3.4.1 THE SPACE OF LEARNING.
Another aspect of variation theory that is salient to the teaching and learning of functions is the space of learning. A space of learning is the space created by the teacher in the classroom or a learning environment by using different approaches to the teaching of a concept. This space consists of examples that are given to the students in a learning environment. In order for learners to experience variation in mathematics, a space for such experiences to occur has to be created and is called the space of learning. The space of learning involves the use of examples that can help the learners to experience the patterns of variation as referred to by Marton et al. (2004). These are contrast, generalisation, separation and fusion.
A space of learning is formed when a critical aspect of the topic is brought to the foreground of the learners’ awareness. This is similar to when a noise is discerned against a background of silence or a tall person is contrasted with a short person, then the critical aspect of tallness and shortness is discerned. Learning according to variation theory involves the ability to discern differences in the critical features of what is to be learned. What is learned depends on taking other conditions into consideration as well (Runesson, 2006, p.402).
Rowland (2008, p.150) contends that there are two types of example spaces that can be used when dealing with variation theory. The first type of example space is inductive in nature and can be created by motivating students to give examples of “something” Rowland (2008, p.150) .The examples are usually “particular” instances of the general Rowland (2008, p.150). The second type of examples can be through exercises. The aim of exercises is to facilitate retention of the procedure by repetition. The teacher’s choice of examples can
facilitate the learners’ awareness of the nature of the concept through variation of examples within the chosen category.
Skemp (1979) posits that the role of examples helps learners in their formulation and comprehension of the concept as well as the assimilation of the concept through subsequent examples may occur in future. The space of learning involves the meticulous planning of the lessons and the proper use of appropriate examples that facilitate the discernment of the object of learning by the learners.
The examples used in the study at the beginning of the intervention lessons (which is
explored in Chapter 5) constitute the space of learning for this study. The choice of examples and how they are introduced to the learners at the beginning of each intervention lesson provides us with sufficient information on whether the particular choice of examples was meaningful for the study or not. The choice of examples is discussed in the following section.
3.4.2 CHOICE OF EXAMPLES.
Research conducted by Rowland (2008) shows how the choice of examples by 12 elementary trainee teachers during their final school placement influenced the discernment of the object of attention by the learners who were aged between 4 and 7 years. The research emphasises the diversity of knowledge required from teachers in order for variation theory to be
effective.
Schulman (1987) as cited in Rowland (2008, p. 15) emphasises three categories of teacher content knowledge that are important to the application of variation theory. These categories are subject matter knowledge (SMK), pedagogical content knowledge (PCK) and curricular knowledge (CK). In order for variation theory to be effective in a learning situation, the teacher should be well-vested in these three types of knowledge. The knowledgeable teacher should be able provide learners with a robust (meaningful, relevant) choice of examples that will enhance learning and discernment of the object of learning.
A learning study was used by Runesson (2006) whereby she asserts that variation theory accounts for the differences in learning. She provides ways to describe the conditions
necessary for learning and involved the experiencing and the understanding of perceiving and seeing something in a different way, which could lead to learners’ ability to interpret
equation form. A version of a learning study is used in this study as a catalyst to enable the learners’ discernment of the object of learning. This is discussed further in Chapter 5. The author further states that variation is a necessary condition for the discernment of the object of learning, the critical features of what is to be learned, seeing the differences in the object of learning and essentially forms a space of learning. In order to provide learners with the opportunities to discern the critical features of the object of learning variation could be used.
When using variation theory, there must be something that varies while another one remains invariant. In choosing examples, one should be cautious to use systematic variation. The chosen examples should follow a carefully-planned variation that enables the learners to discern the object of learning. Carefully-chosen examples and their structure promote mathematically significant variation (Watson & Mason, 2005).
For learners to be able to discern the chosen object of learning, examples given to the learners should involve the variance and the invariance aspect of variation. There are different
patterns of variation theory that create the space of learning. These patterns of variation involve the variance and the invariance aspect of the theory of variation which is discussed in the following section.