Teaching students to think mathematically

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Teaching students to think mathematically

• Kaye Stacey
• University of Melbourne
• k.stacey_at_unimelb.edu.au

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Teaching students to think mathematically.

• I will discuss a mathematical problem which can
  be used to teach students to think mathematically
  and to solve mathematical problems that are
  unfamiliar and new to them. The processes of
  looking at special cases, generalising,
  conjecturing and convincing will be highlighted
  through these examples, These are key processes
  in thinking mathematically.

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Principles

• Mathematical thinking is an important goal of
  schooling
• Mathematical thinking is important as a way of
  learning mathematics
• Mathematical thinking is important for teaching
  mathematics
• Mathematical thinking proceeds by
• specialising and generalising
• conjecturing and convincing

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Andrew Wiles Doing mathematics is like a journey
through a dark unexplored mansion.

• One enters the first room of the mansion and its
  dark. One stumbles around bumping into furniture,
  but gradually you learn where each piece of
  furniture is. Finally, after six months of so,
  you find the light switch, you turn it on, and
  suddenly its all illuminated. You can see
  exactly where you were. Then you move into the
  next room and spend another six months in the
  dark. So each of these breakthroughs, while
  sometimes theyre momentary, sometimes over a
  period of a day or two, they are the culmination
  of, and couldnt exist without, the many months
  of stumbling around in the dark that precede
  them.

Andrew Wiles proved Fermats Last Theorem in
1994. First stated by Pierre de Fermat, 1637.
Unsolved for 357 years. Quoted by Simon Singh
(1997)
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What is mathematical thinking?

• Used in solving problems using mathematics and in
  conducting a mathematical investigation
• It is useful for teachers and students to think
  about its structure and parts

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Solving problems successfully requires a wide
range of skills
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Deep mathematical knowledge
General Reasoning abilities
Solving problems successfully requires a wide
range of skills
Personal Attributes e.g. confidence,
persistence, organisation
Heuristic strategies
Problem solving strategies
Abilities to work with others effectively
Helpful Beliefs and Attitudes e.g.
orientation to ask questions
Communication Skills
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Deep mathematical knowledge
General Reasoning abilities
Mathematical thinking involves a wide range of
skills
Personal Attributes e.g. confidence,
persistence, organisation
Problem solving strategies
Abilities to work with others effectively
Helpful Beliefs and Attitudes e.g.
orientation to ask questions
Communication Skills
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Teaching can improve all of these components if
it contains

• EXPERIENCE - solving non-routine problems in a
  supportive classroom environment
• REFLECTION - active reflection so that students
  learn from these experiences
• STRATEGIES learning about effective problem
  solving strategies and good habits, and the
  processes of mathematical thinking (e.g.
  importance of reasons why).

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Thinking Mathematically

• proceeds by alternating between 4 fundamental
  processes
• Specialising Generalising
• Conjecturing Convincing

Mason, Burton, Stacey Thinking Mathematically
Pearson
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Thinking Mathematically

• proceeds by alternating between 4 fundamental
  processes
• Specialising Generalising
• Conjecturing Convincing

Trying special cases, looking at examples
Mason, Burton, Stacey Thinking Mathematically
Pearson
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Thinking Mathematically

• proceeds by alternating between 4 fundamental
  processes
• Specialising Generalising
• Conjecturing Convincing

Looking for patterns and relationships
Mason, Burton, Stacey Thinking Mathematically
Pearson
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Thinking Mathematically

• proceeds by alternating between 4 fundamental
  processes
• Specialising Generalising
• Conjecturing Convincing

Predicting relationships, results
Mason, Burton, Stacey Thinking Mathematically
Pearson
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Thinking Mathematically

• proceeds by alternating between 4 fundamental
  processes
• Specialising Generalising
• Conjecturing Convincing

Finding and communicating reasons why
Mason, Burton, Stacey Thinking Mathematically
THAILAND Pearson
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The Circle and Spots Problem

• Some spots are placed anywhere on the
  circumference of a circle and every pair of spots
  is joined by a straight line. Into how many
  regions is the circle then divided?
• Martin Gardner, Scientific American, 1969

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Circle divided into 8 regions
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Specialise try examples, looking to see what
happens and why
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Labelled tables to organise information
systematically
Thats a funny region
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Generalise Look for a pattern AHA!
2n
Thats a funny region
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Conjecture R(n) 2n-1
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6 spots31 regions (max) 30 regions
(min)Discuss difficulty of accurate counting
with students
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Define R(n) more carefullyR(n) max number of
regions with n spotsCheck conjecture by
collecting more data (more specialising) - not 2n
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Define R(n) more carefullyR(n) max number of
regions with n spotsCheck conjecture by
collecting more data (more specialising) - not 2n
STUCK!
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• Fit a polynomial
• R(n) (n4 6n3 23n2 18n 24)/24
• Prove by mathematical induction - OK
• This works,
• . . . . . . . . .but it isnt interesting!

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What might be going on?

• Renewed specialising looking for REASONS
• What happens when I add a point?

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What might be going on?

• Renewed specialising looking for REASONS
• What happens when I add a point?

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What might be going on?

• Renewed specialising looking for REASONS
• What happens when I add a point?

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What might be going on?

• Renewed specialising looking for REASONS
• What happens when I add a point?

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What might be going on?

• Renewed specialising looking for REASONS
• What happens when I add a point?

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What might be going on?

• A new region for every chord
• Another region when the new chord intersects an
  existing chord

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What might be going on?

• nC2 is the number of chords
• nC4 is the number of intersections of chords

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Counting intersections of chords

• nC4 is the number of intersections of chords

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Counting intersections of chords

• nC4 is the number of intersections of chords

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Counting all regions

• One region to start (1)
• Every chord creates one new region (nC2 chords)
• Every point of intersection of chords creates one
  new region (nC4 intersections)

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• Fit a polynomial
• R(n) (n4 6n3 23n2 18n 24)/24
• This works, but it isnt interesting!
• R(n) 1 nC2 nC4
• This works and it is interesting!
• (Many small cycles of conjecturing and convincing
  to get here!)

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Im convinced
Fantastic! R(n) is sum of first five terms of
Pascals triangle!
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Mathematicians love to generalise!

• Does it have to be a circle?
• Do I need all the spots joined?
• What is the general result here?

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Brousseaus formula (1966)Number of regions
1 number of chords number of intersections
inside (count multiplicities)
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Brousseaus formula (1966)Number of regions (9)
1 number of chords (5) number of
intersections inside (3)
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Minimum number of regions (unsolved problem
last time I looked!)
?
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Teaching can improve all of these components if
it contains

• EXPERIENCE - solving non-routine problems in a
  supportive classroom environment
• REFLECTION - active reflection so that students
  learn from these experiences
• STRATEGIES learning about effective heuristic
  strategies, good problem solving habits, and the
  processes of mathematical thinking (e.g.
  importance of reasons why).

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• Patterns in mathematics are marvellous
  sometimes even when they dont work!
• Students need to understand that it is not enough
  to spot a pattern you have to find a reason.
• Connections are everywhere in mathematics
• These processes can be part of the mathematics
  education of all students.
• Specialising and Generalising
• Conjecturing and Convincing
• Thinking Mathematically

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Thank you