1. Thinking in Mathematics and Statistics Wellington Mathematical Association 14/10/2006 Jim Hogan

2. My keynote aims to:- • describe how thinking develops in mathematics
• give meaning to the language of the Draft Curriculum
• confirm it is all about “what happens in your classroom”.

3. What is happening? 1994 was the year! Teaching became complex.
Enter the 21st century.
Enter TIMS
The Third International Study of Mathematics and Science.

4. NZ Numeracy Project Born out of failure We did something
And changed that which we found.

5. Main Research Findings
Students learn in steps or stages
Major misconceptions
Content knowledge issues
All that is taught is not learned
All that is learned is not taught
See Numeracy Book 3 (research)

6. Darft Curriculum
An opportunity to link modern mathematics education research and practice with curriculum.
Numeracy and Curriculum share a language.

7. Today Expect change this is a very good read
It’s About Learning…and it’s about Time!…Stoll et al

8. Stages of Learning to Think
I solve problems by:-
• Counting Stages 1 to 4
• Adding Stages 5 and 6
• Multiplying Stages 6 and 7
All with whole numbers
• Proportionally Stage 8
But now with fractions and decimals.

9. Numeracy Stage Language
Counting
Additive Thinking
Multiplicative Thinking
Proportional Thinking
And in there is ALGEBRAIC thinking at all levels.

10. Numeracy Stage Ideas Counting • Zero dimensional
Additive Thinking • 1D
Multiplicative Thinking • 2D
Proportional Thinking • multi-D
And in there is ALGEBRAIC thinking at all levels…generalizing…abstracting

11. Higher Stages I think of the higher forms of thinking as multiplexing.
I am using all that I know along the same line. I am connecting, synthesising and creating.
I am thinking and I am solving problems all at once.

12. The Ideal Data? Years 1,2,3 •Stages 1,2,3,4 •CL1
Year 9 •Stages 7 •CL4
Years 10 •Stages 8 •CL5
Year •CL6
Year •CL7
Year •CL8

13. Could be this?

14. Houston, we have a problem.

15. A big mismatch problem

16. Reality of Year 9
In 2002/3/4/5 60% of Year 9 students were measured to choose additive solutions to problems. (New to SNP)
What does this mean for the Year 9 math programme in your school?
Flexibility at least

17. Year 9 Programme additive or multiplicative?
Factors, multiples, primes,
Four operations on fractions?
Decimals
Squares, square roots and other powers
Transformations
Area and perimeter
Solution of equations

18. Numeracy Stage Ideas Counting • Zero dimensional
Additive Thinking • 1D
Multiplicative Thinking • 2D
Proportional Thinking • multi-D
And in there is ALGEBRAIC thinking at all levels.

19. Teaching Model • strategy • existing ideas • materials
• imaging the model
• using properties
• fold backs
• constructing new knowledge - links
• discovering Book 3
Research supports using materials. This model says find out what the student knows and build. Get them to generalising number properties. Create ways to allow them to discover links. Take a careful look at Book 3.

20. From AssTLe What are you learning? Why are you learning this?
How does you teacher help you?
How will you know when you have learned it?

21. From Numeracy How did you get that answer? Can you do it another way?
Give me another..and another..and another
How do you know that?

22. This mathematics project
Called NZ Numeracy
Stresses understanding
Is THINKING • Key Competency

23. What is additive thinking?
What is the next three terms?
__ __ __
___ ___ ___

24. What is multiplicative thinking?
Same question!
__ __ __
___ ___ ___
So what is the nth term?

25. Solve this
There are 36 birds in each of 8 cages. How many birds all together?
Record on BB

26. Did you ? 30 6
120 24 4 60 12 2
30 x x 8
36 double double double
72 x 4 or 72 double double
40 x x 8
10x x 36 3 6 x 8 2

27. We presume, generally speaking
That all students are ready for Year 9 and are capable of figuring:-
18x9 factors of 24 1/2 +1/3
volume 2/3 of 12 ratio
area 2 cubed draw a 2cm grid

28. Try this What is the ratio that is exactly midway between 1:1 and 1:2
PAINT MIX
How do I get the halfway colour?

29. And there are issues Barriers form between Counters and Adders
If we dilly dally along the way.
Likewise between
Adders and Multipliers
If we do not understand multiplication.

30. Student Centered Learning
What is the thinking I want my students to develop?
It is all about “what happens in the classroom”.
It is all about learning.

31. My Workshops
Math Investigations • do these and discuss, how could you use them.
Proportional Thinking • I have you confront the difficult.

32. BOPMA Math Conference 24/11/2006 • Friday Rotorua Convention Centre
More details on my unfinished and very drafty newsletter that I was forced to publish early for you.
Thank you.