Students and Teachers learning together

1. Students and Teachers learning together Dr Barbara Kensington-Miller University of Auckland

2. Learners do not always attend… • to what the teacher is attending to, or • in the same way as the teacher We need to become aware of how we attend as a learner as this will help us to become aware of how our students attend.

3. As teachers we need to …. • notice what happens inside us as we work • suspend how we see something until we have seen how our students see • provide variations such that you can ask “what would happen if …”

4. What is the same?What is different? If two things are compared then we can ask these questions. This produces a tension which produces an action such as: • Gazing at the whole • Discerning relationships • Perceiving properties

5. What do you see?What do you notice?

6. What is the same?What is different?

7. What is the same?What is different?

8. What do you notice?What do you see? 12 13 16 21 28 37 10 11 14 19 26 -- 8 9 12 17 24 -- 6 7 10 15 -- -- 4 5 8 13 -- -- 2 3 6 -- -- -- 0 1 4 -- -- --

9. What is the same?What is different? -- 43 52 63 76 91 --25 31 39 49 61 75 9116 21 -- 37 48 61 76 9 13 -- 27 37 49 63 4 7 12 -- 28 39 52 1 3 7 13 21 31 43 0 1 4 9 16 25 --

10. What do you notice? What do you see?What do you want to know? a b a ab b b2 ab

11. What happens if I change the variables on the side? What do you notice? a b c (a+b)(c+d) ab =ac+ad+bc+bd Assume a ≠c and b ≠ d b2 ab d

12. What happens if I use this methodology? What do you notice? [(s-a)+a][(s-b)+b] s-a a s-b ab b b2 ab

13. Is this the same? What is different? a [a+(a-b)][a+(a-b)] a b a b a

14. What do you notice? What do you see? q p p

15. To motivate thinking in the classroom, there are different types of activities that can be used: • Classifying – Learners devise their own classifications for mathematical objects, and apply classifications devised by others. – They learn to: • discriminate carefully • recognise the properties of objects • develop mathematical language • develop mathematical definitions

16. In the triplets below, how can you justify each of (a), (b), (c) as the odd one out? (a) a fraction (b) a decimal (c) a percentage (a) sin 60 (b) cos 60 (c) tan 60 (a) 20, 14, 8, 2, … (b) 3, 7, 11, 15, … (c) 4, 8, 16, 32, …

17. a b c

18. 2. Interpreting multiple representations Learners match cards showing different representations of the same mathematical idea. They draw links between different representations and develop new mental images for concepts.

19. Learners are given a large collection of objects on cards and asked to sort them into 2 sets according to criteria of their own choice.

21. They then subdivide each set into two subsets using further criteria. Through sharing criteria, mathematical language is developed.

24. Learners are then given two-way grids on which they can classify the cards. They try and fill every cell or explain why not.

25. Another example is using quadratic functions. Learners divide them into 2 sets according to their own criteria and then subdivide each set into 2 subsets.

26. 3. Creating problems: Doing and undoing processes Learners devise their own problems or problem variants for other learners to solve. This offers them the opportunity to be creative and ‘own’ problems. Example: Doing – the problem poser creates an equation step by step, starting with a value for x and ‘doing the same to both sides’ Undoing – the problem solver solves the resulting equation

27. 4. Evaluating mathematical statements Learners decide whether given statements are always, sometimes or never true. They are encouraged to develop rigorous mathematical arguments and justifications. Example: Numbers with more digits are greater in value. When you cut a piece off a shape you reduce its area and perimeter. Quadrilaterals tessellate. 3 + 2y = 5y If a square and a rectangle have the same perimeter, the square has the smaller area.

28. 5. Problem solving collaboratively Learners work together on a problem trying to solve it and then try and find the simplest solution. Example: A farmer spent $700 on stock that included lambs and ewes. Lambs cost $16 each and ewes cost $25 each. How many lambs and how many ewes did the farmer buy?

29. Example:Tom has8 bags of $1 coins. One of the bags has less coins than all the others. Using only a balance-scale and without counting the contents of each bag determine which bag has less money in it.