1. Thinking & Working like a Mathematician
Darebin Arts & Entertainment Centre
20th Feb, 2009
Northern Metropolitan Region, DEECD
Adrian Berenger
Linda Dimos
Antoinette Hooper
Theresa Hsieh
Silvia Kalevitch

2. Session1: Closed to Open Tasks Group work activities.
Overview
Session1: Closed to Open Tasks Group work activities.
Session 2: Working Mathematically
Structured activities around designing, assessing and adapting learning activities

3. Purpose Build teacher capacity in responding to student diversity
Create Open-Ended Activities to engage students
Build upon knowledge of the VELS Dimension: ‘Working Mathematically’
Link with Progression Points and the Mathematics continuum
Link with current research into Multiplicative Thinking

4. Types of Questions
Closed Questions are those which require an answer or response to be given from memory.
Open Questions Require students to think more deeply and to give a response which involves more than recalling a fact or reproducing a skill. Usually more than one possible answer.

5. ‘Our goals in education are for our students to think, to learn, to analyse, to criticise and to be able to solve unfamiliar problems.’
Peter Sullivan and Doug Clarke

6. Bloom’s Taxonomy
Can the student GENERATE new products, ideas or ways of viewing things?
Can the student JUSTIFY a decision or course of action?
Can the student DIFFERENTIATE between constituent parts?
Can the student USE the new knowledge in another familiar situation?
Can the student EXPLAIN ideas or concepts?
Can a student RECALL information?

7. Example Calculate the Perimeter and Area of the rectangle.
12 cm
3 cm
Calculate the Perimeter and Area of the rectangle.
I want to make a garden in the shape of a rectangle. I have 30 metres of fence for my garden. What might be the area of the garden?

8. Creating Open-Ended Tasks
‘Any closed questions can be reformulated to create an open ended questions using one of two methods’.
(Sullivan & Lilburn 1997)

9. Method 1
Omit enough information so that, although the answer remains the same, the digits required to achieve the answer becomes variable.
TRADITIONAL
OPEN-ENDED
249
+ 173
2**
+ *7*
422

10. Method 1 (continued) TRADITIONAL
Two fifths of 250 students borrow books from the library each day. Calculate the number of students who borrow books each day.
Find the missing angle on this trapezoid
OPEN-ENDED
Two fifths of the students in a school borrow books from the library each day. How many students might there be in the school and how many of them borrow books each day?
What might the angles on this trapezoid be?
40◦
140◦

11. Method 2
Work backwards from the answer. Begin with a closed task. Calculate the answer, then work backwards and using the context of the question, create a question that would allow multiple responses to achieve the same answer.
TRADITIONAL
The following numbers represents the temperature of 5 consecutive days in Melbourne: 44◦C, 42◦C, 36◦C, 22◦C, 29◦C. Find the average temperature.
OPEN-ENDED
The average temperature over five consecutive days in Melbourne was 35◦C. The highest temperature was 44◦C. What might the temperature have been on the other days?

12. Method 2 (continued) The answer is 17.5 What might the question be?
TRADITIONAL
What is the volume of the cylinder?
OPEN-ENDED
The answer is 17.5 What might the question be?
What might the dimensions of a cylinder prism that has a volume of 300 cubic centimetres?
35.0 X
4cm
6 cm

13. Activity 2: Creating Open-Ended Questions
Group work: On the A3 pages, there are 8 closed questions. Change each closed question into and Open-ended question. (20 minutes)
At the duration of the activity, groups will be asked to share their work with the whole group (15 minutes)
Display work around the room

14. Summing Up! Questions? Open Ended Maths Activities
2nd Edition
Peter Sullivan and Pat Lilburn
Oxford
Thinking Tools for the Mathematics Classroom.
Sue Gunningham
Hawker Brownlow Education

15. Exit Pass
Using an ‘ Exit Pass’ identify one thing that you have learnt.

17. Parking Lot

18. Session 2: Working Mathematically

19. Developmental Overview Jigsaw
Each group will need to assign a scribe, presenter and time-keeper
In your groups, assemble the ‘Developmental Overview Jigsaw’. Use the VELS Standards and Progression Points document to assist.
After 20 minutes, groups will be given an A3 answer sheet to be used in the next activity.

20. Activity 3: Chocolate Partitioning
If a 250g block of chocolate has 60 pieces, how might the pieces be arranged?
What might a 400g block look like?
Use butcher paper to record your group’s solution processes to the two questions.
Extension: What might a 190g block look like?

21. What does this task address?
Using the development overview A3 sheet, identify the areas of ‘Working Mathematically’ addressed by the chocolate partitioning activity (10 mins).
On your butcher paper, identify how this problem may be modified or extended to cater for a range of ability levels (5 mins).
Selected groups will be invited to present their work to the whole group. (15 minutes)

22. Summing Up!
Questions?
for all information regarding progression points, the continuum and scaffolding numeracy in the Middle Years.

23. Evaluation
6 ‘thinking hats’ survey (De Bono)

24. Video: 7X13=28

25. The End