These people are
the Principal Investigators in
The Rational Number Project
publications of the Rational Number Project
are available in two formats.
Click the links below to go to the publications.
Both links contain hyperlinks to
versions of each citation.
The Rational Number Project (RNP)
advocates teaching fractions using a model that emphasizes
multiple representations and connections among different
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,
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
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:
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 (firstname.lastname@example.org), co-PI on our
latest grant; Seth Leavitt (email@example.com),
middle school mathematics teacher; Debbie Monson and
Stephanie Whitney, doctoral students in mathematics
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:
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
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.
Please feel free to write with any questions about this
Colleagues in Mathematics Education:
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
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.
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.
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.
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
hope that you will find these materials both interesting
and useful in your future work with children and teachers
in the Rational Number Domain.
J. Behr - Northern Illinois University (Deceased 1995)
Kathleen A. Cramer - University of Minnesota firstname.lastname@example.org
Harel - Univ. of California @ San Diego email@example.com
Lesh - Purdue University firstname.lastname@example.org
R. Post - University of Minnesota email@example.com
Rational Number Project: Achievements and Funding
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
RNPs 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-PIs 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.
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
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.
the years the RNP has had a series of project associates
and Co-PIs in addition to the three initial
founders of the project Merlyn Behr, Richard
Lesh and Thomas Post.
Alphabetical order here are their names and present
Behr. Professor, Department of Mathematics @ Northern
(Years with Project 1979 95)
Cramer, Associate Professor Mathematics Education,
University of Minnesota.
(Formally Professor Mathematics Education @ University
of Wisconsin - River Falls.)
with Project 1987-2002)
Harel, Professor, Department of Mathematics @
University of California - San Diego.
(Years with Project 1990-98)
Lesh, Professor, Mathematics Education and Associate
Dean for Research, College of Education @ Purdue.
(Years with Project 1979- 2001)
Post, Professor, Mathematics Education - @ University
(Years with Project 1979- 2002)
Silver, Professor, Mathematics Education
College of Education @ University of Michigan
(Years with Project 1980-81)
Wachsmuth, Professor, Computer Science @ Universite
of Osnabruck, Germany
(Years with Project 1979-83)
addition of course, there were numerous graduate
students at each of our universities, many of which
completed the PhD with RNP related theses.
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,
(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.
-- The Rational Number Project: Research on
the Teaching and Learning of Multiplicative
Structures. (Behr, Cramer, Harel, Lesh, &
Post). MDR 8955346 & MDR 9144969
- Developing LeadershipMiddle School Mathematics
Teacher Education Project -The Rational Number
Project ( Behr, Lesh & Post) NSF/TEI-8652341
- The Role of Rational Number Concepts in the
Development of Proportional Reasoning Skills
( Behr, Lesh & Post). (NSF DPE-8470077)
- Manipulative Aids, Construct Analysis and
Learning of Rational Number Concepts: The Rational
Number Project - (Behr, Lesh, Post) RISE
(NSF SED 81-12643)
- 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)
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.
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.
all of my colleagues in the RNP - we hope you will
find something of interest and importance to your
work in this collection,
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.
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: 7/31/12.
For questions or comments, contact Seth