BACKGROUND DOCUMENT—WORKING DRAFT
COLLOQUIUM #2: WHY DO RATIONAL NUMBERS DRIVE STUDENTS CRAZY? 11-28-06
This document is divided into two sections: The first section provides evidence from two sources
concerning the low levels of US students’ understanding of rational number concepts. The second section
provides briefly annotated references to influential analyses of, and some proposed remedies to, fractionrelated learning challenges, with brief annotations. Colloquium participants are invited to contribute
additional information and references to this document. Please send suggestions to
[email protected].
I. How Widespread Is the Problem?
COLLEGE MATH STUDENTS
Susan J. Lamon, Mathematics Professor, Marquette University
. . .Over the last ten years, 90 percent of the eighteen- and nineteen-year-old students in my mathematics
classes [at Marquette University] have been unable to answer 50 percent of the questions (below) . . .
[which illustrate the broad base of meaning that is associated with fraction symbols.] We can only
conclude that they have not yet had enough experience to understand rational numbers. It is difficult to
imagine that they will gain that experience by taking courses that assume a knowledge of rational
numbers. Ongoing research in university-level calculus classrooms (Pustejovsky 1999; Lamon
forthcoming) suggests that students who begin university mathematics with only a part/whole
interpretation of a/b may have missed their window of opportunity. Although poor algebra skills are most
often blamed for the lack of success in calculus, we are finding that having little or no understanding of
rational numbers accounts for most students' conceptual difficulties when trying to understand the
derivative.
1. Does the shaded area below show a) 1 (3/8 pie)? b) 3 (1/8 pies)? c) 1 1/2 (1/4 pies)? Does it matter?
2. You have 16 candies. You divide them into 4 groups, select one group, and make it three times its size. What
single operation would have accomplished the same result?
3. You have taken only one drink of juice, represented by the unshaded area in the figure below. How much of your
day's supply, consisting of two bottles of juice, do you have left?
4. If it takes 9 people 1 1/2 hours to do a job, how long will it take 6 people to do it?
5. Without using common denominators, name three fractions between 7/9 and 7/8.
6. Yesterday Alicia jogged 2 laps around the track in 5 minutes, and today she jogged 3 laps around the track in 8
minutes. On her faster day, assuming that she could maintain her pace, how long would it have taken her to do 5
laps?
7. Here are the dimensions of some photos: a) 9 cm x 10 cm, b) 10 cm x 12 cm, c) 6 cm x 8 cm, d) 5 cm x 6.5 cm,
and e) 8 cm x 9.6 cm. Which one of them might be an enlargement of which other one?
SOURCE: Lamon, Susan J. (2001). Presenting and representing: From fractions to rational numbers. In A. Cuoco
(Ed.), The Roles of Representation in School Mathematics (2001 Yearbook). Reston, VA: National Council of
Teachers of Mathematics.
8TH AND 12TH GRADERS IN A NATIONAL SAMPLE
1996 National Assessment of Educational Progress (NAEP) (based on responses of 2,000 schools)
Less than one-half of the 12th graders [in the NAEP sample] gave even partially acceptable answers to
Problem 1 (below); about one-fifth of 8th and one-fourth of 12th graders gave partially correct responses
to Problem 2 (below). NAEP students’ performance was consistent with findings from a long line of
studies that examined students’ abilities to reason about ratios and relative amounts (Harel, Behr, Lesh, &
Post, 1994; Harel, Behr, Post, & Lesh, 1992; Hart, 1978; Karplus, Pulos, & Stage, 1979, 1983; Noelting,
1980a, 1980b; Tourniaire & Pulos, 1985). U.S. students have done poorly on such items in comparison to
students in other countries (Dossey, Peak, & Nelson, 1997; McKnight et al., 1987).
1. Luis mixed 6 ounces of cherry syrup with 53 ounces of water to make a cherry-flavored drink. Martin mixed 5
ounces of the same cherry syrup with 42 ounces of water. Who made the drink with the stronger cherry flavor? Give
mathematical evidence to justify your answer. (NAEP M070401)
2. In 1980 the populations of Towns A and B were 5000 and 6000, respectively. In 1990 the populations of Towns
A and B were 8000 and 9000, respectively.
•
Brian claims that from 1980 to 1990 the two towns’ populations grew by the same amount. Use mathematics to
explain how Brian might have justified his answer.
•
Darlene claims that from 1980 to 1990 the population of Town A had grown more. Use mathematics to explain
how Darlene might have justified her answer. (NAEP M069601)
SOURCE: NAEP as described in Thompson, P.W. & Saldanha, L.A. (2003). Fractions and multiplicative
reasoning. Draft of a paper later published in Kilpatrick, J., Martin, G., and Schifter, D. (Eds.), Research
companion to the Principles and Standards for School Mathematics. http://patthompson.net/Publications.html
II. Selected & Annotated Bibliography
Following are references to influential analyses of students’ learning difficulties with rational number.
Annotations point to some important ideas contained in the research but do not constitute a full summary.
Behr, M.J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. Grouws, D.A.
(Ed.), Handbook on Teaching and Learning Mathematics. Reston, VA: National Council of Teachers of
Mathematics.
The traditional school curriculum is deficient in the study of composition, decomposition, and
conversion of units. The unit concept can serve as a conceptual “bridge” across the varied
mathematical meanings of the fraction symbol: part-whole, quotient, ratio, measure, operator.
Kieren, T.E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. In
Carpenter, T.P., Fennema, E., & Romberg, T.A. (Eds.), Rational numbers: An integration of research.
Hillsdale, NJ: Lawrence Erlbaum Associates.
Kieren’s seminal research in the 70s and 80s laid the foundations for many later studies of the
teaching and learning of rational number and multiplicative reasoning. A central feature of
Kieren’s work, represented in the article cited here, is his identification of three foundations (or
elements) of students’ rational-number knowledge toward which instruction may be directed:
•
•
•
Mathematical: Formally, rational numbers are simultaneously quotients and ratios, extensive
and intensive quantities.
Cognitive: Rational numbers can be construed as quotients, ratios, measures, and operators.
These “subconstructs” are richly intertwined forms of “applicational” knowledge.
Personal (as described by models for knowing and understanding): Learners interweave
intuitive, contextual, and formal understandings in building new knowledge out of, and while
encompassing, prior knowledge.
Lamon, S. J. (2001). Presenting and representing: From fractions to rational numbers. In A. Cuoco (Ed.),
The Roles of Representation in School Mathematics (2001 Yearbook). Reston, VA: National Council of
Teachers of Mathematics.
Lamon tackles the question: Which “subconstruct”—part-whole, quotient, ratio, measure, or
operator—is the most effective primary interpretation for students to use in building a
comprehensive understanding of rational number topics? To develop an answer, Lamon worked
with teachers of five classes of students who progressed from 3rd through 6th grade over the fouryear research period. Each class studied all fraction-related concepts in their curriculum through
the ‘lens’ of a different subconstruct. The two most effective subconstructs proved to be a) partwhole with unitizing, and b) measure—as gauged by student achievement in the areas of
proportional reasoning, computation, understanding of multiple rational number interpretations.
Both of these approaches focused student attention on the inverse relation between the size of the
measurement unit and the number of units that measure a given quantity; and on the successive
partitioning of a unit into finer and finer subunits until one can name the amount in a given
quantity.
Mack, Nancy K. (1993). Learning rational numbers with understanding: The case of informal knowledge.
In Carpenter, T.P., Fennema, E., & Romberg, T.A. (Eds.), Rational numbers: An integration of research.
Hillsdale, NJ: Lawrence Erlbaum Associates.
Mack identifies the limitations of the informal knowledge and strategies that students bring to the
study of rational number concepts and procedures: They initially treat rational number problems
as whole-number partitioning problems. This approach makes it hard for them to reconceptualize
the unit. Their informal knowledge is initially disconnected from knowledge of formal symbols
and procedures. The instructional challenge is to expand students’ informal knowledge of rational
numbers and then tie this growing informal understanding to formal symbols and procedures. The
important research question that Mack poses is “whether students can develop a broad
understanding of rational number by building on their informal conception of partitioning.”
Moss, J. & Case, R. (1999). Developing children’s understanding of the rational numbers: A new model
and an experimental curriculum. Journal for Research in Mathematics Education, 30, 122-147.
Moss proposes an instructional strategy through which “children can be helped to construct a
rapid and serviceable overview of the rational numbers from the time of their first introduction to
them.” Her strategy is to reverse the “fraction-decimal-percent” order in which 10- and 11-yearold students first study rational number. By introducing students first to percent in a linear
measurement context, followed by decimals, and then by fractions, as an alternative form for
representing decimals, Moss’s instructional program seeks to unify children’s intuitions about
ratios with their numerical procedures of splitting. In her research with 4th-grade students, 16
students received the 20-hour experimental curriculum over a 5-month period and 13 carefully
matched students received a traditional curriculum. Students in the treatment group showed a
deeper understanding of rational numbers than those in the control group as indicated in part by
less reliance on whole number strategies when solving non-standard problems. No differences
were found in conventional computation between the two groups.
Ni, Y. & Zhou, Y. (2005). Teaching and learning fraction and rational numbers: The origins and
implications of whole number bias. Educational Psychologist, 40 (1), 27-52.
This is a critical summary of developmental, instructional, and neuropsychological evidence
related to the “whole number bias,” which is widely regarded as impeding student learning of
fraction and rational numbers. The review covers an immense range of research and leads,
ultimately, to the identification of two important questions for further research: What is the
impact of introducing relational reasoning and fraction numbers earlier in the elementary
curriculum? How does the “equal-sharing” approach compare in effectiveness to the
“measurement” approach, as a means of facilitating student learning in the design of long-term
instructional sequences?
Wu, H. (2002). Chapter 2: Fractions. (Draft) http://math.berkeley.edu/~wu/
Wu argues that a major reason for students’ failure to learn fractions is the “mystical and
mathematically incoherent” manner in which the subject has been presented to them—i.e., with
“multiple meanings (part-whole, quotient, ratio) from the outset.” Hence, very few students can
give a definition of a ‘fraction’ that is at all related to the manipulations they are made to learn.
The remedy Wu proposes is to base instruction on one clear-cut definition of fraction as a point
on the number line, and to use reasoning to deduce as logical consequences all other meanings of
this concept.