Exploring boundaries

The following extract is taken from the excellent book Questions and Prompts for Mathematical Thinking produced by the Association of Teachers of Mathematics (ATM) Tel 01332 346599  
ISBN 1 898611 05 X

Quadrilaterals

Draw a quadrilateral
Draw a quadrilateral with one pair of opposite sides parallel
Draw a quadrilateral with one pair of opposite sides parallel, and one pair of opposite angles equal
Draw a quadrilateral with one pair of opposite sides parallel, and one pair of opposite angles equal and diagonals intersecting at right angles

Now go back and make sure that each example is different
Now go back and make sure that your example at stage k does not satisfy condition k+1
Now go back and describe what all xamples at stage k which do not satisfy condition condition k+1 look like.

Integers

Write down number which leaves 1 on division by 7
Write down number which leaves 1 on division by 7, and is a perfect square
Write down number which leaves 1 on division by 7, and is a perfect square, and is a perfect cube

Now go back and make sure that each example is different
Now go back and make sure that your example at stage k does not satisfy condition k+1
Now go back and write down other specific examples, and a general expression for all numbers satisfying condition k but not condition k+1

Decimals

Write down a number
Write down a number, lying between 3 and 4
Write down a number, lying between 3 and 4, which has a rational square root
Write down a number, lying between 3 and 4, which has a rational square root, whose numerator differs from its denominator by a multiple of 3

Now go back and make sure that each example is different
Now go back and make sure that your example at stage k does not satisfy condition k+1
Now go back and write down other specific examples, and a general expression for all numbers satisfying condition k but not condition k+1

Teaching Questioning Skills

Duncan entered a second grade classroom and the regular teachers turned the class over to him. He told the students that their task would be to identify the number he had in mind

The children were obviously familiar with this kind of game and immediately began waving their hands to be called on. "Is it even?" one student asked. "Is it a number (as opposed to a symbol)?" asked another. "Does it have more than three digits?" "Is it black?" "Is it in a row marked with a vowel?"  Click here to read or download the full document.

Fermi Type Questions

Two "typical" Fermi questions are posed and solved in this section.

• How many piano tuners are there in New York City ?
• How many jelly beans fill a one-litre jar?

WARNING: Fermi questions are not really typical and Fermi questions do not have single solutions.
Click here to read or download the full document.

Higher Order Questioning

Several types of questions have the capacity to increase the cognitive level of student responses and shift the environment from one of "show and tell" to one of inquiry and discussion:

- Reasoning questions require students to construct logically organized information, e.g., "How do you know?", "What would happen if...?"
- Open questions allow for more than one acceptable answer, e.g., "Tell us everything you notice about...."
- Interpretive questions focus on applications, relationships, connections, or evaluations and lead students to analyze facts, e.g., "How would this be different if..."
Click here to read or download the full document.

Advice sheet on types of questions click here
A document on effective teaching strategies click here

Exemplars of probing questions produced by the DfES

Effective Questioning in the Classroom -  A PowerPoint presentation using concepts taken from the Questioning unit from the Pedagogy and Practice materials.

Mark Dawes (Comberton College - Cambridge) has taken some KS3 SATs questions and added some follow up questions in order to develop the original question.  In the following examples Slide 1 is for the teacher and the original question follows.  The files were originally produced in notebook format but have been converted to powerpoint slides

The Mathematical Association produced a report on Rich Questioning in Mathematics - Click here

Teaching Mental Maths from level 5 These resources provide rich questions to use in the classroom.