Theoretical framework
3.4 Instructional design
Instructional Design (ID) is a collection of theories and models that helps one to understand and apply instructional methods that foster learning (Paquette, 2014). It is a method or a process that facilitates strategic planning and organisation of learning and teaching activities, thus bringing order to a chaotic situation. This assists the teacher to assume the role of facilitator and the lesson becomes learner-centred. The adoption of technology supports the instructional design process. Paquette (2014) notes that this could include "web authoring tools and languages, knowledge modeling of instructional design methods, automated and guided instructional design, e-learning standards and social/semantic Web environments"
(p.5).
Accompanying technology based learning are tips or hints which guide and probe one's understanding. These provide assistance in solving the problem and developing understanding which provides scaffolding. The Cognition and Technology Group at Vanderbilt (1997) termed these hints embedded data. This relates directly to the Zone of Proximal Development (ZPD) proposed by Vygotsky (1978). When one reaches a dead end or mental block, one can seek help from the embedded data. The embedded data/ hints draw on one's current understanding, taking one from the known to the unknown. These hints or data play a crucial role when instructional time is limited, allowing an individual to cover
more work. Geometry is allocated three weeks in Grades 10 and 11 and two weeks in Grade 12 (Department of Education, 2011). Thus, the embedded data or hints are useful in the learning process, bearing in mind the number of teaching and learning disturbances such as sports, tours, public holidays and unforeseen events.
Instructional design can take the form of any type of multimedia such as web pages with photos, text, and video streaming, etc. This exposes the learner to multiple perspectives when solving a problem. There three main processes that take place during any learning are:
Assimilation, when one encounters something similar to what they already know, including previous experience; Accommodation, when something new is learnt and this knowledge is accommodated; and Equilibrium, which is the balancing of Assimilation and Accommodation - 50150 (Block, 1982; Piaget, 1952; Simatwa, 2010). Constructivism focuses on the learner and occurs while learning takes place. It is the result of an active process where the learner's prior knowledge is taken into consideration and is built on, thus causing cognitive conflict (Von Glasersfeld, 1989). Careful planning and use of ICTs together with ID could cause cognitive conflict, leading to realisation and the creation of strong, meaningful geometric thoughts. Such conflict would be enhanced by integrating GHOM in the design of the instructional material.
To promote proper implementation of technology in geometry, this study adapted the following instructional design principles by Mark, Cuoco, Goldenberg and Sword (2010) in alignment with GHOM to create instructional materials (DOEs):
Figure 4: Instructional design principles adapted from Mark, Cuoco, Goldenberg & Sword, (2010, pp.
506-507)
The links between the GHOM levels and ID principles are as follows:
Smooth integration is key when using ICTs in Mathematics teaching and learning. Proper planning and development of ICTs is required before they can be utilised in the classroom.
Similarly, Mark, Cuoco, Goldenberg and Sword (2010) note that time is required to develop GHOM and they must be phased into learners' normal day-to-day activities.
Experience experimentation: one must be given the opportunity to experiment with the topic at hand such as a geometry theorem. The design of the learning material should allow for trial and error exploration of concrete problems before the formal mathematical development (Mark, Cuoco, Goldenberg &Sword, 2010), thus allowing for the integration of abstract and concrete concepts. The embedded data allows for exploration or discovery learning. Learners are given control of their learning, thus eliciting learner centeredness and responsibility. This is supported in the Van Hiele levels which indicate that one should engage before arriving at the formulae (Abu & Abidin, 2013). Experience experimentation is present in the GHOM levels, Reasoning with relationships, Investigating Invariants and Balancing Exploration and Reflection.
In the low threshold-high ceiling approach, the designed material is initially easy and becomes progressively more difficult, offering a challenge. For example, abstract concepts are introduced together with experimentation with specific numerical examples, enabling one to extend to deeper understanding (Mark, Cuoco, Goldenberg & Sword, 2010). This is seen when proving a theorem with unknown variables and actual measurements. This exercises one's thinking as one is constantly looking for patterns, spotting relationships, generalising and investigating. This is linked to the GHOM levels, Reasoning with relationships, Generalising geometric Ideas and Investigating Invariants. The experience enables one to feel competent and confident in their learning and at the same time be open to experiences that are more challenging.
Being organised means that development organised around GHOM helps teachers to manoeuvre through an accumulation of teaching methods and techniques gained over the years as well as what is prescribed in the curriculum. Some methods and techniques make for relevant learning only on a particular topic and have no meaning when moving onto another topic or subtopic. It is important that the bits and pieces of mathematical concepts and topics be meaningfully linked, as this can be used to show the interconnected network among the topics in Mathematics. This is more common in geometry, where previous knowledge is drawn on in higher grades. Examples include the use of Theorem of Pythagoras, congruency, similar triangles and the relationship of parallel lines which are all taught at junior grades and later required in senior grades. Meaningful exploration and reflection are promoted, which
can lead to generalisation not only in geometry but other topics in Mathematics. This takes place in the following GHOM levels: Generalising geometric Ideas and Balancing Exploration and Reflection. Furthermore, as noted earlier, this type of thinking (GHOM) is not restricted to geometry, but can be applied to various experiences in one's daily life.