Hideki IWASAKI (Hiroshima University, Japan) & Takeshi YAMAGUCHI (Hiroshima University, Japan)
Doerfler’s model, Separation model, Extensional generalization, Intensional generalization
The relevancy on what and when to learn is the significant issue of mathematics curriculum not only in Japan but also countries in the world. In particular the transition from the elementary school mathematics to the secondary school mathematics sounds indeed important. This lecture has two research objectives. The first is the development of the theoretical framework on the transition from the elementary school mathematics to the secondary school mathematics from the viewpoint of generalization. The second is to establish teaching materials for the transition based on the theoretical framework. The authors regarded division with fractions as a critical teaching material for the transition, and scrutinized its teaching and learning process in terms of the theoretical framework which was mentioned below, and finally developed an alternative material for division with fractions.
Regarding to the first objective, the authors critically considered Doerfler's generalization model. "Symbols as objects" is a key point in that model because it connects abstraction with generalization in mathematics. Doerfler,W., however, sets up "extensional generalization" and "intensional generalization" just before and after "symbols as objects" linearly. He says nothing on its order of two kinds of generalization. That linearity implies "extensional generalization" is in "constructive abstraction". It seems logically contradict to regard generalization as abstraction.
We, then, consider another possibility on emergence of two kinds of generalization in Doerfler's model. They could emerge just after "symbols as objects" separately because both of them depended upon the inner structure of symbols.
The authors, therefore, newly introduced the concept of "synectics"implied by Gordon,W. to clarify the substantial difference between extensional and intensional generalization from the angle of cognition although Doerfler distinguished them logically. The definition of those terms has the following predicates. The former is to make the familiar strange in addition to its logical sense of enlargement of the extent of a set. It gives us the new point for integration of various pieces of old knowledge by making the familiar strange as a consequence. On the other hand, the latter is to make the strange familiar in addition to its logical sense of elaboration of the properties of objects. It enables us to assimilate new knowledge i.e. the unknown into the known. The structure of the known, therefore, is substantially and mathematically strengthened as a result.
As mentioned above, the authors thought both extensional generalization and intensional generalization as the alternative of two cognitive processes, and hence both of them could be separately set up just after "symbols as object". The new cognitive model of generalization is different from Doerfler $B!G (Bs model in the mode after "Symbols as objects". We, therefore, call it "separation model".
Regarding to the second objective, the authors focused on difficulties of division with fractions. As a matter of fact, the National Achievement Survey in Japan showed that there was a big difference between the attainments on making an expression of division with fractions and that of division with decimals from word problems although both of them had the same mathematical structure with proportional reasoning. The cause of this difference could be explained as qualitative difference of generalization after
"symbols as objects" in terms of "separation model" as follows.
Most students could successfully go to intensional generalization which makes the new knowledge familiar. The case of decimals, therefore, could be translated into division over natural numbers with which students are well acquainted. On the other hand, only a few students could go to extensional generalization which makes the familiar quite new. It shows us that division with fractions is not division but multiplication and consequently generates rational numbers from fractions.
The authors designed an alternative teaching of division with fractions for extensional generalization, which could induce the transition from arithmetic to mathematics. The main characteristic of this treatment is summarized as follows. The first point of our proposal is to make an expression based on the schema of "comparison" instead of that of proportion. The second is to deduce the algorithm "invert and multiply" logically based on the properties of fractions or the rules about division. The teaching experiment based on this framework was practiced for the sixth graders in this research at a Japanese elementary school. It showed us not only the effectiveness of an alternative teaching but also the cognitive process in the separation model.