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).
Source: http://www.homeedsa.com/
Showing posts with label Fraction. Show all posts
Showing posts with label Fraction. Show all posts
MISTAKES TEACHERS MAKE
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.
Source: http://www.homeedsa.com
Continued fraction From Wikipedia, the free encyclopedia
In mathematics, an infinite continued fraction is an infinite expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. A finite continued fraction is similar, but the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In either case, all integers in the sequence, other than the first, must be positive.
If arbitrary values and/or functions are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a generalized continued fraction. When it is necessary to distinguish the first form from generalized continued fractions, the former may be called a simple or regular continued fraction, or said to be in canonical form.
Continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number p/q has two closely related expressions as a finite continued fraction, whose coefficients ai can be determined by applying the Euclidean algorithm to (p,q). The numerical value of an infinite continued fraction will be irrational; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions. Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number α is the value of a unique infinite continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values α and 1. This way of expressing real numbers (rational and irrational) is called their continued fraction representation.
The term continued fraction may also refer to representations of rational functions, arising in their analytic theory. For this use of the term see Padé approximation and Chebyshev rational functions.
Source: Wikipedia
If arbitrary values and/or functions are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a generalized continued fraction. When it is necessary to distinguish the first form from generalized continued fractions, the former may be called a simple or regular continued fraction, or said to be in canonical form.
Continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number p/q has two closely related expressions as a finite continued fraction, whose coefficients ai can be determined by applying the Euclidean algorithm to (p,q). The numerical value of an infinite continued fraction will be irrational; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions. Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number α is the value of a unique infinite continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values α and 1. This way of expressing real numbers (rational and irrational) is called their continued fraction representation.
The term continued fraction may also refer to representations of rational functions, arising in their analytic theory. For this use of the term see Padé approximation and Chebyshev rational functions.
Source: Wikipedia
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