problems as an undeniable source of incorrect additive reasoning with ratios.
The relapse at the end of the project, after a period of considerable stability in most of the students' learning processes, can be explained by the search for explicit rules for calculating with fractions in the context of an especially designed theme called Land of Together. Here all numbers still reflected fair sharing. The number 2, for instance, appeared as 4/2 expressing that 4 goodies were shared by 2 people, or as 6/3, and so on. The main question that was tackled was: Which procedures will students in the Land of Together have to apply when calculating? (See Streefland, 1991.) These procedures, within the context of the Land of Together, had not crystallized yet into general methods. Another cause for relapse was the search for a solution in cases where the numerical obstacles were too complex and contact with the acquired insights was broken. As can be seen in the final graph, there was great differentiation in the group with respect to N- distractors.
First and foremost, we feel that little importance should be attributed to computations involving fractions as such. Fractions themselves are not the point. The intention of the intertwining of fractions and ratios was primarily to emphasize the mathematization of connections and relations. In other words, the point of this investigation was to determine how students learn to assimilate mathematically situations in which quantities are brought together functionally in one way or another.
In the mathematization of such relations, the students learn to standardize in such a way that equating in one component is achieved, whereby comparing and ordering then can be simply rounded off by determining the relative difference or by performing the demanded division. If mathematics education will deal with fractions in the suggested manner in the future, the following statement of Goethe will lose its meaning for the generations of students to come:
Remember once and for all
The most important aphorism is:
Whole numbers won't bear a secret for you,
But fractions a big one, sure they do!
(translated from German by Leen Streefland)
Behr, M. J., Post, T. R., & Silver, E. A. ( 1983). "Rational number concepts". In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 92-144). New York: Academic.