HOW CAN TEACHING AIDS IMPROVE THE QUALITY OF MATHEMATICS EDUCATION by Ahmed Afzal

1. HOW CAN TEACHING AIDS IMPROVE THE QUALITY OF MATHEMATICS EDUCATION? by Ahmed Afzal Melanie Schmid summer term 2006 Prof. Schl�ter & Prof. Ludwig

2. Introduction Afzal: �The interplay among and connections between objects, images, language and symbols that lead to mathematical reasoning and the stating of mathematical propositions of very wide generality is well worth closer study.�

3. Index of contents 1. Teaching aids and their use 2. Aims of teaching mathematics influence the use of teaching aids 3. Characteristics of teaching aids 4. Computer as a medium for teaching aids 4.1. Examples from a project on linking algebraic and geometric reasoning 4.2. Making the power of computer tools accessible to teachers

4. 1. Teaching aids and their use UK: teachers tend to look for tools and �good ideas for teaching� ?if they work? repertoire ?if not? look for sth. else

5. example: use of fraction blocks

6. result of the example: Dickson(1984): �One of the difficulties with fractions, decimals and percentages is that they have a multiplicity of meanings.� ? interpretation ? in contrast: �whole numbers, which are used mainly either for counting [�] or for measuring [�]�

7. example: triangle? interior angle sum draw a triangle proof: a+�+?=180� How can we state with a particular triangle such proof? ?�In this case the triangle really is an idea, not an object.� ?Papert (1980): �object-to-think-with� many questions? constant research with the active involvement of teachers

8. 2. Aims of teaching mathematics influence the use of teaching aids �Mathematical procedures are taught to all the school pupils because they will help them in everyday life as well as in application.� BUT: �95% of the population will need less than 5% of the procedures [�] for everyday life or for applying for sciences, industry and commerce.� ?main task of teaching mathematics: not the contents, but processes e.g. abstractions, generalisation, logical thinking�

9. 3. Characteristics of teaching aids Biggs (1972): 5 categories in the process of discovering mathematics: fortuitous, free and exploratory, guided, directed, programmed but: fortuitous cannot be planned, and programmed is a directed learning sequence ?need for material which encourages pupils and supports their mathematical development

10. Dewey (1966): �PLAY as being value at all levels of development and maturation� 2 goals:

11. Characteristics of mathematical tools: They must allow student-centred activity with the student in charge of the process. They utilise students� current knowledge. They help develop links between students� current mental scheme while they are interacting with the tools. They reinforce their current knowledge. They assist future problem solving/mathematical activity through enhancing future access to knowledge.

12. Result BUT: �Tools cannot ensure that a particular understanding will come about.�

13. 4. Computer as a medium for teaching aids The computer is able to provide connections between aspects of mathematics and experiences planted in everyday life. We have to find ways to exploit these linkings for using them in school.

14. 4.1. Examples from a project on linking algebraic and geometric reasoning Examples from a current project for the UK�s Qualifications and Curriculum Authority (QCA) Linking algebraic and geometric reasoning with dynamic geometry software Possibility �to bring images from the outside world into the mathematics classroom�

15. Using geometric software

16. Possible tasks explore �geometric ideas of perspective by drawing lines joining corresponding points� �explore numerical ideas of perspective by taking measurememts from the image� �

19. Possible tasks find out the height of the fountain ?afterwards compare the results with the actual height (use the Internet) velocity with which the water leaves the dragon�s mouth angle at which the water enters the harbour

21. Result That�s �an example where we have the technology, but not yet a clear body of what we would call �best-practice� in its educational use�. seldom and rare use in the classroom

22. 4.2. Making the power of computer tools accessible to teachers UK: between 1999 and 2003: 320,000,000 � for additional training for school teachers of course �they (teachers) know the general advantages, but remain unaware of the potential of specific software and tools� two questions:? How do the technological tools enhance the teaching and learning process?? How do teachers perceive the technology in relation to the mathematics that is being learned?

23. Professional development of activities for teachers

24. Results Most teachers knew the result, but other said that they had never approached the teaching of quadratic functions in this way with their pupils. Laborde (2001): �The role played by technology moved from being a useful amplifier towards being an essential constituent of the meaning of tasks.� Papert (1980): �We are learning how to make computers with which children love to communicate. When this communication occurs, children learn mathematics as a living language.�

25. 5. Final remarks Afzal: �How materials are used is the most important factor, since teachers can use: good materials well, good materials badly, bad materials well and bad materials badly.� ?dependence on the classroom tasks, role of the teacher and the climate and atmosphere of the classroom

26. Questions???