An understanding of the empirical environment wherein this research took place required the researcher to be cognisant of the structure and routine of the Engineering Mathematics 3 curriculum as well as the practicalities of teaching and learning in a CAS environment. Findings advocated in literature were mostly from further afield and its applicability and compatibility to local settings needed to be carefully contemplated.
It was important to consider the status quo learning environment and to systematically connect this world with the envisioned research setting. This empirical setting would ultimately be subjected to new complexities and the researcher needed to holistically predict when, how and to what effect each component of the research would be drawn
161 on. The empirical environment can therefore be described as a setting where theory-meets-practice.
4.5.2 The curriculum
The study played out in an undergraduate course, Engineering Mathematics 3, which is a compulsory module for Electrical, Mechanical and Industrial Engineering students registered for a National Diploma in Engineering at UJ. The module is offered during each first semester to an Electrical Engineering cohort in their second academic year and also offered during every second semester to a cohort of Mechanical and Industrial Engineering students. This module is preceded by Engineering Mathematics 1 and Engineering Mathematics 2 which are core8 semester modules in the Engineering Diploma qualifications at UJ (University of Johannesburg, 2013). These two first year modules are respectively taught in a structured classroom milieu during semester 1 and semester 2 of each academic year. Here, the focus is on theory and calculations are done with a non-programmable pocket calculator. Rudimentary graphs are taught and hand-drawn in Engineering Mathematics 1; included are graphs of the straight line, parabola, hyperbola, circle, ellipse, exponential, logarithmic and trigonometric functions (Seloane, 2013). To some extent, students still make use of the elementary table method to sketch graphs (Mofolo-Mbokane, 2011). This is in defiance of alternative, more progressive methods taught in the Engineering Mathematics 1 curriculum (UJ, 2013; Seloane, 2013).
In the second year Engineering Mathematics 3 curriculum, differential equations (DEs) constitute the core topic. This curriculum is divided into a theory (70%) and a practical component (30%) where DEs are solved analytically and numerically. Analytical procedures are taught in the Engineering Mathematics 3 theory classes and follow strict theoretical procedures. Topics covered in the theory classes are first order DEs, the Laplace transform, systems of first order linear equations, D-operators, Matrix Algebra and Fourier series. Although valued, these classic methods are of limited use in most engineering applications where analytical solutions do not exist. The emergence of computer technology has sparked renewed interest in the numerical
8 A core module is compulsory to a qualification as opposed to an elective module where students have options to choose from.
162 analysis of DEs since the 1980’s (Dubinsky & Tall, 1991). Numerical methods can be used to explore the qualitative properties of DEs by means of numerical tables and graphs. Numerical calculations can be performed in two ways. One, traditional paper-and-pen procedures with lengthy calculations done with pocket calculators; or two, CAS can be employed to produce graphs and tables in a fraction of the time. Since 2000, DEs had been solved with numerical methods by using CAS and in particular, Mathematica software9 (Wolfram Research, 2012). The Mathematica practical component is offered in a computer laboratory and has been designed to complement the theoretical aspects of the module. During the computer laboratory sessions, students learn a new programming language in order to solve DEs numerically.
Numerical procedures are decidedly iterative in nature and therefore an ideal topic to be explored with programming technology. Mathematica is used as the programming language to teach numerical methods to solve first and second order DEs. Once a week, students attend a 150 minute Mathematica session in a computer laboratory.
The Engineering Mathematics 3 module is assessed with two theory tests, two Mathematica tests and two exam papers, one on theory and the other on Mathematica course work.
4.5.3 The curricular approach to numerical DEs
In the Engineering Mathematics 3 module, students learn to programme the Euler, Runge-Kutta order 2 and Runge-Kutta order 4 numerical methods from first principles.
Typically, a generic code is programmed for a specific method (either Euler or Runge-Kutta) which can then be applied to any DE with particular initial conditions. Where possible, the analytical solution is also obtained by using paper-and-pen theoretical methods. With CAS, the analytical solution can also be compared with the numerical solution by means of graphs and numerical tables. Ultimately, graphs and tables contain a fair amount of information that can be used to better understand the phenomenon described by the DE. This current curricular process can be summarised as follows: programme the Euler or Runge-Kutta method in Mathematica → apply this programme to a given DE and initial conditions (IC) → generate tables and graphs → interpret solutions.
9 MATLAB was used till 2012.
163 4.5.4 Setting the scene for experimentation
Several design principles were identified from literature and applied to create a mathematical modelling environment. In particular, tenets from the design theory of Realistic Mathematics Education (RME), actions-processes-objects-schema (APOS) theory, the source-target framework and the Pirie-Kieren model of growth in mathematical understanding (see Section 3.4) were drawn upon. The researcher conducted “an anticipatory thought experiment” to design principles that predict possible learning trajectories and the prerequisite support structures to enact the envisioned learning (Cobb, Stephan, McClain & Gravemeijer, 2011:119). These design principles were divided into two categories: firstly, the design of tasks that could stimulate modelling activities to develop students’ visualisation and cognitive processes, and secondly, the learning milieu in the computer laboratory that could support the development of visualisation and related processes. The balance between the design of tasks and CAS learning environment is illustrated in Figure 4.6.
Figure 4.6: Design principles in the learning environment.
The following principles were considered in the design of modelling tasks:
Design suitable tasks that elicit the integration of multiple fields of mathematics and the diversity of newly-acquired CAS skills (Freudenthal, 1991)
Tasks must be open-ended and invite imaginative, wide-ranging reasoning (Baskan & Alef, 2013; Cai, 2003; Wiliams & Goos, 2013)
Design of modelling tasks - Theory driven
- Eleven design principles
CAS learning environment - Facilities driven
- Five learning milieu principles
164
Situate tasks in reality, tasks must be meaningful and lifelike to students so that they can intuitively relate to the context and draw from their own experiences (Dewey, 1944; Gravemeijer & Doorman, 1999)
Solutions to tasks must be supportive of students’ intuitive understanding of the context, their visual intuition and must be defensible within their own reality (Fischbein, 1987; McDowell, 2006; Nardi, 2014)
Allow for iterations of modelling cycles in which students can progressively express, check and revise their understandings (Lesh & Caylor, 2007)
Sub-tasks must be sequentially structured by setting intermediate goals that can help to un-muddle the complex real world problem and avoid bottlenecks (Galbraith
& Stillman, 2006)
Stimulate metacognitive actions and processes with self-questioning; asking questions like ‘does this make sense?’ or ‘what do I know that could be used?’
(Mevarech et al., 2006)
Provide opportunities to construct personal images and schemas to connect different representations (Noss & Hoyles, 1995)
Motivate a sense of competiveness by honouring assessment criteria for originality, completeness and rigour of tasks (Spandaw & Zwaneveld, 2009)
To observe the Pirie-Kieren levels of understanding, tasks should prompt responses such that students can justify, and reflect on, actions and processes that expose their relevant levels of understanding (Pirie & Kieren, 1992). As observer, the researcher can then assume that a group task reveals a valuable framework of student actions and behaviours which in turn, are regarded as evidence of particular understandings. To this end, tasks should include group reports wherein students can reflect on the difficulties encountered and why certain decisions were made. Records of actions, processes and task reports can help the researcher to observe students’ thinking and doing on varying Pirie-Kieren levels of understanding
Tasks should involve thought-revealing actions, processes, objects and schemas through which the researcher can observe, code and analyse understandings (Lesh, Young & Fennewald, 2010).
165 Additionally, the following principles were considered for the CAS learning environment:
Allow enough time to complete each task in a single session (Cai, 2003; Kaiser &
Schwarz, 2010)
Form groups by combining members with a stratified grade distribution (Berry, 2010; Galbraith, 2012)
Provide a private workspace for each group to allow free come-and-go and the availability of all necessary learning materials and tools within the group (Freudenthal, 1991)
Allow and encourage the use of multiple sources (Flavell, 1976) including the internet, cell phones, tablets, laptops, PCs, calculators and textbooks
Adopt a pseudo-laissez-faire role where the researcher become a spectre-spectator facilitator (Verschaffel, Greer, Van Dooren & Mukhopadhyay, 2009). The role of the researcher is reduced to moving around groups and observing events unobtrusively without interference.
These are fundamental factors in the design of novel experiences that aim to prepare students for their future workplace environment; ultimately, it is the duty of the lecturer to pave the way to such new experiences.