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The Case for CAS

Countries involved and current 'CAS' status


CAS status in Austria

The situation in Austria is characterized by a very early start in the CAS-era. With little tradition in the use of graphic calculators in Austria, the use of CAS in math education and in secondary schools started with the purchase of a national license for Derive™ in the early nineties. This was initiated by the Austrian government, the late Eduard Sziruscek from the Ministry of Education and Helmut Heugl, who continues to be a major force in maths education in Austria.

In 1992 the first DERIVE-conference in Krems brought together math educators from all over the world for the first exchange of experience related to the use of CAS in secondary schools and at university entrance level. Helmut Heugl and a group of enthusiast teachers founded the ACDCA (Austrian Centre of Didactics of Computer Algebra) to encourage teachers to modify their teaching practice to incorporate use of the new tools. Since then four nationwide ACDCA projects have been launched with the aim of supporting teachers in this new CAS-era. In the initial years activities mainly focused on training teachers how to use CAS and on the production of attractive teaching materials. Previous and current ACDCA projects include investigations into the impact of CAS-supported teaching on the culture of problems and teaching, the effect of CAS on assessment and the shift in competencies for students and teachers (e.g. competence of methods, competence of recognizing structures and patterns, competence of designing and grading open question problems, …).

All of these activities were supported by the Austrian government, the Pedagogical Institutes and school authorities in the federal states.

When the TI-92 entered the market the projects were opened for handheld CAS and the number of involved students and educators increased immediately, because CAS could be used without having PCs available. Cooperation with T3™ (Teachers Teaching with Technology™) proved to be very fruitful and made many pre- and in service courses possible.

At the moment we estimate that 25-30% of secondary school students (age 15 - 19) have access to CAS and this percentage is continuously growing. Additionally textbook authors and publishers have started including CAS in their textbooks. The fact that Austria does not have common externally set examinations and that teachers have wide scope in curriculum delivery allows for experimentation in the use of CAS in the classroom and for assessment.

The results of the four ACDCA-projects and numerous CAS-oriented papers can be downloaded from the website www.acdca.ac.at.


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CAS status in Belgium

Secondary education in Belgium is divided into three levels where each level is studied over a two year period. The math curriculum in Belgium has supported the use of graphic calculators in the second level (age 15-16) since September 1999. Nowadays (2003), working with a graphic calculator is widely accepted by teachers and there are textbooks available which incorporate the use of graphic calculators. There are no central math examinations for secondary schools in Belgium.

In 1998, Belgian T3-EUROPE instructors (Teachers Teaching with Technology) commenced CAS-training for teachers in the use of TI-92's. This provided impetus for some didactical projects and experiments related to teaching secondary school mathematics with handheld CAS. However, the extent of CAS use is very dependent on the school and the math teacher. There are some teachers working with CAS (handheld or DERIVE) in a math-oriented education where students have 6 or 8 hours of math weekly.

From September 2004 there will be a new math curriculum for the third level (age 17-18) and it is likely that the use of CAS will be encouraged in that curriculum. Currently students in the second level are allowed to use graphic calculators and textbooks assume that students will have a graphic calculator. It may be more difficult to convince teachers and parents that students in the third level should learn mathematics with access to CAS.


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CAS status in Denmark

General upper secondary education in Denmark (''Gymnasium'') leads to the final examinations, Baccalaureate (''Studentereksamen''). University entrance is based upon results from these examinations. The written examinations are national and the Ministry of Education monitors the examination system. The written examinations in mathematics consist of two parts, paper and pencil (''PAP'') short response questions and extended response questions. For the latter students can use books, notes, tables of formulae, graphing calculators, etc. Since the mid 1990's there has been an alternative written examination, where CAS is allowed for part 2. The majority of questions are identical in the two papers. If a teacher applies to the Ministry of Education and the pupils agree unanimously then permission to take the alternative CAS examination will be given. In the CAS examinations pupils are allowed to use CAS calculators such as the TI-89/92Plus/Voyage 200, or a computer with a CAS programme such as MathCAD/TI InterActive!/Derive.

Although written work is the main focus of The Case for CAS, it is worth mentioning that the Danish Studentereksamen also has oral examinations for mathematics. Orals are considered an important part of the assessment process in the ''Gymnasium'', making a broad assessment possible.

In The Case for CAS we show the 2000 Mathematics Studentereksamen in Denmark, both the PAP paper, and the CAS assumed paper and the solutions are provided for the CAS paper.

In the near future is it expected that CAS will be assumed for all written mathematics examinations in the ''Gymnasium'', however there will still be a PAP component of the examinations.


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CAS status in Scotland

The current situation in Scotland can be summarised in one short statement. Graphing calculators are allowed in examinations but CAS machines are not.

There was a major review conducted in 1998, Advanced Calculators and Mathematics Education (SCCC, 1998)

; which sought the views of interested parties and made recommendations for the future. The review reflected the fact that teachers overwhelmingly endorsed the use of graphing calculators. Opinions were however divided on the use of CAS. The major issue appears to centre over the use of CAS in the central examinations. This despite the fact that these examinations now have two papers, one of which relies on pencil and paper only.

The examination authority, Scottish Qualifications Authority (SQA) is now actively promoting graphing calculator solutions to certain questions in the second paper. It recently (November 2002) issued advice to schools outlining expected solutions to selected questions. There is no apparent movement to relax the ban on CAS. This despite 'home grown' evidence to suggest that use of CAS does not have a detrimental effect on student's 'basic skills' (MacIntyre and Forbes, 2002). It seems that the central examination is the barrier to exploiting the full potential of CAS. Regardless of personal visions and expertise in harnessing the technology, staff is going to be discouraged and students themselves will be reluctant to make use of CAS calculators whilst they remain 'banned in assessments (MacIntyre and Forbes, 2002). This paper aims to provide the necessary weight of evidence to convince those in power to remove the ban.


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CAS status in Switzerland

The Swiss school system has some peculiarities. First of all, a population of about 7 million is split into four official languages and secondly, the country has an extremely decentralised system, resulting in 26 different autonomous education departments (cantons) with only minimal influence from the central government. The process of learning how to use technology in mathematics teaching has a rather lengthy history going back to the 1980s. Essentially there are three phases:

a. In the 1980's there were trials by pioneer teachers developing programs for algorithms to carry out numerical approximation on desktop computers.

b. In 1992 a large conference took place in Baden (3 days). Demonstrations of Mathematica, Maple and Derive at workstations and graphing calculators sparked lively discussions, not only among the participants but also in the wider mathematical education community. Following the conference, several schools ordered site licences for Mathematica or Maple and permitted the use of CAS-systems in the "ghetto of computer rooms". Derive and graphing calculators were only used by a few schools. One notable exception was the vocational secondary schools, which integrated graphing calculators into the mathematics curriculum.

In 1996 the first secondary school classes were equipped with CAS handheld calculators (TI-92). At the same time the program of T3-Switzerland became active. Many Swiss secondary school teachers recognised that CAS handheld technology provides the opportunity for a new challenging way of teaching. The situation after 1996 resembles the spread of an infection. In spite of the immunity of the universities against this infection, the CAS virus spread throughout the country. According to an investigation in 2002 more than 60% of secondary II students in Switzerland are using CAS or graphing calculators making Switzerland the leading European country in this domain. It is clear, however, that an efficient integration lags behind this impressive number in many cases. In order to reach the less innovative teachers there is an urgent need for new textbooks and problem sets adapted to good practice in combining the potential of handheld CAS with the need to shape young brains. Activities in this direction are on the way. We hope that The Case for CAS will be another milestone in this direction.


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