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? -

Wednesday, 19 November 2014

To use or not to use calculators: a false dichotomy

The issue of whether to use calculators when teaching students maths seems to worry a lot of people.  The argument for is often presented as the use of calculators is part of the modern world and, by having access to answers to calculations rapidly, students are more likely to be able to understand relationships.  The argument against is that they reduce students' calculation skills which can impact negatively on understanding and also that they over-rely on them and are unable to spot errors as they believe the number on calculator must always be the correct answer.

I think that this is a false dichotomy that is caused by a misunderstanding of what mathematics is and the low quality of many of the questions that students are asked to do in mathematics classes.

A question

Here's a fairly typical question: what is 36 × 9?

Do you think students should do this with or without a calculator?

My answer is (to both questions) - I don't care!

This is an absolutely pointless question.  What does it matter what 36 times 9 is?  In the absence of any realistic or mathematical context it is meaningless and there is no reason to calculate this product.

The only mathematics worth doing is mathematics that helps us understand the relationship between numbers/shapes or helps us solve a realistic problem.

Better questions:

How much would a worker earning £9 per hour earn for a 36 hour week?

A rectangle has sides length 36 units and 9 units.  What is the area of the rectangle and will it be a square number?

Both of these questions require the product of 36 and 9; however, in both cases the important skill here is to be able to identify that product is required, once a student has identified that the answer will require a product I would hope that they had a range of strategies available to them and that they can use the most appropriate one.


Strategy 1: using a calculator.

In the 21st century most people (at least in the UK) have a powerful calculator to hand most of the time (in the guise of a mobile phone).  It should be part of students' mathematical education to learn this.

Strategy 2: pencil and paper methods

These are often useful; however, I would argue for developing alternative strategies for processes.  I also think it is important to emphasise methods that aid relational understanding over those that are merely efficient.  For example, of the two presented here I think that box method gives a clearer representation of the 2-dimensional nature of multiplication.

Strategy 3:  in your head

If I wanted to multiply 36 by 9 I would probably do 36 times 10 to get 360 and then subtract 36 to get 324.  There are many other strategies that can be used for this but I think it is important to develop a playful nature with numbers through these kind of mental calculations.

Which is the best strategy?

The three different strategies presented here will be useful in different situations - it would depend on the context.  Once a student has correctly identified that a product is needed I would be happy with them using any of these three but I think students should be encouraged to develop their skills in all of them.

Developing skills in using these strategies

For all of these strategies it is important that students develop their skills in using them and, with any skill, the best way of doing this is through practice.  This practice can either be in the form of drills or play.  The problem with much of the mathematical activity that students are asked to do is that the practice of particular skills, such as multiplying a two digit number by a one digit number, becomes an end in itself as opposed to a useful tool to have in one's mathematical toolkit to solve problems.  Over-emphasis on practising performing mathematical operations is boring whether they are on a calculator, pencil and paper or in your head.

If the process of multiplying two numbers by hand is seen as an end in itself then obviously using technology to do this could be considered "cheating".   I would suggest instead that the goal of the mathematics taught to students should always be explicitly about solving realistic problems or understanding mathematical relationships.  If this is the case then where there is a need to develop and practise strategies for performing processes it will be natural to consider both calculator and non-calculator strategies.

In short - if you are asking whether students should be using a calculator or not it's not that you're asking the wrong question it's that you're asking the wrong questions.

Friday, 14 November 2014

Video: Integrating technology into the teaching and learning of mathematics

This a video featuring Charlie Stripp and me talking about integrating technology into the teaching and learning of mathematics, recorded at this year's MEI Conference.