In using a CAS, a particular explicit symbolism is forced: each input requires a particular forced way of viewing things and expressing relationships and the output needs to be interpreted similarly. A CAS is expressive in the sense that one can express ideas (mental objects) in concrete form (visible, public objects). For an observer (teacher, educational researcher), the concrete, syntactically-precise expression of thought in actions (writing and rewriting of computer code, gestures and spoken remarks with respect to code and graphical and numerical displays) offers an unprecedented "window" onto students’ mathematical thinking. In this theme, we examined the nature and some of the practical implications of a CAS as an "expressive medium".
(Revised versions of these papers published in IJCAME vol 9, no 2, 2002)
Discussion of the papers:
Werner Peschek: I couldn't follow Michal Yerushalmy's objection to 'black boxes' -- we are using them all the time, they are basic in everyday life and in mathematics. Michal Yerushalmy: It depends on the situation: at university, if a student isn't comfortable with solving a particular linear equation, it might be OK for them to use a black box "solve" function to solve it. But my goal in beginning algebra is to explain to students what equations are, what does it mean to solve. Then, such a black box is not appropriate. Neil Challis: My response would be that we have a lot more choice about what to do now, and it comes down to an argument about what should or shouldn't be in the curriculum anymore; I agree completely that if you want to develop a 'feel' for symbols, then for that activity you've got to shut the CAS system out of it. For other activities, CAS might be central. CAS makes you very specific about what your aims are, what you want the students to achieve. What I liked about CAS right from the beginning is that it gave a greater choice, we didn't have to say 'we've got to do that before we can do this...' -- there are new pathways through the curriculum.
Allan Hayes: There are different sorts of black boxes, sometimes you want to solve equations, sometimes to manipulate results. When you get a solution, you've still got to understand the algebra to manipulate it. We need both of these. If the student asks what's going on inside the black box, we've got to answer it. John Monaghan: That brings us back to what Michèle Artigue started with in Topic 1.
The discussion involved talking less about CAS itself and more about placing CAS in the context of some more general ideas about computer-based learning environments:
Note: These issues are introduced in a previous paper by Phillip Kent: Expressiveness and Abstraction with Computer Algebra Software [PDF file, 118 Kbytes].
We took the opportunity to have demonstrations of various pieces of software, partly those which Neil Challis and Michal Yerushalmy had referred to in their plenary talks.
Members of the group: Burkhard Alpers, Shinwha Cha, Neil Challis, André Heck, Phillip Kent (chair), Bruce Torrence, Michal Yerushalmy.
(Revised version of this paper published in IJCAME vol 9, no 3, 2002: CAS as Environments for Implementing Mathematical Microworlds)
This presentation reviewed the concept of microworld, its relevance to working with CAS, and an example of a microworld designed for use by higher education students to apply mathematical techniques to the modelling of a "Formula 1" racing circuit.
[Powerpoint slides of the Alpers presentation] (75 KBytes)
Neil began by simply recording sound through a microphone, using a tuning fork to generate an approximately single-frequency, sinusoidal wave (this can also be attempted by singing!). He suggested that this can be good motivation for working on functions (especially for the kind of student he encounters who often needs to see the relevance of mathematics to 'non-mathematical' subjects). With more advanced students, it can open discussion to the problems of noise and discrete sampling in real data.
The second demonstration was a "walking the graph" exercise, where a guinea pig is shown a velocity-time graph and has to reproduce it as closely as possible by walking in front of a motion detector.
This technology has been around since the mid-80s, and is now cheap, reliable and very effective. So why isn't it being universally used?
Curricula are highly resistant to technology. In the UK, graphic calculators are just beginning to be make a significant cultural impression. In universities, they do have an unfortunate reputation (amongst mathematicians) as being "second rate CAS", and students are directed into using Maple, Mathematica, etc.
Discipline boundaries are a big issue. In the Netherlands, maths teachers call this kind of work physics; it's like a split world, where laboratories only exist in science.
André introduced his work at the University of Amsterdam on developing maths/science curriculum for high schools. The principle software tool that they use is Coach, which allows video sequences to be analysed for dynamical data, which can then be graphed and modelled.
[see: Centre for Microcomputer Applications, University of Amsterdam: www.cma.science.uva.nl/english]
This software was mentioned by Michal in her plenary talk. It was a very effective demonstration of things that CAS cannot (easily) do, because it takes standard symbolic and graphical representations for granted. For example, a central idea of Michal's suite of computer programs, and the associated curriculum, is to begin the study of functions without symbols, so that graphs are directly manipulated and constructed using "pieces of curve". Also, these are both good examples of a microworld: a representation of (a small part of) a particular knowledge domain constructed explicitly for learning about that domain.
CAS feels, to us, like a very natural tool for doing mathematics, but to what extent is that due to our 'old-fashioned' education based on pen-and-paper mathematics? It's clear from many sources that students today are not comfortable with the software doing things for them automatically without giving an insight into how they are doing it.
There's a prosaic but important use for CAS, in having student's check their answers to pen-and-paper problems. This can be promoted as being part of developing a professional attitude to getting the right answer. But in fact, this is not as simple as it sounds, because CAS systems can give 'wild' answers for some kinds of calculation.
CAS is becoming more accessible to younger students, and there is an argument that they should start young so that they are familiar with it by the time they reach high school and university. The tool for the 13 year old cannot be identical to that for the 18 year old: there needs to be something analogous to "game levels" in video games, the beginner starts at level 1 and as they develop, more and more of the "game" is revealed to them.
One of the things that is going to happen with new technology-based school curricula is that students will come out of it with less manual facility with symbols, they will have been brought up to rely on software/calculator and they will do so when they enter university. Is that the future we want? Will manipulation need to explicitly taught in university? Yes, probably, but the balance in future between mental and machine calculation is going to be different. Note the point made above that students DO NOT feel comfortable when they can only trust the machine to do something.
Neil wrote in his paper: "We do not take the attitude that you must be fluent in a technique before you apply it". It sounds something like learning a language - where meaning is learnt by using the language, not learning rules about it. Does CAS does make that an option for algebraic techniques?
Response: No, I don't agree with the comparison - in talking a language, I will make many mistakes which will be corrected by interacting with others, BUT a CAS is merciless, it only wants the exact, correct words!
Reply from Challis: this comment came from my experience in teaching mathematical modelling. Good practice used to be that the first part of a course you teach the techniques, and in the second part you apply the techniques to models; there might be a year gap between learning a technique and applying it! CAS changes this. For example, you can appreciate the meaning of a differential equation but not yet know how to solve it. You can use a modelling case study to motivate learning a particular technique - ie. the reverse of the traditional practice. For example, graphical solutions as provided by CAS or calculator are very powerful, but they don't seem to be enough to appreciate the phenomenon of critical damping, that needs the algebraic representation.
This comes down to the issue of explanation - and it is going to be the teacher's job to educate the student how to recognise what is an adequate explanation of a problem: the student and CAS don't work in a vacuum.
CAS quickly reveals to students what hardly ever comes across in pen-and-paper learning, that many algebraic equations do not have algebraic solutions, and nearly all integrals cannot be expressed algebraically.
The achievements of the group were partially summarised in a poster: [PDF file of the Poster] (7 KBytes)