**Year**2023**NSF Noyce Award #**1758368**First Name**Terri L.**Last Name**Kurz**Institution**Arizona State University**Role/Position**co-PI**Workshop Category**Track 1: Scholarships and Stipends**Workshop Disciplines Audience**Mathematics**Target Audience**Noyce Master Teachers, Noyce Teaching Fellows, Undergraduate and/or Graduate Noyce Scholars**Topics**STEM Content Area and/or Convergent Description Skills Development**Session Length**45 minutes minutes**Additional Presenter(s)**Tirupalavanam G Ganesh, ganeshtg@asu.edu

## Goals

We have three goals: (1) Participants will define wholes, construct the dividend and divisor, divide, then justify the quotient using a conceptual explanation; (2) Participants will conceptually explain dividing using the set model (two-sided chips); (3) Participants will describe the affordances and constraints of using the set model for fraction division.

## Evidence

Elementary school teachers sometimes struggle with the division of fractions (Copur-Gencturk, 2021; Lubinski, Thomas & Thomason, 1998). For example, Isik and Kar (2012) studied fractional division error problems of preservice teachers (n = 64). They found that they made 311 errors regarding the division of fractions. The researchers observed that the conceptual dimension of fraction division is often ignored by teachers. Their findings indicate the importance of focusing on conceptual division of fractions with preservice teachers in their university preparation coursework. Oftentimes, the traditional algorithm is used to teach fraction division. However, one of the problems with teaching the traditional algorithm is that it does not help students make sense of the meaning of division in relation to fractions (Li, 2008). Students should have opportunities to visualize what it means to divide fractions, moving beyond the invert and multiply approach (Gregg & Gregg, 2007). Models typically used to demonstrate conceptual understanding for fraction sense often focus on the area or linear model. For example, fraction circles and fraction squares are commonly used to help students make sense of the operations in fractions. Linear models are also used to support the development of operations with fractions (Lo & Luo, 2012).

The set model is not as common as the others. The set model can be defined as a set of objects that represent a fractional value. For example, to show 3/5, a set model might consist of three yellow chips and two red chips. The set is the total number of chips, in this case five. Using a set model, the ideas presented here may be fruitful in helping students make sense of dividing fractions. The nature of using the set model encourages thought, reasoning and problem-solving while encouraging conceptual understanding.

References:

Copur-Gencturk, Yasemin. “Teachers’ conceptual understanding of fraction operations: results from a national sample of elementary school teachers.” Educational Studies in Mathematics 107, no. 3 (2021): 525-545.

Gregg, Jeff, and Diana Underwood Gregg. “Measurement and fair-sharing models for dividing fractions.” Mathematics Teaching in the Middle School 12, no. 9 (2007): 490-496.

Isik, Cemalettin, and Tugrul Kar. “An Error Analysis in Division Problems in Fractions Posed by Pre-Service Elementary Mathematics Teachers.” Educational Sciences: Theory and Practice 12, no. 3 (2012): 2303-2309.

Li, Yeping. “What do students need to learn about division of fractions?.” Mathematics Teaching in the Middle School 13, no. 9 (2008): 546-552.

Lo, Jane-Jane, and Fenqjen Luo. “Prospective elementary teachers’ knowledge of fraction division.” Journal of Mathematics Teacher Education 15, no. 6 (2012): 481-500.

Lubinski, Cheryl A., Thomas Fox, and Rebecca Thomason. “Learning to Make Sense of Division of Fractions: One K–8 Preservice Teacher’s Perspective.” School Science and Mathematics 98, no. 5 (1998): 247-259.

## Proposal

Division is a challenging operation with struggles in conceptually understanding what happens when dividing. And when students divide fractions, challenges continue to exist. When teaching fraction division, the traditional algorithm (invert and multiply) is often used and enforced through repeated problems without any opportunity for students to make sense of the procedure. Area and linear models are sometimes used for fraction division; the set model is used less frequently. A set model is a group of items, in this case two-sided chips, that make up the whole with parts of the set representing a fraction. With the fraction circle area model, the whole is sometimes easier for students to see and is usually defined by the whole fraction circle (but this is also a limitation). The set model requires a little more thinking and analysis because the whole must be built so that it is possible to solve the problem; it is a very flexible model. The whole can be easily manipulated and changed based on the problem thereby addressing some of the limitations often faced using fraction circles and other area models. Presented activities offer an opportunity to fruitfully engage in the meaning of division with fractions. Two-sided chips are an adaptable tool that can support students as they visualize division through the use of sets. In this hands-on, interactive session, participants will explore the concept of dividing fractions using the set model, moving beyond the traditional algorithm and more traditional models. Activities are provided.