A new view of teaching is emerging from the work of the constructivists and mathematics education reform leaders. In particular, we examine here four aspects of teaching that must change: task selection, guidance of classroom discourse, setting the learning environment, and the analysis of teaching and learning. Several national curriculum projects are working to effect these changes and examples of their work are provided. This work has motivated individual teachers to similarly design investigations that engage students in the study of significant mathematics, and two examples are included. Assessment must also change and students must learn to become less dependent on “authority” for the correctness of answers. Finally, our present understanding of constructivism and its implications for teaching/learning must not be static; though that view now may be at the center, we must listen to those who are on the edges and expect to be changed again and again in the years ahead.
The van Hiele model of the learning of geometry currently enjoys both popular and research interest. Hoffer (1981) provides an overview of the model and identifies problems that are appropriate for students at each of the five van Hiele levels, the first three of which will be considered in this paper.
A series of longitudinal studies of mathematical computational ability at grades eight and ten in the state of New Hampshire produced the following results. In the fall of 1963. using the Metropolitan Achievement Test and the Arithmetic Computational subtest of that battery, the median raw score for eighth graders in New Hampshire equalled a grade equivalent of 8.8.
Whole numbers are used to count sets, report measures, and order a set. You can report the sum and difference of two counts, report the product of twice a measure, and divide two whole numbers to determine how many subsets of a given size are contained in a larger set or find how many objects will be in each subset if you divide the given set into a specified number of parts.
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