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Questions and prompts used

We noticed that some teachers would address challenging questions and prompts to the whole class, while others would only set the task in whole class and then go round the room giving differently worded challenges to different students. A few teachers did not pose what we would call challenging questions, but set straightforward tasks and their interactions were about how to complete them. In this section, we report the type of questions and prompts which we regarded as mathematically challenging. In the first year teachers used prompts such as these:

What to do
Be systematic
I want to see...
What are you doing...?
Do like I did
How many ... in ...?
Imagine a...
Imagine you...
Notice that...
Choose
What is your strategy?

Check
How can we check?
Did you get one with ...(a feature)?
That's not quite right is it?
How have you got this/these?

Familiarity
Have you seen anything like this as home, outside school?
Remember when we did this?

Sort, compare, classify
Identify and explain differences
What's the same and different?
Identifying and naming
How many are...?
How do these relate?
Is this ... or ...?
What is the difference?

Purpose
What is it for?
Ask me a question which...?
Why is this useful?

Exemplify
Give me an example
Write down an example

What are you thinking? Can you tell me...?
What are you making of this?
What do you think?
What can you see?
What else can you write down?
What is your theory?
What can you write down?

Anticipate
What is the pattern?
What will happen if...?
How many should...?
What is significant...?
What do you expect to happen?

Asking for reasons
Why?
Why...? Why not...?
Give me a reason
Can you prove?

The questions above were common throughout the project. Most of these indicate to students useful ways to think about mathematics.

In the second year some of the questions and prompts used were more sophisticated, combining two or more aspects of mathematical activity for the purposes of developing a sense that completing a task provides raw material for another thought process towards stronger reasoning, or more complex concepts.

PromptPurpose
Predict what will emerge according to some rules One purpose of identifying rules is to predict further behaviour
Does what you have produced fulfil certain properties? Having created something according to guidelines, then see what other features it has
Notice properties and characteristics of a diagram A diagram is not just a representation of objects, but also can reveal further features
How does the task relate to the objectives? Finishing the task is the start of making connections
Which method is more efficient? Methods have to be chosen wisely
Will it tell us anything we don't know already? Evaluating methods
If we want to find out whether J's hypothesis is true, what do we need to do? Focus on methods of reasoning
Are we clear about what we are looking for? General metacognitive strategies
What have we learnt over this year that we can use? Topics and rules become tools for harder maths
Can you see how close or how far you were from a definition? Being precise, concise, complete
See what X has done Compare methods