This section presents how the teaching and learning of Transformation Geometry can create significant learning experiences for students. Effective teachers support students to make connections by providing them with opportunities to engage in complex tasks and by setting expectations that they explain their thinking and solution strategies, and that they listen to the thinking of others (Anghileri, 2006). Teachers can assist students to make connections by using
carefully sequenced examples, including examples of students’ own solution strategies, to illustrate key mathematical ideas (Watson & Mason, 2007).
Hill, Schilling and Bell (2004) extended Shulman (1986)’s original ideas about pedagogical content knowledge and developed a model for mathematics teachers’ knowledge referred to as mathematics knowledge for teaching. In their model the three knowledge domains most central to mathematics teaching are common knowledge of mathematics, specialised knowledge of content, and knowledge of students and their ways of thinking about the content.
Common knowledge is the knowledge that any adult not necessarily educated needs to possess to provide correct mathematical solutions. Specialised knowledge of content is being able to provide students with multiple representations addressing diverse and learning styles (Hill, Schilling & Bell, 2004). Thus, teachers need to have both common and specialised knowledge to enhance their teaching of transformation geometry along with their pedagogical knowledge. It was in the thrust of this study to examine if Mathematics teacher-participants had common knowledge or specialised knowledge or both in teaching transformation geometry. The next discussion unpacks the interactions between teacher knowledge of subject and knowledge of teaching in a different but related model.
To engage in this discussion a model adapted from Danielson’s (2007) framework for teaching was used. The framework has four domains however a more emphatic version of the framework with three domains was adapted for the purpose of this study as shown in Table 2.2 below.
Table 2.2: Framework for the teaching of Mathematics
Domain Description
1. Planning and Preparation Knowledge of content
Knowledge of related students’ informal mathematical knowledge
Knowledge of resources 2. The Classroom Environment Coherent instruction
Environment of respect and rapport Managing classroom procedure 3. Instruction Clear and accurate communication
Use of question and discussion techniques Nature of tasks and student feedback
The framework alludes to three key domains in the teaching and learning process. In the first domain, a teacher has got to engage in planning and preparation for future lessons through considering the content to be taught, students’ informal mathematical knowledge related to the content, as well as the available resources (Danielson, 2013). Transformation Geometry provides a culturally and historically rich context within which to do mathematics. Teachers have a critical role to play in ensuring that resources used effectively support students to organise their mathematical reasoning and support their sense-making (Blanton & Kaput, 2005).
There are many interesting results in transformational geometry that can enhance students’ learning when relevant resources and approaches are selected for use (Mahanta & Islam, 2013). The teacher must also have a command of the content they teach. In line with the first domain, the current study used document analyses to examine the form of teacher planning and preparation.
The second domain refers to the classroom environment. It is an essential skill for teachers to manage a positive classroom environment. Teachers create and maintain an environment that is conducive for creating significant learning experiences for students. Presenting transformation geometry in a way that increases students’ curiosity and enhances exploration can boost student’s learning in the topic (Chigwiza et al., 2013). Also, patterns of interactions are critical for the overall tone of the class. In this study, lesson observations were used to discern the tone in Transformation Geometry classes.
The third domain is about instruction. Teachers who are competent use clear and imaginative analogies and metaphors to increase the bond between students’ informal mathematical knowledge and the formal-taught mathematics (Danielson, 2007; 2013; Purpura, Baroody & Lonigan, 2013). Demonstrating the links among mathematical topics is important for enhancing conceptual understanding. Teachers and learning must encourage students to make connections with their world of experience. Ready-made tools, effective teachers should acknowledge the importance of students generating and employing their own representations, such as in notation, or graphical form (Chick, Pfannkuch, & Watson, 2005).
When learners discover that they can manipulate mathematics as a tool for solving problems in real life, they start to perceive the subject as of value. The focus of this study is to explore teaching of Transformation Geometry is dependent on life experiences of students. Thus, building on students’ existing understanding of concepts can help teachers emphasise the links between different ideas in
mathematics (Arsathamby & Zubainur, 2014; Gravemeijer, 2013; Posnanski, 2010). Student experiences are an essential element of a rich instructional environment, if not used students continue to guess and not learn.
Mathematical tasks also offer opportunities for students to engage in thought provoking and reasoning activities (Henningsen & Stein, 1997; Stein & Smith, 1998). According to (Stein & Lane, 1996; Stein & Smith, 1998) the greatest learning gains on mathematics assessments occur when students are engaged in high level cognitive tasks. Thus, to improve learners’ performance in Transformation Geometry students must engage in cognitively demanding activities (Boston & Smith, 2009) that foster the development of concepts (Jupri, 2017). It was the thrust of the current study to examine the nature of tasks used in transformation geometry classes and evaluate their contribution to significant learning experiences.