Ngu Bing Hiong, Phan Huy P
School of Education, University of New England, Armidale, NSW, Australia.
Front Psychol. 2020 Sep 25;11:558773. doi: 10.3389/fpsyg.2020.558773. eCollection 2020.
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 , 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., )-whether it is a numerator sin30° = /5 or a denominator sin30° = 5/. More specifically, we contend that some students may have difficulties when solving sin30° = /5, say, despite having learned how to solve a similar problem, such as /4 = 3. For more challenging Trigonometry problems, such as sin50° = 12/ where the pronumeral is a denominator, students have been taught to "swap" the with sin30° and then from this, solve for . Previous research has attempted to address this issue but was unsuccessful. 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., /4 = 3) and a Trigonometry problem (e.g., sin30° = /5) to facilitate analogical learning. , 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° = /5 vs. cos50° = 20/) 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.
数学学科是世界上大多数国家的国家优先事项。人们普遍认为，与其他“软理论”或“硬应用”学科相比，数学被视为“纯理论性的”（贝彻，1987）。因此，数学学习对一些学生来说可能极具困难。的确，作为一门纯理论学科，数学并非那么有趣，对一些学生而言，其学习可能有些艰巨且具有挑战性。一个这样的例子就是 的主题，其理解相对复杂。这个涉及代数变换技巧的三角学问题尤其因变量（例如 ）的位置而变得复杂——它是分子（sin30° = /5）还是分母（sin30° = 5/ ）。更具体地说，我们认为一些学生在求解 sin30° = /5 时可能会遇到困难，比如说，尽管他们已经学过如何解决类似问题，比如 /4 = 3 。对于更具挑战性的三角学问题，比如 sin50° = 12/ ，其中变量是分母，学生们已被教导要将 与 sin30°“交换”，然后据此求解 。先前的研究曾试图解决这个问题，但未成功。 依赖于在已学问题和新问题之间建立类比，二者具有相似的求解过程。我们将一个线性方程（例如 /4 = 3）和一个三角学问题（例如 sin30° = /5）并列起来以促进类比学习。相比之下， 则识别两个问题之间的异同，从而有助于学生理解两个问题的求解过程。我们将变量位置不同的两种三角学问题（例如 sin30° = /5 与 cos50° = 20/ ）并列起来以鼓励积极比较。因此，借助类比学习和比较学习理论的互补优势，我们期望克服学习涉及代数变换技巧的三角学问题所固有的困难。总体而言，这篇概念分析文章试图阐明并厘清这两种比较教学方法，以实现三角学的有效学习。
