Memo 3

           Cramer et al. (1993) claimed, “Proportional reasoning abilities are more involved than textbooks would suggest” (p. 169). Textbooks usually help students master their procedural fluency on the content instead of helping the students gain conceptual understanding in the content. For example, the authors mentioned that the textbook give out an algorithm that is used to set up and solve missing values using cross multiply, but how does that relate to the context of the problems. The cross-product algorithm simply has no meaning to the students because it is just a procedural way in computing the problem and finding the solutions. It shows no conceptual understandings. Hence, teachers must use hands-on activities that are applicable to real life to help the students get engaged in the contents.

            Furthermore, students would be able to develop deeper understanding of mathematics. Cramer et al. (1993) advised, “Research on various middle school content areas supports the value of a conceptually oriented curriculum over a procedurally based curriculum. Since textbooks are generally procedurally oriented, textbook-dominated programs should become less frequent” (p. 173). The authors believed that textbooks may not be the best resources in teaching students for understanding because textbooks are too procedural. Furthermore, the students will not be critically thinking about the content if they are using the textbooks. Hence, lessons should be taught in a way that the teacher can provide scaffolds to help the students to attain more conceptual understanding.

            To be more concise, proportional reasoning is just a way to help students figure out an algorithm to solve the problems. However, in the process of figuring out the algorithm, the students should be able to gain conceptual understanding upon finding an algorithm. In other words, understanding the proportional situations means that the students understand beyond the procedural aspects of the problems.

            When teaching students proportional reasoning, the teachers can ask questions like why to help emphasize for understanding in proportional reasoning. Instead of just writing out the algorithm, ask the students why they write the algorithm that way. Furthermore, the teachers can use hands-on activities and manipulative to help students learn proportional reasoning. For example, using rods can allow students to experience hands-on activities. Thus, it will be more likely to help students have a better foundation of the content.