Proportional reasoning involves the use of ratios in the comparison of quantities. While it is a key aspect of numeracy, particularly in the middle years of schooling, students do not always develop proportional reasoning naturally. Research suggests that many students do not apply proportional methods appropriately and that they often erroneously apply both multiplicative and additive thinking. Further, students cannot always distinguish non-proportional situations from those that are proportional. Understanding the situations in which students mistakenly use additive or multiplicative thinking and the nature of the proportional reasoning that students apply to different problem types is important for teachers seeking to support their students to develop proportional reasoning in the classroom. This paper describes the development and use of a two-tier diagnostic instrument to identify situations in which students could and could not apply proportional reasoning and the types of reasoning they used. It presents data from an Australian study involving over 2000 middle years students (Years 5 to 9) as a means of illustrating the use of the instrument for diagnosing students’ reasoning in different situations. The findings showed that the instrument was useful for identifying problem types in which students of different ages were able to apply correct reasoning. It also allowed identification of the types of incorrect reasoning used by students. The paper also describes useful applications of the instrument, including its use as a diagnostic instrument by classroom teachers and its use in the design classroom activities included in teacher professional learning workshops.
Mathematics Education Research Journal, 2013, Vol 25, Issue 4, p. 523-545
Proportional reasoning; middle school mathematics; diagnostic testing; two-tier testing