Fractions and algebra are critically important components of the
mathematics education of our youth. Unfortunately, however, students
have typically struggled in these areas. For this reason, teachers and
researchers have focused their attention on these topics for at least
the past century. This article discusses what research shows regarding
fractions and algebra, particularly, on issues related to when fractions
should be taught, how fractions should be taught, and how competence
with fractions affects the transition from arithmetic to algebra will be
considered. Suggestions for teacher practice are included throughout the
The case for postponement
The first issue teachers and curriculum specialists must address is
when fractions and rational numbers should be taught. Several
researchers feel that the study of fractions and rational numbers often
occurs before the student is ready. These researchers, who include
Kieren and Freudenthal, suggest postponing the study of rational numbers
until they can be taught within the context of algebraic ideas. Kieren
(1976) feels that the experience base necessary for mature functioning
with the complete rational number concept is best provided in a course
of algebra and that, to sufficiently learn algebraic concepts that are
intrinsic to rational number concepts, a student must experience and
master the diverse interpretations of fractional numbers. Therefore, he
recommends that an in-depth consideration of rational numbers be
postponed until such time as the student studies algebra. Similarly,
Freudenthal argues that teaching addition of rational numbers should be
postponed until the concepts arise from algebraic ideas (as cited in
Other researchers have studied how well students learn fractions
when they are taught, as is usually the case, prior to the teaching of
algebra. One such study tried to gather information about how much
average students can learn about fractions under the best conditions.
The results showed that in well-to-do junior high schools, instruction,
even under optimal conditions, did not provide students with the
necessary fractional skills. While students understood the fraction
concept, they showed a poor understanding of the structure of the
rational number system (Ginther, Ng & Begle, 1976). For example,
only 30% of the students were able provide the correct number to make a
true statement (Ginther et al., 1976, p. 4). Moreover, the results of
the computation test were unsatisfactory and the students did poorly on
simple word problems that involved fractions. This study suggests that
without understanding structure, competent manipulation of fractions
will not occur, since too much of the content depends on rote learning
of algorithms, which make little sense to the learner and are too often
misapplied. Is this because the concepts are being presented too early
in a child's cognitive development? The study does not attempt to
answer this question, but concludes, "Much of the work on fractions
should be postponed to secondary school," (p. 9).
Approaches to teaching that allow students to construct their own
knowledge can be powerful. The writings of Henry Margenau regarding the
scientific method (1961) can shed light on what conditions are necessary
in order for a constructivist approach to succeed and for students
"to really know." He breaks knowledge into the elements of
fact and construct and goes on to describe and redefine each element.
Facts function as protocols, the "first draft of an experience
later to become formalised knowledge" (p. 5). Constructs are the
result of the processes of generalising and logical reasoning that lead
to abstraction and ownership of complex concepts through a "long
chain of activity" (p. 5). Protocols are collections of facts and
related experiences that an individual brings to bear upon a problem. If
the facts and related experiences can be connected effectively, then the
individual is able to construct their own knowledge. If the facts are
isolated and related experiences are not present, then one is unable to
make the necessary connections to form a valid construct. These
connections are like pathways that the learner logically negotiates to
link relevant protocols and established constructs, which can be applied
to a problem or to new learning (p. 16).
Margenau's thoughts can be applied directly to the rich and
complex concept of rational numbers. Kieren (1980) asserts that the
number of disjointed protocols a learner must control to form the
rational number concept is extensive. Too often an algorithm has simply
been taught, providing no connections for understanding, and leaving the
student clinging to a prescribed step-by-step set of instructions.
Algorithms that are taught when the concept is beyond the learner's
cognitive development, force the learner to abandon their own thinking
and resort to memorisation--doing without understanding.
If the algorithm is forgotten, the learner must retreat to familiar
protocols (procedures), which can be applied in the given situation. For
example, the individual may try to apply a natural number protocol for
addition of fractions, adding both numerators and denominators, since
addition of natural numbers arises from the natural activity of children
(Kieran, 1980, p. 102).
Postponement and developmental readiness
Piaget's theory of cognitive development (Wadsworth, 1996)
concludes that, in general, school-age children are either in the
concrete operational stage (ages 7-11) of development or in the stage of
formal operation (ages 11-16). The child in the concrete stage
"must deal with each problem in isolation" (p. 112) and is
unable to construct new knowledge from internal reflection alone. Formal
thinkers are able to generalise and use internal reflection that
"can result in new knowledge--new construction" (p. 118). In
terms of Margenau (1961), this suggests that the individual in the stage
of concrete operations does not progress very far into the
constructional domain, yet is able to develop and connect simple
protocols that are closely related to the individual's experience
The concrete operational child is capable of learning the basic
part-whole relationship of rational numbers, but this is not enough for
complete understanding of the rational number concept (Lamon, 1999). If
instruction proceeds directly to computation procedures, then the child
neither has the time nor the cognitive development to construct
Two important formal operational schemes are proportion and
probability (Wadsworth, 1996). Both of these schemes are elemental to
the rational number concept. Susan Lamon (1999) states "instruction
needs to take an active role in facilitating thinking that will lead to
proportional thinking" (p. 4). Mathematical topics that are related
to proportions are fractions, decimals, ratios, percents, probability,
similarity, linear functions, equivalence, measurement, and many others
(p. 9). Consequently, a sizable gap exists in an individual's
rational number concept, a gap that will become even more apparent as
the individual begins to tackle a course in algebra.
McBride and Chiappetta (1978) investigated the relationship between
proportional thinking and a student's ability to understand
concepts related to simple machines and equivalent fractions. They
reasoned that, in order to understand equivalent fractions, a student
would need to think at Piaget's stage of formal operations.
Consequently, students in the concrete operational stage could not be
expected to demonstrate understanding in equivalent fractions after
studying them in school. The reasoning ability of these students would
limit their understanding of this concept. Since Piaget's stage of
formal operational thinking begins around age 11 or 12, few students
below this age level should be expected to display comprehension of the
concept of equivalent fractions.
Several other studies provide support for this assertion. In one
such study, only 7% of 9 to 12 year old students who had studied
equivalent fractions were able to demonstrate understanding (Novillis,
as cited in McBride & Chiappetta, 1978). In another study of 9 to 12
year olds, only 50 % were able to show comprehension of equivalent
fractions, leading the authors to conclude that formal operational
thinking was necessary for success with this topic (Steffe & Parr,
as cited in McBride & Chiappetta, 1978). Similar findings by McBride
and Chiappetta (p. 8) led them to conclude that proportional thinking is
an underlying factor associated with achievement in equivalent
fractions, supporting the hypothesis that postponement of teaching
certain rational number concepts until secondary school is a viable
An extensive study of common fraction understanding and decimal
fraction understanding was undertaken in upper socio-economic suburbs of
Hobart, Tasmania. This study examined contexts related to diagrams,
algorithms, and problem solving. The results were compiled for three
levels of students, grades five and six, grades seven and eight, and
grades nine and ten. With regard to diagrams, the results indicated that
students at all three levels had a better concrete understanding of
common fractions than of decimal fractions. In the problem-solving
context, students at each level performed essentially the same (very
poorly) for both common and decimal fractions (p. 10). The algorithmic
results, however, indicated that students below grade nine were
substantially better at applying decimal fraction algorithms than common
fraction algorithms (Watson, Collis, and Campbell, 1995). Kieren (1976)
supports these results, since operations on decimal fractions "form
a natural extension to the whole numbers" (p. 102). However, by
grades 9 and 10 the ability to manipulate common fractions was slightly
better than the ability to manipulate decimal fractions (Watson, Collis
& Campbell, 1995), lending credibility to the argument for
postponement of teaching common fraction operations until secondary
Some possible solutions
Currently, there is no indication that the mathematics curriculum
will be modified to accommodate the previously discussed suggestions for
postponement, despite an apparent lack of developmental readiness.
Therefore, the logical alternative is that teachers need to provide
better and more meaningful instruction of fraction concepts. Calls for
improvement in this area are not new. More than twenty-five years ago,
Ginther, Ng, and Begle (1976) suggested that research be done to
discover whether or not better instruction would result in improved
student learning of fractions. Research has been done, methods have been
refined, but there has been little if any improvement as indicated by
the data. If fewer than half of the adult population are able to reason
proportionally (Lamon, 1999), then even the best instruction coupled
with experience can be expected to have little effect on developing
sophisticated mathematical reasoning (p. 5).
One positive finding with regard to the teaching of fractions
suggests that providing increased time on this topic may be the answer.
Studies have shown that if children are given the time to develop their
own reasoning for at least three years without being taught standard
algorithms for operations with fractions and ratios, then a dramatic
increase in their reasoning abilities occurred; including their
proportional thinking (Lamon, 1999). How fractions should be taught is
inexorably linked to when the concepts are being presented and what
impact the learned concepts will have on future mathematics courses such
as elementary algebra.
Re-teaching the definition of fractions is one approach that can be
effective when students experience problems with fractions. De Morgan
(1910) suggests that a student having difficulties with fractions should
return to the original definition and reason upon the suppositions,
neglecting the rules until he or she can cognitively establish them by
reflection upon familiar instances (p. 40). In this brief statement, De
Morgan illuminates two of the major problems with the teaching of
fractions. First, the concept of a fraction is never clearly defined
(Wu, 2001); thus, returning to an original definition is impossible.
Second, more time is needed to allow students to invent their own ways
to operate on fractions rather than memorising a procedure (Huinker,
1998). An awareness of these shortcomings in the present approach to
teaching fractions can be beneficial to teachers. If teachers make sure
they provide a sound definition of a fraction and provide additional
time for student exploration with fractions they may find that their
students perform better.
Based on the research already discussed, it seems clear that
teachers need to reform the pedagogies by which they teach fractions.
Additional support for change is provided by the results of The National
Assessment of Educational Progress (NAEP, Mullis et al., 1990), a United
States report. This study indicated that only 46 % of twelfth grade
students demonstrated success with decimals, percents, and fractions.
Similarly, the 1999 NAEP reports that twelfth grade students responded
correctly to test items related to the operations on fraction numbers
only fifty per cent of the time (NCES, 1999). The remainder of this
section will be devoted to discussing some potential pedagogical reforms
that could serve to improve teaching methods in this area.
A change in emphasis from the development of algorithms to perform
operations to the development of quantitative understanding based on
students' experiences with physical models that emphasise meaning
rather than procedure may be warranted (Bezuk & Cramer, 1989, p.
157). An added focus on problem solving is another potentially
beneficial pedagogical technique. A problem-solving approach to teaching
fractions was tested on fifth-graders in an urban school (low SES,
Huinker, 1998). The students were not taught how to add, subtract,
multiply, divide, or compare fractions, but instead, were left to
develop meaning for fraction operations within the context of solving
problems. The students in this four-week study "constructed
intuitive quantitative understandings of fraction concepts and
operations in the context of solving and posing realistic problems"
(Huinker, 1998, p. 181). Carefully directed lessons can be designed to
encourage students to form their own algorithms for adding and
subtracting fractions. These student-invented algorithms are often very
efficient and, with direction from the teacher, can be generalised to
become powerful mathematical tools (Lappan & Bouck, 1998, p. 184).
Effective methods for the teaching of understanding of fractional
numbers must be concerned with allowing students the time to construct
their own understanding as teachers direct them toward accurate and
meaningful student-invented algorithms. Bezuk and Cramer (1989) offer a
few general recommendations, which are echoed in much of the literature
concerned with the teaching of fraction concepts. These are:
1. the use of manipulatives is fundamental in developing
2. the majority of the time spent on fractions before grade 6
should be devoted to developing a conceptual base of fraction
3. operations on fractions should be delayed until students have a
solid understanding of order and equivalence of fractions; and
4. the size of the denominator for computational exercises should
be 12 or below (p. 158).
Teacher content knowledge and additional pedagogical considerations
A study conducted by Putt (1995) shows that a relationship exists
between teachers' knowledge of mathematics and student learning.
This study found that misconceptions about rational number concepts held
by students were also evident among teachers (p. 11). The error patterns
that are passed from teacher to student year after year create confusion
and math anxiety, which too often begins right after introduction to
fraction computation. Wu (2001) adds that teachers must have the
necessary mathematical knowledge to be able to correctly guide their
students through the subject and that textbooks must be written that
treat fractions logically (p. 6).
Susan Lamon's book, Teaching Fractions and Ratios for
Understanding: Essential Content Knowledge and Instructional Strategies
for Teachers (1999) provides a valuable resource that can help
pre-service teachers acquire the requisite knowledge to teach fractions
effectively. This book is designed to develop the rational number
concept among this audience. Lamon underscores the rich
"constructional domain" (Margenau, 1961, p. 9) for a simple
fraction such as 3/4 supporting the position that fractional numbers
might best be taught in the context of problems. There are many more
interpretations of 3/4 than the simple, and single meaning as three
parts of a four-part whole (p. 32).
Other researchers take a somewhat different position. Wu (2001),
for example, feels that conceptual complexities are too often emphasised
"at the expense of the underlying simplicity of the concept"
(p. 2). When students are led through a multitude of interpretations,
the simplicity is lost and the students are deprived of an essential
component of doing mathematics: the ability to abstract. Wu's
position is that prior to the fifth or sixth grade, children should
become acquainted with fractions in an intuitive way through
explorations, collecting data without concern for meaning; but then Wu
(2001) goes on to state that, "when confronted with complications,
[students] try to abstract in order to achieve understanding" (p.
5). He believes that the processes of abstraction should be introduced
as soon as possible in the school mathematics curriculum, and that the
teaching of fraction computation would be "as soon as
possible," since at the age of eleven or twelve children are moving
into formal operations and are capable of employing "reflective
abstraction" (Wadsworth, 1996). "By giving abstraction its due
in teaching fractions, we would be easing students' passage to
algebra as well" (Wu, 2001, p. 6).
Sharp (1998), like Wu, believes that algebraic thinking can be
developed as students are taught fractions. She suggests a method for
teaching division of fractions that uses an algorithm that follows
directly from whole number operations and fraction concepts. Since much
of algebra is generalised arithmetic, prior practice in generalising
previously developed algorithms can begin to build the type of thinking
that is necessary for the transition from arithmetic to algebra (p.
203). If the logical development of algorithms for rational number
operations, supported by fraction concepts, promotes algebraic thinking,
then it would follow that students who have constructed a viable
rational number concept would be successful in algebra.
The relationship between fractions and algebra
There are at least three critical achievements in the mathematical
life of a student: mastering the idea of ten as a unit, understanding
fractions, and grasping the concept of the unknown. Consequently, when
attempting to learn algebra without the aid of understanding fractions,
"it is no wonder that many students' seeming mastery of
fractions begins to fall apart" (Driscoll, 1982, p. 107).
Rotman (1991) contends that although an arithmetic course need not
be prerequisite for a first-year college algebra course, the
understanding of "fraction concepts deserve[s] to be singled out,
because algebra typically uses fractional notation to indicate a
quotient" (p. 8). Similarly, Wu (2001) asserts that the study of
fractions has the potential for being the best kind of pre-algebra and
argues that unless the way in which the teaching of fractions and
decimals is radically changed, then the failure rate in algebra will
continue to be high (p. 10). He claims that adding fractions has become
a conceptual preoccupation, but that understanding the concept is not
sufficient; there is the need for fluency in computation. Wu insists
that such fluency--the ability to efficiently manipulate fractions--is
"vital to a dynamic understanding of algebra" (p. 17). Vague
fraction concepts and misunderstood fraction algorithms will ultimately
be generalised into vague algebraic concepts and procedures. The lack of
precise definitions and reliance upon shortcuts that are thoughtlessly
given to students are likely to hinder performance in algebra.
Additional support for this position is provided by Laursen (1978) who
found that many of the errors that students make in first-year algebra
are due to an incomplete understanding of fraction operations and the
subsequent misapplication of imprecise algorithms, which were previously
taught as shortcuts.
Kieren (1980) suggests that there are algebraic aspects of
operations on fractions, but that most school curriculum materials
simply treat fractions as objects of computation. Rational numbers
present the student with algebraic problems. The student must:
1. understand the notion of equivalence;
2. deal with an addition operation based in axiomatic reasoning
rather than the natural extension of whole number addition;
3. work with a multiplication operation that is distinct from
addition and is abstractly defined; and
4. cope with abstract properties and the concept of an inverse (p.
If students have not had opportunities to learn how to abstract
prior to an elementary algebra course, then they may opt for rote
learning--the memorisation of algorithms without any conceptual
basis--that allowed them to appear to be successful with fraction
Generality and abstraction are characteristics of algebra that must
eventually be expressed in symbolic notation (Wu, 2001). Fluent
computation with numbers lies at the foundation of the ability to
perform symbolic manipulations (p.13). When teaching addition of
fractions, without the concept of the lowest common denominator, Wu
suggests that the operation be clearly defined as . This formula can be
first used for cases where a, b, c, and d are small numbers and then
slowly built up from these specific cases to the general case. Wu
insists that without such a foundation in fractions, students will be
severely hindered when they come to study rational expressions in
algebra (p. 14).
There is no escaping fractions in algebra. From linear equations to
completing the square, from solving systems of linear equations to
solving rational equations, and from simple probabilities to the
binomial theorem, algebra is replete with examples that are directly and
indirectly related to fractions. Much of the basis for algebraic thought
rests on a clear understanding of rational number concepts (Kieren,
1980; Driscoll, 1982; Lamon, 1999; Wu, 2001) and the ability to
manipulate common fractions. "With proper infusion of precise
definitions, clear explanations, and symbolic computations, the teaching
of fractions can eventually hope to contribute to mathematics learning
in general and the learning of algebra in particular" (Wu, p. 17).
As this article suggests, extensive research has been done in the
areas of fractions and algebra, much of which considers the
relationships between these two difficult, but important, topics. Since
no definitive conclusions can yet be drawn, it is incumbent upon
teachers and researchers alike to implement innovative strategies and to
study the efficacy of these strategies with the ultimate goal of
improving instruction in these critical areas. We continue this argument
in a subsequent article that will discuss data from our research, and
consequent implications for classroom teaching.
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Cold Springs Middle School, Reno, USA
Robert J. Quinn
University of Nevada, Reno, USA