INTERACTIVE MODELS IN TEACHING AND LEARNING

Dynamic models can be used to advantage in all kinds of educational activities. In this chapter, we present a number of examples of effective integration of interactive models into learning materials and class work.

Independent Learning

1. A Classic Textbook

The effectiveness of a (digital) textbook as the main type of resource supporting self-study can be dramatically enhanced by embedding links to models that illustrate its contents. By so doing, we enable students to immediately apply and test their newly acquired knowledge.

2. Mathematics After School

The educational advantages of using our dynamic mathematical models can vividly manifest themselves across a range of scenarios from elementary to more complex mathematical challenges, that are found beyond the standard school curriculum. For example the educational benefits of our dynamic software models have been demonstrated in:
– Recreational mathematics: where there is less emphasis on the formal rigour and grading of a curriculum based approach.
– “Mathematical Circles” which is an innovative Russian outreach concept. Here the emphasis is placed on linking professional mathematicians with secondary school students on a regular basis to solve mathematical problems. Such scenarios are often more suitable for a presentation that uses animation, interactivity, and experiment.

Teacher-Guided Lesson

3. A Model with Guidelines

Most models intended for use in school are accompanied by detailed guidelines for teachers. The guidelines describe the model (its place in the school course, the wording of the task, the model toolbar, action buttons, etc.) and advise the teacher on how to use it. An example of a model with guidelines can be found in this section.

4. A Standard Lesson

Many approaches to mathematical topics call for active student participation with an emphasis on practical experimentation to re-inforce the learning process and make it more rewarding and enjoyable. A perfect, but not unique, example is probability and statistics. MathKit provides a special set of tools for conducting statistical experiments, and we give a detailed scenario of a lesson on classical probability during which a teacher runs a number of such experiments.

5. Guided Discovery: a New Type of Lesson Based on Modeling

MathKit, like many other dynamic mathematics systems, allows the user to conduct experiments and apply the results to discover new facts; actually, this is the prime objective of such software. As a rule, the facts discovered are purely geometric in the sense that one can see them directly by watching the varying figures. Here we illustrate this approach by a less common example of the discovery of a rather complicated formula. In this scenario students are required to start from scratch, create themselves the model to be explored (quite a simple one), as well as the tools for processing the results of their experiment and make their appropriate conclusions.

6. Project Work

The project work described in this section is devoted to caustics, that is, the curves formed by a beam of light reflected in a given curve. There are many practical applications of caustics, for example, in the rendering systems of computer graphics. The dynamic examples we provide start with universally known cases of reflection in a conic with the source in its focus. The source is then moved to produce new curves. A real-life example of a caustic is considered. In the end, a series of problems about reflection and refraction is proposed.