Algebra, Trigonometry,andMathematica |
Barry Brunson and Claus
Ernst
Department of Mathematics
One Big Red Way
Western Kentucky University
Bowling Green, KY 42101-5730
email: barry.brunson@wku.edu
claus.ernst@wku.edu
To B. Brunson's
home page
Algebra, Trigonometry, & Mathematica (AT&M) is interactive courseware for algebra and trigonometry or for precalculus. AT&M is appropriate for self-paced individualized instruction in a distance-education format and for group instruction in a computer laboratory environment. A distinguishing feature of AT&M is its incorporation into the Help Browser of Mathematica, which allows for navigation through the courseware based on hyperlinks. With AT&M, the student is encouraged to (and is able to) ask the question "What if ..?", and find out immediately.
Talking isn't teaching.
Listening isn't learning.
(Motto of the Interactive Mathematics Text Project)
Algebra, Trigonometry, & Mathematica (AT&M) consists of interactive courseware written in the form of Mathematica notebooks. This approach allows easy integration of graphical, numerical, and symbolic aspects of key mathematical concepts. AT&M emphasizes an exploratory approach; the student is encouraged to (and is able to) ask the question "What if ..?", and find out immediately. Using the power of Mathematica to handle calculations that would be difficult or impossible to perform by hand, encourages individual discovery of patterns; one example becomes as many examples as the student desires. Furthermore, the student can insert explanations and comments at will, and can maintain a permanent record of investigations made. Students are actively engaged in the learning process, and encounter a strong emphasis on applications, real-world data, and scientific literacy. Writing is an important component of AT&M; most exercises require an explanation in English of the process.
Most of the student's time is spent actively solving problems. Hyperlinks facilitate reference to needed earlier concepts. A distinguishing feature of AT&M is its incorporation into the Help Browser of Mathematica. The Mathematica Help Browser is an environment that permits navigation by the simple clicking on links to the different parts of the AT&M materials, as well as to all of the Mathematica help files. This makes browsing through the AT&M materials similar to surfing on the web. Students learn to use Mathematica on a "just-in-time" basis, and do not need to memorize complicated details of Mathematica syntax; built-in palettes and buttons are available for often-used items, and code can be copied, pasted, and edited as needed. The focus is on the mathematics, not on Mathematica itself.
The AT&M approach is appropriate for self-paced individualized instruction by motivated students in a distance-education format, and for group instruction in a computer laboratory environment. AT&M offers enough content to satisfy requirements of a five-semester-credit hour course at either the beginning college level, or for secondary students.
AT&M requires Mathematica version 3 or higher, and can be used on either a Macintosh or a Windows platform..
AT&M stands for Algebra, Trigonometry and Mathematica. With AT&M, the student learns mathematics mostly by doing mathematics, rather than just by reading about it.
We call this an "interactive" approach, where the interaction is between the student and mathematics. The vehicles that help make this interaction possible are the computer, the student's brain, and Mathematica.
One of the main differences the student experiences with AT&M is this: Every example is many examples.
In a traditional printed textbook, the student gets to read the examples that the authors chose, and that's that. Maybe they have one example, maybe they have several, but in any case the student is limited to the examples the authors chose.
With AT&M, the student is in control. Whenever the student goes through an AT&M example, he or she has the power to go back to that example and change it: change the numbers, change the variables, change the functions. This is painless to do because the student can "cut and paste" just the changes desired.
With AT&M, every example is many examples. The point of doing many examples, however, is not just to take up time or to keep the student off the streets. Doing many examples also is not the same as "drill and practice". Almost every example and exercise in AT&M has behind it a pattern that it should illustrate. With AT&M, the student can change the example as often as he or she needs, until the pattern is clear in the students mind. Sometimes the student will see the pattern immediately. Sometimes it will take a while. Different people will need different numbers of examples at different times. The point is, there should be a pattern, and the student can (quickly and easily) check out examples until the pattern becomes clearer.
The AT&M course is broken down into six chapters, and each chapter has several sections. Each section consists of the main text, exercises, and "Test your understanding" questions.
The main text presents new mathematical concepts, and gives examples and exercises complete with solutions. The "Exercises" comprise examples and questions without answers. The student will be able to work some of the exercises without any help, except perhaps Mathematica for grungy calculations (Mathematica is always available for grungy calculations). Other exercises will be harder, or may require tools the student doesn't already have. If the student encounters an exercise that stops him or her cold, then he or she should check over the part of the corresponding section in the main text. Often the student will encounter a link that points in the appropriate direction.
Finally, the "Test your understanding" section contains questions that the student should be able to answer without the help of Mathematica. These questions address basic concepts and reinforce a reasonable level of hand manipulative skills.
Each student using AT&M must have his or her own computer: watching someone else move the mouse and hit the keys is not effective learning. However, we strongly encourage students to discuss the problems with each other.
AT&M can be used to teach any algebra, algebra and trigonometry, or precalculus course in a college or high school setting. For several years the authors have tested AT&M in a five-credit-hour College Algebra and Trigonometry course at Western Kentucky University. These classes use the AT&M materials as their only text. Students meet in a computer laboratory where the traditional lecture is replaced by individual assistance and guidance. AT&M is also used by off-campus students who take College Algebra and Trigonometry as a distance-learning course. These students communicate with their instructors via e-mail and send in their homework electronically for grading.
AT&M can be used to teach any algebra, algebra and trigonometry, or precalculus course.
An AT&M course is designed to be interactive, which definitely includes, and emphasizes, interaction between the student and Mathematica. Interaction between the student and the instructor also is crucial, as is interaction among the students themselves. In the classroom, students can be encouraged or required to work together in groups at least part of the time.
While the role of instructor as "Dispenser of Truth" is de-emphasized, and traditional lectures should be minimized or eliminated, the instructor is vital to the success of the student and, therefore, of the course.
For several years earlier versions of AT&M have been tested in a five-credit-hour College Algebra and Trigonometry course at Western Kentucky University by the authors. In this format, students work as much as 90% of the time in a computer laboratory. How much students work in the computer laboratory depends on the style of the instructor. Some sections of AT&M require extensive direct involvement of the student with Mathematica. In other sections of AT&M, the preferences of the instructor may be decisive. Where time in the computer laboratory is limited, the students may meet in a traditional classroom. Under these circumstances, the most appropriate format is an open class discussion, guided by the instructor.
Discussion by the students, perhaps in response to questions posed by the students (or taken from the "Test your understanding" sections) is preferable to an "instructor asks, students respond" mode. However, "instructor asks, students respond" is preferable to a traditional lecture mode in which the instructor tells the students what they should know, and the students passively take notes about those things.
A five-credit-hour College Algebra and Trigonometry course over the internet is offered at Western Kentucky University by the authors. In this format, a student must have his or her own computer and a personal copy of Mathematica installed on the computer; the student is mailed a set of installation disks for AT&M. The student completes about one homework set per week and sends it to the instructors as an e-mail attachment. For questions the student e-mails the instructors or visits a web site with commonly asked questions. This approach works successfully provided the student possesses sufficient motivation and self-discipline.
Graphing calculators are handy devices. Many things that Mathematica does in AT&M can be done by a graphing calculator. However, many other things in AT&M cannot be done by a graphing calculator, and Mathematica is necessary.
As technology improves, the things that a graphing calculator can do will include more and more of the things that Mathematica can do. In fact, these calculators will become more and more like laptop computers. In time, it will be harder to distinguish between a small computer and an advanced graphing calculator.
We strongly encourage using a graphing calculator along with the computer.
Barry Brunson and Claus Ernst have been developing AT&M since 1992 at Western Kentucky University (WKU). The development is an ongoing project since, unlike a traditional text, AT&M is software that has to change with new versions of Mathematica. The initial seed of AT&M was made possible by a grant from the National Science Foundation which funded a computer laboratory in the Department of Mathematics. In 1993 AT&M became nationally recognized through the Interactive Mathematics Text Project (IMTP), which selected Barry as an IMTP "Developer". IMTP was funded by grants from IBM and the National Science Foundation and administered through the Mathematical Association of America. The purpose was to improve the teaching of mathematics by encouraging development of courseware for interactive learning.
The first truly interactive mathematics courseware is Calculus&Mathematica, which was developed by Bill Davis, Horatio Porta, and Jerry Uhl at Ohio State University and the University of Illinois. This was the prototype that led to the Interactive Mathematics Text Project grants. The AT&M authors gratefully acknowledge the significant influence of Calculus&Mathematica on the development of interactive learning.
We have used AT&M in one section of the five-credit hour College Algebra and Trigonometry course at WKU each semester for the last several years, and a handful of students have taken the course in a distance learning environment. The number of students involved so far is too small to support meaningful comparative statistical analysis. However, anecdotal data from our experience suggest:
Grade distributions tends to be slightly higher than in traditional sections of College Algebra and Trigonometry that we have taught in the past.
Some students have difficulty adapting to the nontraditional format, and switch to other sections, or say they wish they had switched. However, others thrive with AT&M, including some who had made earlier unsuccessful attempts in a traditional course.
If the instructor of a calculus course, for instance, heavily emphasizes hand manipulative skills, then an AT&M student may be at some disadvantage. In general, however, we expect that AT&M students will tend to do no worse in hand calculation, and to do better conceptually, than students in a traditional course. This is consistent with the experience of the Calculus&Mathematica project. For instance, see Kyungmee Park and K. J. Travers, CBMS Issues in Mathematics Education, Volume 6, pp 155-176.
We eagerly solicit colleagues interested in using AT&M elsewhere.
Back to my home page