Deep progress in mathematics Agder, Norway Anne Watson September 2006

 Cut out as many 2 x 1 blue rectangles as you can  Cut out as many 3 x 1 red rectangles as you can  Cut out as many 5 x 1 green rectangles as you can  Cut out as many 1 x 1 white squares as you can

Talk about  how many you made of each  how you chose to cut them out  anything else which occurs to you

What lengths can you make using only red pieces only green pieces only blue pieces blue and red pieces blue and green pieces other combinations?

Why?

In how many ways can you make a snake of length 21?

Ways of recording 21-snakes  21 x 1  10 x  x 2  3 ( )  3 x x 2  7 ( )  7 ( ) …

What else can you do?

What is the underlying thinking?  Emotional – feeling better about learning mathematics (belonging)  Thinking – being better at learning mathematics (becoming)  Knowing – knowing more mathematics (being)

Deep Progress in Mathematics  Learn more mathematics  Become better at learning mathematics  Feel better about learning mathematics

Choose an expression  n + 2  (5n + 1)/2  2n – 3  Now work out the value of your expression when n is 3

Building on confidence  Find other values for n which alter the order  Can you choose a harder expression?  Choose your own value of n for 2n-3  Can you make up an expression for your own use?  Keep that value of n: can you make up an expression which always leaves you at the right-hand end/left- hand end?

Developing proficiencies  Looking for patterns is natural so can I present concepts using patterns? so can I control variables so the underlying ideas are easy to see?  Matching ideas to other people’s is natural so can I use matching different perceptions in lessons?  Creating own examples is a natural exploration method so can learners’ own examples be incorporated into lessons?

Improving Attainment in Mathematics Project  Ten teachers who wanted year 7 students who were ‘below level 4’ to do better  They believed that all students can think hard about mathematics, and thus do better at mathematics

 They can’t …..  They don’t …..  They don’t, so how can I give opportunities and support so that they do …..

Becoming independent Teachers asked learners to:  Make something more difficult  Make comparisons  Pose their own questions  Predict problems  Give reasons  Work on extended tasks over time  Share their methods  Deal with unfamiliar problems Learners took the initiative to:  Make something more difficult  Make extra comparisons  Generate their own enquiry  Predict problems  Give reasons  Spend more time on tasks  Create methods and shortcuts  Deal with unfamiliar problems  Initiate a mathematical idea  Change their mind with new experience

 Watson: Raising Achievement in Secondary Mathematics. OPEN UNIVERSITY PRESS  Watson, De Geest & Prestage: Deep Progress in Mathematics ATM website (MT157) or my website:  Prestage & Perks: Adapting and Extending Secondary Mathematics Activities: New Tasks for Old FULTON  Watson, Houssart & Roaf: Supporting Mathematical Thinking FULTON  Bills, Bills, Watson & Mason: Thinkers ATM  Watson & Mason: Questions and Prompts for Mathematical Thinking ATM  Ollerton & Watson: Inclusive Mathematics CONTINUUM