Computer simulations in lectures in mathematics and in the natural sciences: learning‐theoretical aspects

This article presents an overview of results from the mathematics education literature about the usefulness of using computer simulations in lectures in mathematics and in the natural sciences. Therefore, it justifies (some) established teaching practices in mathematics lectures. This arcitle discusses some key differences between computer simulations used for scientific research and simulations used for teaching. We describe some aspects of learning theory which are useful for analyzing computer simulations of mathematical phenomena.


Mathematics-based computer simulations and their applications
Every applied mathematician knows the value of computer simulations for conducting applied research. In education (especially university teaching) computer simulations are also increasingly used.
Some computer simulations (both very new ones and ones which have been used for a while for educational purposes) from various areas of mathematics and natural sciences are discussed here, along with ideas of how to use them in lectures and with an analysis of their advantages and possible shortcomings. Mathematical modelling and computer simulations based on it have been used for decades. The list of potential applications is very long. It includes modelling of discrete and continuous dynamical systems, ODEs and PDEs, problems in mechanics, engineering, natural sciences, artificial neural networks, medicine, finance, chaos theory, and many more. In particular, this includes discrete dynamical systems of the form f : x n → x n+1 , where the current state of the system is encoded as point x in some suitable space X and the discrete-time evolution process is represented by a map f : X → X , and continuous dynamical systems of the form d dt y(t) = f (y, t), where time is continuous and the map f represents the direction of change at each point x at time t. See [6] for a general introduction to the theory of dynamical systems. Applied mathematicians have studied all of these topics, often with the help of computer simulations. The users of such simulations are often those who created the simulations and thus are deeply familiar with the underlying science.

Computer simulations for lectures differ from simulations for research
Using computer simulations in lectures in mathematics, engineering and in the natural sciences means that the potential users of these simulations are usually neither the creators of the simulation nor necessarily familiar with the relevant parts of the underlying science. On the contrary, if the goal of using a computer simulation in a lecture is to teach students certain aspects of the underlying science, then students will be a priori unfamiliar with them. Thus using computer simulations in lectures is fundamentally different from using computer simulations to solve scientific problems (even though sometimes students can be encouraged to re-discover scientific facts that are known to the creator of the simulation, an in very ideal simulations, students can attempt to actually discover facts that were previously unknown). Those simulations used as teaching tools thus have different requirements: they must be easy to use; they should produce (almost) immediate feedback; their output should be easy to understand and interpret; they should produce interesting visualizations; they should preferably run on a variety of different hardware platforms. This is substantially different than the requirements for tools which are used for academic research. However, when students become more advanced (having more prior knowledge, and pursuing more advanced steps in an academic education) the difference between these two types of computer simulations becomes smaller.

Computer simulations for students: working with fewer prerequisites
Computer simulations of mathematical models may make extensive use of the theory of ODEs or PDEs. Simulations of waves in physics are an example of the latter. In academic research, use of PDEs is not a problem, since such underlying theory would be either known to the researcher, or they could learn the relevant aspects of it in reasonable time. For students, the situation may be different: students in the first semesters of tertiary education (university) have not yet learned such theory. Students in secondary education certainly have not, and cannot be expected to do so. Thus computer simulations of phenomena that 2 of 2 Section 24: History of fluid mechanics and history, teaching and popularization of mathematics involve ODEs or PDEs must be carefully designed if they are to be useful for teaching. Note that, while students may not know the theory of equations such as d dt y(t) = f (y, t) or the wave equation ∂ 2 ∂t 2 y(x, t) = n k=1 ∂ 2 ∂x 2 k y(x, t), they can understand the concepts of change, velocity, increase, gradient, amplitude, kinetic energy, etc. which are relevant to the latter equation.

Learning-theoretical considerations
To understand why computer simulations are useful for teaching and learning, we can analyze them with concepts from learning theory.

Cognitive load theory
The cognitive load theory of Chandler and Sweller ( [2,3]) describes the mental processes involved in learning by distinguishing intrinsic load (which is caused by the actual content, i.e. mathematical phenomena), germane load (which includes formation of mental structures) and the extraneous load (unhelpful distractions, which decrease the mental ressources remaining for learning). In this framework, it bemes clear that well-made computer simulations can be useful for teaching and learning of some aspects of mathematical theory, in particular easily observable phenomena such as periodicity, symmetry, wave propagation, and the effects of relevant parameters (such as wavelength in the case of wave simulations). Such observable phenomena can easily be taught to students without prior knowledge of differential equations, and can be demonstrated more easily to students even with such prior knowledge, because simulations may cause particular focus (intrinsic and germane load) on these phenomena. We also see that, for teaching and learning purposes, it becomes important to minimize distractions while using the simulations; e.g. an interactive graphical method for choosing parameters is preferable to having to recompile the source code.

Memory theories
Another learning-theoretical framework are theories of the learner's memory ( [1]). Roughly speaking, the theories distinguish between short-and long-term memories (often there are 3 types: sensory, working, and long-term memory), and they distinguish (the working memory) between visual, auditive, and central executive. Computer simulations of mathematical phenomena can be visually very impressive and thus helpful for learning. They can also encourage the learner to apply previously learned theory and to look for strategies for solving particular problems.

Other aspects of learning theory
A long consideration of reasons why computer simulations may be useful, as well as a long bibliography, is included in [5]; see also [4].