Thinking Mathematically Chapter 11: Counting Methods and Probability Theory.
Thinking, reasoning and working mathematically
-
Upload
channing-tyler -
Category
Documents
-
view
42 -
download
0
description
Transcript of Thinking, reasoning and working mathematically
![Page 1: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/1.jpg)
Thinking, reasoning and working mathematically
Merrilyn Goos
The University of Queensland
![Page 2: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/2.jpg)
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)
![Page 3: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/3.jpg)
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?
![Page 4: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/4.jpg)
What is “mathematical thinking”?
![Page 5: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/5.jpg)
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?
![Page 6: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/6.jpg)
How far is it around the moon?
diameter = 3445kmcircumference = π 3445km
= 10,822km
![Page 7: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/7.jpg)
How many cars?
Number of cars
= 10,822 1000 (average length of one car in metres)
= 2.7 million cars
![Page 8: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/8.jpg)
How long to advertise?
time to advertise
= (2.7 106 cars) (2.7 103 cars per week)
= 1000 weeks
= 19.2 years
![Page 9: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/9.jpg)
What is “mathematical thinking?”
Cognitiveprocesses
knowledgeskills
strategies
![Page 10: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/10.jpg)
What is “mathematical thinking?”
Metacognitiveprocesses
awareness regulation
Cognitiveprocesses
knowledgeskills
strategies
![Page 11: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/11.jpg)
What is “mathematical thinking?”
Dispositionsbeliefs affects
Metacognitiveprocesses
awareness regulation
Cognitiveprocesses
knowledgeskills
strategies
![Page 12: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/12.jpg)
Mathematical thinking means …
… adopting a mathematicalpoint of view
![Page 13: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/13.jpg)
How do you know when you understand something in mathematics?
![Page 14: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/14.jpg)
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
![Page 15: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/15.jpg)
Mathematical understanding involves …
knowing-that (stating) knowing-how (doing) knowing-why (explaining) knowing-when (applying)
Understanding means making connections between ideas, facts and procedures.
![Page 16: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/16.jpg)
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
![Page 17: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/17.jpg)
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
![Page 18: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/18.jpg)
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?
![Page 19: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/19.jpg)
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
![Page 20: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/20.jpg)
How were calculators used?
Exploring number concepts: Counting
9 + 9 = = =
Counting by 9s and recording the output on a number roll
91827364554637281
![Page 21: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/21.jpg)
How were calculators used?
Exploring number concepts: Counting backwards
Underground numbers!
![Page 22: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/22.jpg)
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!”
![Page 23: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/23.jpg)
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.”
![Page 24: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/24.jpg)
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.
![Page 25: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/25.jpg)
Open and closed mathematics
Amber Hill School Textbooks Short, closed questions Teacher exposition every day Individual work Disciplined
![Page 26: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/26.jpg)
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
![Page 27: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/27.jpg)
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?
![Page 28: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/28.jpg)
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
![Page 29: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/29.jpg)
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
![Page 30: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/30.jpg)
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
![Page 31: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/31.jpg)
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
![Page 32: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/32.jpg)
Years 1-10 syllabus Rationale
Mathematics is a unique and powerful way of viewing the world to investigate patterns, order, generality and uncertainty.
![Page 33: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/33.jpg)
Years 1-10 syllabus organisation
Attributes of a life long learner
Key Learning Area outcomes
Core and discretionary learning outcomes
![Page 34: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/34.jpg)
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
![Page 35: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/35.jpg)
Years 1-10 syllabus organisation
Attributes of a life long learner
Key Learning Area outcomes
Core and discretionary learning outcomes
![Page 36: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/36.jpg)
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 …
![Page 37: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/37.jpg)
Years 1-10 syllabus organisation
Attributes of a life long learner
Key Learning Area outcomes
Core and discretionary learning outcomes
![Page 38: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/38.jpg)
Core Learning OutcomesLevels
Strands 1 2 3 4 5 6
Number
Measurement
Patterns & algebra
Chance & data
Space
![Page 39: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/39.jpg)
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
![Page 40: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/40.jpg)
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
![Page 41: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/41.jpg)
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
![Page 42: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/42.jpg)
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
![Page 43: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/43.jpg)
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.
![Page 44: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/44.jpg)
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
![Page 45: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/45.jpg)
How big are the pyramids?
If Khafre’s pyramid were as tall as this room, how tall would you be?
![Page 46: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/46.jpg)
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
![Page 47: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/47.jpg)
Pyramids of Egypt investigation
SOSE syllabus strand Time, continuity and
change
Mathematics syllabus strands Measurement Chance and Data Number Space
![Page 48: Thinking, reasoning and working mathematically](https://reader035.fdocuments.in/reader035/viewer/2022062408/56813797550346895d9f3cb8/html5/thumbnails/48.jpg)
Thinking, reasoning and working mathematically
Merrilyn Goos
The University of Queensland