Asking the right question Francis Bove Further Mathematics Support Programme Coordinator East London

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.

Questioning In your groups Look at the questions on the cards Sort the questions into different categories You might like to arrange them on a poster in diagrammatic form

Why ask questions? In your groups: Brainstorm reasons why a teacher might ask a question in a lesson…..

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.

The killer question…… I’ll take five?

What is effective questioning In your groups You are an Ofsted inspector……. What does effective questioning look like? What is a good question? What is good practice?

What is effective questioning Questions are planned, well ramped in difficulty. Open questions predominate. A climate is created where learners feel safe. Pupil’s questions are often ‘answered’ by asking another question A ‘no hands’ approach is used, for example when all pupils answer at once using mini-whiteboards, or when the teacher chooses who answers. Probing follow-up questions are prepared. There is a sufficient ‘wait time’ between asking and answering a question. Pupils are encouraged to collaborate before answering. Pupils are encouraged to ask their own questions. From: Improving Leaning in Mathematics: Challenges and Strategies Malcolm Swan. DFES 2005

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.

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

Creating examples and special cases Show me an example of: A quadratic with both a positive and negative root; a function that has a derivative of 3x 2 - 2; An arithmetic progression that has 4 and 6 as two of its terms; 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

Evaluating and correcting What is wrong with these statements? How can you correct them? + = Squaring makes it bigger. When you multiply by 10, you add a nought. 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.

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

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

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?

Generalising and conjecturing What are these special cases of? 1, 4, 9, 16, 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.

Explaining and justifying Use a diagram to explain why: a 2 − b 2 = (a + b)(a − b) Give a reason why: Differentiating y = 4x 2 - 3x + 2 and y = 4x 2 - 3x - 5 gives the same result. How can we be sure that: this pattern will continue: = 2 2 ; = 3 2 …? Convince me that: sin 2 x + cos 2 x = 1

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.

Optimization Problems: Boomerangs

Evaluating Sample Responses to Discuss What do you like about the work? How has each student organized the work? What mistakes have been made? What isn’t clear? What questions do you want to ask this student? In what ways might the work be improved? P-20

Alex’s solution P-21

Danny’s solution P-22

Jeremiah’s solution P-23

Tanya's solution P-24

Assessing Pupils’ Progress Probing questions Linked to National Curriculum Always challenging Level 4 – level 8 I will them

Assessing Pupils’ Progress Level 6 Fractions

Assessing Pupils’ Progress Level 6 trial and improvement

Assessing Pupils’ Progress Level 6 quadrilaterals

Reflections Make a note of THREE things that you have learnt in this session Make a note of anything that you have met in the session that might require some further study or investigation What were the main messages for a teaching assistant in using questioning?

Useful references ‘Thinkers’ ISBN available from the Association of Teachers of Mathematics Improving Learning in Mathematics ths/collection/282/improving-learning-in- mathematics Improving Learning in Mathematics: Challenges and Strategies. Malcolm Swan ISBN: X