Improving learning in mathematicsImproving learning in mathematics

PD5: Developing questioning

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.

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.

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.

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.

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 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

3

17

2

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.

10

2

10

3

20

5

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

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 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.

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.