ABSTRACT
In our study, opportunity to learn (OTL) has been interpreted more broadly than its more typical use as a gauge of content coverage. When the aim of the lesson is primarily coverage of content, the emphasis on (usually) unconnected pieces of information reduces the cognitive demand on students. To learn mathematics in ways that enable students to use that knowledge outside familiar problem contexts, students need the time and opportunity to explore relationships among mathematical ideas, to extend and apply these ideas in new situations, to reflect on and articulate their thinking, and to make mathematical knowledge their own (Carpenter & Lehrer, 1999). In other words, they need to understand the algorithms they apply-to understand, for instance, that although comparing shadows of two vertical objects would give a good estimation of height ratio, comparing shadows of the leaning Tower of Pisa and a vertical flag pole next to it would not.
