The adage that ‘a picture is worth a thousand words’ is certainly true in the teaching and learning of mathematics.  The widespread availability of free dynamic geometry software (DGS) packages such as Autograph, Desmos and GeoGebra means that maths teachers have incredible illustrative power at their disposal.  Effective use of DGS can aid students in developing a deep understanding of mathematics and this post gives some simple suggestions for embedding the use of technology into your pedagogy.

The current A level specifications are based on the important DfE document ‘Mathematics AS and A level content’ (2014) which states, ‘The use of technology, in particular mathematical and statistical graphing tools and spreadsheets, must permeate the study of AS and A level mathematics.’  In my view, the key word in the above quote is ‘permeate’ which provides both permission and challenge.  The permission means that no longer do we as classroom practitioners have to apologise for using graphing software as some sort of indulgence of a niche interest.  The challenge comes because embedding any effective practice in our teaching is not straightforward, especially when that practice involves significant change.

In this post I focus on how use of DGS can enhance the teaching of pure maths topics within AS/A level Mathematics and will not comment on important issues such as the use of technology in working with the various Large Data Sets which form part of the A level specifications.  In what follows I look at the use of DGS to represent, to illustrate and to question.

1. Representation

One of the keys for students being successful at A level is that they develop the ability to move fluently between multiple representations including algebraic, graphical, pictorial and numerical.  DGS provides the particular opportunity for students to connect the abstract algebraic with the concrete graphical representations. 

For example, displaying the graph of  a along with various algebraic representations can lead to a discussion where students identify various features of the function by connecting an algebraic representation with the graphical. 

 

Alongside the display prompts such as these could be offered:

• Explain why these are all equivalent.
• What are the key features you can see in the graph?  Which algebraic representation highlights the same feature?
• What does the graph tell me? What does the algebra tell me?

Such prompts would lead to a discussion of features such as the co-ordinates of the vertex, equation of the line of symmetry, locations of the roots, and the value of the y-axis intercept.  Discussion could also include relating this function with and drawing out the similarities and differences.

Virtually every pure maths topic in AS/A level maths provides equivalent opportunities for students to move between algebraic and graphical representations, and digital technology opens the door for this to be done effectively.

2. Illustration

It is too easy for AS/A level students to become manipulators of algebraic equations without giving thought to what they are doing leading them to become expert processors rather than thinking mathematicians.  This is a ‘Starter’ activity I saw in a classroom recently:

Q1: Simplify           Q2: Solve simultaneously.  .

Students confidently tackled these algebraically, and there is an argument that this is sufficient since it what they would need to do in an exam.  However, this is an example of where simply illustrating these algebraic questions graphically would have enhanced students’ appreciation of what they were calculating and caused them to pause and think for a moment: 

 

Illustrating the questions students are working on helps to provide them with an image to go alongside the algebraic prompt they are given.  As described above this gives them another representation to consider, but more than that, it helps them to develop a more rounded view of mathematics.  This does not need to be an onerous or time-consuimng activity which eats up valuable classtime.  It simply means having the software available and typing in the equations as the questions are being discussed and then being prepared to use the illustration as a basis for discussion.

Additionally, using technology to illustrate rather than my dubious sketch on the whiteboard means that the image the students are looking at is accurate.  It means that tangents appear as tangents, and intersections actually appear at the correct locations!  The illustration then becomes helpful rather than a distraction.

3. Question

A display involving digital technology provides the opportunity for asking questions which promote deeper mathematical thinking.  This is where the dynamic nature of software packages can be utilised and tools such as sliders and constant controllers come into their own.  Students can be asked to consider questions such as, ‘What happens if I change the value of this coefficient?’ or ‘What do I need to change in order that such-and-such happens?’

For example, this GeoGebra file shows a circle with centre and radius which can be adjusted using sliders.  I like to ask, ‘Change one slider so that the circle lies in exactly three quadrants.’ 

By having the digital display it means that students can reason geometrically as well as algebraically and can ‘see’ the effect a changed value can have.  It means that any conjecture they offer can be quickly checked.  In this question, the surprise the students express as they see a vary and generate another set of solutions they hadn’t spotted previously is well worth it and shows how the use of a dynamic image can stimulate ‘light bulb’ moments.

Use of sliders such as this also means that multiple examples can be viewed quickly in order to enable students to get a sense of the collection of objects with a particular property.

There is so much more that could be written about this important pedagogical theme of effective use of technology.  However, in closing I’d like to mention a couple of points.  Firstly, be aware of the difference between static and dynamic images.  Both are important and both have their place in the classroom.  I often find myself starting with a static image, asking students what they notice and then asking them to consider what might change or vary under certain conditions.  Then I might reveal the dynamic image to see if their conjecture was correct.  On other occasions I might ask students to observe a dynamic image and describe or explain what they are observing.

Secondly, the battle for effective use of technology in your lessons is fought at the planning stage.  During the planning of sessions I ask myself questions such as, ‘What does this maths look like?’, ‘Is there a way of using DGS to show this?’, ‘What different representations are available for this maths?’ and ‘How can students access these representations?’  If I don’t plan for something like this to happen, then I can guarantee it won’t happen!  In contrast, if I spend time carefully considering how use of DGS can shape the lesson, provide a prompt for stimulating discussion and form a basis for questioning then I am much more likely to use it.

Now, just because the battle is fought during planning doesn’t mean it is won there!  There are still issues such as the reliability of internet connections, access to computers and projectors, receptivity of students to having to think more deeply about maths, confidence in using the software, and a whole host of other issues.  These are important issues to consider.  However, it is my belief that the majority of students (and people in general) find images compelling, intriguing and attractive and therefore introducing such stimuli into our lessons will generate interest and motivate learning. 

It can be daunting to embrace and incorporate use of technology into our lessons, but I think we owe it to our students to try.