Is 'More' necessarily 'Better'?

Tom Macintyre, University of Edinburgh

This International event in Austria provided the opportunity to meet with a wide range of practitioners with an interest in the role of technology within the learning and development of mathematics education. As with any conference of this magnitude, the scientific programme was divided up into 'strands' to make it manageable for delegates and to ensure individuals could at least try to devise a coherent programme of their own. There were also Special Groups and Working Groups focused on more specific issues or technologies, providing a suitable forum for those with a narrower interest in a particular aspect of ICT in mathematics education. On this occasion there were seven 'strands' (S1-S7) providing the underlying structure to the conference, and five Special Groups (Sp 1- Sp 5) namely:

The Working Groups were very much discursive sessions, tending not to have any presentations as such, and offering a better opportunity to fully engage with an issue in education and to seriously look to the future developments and their impacts on learning. There were 6 such groups (W1-W6), with delegates mainly sticking to one area over the course of the week:

As the names of the different sessions might suggest, there were opportunities to concentrate on the technical dimension of ICT or alternatively, delegates could take a more pragmatic interest in the developments of technology within their domain. It did not need to be a highly-charged technological experience and indeed some of the plenary sessions were led using the minimum of technology - a somewhat surprising outcome considering the range of visual aids that were on hand in such a modern and well-equipped University setting!

This report reflects my personal involvement in the conference, highlighting some of the issues I felt emerged for the future development of technology in the Scottish educational setting. For a broader review of all the different sessions presented during the event there is a web site providing abstracts from presenters (www.uni-klu.ac.at/ictmt5) and a forthcoming publication offering fuller academic papers in the form of conference proceedings (Borovcnik & Kautschitsch, 2002).

On reviewing my notes of the conference I wish to pick out a selection of comments that impinge on teacher development of ICT in the classroom setting. The idea here is to provide a flavour of the range of issues addressed and to offer some reference point for the reader to access for further information on the same theme. For reference purposes, the Strand or Working Group is indicated.

Strand 3: Innovations in mathematics, Science and Technology (IMST2)- First outcomes of a nationwide initiative for upper secondary schools in Austria.
(Konrad Krainer, Austria)

This initiative of working with teachers was developed through government funding following the TIMSS publication - supporting maths and science teachers' efforts for raising quality of learning and teaching. A teacher-directed development was envisaged, with teachers writing about their experiences and innovations; not 'expert' leadership but an opportunity to support teachers as they integrate the changing technologies. This study relates to the theme of 'Teachers and Change' and how the implementation of any new development might best be achieved. There are two considerations here - the central support for such an initiative on the back of concerning data from the TIMSS report and the cultural issue of having practising teachers very much at the centre of new developments.

The emphasis is on the need to have different teaching styles when integrating technology - just like a tennis player has to be able to play different strokes for different situations. The development team discouraged a monocultural approach. Teachers must be prepared to work in a manner that does justice to the method adopted:

Teaching through questioning, open to all views and comments

v

Formal method, with a clear goal of his own (teacher), focusing on the teacher at the front

The analogy of a sieve was presented:
Avoid the sieve approach where the open questioning is funnelled into the teacher's own goal at the end of the lesson. They argue that you cannot sit between the two styles.

The aim is to shift towards the more reflective and collaborative style of teacher development on the quadrant model of Reflection-Action and Autonomy-Networking axes:

ICT changes the relationship between Teacher and Pupil, accelerating the transition towards the reflective practitioner. With the 'teacher as a learner', it makes for a different relationship in the classroom. Given the pace of technological developments today, it certainly places the teacher in that learner mode. It should be noted however, that the structure of the educational system in Austria does make this approach more accessible, in that the teachers have a lot more autonomy over activities and assessment throughout the stages. The teachers' autonomy over assessment enables them to be more flexible over how they prepare their students - no national tests as such or an assessment framework as we have at various stages throughout the system.

Strand 4: Integrating MUPAD into the teaching of mathematics
(Fred Szabo, Canada)

This session also had an emphasis on getting the students into the mode of learning. No amount of good teaching has the desired effect if you teach to the 'wall' or teach to the 'internet' through on-line learning - you must have the recipients engaged in the process and work off their reactions to enhance the learning experience. A collaborative approach was preferred in using MUPAD - avoiding the punitive stance of:

• if you don't do well in your SAT...
• if you don't get your H grade, can't go to University ...

The favoured route instead, was a more collaborative approach, with the idea of working in teams being promoted. When working with MUPAD, or indeed any other related technology, it's not just about pressing buttons - there is an immediate need to talk about things including interpretational and judgmental issues that emerge from using the technology. Programming was encouraged. Small programs and routines using lists, sub-lists and subscripts; loops and repetitions, recursive definitions; domains defined etc. all of which provided a good basis for exploring the concepts and technology. Such approaches also provided further opportunities for discussion and collaborative work. Szabo agues that technology makes mathematics more accessible, more fun and easier. It also enables the curriculum to be made more relevant.

Strand 1 'Functional algebra' with the use of the graphing calculator.
(Henk van der Kooij, Netherlands)

This session looked at ways of overcoming students' algebraic anxieties. Solving equations using graphs and intersects, emphasising the concepts. Van der Kooij de-emphasised training on lower order skills to allow more to be made of the conceptual development. He made a strong point that supports the concerns of many at home; namely that accuracy depends on context. For example trace may give many decimal places, but the student must appreciate that for our particular result/context it would be inappropriate to use more than 1 d.p. The coursework task 'Which Container?' highlights this scenario, where greater accuracy would be meaningless in the context of manufacturing a container.

Concern remains over students' ability to make such decisions and whether the calculator is really used sensibly in some graphical contexts - pixel interpretations and the trace facility need to be used with great care! A final point raised by van der Kooij concerned the desire to work with the GC, along with its limitations, rather than using CAS. His fear is that if CAS is available, it will be used - regardless of whether more might be gained through a graphical approach to the same problem. 'More' is not necessarily better.

Plenary 6: The use of technology in developing mathematical modelling skills
(John Berry, Plymouth)

It is time to stand back and reflect on where we're going with technology, moving away from a view of mathematics as 'a given body of knowledge' to one of modelling and an introduction to the work of mathematicians. We mustn't lose sight of:

• the importance of modelling
• working styles
• importance of traditional skills

We as mathematicians have an image problem. Berry shared findings from his research into pupils' views and images of 'mathematicians' and 'mathematics' as reported in Berry & Picker (2000). The image, as portrayed by those consulted, was one of pencils in pockets, maths writing on arms, scruffy and untidy, unshaven, but ... aged 30! How can we improve on this image of mathematicians as well improving the pupils' perception of mathematics?

The image of being a 'magician with your head' needs to be guarded against developing into one of being 'a magician with technology'. Progress needs to be made WITH the learners and not the dazzling magical tour of what technology can do, or what I can do with technology.

Berry went on to stress the need to avoid any misuse of methods - e.g. data logging activities taking over from the actual mathematics involved. Data logging focuses on shallow skills and can lead to poor student working styles. The example quoted was on 'Tea Cooling in a Mug' - students electing to collect the data and then 'do something with it' - but what? Surely the decision about what one plans to do will come before the data collection, if indeed it is desirable or necessary to collect empirical data for the given problem. A second example was the 'Traffic Flow' problem with stopping distances etc. - similar to the Higher Grade coursework task 'Keep Your Distance'. Here it is inappropriate to go and gather data on the streets ... even though this would be perfectly possible with the available technology! Reflection is very important - step back and only then go on to consider the data. Spend more time on the formulation and interpretation stages of mathematical modelling.

A demonstration of a solution to the traffic flow problem was provided, highlighting the fact that it is independent of 'thinking time'. At this point John's transparency of traffic travelling along a highway was incorrectly placed on the projector - resulting in an excellent chat line about the international state of this problem with traffic travelling on either the LHS or RHS of the road according to the orientation of the acetate. The mirrored English also introduced a foreign language! We need those lighter moments.

In conclusion, he raised the following points when working with students, stressing the need to:

• be taught how to use the technology properly
• provide good opportunities for modelling challenges
• teach appropriate use of technology
• avoid the 'magician' style of image

Remember that just because things can be tackled more quickly when using technology, do not let time allocation to mathematics be eroded - this is the ideal opportunity to enrich the curriculum through appropriate modelling situations.

Working Group 5: More is more? More is less?
Dynamic Display = Dynamic Thinking?
New Chances or New Barriers
(Claus Meyer-Bothling & Hartmut Kohler, Germany)

These themes were developed and explored over three sessions of a fairly discursive nature - a short introduction was given by one of the leaders and contributions flowed from the floor as delegates put forward experiences, successes and concerns based on their home-based research. As the title of the overall working group indicates - Dangers and Limitations - this session was primarily about potential pitfalls.

Does IT really improve the educational process or does it merely get in the way of communication between teacher and student? What criteria might be used to measure the impact of IT in order to distinguish between success and failure with IT in teaching? Whatever we come up with must be sufficiently complex to be useful, but sufficiently simple to be practical. Examples cited included the evaluation of school projects from Baden-Wurttemberg, taking account of the successes and failures - but what has IT offered to the process? We must remember the art of teaching and to acknowledge that new media is but one method.

Although IT has been explored widely and promoted in schools it accounts for about 5% of actual lesson time - aside from any staff time in preparation of materials or researching of their curriculum, we are talking about actual class instruction through IT. For example a group of students were engaged in 'Making a CD'. This entailed accessing the Internet, with lots of information that had to be evaluated and reduced. The 'product' was achieved but the teacher played a major part for this to be the case - students needed to do it, as the 'product' was not the goal; if it was you could get a commercial firm to produce it! This project did not therefore achieve what it set out to achieve as the students did not ultimately 'do it'.

Another example concerned accessing the Internet to create documents on particular topics. There were difficulties regarding 'selection' and 'evaluation' of materials. Clearer evaluation skills need to be explored - what criteria are used for the inclusion of material or indeed the value of such material in the end product. Further difficulties rest with the availability of readable materials on the www. A project on Camels in the dessert ran into difficulties - how many of the sites are in English? Arabic? French? German? A lot of material was possibly missed or didn't exist in this medium, bearing in mind that a lot of the English language 'hits' were nothing to do with camels in the desert - Camel cigarette brand featured highly! Gaining permission for the inclusion of photographs in a final product can also be a problem. Things to consider in planning projects and work with technology:

• motivation
• demands on student
• teacher expectations

Criteria to be considered for the evaluation of IT in the classroom might be on the following categories:

• tools (BB, OHP, library, IT or otherwise)
• methods (exposition, interaction, groups, projects, collaborative or otherwise)
• contents (careful tailoring of topics to the approaches with or without IT)

Success or failure depends on the decisions of the teacher, influenced by pupils, other teachers and SOCIETY. We must always aim to go beyond the central issue of the development and ensure the wider picture is considered - too strong an emphasis on IT may lead to a collapse:

Maths structure	projects, applying knowledge in context (Potentially a good experience)

The message here is: do not rely on one source [tool] or one method.

Some other issues raised in our deliberations included:

• What are the limitations of the Internet, different search engines and meta-engines [categories, key words, pictures included] - how can these be evaluated?
• Email needs a personal relationship beyond the electronic medium for it to be sustained - think carefully about links with other schools, e-friends etc. This also highlights the need for email and conferencing to be used and developed with individuals who have the opportunity for personal contact beyond the electronic communication through comments on a bulletin board, workshop conference on PGCE courses, Probationers On Line (Moray House, 2001).
• Consider pupil development holistically - what are we losing through ICT? What compensation is there through ICT? E.g. 'Success' in Primary stages [passing a National Test] may lead to gaps in secondary curriculum - strive for global horizons if at all possible.
• Is the experience contracting or expanding? Consider the analogy of reading Shakespeare or making a dramatic production. How does this tie in with mathematical experiences? What chance is there for presentations with ICT in mathematics that might make the experience more memorable?
• What remains after the learning? What are the learners' memories of mathematics past? Is there anything of the inside mathematics?
• Beware of misconceptions being propagated. ICT can be a help in the constructivist paradigm but the basis of understanding is open to criticism through (excessive?) Black Box operations.
• Exams. How influential and necessary are they - very varied experiences across different nations, ranging from a lot of individual autonomy to none!

This debate was very thought provoking and encouraged all present to challenge their own use of technology in a variety of ways - is 'more' necessarily better?

Plenary 7: Chances and limits for teaching in the information age - human mind models and society demands.
(Walter Oberschelp, Germany)

This plenary lecture from strand 7 (The global perspective of information technology) was a fairly negative stance, bearing in mind the overall title and purpose of the conference. That having been said, the sentiments rang home truths and certainly warned against excessive and inappropriate use of technologies as we prepare youngsters for their futures in society. The expectations of society are different depending on the sub-groups and their different goals. Media is used to influence and to meet human demands in a variety of contexts:

• entertainment
• communication
• playing
• implicit learning
• explicit learning

The structure of learning needs to be fully considered. Extreme cases of 'e-learning' lack human contact. Oberschelp requests that proponents of such situations should be condemned to sitting in front of a multi-media screen in air-conditioning environment for extended periods of time! [Maybe that would be welcomed by some learners today?!]

IT is but a substitute for personal interaction. Is it possible to interact with IT? Some claim that dynamic packages take on some form of interaction, responding to actions by user, but it is nothing to replicate the human interaction that can take place through a face-to-face learning experience.

On a slightly different note, to what extent has the availability of technology led to some 'skill' being weakened in the course of time? Suggestions for this proposal include:

A number of bigger philosophical issues were also touched on, basically calling into question the pace and direction of change - is it all for the good? What will the future bring in the way of demands?

The special problems of math teaching are the alienation by formalism, the lack of personal appeal and the somewhat metaphysical nature of mathematics, whereas its pragmatic value is often invisible. Since mathematical ideas are often very compact, the abundant information of the Internet is hard to combine with mathematical thinking. And yet, mathematical thinking establishes useful tools for the complex existence in the global society. We exemplify problems in private and global economy and in our real physical world and discuss essential and obsolete material. We sketch, how methods for self-guided instruction may be improved. But we emphasis, that, due to the anthropological situation, personal instruction and care will become even more important in the future. The satisfaction of really understanding an argument from scratch and the experience of responsibly solving problems without the assistance of non-transparent tools will become a source of creativity and a well accepted motive in the education of independent and mature citizens. [Oberschelf, 2001 (extract from published abstract)]

Let's not lose sight of the role played by the teacher in promoting interaction and providing that personal instruction within a technological environment. More advanced technologies do not automatically mean a better learning experience. The teachers' role is crucial in ensuring the positives outweigh the potential dangers and limitations.

I would like to thank the University of Edinburgh and Texas Instruments for providing financial support to enable me to attend ICTMT 5 in Austria. My own presentation on 'Improving maths skills with CAS technology', which featured in Strand 4 of the proceedings (Borovcnik & Kautschitsch, 2001), provided me with excellent opportunities to discuss the potential of such technology in classroom settings. This is undoubtedly an area for further research that I hope to pursue with both home-based and international colleagues. I'd be pleased to hear from anyone interested in developments with technology in the classroom, with a view to building on the research evidence available - please contact me through the SMC.

References

Berry, J & Picker, S You Pupils' Images of Mathematicians and Mathematics, Mathematics in School, Vol. 29.2, p 24-26, March 2000

Borovcnik, M & Kautschitsch, H (Eds.): Technology in Mathematics Teaching. Proceedings of the ICTMT 5 in Klagenfurt 2001, Schriftenreihe Didaktik der Mathematik v. 25. Oesterreichischer Bundesverlag and H†lder-Pichler-Tempsky, Vienna 2002

Moray House Probationers On Line, http://probationersonline.org.uk, 2001