1. Noel-Ann Bradshaw University of Greenwich Encouraging Maths Students to Think

2. Outline Can maths students think? What we have done to encourage this? How students have reacted to this? What else we can do? Discussion

3. Motivation Thinking Mathematically by John Mason What to do when stuck Specialisation Generalisation Improved grades! Mathematical Technology and Thinking

4. Specialisation Differentiate the following w.r.t.

5. Specialise then Generalise A number like 12321 is called a palindrome because it reads the same backwards as forwards. A friend of mine claims that all palindromes with four digits are exactly divisible by eleven. Are they?

6. General form A four-digit palindrome is of the form: abba 1000a +100b+10b+a = (1000+1)a + (100+10)b = 1001a + 110b = 11(91a+10b) Proved with a general form

7. How does it feel to be stuck? What does it feel like to be faced with a problem you can’t answer? What does it feel like to be stuck? How can you begin to specialise? How can you turn this into generalising? Do you always have to do this? Does it help – why? Record your feelings

8. What do you know and what do you want to know? Our three ages combined amount to just seventy years. As I am just six times as old as you are now, it may be said that when I am but twice as old as you, our three ages combined will be twice what they are at present. Now let me see if you can tell me how old your mother is?

9. Response to being stuck Stare at a blank page Do anything except the problem you are trying to do Get stressed / panic Feel frustrated / useless etc Do what has been suggested, take a break and try again

10. Write down your thoughts What numbers have an odd number of divisors?

11. Consecutive sums Some numbers can be expressed as the sum of a string of consecutive positive numbers. Exactly which numbers have this property? Eg 9=2+3+4 11 = 5+6

12. Maths Café Question {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} Can you separate the integers 1 to 16 into two sets of equal sizes so that each set has the same sum, the same sum of squares, and the same sum of cubes? If yes – list the two sets!

13. Other tools Maths busking Visual demos

14. Student feedback Not entirely positive! Introduce more across all four first year courses Build on this in the second year See the point in final year

15. Discussion Do you find that your students are resistant to ‘thinking’? If so, how do you encourage them to think?