# Tag Archives: worked examples

## Teaching Ratio

Novices and experts see problems differently.  Whereas a novice sees superficial features, an expert notices deeper underlying patterns, discarding the often irrelevant and distracting contextual information.  Here’s an example:

Filed under Maths

## Before, then, now – modelling additive reasoning

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:

Before, there were four people on the bus. Then, three people got on the bus. Now there are seven people on the bus.

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:

Before, there were ten people were on the bus.

Then, some people got off the bus.

Now, seven people are on the bus.

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.

Filed under CPD, Curriculum, Maths

## Modelling percentage change with Numicon

Percentage changes can be effectively modelled using Numicon.  Here’s an example: reducing a price by £30:

Another:

Even for more complex ideas, like working out the percentage change, using Numicon can clearly model the process:

Further opportunities for children to demonstrate depth of understanding could include asking them to represent the same question using multi-link cubes, rather than using ‘more difficult numbers’. When they have a reliable understanding of what happens during percentage change, then they can work on the abstract formulae.

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Filed under Maths

## Throwing out that old RUCSAC

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…