Teachers

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Teachers Mathematical Thinking

• Kaye Stacey
• University of Melbourne, Australia

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Mathematical thinking

• is an important goal of schooling
• is important as a way of learning mathematics
• is important for teaching mathematics
• in planning lessons
• for analysing subject matter
• for planning lessons with a specified aim
• for anticipating students responses
• in conducting lessons, minute by minute

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Teachers make decisions during lessons very
evident in yesterdays lessons
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Teachers make decisions during lessons very
evident in yesterdays lessons
What is the role of mathematical thinking at
these decision points?
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Data collected by Helen Chick

• Grade 5 and 6 teachers in Australia
• Pairs of lessons on their choice of topic (up to
  8 lessons per teacher)
• Lessons were video-taped and observed
• Follow-up interview about the lessons
• Lessons analysed to examine teachers use of
  pedagogical content knowledge (Chick) and (here)
  mathematical thinking

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The Spinners Game

• Two teachers chose to use this game, described in
  a teachers resource book
• Two players
• Two spinners numbered 1 to 9
• Spin both spinners
• Add the two numbers
• If sum even, Player 1 gets a point
• If sum odd, Player 2 gets a point
• Winner is first to 10
• Students instructed to play a fewtimes and
  determine the fairness

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Spinners Game Rules
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
Spinner 1
Spinner 2

• Spin each find if sum is even or odd
• Play ten times
• Which is more likely ?
• more even sums
• more odd sums

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Spinners Games Rules
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
Spinner 1
Spinner 2

• Spin each find if sum is even or odd
• Play ten times
• Which is more likely ?
• more even sums
• more odd sums

3 5 is even
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Spinners Game Rules
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
Spinner 1
Spinner 2

• Spin each find if sum is even or odd
• Play ten times
• Which is more likely ?
• more even sums
• more odd sums

7 8 is odd
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Your Turn Which is more likely?
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
Spinner 1
Spinner 2

• Spin each find if sum is even or odd
• Play ten times
• Which is more likely ?
• more even sums
• more odd sums

7 8 is odd
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Mathematics in the Spinners Game

• Game can support lesson goals about
• Sample space
• Likelihood (probability)
• Short-term and long-term behaviour
• Theoretical versus experimental probability
• Fairness
• Reasoning about addition of odd and even numbers

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Key Features of Spinners Lessons

• Teachers value
• practical experience of probability before theory
• link to everyday life (e.g. fair games, odds)
• building concepts through class discussion
  (socio-constructivist)
• Lessons generally follow Isodas problem
  solving, child-centered approach
• Curriculum not prescribed much freedom
• Ordinary teachers with ordinary lessons and
  ordinary preparation

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Whats wrong with the Spinners Game

• Teachers guide gave virtually no guidance
• Large sample space to enumerate (81 outcomes)
• Likelihood of odd sum is 40/81, even sum is 41/81
  This difference is not perceptible in classroom
  experiences of first to 10 game.
• A hard problem probably only meant to be
  experiential!
• Difficulties added to richness to study teachers
  mathematical thinking!

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Irenes lesson

• Started spinners game late in lesson
• Students played for a few minutes (so just a
  little empirical evidence about fairness) before
  Irenes discussion
• Two short class discussions
• fairness
• how a small difference in likelihood would effect
  the first to ten game.

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Discussion on fairness of game

• Some students talked about following the rules of
  the game.
• Many students talked about numbers of odds and
  evens 5 odds and 4 evens on each spinner
• Student gave erroneous parity argument
• odd odd even
• even even even
• odd even odd

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Discussion on fairness

• Some students talked about following the rules of
  the game.
• Many students talked about numbers of odds and
  evens 5 odds and 4 evens on each spinner
• Student gave erroneous parity argument
• odd odd even
• even even even
• odd even odd

two out of three chances for even sum to win
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How could Irene respond?

• Respond to
• the whole argument?
• the conclusion (even sum more likely)?
• Respond to
• just that student?
• the class, drawing them into this argument?
• Many considerations at this decision point
• mathematical content or processes of mathematical
  thinking
• social establishing social or
  socio-mathematical norms for how the class is to
  work
• practical aspects of the lesson eg how much time
  left

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Many factors to consider

• Lampert extensively illustrates
• problems of teaching occur simultaneously
• single teaching actions must therefore address
  more than one problem
• A few examples of problems of teaching
• mathematical content and processes
• connecting mathematical ideas
• various socio-mathematical norms eg expectation
  for reasoning
• to be civil to classmates, to complete assigned
  tasks on time
• to demonstrate that they can all learn
  mathematics (eg when a student makes a public
  mistake)
• engage students with diverse interests
• M. Lampert (2001) Teaching Problems and the
  problems of teaching Yale University Press.

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What mathematical thinking is involved in Irenes
response?

• Is the conclusion right?
• Is the argument right?
• How could I show the argument is not right in a
  way that the student or class would understand?
• Is it important that students understand there is
  an error in the argument ?
• which error in the argument is the one I should
  highlight (3 events with unequal probability, not
  considering even odd and odd even, ..)
• etc

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Possible response to the argument

• Given erroneous parity argument
• odd odd even
• even even even
• odd even odd
• Present another erroneous parity argument
• odd odd even
• even even even
• odd even odd
• even odd odd

two out of three chances for even sum to win
two out of four chances so same chance for even
and odd to win
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How did Irene respond?

• Responded to conclusion, not to the argument
• said she was not sure about the two out of
  three
• agreed with conclusion game not fair and even
  sum more likely
• Moved onto the next part of the discussion

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Irene had only a few seconds to see these
possibilities
These are critical factors too
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Irenes second discussion

• Deliberately delayed good students response
  until last this boy said he mathsed it
  instead of playing
• Explained he counted possibilities - 38 even sums
  and 35 odd sums (wrong)
• Irene commented the game is not terribly
  weighted but it is slightly weighted to the
  evens
• Asked class if empirical results agreed, and
  highlighted that bias did not mean even sums
  always win
• So thats very interesting, that although we
  thought it was weighted towards player 2, they
  didnt win the whole time without player 1 having
  a chance.

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More demands for mathematical thinking

• Is 3835 correct? He is a very good student!
• How to reconcile the unequal odds with having no
  clear experimental result around the class?
• What is the main mathematical point to make in
  these final few minutes of the lesson
• see the 3835 possibilities
• highlight that probability does not determine
  outcomes
• link students experience that even sum does not
  always win the first to ten game with small
  bias in probabilities
• .

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Irene finished on time, with main point made
Reconciling small bias with class results
Finished on time, with main point made
Wrong parity argument
38 35
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Gregs lesson on the Spinners Game

• Played the game in the first lesson explored
  sample space in the second
• Enumerated sample space with class
• Idea to do this occurred in lesson 1
• Very directed exploration (restricted content
  knowledge did not consider alternative
  approaches suggested by students)
• Fairness
• Finding sample space led to unqualified even is
  more likely conclusion
• Implication of small difference not discussed

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Complete enumeration of 81 outcomes
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Gregs decisions to ignore alternatives suggested
by students

• Maybe judged complete systematic enumeration best
  for these students at this time? Is it good to
  see it once?
• Maybe constrained by lack of mathematical
  knowledge and confidence in correctness of
  alternatives?
• Maybe he did not see the mathematical
  possibilities in the students (erroneous)
  suggestions?

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Greg dismisses Lukes proposal that there are 45
odd sums and 36 even sums

• Luke What I think is, with the odds and evens, in
  our spinners we have five odds on first
  spinner, times nine numbers on the second
  spinner, equals 45.
• Greg All right.
• Luke And we have four evens on first spinner..
• Greg Interrupts Ive written gestures to a
  sheet on the table containing his solution
• Luke Four times nine equals 36. And 36 plus 45
  equals 81.
• Greg Gestures again at his solution Doing it
  manually, you count up the combinations, and
  its not.
• Luke inaudible

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Complete enumeration

• Important to teach students to work
  systematically
• Useful to see the patterns emerging from the
  systematic listing
• For me, it is extremely important to show how we
  move from this to a more elegant and curtailed
  argument

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Generalise to spinners with any numbers
Not for Irene and Gregs classes !!
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Greg tried to organise sample space in array
1 2 3 4 5
1 2 3 4 5 6
2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 10

• Very important tool for systematic work
• Important representation for multiplication
• Can be used to highlight patterns and as a bridge
  to generalisation and explanation

Katagiri APEC 2006 Mathematical Attitude of
attempting to seek better things
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Looking at one pattern at a time
1 2 3 4 5
1 2 3 4 5 6
2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 10
1 2 3 4 5
O E O E O E
E O E O E O
O E O E O E
E O E O E O
O E O E O E
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Abstracting the key feature (even or odd sum)
1 2 3 4 5
O E O E O E
E O E O E O
O E O E O E
E O E O E O
O E O E O E
O E O E O
O E O E O E
E O E O E O
O E O E O E
E O E O E O
O E O E O E
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Shuffle the rows (try this with numbers first)
O E O E O
O E O E O E
E O E O E O
O E O E O E
E O E O E O
O E O E O E
O E O E O
O E O E O E
O E O E O E
O E O E O E
E O E O E O
E O E O E O
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Just checking that shuffling rows is OK
1 2 3 4 5
1 2 3 4 5 6
2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 10
1 2 3 4 5
1 1 2 3 4 5
3 4 5 6 7 8
5 6 7 8 9 10
2 3 4 5 6 7
4 5 6 7 8 9
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Shuffle the columns
O E O E O
O E O E O E
O E O E O E
O E O E O E
E O E O E O
E O E O E O
O O O E E
O E E E O O
O E E E O O
O E E E O O
E O O O E E
E O O O E E
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See the general solution n2, m2 2nm
m odd
n even
O E O E O
O E O E O E
O E O E O E
O E O E O E
E O E O E O
E O E O E O
O O O E E
O E E E O O
O E E E O O
O E E E O O
E O O O E E
E O O O E E
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See the general solution n2, m2 2nm
m odd
n even
O O O E E
O E E E O O
O E E E O O
O E E E O O
E O O O E E
E O O O E E
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Katagiri APEC 2006List of 3 types of
mathematical thinking

• First type Mathematical Attitude (maybe
  Disposition or Value? )
• 1.
• 2.
• 3. .
• 4. Attempting to seek better things
• Attempting to raise thinking from the concrete
  level to the abstract level
• Attempting to evaluate thinking both objectively
  and subjectively, and to refine thinking
• Attempting to economise thought and effort

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Helens lesson to pre-service teachers

• Lesson aim for student teachers to analyse what
  mathematical learning the game could generate and
  how
• Helen generalised problem allowing any numbers on
  spinners looking for general reasons
• Helen chose to use fewer numbers on spinners
  (hence smaller sample space) as a special case to
  make conjectures and develop convincing arguments

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Helens lesson to pre-service teachers

• Students working with 2 spinners 0, 1, 2, 3
• New complication arose is 0 even or odd? A
  quick check of class showed many pre-service
  teachers were unsure.
• Decision point what to do about this?

Next time, Helen will be prepared for this ! All
students know 2, 4, 6, 8, are even
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Helens decision making

• Options
• Say 0 is an even number
• Explain why 0 is even number
• Have students discover why 0 is even number
• Decision made on
• mathematical possibilities perceived
• mathematical priorities and values perceived (eg
  Helen can show she values rationalism - reasons
  over rules)
• PCK - knowledge of students thinking etc
• practical aspects of lesson (time, needs of other
  students, .)

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Helens chose to explain why 0 is even

• Used this to draw attention to nature and use of
  a mathematical definition
• Helen to choose which of the possible definitions
  would be best
• an integer exactly divisible by 2
• an integer that is 2 times an integer
• other, e.g.

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Helens chose to explain why 0 is even

• Used this to draw attention to nature and use of
  a mathematical definition
• Helen to choose which of the possible definitions
  would be best
• an integer exactly divisible by 2
• an integer that is 2 times an integer
• other, e.g.

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Why not an integer exactly divisible by 2

• Helen avoided division because of her knowledge
  of students difficulties (PCK)

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Why not this pictorial representation?
If 0 children go for a walk holding hands in
pairs, is there one left over? NO not odd But
there are no pairs either! Kaplan (1999) The
Nothing That Is
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One spinner game 3 different lessons
Shimizu The lesson is not to follow the lesson
plan.Unexpected treasures found in the lesson.
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One spinner game 3 different lessons
What is the role of mathematical thinking at
these decision points?
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Mathematical thinking

• Provides possibilities for responses
• Assists in prioritising these possibilities, by
  considering mathematical processes, values,
  connections, Katagiris mathl attitudes etc
• Many different factors affect final choices

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Characteristics of mathematical thinking during
teaching

• Needs to be quick see possibilities fast!
• Needs to be confident
• Prior thinking about problem and lesson
  important, but cannot comprehensively cover all
  ideas and students suggestions
• Decisions require simultaneous consideration of
  many different factors

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Analogy of teaching with problem solving

• Is teaching a mathematics lesson like solving a
  real world problem with mathematics?
• YES
• maths used in conjunction with considerations
  from the problem field
• problem environments typically rich in
  constraints, as is teaching
• NO
• unusual time pressure for teachers mathematical
  decisions

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Time for careful thought in lesson preparation
Hosomizu Seiyama
Immediate, high pressure decisions while teaching
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Teachers mathematical thinking

• Within a lesson, mathematico-pedaogical thinking
  occurs on a minute-by-minute basis
• Significant contribution to the path the lesson
  takes and what students learn from it
• Responding to students in a mathematically
  productive way places high demands on teachers
  mathematical thinking
• Good lesson materials and preparation helps, but
  is not sufficient

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Teachers mathematical thinking

• Strong mathematics enables teachers to
• see more possibilities
• make decisions on mathematical grounds
• link lessons to students mathematical thinking
• develop stronger pedagogical content knowledge by
  helping them be more alert to students ideas
• Still only one factor in a good lesson and in
  good teaching

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Strategies for moving to lessons that focus on
mathematical thinking

• Changing teaching to respond more to students
  thinking is
• very important for better student outcomes
• difficult to achieve on a large scale
• will need strong support (human, material)
• Need long term plans for
• gradual changes in teaching methods
• incremental gains in student learning
• sensitive evaluation of improvements.

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Thank you

• Kaye Stacey
• University of Melbourne
• k.stacey_at_unimelb.edu.au
• Acknowledging the work of Dr Helen Chick, Monica
  Baker and Kiri Harris, Australian Research
  Council Grant DP0344229 and the teachers who
  generously made their work available for study.

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Pedagogical Content Knowledge

• Teachers need Pedagogical Content Knowledge
  (Shulman) to see affordances in examples and
  develop them as didactic objects
• Many authors have studied aspects of PCK
• Framework for PCK
• Clearly PCK (pedagogy and content inseparable)
• Content knowledge in a pedagogical context
• Pedagogical knowledge in a content context

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Creating a Better spinners Example

• What can be changed?
• Numbers on the spinners
• The number of numbers on the spinners
• The operation
• The first to ten rule
• What changes as a consequence?
• What do you gain/lose (i.e., do the affordances
  change)?
• Should we expect (primary) teachers to be able to
  make these changes?

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Implications for Teacher Education

• So much for them to learn
• Content itself (in many cases), understood
  conceptually
• PCK
• Recognising affordances
• Turning things into didactic objects
• In short, teachers must be able to
• identify the point,
• pick examples that convey the point, and
• implement the example so that the point is
  learned in the classroom

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Implications for Teacher Education

• Take care to identify the point of the examples
  WE give
• Be explicit about what is critical in the
  examples and how they affect the point
• Discuss the implications of varying values/other
  attributes
• Ask for harder/easier versions
• Examine questions such as How is Compare 2/5
  and 1/3 different from Compare 3/7 and 5/8?
• Discuss how to use the example as a successful
  didactic object

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Acknowledgements

• This research was supported by Australian
  Research Council grant DP0344229.
• Thanks to Monica Baker and Kiri Harris who were
  research assistants on this project.
• Thanks to the teachers in the MPCK project from
  whose examples I have learned much.

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Pedagogical Content Knowledge

• Clearly PCK, e.g,
• Knowledge of typical student thinking
• Knowledge of cognitive demand
• Knowledge of useful representations and examples
• Content knowledge in a pedagogical context, e.g,
• Profound Understanding of Fundamental Mathematics
• Knowledge of mathematical structure and
  connections
• Knowing the significance of a mathematical topic
• Pedagogical knowledge in a content context, e.g,
• Getting and maintaining student focus
• Goals for learning