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.