Figure 1: A great place to visit
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This year, it seems, is the year of the Internet. Everywhere we look we are confronted by images of electronic communication: in advertising, sport, government, and especially in schools. Teachers, of course, have known about electronic mail and the world wide web for some time now, but these have largely been things that "other people do" since few schools have had access to such facilities. So much has changed in just the last twelve months that there are now few schools which do NOT have access to the Internet in at least some form - even if that form is a single computer linked to a modem in the school library. While we may be still a long way from the vision of full-time access for every classroom (let alone every student) we must nonetheless face the reality: the Internet is here, so what does it mean for students and their teachers? More importantly, what can it mean?
I will begin with a story told to me by Michelle Williams from Queensland University of Technology. Michelle coordinates Oz-TeacherNet (http://rite.ed.qut.edu.au/oz-teachernet/), providing a wealth of Internet-based resources for schools and teachers. The story concerns an Australian class which became interested in the Grand Canyon (Figure 1). In particular, one group wanted to research the effects of tourism upon the environment and ecosystems of the Grand Canyon. Turning to the Internet, they tried a variety of searches centred around "Grand Canyon" and "tourism" and "environment" and anything else they could think of that seemed related to their question. Unfortunately, the closest they got was the "Grand Canyon Home Page" which featured full-coloured pictures of the canyon, and advertisements for such activities as donkey rides and joy flights. Nowhere could they find the answers to their questions.
Their problem was that they were stuck in the wrong metaphor. The metaphor most people have of the Internet is that of a "look it up" place: a place where people go to find information. Certainly, it offers a huge resource in this regard, but if this is your main image of the Internet, you may be missing the point. There are many sources of information available concerning wonderful web sites for all manner of learning experiences. This article, however, seeks to explore other metaphors of the Internet which may be even more important for schools and learners, especially as these apply to mathematics.
The first of these metaphors concerns a view of the Internet, not as a "look it up place" but as a place where people go to communicate, where communities with common interests come together and share their knowledge, their problems and their resources. Long before there was a "world wide web" this was the dominant model for use of the Internet. It seems to have been overshadowed by the colour, glamour and the hype of the multimedia world wide web experience, and yet it remains a most valuable resource for learning.
Figure 1: A great place to visit
A place to talk
Table of ContentsReturning briefly to our small class group, it was noticed that there were electronic mail ("email") links to the advertisements featured on the Grand Canyon Home Page. The group then sent Emails to the operator of the donkey rides and the joy flights, asking what these people could tell them about changes in the Grand Canyon resulting from tourism. The replies they received not only answered their questions but gave some real insights into this wonderful place, as seen through the eyes of those that live and work there.
The real power of the Internet lies in its role as a place where people go to "talk" to others. Electronic mail may be just the tip of the iceberg in this regard. More important for teachers are the variety of ways in which electronic mail can be used as a tool for learning.
Oz Projects provides a good place to start. Each year, every Australian school is sent a glossy calender for the year which features a wide variety of Internet-based project activities designed for students from K-12. A joint initiative of the South Australian NEXUS Information Service (http://www.nexus.edu.au) and Queensland's Oz-Teachernet, these projects are available to any students or classes who wish to participate. They vary from the popular "Travel Buddies" project (in which, this year, students send and receive Email messages to and from their soft toy "buddies" aboard the Endeavour replica, while it is sailing around the world) to Project Atmosphere, from CSIRO Scientists OnLine to the Net Gig, where students create their own music and then have it performed via the Internet to a world wide audience. "Tag-along excursions" allow students who cannot actually go to a particular place of interest (such as the Zoo) to ask questions before and after the excursion of those that can go. Both groups benefit from the focus provided by the questions, the added interest of looking on behalf of others, and the language skills needed to ask and answer questions which are often far more challenging than teachers would ask. For more information, the excellent book by Downes and Fatouros (1995) provides more detail on Internet-based projects (and Oz Projects in particular). It also offers some very useful references for further reading.
For mathematics learning, immediate possibilities emerge centred on cooperative projects which bring much larger groups of students together. For chance and data activities, this increases the size of the sample and leads to more realistic results. But the possibilities do not stop there. In ancient Greece, Eratosthenes came up with a pretty good estimate of the circumference of the earth by measuring the shadows cast by sticks in different locations at the same time. The same may be achieved by groups of students from Far North Queensland to Tasmania taking their measurements simultaneously and immediately reporting their results. We are limited only by our own imaginations as teachers.
Chatting with friends
Internet Relay Chat (IRC) has long been a feature of the Internet. It allows people to type text which is read simultaneously by others sharing the same "room." While such activity can be the most frightening and uncontrolled part of the Internet, it may also be a powerful medium for learning and for teachers to "talk" to other teachers.
In the mid 1980s, network-based writing was explored as a medium for deaf learners: groups of students at computer terminals wrote and discussed their learning. Somewhat surprisingly, it was found that this form of learning was not only effective for the deaf but was a powerful medium for all students. Writing became an immediate and interactive experience which was suddenly powerful for the learners. Bruce, Peyton and Batson (1993, pp.4--5) discuss the vision which has emerged from these beginnings in terms of several key elements:
- New social dimensions in the classroom, based on the premise that "...writing on a computer network would blur social distinctions in the classroom...The role of the teacher would shift from lecturer and director of discussion to collaborator in writing, and student participation would be more equally distributed." (p. 4)
- Writing for authentic purposes: "Whereas students previously wrote to a single audience--the teacher--now they would participate in a full-fledged writing community that included their peers." (p. 4)
- Immersion in a writing community, increasing opportunities to write, offering immediate feedback, making the writing process visible to students in many varied ways.
- Collaboration in writing, as peers not only think and plan and critique together, they actually write together, producing a joint product to which no one member can claim ownership but to which all have contributed.
- Writing across the curriculum, moving beyond the English classroom to communicating in all areas. In the case of mathematics, putting concepts into words can be the most powerful aid, both to learning and to evaluation of the quality of learning.
Figure 2: Chatting with friends
IRC software is freely available from the Internet and there are a growing range of sites to which teachers and students can "log on" and create private discussion "channels." Text-based chat is fast and simple to use. Even more exciting are free programs such as Microsoft's NetMeeting which offer full conferencing facilities: electronic whiteboards which allow all chatters to display their ideas, audio and video-conferencing facilities, allowing dynamic interaction as well as text exchange: all these features are freely available now for use by teachers and their students. Chatting with friends was never easier, nor more exciting.Table of Contents
The other model of the Internet of interest here is very new, unfolding and changing even as you read this. This is a vision of the Internet as a place, not to "look up" mathematics, nor even to "talk about" mathematics--the Internet has become a place we can go to do mathematics. The role of technology in mathematics learning at all levels has always been restricted by the physical problems of access, affordability and appropriateness. For schools, the problem with technology has never just been about getting kids in front of computers; equally important with the hardware has been the software: rarely affordable and perhaps even more rarely appropriate. An important aspect of the growth of the Internet has been the development of a new type of "super-application"--the web browser. What began as a simple tool to read HTML pages, the browsers now available are not only free, multimedia-capable and cross-platform (running the same files identically on any computer); they are fast becoming "all things to all people." Reading a word processor document, working on a spreadsheet, even doing mathematical calculations--more and more we will be turning to our web browser to do these things. We can even write our own programs to have them do things that they cannot do at present.
A couple of years ago, my daughter was in Year 4 and having trouble with the long multiplication algorithm which was being studied, presumably as an extension activity. We spent some time working on the process at home, and decided that it would be helpful to have a computer program which would not only produce the final answer, but support the steps along the way. Originally written in HyperCard for the Macintosh, Nomi's Operations has been rewritten using JavaScript so that it can be run on a web browser such as Netscape Navigator 3 or the Internet Explorer 3.01, which support this language. The page may be downloaded from the web and run locally on or offline.
Study Figure 3. The object of scaffolding is to support learners in such a way that they gradually become less dependent and more proficient. In this example, several scaffolds are provided. Clicking in any entry field displays the appropriate product in the status bar at the bottom of the screen. A product greater than 9 can be entered directly into the field and the first digit will be automatically transferred to the next field. If a student does not know a particular product, they may enter it as, for example, 7*8, and the calculation will be carried out for them. For a "sneak preview" of the final result, they may click on the "Answer?" button. Students may use these supports for as long as they are needed, but gradually build up the required skills and may then do away with the scaffolding. This is not offered as a "perfect example" nor even as a finished product. It does, however, illustrate two things: elements of good software design for learning, and the fact that this program may be used identically on Macintosh or Windows platforms, online or offline, by anyone at any time. It provides a taste of the potential of the Internet to flow through and change many of our assumptions regarding access to technology.
Figure 3: Scaffolding number learning
Possibly even more exciting as an example are the live geometry pages which have been developed by David Joyce in the United States using another web programming language, Java. Picture a triangle which you can move around, twist, stretch, play with. Do this with lines, circles and shapes, and students from a very early age can build up much more powerful understandings of space and shape than was ever possible with paper and pencil.Table of ContentsFigure 4 provides a sample of a dynamic figure which students may twist and turn and manipulate themselves. Students and teachers may now get a real "taste" of the power of this medium using their web browsers, both onLine and offLine. They may then explore the excellent dynamic geometry software tools (such as Cabri Geometry, Geometer's SketchPad and the free WinGeom) which have been available for some time now.
The two examples given here are just the beginning of what is becoming available for live mathematics using a web browser, and remember: you do not have to be connected to the Internet to use these wonderful tools. Any computer in a classroom that can run Netscape can now be running mathematical software for number, space, chance and data and much more.
The ideas and examples discussed here may all be accessed through the Australian Association of Mathematics Teachers web site, at http://www.AAMT.edu.au. Following links such as Teachers talking to teachers and Live mathematics on the web will provide further information and the options to try out the various features which are likely to change the ways in which we think about, teach and learn mathematics in the very near future. If you are interested, begin by joining the AAMT List Community, and become part of the rapidly growing electronic mail discussions which are taking place there. Download the free IRC "chat" software and join in one of the regular "chat sessions" organised for teachers around the country. Explore the amazing implications of "live mathematics on the web" and then download the files to your own computer and use them freely within your school. The Internet has always been about breaking down barriers; as teachers of mathematics, we are just beginning to see what this can mean for us.
Bruce, B., Kreeft Peyton, J. and Batson, T. (1993). Network-Based Classrooms: Promises and Realities. Cambridge University Press, Cambridge.
Downes, T. and Fatouros, C. (1995). Learning in an electronic world. Primary English Teaching Association (PETA), Sydney.
