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ERIC Number: EJ1035862
Record Type: Journal
Publication Date: 2014-Feb
Pages: 8
Abstractor: ERIC
Reference Count: 11
ISSN: ISSN-1072-0839
Sharing Cookies: A Case Study
Salls, Jenny
Mathematics Teaching in the Middle School, v19 n6 p368-375 Feb 2014
Rational number interpretations can include part-whole, measure, ratio, quotient, and operator. These are all subconstructs of partitioning (Barnett-Clarke et al. 2010; Behr et al. 1980; Clarke, Roche, and Mitchell 2008; Flores, Samson, and Yanik 2006). Each of these subconstructs uses different cognitive skills (Driscoll 1984), so it is important that students experience all the different forms of interpretation. "Part-whole" involves having students partition a whole into equal-size parts. "Measure" involves iterating a unit, such as iterating 1/8 three times to create 3/8. "Ratio" indicates a comparison between two quantities. The "quotient" interpretation is used when objects are shared among a number of groups. A common use is sharing the 8 biscuits in a can of refrigerated biscuits among 5 people. The "operator" interpretation is closely related to the quotient interpretation and gets at the idea that a ÷ b = a/b. (Curcio and Bezuk 1994; Behr and Post 1992). These interpretations all play a critical role in the conceptual understanding of rational numbers, but students seldom experience the quotient and operator interpretations (Flores, Samson, and Yanik 2006). Because understanding division of fractions and simplifying complex numbers are both related to a comprehension that a ÷ b = a/b, mathematics teacher Jenny Salls wanted a quick task that would focus on the quotient interpretation while pushing students to make the connection to the operator interpretation. In addition to deepening students' understanding of these concepts, she also gave them opportunities to listen to their peers and value their peers' contributions to a whole-group discussion. Very few of these students have been in mathematics classes with other gifted learners. In a heterogeneous classroom, gifted students either do not take part in classroom discussions or dominate the discussion. As a result, they are unaccustomed to critically listening and responding to their peers during whole-group discussion. In this article, Salls describes an activity that can be approached in several different ways so that students find value in listening to their classmates' strategies and reasoning. She concludes that this simple activity showed even highly capable students may not necessarily have a deep understanding of rational number concepts, a lack that could impact their ability to make sense of algebraic concepts in later classes. Giving students the opportunity to make the connection between the operation a ÷ b and the fraction a/b gave Salls insight into what her students understood and prevented potential problems later in the course.
National Council of Teachers of Mathematics. 1906 Association Drive, Reston, VA 20191-1502. Tel: 800-235-7566; Tel: 703-620-3702; Fax: 703-476-2970; e-mail:; Web site:
Publication Type: Journal Articles; Reports - Descriptive
Education Level: Grade 7; Junior High Schools; Middle Schools; Elementary Education; Secondary Education
Audience: N/A
Language: English
Sponsor: N/A
Authoring Institution: N/A