Digital technologies for learning mathematics

An earlier version of this article appeared in the Mathematical Association Journal: Mathematics in School (November 2020). In July 2019 I was fortunate to be able to speak at the GeoGebra Global Gathering in Linz, Austria. I gave a presentation about how allowing students to use mathematical software on their own devices, both to support their learning in the classroom and as a tool they can use within assessment, is a potential solution to: Harnessing the power of technology to improve the teaching and learning of mathematics Increasing equality of access to technology for mathematics students. This article is a summary of the talk I gave there. Why should students use technology in mathematics? When I first started teaching there was an activity that I used with students to give them a strong conceptual understanding of differentiation and what the derivative of a function expresses. The activity was for them to plot the curve for y=x² on graph-paper, draw some tangents at diff...

One of the fantastic features of dynamic graphing software is that you can animate how a graph changes as some aspect of it is varied. I'm particularly keen on making short GIFs of animated graphs that demonstrate interesting properties.  One advantage of animated GIFs is that they are really easy to distribute via Twitter. There are loads of them if you have a look at the hashtag #mathgif .  Parabolas I've been posting animated Maths GIFs to Twitter for quite a while now. Here are some of my favourites featuring parabolas: The midpoint of the points of intersection of y  =  x ² and y  = mx  + c lies on the line x = m /2. The tangent at point P to a parabola with vertex Q can be constructed by finding R, the point of intersection of the vertical through P and the horizontal through Q, finding the midpoint, M, of QR and then drawing the line through P and M. If the tangents at two points A and B on a parabola are perpendicular then the line AB goes through t...

The past few months have been a very chaotic time for teachers and students. Many schools and colleges are now back to teaching full time but this year is likely to be very challenging and teachers want to know that they can provide opportunities for students to learn, even if the normal day-to-day routines are disrupted. For me, one tool stands out as particularly effective in ensuring that students have interesting and engaging activities that will develop their mathematical understanding: Desmos Classroom. Desmos Classroom Activities  Desmos Classroom activities are collections of screens that provide students with the opportunity to interact and respond to mathematical questions. They can feature text answers, mathematical answers, graphical features, multi-choice, card sorts and many other types of elements.  There is a large selection of pre-made activities and it's really easy to adapt these or create your own. These activities are great for supporting students in think...

Most graphing tools have a feature to add a variable constant in a way that can be changed dynamically. In Desmos and GeoGebra this is by using sliders; in Autograph it is via the constant controller. When the parameter is changed you can view its effect on the graph and this can be discussed with students. You can even animate the change so you don't need to do it manually! This feature of being able to dynamically change a variable constant is a very powerful tool to help students build their understanding; however, to make the best use of it, it is useful to have some questions to ask to direct students' thinking. Effective questions when using dynamic graphs When I vary … how does it move and why? For what value of … will …? If I vary … how will it move? Examples using the intersection of a line and a parabola To exemplify this I'm going to use the intersection of the parabola y = x 2 with the line y = mx+c . To do this I've plotted: y = x 2   y...

My article for NCETM: "Ten calculations that A level students should be doing on a calculator: can yours?" can be read at: https://www.ncetm.org.uk/resources/54231

I've been asked again this year to contribute to Craig Barton's end of year podcast: Slice of advice - what have I learned this year:  http://www.mrbartonmaths.com/blog/slice-of-advice-2019-what-did-you-learn-this-year/ The main thing that I've learned this year is how few teachers are familiar with Richard Skemp's ideas on instrumental versus relational understanding. I meet a lot of teachers in my job and I will often have only one or two teachers in the room who have read it. These ideas have been fundamental to my development as a mathematics educator and I see it as the key concept that is most illuminating when considering ideas about teaching and learning in maths. It is also one of the main reasons why I am so passionate about the use of technology in maths. Instrumental versus relational understanding Skemp published his ideas in the 1970s but they are still available via the ATM website at:  https://www.atm.org.uk/write/MediaUploads/Resources/Richard_Sk...

In 2011 I completed an MA in ICT and Education through the University of Leeds. My critical study was on "Using Dynamic Geometry/Graphing software to set students open-ended mathematics investigations". It can be downloaded from:  https://drive.google.com/file/d/1pgFvIjkuEQRpzUm4uJ_KZ1mnql6UZfJ9/view?usp=sharing

Over the past week I've been exploring CloudReady from Neverware. The idea behind it is that it takes an old laptop and effectively converts it into a Chromebook. I took an old Windows laptop (I'm not sure of the exact age but I think it was about 10 years old). It was running so slowly that it was practically unusable. I replaced it with their operating system, which is based on the Chromium OS, to turn it into a Chromebook. The process was pretty straightforward - it required me to make an installer on a USB stick and then I ran the installation. Overall it took less than an hour but it would be quicker if I were to repeat it. It was a full replacement of the OS, but given that the laptop was destined to be disposed of, this wasn't a problem. I've now got a fully working Chromebook and I'm really impressed with it. It runs really quickly - the browser is at least as fast as my current Windows laptop and I can watch/stream videos on it clearly. You can see a pi...

My article on Why Smartphones are a Really Useful Tool in the Maths Classroom has been published in Teachwire https://www.teachwire.net/news/why-smartphones-are-a-really-useful-tool-in-the-maths-classroom

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 ...

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: 3 x + y = 11 x + 5 y = 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...

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 teachi...

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 w...

I was recently asked the following two questions in an email.  Teachers: What can the mathematics education community do to increase the use of digital resources by KS5 teachers? 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 fron...

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 = x ² – 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 mathemat...

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: https://www.gov.uk/government/publications/gce-subject-level-guidance-for-mathematics 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...