Thinking, reasoning and working mathematically

Merrilyn Goos

The University of Queensland

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)

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?

What is “mathematical thinking”?

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?

diameter = 3445kmcircumference = 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?”

Cognitiveprocesses

knowledgeskills

strategies

What is “mathematical thinking?”

Metacognitiveprocesses

awareness regulation

Cognitiveprocesses

knowledgeskills

strategies

What is “mathematical thinking?”

Dispositionsbeliefs affects

Metacognitiveprocesses

awareness regulation

Cognitiveprocesses

knowledgeskills

strategies

Mathematical thinking means …

… adopting a mathematicalpoint 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

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

Mathematical understanding involves …

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

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 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?

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!

10 + 10 = = = =

Exploring number concepts: Counting

How were calculators used?

Exploring number concepts: Counting

9 + 9 = = =

Counting by 9s and recording the output on a number roll

91827364554637281

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.”

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.

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 Textbooks Short, closed questions Teacher exposition every day Individual work Disciplined

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

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?

What were students’ views about school mathematics?

“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

What were students’ views about school mathematics?

“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

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-thatknowing-how

knowing-whyknowing-when

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

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) 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 OutcomesLevels

Strands 1 2 3 4 5 6

Number

Measurement

Patterns & algebra

Chance & data

Space

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

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

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

Pyramids ofEgypt investigation

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

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?

If Khafre’s pyramid were as tall as this room, how tall would you be?

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

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