Interactive mathematics online for school and home

A paper by Bryan Dye. This has been submitted to the International Conference ICME-9 in Tokyo on July 31st to August 6th 2000. The section on dynamic geometry forms the basis of a presentation at the conference "Good Practice in the Use of ICT in Schools" at the RSA, London on March 6th 2000.

Abstract
1. A vision for future online materials
2. Current examplars: A Techologies, B Websites
3. Issues affecting the success of Online education
4. Conclusion

(A) Exemplar Technologies
2.1 Dynamic geometry
Dynamic geometry software began some years ago with the introduction of Cabri, Geometer's Sketchpad , Geometry Inventor, and others. The essential idea is that the user can construct geometric objects such as points and lines by using menu options and icons and then from these build up perpendicular or parallel lines or circles or bisectors and so on. The results can then be manipulated by using point and click mouse operations. Any correct geometric property will remain true throughout all further manipulation. For example, two lines constructed as parallel will remain parallel however the points that constructed them are moved. Figure 1 shows a typical screen from Geometer's Sketchpad. Cabri and other dynamic geometry programs are broadly similar in appearance.


Figure 1: Geometer's Sketchpad

A typical task in dynamic geometry could be "Construct a square". These steps would be a typical response to this task:
1. Construct two points, A and B, and a line through them.
2. Construct lines through both points perpendicular to that line.
3. Measure the distance AB and transfer that distance to the two perpendicular lines, measured from A and B. This constructs two further points C and D.
4. The square ABCD is constructed.

Constructing a rectangle is marginally harder. Though the software has always been sophisticated and educationally pure, one essential problem remains. It is far more difficult for a student to learn the skills necessary to complete steps 1 to 4 than it is for them to learn what a square is. These steps are certainly not accessible to the less able or younger student. Unless a lot of curriculum time is devoted to the technological aspects of the software, and to the ideas of points and lines, parallel and perpendicular, little progress can be made in school; extremely frustrating for all involved. Consequently, at best the software becomes used for trivial and isolated tasks that do not warrant the time or expense involved, and at worst is not used at all. A great step-forward is provided with Online dynamic geometry. We consider two current examples.

2.1.1 JavaSketchpad
JavaSketchpad is the web version of Geometer's Sketchpad, and can be produced automatically by using software available from the publishers of Geometer's Sketchpad.. See Figure 2 for an example. (This page is also available on the web.)


Figure 2: JavaSketchpad

In JavaSketchpad, an interactive geometric construct is embedded in the browser (Netscape in the above case). Points can be dragged with the mouse and the geometry in the diagram will respond appropriately. The icons and menu options from Geometer's Sketchpad are not present. In fact the student cannot introduce any new constructs but can manipulate only what is already there. For anyone who wishes to develop their own expertise, they can purchase the software package and create their own dynamic geometry.

A suitable activity to replace "Construct a square" would be "Here is a geometric shape. By dragging the points, identify what shape it is and what properties the shape has."

The student does not need to know any technological skills relating to the software but is free to concentrate on the mathematics, and through observation of what changes and what stays the same can gain insight into the "squareness" of squares. Thus the problem becomes accessible to a wide range of abilities and ages.
JavaSketchpad does meet three of our basic criteria for Online resources: technology, design and development. With reference to the fourth, content, Many educational websites are using JavaSketchpad with outstanding results. For examples, see JavaSketchpad's home site, MathsNet and Nrich .

2.1.2 Cinderella
The second example of online dynamic geometry is Cinderella . This relatively new software was developed specifically for Internet use. In many ways it is similar to JavaSketchpad and provides the same level of interactivity within a browser. See Figure 3 (and on the web ) for an example. In this figure, all the labeled points can be dragged so that the user should be able to identify, by its geometric properties, what kind of quadrilateral each one is.


Figure 3: Cinderella

Where Cinderella shows the way forward in terms of Online interactivity, is in its facility to create a full dynamic mathematical exercise, along with icons to increase available options and interactive hints and comments. Figure 4 (also available on the Web ) shows such an example. There are three parts to the screen. The main part shows the geometric construct, here a line with two points on it. Lower left is a text window describing the task to be tackled. This window will also display hints and feedback as the task progresses. Lower right is a set of icons that enable constructions to be made. Which icons are available is within the control of the person creating the webpage.


Figure 4: a Cinderella exercise

As the user progresses through the task, so the construct and text windows react accordingly. The task can also be pre-programmed to display a hint in the text window after an amount of time has passed with no progress being made. The flexibility afforded by this interface will allow the educator/programmer to create pages that assume a whole range of levels of expertise on the part of the student. Our original task "Construct a square", rejected before due to the conceptual demands it places on students, can be re-addressed. Now the task can be presented at various levels, anywhere from an initially blank window, through to a square completed apart from one side or one vertex. In this way, students can tackle exercises like construct a rhombus, reflect a triangle, rotate a square, construct the centre of a rotation (see Figure 5), and so on.


Figure 5: a Cinderella exercise (viewed in Internet Explorer)

As with JavaSketchpad, Cinderella meets our criteria regarding technology, design and development. For content, see MathsNet's rotations site (and its full version at the educational website AngliaCampus). Cabri, perhaps the first dynamic geometric software, also has a web development at Cabri-Geometre .

2.2 Dynamic algebra
Dynamic algebra software has been available for many years. Mathematica from Wolfram Research has recently announced a Java package to extend its capabilities and thus make it more Web-friendly. Mathcad and Derive are other examples. Such software can perform many algebraic tasks, such as simplifying, factorising, solving, differentiating, integrating and graphing. As with dynamic geometry, its use in schools - in the UK at least - has been limited, partly due to technological barriers and partly because in many ways this software "does" mathematics for you and leaves the student as a passive observer. However, new possibilities are developing with software such as LiveMath (previously known as MathView) and the associated plug-in for Netscape and Internet Explorer. Besides providing the user with the means to investigate algebra and graphing interactively, LiveMath also allows the educator/programmer to control exactly the degree of interactivity that the user is permitted. LiveMath is a standalone program - see Figure 6 - wherein various components (the 36 in this simple example) can be altered and lines following will change dynamically.


Figure 6: LiveMath

LiveMath can also be embedded in a web page with the same interactivity - see Figure 7 (also available on the Web ).


Figure 7: LiveMath embedded in a browser.

Graphs can be investigated in the same way (Figure 8 - also on the Web ). In this figure the quadratic equation, x²+5x+6=0, can be edited and the graph view itself can be changed by zooming in or out or moving the axes. As occurred earlier with dynamic geometry, the display in Figure 8 is split in two halves. The right-hand side contains the programmed mathematics, but the left contains text available to anyone to edit to suit the expected audience. Further examples of dynamic algebra using LiveMath can be found at MathsNet's algebra site and at Angliacampus. As with dynamic geometry, the student or teacher who wishes to develop this area of mathematics further can purchase and download the software and create their own dynamic geometry.


Figure 8 Graphing with LiveMath

2.3 Spreadsheets
Spreadsheets are mentioned explicitly in the new UK National Curriculum for Mathematics (within "Number and Algebra"). In many schools with networked computers, this is will probably imply use of Microsoft Excel. Though there is no doubting the power of Excel to perform varied mathematical tasks and to enable data to be analysed efficiently, the program remains a technology barrier in schools. Even at the BETT exhibition in London this year (a showcase for state of the art use of ICT in Education), the presenters on the Microsoft stand were taking their audience slowly step by step through each detail in the precise process of setting up a file. You cannot simply hand a student over to a spreadsheet and tell them to get on with it. The Internet should be offering a way forward, but so far there is little sign of an educational site providing an online spreadsheet. This may be due to the fact that spreadsheets are the most applicable of all mathematics software to the "real" world and therefore have great financial worth. Spreadsheets are too valuable to give away. One partial exception to this is Formula One.

2.3.1 Formula One
Formula One is a Java program, produced by TideStone Technologies , primarily for the American business market. As with dynamic geometry and algebra, the educator/programmer can create a spreadsheet embedded in a web page with interactivity limited to whatever they require. Formula One is expensive and not as yet clearly projected towards education. See Figure 9 (also on the web ) for an example.


Figure 9 Formula One

2.4 VRML
VRML, or virtual reality markup language, is the means by which 3 dimensional images can be displayed and manipulated with a browser. Usually this manipulation will include the ability to zoom in and out, to move the object from side to side and to rotate the object. All movements will occur on screen in a believably 3 dimensional way. There are many "flavours" of VRML programming available currently on the Internet, most of which require a plug-in to be downloaded by the user and installed on their computer. One version, shown in Figure 10, that does not require a plugin, is called JGV . Another version, from Internet Explorer is shown in Figure 11. The creation of a typical VRML file is not for the faint-hearted programmer, but the results often provide a stunningly intuitive method of appreciating the nature of solid geometric objects. In this respect VRML fits perfectly into this paper's view on ideal web pages. Far more elaborate polyhedra are catalogued at the Virtual Polyhedra website.


Figure 10: JGV


Figure 11: VRML

2.5 Logo
Microworlds, from LCSI is a interactive software package based on the concept of logo turtle graphics, but providing, with the aid of a browser plugin, interactive projects embedded in web pages. See Figure 12 (also on the Web ) for an example where the user can drag and drop the tangram pieces onto the silhouetted shape, display the solution and access more problems. For further development the school can buy the software and create their own interactive pages.


Figure 12: Microworlds

2.6 Applets
The above sections summarise good practice in specific areas of mathematics, produced by business enterprises working in education. Scattered around the World Wide Web are many isolated examples of excellent interactive mathematics. Most will use Java programming to create material embedded in webpages. (In fact almost all of the above uses Java too.) It is this scattered nature of the web that makes some excellent material practically impossible to use, since both at school and at home, the user needs a coherent interface; they need to find their resources gathered together within an organised whole. Often these applets are there as demonstrations of the medium or of an amateur enthusiast's skills rather than part of coherent curriculum content. The fluid nature of the Internet means that, whilst the quality may be outstanding, the material may well vanish from the Internet when interests or financial commitments change. To solve this problem, it may require commercial enterprises to invest money in collecting together such scattered resources. Here are a few selected highlights. If any are not found at the address given, then the above point is proven! The Manipula site contains some excellent Java applets. Figure 13 shows an example of a construction which enablers the use to dissect a quadrilateral into two parts of equal area. Figure 14, from Math Cove , shows a typical simple applet illustrating aspects of transformations and angles that are immediately appealing.


Figure 13: Manipula


Figure 14: Math Cove

Other examples that show the variety available on the Web from using Java are: Buffon's needle , Central limit Theorem , Cubic polynomials , Integration and Vectors.

2.7 Flash & Shockwave
Flash technology, developed by Macromedia , particularly in version 4, (and Shockwave from Macromedia too) allows the educator/programmer to produce interactive mathematical activities that are quick to load and understood by both Netscape and Internet Explorer - and even Dreamcast and Playstation 2 games consoles too! This is a cutting-edge technology, heavily promoted in the e-commerce world. Developments in education should be expected soon. Figure 15 shows a basic Flash file. Clicking on the buttons will cause the elephant to undergo the requested transformation. Another early example illustrates tangrams . A primary school, Ambleside , in the UK has also developed material on numeracy using Flash.


Figure 15: Flash