Friday, 26 January 2018

Using graphing software for multiple representations (FMSP PD Video)

In March of 2017 I was involved in creating a set of videos looking at aspects of using technology in A level Maths teaching. The full list of videos can be found at:

The first of these videos is on using graphing software for multiple representations.

The video features an example of me using graphing software to highlight the link between graphical, algebraic and numerical representations so students can understand the ideas behind differentiation.

Questions for reflection

The video suggests three questions for reflection:
  • What topics would you use graphing software for?
  • What are the advantages of using prepared files?
  • What questioning strategies are effective when using graphing software?
Here are my responses to these questions:

What topics would you use graphing software for?

When teaching A level Maths, especially Pure, I can't think of a single topic where using software to display multiple representations won't enhance students' understanding. This video shows how graphical, and numerical representations can be used to show connections for differentiation but similar could be used for coordinate geometry, trigonometry, series, integration, vectors and almost all other topics.

When using multiple representations the numerical representation is often considered as much as the link between algebraic and graphical but it is a very powerful representation for students to hang their understanding on. A table of values is very concrete and easily understandable for students. When first meeting graphs or functions this will often be the representation that students start with and so referring back to it in more advanced topics will help them appreciate how these topics have be constructed by extending ideas they are familiar with.

What are the advantages of using prepared files?

For teachers who are not as confident with using software it can be very reassuring to use a file that has been prepared in advanced. If you know that there are only one or two sliders or points on the screen that you need to move, and everything else will be displayed, then there is less chance that it will go wrong or that the software will display something strange. Although I am now confident to produce files "on the fly" with software, building them up as I go along, I did not start out with this confidence. For me, using prepared files was a necessary first step in building my confidence in using technology in front of a class of students and I would recommend this to other teachers starting out using technology. 

There are still cases where I use prepared files now. This is mainly when I want to emphasise a certain concept and the construction would take too long, and be an unnecessary distraction, to do it live.

What questioning strategies are effective when using graphing software?

One of the best questions to ask, especially when an object is moved in dynamic software, is "Describe how this object is moving and explain why is is moving in this way". The connections between the representations are often best expressed in terms of how a change in one representation will affect a change in another one. In this example here I used the change in the gradient of the tangent to the curve as the point moved along y=x² and y=x³. For the latter of these it helped the students see that the gradient would always be positive (or 0) and that this was consistent with an algebraic version of the derivative dy/dx=3x².

As a teacher it's a common to hear the observation that you understand something better if you have to explain it yourself.  "Describe how this object is moving and explain why is is moving in this way" is an opportunity for the students to engage in the activity of explain a mathematical relationship which can improve their understanding. This question works especially well because an object is moving - it feels very natural when observing the motion of an object to want to explain why it is moving in a certain way.

An alternative way to ask a similar question is to tell student that you are going to change something and ask them to describe how the objects will move before you do so. You can then check the answer. Here's one to try:
  • Plot y=x² and y=bx+1.
  • Describe how the midpoint of the points of intersection of the two curves will move as b is varied.

Friday, 16 June 2017

What can the mathematics education community do to increase the use of digital resources by KS5 teachers

I was recently asked the following two questions in an email. 
  1. Teachers: What can the mathematics education community do to increase the use of digital resources by KS5 teachers?
  2. Students: How can time be found in the KS5 teaching programme to bring up the students’ skills in digital resources?
I've reposted my response here:

1. Teachers: What can the mathematics education community do to increase the use of digital resources by KS5 teachers?

I have a model for this of 10:80:10 for teachers who are really interested in technology/will use technology if it helps/will rarely use technology.  I think the most effective strategy is to concentrate on the middle 80%.  For these the single most important criterion to judge any resource on, including a digital one, is whether it is the best tool to help students understand the concept.  Many of these teachers don’t have the time to learn complicated software, such as Mathematica, but would use a graph-plotter in front of students if it’s user-friendly and demonstrates the mathematics better/more efficiently.  The important role for the technology enthusiasts in the maths education community is to communicate to teachers how some very easy to use digital tools will have a positive impact on their students’ understanding.

This needs to be done on an almost topic-by-topic basis.  I’ll flesh it out with a couple of examples – the details aren't important - it’s more to demonstrate that for me there occasions where a graphing tool is simple, and adds something significant, and others where it isn’t:
  • I would definitely use a graphing tool early on in students’ learning of differentiation.  The concept that the gradient of the tangent to a curve varies and that the numerical value can be expressed as a relationship to the point on the curve is something that it significantly quicker to show in a graphing tool. Additionally, the dynamic element of seeing the tangent moves gives a very physical representation of why we are considering the relationship to where we are on the curve.  Pencil and paper or purely algebraic approaches here are markedly inferior (I’ve tried them with students in the past!).
  • I probably wouldn’t use a graphing tool for the initial learning of binomial expansions.  I’m aware I could do some expanding using technology (such as CAS) and/or graphing but I this is a bit fiddly and it’s not going to convince many of the teachers in the middle 80%.
So our role is to demonstrate these ideas for appropriate cases and show teachers how easy various tools are to use.  We need to be cognisant of the fact that many teachers are directed to be assessment driven and consequently tie this way of teaching into how it better prepares students for the assessments.  One thing I’ve been doing recently is drawing teachers attention to questions such as qn 3 on paper 1 of the Edexcel sample A level:
"A circle C has equation x²+y²–4x+10y=k …  State the range of possible values for k."
This is one where seeing it vary dynamically gives a really good way in to the question so teachers are tempted to use something dynamic in the classroom.  I also think we need to meet the challenge head-on of teachers saying that they won’t have a computer with a graph-plotter in the exam by pointing out that they do lots of things in the classroom that aren’t allowed in the exam: asking the teacher a question, discussing with other students, checking the answers in the back of the book …

All of this is an attempt to answer the question: “Why should KS5 teachers use digital resources?”.  If we have a clear answer to this then it will be easier to suggest what we should do.

2. Students: How can time be found in the KS5 teaching programme to bring up the students’ skills in digital resources?

This is a trickier question to answer.  I’m personally convinced that students’ understanding of mathematics can be enhanced by using digital technologies for mathematics.  I’ve observed in my own learning, as well as students’ learning, that the process of constructing mathematical objects in digital technologies very closely models the internal process of constructing the understanding of mathematical relationships.  It requires rigour, the ability to understand how objects relate to each other (such as through dependencies) and an understanding of the nature of different mathematical objects.  There are many times where I have found constructing something in mathematical software has given me an insight that has allowed me to solve a problem.  There is a democratising element to digital tools – they remove many routine aspects allowing time to focus on the bigger picture.  This is evident with the way that numerical calculations are not a barrier to understanding when one has a calculator (and also in music technology too where it’s possible to create tunes without much formal musical training!).

So this question again comes down to the “Why?”.  If we present this as just something that’s needed because the curriculum "says so" then the sensible response would be to do the bare minimum.  Again, as with question 1, I think it is our role to present opportunities where students will learn more efficiently and in a deeper way, by using digital technologies themselves.  If we can do this then the time question becomes irrelevant.  One thing I’ve seen recently that makes me think this is possible is the improvement in smartphone maths apps in the last year or two.  I’ve observed students  using Desmos on their phones in class and ease of use, the mathematical structure of the software and the fact it was on a device that was personal to the students meant that they engaged with it and were able to very quickly explore ideas that deepened their understanding.

Within MEI we’ve been attempting to map the use of technology to the new A level curriculum including lots of examples of tasks focussing on student use of technology:  

Thursday, 19 January 2017

The new Maths A level: Graphing families of curves

Last week the MEI new A level for Maths was accredited – the first full A level to be accredited of all the specifications.  The specification includes advice on using technology and the sample assessment materials have questions which lend themselves to the use of technology when teaching the topic.

Use of graphing tools for families of curves

The MEI specification includes guidance for activities that should be carried out during the course.  The first, and possibly most important, of these is:

"Graphing tools: Learners should use graphing software to investigate the
relationships between graphical and algebraic representations, e.g. understanding the effect of changing the parameter k in the graphs of y = 1/x + k or y = x² kx "

The ability to plot a family of curves, and observe the effect on the graphs of dynamically changing a parameter, is an incredibly powerful tool in helping students understand mathematical relationships.  Understanding how a mathematical object changes is greatly enhanced by considering how its graphical representation moves.

An example from the sample assessment materials

The sample assessment materials include the following question:

Determine the values of k for which part of the graph of y = x² kx + 2k appears below the x-axis.

Plotting this in GeoGebra gives the option to vary k to see how the curve changes. 

This immediately gives some insights into what is happening:
  • How the graph changes for different values of k
  • The graph is sometimes above the y-axis for all values of x
  • For many values of k the graph will be below the x-axis between two values of x

Using the software does not answer the question for the students but it does give them a picture which they can use to understand what the question in asking them.  I think this is Ofqual's intention with their statement that the use of graphing tools should permeate the study of A level Maths - these tools are easily accessible for all students and can have a massive positive impact on the way they build their understanding of mathematical objects.

Wednesday, 13 April 2016

Use of Technology in the new A level Mathematics qualifications

Last Friday (8th April) the DfE published the GCE subject-level guidance for mathematics.  This guidance is for awarding bodies to help them in designing their specifications and assessments.  The full document can be found at:

Requirement for awarding bodies to explain how use of technology will permeate the study of mathematics

In the Overarching themes and use of technology section:

Paragraph 8 of the Content Document states that –

8. The use of technology, in particular mathematical and statistical graphing tools and spreadsheets, must permeate the study of AS and A level mathematics.

This statement should be interpreted primarily as indicating the desired approach to teaching GCE Qualifications in Mathematics.

However, this statement also has implications for assessments. Consequently, in respect of each GCE Qualification in Mathematics which it makes available, or proposes to make available, we expect an awarding organisation to explain and justify in its assessment strategy for that qualification how this statement has been reflected in the qualification’s design.

I think this is very good news in terms of the design brief given to the awarding bodies and, if it applied in the way it is intended, should result in greater and more effective use of technology in the A level mathematics classroom.  I look forward with interest to seeing how the awarding bodies justify that their assessment strategies are ensuring that technology permeates the study.

Strategies I would like to see

There are two main strategies that I would like to see employed: an explicit and an implicit one.

I expect to see questions that explicitly refer to the use of technology.  This could be through means of a statistical test that a candidate would perform on their calculators or by referencing spreadsheets in the questions.  It will be clear to teachers that in order to prepare candidates for the assessment they should be using technology in the teaching and learning.

In addition to this I would like to see questions where, although there is no requirement for the candidates to use technology in answering them, they will be better prepared for them if they have using technology in their studies.  For example a question asking a candidate to explain the impact of the parameter b on the graph of y=x²+bx+4 is likely to be answered better by students who’ve been using graphing tools to explore curves in their study.  This is an implicit strategy but can still be very powerful in encouraging use in the classroom.

I look forward to seeing the specifications and sample assessments when they are produced!

Tuesday, 15 March 2016

Maths on a Smartphone

I recently gave a talk about doing Maths on a smartphone.  I chose four of my favourite apps.  I like these apps because in all of them there are opportunities to think and work mathematically, not just passively observe prepared material.

MyScript Calculator

MyScript is really easy to use – you just write the calculations with your finger.


Problem to try: What’s the maximum product of a set of positive numbers that sum to 19?


Desmos is a very user-friendly graphing calculator.  In my experience most people find the interface intuitive and are able to work with it very quickly.


Problem to try: What’s the effect of varying a in the graph of y=x^3+ax+1 ?


GeoGebra is a very powerful mathematical package that I’ve discussed many times on here.  Currently the app is available for Android but not iOS.  You can see my thoughts on the app at 

In addition to the app you can also open files from the extensive set of materials at using a browser.  At the time of writing there are over 360,000 materials on there.

iOS: not currently available for iOS phones

Problem to try: Add the points A and B on the x-axis and C on the y-axis.  Find the equation of the quadratic that will always go through A, B and C wherever they are moved to.


Sumaze! is a mathematical puzzle app that requires you to move a block around a maze with various routes involving operations or restrictions on the value of your block.  It’s a great puzzle and features lots of maths including arithmetic, inequalities, the modulus function, indices, logarithms and primes.


Thursday, 14 January 2016

GeoGebra App for Android phones

GeoGebra have recently released a version for Android phones. Having played around with it it seems very responsive. The ability to select/drag objects and the speed that it updates appears to be really good - much better than when viewing GeoGebra worksheets via a browser.

An example: gradient of the tangent to a curve at a point


This is an example that shows how the gradient of the tangent to a curve at a point varies with the point.  The app is so quick and easy to use that this took me 17 seconds to create (I timed myself!).

Use of smartphones in classrooms

This app presents a fantastic opportunity to put dynamic maths software into the hands of students.  As I've commented on before, I think the real benefits of technology come when students are using it.  In addition to this there are significant advantages when this is on a device that students have an attachment to and feel ownership of.  Most people feel their own phone is a device that is very personal to them and this means students are more likely to be well-disposed to software on it.

Having GeoGebra on their phones means that students can harness the power of the software wherever they are: at home, on the bus, ...  However, many teachers have reservations about students using phones in class.  There are concerns that this presents a major classroom management issue.  This is an ongoing debate and there is some evidence that banning phones  in schools has a positive impact on achievement: A lot of these arguments focus on general mobile phone use in class and it would be interesting to see some experiences based on students using subject-specific apps such as GeoGebra with effective tasks designed to improve their understanding using the software.

Downloading the the Android app

The Android app can be downloaded from:

Friday, 24 July 2015

GeoGebra Global Gathering 2015

Last week I attended the GeoGebra Global Gathering in Linz, Austria.   There were lots of fantastic ideas being demonstrated but these are a few of my highlights that I think I’ll be making use of …

Students creating animations

It was great to see some ideas from Fabian Vitabar from Uruguay about student tasks that involve them creating animations in GeoGebra.  There were two different suggestions for doing this – one was to get the students to create an animation such as a bouncing ball by animating points appropriately and then adding images to make the animation look like a real scene.  The other was to create some more pure maths based animations such as moving points around a polygon.  You can see a GeoGebraBook of his talk and some examples at:  I think it could be very motivating to students to create animations like these and there is a lot of maths that they’ll need to sort out for themselves to get them to work correctly.

Designing GeoGebra Tasks for Visualization and Reasoning

Anthony Or from Hong Kong gave a fantastic talk on Designing GeoGebra Tasks for Visualization and Reasoning.  You can see a GeoGebraBook of his talk at  There’s much more in it than I’ll be able to do justice to here but the main theme was linking the process of students constructing objects in GeoGebra to enhancing their mathematical reasoning skills.  One useful idea was the contrast between robust and soft constructions: a robust construction maintains properties when objects are dragged and a soft one doesn’t but can be used for investigating.  I’d not really thought about emphasising soft constructions as a way in to robust constructions before but I think it could be a useful technique. A final highlight from this session was the challenges section at the end.  Constructing an equilateral triangle on a set of three parallel lines is a particular favourite – I managed to solve it on the train journey back from Linz to Vienna!

Problems that Challenge Intuition

Diego Lieban from Brazil had some examples of very nice problems that challenge intuition:  The first one about the shape of a net of a tube when sliced diagonally is particularly tricky as it’s hard to make the cut on a real cardboard tube without creasing it.

New features on their way – Phone apps, Badges and Groups

Some upcoming features were presented that are very exciting.  The work on the phone apps is developing and there is a beta version of the Android one available for testing now.  I’ve tried this and I’m really impressed – it’s very responsive when objects are dragged and, contrary to what I was worried about, it seems to work well on such a small screen when selecting and dragging objects.  Badges are coming soon for GeoGebraTube accounts – these will automatically display when users have created specific objects or used specific tools/views.  I can see these as being very effective in encourage people to continually develop their skills.  The last feature that I’m excited about seeing is being able to create groups of GeoGebra users.  This will be especially useful for Professional Development workshops – it will mean that teachers from these can form a network and continue to support each other afterwards.

Smart Board software

I’d been aware for a while that GeoGebra was built-in to Smart Board software but hadn’t seen it in action.  The ability to hand write an equation of a curve and then drag this into a GeoGebra widget looks like a handy tool and one that I plan to investigate further.  There’s more instructions online about how to do this at: 

Other sessions

There were lots of other great sessions and plenaries at the gathering.  A full list of GeoGebraBooks for the talks is at:

Ben Sparks and I gave two sessions too:
Professional Development for practising teachers including live online sessions -
Creating effective teaching resources and using GeoGebra in examinations? -