The Formation of Mathematical Concepts

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The Formation of Mathematical Concepts
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Abstracting and Classifying

• Abstracting
• Invariant properties of examples are abstracted
• these memories persist longer in the memory than
  particular instances
• an activity by which we become aware of
  similarities among our experiences
• product of abstraction -- concept
• Classifying
• similarities of an instance with a class are
  noted
• collecting together our experiences on the basis
  of these similarities

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Abstraction
Ch
Concepts
C
C
C
Abstraction
Classification
C1
C2
Cn
C3
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Classifying

• Naming an object classifies it.
• We then know how to behave in relation to it.
• Once classified, less open to other
  classifications

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Concept

• Requires for its formation a number of
  experiences which have something in common
• everyday concepts come from everyday experience
• process involved
• generalization
• discrimination

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Naming A Concept and Its Name

• Example given by Vygotsky
• Children were told that in a game a dog would be
  called cow. The following is a typical sequence
  of questions and answers
• Does a cow have horns?
• Yes. But dont you remember that the cow is
  really a dog? Come now, does a dog have horns?
• Sure, if it is a cow, if its called cow, it has
  horns. That kind of dog has got to have little
  horns.

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• Another example
• A peasant who, after listening to two students of
  astronomy talking about the stars, said that he
  could understand that with the help of
  instruments people could measure the distance
  from the earth to the stars and find their
  positions and motion. What puzzle him was how in
  the devil they found out the names of the stars.

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• Other Examples
• (1)
• Hong Kong is an island
• Hong Kong is a bisyllable word
• (2) Number and numerals

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Naming

• A concept is an idea the name of a concept is a
  sound, or a mark on paper, associated with it.
• This association can be formed after the concept
  has been formed, or in the process of forming it.
• The use of a name help classifying object e.g.
  This is a kind of bottle opener
• useful and sometimes essential in the formation
  of new concepts -- different names heard together.

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Communication of Concept

• Can a concept be communicated by simply verbally
  defining it?
• Two ways to describe red
• Red is the colour we experience from light of
  wavelength in the region of 0.6 microns
• pointing to various objects and say This is a
  red tie, This is a red tie,...

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Two Kinds of Concept

• Primary concepts those which are derived from
  our sensory and motor experiences of the outside
  world, such as red, motor car, heavy,...
• Secondary concepts those which are abstracted
  from other concepts, such as ....
• Of higher order means abstracted from (directly
  or indirectly)
• more abstract means more removed from
  experience of the outside world

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Concepts of a higher order than those which
people already have cannot be communicated to
them by a definition but only by collecting
together, for them to experience, suitable
examples.
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Use of Definitions

• A way of adding precision to the bounaries of a
  concept, once formed,
• stating explicitly its relation to other
  concepts.
• Communicate new concepts of lower order, e.g.,
  It is a colour, between red and blue, rather more
  blue than red

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The criterion for having a concept is not being
able to say its name but behaving in a way
indicative of classifying new data according to
the similarities which go to form this concept.
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Ways to evoke a concept

• Encountering an example of the concept --
  recognition
• hearing, reading or otherwise making conscious
  the name, or other symbol, for the concept --
  naming
• the ability to isolate concepts from any of the
  examples enable the formation of new concepts of
  greater abstraction

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If we cannot detach concepts from the experience
(using language)

• Each individual has to form their own concepts
  directly from the environment
• without language, the primary concepts cannot be
  brought together to form concepts of higher
  order.
• Concepts of the past not available

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The Power of Conceptual Thinking

• Ability to combine and relate many different
  experiences and classes of experience. The more
  abstract the concepts, the greater their power to
  do this.
• Dont worry me with theory -- just give me
  the facts ?
• Appropriate theory enables us to explain, predict
  and control a great number of particular events
  in the classes to which it relates.
• Learning by using abstract symbols is meaningless
  to most students.

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The Leaning of Mathematical Concepts

• The particular problem (but also the power) of
  mathematics lies in the great abstractness and
  generality, achieved by successive generations of
  particularly intelligent individuals each of whom
  has been abstracting from, or generalizing,
  concepts of earlier generations
• mathematics cannot be learnt directly from the
  everyday environment, but only indirectly from
  other mathematicians, in conjunction with ones
  own reflective intelligence.

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Principles of learning mathematical concepts

• Concepts of a higher order than those which
  people already have cannot be communicated to
  them by a definition, but only by arranging for
  them to encounter a suitable collection of
  examples
• since in mathematics these examples are almost
  invariably other concepts, it must first be
  ensured that these are already formed in the mind
  of the learner

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Examples
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Learning and Teaching

• Although we have to create all the concepts in
  our own minds, we are only able to do this by
  using the concepts arrived at by past
  mathematicians.
• Very dependent on good teaching
• to know mathematics
• to communicate it to those at a lower conceptual
  level.