A Proposal to Aid and Improve Mathematics Learning in Schools
By Arthur N. DiVito, Ph.D. (copyright © 2000 by A. N. DiVito)
Here is a simple example:
Two battery operated model cars and stopwatches are presented to the students. A team (generally, a group of three to four students such as typically assigned in cooperative or collaborative learning environments) is appointed the task of determining the velocity of each car (say in centimeters per second). This is performed in front of, and may employ the aid of, the other students.
Once these velocities are known, the class is asked the following question: If each car begins a trip at the same moment, facing each other, at a distance of twelve meters, at what time and at what position would they collide?
After a certain time, answers and procedures may be solicited from the class. Following that, the actual experiment is performed and the results are compared to the suggested responses. Appropriate discussion and variations would follow. Finally, a Lab Report would be completed as homework by each member of the class, and each of these reports is graded by the teacher.
Are there any general guidelines or components of the Lab Reports which are required? If so, what are they?
Yes.
The Lab Report will consist of five components, three mandatory and two optional.
The mandatory components are:
An explanation of what was done and what questions were posed.
An accounting of what the involved mathematics looks like.
A statement of how confident the student feels that he or she could solve similar or related mathematics problems.
The optional components are:
The student may suggest improvements or alternative activities (hopefully, that are impressive enough to be used in future MathLabs) which address the same mathematical content.
The student may propose reasons to account for whatever differences appear between the mathematical solutions to the involved questions and the empirical answers provided by the actual experiment.
Essentially, these reports are graded according to the quality of the mandatory components, although extra credit should be assessed for good ideas in the optional components.
What else?
Nothing. That’s it. It’s a clear and simple strategy to advance mathematics learning. Teachers aren’t told what to do in their classes. Teachers aren’t advised how their courses could be improved. All we’re asking is that time be found to have, at most, a once-a-week separate, outside-of-the-regular classroom, mathematics lab just as we have physics, chemistry, and biology labs. Most teachers do an excellent job with the resources they have and within the constraints they are given. This proposal is intended only to further support their already fine efforts. This is more about learning than about teaching.
The only remaining matter regards the MathLab activities. Do enough currently exist? Yes. Can the teachers learn to conduct these pretty well? Yes.
Why do you say “yes” in the two instances above?
For many years now there have been some very fine programs in mathematics which emphasize such approaches as hands-on, collaborative, real-world, inquiry, phenomenological, process, concepts, portfolio writing, having fun, etc. Two outstanding local examples have been the SMILE (Science and Mathematics Initiative for Learning Enhancement) Program at IIT, and TIMS (Teaching Integrated Mathematics and Science) at UIC. Such programs are generally well received and are deemed quite effective by the teachers who participate in them.
However, many believe that very little of the above is actually being used inside the schools. This is simply because there isn’t enough time to implement such methodologies in the regular classroom. Program directors are talking to teachers, but the teachers don’t have enough time to do many of these things with their own students. Why? Because content has to be covered whether pedagogical theorists will admit it or not. As long as there is as much content as mathematics possesses, it will remain a Herculean task for classroom teachers to find enough time to incorporate the activities and modes of delivery such programs have been espousing these many years. It’s not because these excellent methods do not have value; it’s because the methods require some time. The obvious resolution, of course, is to remove these activities from the classroom. That is why the sciences have separate labs; and that is why mathematics needs its own lab.
Upon the pages which follow, a number of MathLab-like activities are delineated. This is so that the reader can get a feel for the kinds of activities being considered.
Battery operated model car experiments