PD5: Developing questioning

1. PD5: Developing questioning

2. Aims of the session This session is intended to help us to reflect on: • the reasons for questioning; • some ways of making questioning more effective; • different types of ‘thinking questions’ that may be asked in mathematics.

3. Why ask questions? • To interest, challenge or engage. • To assess prior knowledge and understanding. • To mobilise existing understanding to create new understanding. • To focus thinking on key concepts. • To extend and deepen learners’ thinking. • To promote learners’ thinking about the way they learn.

4. Ineffective questioning • Questions are unplanned with no apparent purpose. • Questions are mainly closed. • No ‘wait time’ after asking questions. • Questions are ‘guess what is in my head’. • Questions are poorly sequenced. • Teacher judges responses immediately. • Only a few learners participate. • Incorrect answers are ignored. • All questions are asked by the teacher.

5. Effective questioning • Questions are planned and related to session objectives. • Questions are mainly open. • Teacher allows ‘wait time’. • Both right and wrong answers are followed up. • Questions are carefully graded in difficulty. • Teacher encourages learners to explain and justify answers. • Teacher allows collaboration before answering. • All participate e.g. using mini-whiteboards. • Learners ask questions too.

6. Different types of questions • Creating examples and special cases. • Evaluating and correcting. • Comparing and organising. • Modifying and changing. • Generalising and conjecturing. • Explaining and justifying.

7. Creating examples and special cases Show me an example of: • a number between and ; • a hexagon with two reflex angles; • a shape with an area of 12 square units and a perimeter of 16 units; • a quadratic equation with a minimum at (2,1); • a set of 5 numbers with a range of 6…and a mean of 10…and a median of 9

8. Evaluating and correcting What is wrong with these statements? How can you correct them? • When you multiply by 10, you add a nought. • + = • Squaring makes bigger. • If you double the radius you double the area. • An increase of x% followed by a decrease of x% leaves the amount unchanged. • Every equation has a solution.

9. Comparing and organising What is the same and what is different about these objects? • Square, trapezium, parallelogram. • An expression and an equation. • (a + b)2 and a2 + b2 • Y = 3x and y = 3x +1 as examples of straight lines. • 2x + 3 = 4x + 6; 2x + 3 = 2x + 4; 2x + 3 = x + 4

10. Comparing and organising How can you divide each of these sets of objects into 2 sets? • 1, 2, 3, 4, 5, 6, 7, 8, 9,10 • , , , , , • y = x2 - 6x + 8; y = x2 - 6x + 10; y = x2 - 6x + 9; y = x2 - 5x + 6

11. Modifying and changing How can you change: • this recurring decimal into a fraction? • the equation y = 3x + 4, so that it passes through (0,-1)? • Pythagoras’ theorem so that it works for triangles that are not right-angled? • the formula for the area of a trapezium into the formula for the area of a triangle?

12. Generalising and conjecturing What are these special cases of? • 1, 4, 9, 16, 25.... • Pythagoras’ theorem. • A circle. When are these statements true? • A parallelogram has a line of symmetry. • The diagonals of a quadrilateral bisect each other. • Adding two numbers gives the same answer as multiplying them.

13. Explaining and justifying Use a diagram to explain why: • a2 − b2 = (a + b)(a − b) Give a reason why: • a rectangle is a trapezium. How can we be sure that: • this pattern will continue: 1 + 3 = 22; 1 + 3 + 5 = 32…? Convince me that: • if you unfold a rectangular envelope, you will get a rhombus.

14. Make up your own questions • Creating examples and special cases. • Evaluating and correcting. • Comparing and organising. • Modifying and changing. • Generalising and conjecturing. • Explaining and justifying.