Algebra - major stumbling block

Algebra is known to be a major stumbling block in school mathematics, both in the past and at present. Historical studies on the developments of algebra education in the twentieth century show that the algebra studied in secondary school has not changed much over the years. Unintentionally algebra has functioned as a means of selecting the more capable learners – the ‘happy few’ who understand and enjoy the powers of algebra – from the rest, who experience and remember it as an elusive interplay of letters and numbers. Problems with algebra can be ascribed to external factors like the teaching approach and a poor image, but also to intrinsic difficulties of the topic.

Researchers have reported that grown-ups often have a negative image of school algebra, and many students can make no sense of it. There is a plausible explanation for this. Traditional school algebra is primarily a very rigid, abstract branch of mathematics, having few interfaces with the real world. It is often presented to students as a pre-determined and fixed mathematical topic with strict rules, leaving no room for own input. Traditional instruction begins with the syntactic rules of algebra, presenting students with a given symbolic language which they do not relate to. Students are expected to master the skills of symbolic manipulation, before learning about the purpose and the use of algebra. In other words, the mathematical context is taken as the starting-point, while the applications of algebra (like problem solving or generalizing relations) come in second place. Students are given little opportunity to find out the powers and possibilities of algebra for themselves. One can imagine that an average or below-average learner finds little satisfaction in practicing mathematics without a purpose or a meaning. Another characteristic of the traditional apLearning and teaching of school algebra proach is the rapid formalization of algebraic syntax. School algebra has always had a highly structural character, where algebraic expressions are conceived as objects rather than computations or procedures to be carried out. The procedural (or operational) aspects of algebra, which are more closely related to the arithmetical background that early algebra learners have, are usually cast aside soon after the introduction. This procedural-operational duality of algebra .

Even though we all have an immediate idea what students learn when they learn school algebra, it is not an easy task to give a cast-iron definition. In an attempt to capture ‘school algebra’ in one sentence, we might suggest it is the mathematical domain dealing with (general) relationships between quantities on a symbolic level. Still, this description does not do justice to the multiple roles and utilities of algebra. Typical topics of school algebra include simplifying algebraic expressions, the properties of number systems, linear and quadratic equations in one unknown, systems of equations in two unknowns, symbolic representations and graphs of different kinds of functions (linear, quadratic, exponential, logarithmic, trigonometric), and sequences and series. In most of the core activities we find aspects of algebraic thinking (mental processes like reasoning with unknowns, generalizing and formalizing relations between magnitudes and developing the concept ‘variable’) and algebraic symbolizing (symbol manipulation on paper). Generally it is agreed that students must acquire both competencies in order to have full algebraic understanding.