Children’s competency about grasping some elementary mathematical principles and generating reasonable problem-solving procedures to solve arithmetical problems involving objects continues when children begin to school, although the focus of activity becomes increasingly symbolic and abstract as children progress through school (Rasmussen, Ho and Bisanz, 2003). A grasp of inversion is critical for understanding the additive composition of number (Bryant, Christie, & Rendu, 1999; Piaget, 1952) and for enabling shortcuts that render computationally difficult problems easy (Bisanz & LeFevre, 1990) (in Rasmussen, Ho and Bisanz, 2003). Thus the concept of inversion is an important acquisition in the development of mathematical cognition (Rasmussen, Ho and Bisanz, 2003). Inversion might be a powerful aid in the often-used informal procedure of decomposition (Nunes, Schliemann, & Carraher, 1993) in problems with an addend and a subtrahend that are unequal but close to one another (in Bryant, Christie, and Rendu, 1999).
One aspect of the relation between addition and subtraction is the principle of inversion, that a + b –b must equal a; calculation is not required (Sherman and Bisanz, 2007). This study aims to investigate 2nd and 4th grade children’s adequateness about understanding and using “Inversion Principle and Inversion Strategy” in relation to additive and subtractive expressions and to overcome the deficiencies of the elementary school mathematics curriculum in Turkey and the teachers’ implementation. Understanding of that in which situations or steps of the problems including additive, subtractive (a + b-b = a), and multiplicative (d x e / e = d) expressions, the students need to use the principle or why they do not need to use the strategy and also understanding of determining whether the students’ academic achievements at math varies to their awareness of “the Inversion Principle” and usage efficacy of the strategy are the lower dimensions of the study.
Drawing on the principle of inversion from the point of both qualitative and quantitative views, I investigated the participant elementary school children’s mental process when they come across inversion problems as basis for understanding how the students solve this kind of problems and which steps they follow in their minds.
The following hypotheses show the uncertain results of the study;
• The students can understand the inversion principle and use the inversion strategies when they solve the problems having additive and multiplicative expression effectively and beneficially are better at math than the students can not understand the inversion reason and use the related strategies.
• In parallel with math teaching teachers’ (math teachers or classroom teachers) understanding of the inversion principle and its importance, the teachers’ proficiencies on teaching the principle affect the students’ reasoning on the inversion problems and math positively.
