Using Meredith’s models to reason about comparing and ordering unit fractions
Creators: name: Horwitz, Kenneth
name: Schmeelk, Suzanna
Publisher: name: No Publisher Supplied
Year: 2013

description: This VMCAnalytic focuses on a set of activities taken from a yearlong study designed to investigate how 4th grade students build fraction ideas (Schmeelk 2010). The activities on this day were partitioned into two analytics: the first focusing on the placement of fractions on a line segment and the second focusing on placement of fractions on a number line.  The first task that is shown in this first VMCAnalytic involves comparing and ordering unit fractions and placing them on a number line segment between zero and one, later extended by one of the students to “2”.  The depiction of comparing and ordering fractions presented in this VMCAnalytic is consistent with the grade 4 Standards for Mathematical Content which indicate that students should be able to extend understanding of fraction equivalence and ordering, decompose a fraction into a sum of fractions with the same denominator in more than one way and justify decompositions, for example, by using a visual fraction model. (CCSSI, 2010). The events show students comparing and ordering fractions and eventually placing them on a line segment. The intervention reported here was conducted with a fourth grade class at the Conover Road School in Colts Neck. The Colts Neck Study took place during fifty-six 60 to 90 minute classroom sessions over the course of one year.  The fourth grade class consisted of twenty five heterogeneously grouped students.  Fourth grade was selected since it was not until fifth grade in the New Jersey State Standards, which schools followed at that time, when students are formally introduced to operations with fractions in their curriculum. It was of interest to explore what understandings the students had and could build prior to formal instruction in grade 5. Other data from the Colts Neck Study research project have shown that students as young as nine and ten years old were able to build on their existing ideas about fractions and extend their knowledge to equivalent fractions (Steencken 2001), comparing fractions  (Reynolds 2005),  dividing fractions (Bulgar 2002) and reasoning about fractions (Yankelewitz, 2009).   The events depicted on this VMCAnalytic occurred on November 10, 1993 and are taken from the third month of these sessions. This is the 17th session of 44 recorded sessions of the fraction intervention (Yankelewitz, 2009). Prior to this, the students had spent seven sessions studying representations of fraction ideas, seven sessions studying the comparison of fractions and two sessions studying placing fractions on a number line segment.  Following this session, the students continued to study the placement of fraction on the number line for one additional session. Those subsequent events are described in the second VMCAnalytic by these authors. In the interventions leading up to the events in this VMCAnalytic, students had explored various problems concerning the relationship between whole numbers and fractions. Some examples of the tasks students were asked to do include using Cuisenaire rods to identify number names for the various fractions in relation to the particular rod that was given the number name 1. Students also explored the concept of “infinitely many”. The concept of comparing and ordering fractions began with a problem that involved sharing candy bars that provided a metaphor for the importance of retaining the same unit if the candy was to be equitably shared. The idea of “fair sharing” triggered a discussion of which fraction of the candy bar was bigger or smaller.  The research was supported by a grant from the National Science Foundation: MDR-9053597 directed by R. B. Davis, C. A. Maher. The facilitator on this day was Researcher Carolyn Maher.   ReferencesBulgar, S. (2002).  Through a teacher’s lens: Children’s constructions of division of fractions. Unpublished doctorial dissertation, Rutgers, The State University of New Jersey, New Brunswick.Maher, C. A., Palius, M. F., Maher, J. A. &amp; Sigley, R. (2012). Teachers’ identification of children’s upper and lower bound reasoning. . In L. R. Van Zoest, J. J. Lo, &amp; J. L. Kratky (Eds.), Proceedings of the 34th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 460-467. Kalamazoo, MI: Western Michigan University.Mueller, M., Yankelewitz, D. &amp; Maher, C. (2010). Promoting student reasoning through careful task design: A comparison of three studies. International Journal for Studies in Mathematics Education, 3(1), 135-16.National Governors Association Center for Best Practices, C. o. (2010). Common Core State Standards for Mathematics. National Governors Association Center for Best Practices, Council of Chief State School Officers, Washington D.CReynolds, S. L. (2005).  A study of fourth grades students’ exploration into comparing fractions. Unpublished doctorial dissertation, Rutgers, The State University of New Jersey, New Brunswick.Rowland, T. (2002). Proofs in number theory: History and heresy. In Proceedings of the twenty-sixth annual meeting of the international group for the psychologyof mathematics education Vol. I, Norwich, England, (pp. 230–235).Schmeelk, S. (2010) An Investigation of Fourth Grade Students Growing Understanding of Rational Numbers. Unpublished doctoral dissertation, Rutgers, The State University of New Jersey, New Brunswick.Steencken, E. P. (2001).  Studying fourth graders’ representations of fraction ideas. Unpublished doctorial dissertation, Rutgers, The State University of New Jersey, New Brunswick.Yankelewitz, Y., Mary Mueller and Carolyn A. Maher. (2010). A task that elicits reasoning: A dual analysis. Journal of Mathematical Behavior, 29, 76-85.Yankelewitz, D. (2009) The development of mathematical reasoning in elementary school students’ exploration of fraction ideas. Unpublished doctoral dissertation, Rutgers, The State University of New Jersey, New Brunswick.