La continuidad de un método

The purpose of research question 1 was to explore Mathematics Teachers’ conception of student experiences that have grounding in transformation geometry concepts. This was meant to ascertain the extent to which teaching and learning in Transformation Geometry values students’ out-of- school experiences. The results on research question 1 revealed some concepts in Transformation Geometry linked to students’ out-of-school experiences. In this presentation, such experiences were deduced from objects like mirrors, elastic bands, and catapults. Research has it that most learners’

interest and achievement in Mathematics improve dramatically when they are helped to make connections between new information and experiences they have had (Cord, 1999 & Gravemeijer, 2008). In other words, bringing objects like these that bear the life experiences of students into the teaching and learning context is commensurate with ideal teaching and learning dynamics.

Teachers’ responses highlighted some students’ out-of-school experiences that relate to translation concepts, that is, the general movements of objects and pattern building on some traditional objects. These responses were however coming from teachers who had undergone a teacher training course. In other words, teachers with a bit of pedagogical training were aware of the value of such knowledge for teaching and learning purposes. The unqualified teacher only knew the more mechanical features of teaching concepts, for example, Teacher A2 who says “I don’t have clear experiences on this one related to learner experiences out there. That’s why we end up resorting to theory”. Teacher A2 and B2 described a Translation based on the procedural fluency of the topic. For instance, Teacher B2 illustrated it as the notion of a displacement of an object from point A to point B in terms of changes on the x-coordinate and y-coordinate. They described a ‘Translation’ as a movement in a straight line in a particular direction. While their explanations were correct, they were very mechanical and had difficulties in visualising a translation in the mind of the student or in real life contexts of the learner.

Nevertheless, the qualified teachers’ explanations exhibited a superficial illustration of a translation as resembling a movement of an object without specifying whether the movement is in a straight line or not. According to Jung (2002) a ‘Translation’ is a construction where an original figure is translated or moved or displaced and its original size, shape and orientation is preserved. In other words, a Translation has got to be a movement of a figure in a straight line without altering its compass reading. The four teachers (Teacher B1, B2, C1 and C2) seem to envisage a translation as simply a movement of an object without putting emphasis on preserving orientation. This might mean that some teachers are not fully aware of the meaning of a translation.

In contrast to the above, Teacher A1 mentioned decorations made on traditional objects, where patterns symbolise shapes repeated through translations. The example used by Teacher A1 gives a more precise model of a Translation, which demonstrates a movement where size, shape and orientation are preserved. Teacher A1’s illustration of a translation here resonates well with RME’s principle on guided reinvention through progressive mathematisation, which requires the choosing

of relevant contexts that offer students opportunities to see value in their informal knowledge (Doorman, 2001).

Teaching and learning need to afford students an opportunity to bridge the gap between their informal, what they are used at home, and the formal knowledge, the school mathematics (Barnes, 2004). Learners’ active participation in class and schoolwork is more dependent on teachers’ utilisation of students’ informal knowledge. These, in turn, become intrinsic motivators for further learning and resiliency. Teachers must be able to draw a lot of transformation geometry from the physical environment, pattern repetitions, object movement and photographs (Einsten, 2014).

Rivet and Krajcik (2008) highlights that if students are taught abstract ideas without meaning, they may not develop their understanding. Although participants demonstrated an appreciation of the fact that practically relevant examples are critical in making learners realise the link between Transformation Geometry and students’ world of experience, and hence increase comprehension of the concept, teachers referred did not refer to many examples. In this study teachers referred to the following among other examples; the patterns (Translation); mirrors (Reflection); door movement (Rotation); photographs (Enlargement); pile of books (shear) and catapult (stretch). This is an indication that some teachers, in theory, are aware of the importance of manipulative in their teaching that expose students to real life situations. By using what is real to the learner, the real- world context as a source of concept development and as an area application through process of mathematisation both horizontal and vertical, abstract mathematics become simpler (Freudenthal, 1977).

It also emerged in this study, that the more experienced and more qualified a teacher is (Teacher A1 and Teacher C1 compared to Teacher B1 and Teacher B2) the more likely is the teacher able to value students’ informal mathematics knowledge. There are however many aspects of real life that contain mathematics involving Transformation Geometry, such as in music. A classroom practitioner should be able to build a vast knowledge base on real world elements containing transformation geometry, which according to this study can build over a period of time. For example, a class can discuss objects that rotate. Rotations are also compositions (in the mathematical sense) of reflections (Usiskin et al. 2003, p. 315). Only Teacher A1 and C1 gave an example of Rotation.

The music connection also holds great potential for the high school geometry classroom. A geometric translation is like sliding an object from one place to another without changing orientation (Usikin et al., 2003). In music there is a horizontal translation where a melody shifts to later time. Anytime a melody line repeats in music a translation in time has occurred (Cooper and Barger, 2009). That’s the translation will be noted in the tunes. During interviews with teachers, Teacher A1 gave an example of decorations on clay pots as a representation of geometric translations. using examples that directly have to do with student experiences, be it in their play or in certain menial tasks they do outside of school (Naidoo, 2012) makes students more likely to experience academic success (Ladson-Billings, 1995).

Teacher inability to imagine and use students’ world of experience attribute to low student performance in the topic. Freudenthal (1991) says if learners are made to process new information in a way that does not make sense to them mastery of concepts will be a challenge. According to literature, use of learners’ experiences is pivotal for student success in Transformation Geometry (Walkerdine, 2003). Teachers need to choose and design learning environments that incorporate as many different forms of learner experience as possible – social, cultural, physical and psychological. In line with the study’s Theoretical Framework and Realistic Mathematics Education Model, it is pivotal for teaching and learning to encourage students’ comprehension of concepts through recognising connections. This is what Loewenberg et al. (2008) refers to as SCK (specialised Content Knowledge) for teachers.