With thanks to Nick Hart, Deputy Headteacher at Penn Wood Primary School in Slough, for this really interesting article on modelling mathematical concepts
The simple answer is ‘Yes, but…’. Real life problems are a valuable task to set but where they sit in a unit of work is a vital decision for teachers to make.
Let’s take the example of adding fractions. Proponents of real life maths may jump straight in with pizzas, with the intention of making the learning relevant. For children who are new to learning how to add fractions, this could cause a distraction rather than help them to learn what is intended. It’s a fairly common conversation, I would imagine: Son, what did you learn in maths today? Errrr… pizza?
There are two issues here. The first is that the context can often be remembered at the expense of the actual maths. Memories are formed more securely when they are associated with a strong emotion, when there is something unusual about the scenario or when the learner already knows something about the content. With those conditions in mind, what’s more likely to be remembered?
It’s partly because of scenarios like this that teacher up and down the country have conversations with their classes that go something like this:
Teacher: We’re going to be carrying on from our work on adding fractions last Friday. Let’s see what you can remember…
*Teacher displays the question 3/4 + 1/8.
*Blank looks from some children
Teacher: We only did this on Friday! You must remember. We cut up pizza and everything!
Class: Oh yes! Pizza!
*Excited chatter followed by the return of the blank looks.
There is also the risk that such strong context as a starting point can hinder the transfer of learning to other contexts. For example, children might well succeed at solving problems about pizza but in the process, form the misconception that adding fractions and pizza are exclusively linked. They may then come across a problem about cartons of juice and not be able to see the link because pizza has been so ingrained for this area of maths.
If the first issue about making maths real life is that children remember the context and not the maths (another example: football transfer fees for reading large numbers) then the second issue is that for a novice, there is just too much to think about when there is all the context to process too. The teacher is talking about fractions and then pizza. Some children’s minds will naturally wander to the merits of different vendors or toppings or the last time they ate the doughy delight. By this time the child may well have missed some vital part of an explanation or not practised with enough concentration to have understood how to actually add fractions and the lesson is lost. Working memory is finite and for novices, who may be less able to filter out necessary and unnecessary information than skilled peers, lessons like this could well set children up to fail.
At this stage it is important to clarify my position. Real life problems are a way of taking the abstract world of maths and making it more concrete, but there are more efficient ways of doing that in the early stages of a unit of work that keep cognitive load low and ensure that children are thinking about the right thing. It’s down to us to structure a sequence of lessons to increase the likelihood of children being successful. Being able to solve contextual problems, or real life maths, is certainly a desirable outcome of maths education, but children don’t necessarily get better at solving these problems simply by being presented with them. Put another way, novices don’t become experts by behaving like an expert.
There are a number of prerequisites that novices must acquire in order to solve contextual problems and in the case of adding fractions, these are:
- An understanding that a whole is made up of parts.
- An understanding that a whole can be made up of equal parts.
- An understanding of how fractions are written, i.e., what the numerator and denominator refer to.
- Combining two fractions and how the numerators / denominators are affected:
- 2 bananas + 3 bananas = 5 bananas
- 2 tenths + 3 tenths = 5 tenths
- What happens when we end up with more parts than there are in the whole.
- What happens when we add fractions with a different number of equal parts.
- The varied language that could be used to present a problem about adding fractions, including where the inverse is presented (total, difference, sum etc).
That’s a lot to master for children to be successful at solving contextual problems. If there are weaknesses in those prerequisites and there are more things to think about, such as pizza, where the scissors are to cut my paper plate etc, then children will unlikely succeed.
Instead, the maths should be made concrete with appropriate manipulates and pictorial representations, initially with very simple language, so that children can really focus on the mechanisms behind adding fractions together. When they are secure with the mechanisms, we can show children the underlying structures that are possible when adding fractions before teaching them specific mathematical and contextual vocabulary and how those words and phrases can reveal what is required in a problem.
Finally, with all those building blocks, we can introduce varied contextual problems to give children a chance to apply their learning.