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