**ERIC Identifier:** ED478711

**Publication Date:** 2002-06-00

**Author: **Meagher, Michael

**Source: **ERIC Clearinghouse for
Science Mathematics and Environmental Education Columbus OH.

## Teaching Fractions: New Methods, New Resources. ERIC Digest.

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.

### 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 "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.

### 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 "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.

### 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"

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

### 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
http://ericir.syr.edu/Eric/adv_search.shtml.

### 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.