Experts, say Hattie and Yates, see and represent problems on a deeper level, whereas novices focus on superficial aspects. With this in mind, take a closer look at one of the most prevalent strategies for solving problems:
In some variants of the acronym, the U even stands for ‘understand the problem’! If children practise solving problems in this way, they can only get better at analysing the superficial structures of the problems. Some of the advice leads children to develop near useless strategies when problems get trickier. At its worst, I’ve seen (and probably set up myself) lessons where children are told that they are doing subtraction word problems. Every problem has the same sort of language pattern and children could feasibly get by simply by picking the numbers out and subtracting one from the other.
Take the ‘underline’ and ‘choose which calculation’ advice. Underlining key words may well be useful by there is often ambiguity in the wording used. ‘How many more than X is Y?’ is different to ‘What is X more than Y?’ Although the wording is similar, they require different calculations.
If children are to understand the deeper structures, then they need to know the deeper structures. And of course they’ll need to practise analysing problems to classify those structures.
In the example of multiplicative reasoning we can start by suggesting that there’ll be three basic structures.
By modelling the thinking behind this and relating the wording used in the problem to the bar models, children can be shown the three deep structures. Certainly, they’ll need to practise over time analysing problems like this to become skilled at it. In the first instance, a little guided practice is on order: sort these problems by underlying structure:
Note that a deliberate difficulty built into this practice is that the problems are all similarly worded. There’s a decision to be made about the work that children will do to practise further. Do they solely work on sorting by deep structure or do they solve the problems too? In the example below, some children worked on just the first column. Novices have comparatively weaker short term memories than experts so may only be able to deal with the sorting. It makes some sense to provide a scaffold to help them remember important information: the deeper structure of the problem.
Here is a similar approach when showing children the structures of problems involving ratio:
Once children are aware of the possible structures, the teacher can show them how to represent the problem. Bar models are great here. Plus, worked or partially worked examples are powerful in showing children how to grapple with a problem:
Children can then have a go on their own. Here are some example questions for them to sketch out with bars and solve:
And here is an example of what children who already understand the basics would be up to – much trickier problems with more layers that might not entirely fit the basic structures described to most children:
This work on deeper structures could start from an early age. Too often, problem solving like this is bolted on to work on calculation with the assumption that if children know which operation a word or phrase means, they can solve problems. This however, is only superficial analysis. Children need knowledge of the structures of problems, just like we teach them the structures of stories. Perhaps we should approach problem solving like this more along the lines of teaching reading comprehension…