Teaching
Fractions:
New Methods, New Resources
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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 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.
The domain of skill and knowledge referred to as "fractions" or
"rational numbers" has been parsed in various ways by researchers in
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 "prefraction 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.
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 "prefractional 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.
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).
* "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"
http://www.highwired.com/Classroom/Project/0,2069,23713-68258,00.h
tml
"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 kindergarteners 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
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 http://ericir.syr.edu/Eric/adv_search.shtml.
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.
----- 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:
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Descriptors: Arithmetic, Concept Formation, Elementary Secondary
Education, Fractions, Mathematics Instruction, Teaching Methods
Identifiers: ERIC Digests
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