The Other Role of Technology: Communication Between People Paper Presented at ATCM 2012
For the most part, the traditional role of technology in the teaching of mathematics has been focussed on its ability to solve, to evaluate, to produce images and even to guide students through stereotypical working steps that are supposed to be employed as one works through exercises that appear in the textbooks. But technology has another role that is sometimes neglected. It gives us new and efficient ways to talk with one another. It gives us opportunities to convey mathematics in a much more friendly form than we can find in any ordinary textbook. It allows instructor and student to exchange mathematical ideas even when they are not standing face to face in the same room. It can be a means of communication between people. In some of my earlier presentations at ATCM and elsewhere, I have emphasized the value of screen capture videos in the creation of mathematical learning materials and the role of video will again be featured this year. I shall refer both to classroom video and to the video in products like my Virtual Calculus Tutor. However, the production of video is not the whole story and I shall not be confining my attention to this topic. My presentation will demonstrate some of the unusual textbooks that I have designed for reading on the computer screen and it will show how, in my two main video products, Virtual Math Tutor and Virtual Calculus Tutor, a balance between video material and document material can provide a very successful medium for helping students to understand mathematics. I shall show how an on screen book can be just as easy to browse through as a printed bound text and provide more help, more solutions to exercises, and more development of the material. It can do so without becoming cluttered and at a modest price. I should add that several of my works are provided for free as public domain items. Some of my products involve coordinated use of on screen documents and video and I intend to demonstrate how these two media can work with each other to help a student to understand mathematical ideas. The Different Roles of Technology Introduction Over the past 20 years, the use of technology in the teaching and learning of mathematics has come to be identified largely with the computer algebra system (CAS) technology that has provided us with some exciting opportunities to explore mathematics and to experiment with it. CAS technology also allows us to extend our activity to more useful and realistic problems than we could have brought into the classroom in the old pencil and paper days. There is no doubt that CAS technology has changed the face of mathematical study and it is to be expected that a focus on CAS technology should be a principal ingredient in a mathematics technology conference such as ATCM. In the same way, it is to be expected that the majority of commercial products that bring technology into the mathematics classroom are focussed on CAS technology. There are, however, other ways in which access to technology can change the face of mathematical study and I believe that these other ways are being somewhat neglected by the community. One of these other ways is to use technology as a communications medium rather than for its CAS ability and it is this kind of application of technology that is the subject matter of my paper. I am in no way denying the immense value of CAS technology in the mathematical learning process but I do want to add that I believe that CAS technology has received more than its fair share of attention. While I believe that CAS technology, used in the proper context when it is relevant, is a wonderful tool in the teaching and learning of mathematics, I also see evidence of some slavish over use of CAS technology that may be doing a disservice to mathematical learning. With this thought in mind, I shall begin my presentation by describing some ways in which I believe that CAS technology may have been misused. Some Undesirable Effects on Mathematical Learning when CAS Technology is Misused From the time when I began teaching university courses in mathematics 48 years ago, I have often expressed my concern that too many undergraduate mathematics courses confine themselves largely to the methods that must be used to obtain "answers" to problems. The courses are replete with problems that invite the students to evaluate, and to crank out "answers" rather than to demonstrate understanding and appreciation of mathematical principles. Such problems are largely unconcerned with the logical flow that is the bedrock of mathematics and the courses pay little or no attention to mathematical writing. All too often, the students learn to solve a problem without acquiring any appreciation of the relevance of the problem to the course being studied. When CAS technology burst upon the scene about 20 years ago, I heard grand promises that all this was going to change. We were promised that, now that computers were available to do the manipulations, we could give greater emphasis to logical connections between ideas and to mathematical writing. However, I am sorry to have to report that, as far as I can see, many of those promises have proved to be empty. Far from tilting the thrust of classroom activity away from the cranking out of answers and towards understanding of ideas, the effect of CAS technology has, all too often, been an even stronger emphasis on mechanical steps. To some extent, we have been turning our students into button pushers and spectators instead of students of mathematics. I think that the worst aspect of this misuse of computer technology involves the use of certain software products that are designed to steer students through stereotyped sequences of working steps that must be followed to obtain the desired "answer" to a problem. There are several such products on the market. They are raking in millions of dollars but they are doing nothing to help the cause of mathematical learning. One of the symptoms of our neglect of mathematical writing and the understanding of mathematical principles is the abundance of multiple choice and "put your answer in a box" testing that we see in so many mathematics courses. There is no doubt that, used in moderation, CAS technology is a wonderful tool but, unfortunately, I do not believe that it is always used in moderation. I believe that the degeneration of our undergraduate courses into mindless button pushing sessions is counter productive to the learning process of mathematics and serves only to line the pockets of the companies that produce the software products that are supposed to teach students how to crank out "answers" to problems. I offer the following examples as illustrations of the way in which I think that a great many mathematics courses are tilted towards mechanical steps and away from the learning of mathematics. CAS Technology and the Box Problem I had an interesting experience, several years ago, at the Perimeter Mathematics Technology Conference that takes place each year near Atlanta, Georgia where I live. I attended a presentation there that was given by the representative of a software company and discovered that the sole objective of that presentation was to market a software product. I was surprised to see that, in spite of the fact that the presentation was directed to the level of high school algebra, a principal item in that presentation was the well known "box problem" that we often teach in a beginning calculus course.