PurposesEffective teaching; Professional development activity; Student collaboration; Student engagement; Reasoning; Representation

DescriptionThis RUanalytic 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 related to comparing and ordering fractions have been partitioned into three analytics. The analytics preceding this one in the classroom investigations are entitled: (1) Imagining the Density of Fractions; and (2) Using Meredith’s Models to Reason about Comparing and Ordering Unit Fractions. These analytics focus on comparing and ordering fractions as well as student placement of fractions on a line segment. This analytic extends students’ learning about ordering fractions on a line segment to ordering fractions on an infinite number line. It identifies some obstacles that students encounter as they work to coordinate their understanding of fraction as operator building from rod and line segment models, to fractions as number as they extend their problem solving to deal with the disequilibrium that they confront in working out the placement of fractions on an infinite number line.

The events depicted in this RUanalytic occurred on November 10, 1993 and are taken from the third month of the study. This is the 17th session of 44 recorded sessions of the fraction intervention (Yankelewitz, 2009). Prior to this, the students spent seven sessions investigating different representations of fraction ideas, seven sessions investigating the comparison of fractions and two sessions placing fractions on a number line segment. The number line segment explorations are described in the first two RUanalytics mentioned above.

The events described in this analytic follow immediately after the analytic, Using Meredith’s Models to Reason About Comparing and Ordering Unit Fractions, in which students made use of Meredith’s models of fraction placement on a line segment. During the last 15 minutes of the November 10, 1993 session we observe the students placing fractions on an infinite number line. The number line that is shown in the introduction to this analytic is labeled with numbers from -3 to +3 with arrows at each end denoting a number line. This is an extension of the earlier task where students placed fractions on a line segment with endpoints at zero and one, which, later in the activity, was extended by one of the students to an endpoint at the positive two. Certain cognitive obstacles can be observed during the events and evidence of how students resolved the obstacles and dealt with both equivalent and improper fractions.

The intervention reported here was conducted with a fourth-grade class at the Conover Road School, Colts Neck, N.J. The Colts Neck Study included 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 this district at this time, that students were 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 the formal instruction in grade 5.

It might be useful to consider some activities in which the students were engaged prior to those shown in this analytic. During the sessions prior to these three analytics students explored various problems concerning relationships among whole numbers and fractions. Examples of the tasks include using Cuisenaire rods to identify number names for the various fractions in relation to a particular rod that was given a specific number name. Students also explored the concept of “infinitely many”, as they investigated how many unit fractions might exist between zero and one. 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 comparing fractions and “fair sharing” was triggered by a discussion of sharing two candy bars of different sizes in which the fraction one half of a smaller bar was smaller in size than one half of a larger candy bar, emphasizing the importance of keeping the unit constant in making fraction comparisons.

The research was supported by a grant from the National Science Foundation: MDR-9053597 directed by R. B. Davis and C. A. Maher. The facilitator on this day was Researcher Carolyn Maher.

References

Bulgar, 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. & Sigley, R. (2012). Teachers’ identification of children’s upper and lower bound reasoning. . In L. R. Van Zoest, J. J. Lo, & 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. & 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.

Reynolds, 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 psychology

of 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.

The events depicted in this RUanalytic occurred on November 10, 1993 and are taken from the third month of the study. This is the 17th session of 44 recorded sessions of the fraction intervention (Yankelewitz, 2009). Prior to this, the students spent seven sessions investigating different representations of fraction ideas, seven sessions investigating the comparison of fractions and two sessions placing fractions on a number line segment. The number line segment explorations are described in the first two RUanalytics mentioned above.

The events described in this analytic follow immediately after the analytic, Using Meredith’s Models to Reason About Comparing and Ordering Unit Fractions, in which students made use of Meredith’s models of fraction placement on a line segment. During the last 15 minutes of the November 10, 1993 session we observe the students placing fractions on an infinite number line. The number line that is shown in the introduction to this analytic is labeled with numbers from -3 to +3 with arrows at each end denoting a number line. This is an extension of the earlier task where students placed fractions on a line segment with endpoints at zero and one, which, later in the activity, was extended by one of the students to an endpoint at the positive two. Certain cognitive obstacles can be observed during the events and evidence of how students resolved the obstacles and dealt with both equivalent and improper fractions.

The intervention reported here was conducted with a fourth-grade class at the Conover Road School, Colts Neck, N.J. The Colts Neck Study included 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 this district at this time, that students were 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 the formal instruction in grade 5.

It might be useful to consider some activities in which the students were engaged prior to those shown in this analytic. During the sessions prior to these three analytics students explored various problems concerning relationships among whole numbers and fractions. Examples of the tasks include using Cuisenaire rods to identify number names for the various fractions in relation to a particular rod that was given a specific number name. Students also explored the concept of “infinitely many”, as they investigated how many unit fractions might exist between zero and one. 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 comparing fractions and “fair sharing” was triggered by a discussion of sharing two candy bars of different sizes in which the fraction one half of a smaller bar was smaller in size than one half of a larger candy bar, emphasizing the importance of keeping the unit constant in making fraction comparisons.

The research was supported by a grant from the National Science Foundation: MDR-9053597 directed by R. B. Davis and C. A. Maher. The facilitator on this day was Researcher Carolyn Maher.

References

Bulgar, 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. & Sigley, R. (2012). Teachers’ identification of children’s upper and lower bound reasoning. . In L. R. Van Zoest, J. J. Lo, & 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. & 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.

Reynolds, 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 psychology

of 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.

Created on2015-04-16T22:35:42-0400

Published on2015-08-31T09:20:20-0400

Persistent URLhttp://dx.doi.org/doi:10.7282/T39Z96SR