Contributed by:

The basic outline of this presentation is:

1. What is mathematical thinking?

2. What teaching approaches can develop students’ mathematical thinking?

3. How does the syllabus support current research on mathematical thinking?

4. How can we engage students in thinking, reasoning, and working mathematically?

1. What is mathematical thinking?

2. What teaching approaches can develop students’ mathematical thinking?

3. How does the syllabus support current research on mathematical thinking?

4. How can we engage students in thinking, reasoning, and working mathematically?

1.
Thinking, reasoning

and working

Merrilyn Goos

The University of Queensland

and working

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)

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.
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 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?

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 = π 3445km

= 10,822km

diameter = 3445km

circumference = π 3445km

= 10,822km

7.
How many cars?

Number of cars

= 10,822 1000 (average length of one car in metres)

= 2.7 million 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

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

Cognitive

processes

knowledge strategies

skills

10.
What is “mathematical thinking?”

Metacognitive regulation

awareness

processes

Cognitive

processes

knowledge strategies

skills

Metacognitive regulation

awareness

processes

Cognitive

processes

knowledge strategies

skills

11.
What is “mathematical thinking?”

beliefs affects

Dispositions

Metacognitive regulation

awareness

processes

Cognitive

processes

knowledge strategies

skills

beliefs affects

Dispositions

Metacognitive regulation

awareness

processes

Cognitive

processes

knowledge strategies

skills

12.
Mathematical thinking means …

… adopting a

mathematical

point of view

… adopting a

mathematical

point of view

13.
How do you know when you understand

something in mathematics?

something in mathematics?

14.
How do you know when you understand

something in mathematics?

Category Frequency Proportion

I Correct answer 234 0.71

II Affective response 35 0.11

III Makes sense 52 0.16

IV Application/transfer 27 0.08

V Explain to others 24 0.07

something in mathematics?

Category Frequency Proportion

I Correct answer 234 0.71

II Affective response 35 0.11

III Makes sense 52 0.16

IV Application/transfer 27 0.08

V Explain to others 24 0.07

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.

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

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

6 Melbourne schools:

1000 children & 80 teachers

Prep-Year 4

Children given their own

arithmetic calculators

Teachers not provided with

activities or program

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

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 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?

19.
How were calculators used?

Exploring number concepts:

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!

Exploring number concepts:

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!

20.
How were calculators used?

Exploring number concepts: Counting

9 + 9 = = = 9

18

27

Counting by 9s and

36

recording the output 45

on a number roll 54

63

72

81

Exploring number concepts: Counting

9 + 9 = = = 9

18

27

Counting by 9s and

36

recording the output 45

on a number roll 54

63

72

81

21.
How were calculators used?

Exploring number concepts: Counting backwards

Underground numbers!

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!”

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

“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.”

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.”

24.
What were the effects on children’s

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.

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

Amber Hill School

Textbooks

Short, closed questions

Teacher exposition every day

Individual work

Disciplined

26.
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

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

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?

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

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.”

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.”

29.
What were students’ views about school

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.”

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.”

30.
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-why

knowing-how knowing-when

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-why

knowing-how knowing-when

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

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.

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

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

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

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 …

(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

Attributes of a life long learner

Key Learning Area outcomes

Core and discretionary

learning outcomes

38.
Core Learning Outcomes

Levels

Strands 1 2 3 4 5 6

Patterns &

Chance & data

Levels

Strands 1 2 3 4 5 6

Patterns &

Chance & data

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

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

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

Pyramids of

across KLAs Egypt investigation

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

Pyramids of

across KLAs Egypt investigation

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

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.

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 Mathematics Content

When were the pyramids Measurement of time,

built? (dating methods) length, mass, area, volume

Political/social structure of Data presentation and

ancient Egypt interpretation

Geography of Egypt Ratio and proportion (scale)

Religious/burial practices Angles, 2D and 3D shapes

Pyramid construction

methods

SOSE/History Content Mathematics Content

When were the pyramids Measurement of time,

built? (dating methods) length, mass, area, volume

Political/social structure of Data presentation and

ancient Egypt interpretation

Geography of Egypt Ratio and proportion (scale)

Religious/burial practices Angles, 2D and 3D shapes

Pyramid construction

methods

45.
How big are the pyramids?

If Khafre’s pyramid were as tall as this room,

how tall would you be?

If Khafre’s pyramid were as tall as this room,

how tall would you be?

46.
How were the pyramids built?

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

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

47.
Pyramids of Egypt investigation

SOSE syllabus strand Mathematics syllabus strands

Time, continuity and Measurement

change Chance and Data

Number

Space

SOSE syllabus strand Mathematics syllabus strands

Time, continuity and Measurement

change Chance and Data

Number

Space

48.
Thinking, reasoning

and working

Merrilyn Goos

The University of Queensland

and working

Merrilyn Goos

The University of Queensland