Discussion and Reflection on Skemp's Article

 

Education is for every global citizen. So are the discussions on the education. (1) However the jargon heavy beginning part of the article and the use of music lesson example as analogy to ‘relational mathematics’ vs ‘instrumental mathematics’ could make its accessibility limited. (2) Author uses two definitions of ‘football’, namely soccer and rugby, as a supporting example to says it matters when we have two different concepts of understanding. Football example can be viewed as a definition problem in which having mutually acceptable and consistent definition would avoid such problems. (3) The three examples demonstrate how a teacher would teach area of a rectangle, multiplication of two rational numbers and circumference of a circle. They may be a problem of insufficient information. It may be because the term ‘instrumental understanding’ is not a well-accepted and consistent definition yet. These made me stop and think and the next paragraph is a generalization out of (3).

The issue the author raised looks like a part (or parts) of a larger problem. Mathematics is a growing body of knowledge and people contribute their parts in this construction process without a detailed drawing of the entire mathematics building. The builders get more mature as the building gets better and humans gather more experiences. The learners of mathematics also more or less follow this road from behind the builders. In each topic of interest, it is important to declare the set of objects we work with and the consistent definitions along with validity, conditions and logical reasoning. However, due to different maturity levels among learners, teachers would learn to know where/how to start. For example, a definition of the area of a bounded region that is part of a Euclidean two dimensional space is a measure of its extent that we humans want in number. One agreed definition is a multiple of a standard square-unit. This is how I understand from learning, not yet what a pure mathematician would say. Whether we tell such story to a grade 5 student is debatable. According to the mathematical maturity of the students we may start with:

Length and width must be measured in the same unit and are non-negative numbers. This formula is valid only for the rectangular regions laid on a flat surface. Some of its applications are measure of a land plot for pricing, measure of floor area for estimating materials to buy. This can be used as definition for beginners. From that starting point, students can continue to learn areas of different shapes in flat plane using logical reasoning. After having familiar with areas on flat surface, they could generalize the concept, again using logical reasoning, to cover areas of cylinders, spheres and so on. Students should be offered a safe sufficient amount of information depending on their maturity.