Although one of these sets of understandings—proportional estimation—is primarily visual and nonnumerical, while the other, halving and doubling, is numeric, both have their grounding in multiplicative operations. It was our proposal that if we could help students merge these separate kinds of multiplicative understandings, we would allow them to construct a core conceptual grounding for rational numbers.50

Our strategy from the beginning was to develop what we called a “bridging context”51 to help students first access and then integrate their knowledge of visual proportions and their flexibility in working with halving numbers. The context we chose was to have students work with percents and linear measurement. As will be elaborated below, students were engaged from the start of the instructional sequence in estimating proportional relations based on length and in using their knowledge of halving to compute simple percent quantities. In our view, the percent and measurement context allowed students to access these initial kinds of understandings and then integrate them in a natural fashion. We regarded the integration of initial intuitions and knowledge as a foundation for rational-number learning.

Why Percent As a Starting Point?

While we found that starting with percent was useful for highlighting proportionally, we also recognized that it was a significant departure from traditional practice. Percent, known as the most difficult representation for students, is usually introduced only after fractions and decimals. Several considerations, however, led to this decision. First, with percents students are always working with the denominator of 100. We therefore postpone the problems that arise when students must compare or manipulate ratios with different denominators. This allows students to concentrate on developing their own procedures for comparison and calculation rather than requiring them to struggle to master a complex set of algorithms or procedures for working with different denominators.

Second, a further simplification at this beginning stage of learning is that all percentages have a corresponding decimal or fractional equivalent that can be relatively easy to determine (e.g., 40 percent = 0.40 or 0.4 = 40/100 or 4/10 or 2/5). By introducing percents first, we allow children to make their preliminary conversions among the different rational-number representations in a direct and intuitive fashion while developing a general understanding of how the three representations are related.

Finally, children know a good deal about percents from their everyday experiences.52 By beginning with percents rather than fractions or decimals, we are able to capitalize on children’s preexisting knowledge of the meanings of these numbers and the contexts in which they are important.53



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