Transitioning from additive to multiplicative thinking:a design and teaching experiment with third through fifth graders.

Abstract

The maturation of multiplicative thinking is key to student progress in middle school as rational number, ratio, and proportion concepts are encountered. But many students arrive from the intermediate grades and falter in developing this essential disposition. Elementary students have historically learned multiplication and division as operation procedures, but this does not mean the students are thinking multiplicatively. Understanding how intermediate elementary students begin to shift from additive approaches to multiplicative thinking in scale and proportion is not fully understood. This study asks interrelated questions of how the development of additive to multiplicative thinking evolves over time and what students' thinking looks like at various points of transition as they move through third, fourth, and fifth grades. Secondly, what integrated and coordinated series of tasks can teachers use to reveal the emergence of multiplicative thinking? The study took place over the first six months of the school year. Using the structures of a design study and teaching experiment, the principal investigator and classroom teachers collaborated using instructional tasks conjectured to reveal the development of multiplicative thinking among students. Reported here are results from the pre and post interviews of a sample group at each grade level (n =11, 12, 11, respectively). Student transitions over "unit confusion," the justified emergence of a place value cover pattern, the role of number size in influencing multiplicative thinking, and role of language in monitoring students' level of thinking is discussed. The development and effectiveness of the interview protocol and scoring rubric used to analyze student level of multiplicative thinking is included.