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

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

• Melanie Schmid
• summer term 2006
• Prof. Schlüter Prof. Ludwig

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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.

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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

• 5. Final remarks

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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

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example use of fraction blocks
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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
  

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example triangle? interior angle sum

• draw a triangle
• proof aß?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

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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

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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

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• Dewey (1966) PLAY as being value at all levels
  of development and maturation
• 2 goals

short-term goal complete freedom of the solvers
long-term goal solution of the problem
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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.

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Result

• BUT Tools cannot ensure that a particular
  understanding will come about.

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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.

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4.1. Examples from a project on linking algebraic
and geometric reasoning

• Examples from a current project for the UKs
  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

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Using geometric software
Picture of the roof structure of Stockport
railway station, near Manchester
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Possible tasks

• explore geometric ideas of perspective by
  drawing lines joining corresponding points
• explore numerical ideas of perspective by taking
  measurememts from the image

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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 dragons
  mouth
• angle at which the water enters the harbour

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• Quadratic function f(x) ax2
• utilities to vary a

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Result

• Thats 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

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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?

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Professional development of activities for
teachers

• two linear functions y1(x) and y2(x)
• Teachers are asked to predict what the resulting
  graph of y1(x)y2(x) might look like.

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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.

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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

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• Questions???

Thanks for your attention!