Thinking, reasoning
and working
mathematically
Merrilyn Goos
The University of Queensland
Why is mathematics important?
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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)
Outline
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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?
What is “mathematical thinking”?
Some mathematical thinking …
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How far is it around the moon?
How many cars does this represent?
How long would it take to advertise this number
of cars?
How far is it around the moon?
diameter = 3445km
circumference = π  3445km
= 10,822km
How many cars?
Number of cars
= 10,822  1000  (average length of one car in metres)
= 2.7 million cars
How long to advertise?
time to advertise
= (2.7  106 cars) 
(2.7  103 cars per week)
= 1000 weeks
= 19.2 years
What is “mathematical thinking?”
Cognitive
processes
knowledge
skills
strategies
What is “mathematical thinking?”
awareness
Metacognitive
processes
regulation
Cognitive
processes
knowledge
skills
strategies
What is “mathematical thinking?”
beliefs
awareness
Dispositions
Metacognitive
processes
affects
regulation
Cognitive
processes
knowledge
skills
strategies
Mathematical thinking means …
… adopting a
mathematical
point of view
How do you know when you understand
something in mathematics?
How do you know when you understand
something in mathematics?
Category
Frequency
Proportion
234
0.71
35
0.11
III Makes sense
52
0.16
IV Application/transfer
27
0.08
V
24
0.07
I
Correct answer
II
Affective response
Explain to others
Mathematical understanding involves …
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knowing-that (stating)
knowing-how (doing)
knowing-why (explaining)
knowing-when (applying)
Understanding means making connections
between ideas, facts and procedures.
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
Calculators in Primary Mathematics
project
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6 Melbourne schools:
1000 children & 80 teachers
Prep-Year 4
Children given their own
arithmetic calculators
Teachers not provided with
activities or program
Calculators in Primary Mathematics
project
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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
children’s learning of number concepts?
How were calculators used?
Exploring number concepts: Counting
10
+
10
=
=
=
=
Alex (5 yrs): I’m counting by tens and I’m up to 300!
Teacher: And what would you like to get to?
Alex: A thousand and fifty!
How were calculators used?
Exploring number concepts: Counting
9
+
9
=
=
Counting by 9s and
recording the output
on a number roll
=
9
18
27
36
45
54
63
72
81
How were calculators used?
Exploring number concepts: Counting backwards
Underground numbers!
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!”
What were the effects on teachers?
More open-ended teaching practices
“I’m not so worried about them finding out things they
won’t understand any more … I think I’m being a lot more
open-ended with their activities.”
 More discussion and sharing of children’s 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 they’re doing.”
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What were the effects on children’s
number learning?
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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.
Open and closed mathematics
Amber Hill School
 Textbooks
 Short, closed questions
 Teacher exposition every day
 Individual work
 Disciplined
Open and closed mathematics
Amber Hill School
Phoenix Park School
 Textbooks
 Projects
 Short, closed questions
 Open problems
 Teacher exposition every day  Teacher exposition rare
 Individual work
 Group discussions
 Disciplined
 Relaxed
Open and closed mathematics
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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?
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 there’s a certain formula to get
from A to B, and there’s no other way to
get to it.”
“In maths you have to remember; in
other subjects you can think about it.”
What were students’ views about school
mathematics?
Phoenix Park: thinking and connections
“It’s more the thinking side to sort of look
at everything you’ve got and think about
how to solve it.”
“Here you have to explain how you got [the
answer].”
“When I’m out of school now, I can connect
back to what I done in class so I know what
I’m doing.”
What mathematical knowledge did the
students develop?
% of Students
Amber Hill
Phoenix Park
Investigation task
55%
75%
GCSE: A-C grade
11%
11%
GCSE: pass
71%
88%
knowing-that
knowing-how
knowing-why
knowing-when
How does the syllabus support current
research on mathematical thinking?
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Syllabus rationale: what is mathematics?
Syllabus organisation: three levels of outcomes
Planning with outcomes: using investigations,
making connections
Years 1-10 syllabus Rationale
Mathematics is a unique and
powerful way of viewing the
world to investigate patterns,
order, generality and uncertainty.
Years 1-10 syllabus organisation
Attributes of a life long learner
Key Learning Area outcomes
Core and discretionary
learning outcomes
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
Years 1-10 syllabus organisation
Attributes of a life long learner
Key Learning Area outcomes
Core and discretionary
learning outcomes
Mathematics KLA Outcomes
(thinking, reasoning and working mathematically)
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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 …
Years 1-10 syllabus organisation
Attributes of a life long learner
Key Learning Area outcomes
Core and discretionary
learning outcomes
Core Learning Outcomes
Levels
Strands
Number
Measurement
Patterns &
algebra
Chance & data
Space
1
2
3
4
5
6
Planning with outcomes:
Making connections
When planning units of work, teachers could
combine learning outcomes from:
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within a strand of a KLA
across strands within a KLA
across levels within a KLA
across KLAs
Planning with outcomes:
An investigative approach
The focus for planning within and across key
learning areas can be framed in terms of:
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a problem to be solved
a question to be answered
a significant task to be completed
an issue to be explored
How can we engage students in thinking,
reasoning and working mathematically?
An investigation that combines outcomes:
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within a strand of a KLA
across strands within a KLA
across levels within a KLA
across KLAs
Pyramids of
Egypt investigation
Investigations across KLAs:
The curriculum integration project
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The impact of the mediaeval plagues
The mystery of the Mayans
Managing the Bulimba Creek catchment
Building the pyramids of Egypt
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.
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
How big are the pyramids?
Pyramid
Khu fu
Khafre
Menkaur e
Side
(m)
230
216
108
Height
(m)
146.5
140.5
66.5
Base Ar ea
(m2 )
If Khafre’s pyramid were as tall as this room,
how tall would you be?
How were the pyramids built?
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Volume of Khufu’s pyramid = 2,583,283m3
If the density of limestone is 2280 kg/m3, what is the total
weight of Khufu’s pyramid?
Weight of pyramid = 5,889,886 tons
If the average weight of a limestone block is 2.5 tons, how
many blocks comprise Khufu’s 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
Pyramids of Egypt investigation
SOSE syllabus strand
 Time, continuity and
change
Mathematics syllabus strands
 Measurement
 Chance and Data
 Number
 Space
Thinking, reasoning
and working
mathematically
Merrilyn Goos
The University of Queensland