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 3x2 - 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.

10

2

10

3

20

5

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 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

1, 2, 3, 4, 5, 6, 7, 8, 9,10 , , , , ,

a

y = x2 - 6x + 8; y = x2 - 6x + 10;

y = x2 - 6x + 9; y = x2 - 5x + 6

How can you divide each of these sets of objects into 2 sets?

2

13

2

4

3

5

4

6

5

7

6

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 conjecturingWhat 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.

Explaining and justifyingUse a diagram to explain why: a2 − b2 = (a + b)(a − b)Give a reason why: Differentiating y = 4x2 - 3x + 2 and y = 4x2 - 3x - 5

gives the same result.How can we be sure that: this pattern will continue:

1 + 3 = 22; 1 + 3 + 5 = 32…?Convince me that: sin2x + cos2x = 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 email 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 1-898611-26-2 available from the Association of Teachers of Mathematics

http://www.atm.org.uk/

Improving Learning in Mathematics

http://www.nationalstemcentre.org.uk/elibrary/maths/collection/282/improving-learning-in-mathematics

Improving Learning in Mathematics: Challenges and Strategies. Malcolm Swan ISBN: 1-84478-537-X