Thinking, reasoning and working mathematically

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Thinking, reasoning and working mathematically

• Merrilyn Goos
• The University of Queensland

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Why is mathematics important?

• Mathematics is used in daily living, in civic
  life, and at work (National Statement)
• Mathematics helps students develop attributes of
  a lifelong learner (Qld Years 1-10 Mathematics
  Syllabus)

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Outline

• What is mathematical thinking?
• What teaching approaches can develop students
  mathematical thinking?
• How does the syllabus support current research on
  mathematical thinking?
• How can we engage students in thinking, reasoning
  and working mathematically?

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What is mathematical thinking?
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Some mathematical thinking

• How far is it around the moon?
• How many cars does this represent?
• How long would it take to advertise this number
  of cars?

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How far is it around the moon?

• diameter 3445km
• circumference p ? 3445km
• 10,822km

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How many cars?

• Number of cars
• 10,822 ? 1000 ? (average length of one car in
  metres)
• 2.7 million cars

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How long to advertise?

• time to advertise
• (2.7 ? 106 cars) ? (2.7 ? 103 cars per week)
• 1000 weeks
• 19.2 years

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What is mathematical thinking?
Cognitive processes
knowledge
strategies
skills
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What is mathematical thinking?
Metacognitive processes
regulation
awareness
Cognitive processes
knowledge
strategies
skills
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What is mathematical thinking?
beliefs
affects
Dispositions
Metacognitive processes
regulation
awareness
Cognitive processes
knowledge
strategies
skills
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Mathematical thinking means
adopting a mathematicalpoint of view
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How do you know when you understand something in
mathematics?
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How do you know when you understand something in
mathematics?
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Mathematical understanding involves

• knowing-that (stating)
• knowing-how (doing)
• knowing-why (explaining)
• knowing-when (applying)

Understanding means making connections between
ideas, facts and procedures.
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What teaching approaches can develop mathematical
thinking?

• Develop a mathematical point of view
• Knowing that, how, why, when
• Making connections within and beyond mathematics

Investigative approach
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Calculators in Primary Mathematics project

• 6 Melbourne schools 1000 children 80 teachers
• Prep-Year 4
• Children given their own arithmetic calculators
• Teachers not provided with activities or program

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Calculators in Primary Mathematics project

• How can calculators be used in lower primary
  mathematics classrooms?
• What effects will the calculators have on
  teachers beliefs, classroom practice, and
  expectations of children?
• What effects will the calculators have on
  childrens learning of number concepts?

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How were calculators used?
Exploring number concepts Counting
10


10

• Alex (5 yrs) Im counting by tens and Im up to
  300!
• Teacher And what would you like to get to?
• Alex A thousand and fifty!

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How were calculators used?

• Exploring number concepts Counting

9 18 27 36 45 54 63 72 81
9


9


Counting by 9s and recording the output on a
number roll
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How were calculators used?
Exploring number concepts Counting backwards
Underground numbers!
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How were calculators used?
Exploring number concepts Place value
Put on your calculator the largest number you
can read correctly.
9345
Nine thousand three hundred and forty-five
6056
Six thousand and fifty-six
9000000000
Nine billion!
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What were the effects on teachers?

• More open-ended teaching practices
• Im not so worried about them finding out things
  they wont understand any more I think Im
  being a lot more open-ended with their
  activities.
• More discussion and sharing of childrens ideas
• It certainly encouraged me to talk to the
  children much more and discuss how did they do
  this, why did they do that, and getting them to
  justify what theyre doing.

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What were the effects on childrens number
learning?

• Interviews and written tests with project
  children and control group in Years 3 and 4.
• Two types of test (1) paper pencil (2)
  calculator.
• Two types of interview(1) choose any
  calculation method or device(2) mental
  computation only
• Project children had better overall performance.

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Open and closed mathematics

• Amber Hill School
• Textbooks
• Short, closed questions
• Teacher exposition every day
• Individual work
• Disciplined

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Open and closed mathematics

• Amber Hill School
• Textbooks
• Short, closed questions
• Teacher exposition every day
• Individual work
• Disciplined

• Phoenix Park School
• Projects
• Open problems
• Teacher exposition rare
• Group discussions
• Relaxed

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Open and closed mathematics

• How do students view the world of the school
  mathematics classroom?
• How do their views impact on the mathematical
  knowledge they develop and their ability to use
  this knowledge?

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What were students views about school
mathematics?
Amber Hill monotony and meaninglessness

• I wish we had different questions, not three
  pages of sums on the same thing.
• In maths theres a certain formula to get from A
  to B, and theres no other way to get to it.
• In maths you have to remember in other subjects
  you can think about it.

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What were students views about school
mathematics?
Phoenix Park thinking and connections

• Its more the thinking side to sort of look at
  everything youve got and think about how to
  solve it.
• Here you have to explain how you got the
  answer.
• When Im out of school now, I can connect back
  to what I done in class so I know what Im doing.

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What mathematical knowledge did the students
develop?
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How does the syllabus support current research on
mathematical thinking?

• Syllabus rationale what is mathematics?
• Syllabus organisation three levels of outcomes
• Planning with outcomes using investigations,
  making connections

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Years 1-10 syllabus Rationale

• Mathematics is a unique and powerful way of
  viewing the world to investigate patterns,
  order, generality and uncertainty.

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Years 1-10 syllabus organisation
Attributes of a life long learner
Key Learning Area outcomes
Core and discretionary learning outcomes
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Attributes of a lifelong learner

• A lifelong learner is
• A knowledgeable person with deep understanding
• A complex thinker
• A responsive creator
• An active investigator
• An effective communicator
• A participant in an interdependent world
• A reflective and self-directed learner

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Years 1-10 syllabus organisation
Attributes of a life long learner
Key Learning Area outcomes
Core and discretionary learning outcomes
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Mathematics KLA Outcomes (thinking, reasoning
and working mathematically)

• Understand the nature of mathematics as a dynamic
  human endeavour
• Interpret and apply properties and relationships
  
• Identify and analyse information
• Create mathematical models
• Pose and solve mathematical problems
• Use the concise language of mathematics
• Collaborate and cooperate, challenge the
  reasoning of others
• Reflect on, evaluate and apply their mathematical
  learning

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Years 1-10 syllabus organisation
Attributes of a life long learner
Key Learning Area outcomes
Core and discretionary learning outcomes
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Core Learning Outcomes
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Planning with outcomes Making connections
When planning units of work, teachers could
combine learning outcomes from

• within a strand of a KLA
• across strands within a KLA
• across levels within a KLA
• across KLAs

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Planning with outcomes An investigative approach
The focus for planning within and across key
learning areas can be framed in terms of

• a problem to be solved
• a question to be answered
• a significant task to be completed
• an issue to be explored

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How can we engage students in thinking, reasoning
and working mathematically?
An investigation that combines outcomes

• within a strand of a KLA
• across strands within a KLA
• across levels within a KLA
• across KLAs

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Investigations across KLAs The curriculum
integration project

• The impact of the mediaeval plagues
• The mystery of the Mayans
• Managing the Bulimba Creek catchment
• Building the pyramids of Egypt

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Pyramids of Egypt Investigation

• You have been declared Pharaoh of Egypt! As a
  monument to your reign, you decide to build a
  pyramid in your honour. Prepare a feasibility
  study for the construction project, including a
  scale model of your pyramid.

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Pyramids of Egypt investigation

• SOSE/History Content
• When were the pyramids built? (dating methods)
• Political/social structure of ancient Egypt
• Geography of Egypt
• Religious/burial practices
• Pyramid construction methods

• Mathematics Content
• Measurement of time, length, mass, area, volume
• Data presentation and interpretation
• Ratio and proportion (scale)
• Angles, 2D and 3D shapes

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How big are the pyramids?

• If Khafres pyramid were as tall as this room,
  how tall would you be?

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How were the pyramids built?

• Volume of Khufus pyramid 2,583,283m3
• If the density of limestone is 2280 kg/m3, what
  is the total weight of Khufus pyramid?
• Weight of pyramid 5,889,886 tons
• If the average weight of a limestone block is 2.5
  tons, how many blocks comprise Khufus pyramid?
• Number of blocks 2,355,954
• Khufu reigned for 23 years. How many blocks of
  limestone needed to be delivered to the pyramid
  every hour for it to be completed within his
  reign?
• 12 blocks/hr all year or 35 blocks/hr during
  inundation period

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Pyramids of Egypt investigation

• SOSE syllabus strand
• Time, continuity and change

• Mathematics syllabus strands
• Measurement
• Chance and Data
• Number
• Space

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Thinking, reasoning and working mathematically

• Merrilyn Goos
• The University of Queensland