Learning to Solve Trigonometry Problems That Involve Algebraic Transformation Skills via Learning by Analogy and Learning by Comparison

Abstract

The subject of mathematics is a national priority for most countries in the world. By all account, mathematics is considered as being “pure theoretical” (Becher, 1987), compared to other subjects that are “soft theoretical” or “hard applied.” As such, the learning of mathematics may pose extreme difficulties for some students. Indeed, as a pure theoretical subject, mathematics is not that enjoyable and for some students, its learning can be somewhat arduous and challenging. One such example is the topical theme of Trigonometry, which is relatively complex for comprehension and understanding. This Trigonometry problem that involves algebraic transformation skills is confounded, in particular, by the location of the pronumeral (e.g., x)—whether it is a numerator sin30° = x/5 or a denominator sin30° = 5/x. More specifically, we contend that some students may have difficulties when solving sin30° = x/5, say, despite having learned how to solve a similar problem, such as x/4 = 3. For more challenging Trigonometry problems, such as sin50° = 12/x where the pronumeral is a denominator, students have been taught to “swap” the x with sin30° and then from this, solve for x. Previous research has attempted to address this issue but was unsuccessful. Learning by analogy relies on drawing a parallel between a learned problem and a new problem, whereby both share a similar solution procedure. We juxtapose a linear equation (e.g., x/4 = 3) and a Trigonometry problem (e.g., sin30° = x/5) to facilitate analogical learning. Learning by comparison, in contrast, identifies similarities and differences between two problems, thereby contributing to students’ understanding of the solution procedures for both problems. We juxtapose the two types of Trigonometry problems that differ in the location of the pronumeral (e.g., sin30° = x/5 vs. cos50° = 20/x) to encourage active comparison. Therefore, drawing on the complementary strength of learning by analogy and learning by comparison theories, we expect to counter the inherent difficulty of learning Trigonometry problems that involve algebraic transformation skills. This conceptual analysis article, overall, makes attempts to elucidate and seek clarity into the two comparative pedagogical approaches for effective learning of Trigonometry.