Literature review
Level 4 Rigor
2.6 Self-regulating and metacognition
2009). Hence, PSTs that are the focus of the current study were exposed to a dynamic online environment, making this technology available to them as a resource for future lessons, and encouraging them to investigate other technology based avenues when teaching mathematics, especially geometry. It is important to note that when learners work on geometry problems at home or during an examination, the teacher is not present to provide assistance. Thus, self- regulation is important when learners solve a problem.
Technology can play a vital role in allowing one to control one's learning. Wilson, Fernandez and Hadaway (1993) refer to this process as metacognition, which involves thinking about one's own cognition and therefore promoting orderly, constructive reflection on a problem.
The use of technology can bring about such order (Clapp & Swenson, 2013; Lim, Zhao, Tondeur, Chai &Tsai, 2013) Metacognition is the type of thinking that is required to connect pieces of knowledge in order to solve a problem effectively, especially in geometry, when prior theorems are used to solve current theorems. Learners that lack SRL are not able to deeply comprehend complex problems (Chen &Huang, 2013). Metacognition is knowledge of one's own thinking and the ability to monitor one's own understanding and problem- solving activity (Kilpatrick, Swafford & Pindell, 2001). It has multiple and somewhat disjointed meanings that include knowledge of one's thought processes or self-regulation during problem solving (Schoenfeld, 1992). While there are numerous definitions of metacognition, knowledge and monitoring of one's own cognitive process are common to most. Technology gives one control of one's learning and exercises one's metacognition in becoming self-regulated. Thus, the manner in which technology is used in the classroom becomes important. In addition, the teacher needs to be knowledgeable of the subject content, ensuring the accuracy of the information provided. As Bransford, Brown, Cocking, Donovan, and Pellegrino (2000) remark, "teachers must come to teaching with the experience of in- depth study of the subject area themselves" (p.20). The teacher is the only person in the classroom who can model complex procedures (Schraw, Crippen &Hartley, 2006). However, when learners self-regulate, it helps them to think like mathematicians, where they pose their own mathematical questions and endeavor to solve them. In this way, they construct new knowledge.
In general it has been found that little time is spent on learning geometry as compared to algebra, although geometry is regarded as difficult (Department of Basic Education, 2014;
Hansen, Gustafsson, Rosen, Sulkunen, Nissinen, Kupari, Olafsson, Bjornsson, Gronmo, Ronberg, &Mejding, 2014; Mji &Makgato, 2006). Learners become anxious and frustrated as they are expected to master selected theorems in the short space of time set by the curriculum. Being able to self-regulate is critically important; Zimmerman (2002) points out that a lack of self-regulation results in learners with vague self-evaluative standards that cannot gauge the level of academic preparation required for tasks like tests and examinations.
When the learner is able to generate thoughts and reflect on their own thinking, they become critical thinkers. Schraw, Crippen and Hartley (2006) observe that critical reflection plays an important role in self-regulation. If learners can criticise their thinking, it is easier for them to
identify ways to help them to achieve their goals. This results in a positive outlook on life since they know exactly what needs to be done to achieve their objectives. Critical reflection can assist in testing hypotheses and discovering theorems. Taken together with a dynamic geometry environment, it enables learners to take a second look at the role of proof (Mudaly, 2002).
Middleton and Spanias (2002) maintain that intrinsic motivation results in a person enjoying involvement in an activity, wanting to develop skills; and always applying themselves in order to achieve their goals. These are the qualities of a person who has a positive outlook - they always apply themselves and desire self-development. As Zimmerman (2002) points out, a self-regulated learner is likely to achieve better understanding of the subject matter and will display higher levels of self-efficacy. A learner that is intrinsically motivated will appreciate geometry. In terms of the five strands of mathematical proficiency, a learner with a productive disposition, is described as having the habitual inclination to regard mathematics as sensible, useful; and worthwhile, coupled with a belief in diligence and one's own efficacy (Kilpatrick, Swafford &Pindell, 2001) will regard geometry as valuable. A learner who is not intrinsically motivated would not appreciate geometry and give up more easily when posed with a problem that is underpinned by their geometry knowledge. Self-regulated learners are self-aware and self-motivated, and would apply such knowledge appropriately (Bodrova, Germeroth &Leong, 2013; Zimmerman, 2002).
Self-regulated learners sometimes adopt a defensive position during the self-reflection phase.
This leads to withdrawal or avoiding opportunities to learn and perform, such as dropping a subject or being absent (Zimmerman, 2002). The learner behaves in this manner to protect their image (they might have excuses), thus giving them a positive outlook despite the fact that they have not admitted to failure. However, with the use of technology, this is less likely to occur since the learner will engage with the technology and make mistakes without being looked down upon by others. Self-regulated learners seek help as a corrective measure rather than the help which the teacher provides. These learners would approach the teacher when they have complex model queries. As proposed by Zimmerman (2002), "self-regulation is not a mental ability or an academic performance skill; rather it is the self-directive process by which learners transform their mental abilities into academic skills" (p. 65). In order for learners to self-regulate they need to adopt specific learning strategies or methods as they concurrently learn or complete a task. This justifies the use of technology, especially
instructional technology, which is a dynamic online environment that offers the user control and can help in adopting such strategies or methods.