Using ICT to bring Mathematics to Life
Prof. Adrian Oldknow
Visiting Fellow, London University Institute of Education [email protected]
http://adrianoldknow.org.uk
The Mathematical Association http://www.m-a.org.uk
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
www.royalsoc.ac.uk/education/ ) gives an example of the use of the MS Windows Paint
program to collect pixel coordinates from points on the Sydney Harbour bridge, and their
analysis with a graphical calculator.
One tool for collecting data using a coordinate system defined by the user is the free
DigitiseImage program which can be found at
http://maths.sci.shu.ac.uk/digitiseimage/ .
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. TI83) 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 screenshot is from the package
called Vidshell.
Vidshell can be downloaded from: http://webphysics.tec.nh.us/vidshell/vidshell.html
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 Singapore
Harbour. 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 firsthand 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?
If we extract data for the
heights and times of
successive bounces, what
sort of function do you think
will model the data?
N
0
1
2
3
4
5
xn (s)
0,00
0,84
1,68
2,45
3,10
3,69
yn (m)
1,30
1,02
0,78
0,60
0,46
0,37
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 (www.cpd4maths.co.uk ) – 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 Image
free from Jeff Waldock at: http://maths.sci.shu.ac.uk/DigitiseImage/
The Geometer’s Sketchpad
from Chartwell-Yorke or QED Griffin
TI Interactive!
from Oxford Educational
TI-83 graphical calculator
from Oxford Educational
TI Voyage 200 handheld computer
from Oxford Educational
TI Calculator Based Ranger CBR data-logger
from Oxford Educational
Vidshell 2000 free from Doyle V. Davis at: http://webphysics.tec.nh.us/vidshell/vidshell.html