in advanced math classes to reinforce the concept of functions with graphical representations.
However, she added, teachers encountered the same difficulties that the Ohio State researchers had discovered years earlier. Some teachers, meanwhile, were teaching pilot courses using the early Casio calculators.
"We were convinced that we really wanted to incorporate that technology into our courses," she said.
Ms. Briars then wrote into the district's math budget a funding request for 1,000 of the Texas Instruments machines, as well as several classroom sets of the calculators to be used in algebra and other lower-level math classes.
The district also adopted a textbook, written by Mr. Demana and Mr. Waits, that specifically incorporates the use of graphing calculators into its pedagogy.
"Having an integrated book makes all the difference in the world," she said.
But even more important to spreading the new gospel, proponents agree, is the word-of-mouth advertising by math teachers themselves, who often learn about the new devices at seminars like the annual meeting at Ohio State.
Phillips Academy, for example, recently held a weekend seminar for graduates of its annual summer math institutes.
Graphing calculators first appeared at the institutes in 1989 and last summer became required equipment for every participant.
"We became convinced that these things were very, very powerful," said Mr. Eccles, the seminar's organizer.
The school convened the special seminar earlier this month to assess what changes the machines were making to mathematical pedagogy.
"We decided that it would be good to bring together a group of people who have had some experience and see where this is taking us," Mr. Eccles said.
Participants were chosen for their leadership ability and were expected to pass on their experience to their colleagues back home in Atlanta, Baltimore, and other major urban areas.
"We wanted the inner cities to be on the cutting edge of major changes that are taking place, rather than [to] be in the position--[which] so many of our cities are--of following the suburbs," Mr. Eccles said.
Advocates of the new technology, who frequently could be heard here referring to their uninitiated colleagues as "dinosaurs," also concede that the new approach is not without its disadvantages.
The Pittsburgh district, for example, has been using the graphing calculator in algebra courses, but has been unable to expand the use of the device into other math classes because there is no textbook that incorporates the new technique, Ms. Briars of the Pittsburgh schools said.
"There is a perception among publishers that there is not a market for textbooks that [emphasize] technology," she said.
And, as is the case with many new technologies, teachers are hesitant to adopt major changes.
"The teachers are very reticent," said Ms. Hook, who has conducted in-service programs at other Norfolk high schools. "They have a little anxiety toward technology. But they are succeeding."
There is also the concern that students will use the devices to circumvent learning the mathematical concepts involved. While she is a staunch proponent of the graphing calculator, Ms. Hook of the Norfolk schools also requires her students to learn the basic concepts of the course before turning to the machine.
"We can't allow our students to become totally dependent on the calculators," she said. "We want to make sure [it] is used as a tool."
Nonetheless, proponents of the graphing approach argue that, once they have developed the appropriate proficiency, new challenges will arise for both students and teachers.
Students "explore, experiment, investigate, and draw conclusions," Mr. Demana said.
At Phillips Academy, students this year are required to own graphing calculators. The biggest change as a result of that decision probably will be teachers' perceptions of what they are teaching, Mr. Eccles said.
"Perhaps one of the dangers is that it is going to emphasize real thinking and theory, which may mean that teachers and students may find it harder to cope," he said.
Vol. 10, Issue 29, Page 1, 16