Discussion Draft (Standards 2000)
2.5 Mathematical modeling in schools
2.5.2 Teaching experiments involving modeling at schools
J ones also argues that : "What follows is a pedagogical approach designed to increase opportunities for students to engage in mathematical modeling but reduce the time needed by the teacher to reestablish the background for the problem. This approach uses the same real problem but varies the conditions or constraints on the problem in a number of ways.
From a teaching-learning perspective it not only follows the initial exploration of the problem but is an extension of it".
Von Essen (1991 : 196 - 202) describes two curriculum projects that were undertaken by learners in a Danish Secondary school. The first involved a class of learners in the second/third form in 1983-1985. The learners in this project had to learn about modeling especially in industry. In the second form the learners learnt the basic mathematics (like differential calculus) required for the modeling process and aspects of the actual modeling.
In the third form the learners worked with the modeling of actual problems in the fishing
industry. The second project also involved a class of second/third form learners in 1987- 1989. Here again the learners were introduced to the concept of modeling, but instead of working with only modeling over a prolonged period, the time was divided in a way where learners had to work with two different models (two thirds of the time) and then they worked in groups for the remainder of the time. For the two models the learners worked with the models of the structure of the structure of the stars and the other dealt with fishery models (one project had to have some social significance). In the last third of the time the learners worked in groups and they could choose from any theme from the national economy, business economy, population dynamics, fisheries and chaos.
At the end of both the projects the learners had to answer a questionnaire. Almost every learner expressed high degree of satisfaction with the modeling process and stated that they had benefited from the instruction. The learners were happy to have been able to work on their own. In the first project the learners felt that there was too much independence. This resulted in the second project being changed so that learners could work together as well.
The learners also commented that they gained sufficient skills and insight during this process and they found the study of models relevant and interesting. An added comment on their questionnaires was the fact that they found the application of mathematics on location
important and relevant (they visited the Department of Fisheries and the Astronomical Institute ofthe University of Aarhus).
Von Essen (1991 : 201) also makes the following cautionary comments regarding the teaching of modeling and the use of projects:
1. "The learners learn too little compared with the time invested". This problem could be overcome by proper planning and by incorporating modeling into the normal syllabus. This would imply that as the learners work in the different sections in mathematics, real-life questions must be posed. This will force learners to use the mathematical knowledge they already learned.
2. flit takes too much time from the teaching of (pure) mathematics". There is little doubt that teaching modeling will impact on the length of the traditional mathematics syllabus. But when one measures the relevance of modeling in the real world some aspects of the traditional syllabus has to be sacrificed. Perhaps a more relevant suggestion would be to increase the amount of time allocated to mathematics instruction.
3. flIt cannot be tested in a reliable way". In real life, modeling is conducted in many aspects of society, for instance, building of bridges and so on. How does one test the success of the modeling? The answer lies in the actual product - the bridge that is built. However, there may be many different solutions and assessing learners' work therefore becomes difficult.
4. "The teachers' knowledge of different applications is inadequate". Educators ought therefore to be trained in the various aspects of modeling and applications on a continuous basis.
In another experiment conducted to understand children's ways of modeling non-standard (or real world) problems, Nesher and Hershkowitz (1997 : 281 - 287), worked with 480 children of grades 4, 5 and 6 from 15 different schools in Israel. Different children had to work with a single problem and there were six problems altogether. The second problem given to 84 of the children was: For dinner in a summer camp 17 pizzas were ordered.
Some were large and others were small. Each large pizza was divided among four children, and each small pizza was divided between two children. How many children were in the camp, and how many pizzas of each kind were ordered? The problem had to be solved by each child individually and they were instructed to find different possibilities for the solution. The complete instructions were to draw the story as described in the text. Also to explain in detail the line of thinking, either verbally, by drawing or by mathematical sentences and to write how they would explain the solution to another friend.
The results showed that 79% of the children coped with the problem offering some solutions to the problem. In determining the degree to which they coped, Nesher and Hershkowitz discovered that 47% of the children were able to give one correct solution, 17% were able to give several correct solutions and 2% were able to give solutions that showed a systematic method of inquiry. It was therefore clear from this experiment that even young children in primary grades are able to cope with the non-standard problem given to them. More importantly, the children were able to bring real life considerations into their solutions. For example, they made assumptions that some children would get more than others or they talked about different types of pizzas (that is, different toppings) and for the problem where the number of pizza portions was unknown, the children immediately assumed that the large pizza yielded eight slices, and the small one yielded four, as is common knowledge in Israel. A possible explanation forwarded by Nesher and
Hershkowitz for the large percentage of children who gave only one solution is that in real life one makes only one order at a pizza shop - no one places more than one order for the same number of pizzas.
In an Australian report based on the evaluation and monitoring of a modeling course, Galbraith and Clatworthy (1990 : 143 - 162) made a few significant discoveries. The problem that the learners had to investigate is summarized as follows. The State Government in Brisbane planned to build a road under one of the hills in a place called Bardon. Due to financial reasons, only a single lane (in both directions) could be built. This meant that during peak hour traffic (in the mornings and afternoons) there would be severe traffic flow problems. In order to reduce traffic congestion it was decided that road signs should be erected, advising the road users about the maximum speed that a motorist may travel at and the safe distance to be observed between cars. The learners were then asked to make recommendations.
The written responses of the learners are significant in motivating for teaching modeling at schools. Despite finding the idea of changing the real life problem to a mathematical one difficult, the learners found the act of finding a solution to a real life problem most interesting. They felt that it was a great sense of achievement at the end. There is little doubt that learners feel motivated when they feel that their efforts have attained some success. The learners also felt that the most useful aspect of the entire exercise was the modeling process itself - "modeling itself because I can now look at a problem and express it mathematically" or "the use of modeling or scaling down a problem from real life to a workable situation are both useful in the world outside math". Many of the learners had
changed their opinion of mathematics after going through this course because "maths is useful and I've learned the human nature ofmaths".
These responses indicate that given modeling opportunities together with some guidance, learners enjoy working through real life problems. One of the unexpected findings of this investigation was the fact that the learners were able to use the modeling experience gained in mathematics in other subjects and in their hobbies.