Acknowledgment of Country

1. Acknowledgment of Country

2. We acknowledge the elders and people, past and present, of the Aboriginal people, as the traditional owners of the land on which we meet today; and recognise their strength, resilience and capacity.

3. Open-ended Enquiry in MathsDay 1 • Working Mathematically and Authentic Enquiry • Questioning models • Problem solving • Teacher sharing session • Internet sites

4. Open-ended Enquiry in MathsDay 2 • Extending Open Investigations • Articulating Mathematical Thinking and Learning - Maths Blogs and Reflection tools • The Open-Ended Approach and Lesson Study (Japan et al) • Learning styles (M I) and programming • Technology in Mathematics (research and shared experience) • Quality Teaching and Assessment • Final Reflection

5. A Game to Start our Thinking Double, Halve or Stay An activity for two to four players Equipment: two different coloured dice Decide on one coloured dice to represent the tens and the other one to represent the ones. Choose a target number between 5 and 122. Players take turns to roll the dice. Once the dice are rolled a number is formed, then may be doubled, halved or kept unchanged to create a score. The players repeat the process and continue to add or subtract their scores, until the target number is reached exactly, or after a number of rounds the player closest to the target number wins.

6. Numeracy • Numeracy is the ability to effectively use the mathematics required to meet the general demands of life at home and at work, and for participation in community and civic life. • As a field of study mathematics is developed and/or applied in situations that extend beyond the general demands of everyday life.

7. Why Mathematics ? • Why Mathematics? • Making sense of the world

8. Why Mathematics ? • Think Pair Square • Make a list of contexts or occupations in the real-world where mathematics needs to be understood and used well

9. Working Mathematically • Affinity Diagram • Write one aspect of what it means to be working mathematically on each post-it note • Group sort in silence (individually) • Move post-it notes in silence (individually) • Refine the group sort with collaborative discussion and decision-making • Assign headings to each group

10. Working Mathematically Syllabus definition • questioning • applying strategies • communicating • reasoning • reflecting • Make connections to the double, halve or stay game and to the world relevance of mathematics list • (HO) - Working Mathematically Outcomes overview

11. Working Mathematically • Teaching Mathematics • stages and outcomes • content • process Table group reflection Implications for teaching from the affinity diagram

12. Jigsaw Reading - three groups Table group reflection If …. Then ? Reforming Education: The Pursuit of Learning Through Authentic Inquiry in Mathematics, Science and Technology Watters and Diezmann, QUT, Brisbane Authentic Inquiry

13. Examples of open, closed and extended investigations (HO) - Working Mathematically Activity In pairs select several closed questions and write open questions and extended investigations for these (topics on cards) Journal Writing - Five Whys? Why is it important to provide students with open questions and extended investigations? Closed and open questions Questioning Models

14. overview of models divergent questions Bloom’sTaxonomy - question stems Thinker’s Keys (Tony Ryan) De Bono - 6 Thinking Hats Socratic Dialogue Weiderhold’s Matrix Group Activity - each group studies one questioning model and identifies areas of mathematics teaching and learning where this model could be used Whole Group Sharing Understanding questioning models Questioning Models

15. Collaborative problem solving Group structures and roles Problem Solving Timekeeper Resource gatherer Recorder Encourager Observer Manager Reporter Clarifier

16. The Problem Working in groups of 4 find a solution to the following problem: A farmer takes pigs and cows to the local stock sale. Each pen at the saleyards is the same size holding either 2 cows or 5 pigs. If he delivers a total of 81 animals to be sold and fills 30 pens, how many cows did he have to sell? Discuss a variety of ways in which you could solve this problem Recorder / Reporters and Observers to report to the whole group (HO) - Cooperative Learning Structures Collaborative problem solving Group structures Group roles - manager - recorder / reporter - clarifier / encourager - observer Problem Solving

17. Table group brainstorm What are the elements of problem solving? Structured Brainstorm Whole group thinking and ideas captured on an A1 chart Collaborative problem solving Cooperative Learning Strategies (HO) Problem Solving

18. make a table act it out simplify the problem trial and improve work backwards look for a pattern draw a picture of graph try all possibilities write an equation guess and check make a model process of elimination …. whatever works for me! (HO) - Problem Solving Strategies Strategies for problem solving Problem Solving

19. Problem Solving • TAPE DIAGRAMS • A tape diagram offers students a thinking tool to visually represent a mathematical problem and transform the words into an appropriate numerical operation • I had 3 apples, how many more do I need to buy to make 10? • 22 sweets are divided among three children in a family. The twins have the same number each, while their younger sister has 6. How many sweets does each twin have?

20. Problem Solving • TAPE DIAGRAMS • Activity • Solve problems using tape diagrams • Create a tape diagram, then draft three different problems that could be represented by that tape diagram

21. Newman’s Error Analysis Transformation Skills (HO) - overview Newman’s Error Analysis Professor Anne Newman’ research into children’s problem solving barriers Problem Solving

22. . Reflection Journals Ah - ha !!!

23. . Teacher Sharing Session Table groups by Stages Strategies/ processes that work for me!

24. Internet Sites . Websites that have a focus on interactive activities and problem solving in mathematics • CAP website - Maths on the Net - Research Modules • TaLe • http://www • Diane’s Favourites (HO) • Others to share …..

25. Reflection . Reflecting on today From the black and white photographs provided, select one photograph that says something to you about your learning and participation in the workshop sessions today.

26. . Multo Fay’s Nines Number tiles Is it fair? Possible prisms and wrapping The traffic survey Happy and sad numbers Extending Open Investigations RICH TASKS

27. . Articulating mathematical thinking and learning through writing Maths Blogs Articulating Mathematical Thinking and Learning MATHS BLOGS A middle years project for PSP school communities Amanda Schofield - QT Numeracy Consultant - North Coast Region

28. . Some Examples: Bloom’s verbs and stems Apollo Parkways Primary School reflection stems (on cards) Quality Improvement tools (Tool Time book) Visual imagery eg photographs Free writing Articulating Mathematical Thinking and Learning REFLECTION TOOLS

29. “The Open-ended Approach” • “…provides students with experience in finding something new in the process …” (Shimada - 1970s) • “ In the open-ended approach students are often asked to not only show their work, but also to explain how they got their answers or why they chose the method they did.” (Schoenfeld 1997)

30. “The Open-ended Approach” PROBLEM solution solution solution solution solution COMPARING AND DISCUSSING IDEAS / QUESTIONS / PROBLEMS

31. Advantages of “The Open-ended Approach” 1. Students participate more actively in lessons and express their ideas more frequently 2. Students have more opportunities to make comprehensive use of their mathematical knowledge and skills 3. Every student can respond to the problem in some significant ways on his/her own 4. The lesson can provide students with a reasoning experience 5. There are rich experiences for students to have the pleasure of discovery and to receive the approval from fellow students.

32. The Open-ended Approach and Lesson Study Lesson Study is an ongoing, collaborative, professional development process that was developed in Japan. Many teachers in other countries are interested in this process, particularly in light of TIMSS (Third International Mathematics and Science Study) results, in the 1990s, which highlighted the advanced performance and deeper thinking in mathematics by Japanese students.

33. The Lesson Study Process • Groups of teachers identify an area of need in student learning and progress in their classes that is in need of improvement. • They then enquire into developments in teaching that are likely to have an impact on this aspect of student learning • The group spends between one and three years working together: • - planning interventions in lessons that may improve student learning • - teaching and collaboratively closely observing these ‘research lessons’ • - carefully discussing the outcomes • - writing up what happens - ‘failures’ as well as ‘successes’

34. The Lesson Study Process • Choose a research theme • Focus the research • Create the lesson • Teach and observe the lesson • Discuss the lesson • Revise the lesson • Document the findings

35. The Lesson Study Process • Select a broad goal, such as increasing your students’ ability to reason mathematically, or increasing their confidence in their mathematical abilities • Select a unit to focus on and analyse the current abilities and needs of your students • Select a lesson to develop together, being sure to look at how the skills for that lesson fit in the continuum of skills across the grades. Also think about how evidence of student thinking can be observed during the lesson. • Teach the lesson and observe it • Get together and discuss and analyse the lesson • After discussing your observations, work together to revise the lesson, and then have another teacher teach the lesson, then repeat the observation and discussion.

36. The Open-ended Approach and Lesson Study Process • Example • Focus area - multiplying and dividing decimal numbers in Grade 5 (Japan) • The problem: A 2 metre length of wire weighs 24.8 grams. If you have 6 metres of the same wire, how much will it weigh?

37. The Open-ended Approach and Lesson Study Process • Video segment 1 - Introduction • Mr Masahiro Seki sets up the problem by showing students a wire. After helping students think about things that change as the length of wire changes, he poses today’s problem.

38. The Open-ended Approach and Lesson Study Process • Video segment 2 - Individual problem solving • Mr Seki circulates around the classroom, asking questions and making suggestions. He also records students’ responses on the seating chart he carries with him. After the individual problem solving time, Mr Seki calls on some students to write down their solutions on the blackboard.

39. The Open-ended Approach and Lesson Study Process • Video segment 3 - Whole class discussion • The class discusses the solutions written by their classmates. They note that some strategies involve multiplication by 3 while others do not. Mr Seki orchestrates the discussion as the class tries to articulate the reasons behind both approaches.

40. The Open-ended Approach and Lesson Study Process • Video segment 4 - Summarising and consolidating students’ understanding • Mr Seki summarises the whole class discussion. He then asks the students to write a journal entry

41. The Open-ended Approach and Lesson Study Process • Video segment 5 - Final comment • Final comment at the end of the post-lesson discussion by Professor Toshiakira Fujii, Tokyo Gakugei University

42. The Open-ended Approach and Lesson Study Process PERSONAL REFLECTION Consider the implications and application of these processes to your own teaching Reflection Journal writing How does this learning impact on my reflection on my current teaching practice in teaching mathematics?

43. Using an Enquiry-Based Approach National Teacher Research Panel - Conference Summary: Teaching and Learning Mathematics using an Enquiry-Based Approach Mark Richards, Lancaster Girls’ Grammar School, Lancaster UK Activity: Individual reading of the report

44. Learning Styles • Overview of Learning Styles: • Mumford (1996) - theoretical, pragmatic, reflective, activist • Rose (1985) - visual, auditory, kinaesthetic • Gagne (1985) - nine phases of learning • Kroehnert (1990) - nine guidelines for teaching situations • Myers-Briggs - identification of a person’s type (and the impact of • this on learning) • Grasha-Riechmann - categorisation of social indicators and the • classroom preferences of these categories - competitive, collaborative, • avoidant, participant • - Herrmann Brain Dominance Index (HBDI) - four categories of thinkers/ • learners - analyser, organiser, sensor, explorer • Multiple Intelligences - originally 7, now 8 and more being • investigated (eg emotional, financial ….) • Dunn and Dunn (1994)- five major stimuli to which students respond in • learning situations - environmental, emotional, sociological, physical, • psychological • - Global Vs Analytical

45. Learning Styles MULTIPLE INTELLIGENCES It is not how smart are you but how are you smart? Howard Gardner 1983 onwards View video clip from: Multiple Intelligences: Discovering the Giftedness in ALL Thomas Armstrong (USA) (HO) - Smart pizza sheet

46. Learning Styles • MULTIPLE INTELLIGENCES • Programming using multiple intelligences mapped against • Higher order thinking (Bloom’s Taxonomy) - an example • Sample units: • fractions • money • area and volume • numeration • (HO) - copies of sample units

47. Learning Styles • MULTIPLE INTELLIGENCES • Activity: • Using the appropriate mathematics syllabus, select a topic • that may lend itself to using a 48/56 grid and enter some • activities in the appropriate cells

48. Technology in Mathematics • Research on the effects of technology use on the teaching and learning of mathematics - ppt • CAP Maths in Technology workshops • Table group activity - • Share with others the technology used in your mathematics teaching and learning activities

49. Technology in Mathematics • Mathematics learning opportunities in virtual environments or virtual worlds • Game environments • Software packages eg Kahootz • Maths Education in Second Life - ppt

50. Quality Teaching and Assessment Intellectual Quality Quality Learning Environment Table group activity: Discuss and identify the elements of the NSW Quality Teaching framework that will be supported directly by the use of open-ended enquiry in mathematics teaching and learning. Significance