Developing (Mathematical) Habits of Mind

Mathematicians need to be Curious Thoughtful Collaborative Determined

Encouraging Curiosity

mathematics The most exciting phrase to hear in science, the one that heralds new discoveries, is not Eureka!, but rather, “hmmm… that’s funny…” Isaac Asimov

Mind Reader? http://www.flashlightcreative.net/swf/mindreader/ Choose any two digit number, add together both digits and then subtract the total from your original number… http://www.flashlightcreative.net/swf/mindreader/

Another starting point… Work out the mean, median, mode and range of: 2, 5, 5, 6, 7 What do you notice? Are there other similar sets?

May lead to: Is it possible to find five positive whole numbers where: Mode < Median < Mean Mode < Mean < Median Mean < Mode < Median Mean < Median < Mode Median < Mode < Mean Median < Mean < Mode

This rectangle has an area of 24 cm² and a perimeter of 20 cm

Some questions that might emerge: Is it possible to have an odd number perimeter and an area of 24 cm²? What is the smallest possible perimeter? And the largest? More generally, is it possible to have a fractional perimeter but a whole number area? And vice versa? If I give you the area and perimeter of a rectangle, will you always be able to work out its dimensions?...

Thinking (Mathematically)

“You can know the name of a bird in all the languages of the world, but when you're finished, you'll know absolutely nothing whatever about the bird... So let's look at the bird and see what it's doing - that's what counts.”  Richard P. Feynman

What learning behaviours do we value and how can we encourage them in our classrooms? Possible examples: Make 37 Odds and Evens Mention TIRED

Higher Order Thinking Exploring Working systematically Looking for connections Conjecturing Explaining Generalising Justifying Transition from tentative knowledge – conjecturing, to concrete knowledge.

Forwards add Backwards 747 is an interesting number… Can you find any other interesting numbers between 700 and 800?

Dicey Operations With a partner roll a die 9 times. Fill in a digit in the grid after each roll. Who can get the answer closest to 1000? Powers. Justification and proof. Show the Place Value and Powers teacher feature, and the Powers and Roots collection and the Factors, Multiples and Primes collection.

Cryptarithms

Gabriel’s Problem

Working Collaboratively

Rules for effective group work All students must contribute: no one member says too much or too little Every contribution treated with respect: listen thoughtfully Group must achieve consensus: work at resolving differences Every suggestion/assertion has to be justified: arguments must include reasons Neil Mercer

The Factors and Multiple Challenge You will need a 100 square grid. Choose which numbers you want to cross out, always choosing a number that is a factor or multiple of the previous number that has just been crossed out. Each number can only be crossed out once. e.g. 40, 10, 30, 60, 6, 3, 33, 66, 22, 11… Try to find the longest sequence of numbers that can be crossed out.

Reflecting Squarely In how many ways can you fit all three pieces together to make shapes with line symmetry?

Buckminster Fuller, Inventor “If I ran a school, I’d give all the average grades to the ones who gave me all the right answers, for being good parrots. I’d give the top grades to those who made lots of mistakes and told me about them and then told me what they had learned from them.” Buckminster Fuller, Inventor 22

Encouraging a growth mind-set

… our studies show that teaching people to have a “growth mind-set”, which encourages a focus on effort rather than intelligence or talent, helps make them into high achievers in school and life. Carol Dweck

Product Sudoku Like a conventional Sudoku, but this time the numbers in the cells are the product of the digits in the cells horizontally and vertically adjacent to the cell…

Product Sudoku The numbers in the cells suggest the order in which they could be filled; it is just one possible route through the problem…

What next? Enriching the Secondary Curriculum: http://nrich.maths.org/enriching What we think and why we think it: http://nrich.maths.org/whatwethink

I don't expect, and I don't want, all children to find mathematics an engrossing study, or one that they want to devote themselves to either in school or in their lives. Only a few will find mathematics seductive enough to sustain a long term engagement. But I would hope that all children could experience at a few moments in their careers ... the power and excitement of mathematics ... so that at the end of their formal education they at least know what it is like and whether it is an activity that has a place in their future. David Wheeler