Post, T., Cramer, K., Harel, G., Kiernen, T., & Lesh, R. (1998) Research
on rational number, ratio and proportionality. Proceedings of the
Twentieth Annual Meeting of the North American Chapter of the
International Group for the Psychology of Mathematics Education, PME-NA
XX Volume I (pp. 89-93). Raleigh, North Carolina.
RESEARCH ON RATIONAL NUMBER, RATIO
AND PROPORTIONALITY
Working Group Organizer
Thomas Post, University of Minnesota, postxOO1@umn.edu
Working Group Panel
Kathleen Cramer
University of Wisconsin-River Falls
Guershon Harel
Purdue University
Thomas Kieren
University of Alberta
Richard Lesh
Purdue University
The purpose of this series of three two hour sessions will be to help frame
current and future research on rational number, ratio and proportionality
within the perspective of earlier important studies and the projects which
have conducted them. This will be accomplished with a series of
presentations providing overviews of several of these earlier studies along
with implications for further study and investigation. Ample time for
discussion (about half) will be provided. Persons interested will receive a
bibliography with suggested readings prior to the conference.
Major areas to be considered include: multiple interpretations of rational
number (subconstructs), children's partitioning schemes, observed
blockages to efficient learning translations within and between modes of
representation, concept of unit considerations, ratio-a measure space
interpretation perceptual distracters, evaluation--criteria for
understanding, and connections between rational number, ratio and
proportionality. Teachers knowledge in these areas will be addressed. A
survey of rational number, ratio and proportionality related research
reported in past PME proceedings will be discussed. Guidelines for the
construction of theory based instructional student/teacher materials will
be considered. Time will also be devoted to understanding innovative
research designs for research on rational number constructs. Importantly,
implications for our future work in these domains will also be explored
relying on participant discussion and contributions from the floor.
We have lived through a series of "content free" programs which have
captured the attention of teachers and school administrators nationwide.
As a result, teachers have not spent the needed time to learn about
actual mathematical content and students struggles to learn it. An
elementary school administrator recently (and proudly) had boasted that
every elementary teacher in her district (over 1000) had 30 hours of
Madeline Hunter. There is nothing wrong with Hunter's ideas, and the
intent here is not to demean those ideas but unless they are applied to
meaningful mathematical (or other) content there is no a-priori reason to
expect students achievement levels to prosper. That time would have
been better spent applying Hunter's ideas to children's learning of rational
number concepts or some other topical domain. Similar things have
happened with cooperative learning and radical constructivism. All sound
ideas, but useful only to the extent they are applied to high quality
(useful and theoretically sound) mathematical and pedagogical content. It
is this latter category of concern which has proven to be elusive to many
teachers and administrators in school situations who are attempting to
improve the school mathematical experience for children. Site based
management where individual school based teachers and administrators
make all curricular decisions does not help the situation either.
This session will attempt to reassert the importance of accumulating
knowledge about teachers and students learning of rational number,
ratio, and proportionality. We will do this by examining the work of
several projects, groups and individuals who have made important
contributions to the field. We will then use these "classical" studies as a
springboard to identify additional investigations to further our knowledge
and understandings in these areas.
The Rational Number Project (RNP) is a program of cooperative research
which has been funded continuously by the National Science Foundation
(NSF) since 1979. It is thought to be the longest lasting cooperative
research project in the history of mathematics education. To date, there
have been six separate multi-year NSF grants involving the Universities of
Minnesota, Wisconsin at River Falls, Northern Illinois, Louisiana State,
Northwestern, Massachusetts at Dartmouth and Purdue. The project
bibliography contains eighty or so entries including books, research
reports, book chapters and technical reports. There also have been a like
number of presentations at regional, national and international meetings.
The RNP bibliography will be shared with participants at one of our
sessions. The RNP is generally considered to have made important
contributions to our understanding of children's rational number thinking.
The first of the RNP grants was obtained in 1979 to examine the impact
of manipulative materials on children's understanding of rational number
concepts. Later grants have extended our study of fractions to the study
of proportionality in the middle grades. We are now working with middle
grade teachers in a teacher enhancement program to facilitate the
implementation of new NSF middle school curricula, including the RNP
fraction lessons.
This latest grant (1994-98) is primarily concerned with the development
and testing of a model for re-educating middle grades teachers. The
model used attempts to integrate teachers mathematical, pedagogical
and psychological content knowledge. (a la Shulman).
Behr, Cramer, Harel, Lesh and Post have been the mainstays of the RNP
over the years although others have worked with us for shorter periods of
time. (Bright, Khoury, Silver and Wachsmuth and many, many graduate
students through the years). In addition, RNP project personnel have
worked with many of our colleagues in mathematics and in psychology on
issues of mutual interest.
1) Tom Kieren has been interested in the domain of rational number
throughout much of his career at the University of Alberta. His work on
partitioning with Pothier have contributed important understandings into
the partitioning related behavior of young children. Tom is fond of saying
that mathematics is "about something'' "About something" is an
important idea which has been lost on many who have over the years
driven students to premature abstraction and to a preoccupation with
computational algorithms. Tom has provided an important conceptual
framework for the RNP with his paper "On the mathematical cognitive and
instructional foundations of rational numbers" (1976) in which he
suggests important components of rational number understanding and his
belief that a mathematically literate person in the rational number domain
has an integrated view as to how the various subconstructs part-whole,
ratio, decimal, indicated division, measure and operator interact and are
related to one another. He argues for a research program which
acknowledges the interrelated nature of these subconstructs. Tom will
comment on his current views of these subconstructs and related rational
number issues.
2) Vergnaud (1983, 1988) coined the term "Multiplicative Conceptual
Field" (MCF) to refer to a web of multiplicatively related concepts, such as
multiplication, division, fractions, ratio and proportions, linearity, and
multilinearity. Similar to Kieren's' subconstructs of rational number, a key
idea of the MCF is the observation that its content is not a mere collection
of isolated concepts but rather an interconnected and interdependent
complex structure. Its complexity is both mathematical and
developmental.
Until the mid 80s, research in mathematics education looked, to a large
extent, at the development of individual multiplicative concepts, without
explicit attempts to deal with the interrelations and interdependencies
within, between, and among these concepts. For example, research on
the rational number concept did not take into account children's
conceptions of multiplication and division, and vice versa. Similarly,
research on the learning of the decimal system was quiet separate from
research on fractions and proportionality. During the middle-to late 80s,
the RNP advanced the research on the MCF by exploring the
interconnectivity of the cognitive development of multiplicative concepts.
As an example, we mention one important result of this research:
In a study with inservice elementary school teachers, we found that
teachers use four solution strategies to multiplicative problems that do
not conform to Fishbein's intuitive models:
(a) The Multiplicative strategy, involving the concept of
proportionality.
(b) The Pre-multiplicative strategy, reflecting an early stage
toward proportionality.
(c) The Operation- search strategy, based on a trial-and-error
approach.
(d) The Keyword strategy.
We found that teachers who solved the problems correctly and
relationally reasoned in terms of ratio and proportion concepts, whereas
teachers who arrived at correct solutions without these concepts did so by
a trial-and error like methods. This suggests that multiplicative problems
that do not conform to Fishbein's models require for their solution a
scheme that includes the concepts of ratio and proportion. This finding
suggests that the formation of ratio and proportion concepts can be
powerful tools in dealing with multiplicative problems. The question of
whether these concepts are necessary tools for multiplicative problems
that do not conform to Fishbein's intuitive models is still open and needs
further research.
Within the enormous structure of the MCF, we have identified smaller
structures that, although interconnected, are, to some extent,
autonomous. An example of such a substructure consists of a three-stage
development of the concept of multiplication: from an early stage of
whole number multiplication, to an operation non-conservation stage, and
to an operation conservation stage. We have called this structure, a
Multiplicative Conceptual Subfield (MCS) because it represents a closed
unit within the greater structure of the MCF. Our semantic analyses of the
different MCS's-drawing heavily on the work of many other researchers,
particularly, the work by Kieren, Steffe, Thompson, and Kaput-revealed
critical deficiencies in the mathematics curricula of elementary and middle
schools. The areas where the teaching of [MCS's] is deficient include
composition, decomposition, and conversion of units, operation on
numbers from the perspective of mathematics of quantity, and
mathematical variability.
Guershon Harel will discuss the implications of these issues for school
rational number curricula and for further research in the area.
3) Innovative Research Designs Needed for Research on Rational
Numbers Constructs
New research designs have been developed are based on new
assumptions about the nature of students' knowledge, problem- solving,
learning and teaching, they frequently involve lines of reasoning that are
fundamentally different from those that applied to industrial-era factory
metaphors for teaching and learning. Therefore, new standards of quality
often are needed to assess the significance, credibility, and range of
usefulness of the results that are produced by such studies. But, in
general, the development of widely recognized standards for research has
not kept pace with the development of new problems, new theoretical
perspectives, and new approaches to the collection, analysis, and
interpretation of data.
High-quality studies may be rejected because they involve unfamiliar
research designs, and because inadequate space is available for
explanation, or because inappropriate or obsolete standards of
assessment are used (similar to a Type I error).
Low quality studies may be accepted in which innovative research designs
are done poorly, or in which traditional research designs even though
they are based on obsolete assumptions about the nature of teaching,
learning and problem-solving (or about the nature of program
development, dissemination, and implementation) (similar to a Type Il
error).
In research on rational numbers and proportional reasoning, as in other
areas of mathematics education research, some of the most important
products of research has involved the development of new tools and
research methodologies. Yet, these products seldom are reported in
research journals. In a series of projects known collectively as the
Rational Number Project, some of the most useful research designs that
we've developed involve the integrated use of qualitative and quantitative
methods. Also, because we often are interested in going go beyond
investigating typical development in natural environments to also focus
on induced development within carefully controlled and mathematically
enriched environments, we have had to develop new approaches to
research that involve:
1) Action research: in which teachers participate as coresearchers.
2) Multi-tiered teaching experiments: investigating the
interacting development of students and teachers often over
time periods involving several months or years.
3) Carefully structured clinical interviews: in which it is
important to minimize uninteresting interventions by the
researcher.
4) Iterative videotape analyses: in which it is important to
take into account interpretations involving a variety of
theoretical and practical perspectives.
5) Ethnographic observations: in which it is important to
avoid needlessly distorting the perspectives of the people
being observed.
6) New approaches to assessment: that focus on deeper and
higher order understandings, and which go beyond simplistic
assumptions that underlie most standardized testing
programs.
Dick Lesh will lead a session which will deal with each of the preceding
approaches to research. Examples will be taken from relevant research on
rational numbers and proportional reasoning, and resources will include
selections from a new book, edited by Kelly & Lesh (in press), on
Research Design in Mathematics and Science Education.
References
To be distributed at or prior to the PME-NA Meeting.