Proportional reasoning refers to “the ability to scale up and down in appropriate situations and to supply justifications for assertions made about relationships in situations involving simple direct proportions and inverse proportions” (Lamon, 2006, p. 3). It is a pivotal concept and serves as the capstone of children’s elementary school arithmetic and the cornerstone of all that is to follow (Post, Behr & Lesh, 1988). It is also a part of the foundation for more complex concepts. For instance, it constitutes the backbone of many mathematical concepts and topics such as algebra, geometry, trigonometry, statistics and probability and measurement (Lesh, Post & Behr, 1988). Despite being a core concept in elementary and secondary school mathematics, numerous studies have shown that students have difficulty with problems involving proportional reasoning concept (Kaput & West, 1994; Noelting, 1980; Vergnaud 1983). Besides, it is estimated that more than half of the adult population cannot be viewed as proportional thinkers (Lamon, 2006). Increasingly complex levels of proportional reasoning require conceptual understanding (Skemp, 1987). Therefore, it is reasonable to elaborate on learners’ conceptual knowledge of proportions when attempting to explore causes of their difficulties with proportionality tasks.

According to Engelbrecht, Bergsten and Kagesten (2009), conceptual learning occurs as a result of a combination of existing knowledge and newly encountered knowledge and it enables individuals to understand and appropriate new knowledge. Therefore, a student who has learned conceptually recognizes and applies definitions, principles, rules and theorems and can compare and contrast related concepts. Likewise, Eisenhart et al. (1993) state that conceptual knowledge refers to the knowledge of the underlying structure of mathematics. In other words, it refers to the relationships and interconnections of ideas that explain and give meaning to mathematical procedures. Hiebert and Wearne (1986) define conceptual knowledge as knowledge of facts, properties, and relations in mathematics. These facts and properties are related to each other significantly. Therefore, they are not isolated pieces of information; instead they are pieces of information interconnecting with one another and with other existing concepts.  The most important feature of conceptual knowledge is that a fact or proposition becomes a part of conceptual knowledge if it is linked to a larger network through relationships that have been constructed in some way.

There is a large body of research on students’ conceptual knowledge of algebra (Baki & Kartal, 2004; Bekdemir & Işık, 2007; Soylu & Aydın, 2006) and more specifically on definite integral concept (Delice & Sevimli, 2010), function of logarithm concept (Tekin, 2008), differential equations (Arslan, 2010), rational numbers (Zakaria & Zaini, 2009), mathematical equivalence (Rittle-Johnson & Alibali, 1999) and decimal fraction concept (Rittle-Johnson et al., 2001). However, limited number of studies dealt with students’ conceptual knowledge of proportions (e.g., Singh, 2000). Therefore, the current study aimed to explore middle school students’ conceptual knowledge of proportions with the following research question: What sorts of reasoning do middle school students make while solving tasks requiring conceptual knowledge of proportions?

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