As part of the teaching of one of our maths subject content modules, a variety of software packages were used in demonstrating concepts to the students, which were then subsequently used by the students in preparing their submissions for the assignment. Technology used included Geogebra, Desmos, spreadsheet software, programming languages and visualisation applets, and as well as being illustrative for learning, they allowed for a rich exploration of topics which gave students more opportunities to demonstrate their understanding.

Motivation

It is possible to study dynamical systems as abstract ideas, using algebraic equations and writing down the results of iteration of functions. However, many of the more interesting and mathematically rich aspects of the topic lend themselves well to computational methods. This can be either through generating visualisations of aspects of the behaviour of functions, or through calculating many iterations of a function – which would be too slow and error-prone to calculate by hand – to explore longer-term behaviour.

Several different software approaches were introduced during class teaching in order to illustrate concepts, with time given for exploration and experimentation by the students.

Already being familiar with Geogebra, this gave them a chance to learn some new applications of the software (since they might previously have mainly used it for geometric constructions) and to explore some of its more advanced features. They may have also seen Desmos before, which meant they would be familiar with the interface.

Other software that was used was less likely to have been seen previously by the students, but it was introduced in a way that let them explore and experiment, and used as part of specific computational sessions within the module in which they were given questions to answer but not specific instructions on how to do this using the software. This meant they were able to build confidence and adapt their understanding to the problems they chose to tackle for the final assignment, and in some cases students went beyond the techniques covered in class and used other software packages to generate their own visualisations, when existing applets were not available.

Approach

The module Infinite Processes runs in the third year of the maths education degree, and covers topics including fractals, dynamical systems and chaos. The assessment for the module is in the form of a two-part investigation, one studying a specified simple two-dimensional family of dynamical systems, and the other a free investigation covering a topic chosen from a range of starting points, including fractals arising from complex quadratic maps, behaviour of maps on the 2-torus and variations on the Tent map.

During the module content teaching sessions, various software platforms were used to introduce and explore these topics, which the students could later make use of in conducting their investigation, and for generating images and data to include in their writeup. The marking grid for the assessment includes specific marks for ‘use of digitally produced diagrams and computational tools’, rewarding fluent use of advanced computational tools and clear, illustrative digitally produced diagrams.

Software used included:

• Geogebra, dynamic geometry software, which was used for generating visualisations of the behaviour of functions, and allowing parameters in the family of functions to be varied (using a slider) to explore how this changes behaviour. This was used by students to study a two-parameter family of maps, generating a plot of the image of the function for a given pair of parameter values, to identify fixed points and periodic orbits. Geogebra implementations were also used as part of the taught content to illustrate concepts like bifurcation diagrams, iteration, Julia sets and other fractals, to help develop a better understanding of the nature of these types of visualisations
• Desmos, graphing calculator software used to demonstrate cobweb plots
• Spreadsheet software, including Excel and Google Sheets, to iterate functions, demonstrate the existence of attracting fixed points, record data and perform simple calculations
• PARI-GP, a computer algebra system (programming language) designed for fast computations in number theory, which can be accessed via a web browser and given code samples run, adapted and rewritten to address specific problems; used to iterate functions and generate precise numerical values when spreadsheets were not accurate enough
• A variety of online applets to generate visualisations of fractals, including Mandelbrot sets, Julia sets, bifurcation diagrams and isoperiodic diagrams.

Students were encouraged to use these packages to create plots to illustrate their investigation, to find particularly interesting parameter values to study, and to explore the behaviour of functions. This type of free investigation is something students generally tend to struggle with, since it requires a high level of mathematical writing skill combined with a broader and deeper understanding of the content. Being able to explore using the software allows for investigations to be conducted more quickly than they would using hand-calculations, and findings from computational exploration can be backed up with algebraic working-out. It also allowed for students to create illustrative plots which backed up their written work effectively, and many students made good use of this.

Outcomes

Students produced well-written, creative investigations which showed their understanding of the material and included illustrative diagrams, feeding into their development of mathematical communication skills

Working with the software in class allowed them to develop confidence and move on more easily to their own explorations later in the module, equipped with effective tools to do this

Allowed students to see new ways to use tools they were already familiar with, including Geogebra and spreadsheet software, which they could implement in their teaching practice

Discussion

Some students found the technological aspects of the module easy to grasp, as there is a reasonably high level of technological literacy, but others needed more support with understanding the software. There are also aspects of visualisations of some dynamical systems concepts which can be difficult to grasp, and to understand exactly what a visualisation is showing.

For example, the Geogebra visualisations of the 2D dynamical systems were showing only a slice through the space of points (which is actually four-dimensional), plotting the behaviour for the x and y coordinates as two separate lines, meaning that fixed and periodic points could be found where these lines intersect.

Similarly, visualisations like bifurcation diagrams are plotted across a parameter space rather than a space of possible input values, and getting these ideas across to the students was made much easier by having access to interactive visualisations – and in Geogebra, building the plots by typing in the equations made a real tangible link between the algebraic interpretation of the system and its visual presentation.

The choice to use PARI-GP was mainly for practical reasons, since it has an in-browser implementation and the code snippets used to implement the function iteration were pre-existing from an earlier version of the course; if I were to run this module again, I would prefer to rewrite these parts of the course to use a more widely used programming language such as Python, which might be of more use to the students in other aspects of their work and in future.

Resources

Cobweb plots in Desmos
Video of Ben Sparks constructing a bifurcation diagram in GGB
Bifurcation applet (Khan Academy)
Hénon Map in GGB
PARI-GP
Isoperiodic diagram visualisation