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Thinking, reasoning and working mathematically

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Title: Thinking, reasoning and working mathematically


1
Thinking, reasoning and working mathematically
  • Merrilyn Goos
  • The University of Queensland

2
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)

3
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?

4
What is mathematical thinking?
5
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?

6
How far is it around the moon?
  • diameter 3445km
  • circumference p ? 3445km
  • 10,822km

7
How many cars?
  • Number of cars
  • 10,822 ? 1000 ? (average length of one car in
    metres)
  • 2.7 million cars

8
How long to advertise?
  • time to advertise
  • (2.7 ? 106 cars) ? (2.7 ? 103 cars per week)
  • 1000 weeks
  • 19.2 years

9
What is mathematical thinking?
Cognitive processes
knowledge
strategies
skills
10
What is mathematical thinking?
Metacognitive processes
regulation
awareness
Cognitive processes
knowledge
strategies
skills
11
What is mathematical thinking?
beliefs
affects
Dispositions
Metacognitive processes
regulation
awareness
Cognitive processes
knowledge
strategies
skills
12
Mathematical thinking means
adopting a mathematicalpoint of view
13
How do you know when you understand something in
mathematics?
14
How do you know when you understand something in
mathematics?
15
Mathematical understanding involves
  • knowing-that (stating)
  • knowing-how (doing)
  • knowing-why (explaining)
  • knowing-when (applying)

Understanding means making connections between
ideas, facts and procedures.
16
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
17
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

18
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?

19
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!

20
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
21
How were calculators used?
Exploring number concepts Counting backwards
Underground numbers!
22
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!
23
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.

24
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.

25
Open and closed mathematics
  • Amber Hill School
  • Textbooks
  • Short, closed questions
  • Teacher exposition every day
  • Individual work
  • Disciplined

26
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

27
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?

28
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.

29
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.

30
What mathematical knowledge did the students
develop?
31
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

32
Years 1-10 syllabus Rationale
  • Mathematics is a unique and powerful way of
    viewing the world to investigate patterns,
    order, generality and uncertainty.

33
Years 1-10 syllabus organisation
Attributes of a life long learner
Key Learning Area outcomes
Core and discretionary learning outcomes
34
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

35
Years 1-10 syllabus organisation
Attributes of a life long learner
Key Learning Area outcomes
Core and discretionary learning outcomes
36
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

37
Years 1-10 syllabus organisation
Attributes of a life long learner
Key Learning Area outcomes
Core and discretionary learning outcomes
38
Core Learning Outcomes
39
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

40
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

41
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

42
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

43
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.

44
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

45
How big are the pyramids?
  • If Khafres pyramid were as tall as this room,
    how tall would you be?

46
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

47
Pyramids of Egypt investigation
  • SOSE syllabus strand
  • Time, continuity and change
  • Mathematics syllabus strands
  • Measurement
  • Chance and Data
  • Number
  • Space

48
Thinking, reasoning and working mathematically
  • Merrilyn Goos
  • The University of Queensland
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