ICT in Teaching and Learning Mathematics Adrian Oldknow April 2004 a_oldknow@compuserve.com www.adrianoldknow.org.uk

This year sees the 20th anniversary of the ATM’s Micromath journal and also of the original little Apple Macintosh personal computer. I’m glad to say that the former is still going strong, as is the influence of the latter! In this session I want to try to draw on some of the lessons we have learned in these twenty years in the context of the new range of ICT tools available to us and the framework for funding and support which is emerging. More than 20 years ago the Shell Centre at Nottingham University, with St. Mark & St. John Plymouth, had an extensive project on the use of one computer in the mathematics classroom. Since then we have had the Cockcroft report, the National Curriculum, the Key Stage 3 Strategy and now the Smith report. Not for the first time we are now seeing the pendulum swinging back to ICT to support whole class teaching!

1. ICT hardware to support interactive whole class teaching.

These are conventionally based on a computer and a projector. Either or both can be portable or fixed. If you use a PC it is helpful to be able to split the video signal (VGA) so that the output is visible both on the normal PC monitor screen as well as through the projector. If you use a laptop there is normally some combination of keys (like Fn and 5) which cycles through the display being shown on either the laptop’s screen, or the projector or both. Projectors (called `video’ or `data’) have become items of home entertainment and are now widely available in high street stores. They can be permanently mounted (preferably from the ceiling) or portable. They usually have several type of input sockets – the one illustrated has VGA (for computer), video and S-video sockets. If the projector is ceiling mounted then you will need the remote control to switch between inputs, and you will need suitable wiring from the projector to matching sockets in a convenient position to attach the computer and other devices.

You can project the display onto an Interactive WhiteBoard (IWB), an ordinary whiteboard, a clear patch of wall or a screen. Each have their pros and cons! There are two main kinds of IWBs: analog and digital. An analog board, such as the Smartboard illustrated, detects the presence of any object which comes in contact with it – such as a finger – whereas a digital board, such as a Promethean ActiveBoard, requires a special stylus.

Each kind of IWB comes with its own pack of software tools. These enable you to use the board like a `flipchart’ and save pages of written notes, drawings etc. They also provide useful `tools’ such as graph-paper, grids, number lines, music staves etc. If you are using some other software, such as a graph-plotter, you can use the IWB software to annotate on a virtual `overlay’. If you have a permanently mounted board, or one on a wheeled stand, you will have to give careful thought to the height to mount it so that all potential users can reach it, and all potential viewers can see it. With an analog board, shorter users can be given something like a drumstick to enable them to reach parts of the board. The IWB software will provide a `registration’ tool which you should use from time to time to make sure the IWB `knows’ where you are pointing.

Whether or not you use an IWB for the display, there are other – cheaper and possibly more convenient - means for people in a class to interact with the computer than by standing at the board. The Compaq laptop illustrated below is one of the new generation of so-called `tablet PCs’. These are like ordinary laptops except that the screen is also able to sense the position of a stylus – and so they work as a sort of private IWB – including their own built-in software which converts hand-writing into text! Connected to a projector they allow anyone within cable reach of the projector to control the whole class display. Quite soon there will be wireless data projectors which will enable a tablet (or any other) PC to be used anywhere in the classroom. A much cheaper wireless alternative is provided by buying a wireless mouse or a wireless keyboard/mouse set. With the tablet PC illustrated is a Gyration Ultra mouse/keyboard set. The wireless transmitter is connected to the USB port. The mouse and keyboard can be used within a range of about 10m from the transmitter. This way the computer and projector are cable-linked and stay fixed, while the mouse and/or keyboard can be moved around the class.

A much cheaper alternative to a PC or laptop is provided by the use of a teacher’s graphical calculator (which is, after all, more powerful than the classroom computers of 20 years ago). Its display can be projected using an external LCD pad, such as a `Viewscreen’, on top of a conventional overhead projector (OHP). It can also be connected to a conversion box, such as a `TI Presenter’, which converts the output to a standard video signal for connection to a projector or TV. Another item of ICT hardware which greatly enhances mathematics teaching and can be used in conjunction either with a graphical calculator or PC is a data-logger, such as the Calculator Based Ranger (CBR). With this item you can capture distance-time graphs in real-time. Examples will be shown later of how this can be used to enliven mathematics teaching.

Finally we come to digital images. We are now used to sources of digital images e.g. from the Web, via a scanner, using a digital camera or web-cam. A useful piece of equipment is called a `document camera’, like the AverVision one illustrated here. This is a stand-mounted high-quality digital camera which provides a simple way of displaying pages from a book, handwritten pages, student work, physical models etc. My version has both video and S-video outputs. In my classroom I have the projector attached to my tablet PC (VGA), to my TI-83 graphical calculator (video) and to my document camera (S-video) – so I can easily switch between them.

2. Mathematics software to support interactive whole class teaching (and beyond).

I am not going to make any attempt to survey the range of possible uses, but instead I will concentrate on two `generic’ styles of mathematical software `tools’. These were ones we selected for use with analog IWBs for the DfES year 7 project called `RM MathsAlive!’ in 2000/1 and which were used with 20 pilot teachers and classes.

2.1 Dynamic Geometry Software – The Geometer’s Sketchpad and Cabri Geometry

The first is an example of `dynamic geometry software’ (DGS) – in this case The Geometer’s Sketchpad V.4. Here are some screen shots of its use in conjunction with some digital photographs – in this case they all show bridges. The first is of the well-known `mathematical bridge’ at Queen’s College Cambridge. Using the simple construction tools we can place points on the bridge and construct rays joining them to show how parallel lines appear in a photograph. As well as learning something about perspective and vanishing points we can also tie this in with enlargement, ratio and proportion.

In the second example we have a view of two different bridges over the Cam. The far one looks as if it may have a circular arch, but probably not a semicircular one (look how the piers enter the water – would the tangents be perpendicular to the water?). We know that the centre of a circle lies on the perpendicular bisector of a chord (fold symmetry?) – so all we need to do is to construct the perpendicular bisectors of two chords – which is the same as finding the circumcircle of a triangle.

Finally I might guess that the bridge nearer the viewer has the shape of a quadratic function. So we can create axes, enter the equation of a function and plot its graph. We may then need to adjust the equation of the function to find a suitable model. This example shows how we can use digital images to bring the outside world into the mathematics classroom - in this case as vehicles to use with a range of geometrical tools for pure, transformation and coordinate geometry.

Within a project for QCA directed by Kenneth Ruthven we have also been exploring the use of ICT as a bridge between algebra and geometry. One of the examples used here is also the basis of a sample lesson on the Mathematics Consortium’s on-line CPD programme (www.cpd4maths.co.uk - the partners are The Mathematical Association, University College Chichester and New Media) . Here a dynamic number line has been created to give a visual illustration of an algebraic rule connecting one value dependent on another. The materials provided show additive and multiplicative models. Of course it quite possible to create such illustrations in `Flash’ or to export them as `Java applets’. However then the user has no option but to interact with them in the way the designer has dictated. If teachers (and pupils) have access both to the source software, such as Cabri or Sketchpad, and to sets of useful files then they can make decisions whether: to use files `out of the box’ – the adopter model to make slight changes to the files – the adapter model to build up their own files – the innovator model.

An important feature of dynamic geometry software is that it can be used as an aid to teaching and learning aspects of mathematics at all levels – from primary through to Further Mathematics. Another is that contains within itself the necessary tools to support communication, such as writing text, showing results, making tables. These support both the teacher in creating materials and the learner in communicating what they have done. Perhaps most importantly of all, the major packages such as Cabri and Sketchpad qualify for e-learning credits (e-LCs) and there are also trial, demo and free versions such as WinGeo – so the software is affordable. Versions of DGS also appear built-in to hand-held PDAs like the TI Voyage 200 (which has both Cabri and Sketchpad) and for the TI-83/84 range of graphical calculators with the free Cabri Junior application.

In summary, then, we can say that dynamic geometry software passes the following tests: breadth of styles of use width and depth of curricular application ease of communication affordability.

So why isn’t it in everyday use in all mathematics departments? Perhaps because its use just isn’t well enough known – and maybe because sources of funding, CPD, pre-written materials etc. are also not well enough publicised. But in the end, just what is the incentive for a mathematics department to improve its ICT use?

2.2 Graphical calculators and data-loggersGraphical calculators have been around for a long time. They have been integrated into courses such as SMP 16-19, MEI, IB, FSMQs… Their use is very well illustrated in many of the pages of examples in the National KS3 Strategy’s Framework for Mathematics Years 7,8,9. The new AS/A programmes being introduced this September provide encouragement for their use. But why, then, are they not in more frequent use at levels at least in 11-19 mathematics? One reason may be that they have acquired a reputation for not being very easy to use – at least by teachers. Another is that they are not seen as `genuine’ ICT tools, but something so maths specific that they can only be bought from (inadequate) mathematics departmental budgets. But graphical calculators have come a very long way in the 15 or more years they have been available. The TI-84 Plus Silver Edition graphical calculator illustrated above has a 15MHz Z80 processor, a USB port, 32k RAM (24k available to user), 1540 Mb Flash ROM and comes pre-loaded with 21 Apps. This is the modern equivalent of a hand-held computer with a hard-disk. It connects easily to computer. There is a wealth of supporting materials and resources for graphical calculators on the Internet, such as free Apps which can be easily downloaded into graphical calculator. The illustrations below show screens from Cabri Junior, Smile Mathematics and CellSheet - all of which are free Apps from TI.

Used together with a Calculator Based Ranger (CBR) the pre-loaded CBL/CBR App provides a simple way of obtaining distance-time graphs – perhaps by asking a student to walk up and down in front of the CBR (whose range is 0.5m to 6m.). Here we see that data once collected are displayed as a scattergram, and stored in the Data Editor in lists. Parts of the data sets can be modelled analytically by graphing functions on top of the scattergram. Here is an example of fitting a straight line graph `by eye’.

Of course sets of graphical calculators are also a very practical way of giving hands-on access to students in ordinary mathematics classrooms.

2.3 Integrated mathematics software – TI InterActive!In June 2002 the TTA funded a group drawn from the ATM, MA, NAMA, KS3 strategy etc. to write a guide on ICT and Mathematics. It includes the following:

ICT and Mathematics: a guide to learning and teaching mathematics 11-19

Integrated mathematics packages

.. Texas Instrument’s TI InterActive! was developed explicitly for education and contains a blend of functions found in (a) MathCad, (b) Derive, (c) the TI-83 graphical calculator and (d) MS Office! It was this range of functions that led to TI InterActive! (TII!) being adopted as the main mathematics tool, along with graphical calculators and the Geometer’s Sketchpad, for the DfES Year 7 pilot project: MathsAlive! Its features include: a graph-plotter for functions and data; a spreadsheet – a subset of MS Excel; a symbol manipulator – very similar to Derive; a data editor with facilities for data-handling, probability and statistics with lists; a word-processor – a subset of MS Word; a web-browser with easy means of capturing data into lists – a subset of MS Internet

Explorer; an e-mailer – a subset of MS Outlook Express; live mathematics boxes where values and functions may be defined and altered so that any

dependent boxes, graphs etc. are recomputed when changes occur; an easy interface with data-loggers (CBR, CBL) and graphical calculators (TI-83) for

capture and exchange of data; easy ways to transport data and documents using a variety of compatible file formats.

You can click on any live part of a TII! document to edit it – and used with large fonts, thick lines etc. it is an ideal medium for whole class displays of mathematics such as with an interactive whiteboard. Put simply, the view of the group which produced this advice document is that “it is the best thing since sliced bread!” – especially since a single copy costs around £30 and a licence about £10 per station.

You can download the whole guidance document either from my own website (www.adrianoldknow.org.uk) or from the MA (www.m-a.org.uk). The DfES has funded the updating and revision of the document which will be made freely available to schools in the New Year via Becta.

Some examples of the use of TII! are to be found in the Micromath article `Mathematics from still and video images’ from Vol 19 No2 (also available from my website). The screen shot shows an extract from a new book shortly to be published by T-cubed Europe called `TI InterActive!™ in the Classroom’

This shows an example of using TII! to produce an interactive document to support student investigation. Only the mathematical toolbar of TII! is currently shown. The page contains a mixture of text, graphics and a portion of data table. If you double-click in the data table this opens up TII!’s data editor which you can then use for a variety of statistical, graphical and other analysis on the data. Below is an example of the use of TII! to produce scatterplots, graphs of functions and regression calculations from the data provided. Above all it is being used as the vehicle for students to communicate what they have found. So it is an excellent tool to support the ICT based coursework in GCSE data-handling or to use as a spreadsheet in FSMQs! The results can be exported either as an RTF or HTML file (but of course the documents are then no longer `live’.)

You can download a 30-day free trial version of TI InterActive!TM from: http://education.ti.com/us/product/software/tii/down/download.html As well as a personal tool for developer, teacher or student, the software is ideal for use for interactive whole class teaching – such as with an Interactive WhiteBoard. The screen below shows how the `Maths Box’ tool brings up a mathematical keyboard which bears more than a passing resemblance to a TI-83 graphical calculator. The `Math’ and `Statistics’ menus open up a wealth of functions from simple numerical ones through to calculus, matrices and complex numbers. The screen also shows the resemblance to Computer Algebra Systems (CAS) like Derive. Here we have 25! evaluated as an integer, and some of the steps which could be used to explore how to solve simultaneous equations.

As a final illustration here is an example of the use of TII! for data collection using a Calculator Based Ranger (CBR) data-logger – attached to the PC with a USB cable. Data are captured into lists in the List Editor, and these can be used to create the scatterplot for the distance-time graph. In this example two `sliders’ have been used to define the variables m and c . The graph of y = mx + c is superimposed over the scatterplot. By dragging either slider you can quickly see the effect that each parameter has on the graph – as well as having a context in which to explore linear (or any other) functions.

3. Current initiatives to support ICT in mathematics

In summary the new developments in ICT for secondary schools are the following.

Charles Clarke instituted a new strategy for the DfES ICT in School (ex NGfL) just over a year ago. The buzz word is `embedding’ and the mantra is `Enhancing Subject Teaching Using ICT’ (ESTUICT). There is a new DfES ICT in Schools website.

The projected `NOF Phase 3’ CPD for teachers using ICT has been taken back into the DfES (from TTA) and became a pilot on-line CPD project for KS1/2 numeracy and literacy, and for KS3 science. Contracts were let to Indigo Visions (all 3 subjects), SfE (KS1/2) and the Science Consortium (KS3).

The DfES has since extended these to nearly all KS3 subjects – but just to sole providers in all but science. They have renamed these as ESTUICT(CPD). These each now provide an element of face-to-face support, a web-site of lesson plans, tutorials, guidance and other resource materials, and access to a mentor by e-mail. The provider for mathematics is The Mathematics Consortium, which consists of the Mathematical Association (MA - responsible for tutorial support), University College Chichester (UCC - responsible for website content) and New Media (responsible for technical operations and admin.). There is also an opportunity to develop a portfolio for accreditation for an M-level module via UCC. The expected costs per school are c. £480 + £120n where n is the number of teachers enrolled.

The BBC announced a while ago that it intended to develop `digital curriculum’ resources across all subjects and key stages. A concerted reaction from educational publishers has led (a) to the postponement of the launch of these materials until 2006, (b) a restriction to 50% of the maximum curriculum coverage in each subject and (c) the establishment of `e-Learning Credits’ and `Curriculum Online’ as a means of providing schools with resources to purchase `curriculum focused digital content’. There is £100m for e-LCs at KS1-4 in 2004/5 and 2005/6. For a secondary school the formula is about £1000 + £10n where n is the number of pupils.

The earmarked money for schools to spend on ICT is in section 31a of the Standards Fund. This gives freedom for schools to determine their own spending priorities for ICT which may include hardware, CPD and generic software. The hardware could include laptops for teachers, more PCs and networking, IWBs and subject specific tools (e.g. graphical calculators and data-loggers). The CPD can include funding the ESTUICT(CPD) and a new scheme called `Hands-on Support’ – for which an extra £60m has been provided. Schools have much greater flexibility over how this money is used than previously. They may also spend some of their capital grant on ICT if they chose.

Hands-on Support (HOS) is supposed to provide schools with funds to buy in peer-to-peer support from successful practitioners to help a group of teachers use their own ICT resources more effectively. Schools can buy in from LEAs, or from external providers or from each other. A possible source for HOS in KS3/4 maths is the network of ASTs and LMTs. The funding has been provided for 2004-6.

The DfES is also funding Subject Associations to develop a range of materials to be offered free to schools from Jan 2005, called the `Key Stage 3 Offer’. This is being focused on maths, science, English and MFL in the first instance. Materials will include lesson plans, software, resources, guidance etc. They will usually be available both in physical form (CDs and books) and in digital form (via web-sites).

Becta is undergoing a transformation currently and has been given a new government remit which includes providing strategic advice. It is redeveloping the Becta ICT Advice website to provide better support for subjects. Subject Associations have been working with Becta this year with quite small budgets to provide online newsletters, `ask an expert’

services, needs analysis surveys and face-to-face events. The work is being extended and ATM/MA are working together on this.

The DfES have produced a 2 CD set of videoed case studies in KS3 and post-14 use of ICT in subjects – the maths case study uses dynamic geometry software and an IWB. (The 3 CD set of case studies produced by the TTA in preparation for the NOF-funded training is still available – it contains four maths examples including dynamic geometry software, spreadsheets, small software and graphical calculators.) DfES have also produced exemplar packs with CDs giving tasters of the ESTUICT(CPD) provisions at both KS3 and KS1/2. Further CD case studies are expected soon – possibly with three in maths.

Interactive WhiteBoards (IWBs) are very much `flavour of the month’ with, for example, funding going into London schools, the National Primary Strategy and now ITT (via TTA). A NESTA funded project at Hull University has produced a 1 hour self-instructional DVD and supporting resources which will be widely distributed to LEAs, schools and ITT. (For a really bad example of IWB use in KS3 mathematics see the Becta ICT advice website!)

4. References

ATM, Special edition on dynamic geometry, Micromath V18 N3 2002DfES, Fulfilling the Potential – transforming teaching and learning through ICT in schools,

DfES/0265/2003 Oldknow, A. (ed), ICT and Mathematics: a guide to learning and teaching mathematics 11-19,

MA 2002 (www.m-a.org.uk) Oldknow, A. Geometric and Algebraic Modelling with DGS, Micromath V19 N2 2003Oldknow, A. Mathematics from still and video images, Micromath V19 N2 2003Oldknow, A. What would it take to get ICT established in a maths department? Micromath

V20 N1 2004 Oldknow, A. What a picture, what a photograph, Teaching Mathematics and its Applications

V22 N3 2003Oldknow, A. Here we go round the merry-go-round, Micromath V20 N1 2004Oldknow, A. Maths alive – inside 20 mathematics classrooms for a year in Johnson-Wilder, S. & Pymm, D. (eds) `Teaching Secondary Mathematics with ICT’, Open University 2004 Oldknow, A. & Taylor, R. Teaching Mathematics using ICT – 2nd edition, Continuum 2004 ISBN 0-8264-7059-9QCA, Developing reasoning through algebra and geometry, QCA/04/1289 2004Royal Society, Teaching and Learning Geometry 11-19, 2001Ruthven, K. Linking algebraic and geometric reasoning with dynamic geometry software,

QCA March 2003 (unpublised)Smith, A. Making Mathematics Count, Stationery Office 2004 T3 Europe, TI InterActive!™ in the Classroom, Westfälische Wilhelms-Universität Münster 2004 ISBN 3-934064-51-5The Review project, The Good Guide to Interactive Whiteboards, Nesta 2004

5. Sources of materials

AverVision 110 document camera: Matrix Display http:// www.matrixdisplay.com 01784 439044Digitise Image: free from Jeff Waldock at: http://maths.sci.shu.ac.uk/DigitiseImage/ Gyration Ultra cordless keyboard and mouse: After Hours Computers http://www.afterhours.co.uk Tel: 01892 618497 Fax: 01892 618497Problem pictures : with e-LCs from Richard Phillips http://www.problempictures.co.uk/ Texas Instruments UK: http://education.ti.com/uk/index.html The Geometer’s Sketchpad: Chartwell-Yorke http://www.chartwell-yorke.co.uk/ Tel: 01204 811001 Fax: 01204 811008 QED Griffin http://www.qedgriffin.com/door/index.html Tel: 01494 772973 Fax: 01494 793951TI InterActive! :TI-82/83/84 graphical calculator:TI Voyage 200 handheld computer:TI Calculator Based Ranger CBR data-logger: Oxford Educational Supplies www.studentcalculators.co.uk www.oxford-educational.co.uk Tel : 01869 344500 Fax: 01869 343654 Vidshell 2000: free from Doyle V. Davis at: http://webphysics.tec.nh.us/vidshell/vidshell.html Wingeom: free from Peanuts software http://math.exeter.edu/rparris/wingeom.html