I think that the ability to create well-thought-out questions with varying degrees of complexity that allow pupils to access content and explore and develop ideas further is fundamental to good teaching. I would like to say I always think out these questions in advance, but I often find that I need to create a line of questioning on the spot as I adapt to the needs of students during a lesson. Is there a simple way to do this?
One simple method is to extend or alter a given problem in various ways. In his book Learning and Doing Mathematics, John Mason talks about creating new problems from old ones by stressing various words in a statement or problem and considering what would happen should those words change. He gives the following example:
Three points determine a circle.
Now, read this statement aloud emphasising a different word each time. Now ask yourself - what would two or four points determine? What if we changed the word points to lines? What do we mean by determine? Do three points only determine one circle - how many points would be needed to determine n circles? Change the word circle: what would determine other shapes (e.g. a square)?
This is a lovely idea. Try it for yourself with a statement or formula of your choice, such as:
The circumference of a circle is pi x diameter.
What if we change the word circumference? What else can we work out about a circle? How about changing circle to a different shape, like a square? Does the formula still hold? We don't have circumference for non-circles, but is there a similar formula for perimeter? If there is, what is the equivalent of pi for a square? What is pi anyway? Where does it come from? Do we have to use pi, or can we use something else? Can we express the circle circumference formula in other ways, maybe in terms of the area?
This idea of extending a problem to create new and interesting areas for investigation is something I have been developing when creating these problem sets. Here are some examples of this method in practice, from Problem Set 4.2 which is about probability .
Here is a nice problem (or an exercise for some!) that involves uniform distributions (though not formally) and multiplying negatives:
Here's another nice problem from the same problem set, that encourages pupils to consider inequality regions and probability as area:
Here's a third problem from the same problem set that is rich in possibilities for extension:
As you can see, this is a simple, powerful method to create questions that will challenge pupils by encouraging them to think more deeply and develop their ideas.
I have only included examples here of extending problems. If you look at the problem sets, you will see that in general for the first step for each problem I have used these ideas to simplify the problem in order to make it more accessible, with each subsequent step increasing in complexity to allow pupils to solve the original problem and perhaps go further to solve an even harder one, as shown above.
In each of these cases we could go further still and ask students to generalize their findings, as in example 3(d).
Try this method out for yourself. If you have any thoughts on this, or know other good methods for creating problems, please let me know via twitter @dannytybrown.