Presenting a task is a major factor in how the task is taken up and engaged in.
There are many different ways of presenting essentially the same task or initiating mathematical activity.
- Start simple and complexify
- Start with the complex and then publicly simplify
- Start moderately complex, expect learners to simplify and then re-complexify and extend for themselves
- Present a single ‘example’ and invite them to tackle their own variations
- Present several ‘examples’ and use these to highlight important features [see dimensions of possible variation]
- Start with a phenomenon to make sense of or explain
- Start with a context where the topic is used authentically, or which learners can relate to their own experience.
- Start with individual work, then in a plenary discuss options and possibilities before moving to group work
- Start in silence [silent lessons]
- Start with a brain shower: people contributing ideas in plenary
- Start with a brain shower in groups (pairs, larger groups) before moving to plenary [see talking in pairs]
What is probably most important is to vary the way you present tasks, because if learners recognise a habit, they will come to depend on that format, and are likely to be thrown by any sudden variation.
Starting simply and gradually complexifying or generalising appears to be a safe and secure strategy which enables everyone to work to their own strengths (so quicker thinkers progress more quickly). However it has many weaknesses as a strategy. First, it fails to make use of learners' powers because all the simplifying and specialising has been done for them; second, it conforms with a ‘gentle ramp’ metaphor for learning which imagines that it is possible for learning to be gradual and unproblematic. There is plenty of evidence that learning is a response to challenge, so that if learners are not being challenged, they are unlikely to be learning. Put another way, if all learners are doing is what they can already do, without needing to modify their behaviour or their ways of thinking, then they can not be learning anything useful (though the may be learning that mathematics is not for them!).
Staring with the complex or the general, and then publicly simplifying and specialising demonstrates how those powers can be useful in order to understand a problem, before re-generalising and resolving it. By specialising publicly, learners see you ‘being mathematical with and in front of them’. By later reducing the directness of your simplifying and relying on learners to do it for themselves, you provide both the scaffolding and fading needed for learners to be prompted to internalise those processes for themselves.
By always presenting tasks the same way, you create learner dependency, because they come to depend on that particular form of presentation.
Mathematics is primarily a way of making sense of the world. By presenting learners with something happening, whether a physical activity, an animation or simulation, or a mental imagining, it is often possible to provoke at least some learners into wanting to have an explanation for the phenomenon. [see also Structure of a Topic]
You can use a situation in which a challenge has been set to create an opportunity for discovery - and at the same time consolidate and practice understanding of concepts previously learnt. For example, the Factors and Multiples Game on the NRICH web site can give students the chance to use their understanding of factors and multiples to find out how many numbers in the hundred square can be covered. When you play this game you may explore the relationship between factors and multiples and seek a more profound understanding by asking yourself some of the following questions:
What numbers are best to start with?
What's different about square numbers?
How many squares is it possible to cover in one run of numbers?
So this 'game' prompts you to explore the mathematics behind factors and multiples further.