Thinking, reasoning and working mathematically

1. Thinking, reasoning and working mathematically 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)

3. 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?

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?

6. How far is it around the moon? 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

8. How long to advertise? 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

10. What is “mathematical thinking?” Metacognitive processes regulation awareness Cognitive processes knowledge strategies skills

11. What is “mathematical thinking?” beliefs affects Dispositions Metacognitive processes regulation awareness Cognitive processes knowledge strategies skills

12. Mathematical thinking means … … adopting a mathematicalpoint of view

13. How do you know when you understand something in mathematics?

14. How do you know when you understand something in mathematics?

15. Mathematical understanding involves … • knowing-that (stating) • knowing-how (doing) • knowing-why (explaining) • knowing-when (applying) Understanding means making connectionsbetween 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

17. 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

18. 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?

19. 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! Exploring number concepts: Counting + 10 = 10 = = =

20. How were calculators used? Exploring number concepts: Counting 9 18 27 36 45 54 63 72 81 + 9 = 9 = = Counting by 9s and recording the output on a number roll

21. How were calculators used? 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!”

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.”

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.

25. Amber Hill School Textbooks Short, closed questions Teacher exposition every day Individual work Disciplined Open and closed mathematics

26. 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

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?

28. 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

29. 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

30. knowing-that knowing-how knowing-why knowing-when What mathematical knowledge did the students develop?

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

32. Years 1-10 syllabus Rationale 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

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

35. Years 1-10 syllabus organisation 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 …

37. Years 1-10 syllabus organisation Attributes of a life long learner Key Learning Area outcomes Core and discretionary learning outcomes

38. Core Learning Outcomes

39. Planning with outcomes: Making connections • within a strand of a KLA • across strands within a KLA • across levels within a KLA • across KLAs When planning units of work, teachers could combine learning outcomes from:

40. Planning with outcomes: An investigative approach • a problem to be solved • a question to be answered • a significant task to be completed • an issue to be explored The focus for planning within and across key learning areas can be framed in terms of:

41. Pyramids of Egypt investigation How can we engage students in thinking, reasoning and working mathematically? • within a strand of a KLA • across strands within a KLA • across levels within a KLA • across KLAs An investigation that combines outcomes:

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

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.

44. 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 Pyramids of Egypt investigation

45. How big are the pyramids? 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

47. SOSE syllabus strand Time, continuity and change Mathematics syllabus strands Measurement Chance and Data Number Space Pyramids of Egypt investigation

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