Academic journal article International Journal of Applied Educational Studies

Informal Problem Solving Strategies of Fifth Grades in the United Arab Emirates: Indicators of Algebraic Thinking

Academic journal article International Journal of Applied Educational Studies

Informal Problem Solving Strategies of Fifth Grades in the United Arab Emirates: Indicators of Algebraic Thinking

Article excerpt

Abstract: This study investigated how students in grade 5 approached word problems of an algebraic nature. Two problems were used for this purpose, one in the form x + (x + a) + (x + b) = c, and the other in the form ax + bx + cx = d. Data were collected from 82 students from two elementary schools in Al-Ain City, United Arab Emirates. The data consisted of students' written solutions and explanations and follow up interviews with selected students. The results showed that successful and partially successful solutions involved two approache:, the unwind approach and the guess and check approach (both systematic and random). The unwind approach was used in approximately two thirds of the solutions. Unsuccessful solutions involved random guess and check, number manipulation, and arbitrary approaches. Generally, the results support the idea of introducing algebra earlier in school mathematics curricula.

Background

Many secondary school students face difficulties in learning algebra. These difficulties include constructing equations from word problems as well as interpreting, rewriting, and simplifying algebraic expressions (Filloy & Rojano, 1989; Kieran, 1989). Some mathematics educators (e.g., Kieran, 1989; Herscovis & Linchevski, 1994; Bendarz & Janvier, 1996) suggest that the "sudden exposure" to algebra is behind these difficulties. This means that algebra is so inherently different from arithmetic and that elementary curriculum is conceptually far away from algebra. For example, Kieran (1989) highlighted two conceptions of mathematical expressions: procedural (related to arithmetic) and structural (related to algebra). In arithmetic, an expression such as 7 + 8 means "add 8 to 7." This interpretation corresponds to procedural conception. In algebra, however, the same expression (7 + 8) is interpreted as a number. This interpretation corresponds to the structural conception.

Similarly, Herscovis and Linchevski (1994) suggested that algebraic skills require a problem solving approach different from that required for arithmetic. Arithmetic requires retrieval of number facts and direct calculations while algebra requires thinking about the relationships involved in the problem. Elementary mathematics curricula primarily focused on number facts, and computational fluency. It is not until higher grade levels, that letters are used to stand for sets of numbers (Bendarz & Janvier, 1996).

The focus on number facts and direct calculations in elementary mathematics curricula is due to the belief that elementary students are not capable of doing algebra (Filloy & Rojano, 1989; Herscovis & Linchevski, 1994). This belief was enhanced by the traditional view of algebra which has been thought of as "describe and calculate" (Lesh, Post, & Behr, 1987) with a focus on writing and manipulating equations. This traditional view of algebra, however, has started to change which allows new activities and perspectives to be thought of as algebra.

For example, Kieran (1996) suggested three types of algebra-based activities: generational, transformational, and global, meta-level. In generational activities, situations are generated into equations or expressions. Activities such as collecting like terms, factorizing and simplifying expressions are called transformational activities. Global, meta-level activities are those situations in which algebra is used as a tool but this use does not have to be restricted to algebra. In solving a problem, the student needs to understand the relationships among the quantities in that problem. The student also needs to make representations that are not necessarily letter-symbolic to be able to solve the problem in a relational way. The understanding and representations described here are what Kieran (1996) calls "global, meta-level" aspect of algebra. In this regard, he suggested:

```   Thus, algebraic thinking can be
interpreted as an approach to
quantitative situations that emphasize
the general relational aspects with
tools that are not necessarily letter-symbolic,
but which can ultimately be
used as cognitive support for
introducing and for sustaining the