ED478711 2002-06-00 Teaching

Fractions: New Methods, New

Resources. ERIC Digest.

ERIC Development Team

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Teaching Fractions: New Methods, New Resources. ERIC Digest....... 1

STUDENT CONCEPTIONS...................................................2

MISTAKES TEACHERS MAKE.............................................. 3

NEW TEACHING APPROACHES........................................... 3

WORLD WIDE WEB RESOURCES......................................... 4

FRACTIONS IN THE ERIC DATABASE.................................... 5

REFERENCES.................................................................. 6

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.

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ED478711 2002-06-00 Teaching Fractions: New Methods, New Resources. ERIC Digest. Page 1 of 7
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

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Page 2 of 7 ED478711 2002-06-00 Teaching Fractions: New Methods, New Resources. ERIC Digest.
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))

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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

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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

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ED478711 2002-06-00 Teaching Fractions: New Methods, New Resources. ERIC Digest. Page 5 of 7
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.

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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:

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Descriptors: Arithmetic, Concept Formation, Elementary Secondary Education,

Fractions, Mathematics Instruction, Teaching Methods

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