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

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”.

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

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

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)

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

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

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.

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

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

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.

The Ideal Data? Years 1,2,3 •Stages 1,2,3,4 •CL1 Year 9 •Stages 7 •CL4 Years 10 •Stages 8 •CL5 Year 11 •CL6 Year 12 •CL7 Year 13 •CL8

Could be this?

Houston, we have a problem.

A big mismatch problem

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

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

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.

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.

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?

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?

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

What is additive thinking? What is the next three terms? 3 6 9 __ __ __ 28 56 84 ___ ___ ___

What is multiplicative thinking? Same question! 3 6 9 __ __ __ 28 56 84 ___ ___ ___ So what is the nth term?

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

Did you ? 30 6 120 24 4 60 12 2 360 - 72 30 x 8 + 6 x 8 36 double double double 72 x 4 or 72 double double 40 x 8 - 4 x 8 10x36 - 2 x 36 36 + 36 + 36 + 36 + 36 + 36 3 6 x 8 2

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

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?

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.

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.

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

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.