Argumentative discussion - ADVOCATUS DIABOLI
Having now considered and summarised the available research, examined various teaching approaches, and how assessment might be carried out, this final section asks and responds to some searching questions regarding the use of CAS.
a) Mathematics educators in the 80's and 90's misjudged the central role of maintaining a balance between basic skills and use of the calculator. Is there not a danger that we may repeat with CAS the errors we made in taking for granted the use of four function calculators - but on another level?
The mistakes made with four function calculators have been widely recognised and considerable efforts have been made in order to avoid this over-reliance. The worldwide programme of training, via -T3™ for example, ensures that teachers have access to clear guidance as to how best to use the technology and how to integrate it into teaching and learning to ensure it is used in an optimum manner. The extensive research and pilot projects, such as the Austrian ACDCA, Australian CAS- CAT, and the various CAS conferences, help to ensure that classroom practice is well informed. Many Universities now include the use of CAS as part of the initial teacher training programme to prepare new teachers for the challenges which use of CAS presents. Finally, many countries have PAP (Pencil and Paper) tests to ensure that the by-hand skills are kept under review and to the fore.
b) Studies show that students do not lose their basic 'by hand skills' when using CAS but I am less than convinced of this. What evidence is available which proves they do not become over reliant on CAS to the detriment of these skills?
As indicated earlier in the text of this book the research evidence is very clear. Use of CAS does not appear to adversely affect 'by hand skills'. Conversely, evidence points to an improvement of these skills. Students require to be more flexible and need to interpret different algebraic representations when using CAS, often dealing with unexpected outcomes. The issue is very much in the public domain and most teachers split time - doing things without machine as well as with CAS. The Australian CAS-CAT project and the debate started by Kutzler et al, to name but two, help ensure that the issue of what must be done by hand is kept under review. CAS in hand held form allows instant access to CAS but most importantly teachers and students are aware that there are times when they must SWITCH OFF!
c) Enthusiasts tend to be optimistic and identify the benefits and possibilities of using new technology, but what are the dangers or drawbacks of using CAS technology?
Enthusiasts generally want to impress, want to show all that CAS can do. They need however to consider what is relevant (less is more?) They also need to be aware that what is natural and easy for them is not necessarily natural or easy for many teachers. (Monaghan, 2002) An overall awareness is required of these factors and to avoid old problems being made too (artificially) complex to justify use of CAS. Traditional problems can remain - polynomial of degree 3 can now handle one of degree 4 etc. However, if CAS is not used in a suitable way (see teaching sequences?), its use may serve only to widen the gap between gifted and ungifted pupils.
d) As mentioned above, the gap between the weaker student and the more able is already large, will using CAS not widen this gap, since there is more emphasis on Formulation and higher order strategies and less on process skills? Furthermore, is there not a danger that a subject already perceived by students as 'difficult' will be made to seem more difficult and thus discourage students from studying further?
Weak students are those who also have difficulty with process skills. However, before the use of CAS the same students must have been frustrated because many of them didn't like training of skills. Rearrangement of terms for example, often prior to the need, made no sense to them. On the other hand, use of CAS provides many opportunities for weaker students to show their creative side, to involve themselves in discussion about the mathematics, and to find more ways to express themselves.
With CAS more ways to solving problems are made possible. 'Social learning' with problems takes place, helping each other with hardware and software, exchanging devices comparing answers, cooperation and brighter helping weaker - the result being an overall change of 'mood' in class! Weaker students often have good ideas but were 'blocked' by manipulation. CAS now sets them free to explore these, and being no longer frustrated - enjoy maths! Using CAS allows some content to be taught at an earlier stage is a well used argument and can be used in either the black box and white box approaches as required.
One of the authors of this book remarked when asked these questions:
When using CAS you become aware that most of the exercises in our textbooks are prepared to get "good" results in order to be solvable by-hand. With CAS you don't have these restrictions. You can e.g. introduce parameters, use realistic data etc., see the result of changes in parameters and data immediately. This often leads to interesting discussions of mathematical models depending on these parameters. Weaker students are mostly those who have also difficulties with or no interest in process skills. In my CAS classes the gap between weaker and more able students was significantly smaller and I had fewer students who dropped studies. With the time you earn with less process skills, you can treat with more interesting demanding tasks which raises the mathematical understanding and interest of the students.
e) `Where is the evidence to show that students ARE more motivated when using CAS (as opposed to graphing calculators)? Is this only anecdotal or is there clear research evidence to support this?
The evidence for this is well supported in the research section of this document and full analysis of ACDCA reports 1 to 4 and personal evidence from the authors concurs. Perhaps the students should have the last word here as indicated in their answers to a questionnaire.
How much did/do you like Mathematics ....
(i) Before working with the TI-92:
|did not like
|liked very much
(ii) Since working with the TI-92:
|did not like
|liked very much
Evaluation by "Centre for School development" ACDCA - Report
f) What can be done with CAS that cannot be done with a graphing calculator?
There are many answers to this as can be seen in the teaching examples. When using CAS the authors believe that students can: generalise with CAS after exploration and investigating, define functions with more than 1 variable, perform the symbolic part of working with functions which cannot be done with the TI-83 Plus for example. They can see multiple representations of a problem which now includes algebraic as well as tabular, graphical, and numerical ones. Teachers can treat examples differently (earlier) and other didactical approaches become possible. Some interesting subject areas become more accessible because CAS takes care of time consuming work and weaker students can obtain exact results from calculations. Students are able to check results at various stages before progressing on to the next part of the problem. CAS can be used to improve basic manipulation skills - algebraic equivalence and fractions in particular - pattern recognising a strength. CAS sets students 'free' to choose own values - frees them from drudgery- but with the responsibility to interpret results! Tables of formulae are no longer necessary.
Again one of the authors commented:
Solving a demanding task with parameters and CAS needs as much skill as 10 routine tasks without CAS. Routine tasks are often made without reflection. This means you can get more with fewer demanding tasks with reduced skills than with a lot of routine tasks. I compared exercises by-hand with CAS classes which I made with former classes without CAS in the domain of derivations. Astonishing results: CAS classes were better although they had much less training with the subject. By-hand you have to follow a strict order in teaching. The scaffolding method with CAS allows you to treat subjects which are much later in the curriculum or even not possible by-hand.