Using ICT to bring Mathematics to LifeProf. Adrian Oldknow Visiting Fellow, London University Institute of Education
Mathematical Association
Abstract: We are very used now to having easy access to digital images and video. These provide a ready means of bringing aspects from the outside world into the mathematics classroom. There is ready available, powerful and inexpensive software for mathematics such as the Geometer's Sketchpad and Cabri Geometry. These tools have powerful applications in teaching geometry (as recommended in the Royal Society/JMC report on Teaching and Learning Geometry 11-19). They now also contain additional graphing tools to support their use in algebra. Still photographs can be imported so that both geometric constructions and graphs of functions can be superimposed over them - allowing us to perform mathematical modelling directly on the image. Such software can be bought using e-Learning Credits from Curriculum Online and should be part of every mathematics department's ICT resources. The talk will show applications from Y7 to Further Mathematics.
The photographs below are snaps from some of my recent travels:
No – this isn’t a travel quiz! But we all have much greater access to sources of digital images than many of us would ever have thought possible, usually in the form of jpeg files. This is due to at least four kinds of technologies: digital cameras, scanners, CD-Roms and the Internet. Now, within the KS3 Framework for ICT, students are expected to be able to manipulate such images with considerable sophistication. But what use can we make of such images in mathematics, and what ICT tools exist to help us do so? One such use is as a source of data – perhaps it seems odd to think about extracting numerical data from digital images! For example p.83 in Appendix 13 of the Royal Society/JMC report Teaching and Learning Geometry 11-19 (which can be downloaded in pdf format from ) gives an example of the use of the MS Windows Paint program to collect pixel coordinates from points on the SydneyHarbour bridge, and their analysis with a graphical calculator.
One tool for collecting data using a co-ordinate system defined by the user is the free DigitiseImage program which can be found at .
The corresponding data file can be opened in a variety of ICT tools, such as a spreadsheet (e.g. MS Excel), a graphical calculator (e.g. TI-83) or an integrated mathematics software package (e.g. TI Interactive! ).
Here we could see if there was some way we could use the coordinate data to estimate the area of the ivy leaf.
Another powerful way of using ICT to bring life into the mathematics classroom is through the use of digital video clips, this time usually in the form of avi files. Again these can be produced directly with digital video cameras, or through movie editing software from video tape, or from DVDs or from the Internet. There are also sources of free software to help us extract data from such clips. The screen-shot is from the package called Vidshell.
Vidshell can be downloaded from:
Again it enables you to define your own origin and scales. Then you can single step through the clip placing coloured dots at points of interest. The software produces a data-file with three readings per point: time, x- and y-coordinates.
Such files can also be analysed with tools such as the Excel spreadsheet, the TI-83 graphical calculator or the TI Interactive!(TII!) integrated mathematics software. Here is an example of a student exercise written as a live document in TI Interactive!
A scattergram can be produced for any two of the three variables, and different models superimposed.
Here are scattergrams of both x against t (horizontal displacement) and y against t (vertical displacement). What information can you extract from them? Also shown are superimposed graphs of a linear and a quadratic function which have been fitted `by eye’. The software also allows us to calculate `best fit’ (regression) linear and quadratic models for our two scattergrams. Of course we could do exactly the same for the y against x graph (the trajectory of the ball). Can you work out the angles at which the ball was released, and at which it entered the basket? How about the velocities?
I want to finish this part of the talk by giving an example of another powerful piece of software for analysing images – this time a piece of dynamic geometry software (DGS) called The Geometer’s Sketchpad (GSP). This is also illustrated in the Royal Society Geometry report (appendix 13, p. 84) Here we have an example of the analysis of the `Merlion’ water spout in SingaporeHarbour. The jpeg file has been copied to the clipboard and pasted into the background.
Here we are using some of the algebraic and analytic tools which are now to be found in the newest versions of some DGS software like GSP. We can superimpose axes over the image, adjust the coordinate scales, read off coordinates and plot graphs of functions. Knowing an actual measurement, such as the height of the fountain (found from the Internet!), we can not only model the trajectory, but also determine the initial velocity of the spout, the range and fall of the jet, the angle it strikes the harbour and even find an estimate for g !
Of course DGS software also allows geometric constructions to be made over images imported in the same way. For example we can test whether the arch in Lisbon is circular or not!
As well as supporting geometric constructions and algebraic graphing, GSP has many facilities for drawing loci, and for performing transformations. Many packages for photo-editing, drawing and CAD include a tool like a `flexi-curve’ based on so-called Bézier curves – which were invented for designers at Renault cars. These can be produced as the locus of a point controlled by a set of points and using a series of dilations.
Of course we do not have to rely solely on secondary data brought into the classroom using digital images. We can capture data first-hand using simple technology like a sensor for finding distances (CBR) with a graphical calculator. Here for example is a screen captured for a bouncing ball. From this could you estimate both g and the coefficient of elasticity e?
N / xn (s) / yn (m)0 / 0,00 / 1,30
1 / 0,84 / 1,02
2 / 1,68 / 0,78
3 / 2,45 / 0,60
4 / 3,10 / 0,46
5 / 3,69 / 0,37
If we extract data for the heights and times of successive bounces, what sort of function do you think will model the data?
For a final snapshot of ICT bringing mathematics to life here is a simulated bungee jumper captured by a CBR (Calculator Based Ranger) connected to a laptop running TI Interactive!
Well, I hope I have made the case that we are not short of good ICT tools, such as digital cameras, data-loggers, graphical calculators and mathematical software to help bring mathematics to life. The question now is what will it take to see them used effectively by teachers and learners? Obviously the stock reply is “money and time”. But things are looking up in this direction.
The current DfES agenda for ICT in schools is `Enhancing Subject Teaching Using ICT’ (ESTUICT). This includes CPD, face-to-face and on-line, provided by The Mathematics Consortium ( ) – with lesson plans, files, resources, guidance, tutorials etc. supported by a new scheme called `Hands On Support’ (HOS). E-learning credits and Curriculum Online provide schools with the means to buy important curriculum software, such as site licences for GSP and TII!. The Standards Fund grant 31a still provides the main source of funds for the purchase of ICT hardware, and now can be used very flexibly – so there is nothing to stop a subject department, such as mathematics, making its case for subject specific hardware, such as data-loggers and graphical calculators. There is much attention being given to Interactive Whiteboards (IWBs) currently. Having worked with the 20 pilot schools for the DfES/RM Y7 MathsAlive! project I am convinced that they have much to offer in the mathematics classroom. However there are cheaper and more practicable ways of supporting whole class interactive teaching with ICT tools such as wireless mice and keyboards, tablet PCs etc. With a little care in budgeting there should be enough funds available to mathematics departments to bring injections of hardware, software and supporting CPD over the next 2 years. In addition the DfES is funding `KS3 Offers’ to teachers in partnership with Subject Association such as the MA – starting with mathematics, science, English and MFL. So come next January there should be a range of supporting resources such as video case studies, innovative & powerful software, sources of digital images, lesson plans and materials winging their way to your mathematics departments.
Ever the optimist, I hope that this joined-up approach, supported by the KS3 mathematics strategy, can bring the sort of excitement into the mathematics classroom that’s needed if we are to get ICT properly embedded in teaching and learning mathematics,
References
Oldknow, A., Geometric and Algebraic Modelling with DGS, Micromath V19 N2 2003
Oldknow, A. Mathematics from still and video images, Micromath V19 N2 2003
Oldknow, A. What would it take to get ICT established in a maths department? Micromath – to appear
Oldknow, A., What a picture, what a photograph, Teaching Mathematics and its Applications V22 N3 2003
Oldknow, A. & Taylor, R., `Teaching Mathematics Using ICT’, 2nd edn., London, Continuum, 2004
Royal Society, Teaching and Learning Geometry 11-19, 2001
Resources illustrated
Digitise Imagefree from Jeff Waldock at:
The Geometer’s Sketchpadfrom Chartwell-Yorke or QED Griffin
TI Interactive!from Oxford Educational
TI-83 graphical calculatorfrom Oxford Educational
TI Voyage 200 handheld computerfrom Oxford Educational
TI Calculator Based Ranger CBR data-logger from Oxford Educational
Vidshell 2000 free from Doyle V. Davis at: