A conception of between-state ratio in fraction form

Gulseren Karagoz Akar (1)

1 Boǧaziçi üniversitesi = Boğaziçi University [Istanbul]

This study investigated one prospective secondary mathematics teacher's (Jana's) reasoning on between-state ratios in missing value problems and comparison problems. In two one-and-one-half hour written problem solving sessions followed by one hour-long clinical interview, Jana's use of informal and formal strategies and justifications behind those strategies in the context of ratio were examined. Extending previous research, results of this study showed that someone could quantify between-state ratios acting as an operator in fraction form once she has understood ratio as an association of amounts of quantities in within-state ratios. Results indicated a dichotomy within the boundaries of identical groups conception in terms of within-state ratios and between-state ratios prior to an understanding of between-state ratios as a single intensive quantity. Keywords: Between-state ratio, within-state ratio, division, multiplication, extensive and intensive quantity. Theoretical framework Students might approach to a proportion such as a/b=c/d by comparing the first set of ratios a/b or c/d or the second set of ratios a/c or b/d (Noelting, 1980). In the first case, the ratios a/b or c/d are called within (state) ratios, where the ratio represents the original quantities within one state. In the second case, the ratios a/c or b/d are called between (state) ratios, where the ratio represents quantities between two situations (Noelting, 1980). For instance, envision the Recipe 1 Problem in this study. The original quantities of 9 tablespoons of oil and 4 tablespoons of vinegar could be represented by the within-state ratio, 9/4; and, the 4 tablespoons of vinegar and the 7 tablespoons of vinegar from two situations could be represented by the between-state ratio, 7/4. Researchers investigating prospective teachers' conceptions of ratio have revealed how teachers interpreted the relationships between quantities in ratio situations to quantify some attribute of interest such as lemon-lime flavor the Mixture Problem in this study. For this problem, research has shown that one might interpret the relationship between the quantities of 36 grams of pure lemon juice and 32 grams of pure lime juice in the following three different ways: First, someone having a robust conception of ratio conceptualizes that ratio is a single intensive quantity that expresses the size of one quantity (i.e., amount of lemon) relative to the size of the other quantity (i.e., amount of lime) represented by within-state ratios (Simon & Placa, 2012). In this conception one can utilize both partitive and quotitive division of the quantities in within-state ratios to quantify the attribute (i.e., lemon-lime flavor) in the situation. That is, either engaging in partitive or quotitive division, one might interpret the quotient (i.e., 1.125) of the original quantities represented in the within-state ratios, 36/32 , as a single intensive quantity representing the invariant multiplicative relationship between the quantities (Simon & Placa, 2012). This concept of ratio is also called as ratio as measure conception (Simon & Blume, 1994). Secondly, in order to quantify the lemon-lime flavor, one might think of the within-state ratio, 36/32, as representing an association of amounts of two quantities (Johnson, 2015). In this regard, s/he thinks of the quantities making up a particular combination that quantifies the taste of the mixture. This understanding Thematic Working Group 02 Proceedings of CERME10 306

Domains: Mathematics [math]; Humanities and Social Sciences

Gulseren Karagoz Akar. A conception of between-state ratio in fraction form. CERME 10, Feb 2017, Dublin, Ireland. ⟨hal-01873489⟩