Bloom taxonomy descriptor cards: Bloom-descriptors.doc

Problem-solving cycle cards: ProblemSolvingCycleCards.doc

Rich task cards: RichTaskCards.doc

Higher-order thinking skills are not about mathematical content knowledge. Just like it is possible to engage in very hard questions that involve a high level of content knowledge but few problem-solving skills, it is also possible to identify very difficult problems that only need very low levels of mathematical content knowledge. In the former case, you are going to need well-tuned knowledge skills and in the latter, your HOTS.

What is a problem?

A problem is something you do not immediately know how to solve. There is a gap between where you are and even getting started on a path to a solution. This means that something that is a problem to your students is something that they cannot get to grips with immediately and requires thinking and playing time. By playing with the mathematics, patterns and connections often reveal themselves. We need to arm our pupils with a repertoire of skills to help them step into problems independently rather than immediately turning to us as teachers to ask what to do! We can begin by selecting problems with engaging starting points which invite pupils to step in (such as a game). Once they get started, the richness comes from what happens next. Ideas begin to emerge from playing with the initial situation and sometimes from posing problems of their own.

What is problem solving?

The need to apply problem-solving techniques to a problem is an indicator that it has the potential to be a rich task. Problem solving requires you to have a problem to solve, which may be one you have been given or one you have posed for yourself. The activity that we call 'problem solving' is a complex one and can be considered as a cycle of activity (though the cycle often requires us to move backward and forward whilst maintaining a general sense of direction). There are many models of the problem solving cycle. Possibly the most well known is the one described by Polya in his book How to Solve It (1957), which is a must-read for those of us interested in improving our pupils' problem-solving skills. Here is one we use at NRICH:

The application of the problem-solving cycle is a high-order skill. Evidence suggests that few pupils utilise the problem-solving cycle effectively. One important thing to note is the emphasis the cycle places on the high-order thinking skills described by Bloom. It is therefore not surprising that most pupils do not naturally have a sense of where they are and what they might do next. One of our aims when teaching mathematics is to help pupils become familiar with this process and have confidence to use it.

See Polya, G. (1957). How to Solve it, Princeton University Press.

Here is a task to help answer this question.

We feel that any problem has the potential to be a rich task but this depends on us as teachers offering those opportunities to our pupils. We will talk about this in Activity 2.1 .