There are, of course, costs associated with new technologies, just as there were with older technologies that we now take for granted. There are hardly any believable studies of cost-effectiveness of new (or old) technologies -- not because we don't know the costs or won't admit to them, but because we don't have good ways to measure effectiveness of education. Think again about the goals in Section 2 (or any alternate set of goals). Our effectiveness at addressing those goals may not be known until long after the students have left us, and maybe not even then.
A more productive line of inquiry is to examine the costs of not using technology, in light of the current context of education, of reasonable projections about the world our students will live in, and of what we now know about learning.
Technology is a fact of life for our students -- before, during, and after college. Most students entering college now have experience with a graphing calculator, because calculators are permitted or required on major standardized tests. A large and growing percentage of students have computer experience as well -- at home, in the classroom, or in a school or public library. ''Surfin' the 'Net'' is a way of life -- whether for good reasons or bad. Many colleges require computer purchase or incorporate it into their tuition structure. Where the computer itself is not required, the student is likely to find use of technology required in a variety of courses. After graduation, it is virtually certain that, whatever the job is, there will be a computer close at hand. And there is no sign that increase in power or decrease in cost will slow down any time in the near future. We know these tools can be used stupidly or intelligently, and intelligent choices often involve knowledge of mathematics, so this technological environment is our business. Since most of our traditional curriculum was assembled in a pre-computer age, we have a responsibility to rethink whether this curriculum still addresses the right issue in the right ways -- and that is exactly what has motivated some reformers.
But calculus renewal is not primarily about whether we have been teaching the ''right stuff.'' Rather, it is about what students are learning and how we can tell. To review, we have seen that the external world (as represented by employers) has certain expectations of education that turn out to be highly consistent with both learning theories and good practice, as described by cognitive psychologists. In the past decade neurobiologists have provided the biological basis for accepting sound learning theories and practices, while rejecting unsound ones. What does technology have to do with this?
Looking first at the Kolb cycle, we see that computers and calculators can be used as tools to facilitate the concrete experience (CE) and active experimentation (AE) phases -- but not the other two phases, which are right brain and left brain activities. Thus, if the activity allows the student to go directly from CE to AE without engaging the brain -- which means the AE will not be about anything of substance (except perhaps in the instructor's brain -- it may do more harm than good. Well designed learning activities involve the entire cycle, at least most of the time.
Now let's consider how technology can support the Seven Principles of good practice.
Student-faculty interaction: Electronic communication via web pages or e-mail enables exchanges with students that never happen in the classroom or office, where students are often intimidated.
Cooperation among students: Here, a lot depends on the atmosphere created by the teacher, but students are quite willing to share with peers what appears on their calculator or computer screen, and this serves as an ice-breaker to get them engaged in more substantive conversation. They are somewhat less willing to share their calculations on paper -- for which they alone are responsible. If something on the screen does not look ''right,'' there is psychological security in blaming the intervening tool, which is often attributed a personality of its own.
Active learning: This is where the technology comes into its own -- as already noted for the CE and AE phases of the learning cycle. Students can be much more actively engaged in experiences and experiments with substantial mathematics than was ever possible with pencil and paper.
Prompt feedback: Garbage in, garbage out -- immediately. No human teacher can possibly respond as fast or as non-judgmentally.
Time on task: Students are willing to stay focused on substantial mathematical tasks much longer with technology than without. But there are risks here. If the task is not carefully designed and supervised, students are likely to waste a lot of time addressing questions the teacher never thought of. And, even with well-designed tasks, there is a risk of students subverting the learning process by making their own shortcut from CE to AE -- that is, by refusing to think about what they are doing. Even worse, flashy but poorly designed tasks can seduce students into spending lots of time avoiding thought.
High expectations: This is entirely up to the teacher. Technology helps only in the sense that the teacher can expect everyone to carry out tasks that many would be unable to do on their own, e.g., to solve f(x) = 0 for some given function.
Diverse talents and ways of learning: Technology is a ''great equalizer'' in the sense just noted. It's not unusual for a student who lacks skill at symbol manipulation nevertheless to have insight into, say, an emerging pattern of results on the screen. Furthermore, if students are not always told which buttons to push, they will often come up with techniques that are quite different from the instructor's, but just as correct or just as likely to reinforce the correct concept.
| Title page |
| Reform or Renewal? | Students | Problems |
| Cognitive Psychology | Brain Research |
| Technology and Curriculum |
| Renewal in Calculus Courses | References |