## The Rational Number Project

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

 The Rational Number Project has been funded by the National Science Foundation since 1979. (will host and maintain this web site)

These people are the Principal Investigators in
The Rational Number Project

 Merlyn J. Behr Northern Illinois University deceased 1995 Kathleen Cramer University of Minnesota crame013@umn.edu

 Guershon Harel University of California, San Diego harel@math.ucsd.edu Richard Lesh Indiana University Contact Richard Lesh Thomas Post University of Minnesota postx001@umn.edu

The publications of the Rational Number Project
are available in two formats.
Click the links below to go to the publications.
full-text versions of each citation.

 Publications in chronological order. Publications alphabetized by author.

January 2010

The Rational Number Project (RNP) advocates teaching fractions using a model that emphasizes multiple representations and connections among different representations.  This translation model (Cramer, 2003) is based on theoretical work of Jerome Bruner and Zoltan Dienes. The RNP has documented student success using this translation model to teach initial fraction ideas (Cramer, Post, & delMas, 2002; Cramer & Henry, 2002).  Important outcomes from the early RNP work include a deeper understanding of children’s thinking as they develop initial fraction ideas and a curriculum module (RNP: Fraction Lessons for the Middle Grades currently renamed: RNP: Initial Fraction Ideas) for teaching students fractions using a multiple representation approach (Cramer, Behr, Post & Lesh, 1997).  With 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 among different representations.

Important products from this latest work include (a) a companion module of 28 lessons related to fraction operations, decimal concepts, order, equivalence and operations with decimals (addition/subtraction); (b) deeper understanding of students’ understandings and misunderstandings as they develop meaning and procedural skill with fraction operations and initial decimal ideas; (c) burgeoning understanding of the support teachers in urban settings need to teach RNP curricula effectively; (d) an online professional workshop to prepare teachers to use this new module.

The revised RNP 1 curriculum and our newest one are available on this website  at the following locations:
http://www.cehd.umn.edu/ci/rationalnumberproject/rnp1-09.html

We are pleased to have individual teachers download these lessons for their classroom use.  If districts are interested in using these materials we do ask that they contact Kathleen Cramer for more information and assistance on using these materials with large numbers of students.

As PI for the latest NSF grant I would like to acknowledge the important work done by my colleagues and graduate students at the University of Minnesota.  The new RNP team members working on this latest grant include, Dr. Terry Wyberg (wyber001@umn.edu), co-PI on our latest grant; Seth Leavitt (seth.leavitt@mpls.k12.mn.us), middle school mathematics teacher; Debbie Monson and Stephanie Whitney, doctoral students in mathematics education.

Kathleen Cramer
crame013@umn.edu

October, 2006

The Rational Number Project has received funding from the National Science Foundation to create a companion module to the current RNP Fraction Lessons for the Middle Grades.  This module will extend our previous work to include:

• fraction operations

• meaning for decimals and percents

• operations with decimals and percents

The module development will take place within two teaching experiments conducted this year.  A larger study around the final version of this module is planned for the 2007-08 academic year.  As we develop this new module, we will better understand why middle-school students have difficulty working with fractions, decimals and percents in a meaningful way.

To support teacher’s use of this module, we will create an online workshop for teachers. The online workshop will be based on our insights into students’ thinking.  It will be modeled after a workshop we created for Twin Cities Public Television as part of the US Department of Education's Ready to Teach program. For more information about that project, please view the Twin Cities Public Television web site.

Kathleen Cramer
crame013@umn.edu

October 2002

Dear Colleagues in Mathematics Education:

We are pleased to share with you this web site containing each of the Rational Number Project's 86 research publications and think pieces. This site will be updated regularly as new RNP related publications appear. A small number of the articles do not appear in their entirety due to copyright constraints and our inability to get the publishers permission to include them in our website.

As far as we know the RNP is the longest lasting federally funded cooperative multi-university research project in the history of mathematics education. With the exception of one year (1983), it has been continuously funded by NSF since 1979. The project ended officially in August 2002. As a parting gesture we felt it appropriate to accumulate each of the publications in their original format and put them in one place for use by present and future researchers and practitioners interested in middle grades mathematics - its teaching, its learning, and its assessment.

We have been fortunate to have had the ongoing financial support of NSF. Specifically, our various project managers have been very helpful and supportive in a wide variety of ways. Notable among our early project managers at NSF were Tom Cooney, Ray Hanapel and Dick Shumway and most recently Diane Spresser who has overseen our efforts for the past decade or so. We shall always be grateful to each of them for their ongoing support.

We believe that the continuous funding of RNP was a productive and useful model, one that has resulted in a substantial body of related literature representing a variety of perspectives yet having a cohesiveness and more or less singular purpose often not found in the literature. Aside from the PI's listed below we have through the years involved, on a co-author basis, a variety of researchers from our field. A perusal through the list of publications will indicate the depth of their contributions. We are indebted to each and every one of these folks as they have played a significant part in the overall success of the project.

Our efforts through the years have been cumulative. Predictably new questions arose from earlier investigations. To be able to pursue those additional questions over an extended period of time was a luxury which not many have had. We think that the body of literature produced by the RNP is an example of the level of impact which might occur in other areas with such lengthly and continuous external support.

We hope that you will find these materials both interesting and useful in your future work with children and teachers in the Rational Number Domain.

Sincerely,

Merlyn J. Behr - Northern Illinois University (Deceased 1995)
Kathleen A. Cramer - University of Minnesota crame013@umn.edu
Guershon Harel - Univ. of California @ San Diego harel@math.ucsd.edu
Richard Lesh - Purdue University rlesh@purdue.edu
Thomas R. Post - University of Minnesota postx001@umn.edu

The Rational Number Project: Achievements and Funding History

The Rational Number Project (RNP) was the longest lasting cooperative multi-university research project in the history of mathematics education. NSF has, with the exception of 1983-84, funded it continuously since 1979.

The RNP’s most significant accomplishment is the collection of over 90 papers, book chapters, several books and other project publications. The vast majority of these are concerned with the learning and teaching of rational number concepts including fraction, decimal, ratio, indicated division, measure and operator. These studies led naturally to investigations of proportionality with specific attention to the components of proportional reasoning. We have examined the contributions of multiplication and division understandings to these earlier mentioned concepts and then proceeded to concern ourselves with the design of effective professional development programs for teachers and concurrently with appropriate assessment practices in our field. As project interests evolved, various project Co-PI’s cooperated in preparing material for publication. We have been quite successful in this regard and view these publications as a significant body of literature and contribution to what is known about these issues in our field.

These efforts also culminated in the development of three mathematics courses designed specifically for elementary teachers who traditionally do not have significant mathematical backgrounds, but who nevertheless are teaching mathematics to children each and every day. We have also produced two curriculum texts for teachers that reflect our suggestions as to how rational number concepts should be taught to children.

For the past several years we have been assembling a RNP web site that will contain each of these publications in a single location. The publications will appear as they did when they were originally published.

Over the years the RNP has had a series of project associates and Co-PI’s in addition to the three initial founders of the project – Merlyn Behr, Richard Lesh and Thomas Post.

In Alphabetical order here are their names and present institutional affiliations:

Merlyn Behr. Professor, Department of Mathematics @ Northern Illinois University
(Deceased 1995)
(Years with Project 1979 – 95)

Kathleen Cramer, Associate Professor Mathematics Education, University of Minnesota.
(Formally Professor Mathematics Education @ University of Wisconsin - River Falls.)
(Years with Project 1987-2002)

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

Richard Lesh, Professor, Mathematics Education and Associate Dean for Research, College of Education @ Purdue.
(Years with Project 1979- 2001)

Thomas Post, Professor, Mathematics Education - @ University of Minnesota.
(Years with Project 1979- 2002)

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)

In addition of course, there were numerous graduate students at each of our universities, many of which completed the PhD with RNP related theses.

The dates, titles and amounts of the RNP grants in reverse order were as follows:

 2006-2009 -- Instructional Module for Fractions, Decimals, and Percents (ESI-0628005) (Cramer, Wyberg). \$554,418 1995-98 (02) -- Middle Grades Teacher Enhancement: Model Program Development and Evaluation (NSF #9408915) (Behr, Cramer, Harel, Lesh & Post)). Note: There have been four one-year no-cost extensions to this latest, and last, RNP grant. \$1,904,888 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 1987-89 - Developing Leadership—Middle School Mathematics Teacher Education Project -The Rational Number Project ( Behr, Lesh & Post) NSF/TEI-8652341 \$341,823 1984-87 - The Role of Rational Number Concepts in the Development of Proportional Reasoning Skills ( Behr, Lesh & Post). (NSF DPE-8470077) \$718,325 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 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) \$200,000

This website contains a significant body of literature, all of which has been generated by the faculty associates of the Rational Number Project during the past twenty plus years of project activity. We hope that it will prove to be useful to current and future researchers who are or will become involved with research in the areas of rational number learning and understandings, and/or multiplicative structures including proportionality and proportional reasoning.

We have come to appreciate our early and continued reliance on the related work of others and the vastness, complexity and importance of these domains. We also realize just how much there is yet to be learned and to be understood. In the same vein, we hope that researchers will use our findings as a springboard to launch future investigations that will further our present understandings rather than to retread areas, which at present are fairly well understood.

For all of my colleagues in the RNP - we hope you will find something of interest and importance to your work in this collection,

Tom Post

October 2002

Note: The College of Education and Human Development at the University of Minnesota has graciously consented to maintain and update this web site in the years to come. All of us are indebted to the College and to the University for this important contribution.