One of the parts of the NCETM’s Calculation Guidance for Primary Schools is the ‘Before, Then, Now’ structure for contextualising maths problems for additive reasoning. This is a very useful structure as by using it, children could develop deep understanding of mathematical problems, fluency of number and also language patterns and comprehension.
The first stage is to model telling the story. We cannot take for granted that children, particularly vulnerable children in Key Stage 1, will know or can read the words ‘before’, ‘then’ and ‘now’. Some work needs to be done to explain that this is the order in which events happened. Using a toy bus, or failing that, an appropriate picture of a bus, we would talk through each part of the structure, moving the bus from left to right and modelling the story with small figures:
The child could then retell the story themselves, manipulating the people and the bus to show what is happening. For the first few attempts, the child should get used to the structure but before long we should insist on them using full, accurate sentences, including the correct tense, when they are telling the story.
I have chosen a ten frame to represent the windows on the bus, which enables plenty of opportunity to talk about each stage of the problem in greater depth and to practise manipulating numbers. For example, in the ‘Before’ stage, there were four people on the bus: if the child could manage it, it would be interesting to talk about the number of seats on the bus altogether and the number of empty seats. By doing so, they are practising thinking about number facts to ten and building their fluency with recall of those facts. The task could easily be adapted to use a five frame or a twenty frame.
The next stage could be to tell children a story and while they are listening, they model what is happening with the people and the bus. After each stage, or once we have modelled the whole story, they could retell it themselves. Of course, the adult would only tell the ‘Before’ and the ‘Then’ parts of the story as the child should be expected to finish the story having solved the problem.
When the child is more fluent with the language and they understand the structure of the problems, we can show them how it looks abstractly. For the ‘Before’ part, the child would only record a number – how many on the bus. For the ‘Then’ part, we would need to show the child how to record not only the number of people that got on or off the bus but the appropriate sign too – if three people got on they would write +3 and if two people got off they would write -2. Finally, for the ‘Now’ part, they would need not only the number of people on the bus but the ‘is equal to’ sign before the number. Cue lots of practise telling and listening to stories whilst modelling it and writing the calculation.
A more subtle level of abstraction might be to repeat the same problems but rather than the child modelling them using the bus and people, they could use another manipulative such as multi-link cubes or Numicon. They could also draw a picture of each stage – multiple representations of the same problem provide the opportunity for deeper conceptual understanding.
The scaffolding that the structure and the multiple representations provide allows for some deeper thinking too. In the problems described so far, the unknown has always been the ‘Now’ stage or the whole (as opposed to one of the parts). It is fairly straight forward to make the ‘Then’ stage unknown with a story like this:
This could be modelled by the teacher, who asks the child to look away at the ‘Then’ stage. Starting with ten people on the bus and using a ten frame is a deliberate scaffold – deducing how many people got off the bus is a matter of looking at how many ‘empty seats’ are represented by the empty boxes on the ten frame in the ‘Now’ stage. A progression is to not use a full bus in the ‘Before’ stage – it is another level of difficulty to keep that number in mind and calculate how many got on or off the bus.
Another progression is to make the ‘Before’ stage unknown. The child will need a different strategy to those already explained in order to solve this kind of problem. Then story would have to be started with: ‘Before, there were some people on the bus.’ Of course, the adult would not show the child this with the bus and toy people, but they would show the completed ‘Then’ stage: ‘Then, four people got on the bus.’ Finally, the adult would model moving the bus to the ‘Now’ stage and completing the story: ‘Now, there are eleven people on the bus.’ The child would have to keep in mind that four people had got on and now there are eleven, before working backwards. They would have to be shown that if four had got on, then working out how the story started would mean four people getting off the bus. They could be shown to run the story in reverse, ending up with seven people on the bus in the ‘Before’ stage.
This task has the potential to take children from a poor understanding of number facts, calculating and knowledge of problem structures to a much deeper understanding. The familiar context can be used as a scaffold to build fluency and think hard about complex problems with varied unknowns.