Smith and Jones

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Smith and Jones

• Mr Smith and Mr Jones are two maths teachers, who
  meet up one day. Mr Smith lives in a house with a
  number between 13 and 1300. He informs Mr Jones
  of this fact, and challenges Mr Jones to work out
  the number by asking closed questions.
• Mr Jones asks if the number is bigger than 500.
  Mr Smith answers, but he lies.
• Mr Jones asks if the number is a perfect square.
  Mr Smith answers, but he lies.
• Mr Jones asks if the number is a perfect cube. Mr
  Smith answers and (feeling a little guilty) tells
  the truth for once.
• Mr Jones says he knows that the number is one of
  two possibilities, and if Mr Smith just tells him
  whether the second digit is 1, then he'll know
  the answer. Mr Smith tells him and Mr Jones says
  what he thinks the number is. He is, of course,
  wrong.
• What is the number of Mr Smith's house?

www.nrich.maths.org April 2004
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What is mathematics enrichment anyway?

• Jennifer Piggott
• July 2005
• www.nrich.maths.org

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Outline

• Proposals
• Consequences
• Time to reflect

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(No Transcript)
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Some proposals

• Depth
• Breadth
• Balance
• Relevance
• Acceleration
• Extension
• Extra to normal classroom practice
• Provision for the most able

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Depth

• The measurement from the top down, from the
  surface inwards, or from the front to the back
• difficulty, abstruseness
• comprehensively, thoroughly or profoundly
• intensity of emotion.

Extracts from the Oxford English Dictionary
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Breadth

• The distance from side to side of a thing
• extent, distance, room
• freedom from prejudice or intolerance

Extracts from the Oxford English Dictionary
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Balance

• An amount left over
• harmony of design and proportion
• offset or compare one with another
• establish equal or appropriate proportions
• choose a moderate course or compromise
• zodiacal sign.

Extracts from the Oxford English Dictionary
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Possibilities so far

• Harmony of design and proportion (balance)
• Extent (breadth)
• Freedom from prejudice and intolerance (breadth)
• Thorough and comprehensive (depth)
• Emotional involvement (depth)

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Four-points

• There are four points on a flat surface
• How many ways can you arrange those four points
  so that the distance between any two of then can
  be only one of two lengths
• Example

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Relevance

• Bearing on or having reference to the matter in
  hand.
• Real world
• Actually existing or occurring

The Oxford English Dictionary
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Acceleration - Extension

• Acceleration is the intentional exposure of
  pupils to more advanced standard curriculum
  subject matter with the specific aim of
  examination on that material in advance of
  chronological age.
• Extension is the exposure of pupils to content
  not normally found in standard curriculum and
  which might be considered appropriate to that
  chronological age or older
• the opportunity to learn new mathematical content
  or techniques
• application of an area of mathematics to
  different contexts not normally covered within
  the curriculum
• the study of mathematics as a cultural, social or
  historical phenomenon .

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And finally

• Extra to normal classroom practice
• Trips
• Activities
• Clubs
• Aspiration raising

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Aspiration

• Long term gains for pupils in terms of their
  attitudes to and understanding of what it is to
  be mathematical by
• improving pupil attitudes,
• developing an appreciation of mathematics as a
  discipline.

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Consequences a view of enrichment
What I have described involves a level of
engagement with the subject on a personal and
social as well as an intellectual level, which in
turn has implications for

• Content
• Teaching
• Aspiration raising
• Audience

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Content

• Engaging contexts
• Extend knowledge
• Challenging knowledge and conceptions
• Makes connections
• Offers opportunity for a developing interest
• Involves problem solving, problem posing and
  mathematical thinking.

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Teaching

• Encourages pupils to be mathematical by building
  on appropriate content and
• uses effective mediation
• engages with the mathematics as a community
  communicating
• encourages independent, critical thinkers
• values the individual and different approaches
  but also encourages critical evaluation of
  efficient methods
• makes use of metacognition and misconceptions.

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Problem solving

• Understanding the problem
• Devising a plan
• Carrying out the plan
• Looking back
• Polya 1957

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CAPE model

• Comprehension
• Making sense of the problem/retelling/creating a
  mental image,
• Applying a model to the problem,

• Analysis and synthesis
• Applying facts and skills, including those listed
  in mathematical thinking (below),
• Identifying possible mathematical knowledge and
  skills gaps that may need addressing,
• Conjecturing and hypothesising

• Evaluation
• Reflection and review of the solution,
• Are there more questions to answer?
• Self assessment about ones own learning and
  mathematical tools employed,
• Communicating results,

• Planning and execution
• Considering novel approaches and/or solutions,
• Planning the solution/mental or diagrammatic
  model,
• Execution of solution,

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Using Subgoals

• Place the numbers 1-9 in a 3x3 magic square
• How many zeros appear at the end of 100! ?
• Find the sum of all the mulitples of 4 or11 in
  the integers from 1 to 1000
• Consider the groupings (1), (2,3), (4,5,6),
  (7,8,9,10), What is the sum of the digits in the
  kth grouping?
• How many rectangles can be drawn on a 17 x 31
  magic grid?
• Shoenfeld 1985

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Mathematical Thinking

• Mathematical strategies that are employed in
  solving the problems
• Type I examples
• Generalising (as identifying patterns general or
  common patterns formula looking for an
  essential shape or form)
• Being systematic
• Mathematical analogy
• Type II examples
• Introducing variables
• Specialising, looking for a particular case
  (specific action that comes out of the problem
  doing a particular thing to help to simplify,
  e.g. paper folding)
• Solving simpler related problems
• Working backwards.

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Purposes of problem solving

• For- problem solving seen as mathematical
  activity in its own right, often with problems
  designed to extend or connect mathematical
  concepts and undertaken explicitly for the
  purpose of being mathematical
• About- involving the overt teaching of problem
  solving skills, teaching about how to problem
  solving
• Through- teaching mathematical concepts through
  problem posing.

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For - Pentagonal

• Can you prove that the sum of the distances of
  any point inside a square from its sides is
  always equal (half the perimeter)?Can you prove
  it to be true for a rectangle or a regular
  hexagon?Does the hexagon need to be
  regular?Can you show the same is the case for a
  regular pentagon? Does the pentagon need to be
  regular?

www.nrich.maths.org June 2005
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About - Isometrically

• How many unique symmetrical shapes can you make
  by shading four small triangles?

www.nrich.maths.org Oct 2003
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Through Subtended Angles

• Choose two points on the circumference of the
  circle. Call them A and B. Join these points to
  the centre, C. What is the angle at C?Join A
  and B to a point on the circumference. Call that
  point D. What is the angle at D?If the angle at
  D is acute, what do you notice about the angles
  at C and D?If the angle at D is obtuse, what is
  its relationship with the reflex angle at
  C?What happens if you choose a different point
  D?What happens if you choose a different pair of
  points for A and B?Would the same thing happen
  if you started with any two points on the
  circumference of any circle?Can you prove it?

www.nrich.maths.org July 2005
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Descriptions of a good problem situation

• Related to the initial impact of the problem or
  context
• uses succinct clear unambiguous language,
• draw the solver in and offers intriguing contexts
  such that solving them feels worthwhile,
• gives opportunities for initial success but have
  scope to extend and challenge (low thresh hold
  high ceiling problems).
• Related to the experience for the solver
• encourages solvers to think for themselves and to
  apply what they know in imaginative ways,
• gives the solver a sense of slight unease at
  first
• Related to the problem
• allows for different methods which offer
  opportunities to identify elegant or efficient
  solutions,
• opens up patterns in mathematics and leads to
  generalisations,
• reveals underlying principles and can lead to
  unexpected results,
• requires a solution that calls for a good
  understanding of process and/or concept
• draws together different mathematical concepts or
  branches of mathematics.

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The trick
What is the missing term in 6, 11, 12, ,
110?
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Enrichment
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Big wheel
100 mph
100 miles
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Squirty

• Using a ruler and compass only it is possible to
  fit a square into any triangle so that one side
  of the square rests on one side of the triangle
  and the other two vertices of the square touch
  the other two sides of the triangle
•  

www.nrich.maths.org May 2004