Preservice Elementary Teachers Use Drawings and Make Sets of Materials to Explain Multiplication and Division by Fractions

Background: Multiplication and division by fractions are among the most troublesome concepts in the elementary mathematics curriculum. Recent studies have shown that preservice elementary teachers in the United States do not have deep understandings of these concepts. Effective ways to improve preservice teachers\u27 conceptual understanding of these concepts need to be identified. Citation: Rule, A. C., \u26 Hallagan, J., editors. (2006). Preservice elementary teachers use drawings and make sets of materials to explain multiplication and division by fractions. A research study presented at the 2nd Annual Preparing Mathematicians to Educate Teachers (PMET) Conference at Oswego, New York, June 6, 2006. Conclusions: The two activities increased student understandings of multiplication and division by fractions. Although students improved through the activities, many students\u27 understandings were still incomplete. More than two focused activities are needed to ensure deeper understanding of concepts. Preservice teachers need concrete experiences with these concepts in their mathematics classes as well as in mathematics education coursework. 2 Control or Comparison Group: Both the control group and the experimental group consisted of preservice teachers from several sections of the same instructor\u27s undergraduate mathematics methods courses and were matched on pretest scores. Both groups completed the homework assignment in which they used drawings to illustrate multiplication and division by fractions. The instructor did not present lessons on these concepts to the classes until after the posttest had been completed so that the effects of these activities would not be confounded. The experimental group completed the additional activity of making hands-on materials to model these concepts. The study examined the increase in preservice teachers\u27 conceptual understanding of multiplication and division by fractions through the two activities. Data Collection and Analysis: Both control and experimental groups were assessed with identical pretest/ posttest instruments constructed by the investigators to determine both procedural knowledge of solving equations involving multiplication and division by fractions and conceptual knowledge of writing equations for story problems and using drawings to illustrate concepts. Posttest scores, student work on the assessments, drawing assignment, and hands-on materials were examined along with student comments on a survey that asked what subjects learned from participating in the intervention activity. Findings: The two activities improved preservice teachers\u27 understandings of these concepts as revealed by the change in scores from pretest to posttest (50.8% on pretest to 67.5% and 71.4%). Those who completed both assignments scored somewhat higher (71.4% compared to 67.5%) than those who only completed the drawing assignment, but this difference was not statistically significant. Preservice teachers reported that their understandings of these concepts improved through the activities. Intervention: Both control group and experimental group participated in composing story problems with drawings to illustrate multiplication and division by fractions. The experimental group completed the additional activity of making hands-on materials with accompanying story problems to model multiplication and division by fractions. Purpose: The purpose of the study was threefold: 1) to investigate the effectiveness of two activities in helping preservice teachers develop deeper understandings of multiplication and division by fractions; 2) to identify typical errors preservice teachers make and identify difficulties they encounter while learning these concepts; and 3) to provide examples of drawings and hands-on materials that effectively model multiplication and division by fractions for others to use in learning and teaching. Research Design: The study was a pretest - intervention - posttest design with control and experimental groups. Because lower-performing students tended to volunteer for the extra-credit activity (the intervention for the experimental group), blindly matched groups were formed on pretest scores. Setting: Preservice teachers from three mathematics methods classes of college students majoring in elementary education at a mid-sized college in central New York State during the spring semester of 2006. Study Sample: Forty-two white preservice elementary teachers enrolled in a mathematics methods course. The experimental group consisted of 18 females and 3 males; the control group consisted of 16 females and 5 males