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).


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