Objectives
I want my students to be able to deal with fractions effectively, whenever and wherever fractions are encountered in algebra or higher secondary studies.
It would be nice to leave it at that, but, of course, it would also be an impractically simplified statement of objectives. I propose two subdivisions to my overarching objective, as refinements that speak directly to improving effective student work with fractions in algebra studies.
First, students should have a more expansive understanding of fractions than as mere part-to-whole expressions. This idea of fractional notation, most prevalently taught in the elementary levels of math study in the US, becomes so fixated that students have difficulty moving from it to other valid interpretations of the form. So the overarching objective is extended to improve students' understandings of fractions in all their various purposes, as division, ratio, etc., via examples and practice in applying these interpretations. Fortunately algebra studies provide fractions as expressions of these relationships in many contexts, so the content is readymade for such adaptation.
Second, students should have a skill set for management of and communication about fractions that is built upon an extrapolation approach. Not only should students be able to recognize all the interpretations that are possible for a fraction, they should be able to accurately choose the appropriate interpretation to apply within the solution of any given problem, based on their "reasoning out" of the problem context. In other words, as students gain an expanded concept of what "a/b" might mean, they need to gain a complementary, diverse heuristic that allows them to choose the accurate meaning. Fortunately, again, algebra studies provide the opportunity to build this skill set..
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