generally correlated positively with the learning measures, while the dimensions that appeared not constructive were negatively associated with mathematical learning.
Again, we return to the contrast between project, nonproject, and individual differences. In both studies, the project led to increases in some of the desirable dimensions, while not affecting a second group of dimensions which were, at the level of individual differences, quite highly associated with the former. As shown in Table 9.5, the goal of understanding (task orientation 11) and the associated beliefs that success requires collaboration and attempts to understand were both associated with more advanced mathematical performance. But the project classes were superior on only these beliefs, not on task orientation II. Similarly, at the level of individual differences in associations with the different aspects of students' theories, the relational and instrumental scales did not look very different (Table 9.5). But a difference was clear when project and nonproject students were compared.
Conventional achievement tests might seem to produce hard, substantial data. One should not forget, however, that, like questionnaires, they involve marks that students make on paper. Their substantiality, to the extent that it exists, depends on our agreement to construe these marks as important and substantial. If students scored high on tests that are sensitive to higher order thinking but thought mathematics had nothing to do with making life meaningful or interesting, would we have achieved anything of substance or of a high order? What do any of our achievements profit us if they fail to make our lives subjectively meaningful or somehow satisfying? In education, we hope to make learning more meaningful more connected to the rest of students' lives. This does not mean that we can ignore measures of attainemnt or reasoning.
In the case of the classes we studied, the promotion of insightful learning was accompanied by beliefs that appeared to auger well for mathematics education, for students as individuals, and for the social fabric. The project experience appeared to dispose students to see school mathematics as the meaningful activity most of them want it to be. This makes the higher levels of rational reasoning of the project students appear to be a genuinely substantial achievement. The tendency of the project to lead students to reject the notions that conformity leads to success and that learning mathematics makes one likely to conform to whatever others suggest might also foretell of personal and intellectual autonomy. Finally, for a society often accused of extreme individualism and competitiveness, it appears significant that these strengths -- that some might have thought to be the products of competition -- were accomplished in an atmosphere of collaboration, with reduced levels of ego orientation and elevated beliefs in the contribution of collaboration to success.
These results, based on marks made on paper by second graders, converge with experiences of teachers and observers and with the theoretical frameworks within which the measures were conceived. These frameworks, in turn, are nothing more lofty than attempts to make sense of the experiences of being students and teachers. Our measures are somewhat more formal, but, in turn, they seem to help us make sense of those experiences, to re-examine what we are about, and what we should do next. They seem to have some validity.
The research and development project reported in this paper was supported in part by the National Science Foundation under grant nos. MDR 897- 0400 and MDR 885-0560. The development of the instrument to assess conceptual development and computational proficiency in arithmetic was supported by the Indiana State Department of Education. All opinions expressed are, of course, solely those of the authors.
Asch, S. E. ( 1952). Social psychology. Englewood Cliffs, NJ: Prentice-Hall.
Ausubel, D. P., Novak, J. D., & Hanesian, H. ( 1978). Educational psychology: A cognitive view ( 2nd