## The Rational Number Project

The Rational Number Project is an
ongoing research project investigating
student learning and teacher enhancement.

 The Rational Number Project was funded by the National Science Foundation. (will host and maintain this web site)

The publications of the Rational Number Project
are available in chronological or alphabetical order.

### Rational Number Project (June, 2018)

The Rational Number Project (RNP) is a research and curriculum development project involving multiple groups of faculty and graduate students from different universities interested in the teaching and learning of rational numbers. This website contains over 100 publications, the earliest was published in 1979; our most recent research article was published in December, 2017.  The strength of the RNP is its continued focus for over 30 years on teaching and learning of an important content domain by a wide group of educators from multiple universities. We acknowledge the long-term support from the National Science Foundation.

The rational number domain is a significant mathematical structure that spans upper elementary, middle grades and high school mathematics.  It is the first time that students engage in multiplicative reasoning, a paramount cognitive structure important in all of mathematics.  This domain is central to student development of formal operational thought, an overarching psychological concept elaborated upon by Jean Piaget over five decades ago.

The vast majority of our publications are concerned with the teaching and learning of rational number ideas including fractions, decimal, ratio and proportion. Much of our effort involved working closely with teachers and students in classrooms over long periods of time.

We studied these evolving understandings while simultaneously investigating, within a research framework, the effectiveness of a curriculum developed by project associates to teach important ideas within the rational number domain.   Our teaching experiments provided detailed analyses of students’ thinking as they were immersed in these instructional environments.  These environments incorporated the use of multiple representations and translations within and between these representations. The understandings and misunderstandings we documented added to the knowledge base on students’ thinking of rational number and reflect new insights as to what students are capable of doing, and, the misunderstandings that occur during the learning process.

An important outcome from the early RNP work includes this deeper understanding of children’s thinking as they develop initial fraction ideas using a project developed curriculum module (RNP: Fraction Lessons for the Middle Grades currently renamed: RNP: Initial Fraction Ideas).  This curriculum focused on a multiple representation approach and extensive student participation in the learning process (Cramer, K., Behr, M., Post T., Lesh, R., (2009) Rational Number Project: Initial Fraction Ideas).

As the project evolved from its first grant in 1979, various new project investigators became involve in RNP research related activities including preparation of results for publication.

#### History and Founding Members

The original founding members of the RNP are Dr. Merlyn Behr,
Dr. Thomas Post and Dr. Richard Lesh.

 Merlyn Behr Northern Illinois University Years with the project 1979-1985 Thomas Post University of Minnesota Years with the project 1979-2002 Richard Lesh Purdue Years with the project 1979-2001

In addition, there was a series of project associates in the years between 1979 and 1998.

Guershon Harel, Professor, Department of Mathematics @ University of California - San Diego. (Years with Project 1990-98)

Edward Silver, Professor, Mathematics Education – College of Education @ University of Michigan (Years with Project 1980-81)

Ipke Wachsmuth, Professor, Computer Science @ Universite of Osnabruck, Germany
(Years with Project 1979-83)

The dates, titles and amounts of the grants between 1979 and 2002 are listed below.

#### NSF Funding History 1979-2002

1979-81 - The Role of Manipulative Materials in the Learning of Rational Number Concepts: The Rational Number Project. (Behr, Lesh, Post & E. Silver —RISE (NSF SED 79-20591; \$200,00)

1981-83 - Manipulative Aids, Construct Analysis and Learning of Rational Number Concepts: The Rational Number Project - (Behr, Lesh, Post)— RISE (NSF SED 81-12643; \$150,000)

1984-87 - The Role of Rational Number Concepts in the Development of Proportional Reasoning Skills (Behr, Lesh & Post). (NSF DPE-8470077; \$718,325)

1987-89 - Developing Leadership—Middle School Mathematics Teacher Education Project -The Rational Number Project (Behr, Lesh & Post) (NSF/TEI-8652341; \$341,823)

1990-95 -- The Rational Number Project: Research on the Teaching and Learning of Multiplicative Structures. (Behr, Cramer, Harel, Lesh, & Post). (MDR – 8955346 & MDR 9144969; \$819,843)

1995-98 (02) -- Middle Grades Teacher Enhancement: Model Program Development and Evaluation (Behr, Cramer, Harel, Lesh & Post). (NSF #9408915; \$1,904,888) Note:  There have been four one-year no-cost extensions to this RNP grant.

#### RNP: Current Activities and Personnel 2006-2018

Kathleen Cramer, University of Minnesota, has led the RNP activities since 2006. She became involved with the RNP as a graduate student and continued her research under the umbrella of the RNP as a faculty member while at the University of Wisconsin River Falls and most currently at the University of Minnesota.

Several additional colleagues have worked on the RNP since 2006 when we received the final RNP grant: Instructional Module for Fractions, Decimals and Percents 2006-2009(ESI-0628005; \$554,418).  While the grant funding ended, our work continues.

Kathleen Cramer
Associate Professor in Mathematics Education - retired
University of Minnesota
crame013@umn.edu

 Debra Monson Associate Professor Teacher Education University of St. Thomas Minneapolis, MN debbie.monson@stthomas.edu Terry Wyberg Senior Lecturer in Mathematics Education University of Minnesota wyber001@umn.edu Sue Ahrendt Associate Professor Metropolitan StateUniversity of Denver, CO sahrendt@msudenver.edu Seth Leavitt Mathematics Teacher Minneapolis Public Schools Seth.Leavitt@mpls.k12.mn.us

In addition to those listed above who are the currently the most active members of the RNP, several past and current graduate students have also contributed to our work.  Some are presently on faculty at different Universities.

Stephanie Whitney, DePaul University, Illinois
swhitne4@depaul.edu and whitney.stephanie@gmail.com

Karen Colum- Associate Professor Teaching and Learning: Elementary and Literacy Education. Minnesota State University-Mankato.
Karen.colum@mnsu.edu

Christina Miller – Executive Director Math Recovery, Apple Valley, MN
Christina@mathrecovery.org

Bethann Wiley, Assistant Professor Elementary Mathematics, Winona State University, Winona, MN
bwiley@winona.edu

Christy Pettis, University of Wisconsin at River Falls
christy.pettis@uwrf.edu

Chelsey Fagerlund, Ph.D. candidate in Elementary Education, University of Minnesota
cookx809@umn.edu

#### Focus of Our Current Work

The RNP advocates teaching fractions using multiple representations and connections among different representations.  This translation model (Lesh, 1979; Cramer, 2003) model has been key to our research and curriculum development from its inception.  The RNP has documented widespread student success using this model for both teachers as they organize instruction in their classrooms, and for students as they grapple with these new, challenging, and important mathematical ideas (Cramer, Post, & delMas, 2002Cramer & Henry, 2002). Important teacher related outcomes from the early RNP work include a much deeper and widespread understanding of children’s thinking as they develop initial fraction ideas, and, a well defined curriculum module (RNP: Fraction Lessons for the Middle Grades currently renamed: RNP: Initial Fraction Ideas) for student instruction  (Cramer, Behr, Post & Lesh, 1997).  With its most recent NSF support (ESI-0628005) the RNP staff has created a companion module (RNP: Fraction Operations and Initial Decimal Ideas) to extend the content in the original RNP lessons.  This new module provides instruction for fraction operations and initial decimal ideas that utilizes multiple representations and translations within (multiple embodiments) and between different representations of rational number ideas.

In partnership with Minneapolis Public Schools, we created a third RNP module for use with students in third grade.  In addition, we continue our work on the teaching using multiple representations with a new project studying the effectiveness of the number line as a model for fractions (Cramer, Ahrendt, Monson, Wyberg, & Colum, K., 2017; Cramer, Ahrendt, Monson, Wyberg, & Miller 2016; Cramer, Monson, Ahrendt, Wyberg, Pettis, Fagerlund, 2018).

Our latest teaching experiments have led to a fourth RNP curriculum module. This module, Navigating Fractions on the Number line, is currently available on this website, along with the revised RNP 1 curriculum, its companion module, Fraction Operations and Initial Decimal Ideas, and the new third grade module. They are available on this website at the following locations:
Rational Number Project - Initial Fraction Ideas
http://www.cehd.umn.edu/ci/rationalnumberproject/rnp1-09.html
Rational Number Project - Fraction Operations and Initial Decimal Ideas
http://www.cehd.umn.edu/ci/rationalnumberproject/rnp2.html
Rational Number Project - Initial Fraction Ideas
Abridged Edition for use with Third Graders
http://www.cehd.umn.edu/ci/rationalnumberproject/rnp1-13-grade3.html
Rational Number Project - Navigating Fractions on the Number Line
http://www.cehd.umn.edu/ci/rationalnumberproject/numberline_module.html

We are pleased to have individual teachers or districts download these lessons for classroom use.  You are encouraged to contact any current RNP members with questions about these curricula.

#### Closing Comments

We encourage teachers, faculty, graduate students and interested others to investigate the RNP website. The site was designed to be useful in tracing the evolution of our interests and perspectives.  We have published in prominent research journals (i.e. Journal for Research in Mathematics Education (JRME) and also extensively in journals for classroom teachers such as Teaching Children Mathematics and Mathematics Teaching in the Middle School as well as in several NCTM Yearbooks.   We have been regular contributors to national, international and state professional conferences.  Due to copyright regulations, in some cases we were not able to gain permission to include the complete journal articles or book chapters.  In that event, we have included an abstract and reference list for those publications.

The RNP has been successful in part because of its long-term focus on a single conceptual domain and enabled by continued by long term external funding.  Our most recent publications contribute to a better understanding of students’ thinking in the area of rational number and the teaching strategies found to be effective in building both conceptual and procedural knowledge among students in grades 3-8.   We have also been successful because the project has been a collaborative one.  We have been fortunate to have had many productive groups of faculty and graduate students working together at different stages of this project.  We have also been fortunate to partner with so many teachers who opened their classrooms to us so together we can learn better ways to build meaning for such an important topical area.

#### Dedication

We dedicate this website to Dr. Meryln Behr, one of the original founders of the RNP website. He was the heart of the RNP in its early years and has been missed greatly since his passing in 1995.

In addition, we dedicate this website to those teachers who welcomed the RNP faculty and graduate students into their classrooms so we could do this important work together.

Kathleen Cramer
June 24, 2018

This page copyright 2002 by the Regents of the University of Minnesota. The University of Minnesota is an equal opportunity educator and employer. This page subject to change without notice. Last modified: 10/7/18. For questions or comments, contact Seth Leavitt.

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