Revitalized Undergraduate Mathematics with Symbol Manipulating Graphics Calculators
High-level programmable Hewlett-Packard calculators may afford students, for the first time, real graphical, numerical and symbolic computing power that is reliable and portable. But most freshmen and sophomores are still computing with pencil and paper in under-graduate mathematics, even though new computational devices are available. The goal of the project was to integrate these graphics calculators effectively into six service mathematics courses for science and engineering: single-variable calculus (two courses), multivariable calculus, differential equations, linear algebra and statistics.
Clemson's ambitious experiment took aim at the actual character of undergraduate mathematics instruction and learning. Reshaping the service courses for science and engineering students with the use of "supercalculators" would give them the power to do symbolic algebra and sophisticated graphics, and to use interactive operating modes in class. Instruction would move away from stressing manipulation skills and toward enhancing conceptual understanding.
Prototype and pilot versions of the six courses were designed and tested with the following questions in mind: Where is calculator use appropriate or inappropriate? How does calculator use subtract or add to efficient coverage of course material? Can calculators enhance conceptual understanding and allow new topics of study?
Hewlett Packard lent the project 95 graphics calculators, and Clemson's administration provided another 85. Students made almost daily use of them for both classwork and homework, which allowed them to explore and experiment as they studied core theory and methods, to get immediate feedback from instructors, and to interact in class in new ways.
No objective measures of student learning, such as comparison of group performance on common tests, have been undertaken. Matching student groups by ability, math background, and skill levels was very difficult. Because the powerful calculators allowed theoretical questioning as well as computational questioning, students taught in the traditional mode were found to be at a great disadvantage when tested without calculators at their side.
Instead of using objective measurements, faculty observed classes, evaluated course materials (syllabi, handouts, and tests), and interviewed and surveyed 969 students in 39 classes about mathematical understanding, problem solving and exploration using the graphics calculators.
The claims of success for this project, then, are not based on comparative learning scores. Rather, project staff claim that the calculators, more than anything else, changed not only what they taught and how they taught it, but how and what they tested, and how students learned. The unique dynamic introduced into the learning process by these calculators "encouraged the students to learn--and the faculty to teach--the concepts and methods in a more active, constructive environment from analytical, graphical and numerical perspectives."
The integration of the graphical and numerical solve features of these devices have enabled students to make important visual and numerical connections to the analytic presentations in the texts. The technology, then, removed the traditional routine computation and permitted more thinking about the underlying mathematical concepts and theories.
Having observed students in class, faculty claimed that the way students learned was fundamentally changed by the integration of graphics calculators--that they developed strategies to generate, manipulate, and use visual images in the process of understanding math concepts, whereas in the past, visualizations had been largely a product of students' mathematics.
The supercalculators allowed the successful introduction of two modern topics which often could not be covered in previous offerings into the linear course: the interpretation of Gaussian elimination as an LU-factorization and its application to linear systems with multiple right-hand sides, and the interpretation of the Gram-Schmidt process as a QR-factorization and its application to least squares problems.
In these cases, students were able to grasp and handle complex contemporary math concepts, something that had not occurred in the past. However, data from a carefully controlled experiment to this effect are not available. A new level of testing on new topics has been opened up through using graphics calculators that is not possible in a more computationally restricted environment.
In interviews and questionnaires, the majority of students across all experimental courses reported that the graphics calculators helped them understand because they could visualize the mathematical relations by use of graphs. Almost all students claimed that the graphics calculators were useful in solving problems, often because of their ability to store programs to be recalled later for complex solutions. Students also said that they sensed the opportunity that graphics calculators gave them to explore and investigate beyond what they could achieve without them. Class attendance in the calculator-enhanced classes was dramatically higher than in other sections.
Clemson faculty have felt the impact of these reforms as well: the first seminar on teaching within the math department in over 20 years grew out of pedagogical issues surrounding these new courses. Currently, instructors meet in three seminars every week about each of three courses. Growing acceptance of the experiment from Clemson administrators and faculty is reflected in the number of calculator sections over time: 6 in 1988-1990, 33 in 1990-1991, 52 in 1991-1992, and 56 in 1992-1993.
This project was selected in 1990 by the Mathematical Association of America as one of the ten exemplary mainstream calculus reform projects in the nation and was specifically featured in its publication, Priming the Calculus Pump: Innovations and Resources.
By the end of the 1991-1992 academic year, 3,500 students and 26 math instructors had studied calculus, differential equations, linear algebra, and statistics in calculator-enhanced classes. The project is firmly institutionalized within the Clemson curriculum, with plans to substantially expand the program through National Science Foundation funding. A recent award to Clemson and the Georgia Institute of Technology from NSF will extend and adapt the methods and materials across the two campuses.
Currently, the 52 calculator-enhanced class sections represent about one-half of all offerings in science and engineering for undergraduates. By the fall of 1994, the project plans to integrate the calculator technology into two new areas: 31 sections and 1,000 students in college algebra and precalculus courses, and 55 sections and 2,100 students in business calculus courses.
Project faculty have given approximately 135 presentations, many of them by invitation and including the nation's foremost math and technology conferences. A growing list of colleges and universities, including Duke University, the University of South Carolina, Lock Haven University, and several technical colleges in South Carolina and community colleges in New Jersey have adopted the project's methods and materials.
Based on this path-breaking work in improving calculus instruction, FIPSE has recently funded another three-year project at Clemson, this time for revitalizing nonstandard calculus, the calculus taken by students in economics, management, business and the social sciences.
In 1992 the project faculty published a series of five course supple- ment books through Harcourt, Brace Jovanovich and Saunders College Publishing on the pedagogical use of high-level graphics calculators:
Iris Brann Fetta, Calculator Enhancement for Introductory Statistics, 1992.
D.R. LaTorre, Calculator Enhancement for Linear Algebra, Saunders College Publishing, 1992.
T. G. Proctor, Calculator Enhancement for a Course inDifferential Equations, 1992.
J. H. Nicholson, Calculator Enhancement for Single Variable Calculus, 1992.
J. A. Reneke, Calculator Enhancement for a Course in Multi-Variable Calculus, 1992.
Twenty seven articles have been either published or are forth- coming in an array of math and other professional journals on the local and national impact of the experiment. Several manuals on the use of graphics calculators in algebra, precalculus, and statistics will also be published this year by Harcourt Brace. For more information about the project and its publications, contact:
Donald R. LaTorre
Department of Mathematical Sciences
Clemson, SC 29634-1907
803-656-3437 FAX: 803-656-5230
e-mail: latorrd @clemson.clemson.edu