Working Group 6:
Curricular questions
Rolf Neveling
Wuppertal , Germany
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Danish experiences with technology in mathematics teaching in upper secondary schools |
1. Our results were the following
The working group CQ discussed, which aspects concerning learning with new technologies need to be considered when curricula are modified. The participants agreed that at most suggestions can be put forward and that at the moment an attempt at completeness is neither possible nor desirable.
The group first listened to a paper on changes
planned in
In the following elements of present-day concepts of mathematics were discussed which led to some recommendations regarding learning with new technologies.
1 Our results were the following
Mathematics studies patterns. A lot of these are relevant in our society even though nobody sees them. Learning mathematics - among other things - means getting to know some relevant patterns. In this process, however, only such mathematical topics are useful which are recognizably valuable for the later lives of the young people.
A mathematical problem is especially suitable for learning processes if it can be dealt with from different perspectives, e. g. a conceptual, an information technological and a historical perspective. Problems should challenge creativity; they should make pupils to experiment. Furthermore aspects of application are desirable. In short, mathematical problems ought to be chosen so as to enable pupils to realise the importance of these problems.
In their later professional lives our students will use computers largely as black boxes. Therefore working with new technologies at schools should allow an insight into the black box to demystify the machine - for example by conveying an idea of what a programme is, what it can and what it cannot do.
Curricula ought to show clearly that the use of new technologies require a change of the teacher's role. This new mode of teaching is generally characterized as coaching, i.e. the teacher considers himself rather as an adviser than as an instructor of his pupils.
When working with new technologies coaching as a form of teaching develops naturally, besides, at present it is regarded as the relevant form for many learning processes.
2 A final aspect
The use of new technologies in maths teaching directly depends on the teaching objectives. If the only aim when computing the length of
is achieving a numerical value, it is no doubt appropriate to use the black box CAS. If, however, the aim is to understand why the ln occurs in the exact solution CAS will of course not be used. Understanding means realising relations.
New technologies in maths teaching require a discussion, which as yet is just at the beginning. Comprehensive conceptions are not to be expected at present, our work could therefore not go beyond an initial stage.
Danish experiences with technology in mathematics teaching in upper secondary schools
Nils Fruensgaard
Dronninglund, Denmark
3. On-going experiments with CAS
4. New experimental curriculum with enhanced use of ICT
5. ICT-created problems in mathematics curriculum
A co-operation between the Association of Mathematics Teachers and the Ministry of Education has resulted in building a new experimental curriculum, based on enhanced use of ICT, which teachers are encouraged to use in their classes in grade 10-12 in the Gymnasium. The old and new curricula are presented, with emphasis on the national written assignments and the intended use of technology in the experimenting classes. The general curriculum problems that ICT has created in teaching mathematics are discussed.
1 So, - what is the problem?
In the last 5 years it has been affordable for every school to make extensive use of ICT in mathematics teaching. In very many countries the schools are equipped with sufficient machinery, so that students can be working in couples in the PC-lab. And the school has a laptop and a video projector for classroom use. The software could consist of Derive, Cabri-Geometry, PC-Autograph, Excel, Math-editor, Graphmatica, Basic and Logo - all programmes of very high quality and with a total cost of 1500 EURO (1000£) for a lifelong site license. There are a lot of literature and ready-to-use material to support this software. There will be activities for more than 500 lessons in all topics within the mathematics curriculum.
So, - what is the problem? The problem is that nobody has a serious didactical proposal for a new curriculum with ICT that will gain a broad support among teachers. It is a very big task to reformulate the total mathematics curriculum, which has been at work for many decades. And it is not just a mathematical reformulation - today we also have to think in words like student activity, learning goals, project periods, competence building and so on. In Denmark we try to make small steps and gradually change the curriculum and the final written examination problems. I will give a review of the Danish ICT history in the last 5 years.
2 Old curriculum
The Danish primary school consists of 9 compulsory years in unparted mixed-ability classes. Then the student can choose different paths. About 50% choose an upper secondary school for three years. And a total of 20% choose mathematics at the highest level. It is mostly these 20% that have my concern here. The mathematics curriculum for the three years in the Danish Upper Secondary School (Gymnasium) can be viewed in English at the website of the Danish Ministry of Education:
All students have graphics calculators like a TI-83 and the final written examination is based on the use of these calculators. An English version of the examination paper from 1996 can be downloaded from the homepage of our Teachers Association:
In the last couple of years the written examinations have been divided in two. The first part of one hour is made only with paper and pencil and the second part of three hours is made with all help including books and notes, but no CAS-tools. These examination papers from the last few years can be found in Danish at
www.uvm.dk/gymnasie/almen/eksamen/opgaver/
3 On-going experiments with CAS
Every year within the last 5-10 years there has been about 10 schools, which have worked intensively with CAS. The students have been given an alternative examination paper, which included 2-3 problems, which should be solved with their CAS-tool, which could be a TI-92 or MathCad, Derive or another PC-based CAS. These experiments use the old curriculum and have not made extensive use of other PC-based math tools like Spreadsheets or Dynamic Geometry or Internet. From 2001 any school that wishes can use the special CAS examination paper and the students can bring their PC or borrow from the school when they do their final written examination. There will still be a one-hour part only with paper and pencil without any technical tools or books. English versions of examples of problems for the CAS experiment can be found at
4 New experimental curriculum with enhanced use of ICT
A working group under the Association of Mathematics Teachers (Matematiklaererforeningen) has made a proposal for a new curriculum with enhanced use of ICT. Beside CAS there is included work with dynamic geometry, spreadsheets, Internet, layout with math-editor and graphing tools. We have omitted about 10-20% of the traditional subjects from the curriculum and stated that this extra time shall be used for ICT work with optional mathematics subjects. This work will not be assessed at the written examination but at the oral examination. We hope that some schools will use this new curriculum in the next years, so that we can get some experiences that can be used in the coming debate about the future structure of our gymnasium. If mathematics wants to keep its strong position in our educational system we must show that we can adapt to the new demands for knowledge and skills.
5 ICT-created problems in mathematics curriculum
Mathematics is the subject in our Gymnasium where ICT has the greatest impact. Very many of our basic disciplines are not worthwhile teaching anymore because the machine does the job. Calculations of angles and sides in triangles, derivatives and antiderivatives, solving differential equations, solving equations are old-fashioned pastime activities that are of no use any longer. When my students ask me where to use the math they're learning or why they should work with this math I used to have good reasons. But now I often run dry in arguing for the reasons of the topics I'm teaching. So I try to find new activities and use less time on the old topics. I turn to new ways of planning my teaching. Find historical topics, find authentic applications, do games and creative and experimental activities, discuss different types of mathematical reasoning and use ICT massively.
