Tmn3704_TEACHINGMATHEMATICS_STUDY_GUIDE.

Tmn3704_TEACHINGMATHEMATICS_STUDY_GUIDE. WAYS OF PRESENTING KNOWLEDGE AND FACILITATING LEARNING IN MATHEMATICS A C T I V I T Y 1.6.1 Here is an example of how CAPS (page 232) explains the size of an angle: a) Comment on the teaching guidelines given in CAPS. Justify your reasoning. b) Discuss an alternative way in which you can introduce the concept of an angle to Grade 6 learners. c) Provide two suggestions that you can make to facilitate learning in your presentation on teaching angles. A C T I V I T Y 1.6.2 Design an activity to illustrate how you would explain the concept of transformation to Grade 6 learners. 1.7 UNDERSTANDING OF MATHEMATICS Haylock (2010:3) defines the understanding of mathematics as “learning in which the learner is involved in constructing understanding through exploration, problem-solving, discussion and practical experience”. Hiebert and Carpenter (1992:67) define the degree of understanding as being determined by the number and the strength of connections. A mathematical idea, procedure, or fact is understood thoroughly if it is linked to existing networks with stronger or more numerous connections. Skemp (1976) identifies two types of understanding: relational and LEARNING UNIT 1: CONCEPTS RELATED TO THE TEACHING MATHEMATICS instrumental. He describes relational understanding as knowing both what to do and why and the process of learning relational mathematics as being one of building up a conceptual structure. He describes instrumental understanding, in contrast, as simply understanding rules without reasons. Both forms of understanding are discussed separately, and in detail, below. 1.7.1 Instrumental understanding Instrumental understanding is understanding in which all the learners know is what to do. Being dependent on memory, the learners do not necessarily know why they are doing what they are doing, or why what they doing produces the correct answer. A learner might know how to carry out long division using a particular algorithm, but they might not know why they bring down the digits during the procedure. This understanding is thus rather a product of memory than knowing what to do (Skemp 1976). The teaching characteristics that produce instrumental understanding are demonstrating procedures, which emphasise memorisation and drill, pen and computations, and teaching by rules (Orton & Forbisher 1996; Romberg & Kaput 1999). The short-term effects of this understanding (in terms of which the learner knows what to do) are great, and this understanding is what is often rewarded (Lindquist 1989:10). The long-term effects of this form of understanding are negative, as is shown by the present state of mathematical learning, in terms of which the learners remember little of what they have been taught (Barmby, Harries, Higgins & Suggate 2007; Lindquist 1989). 1.7.2 Relational understanding The second form of understanding, relational understanding, is less dependent on memory, as the learner has the ability to know what to do, and why (Skemp 1976). The kinds of teaching characteristics that produce relational understanding emphasise carrying out mathematical procedures, making connections, constructing one’s own mathematical concepts, and problem-solving (Clarke 2002; Haylock 2010; Orton & Forbisher 1996). When given the fraction 6/8, Peter only knows the name of, and the procedure for, simplifying the fraction to ¾ , in other words, he possesses instrumental understanding. In contrast, Alex can illustrate the fraction through diagrams, give examples of it, calculate equivalences and approximate the size of the fraction; in other words, he has relational understanding. PETER ALEX Peter only knows the name of the fraction and the procedure to simplify the fraction to 3 4 –. Alex can illustrate this fraction through diagrams, give examples, calculate equivalences and approximate the size of the fraction. 15 . . . . . . . . . TMN3704 Learning unit 1: Concepts related to the teaching mathematics . . . . . . . . . 16 The effective teaching of Mathematics therefore entails the learner being provided with opportunities to construct understanding through exploration, problem-solving, discussion and practical experience. Kleve (2010:158) observes that “traditional beliefs and practice regarding school mathematics are challenged by reform-oriented curricula and the teachers’ deeply held beliefs can serve as an obstacle in implementing the new reforms”. South African teachers have experienced, and continue to experience, several different curriculum reforms. Recent examples are outcomes-based education (OBE), the Revised National Curriculum Statement (Department of Education 2005), the Foundations for Learning Campaign and, most recently, CAPS. These curriculum changes are driven by the desire to “improve learner numeracy and literacy performance in view of very poor achievement by learners in the national systemic evaluations and in international assessment” (Department of Education 2007:41). The lack of significant change in learner performance might, in part, be due to the teachers’ deeply held traditional beliefs about the teaching and learning of Mathematics serving as an obstacle to implementing the reforms (Kleve 2010:159). Assessment in TIMSS, and other external assessments that are constructed around the TIMSS assessment framework, tend to emphasise high-order thinking and analytic skills (The Education Alliance 2006:11). The challenge of teaching Mathematics in the current decade is that it requires teachers to adopt teaching strategies that develop the relevant skills (Hiebert & Grouws 2007:390; Peterson 1989:6). If teachers continue to use the traditional approaches that yield only instrumental understanding, then the learners will continue to perform badly in tasks that focus on higher-order thinking and on the acquisition of critical analytic skills.