Monday, 16 July 2018

Slice of Advice: What have I learned this year?

I recently submitted a short clip for Craig Barton’s podcast on the theme of Slice of Advice: What have I learned this year?  In this I suggested two things that had had an impact on me this year: the use of graphing apps on smartphones and diagrams representing functions as mappings between number lines.

Graphing apps on smartphones


I’ve been using GeoGebra for over ten years now and it has had a huge impact on how I think about mathematics and teaching mathematics. I’ve also been massively impressed with Desmos though I don’t know it inside-out in the way I do with GeoGebra. My use of these graphing tools, and Autograph before it, was always based on using them on a computer and this means I have a preference for the computer software over the phone apps: I find them much more usable on a large screen with a proper keyboard and mouse.  This works fine for me but, as I’ve commented on here before, I think the greatest impact of technology in the mathematics classroom is as a result of students using it. Consequently I’ve long been an advocate of the use of graphing apps on phones by students, especially in A level classrooms. However, even though I’d promoted the use of phones I had always thought that this was an inferior device for students to be using the software on and that the ideal would be to have them using the software on laptops if it were possible.



What has changed for me this year is that I no longer consider phones an inferior device for using the software for students. Today’s generation of students have grown up with small touch-screen devices and find it completely natural to operate software in such an environment. For many students their phone is the device they are happiest using and so the opportunity to situate such fantastic bits of software as GeoGebra or Desmos in this environment is one that we should be capitalising on. The quality of modern phones is such that the apps work beautifully on them. In particular I think the GeoGebra 3D app on Android phones renders the image better, and is more responsive, than the equivalent computer-based version.

This year we recorded some more PD videos for the Further Mathematics Support Programme – short 5 minute videos that address a particular bit of pedagogy. In one of these we looked at using graphing apps in the Further Maths classroom. In this video you’ll see students using them and hear the teacher comment on how surprised he was that the students didn’t find the small screens an impediment to them using the app effectively. This video, along with questions for reflection, can be seen at furthermaths.org.uk/pd-videos-further-pure#video5 




At MEI we’ve developed a series of classroom tasks that are designed to be compatible with the phone apps for both GeoGebra and Desmos. These can be found at mei.org.uk/geogebra-tasks and mei.org.uk/desmos-tasks.

Representing functions as mappings between number lines


At the BCME conference at Warwick in April I attended a talk by Martin Flashman, a visiting professor from the US, about the use of mappings between number lines for graphical representations of functions. This is in contrast to the usual convention of plotting them on a pair of cartesian x-y axes.

This was an idea I’d seen before but I hadn’t really thought about it in detail. It was a revelation to see how a change of representation could give whole new insights into mathematical ideas I thought I’d understood. In particular I think the representation of composite functions and their inverses is much clearer in this way. I was also amazed at how this could be related to the cartesian system: for example a “linear” function is the set of points that lie on a line; if this is transferred to a mapping between number lines this becomes a set of lines that pass through a single point. What is more, these two representations are the duals of each other!


Martin went on to give a lot more detail to how this system could be extended to represent ideas such as calculus and functions of complex variables represented as mappings between two planes in 3D space. There is a lot more information, including presentations from his BCME conference sessions, at: users.humboldt.edu/flashman/

Overall this has had a wider impact on me than just thinking about representation of functions: it has made me consider how the way an idea is represented impacts on my understanding of it. For example I think setting out long multiplication in a grid helps me focus on the two-dimensional nature of a product and I’m now searching for more such examples…

Thursday, 31 May 2018

Simultaneous equations: an insight from playing with technology


I've noticed a couple of tweets recently about the importance of being playful when doing mathematics and this has brought to mind an insight I had about simultaneous linear equations that occurred when I was being playful with them in technology. As a result of this I now have a different method for solving that I  prefer to the standard textbook approaches.

Playing with simultaneous equations

In trying to construct a pair of linear simultaneous equations where the solution went through a given point. I can't remember the point but I'll use (3,2) for the example. I did this by creating a point A at (3,2) then two more points B and C and finding the lines that went through these and A. This gave me the simultaneous equations:
3x + y = 11
x + 5y = 13


I then moved the points B and C around to find some different equations that would have the same point of intersection. This is probably best displayed here by showing all the different lines I got in a different colour.
 
The equations for these lines are:
4xy = 10
3x + y = 11
2x + 3y = 12
x + 5y = 13
y = 2

By playing around with these I noticed some interesting features:
  • There are infinitely many lines through the point A.
  • The coefficients of these lines are related linearly.
  • One of these lines will be horizontal and one will be vertical (and hence have a simple equation just in terms of x or y).
NB rewriting the final equation as 0x + 7y = 14 makes the linear relationship more obvious.

A simple method for solving linear simultaneous equations 

Putting all this together I realised this could be used to form a simple method for solving simultaneous linear equations. They can be solved by moving linearly along the coefficients until one of the coefficients of either x or y is 0. For example:

3x + y = 11
x + 5y = 13

The gap from 3x to x is –2x, so if you move half this gap again (–x) from x you'll have 0x.


Moving an equivalent amount for the other terms means you need to move half the gap from y to 5y, i.e. +2y, on from 5y to get 7y and half the gap from 11 to 13, i.e. +1, to get 14. 


This gives 0x + 7y = 14 and hence y = 2. Then using the first equation 3x = 9 gives x = 3.

I've tried this method with lots of linear simultaneous equations and I think it's quicker and makes more sense to me. It's the method I know use if I need to solve them.

The importance of being playful

This method only occurred to me because I was being playful with the mathematics in the technology. The dynamic aspect of technology offers a fantastic opportunity to be playful with mathematics as it allows the users to change aspects of a situation and observe the impact on representations (here it was graphical and algebraic but it applies to others such as geometrical or numerical too). The playfulness here is not in the sense of 'lets have fun messing around' but being open to explore a situation and then try to explain what is happening and why.

Friday, 23 March 2018

EEF report: Calculator use has a positive effect on students’ calculation skills

EEF (Education Endowment Fund) have published a report today on "Improving Mathematics in Key Stages Two and Three". The report is a meta-analysis of research into teaching and learning strategies and can be access in full at: https://educationendowmentfoundation.org.uk/evidence-summaries/evidence-reviews/improving-mathematics-in-key-stages-two-and-three/

The report has been widely publicised, mainly for what it has to say about calculator use. Although Key Stages Two and Three are outside my area of expertise I think there are useful reflections that can be made with reference to the use of technology, including calculators as well as other tools, in GCSE and A level Maths.

Calculator use has a positive effect on students’ calculation skills 

The conclusion in calculator use states: "When calculators are used as an integral part of testing and teaching, their use appears to have a positive effect on students’ calculation skills. ... When integrated into the teaching of mental and other calculation approaches, calculators can be very effective for developing non-calculator computation skills; students become better at arithmetic in general and are likely to self-regulate their use of calculators, consequently making less (but better) use of them." 

This emphasises the importance of calculator use being integrated into the teaching and learning (and assessment). As with any technology simply adding in a technology without changing the teaching or activities is unlikely to have positive impact on students' understanding. However, if they are used purposefully, with appropriately designed tasks, this is suggesting that students' calculation skills will improve. The challenge is to design appropriate tasks that take advantage of this opportunity. This point is further highlighted by a suggestion that such tasks can enhance students' problem solving skills.

I think much of this will apply at GCSE and A level too. Some classroom activities that can be tried are:
  • attempting the same problem with and without a calculator and comparing;
  • using a calculator to investigate a function numerically, such as sin(x) or ln(x).

Technology: technological tools and computer-assisted instruction

A separate section of the report discusses technology tools other than calculators. These are split into two categories: technological tools and computer-assisted instruction. I am pleased that this split has been made. I think technology tools for learning mathematics have a much greater potential than computer-assisted instruction.

The analysis looked at three types of tools: dynamic geometry software, exploratory computer environments and educational games. Here the report suggests that dynamic geometry software has huge potential but stresses the importance of how these tools are used if they are to have a positive impact and that there is a need for professional development for teachers to keep pace with this change.

I think there are parallels with the use of technology in GCSE and A level Maths; however, the technology tool that is most widely used (apart from calculators) is graphing software. This is probably due to the nature of the GCSE/A level content: there is a much greater emphasis on the behaviour of functions and understanding them through a combination of graphical and algebraic techniques. For graphing tools it is still essential to use them carefully, and in a structured way, if they are going to have the most impact. This is similar to the use of calculators in that care should be taken to design and use tasks that take advantage of the technology and not just replicate what is done with pen and paper in the software. Detailed consideration is required about how the features of the software, such as sliders, can be used to illuminate mathematical concepts and this then needs to expressed through appropriate tasks (and professional development for teachers).

I would be interested to see a similar analysis of graphing software at GCSE and A level. I have definitely found it more difficult to design classroom tasks for dynamic geometry software than I have for graphing software, though this possibly warrants a different blog post.

It's all about the maths!

Reading through the technology sections of this report I'm left with a strong impression that calculators and other technology tools are useful for doing and learning maths and so should be integrated into students' experiences of the subject. This can then have a positive impact on their understanding and skills. Calculators and technology tools are just one of many strategies that can be employed and they should be judged in the same way as any other strategy - can they help students understand maths better?   

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: http://furthermaths.org.uk/pd-videos-technology

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: https://qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2017/specification-and-sample-assesment/a-level-l3-mathematics-sams.pdf
"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: http://mei.org.uk/integrating-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.