Author: Michael Meagher Source: ERIC Clearinghouse
for Science Mathematics and Environmental Education, Columbus, OH

The teaching of fractions continues to hold the attention of
mathematics teachers and education researchers worldwide. In what
order should various representations be introduced? Should multiple
representations be introduced early, or one representation pursued
in depth once? Does it matter if fractions are introduced as counting
or as measurement? What is the relative importance of procedural,
factual, and conceptual knowledge in success with fractions? These
and other questions remain debated in the literature.

Following is an overview of recent research on teaching and learning
fractions, suggestions are offered for practice, for locating resources
having direct application in the classroom, and for further reading
in the research literature.

STUDENT CONCEPTIONS

The domain of skill and knowledge referred to as "fractions" or
"rational numbers" has been parsed in various ways by researchers
in recent years. Tzur (1999) sees children's initial reorganization
of fraction conceptions as falling into three strands: (a) equidivision
of wholes into parts, (b) recursive partitioning of parts (splitting),
and (c) reconstruction of the unit (i. e. the whole). Recognizing
this division, he suggests that teachers consider one of these strands
at a time in teaching rational numbers.

Taking a psychological
approach Moss and Case (1999) suggest that for whole numbers children
have two natural schema, one for verbal counting and the other for
global quantity comparison. In the realm of rational numbers they
also see children as having two natural schema: one global structure
for proportional evaluation and one numerical structure for splitting/doubling.
They propose, then, as a plan for learning that teachers need to
refine and extend naturally occurring processes.

Hunting's (1999) study of five-year-old children focused on early
conceptions of fractional quantities. He suggested that there is
considerable evidence to support the idea of "one half" as being
well established in children's mathematical schema at an early age.
He argues that this and other knowledge about subdivisions of quantities
forming what he calls "pre-fraction knowledge" (p.80) can be drawn
upon to help students develop more formal notions of fractions from
a very early age. Similarly, based on her successful experience
of teaching addition and subtraction of fractions and looking for
a way to teach multiplication of fractions, Mack (1998) stresses
the importance of drawing on students' informal knowledge. She used
equal sharing situations in which parts of a part can be used to
develop a basis for understanding multiplication of fractions; e.g.
sharing half a pizza equally among three children results in each
child getting one half of one third. Mack noted that students did
not think of taking a part of part in terms of multiplication but
that their strong experience with the concept could be developed
later.

Taking an information-processing approach (Hecht, 1998) divides
knowledge about rational numbers into three strands: procedural
knowledge, factual knowledge, and conceptual knowledge. Hecht's
study isolated the contribution of these types of knowledge to children's
competencies in working with fractions. He made two major conclusions:
(a) conceptual knowledge and procedural knowledge uniquely explained
variability in fraction computation solving and fraction word problem
set up accuracy, and (b) conceptual knowledge uniquely explained
individual differences in fraction estimation skills. The latter
conclusion supports the general consensus in current research that
a holistic approach to teaching of fractions is necessary with recommendations
for a move away from attainment of individual tasks and towards
a development of global cognitive skills.

MISTAKES TEACHERS MAKE

Based on previous research Moss and Case (1999) identified four
major problems with current teaching methods in the area of fractions.
The first is a syntactic rather than a semantic emphasis, which
is to say that researchers have identified that teachers often emphasize
technical procedures in doing fraction arithmetic at the expense
of developing a strong sense in children of the meaning of rational
numbers. The second problem identified is that teachers often take
an adult-centered rather than a child-centered approach, emphasizing
fully formed adult conceptions of rational numbers. As a result
teachers often do not take advantage of students "pre-fractional
knowledge" and their informal knowledge about fractions thus denying
children a spontaneous "in" to their formal study of fractions.
A third issue is the problem of teachers using representations in
which rational and whole numbers are easily confused e.g. students
count the number of shaded parts of a figure and the total number
of parts so that each part is regarded as an independent entity
or amount (Kieran cited in Moss & Case (1999). Finally, researchers
have identified considerable problems in use of notation that can
act as a hindrance to student development. These problems center
around teachers' perceptions that the notation used for rational
numbers is transparent while this has been shown not to be the case,
especially with regard to decimal fractions (Hiebert, cited in Moss
& Case (1999)). Tirosh (2000) conducted a study on teacher knowledge
in teaching of fractions and concluded that teachers needed to pay
considerably more to analysis of student errors.

NEW TEACHING APPROACHES

Moss and Case identified three different proposals on approaches
to teaching of fractions that address the above-mentioned problems
in various ways and then propose a new curricular approach which
they tested themselves in a study involving fifth and sixth grade
students. The first of the older studies conducted by Hiebert and
Warne (as cited in Moss & Case (1999)) was judged to have addressed
primarily the syntactic and notational problems mentioned above
and placed a great deal of emphasis on the use of base 10 blocks.
In the second study Kieran (as cited in Moss & Case (1999)) was
seen to address the syntactic and representational issues and, among
other innovations, used paper folding to represent fractions in
preference to pie charts. The third of the studies, conducted by
Streefland (as cited in Moss & Case (1999)) attempted to address
all four concerns and was based on using real-life situations to
develop children's understanding of rational numbers.

Moss and
Case's (1999) own approach was designed to address all four of the
identified problems and was characterized by several qualities distinguishing
it from previous approaches. They started with beakers filled with
various levels of water and asked students to label beakers from
1 to 100 based on their fullness or emptiness. They emphasized two
main strategies: halving (100 -> 50 -> 25) and composition (50 +
25 =75) in determining appropriate levels. Refining this approach
they developed the notion of two place decimals with five full beakers
and one three-quarter full beaker making 5.75 beakers. Four place
decimals were then introduced with 5.2525 (initially, spontaneously
denoted as 5.25.25 by the students) characterized as lying one quarter
of the way between 5.25 and 5.26. Students eventually went on to
work on exercises where fractions, decimals and percentages were
used interchangeably. Moss and Case found that this approach produced
deeper, more proportionally based, understanding of rational numbers.
They see their approach as having four distinctive advantages over
traditional approaches: (a) a greater emphasis on meaning (semantics)
over procedures, (b) a greater emphasis on the proportional nature
of fractions highlighting differences between the integers and the
rational numbers, (c) a greater emphasis on children's natural ways
of solving problems, and (d) use of alternative forms of visual
representation as a mediator between proportional quantities and
numerical representations (i. e. an alternative to the use of pie
charts).

WORLD WIDE WEB RESOURCES

"Visual Fractions"
This World Wide Web (WWW) site is designed to help users visualize
fractions and the operations that can be performed on them. There
are instructions and problems to work through for the operations
of addition, subtraction, multiplication, and division, first using
fractions and then working with mixed numbers. Number lines are
used to picture the addition and subtraction problems while an area
grid model is used to illustrate multiplication and division problems.
http://www.visualfractions.com

"The Sounds of Fractions: Math in Music"
Overview - You've probably heard that math and music are related,
but you may not have ever heard how or why. Objective: - Compare
math and music to see how mathematical concepts of ratio, proportion,
common denominator, frequency, and amplitude connect with musical
elements such as time signature, pitch, tone, and rhythm"

"No Matter What Shape Your Fractions Are
In"
"Description: These activities are designed to cause students to
think; they are not algorithmic. They do not say, "To add fractions,
do step one, step two, step three." Students will explore geometric
models of fractions and discover relations among them. Appropriate
Grades: 3rd - 6th, maybe. But precocious kindergartners could do
some of it, and middle schoolers needing another look at fractions
could appreciate it as well. 'Drawing Fun Fractions' would be good
for most middle school students."
http://math.rice.edu/~lanius/Patterns

"Flashcards" This
web site was developed to help students improve their math skills
interactively. Students can test their mathematics skills with Flashcards,
which give students practice problems to try and then gives them
feedback on their answers. Students can also create and print your
own set of flashcards online.
http://www.aplusmath.com/Flashcards/fractions-mult.html

FRACTIONS IN THE ERIC DATABASE
There are over 1,000 records in the ERIC database pertaining
to fractions. The best way to locate those records is to search
the database using one or both of the following ERIC Descriptors:
"fractions" or "decimal fractions". You can narrow your search by
combining these two Descriptors with others, such as teaching "methods",
"educational strategies", "instructional materials", "research",
"literature reviews", "mathematics instruction", "mathematics materials",
"mathematics curriculum", or "mathematics skills". You can further
narrow your search by using education level Descriptors, such as
"elementary education", "middle schools", "intermediate grades",
or "junior high schools", or individual grade levels. You can search
the database on the Web at www.eric.ed.gov

REFERENCES

Hecht, Steven Alan. (1998). Toward an Information-Processing
Account of Individual Differences in Fraction Skills. "Journal of
Educational Psychology". 90 (3) 545-59. Hunting, Robert P. (1999).
Rational-number learning in the early years: what is possible? In
J. V. Copley. (Ed.), "Mathematics in the early years", (pp 80-87).
Reston, VA: NCTM.

Mack, Nancy K.
(1998). Building a Foundation for Understanding the Multiplication
of Fractions. "Teaching Children Mathematics". 5 (1) 34-38.

Moss, Joan & Case, Robbie. (1999). Developing
Children's Understanding of the Rational Numbers: A New Model and
an Experimental Curriculum. "Journal for Research in Mathematics
Education". 30 (2) 122-47.

Tirosh, Dina. (2000). Enhancing Prospective
Teachers' Knowledge of Children's Conceptions: The Case of Division
of Fractions. "Journal for Research in Mathematics Education". 31
(1) 5-25.

Tzur, Ron. (1999). An Integrated Study
of Children's Construction of Improper Fractions and the Teacher's
Role in Promoting That Learning. "Journal for Research in Mathematics
Education". 30 (4) 390-416.

This digest was funded by the Office of
Educational Research and Improvement, U.S. Department of Education,
under contract no. ED-99-CO-0024. Opinions expressed in this digest
do not necessarily reflect the positions or policies of OERI or
the U.S. Department of Education.

Title: Teaching Fractions: New Methods,
New Resources. ERIC Digest.
Document Type: Guides---Classroom Use---Teaching Guides (052); Information
Analyses---ERIC Digests (Selected) in Full Text (073); Reports---Descriptive
(141); Target Audience: Practitioners, Teachers Available From: ERIC/CSMEE, 1929 Kenny Road,
Columbus, OH 43210-1080. Tel: 800- 276-0462 (Toll Free); Fax: 614-292-0263;
e-mail: ericse@osu.edu;
Web site: http://www.ericse.org. Descriptors: Arithmetic, Concept Formation,
Elementary Secondary Education, Fractions, Mathematics Instruction,
and Teaching Methods

ERIC Digest - THIS DIGEST WAS CREATED BY
ERIC, THE EDUCATIONAL RESOURCES INFORMATION CENTER. FOR MORE INFORMATION
ABOUT ERIC, CONTACT ACCESS ERIC 1-800-LET-ERIC
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ERIC Identifier: ED478711
Publication Date: 2002-06-00

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