Teaching the Concept of Unit in Measurement Interpretation of Rational Numbers

Abstract

This study investigated middle school students understanding of and unitization concepts in measurement interpretations of rational numbers using the number line as a tool. Fifty-six seventh-grade students were pretested and five consecutive whole-class teaching experiments were developed and administered based on pretest results. Five students were later chosen for semi-structured clinical interviews, based on their conceptions of and unitization. Students' reasoning was induced from the analysis of pre- and post-tests, observations of classroom teaching episodes, videotapes of interviews, and transcriptions and photographs of student artifacts. Results suggested that identification created difficulty for students in locating rational numbers on number lines. SUMMARY Purpose and significance: The purpose of this study was to investigate middle school students' difficulties understanding the unit associated with measurement interpretations of rational numbers using number lines. We showed what kind of unit understanding is necessary for a measurement interpretation of rational numbers and how classroom instruction can support the development of such understanding, especially building on students' part-whole interpretations of unit. Methods: A total of fifty-six seventh grade students from an urban K-8 school located in southwestern United States participated in this study. Students from three mathematics classes were pretested, based on the existing literature concerning length measurement and number lines, to determine their initial understanding. Based on the data gathered, a teaching experiment was conducted to see how students' understanding developed over a three-week period. Data was gathered through classroom observations, student artifacts, and interviews. Students demonstrated several misconceptions when using number lines as a tool for measurement. Results: Results indicated that students had problems locating improper fractions on the number line. For instance, fraction notation was read inverted to make improper fractions into proper fractions, resulting in a wrong answer. Furthermore, students misinterpreted the whole number line as the rather than a connected, continuous composition of units. The majority of students mislocated proper fractions on a number line from 0 to 5. For instance, when locating ae on the number line (ranging from 0 to 5), these students partitioned the number line into four equal pieces and then marked the 3 rd point (partition) to the right of zero. Discussion and Conclusion: Students' difficulties in applying rational numbers as measures on number lines revealed that the and unitization concepts do not develop naturally. When using an abstract tool such a number line, students did not easily see fractions as measures of distance. Using concrete materials as a and focusing on the iteration of that help students to develop a measurement sense. Moreover, transition from concrete representations to more abstract representations (e.g. number line) is critical to fully understanding rational numbers as measures. Several instructional strategies help students see fractions as measures. Effective instruction should focus on the important concepts within measurement such as unit, unitization, and the iteration of length units; and connect key mathematical content to students' previous knowledge and experience. Also, group activities need to actively engage children in meaningful representations and discussions of measurement strategies that encourage increasingly sophisticated strategies and metacognitive thinking. Students need to actively participate in measuring activities and group discussions that involve using different units and pictorial representations.