Assessment of Authentic Performance in School Mathematics

By Richard Lesh; Susan J. Lamon | Go to book overview

the reader can say. But we have become convinced, from studying these videotapes, and from working with teachers in going over analyses of these tapes, that both teachers and students have much to gain from making efforts toward this very fine-grained analysis of how students are actually thinking about mathematical problems and mathematical situations. This is a valuable and important form of assessment, and in the right context it can be carried out successfully.


A Note on Heroes

Clearly, we argue for the importance of looking on a very minute level at how students think about mathematical situations, and at how teachers think that students think about mathematical situations. We see this as one of the most critical questions in mathematics education. It requires careful and insightful analysis--but even more, it requires teachers who will seek out the best possible data, even in instances where they may, given hindsight, wish that they had done something differently.

This is the spirit of the very best in science, seeking truth without placing blame. In our view, the real heroes of these studies are the teachers, people like Linda and Pat, who have worked so hard and so selflessly to get and to share the most complete possible data, even when they wish they had known earlier some of the things they only found out about later on.


NOTES AND REFERENCES
1.
Much could be said in analysis of the teacher's sudden introduction of the idea of boxes to hold the pizza. Pedagogically, we do not like it, since we see it as an intrusion that is not central to this problem. But from a theoretical point of view, it is in fact important. For one thing, it is one instance of the role that previous personal experience and concrete materials can have on shaping one's ideas (see, for example, Dana's concern about making outfits that do not match, Maher and Martino, 1992). It also looks quite different when viewed from the perspective of Lesh's important theory of "model-eliciting problems" (Algebra Group e-mail Forum, JKAPUT@umassd.edu, 19 February 1992). Since these tasks start with real- world activities or situations, many different aspects of these situations can be considered to be relevant. Indeed, just this sort of "reality" consideration has led the teacher into one interpretation, and Brian into a different one. Unfortunately, in this instance they were not able to share their different perspectives effectively.

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