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Using Models to Understand Fractions
(Elementary)

Overview

This workshop provides an opportunity for teachers of grades 3 through 5 to explore how mathematical models can be used to promote a deep understanding of fractions and fractional relationships. All too often, students are taught mechanical ways to make sense out of fractions, whether in the form of an addition algorithm or a procedure that produces equivalent fractions. This course offers alternative ideas for fraction instruction and is premised on the belief that students can come to understand fractions in a number of different ways. Completion of a student interview and a final project are integral parts of this course. A variety of readings, applets, and videos form the content of this course, and participants are expected to share ideas with their online colleagues in the discussion forums.

Goals and Products

This workshop will enable participants to:

• Develop a broader understanding of what fractions are and the mathematics underlying the algorithms
• Identify ways that mathematical models can be incorporated into classroom instruction, for the purpose of increasing student understanding about fractional relationships and fraction operations
• Deepen understanding of student difficulties with fractions through examining student work and conducting a student interview
• Explore specific mathematical models, specifically linear and area models, which can be applied to their teaching practice to foster better understanding of fractions
• Engage in a “learning-by-doing” methodology, which is applicable to students at all ages

In the Final Project, you will plan a fractions lesson that brings together a number of different ideas from this course. As a final course project, participants will design a lesson that teaches students about a fractional concept or operation that uses either an area model or linear measurement model and that targets at least one student misconception about fractions. The lesson will teach students about a fractional concept or operation, will utilize one of the fraction models introduced in this course, and will target specific student misconceptions about fractions. In order to learn what misconceptions students have about fractions, you will be required to conduct a student interview. This interview forms a significant part of your Final Project. (More information about the student interview is provided in Session 3.)
Opportunities for you to work on your Final Project, including the student interview, have been included throughout this course. Your completed Final Project will be due by the end of Session 6.

Suggested Timeline for Final Project:
Session 1: Review requirements for Final Project.
Session 3: Review requirements for student interview, and identify a student that you will interview.
Session 4: Begin planning and writing your lesson plan; conduct your student interview.
Session 5: Complete the Student Interview Write-Up, and incorporate lessons learned from the interview into your lesson plan.
Session 6: Finish writing your lesson plan, and submit the finished Final Project

Format and Requirements

This course is divided into six one-week sessions which each include readings, an activity and an online discussion among course participants. The time for completing each session is estimated to be two to four hours.

The outline for the course is as follows:

This course begins with an exploration of different ways to think about fraction problems, and also a review of how fractions are taught in elementary school. In Sessions 2 and 3, participants will depart from using traditional algorithms and investigate how linear measurement models can be used to promote a deeper understanding of fractional relationships. The use of area models, specifically pattern blocks, is examined in Sessions 4 and 5. Throughout this course participants will have many opportunities to apply linear and area models to actual fraction problems, use virtual manipulatives, look at student work that sheds light on how young students think about fractions, and discuss their experiences and ideas with their colleagues. Participants are expected to complete a Final Project and a Student Interview by the end of Session 6.

Prerequisites

This is an introductory course for teachers, technology specialists, curriculum specialists, professional development specialists, or other school personnel. Participants are expected to have regular access to computers. In addition, participants should be proficient with using email, browsing the Internet, and navigating to computer files.

Content and Technology Standards

This workshop will help teachers to enable their students to meet the following Content Standards as identified by the National Council for Teachers of Mathematics (NCTM) (http://standards.nctm.org/document/chapter5/numb.htm)

• develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers
• use models, benchmarks, and equivalent forms to judge the size of fractions
• recognize and generate equivalent forms of commonly used fractions, decimals, and percents
• develop and use strategies to estimate the results of whole-number computations and to judge the reasonableness of such results
• develop and use strategies to estimate computations involving fractions and decimals in situations relevant to students' experience
• use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals

In addition, this workshop, Using Models to Understand Fractions, will help participants meet the ISTE Educational Technology Standards and Performance Indicators for All Teachers

I. Technology Operations and Concepts
Teachers demonstrate a sound understanding of technology operations and concepts. Teachers:
B. demonstrate continual growth in technology knowledge and skills to stay abreast of current and
emerging technologies.

II. Planning and Designing Learning Environments and Experiences
Teachers plan and design effective learning environments and experiences supported by technology.
Teachers:
B. apply current research on teaching and learning with technology when planning learning
environments and experiences.
C. identify and locate technology resources and evaluate them for accuracy and suitability.

V. Productivity and Professional Practice
Teachers use technology to enhance their productivity and professional practice. Teachers:
A. use technology resources to engage in ongoing professional development and lifelong learning.
B. continually evaluate and reflect on professional practice to make informed decisions regarding the use of technology in support of student learning.

Reproduced with permission from Education Development Center, Inc.,
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