Developing proportional reasoning skills may be accepted as one of the most important goals for middle school math programs across the world.  Learning proportional reasoning is important since it is considered a prerequisite for understanding other subjects in high school and in the following years of education, and for handling daily life problems and situations (Modestou & Gagatsis, 2007; 2008; Spinillo & Bryant, 1999; Van Dooren, De Bock, & Verschaffel, 2010a; Van Dooren, De Bock, Vluegels, & Verschaffel, 2010b; Vergnaud, 1988).  In contradiction with its significance, students have serious difficulties in developing proportional reasoning (Capon & Kuhn, 1979; Lawton, 1993).  For a better understanding of these difficulties, middle school students’ understanding of proportional reasoning needs to be investigated.  

Proportional reasoning is a special form of multiplicative reasoning that requires considering the co-variation between variables, comparing the multiple variables at the same time, and using information as a whole (Lesh, Post, & Behr, 1988).  When problems that students encounter in school and instructions are considered, there is a tendency to limit proportional reasoning as the ability to solve certain set of problems or use of some algorithms (Modestou & Gagatsis, 2008). However, in addition to the ability to solve proportional problems, proportional reasoners need to differentiate proportional and non-proportional situations (Modestou & Gagatsis, 2008; Van Dooren et al., 2010a). Van Dooren et al. (2010a) identified four profiles according to students’ use of erroneous strategies depending on the problem type and number structure. These are (a) additive reasoners, (b) proportional reasoners, (c) number-sensitive reasoners, and (d) correct reasoners.  Additive reasoners overuse additive methods in proportional and constant situations, where it is inappropriate.  Proportional reasoners overuse multiplicative (proportional) method in non-proportional situations, for example, in additive situations or constant situations.  Number sensitive reasoners solve problems including integer ratio proportionally, and problems including non-integer ratio additively.  Correct reasoners are the ones who can differentiate proportional and non-proportional situations and choose the appropriate solution strategy for each situation.  Van Dooren et al. (2010a) observed that almost half of the 5^th and 6^th grade students overused additive and proportional methods. Therefore, research focusing on understanding the development of proportional reasoning should look at middle school years more closely.

The purpose of this study is to examine middle school students’ (5^th and 6^th grade) understanding of proportional and non-proportional situations by analyzing their performance on proportional and non-proportional problems.  More specifically, this study investigated the following research questions:

(1) Can 5^th grade students differentiate between the proportional problems (missing-value and comparison) and non-proportional situations (additive and constant situations)? In order to answer this question, the following questions will be examined: (a) Can 5^th grade students solve proportional and non-proportional problems with the same success rate? (b)What is the distribution of students’ erroneous strategies in each problem type?

(2) Can 6^th grade students differentiate between the proportional problems (missing-value and comparison) and non-proportional situations (additive and constant situations)? In order to answer this question, the following questions will be examined: (a) Can 6^th grade students solve proportional and non-proportional problems with the same success rate? (b)What is the distribution of students’ erroneous strategies in each problem type?

(3) Does the use of integer and non-integer numbers in proportional problems (missing-value and comparison) and non-proportional problems (additive, and comparison) affect 5^th grade students’ solution strategies?

(4) Does the use of integer and non-integer numbers in proportional problems (missing-value and comparison) and non-proportional problems (additive, and comparison) affect 6^th grade students’ solution strategies?

Capon, N., & Kuhn, K. (1979). Logical reasoning in supermarket: Adult female’s use of a proportional strategy in everyday context. Developmental Psychology, 15: 450-452. Harel, G., & Confrey, J. (1994). The development of multiplicative reasoning in the learning of mathematics. Albany, NY: State University of New York. Lawton, C.A. (1993). Contextual factors affecting errors in proportional reasoning. Journal for Research in Mathematics Education, 24: 460-466. Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In J. Hiebert & M. Behr, (Eds.), Number concepts and operations in the middle grades (pp. 93–118). Reston, VA: National Council of Teachers of Mathematics. Modestou, M., & Gagatsis, A. (2007). Students’ improper proportional reasoning: A result of the epistemological obstacle of “linearity.” Educational Psychology, 27(1): 75-92. Modestou, M., & Gagatsis, A. (2008). Proportional reasoning in elementary and secondary education: Moving beyond the percentages. In A. Gagatsis (Ed.), Research in Mathematics Education (pp. 147-162). Nicosia: University of Cyprus. Modestou, M., & Gagatsis, A. (2010). Cognitive and metacognitive aspects of proportional reasoning. Mathematical Thinking and Learning, 12(1): 36-53. Spinillo, A.G., & Bryant, P. (1999). Proportional reasoning in young children: Part–part comparisons about continuous and discontinuous quantity. Mathematical Cognition, 5: 181–197. Van Dooren, W., De Bock, D., Verschaffel, L. (2010a). From addition to multiplication … and back. The development of students' additive and multiplicative reasoning skills. Cognition and Instruction, 28(3):360-381. Van Dooren, W., De Bock, D., Vleugels, K., Verschaffel, L. (2010b). Just answering … or thinking? Contrasting pupils’ solutions and classifications of missing-value word problems. Mathematical Thinking and Learning, 12(1): 20-35. Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 141-161). Hillsdale, NJ: Lawrence Erlbaum Associates; Reston, VA: National Council of Teachers of Mathematics.