Thanks to Nick Hart, Head Teacher at Chalfont Valley E-ACT Primary Academy, for this blog post on teaching ratio!
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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:
In this example, a novice may say that the problem is about apples but an expert will notice that this is a ratio problem where the whole is known and one of the proportions is unknown.
In order to get children to think like mathematical experts when it comes to ratio, they must first understand the idea of unequal sharing. Below are two models for this:
The first represents the whole as the sum of the parts in a much clearer way than the second – this model is the obvious starting point. I would ensure that children see the second model too, which more clearly demonstrates the difference between the two proportions, especially when it comes to more complicated problem solving later on. Once this knowledge is secure, it seems sensible to teach children the small number of possible underlying structures from the outset whilst stripping back the confusing and distracting contextual information. When it comes to ratio, there are 5 underlying structures, each a combination of whether the whole, the ratio and the proportions are known or unknown:
1. Whole known, ratio known, proportions unknown
2. Whole unknown, ratio known, singular proportion known
3. Whole unknown, ratio known, multi-part proportion known
4. Whole unknown, ratio known, proportions unknown, difference between proportions known.
5. Whole known, ratio unknown, proportions known
Teacher’s clarity of thinking about these structures is the critical subject knowledge required for teaching ratio. Knowing the structures enables teachers to plan effective explanations, as well as vary problems for independent practice. I suggest that the first stage of teaching a unit on ratio is to provide plenty of number puzzles, set out on bar models as above, without any mathematical language. This will ensure that children are thinking about the structure of ratio problems with the cognitive load of distracting language or contexts. Making maths 'real life' is a very popular approach and although I'm not contesting that, I do think that context needs to be added after children have a good knowledge of structures to help them to see what is unknown in a problem.
With sufficient time dedicated to deepening children's understanding of solving number puzzles using the underlying patterns of ratio, teachers can then start to add layers of language and contextual information to problems so that they can begin to see how mathematical language reveals what is unknown in a problem. This is not as simple as giving children worded problems though.
Before that, problems need to be presented with minimal language. Here are some examples for each of the underlying structures:
I've used the word 'share' or variations of it in each question deliberately to make the point that individual words do not equate to certain operations. Posters like the one below are too simplistic:
Share doesn't always mean divide because words only have meaning in context.
If children are at a stage where they can solve basically worded problems, they are ready for the next stage of adding more contextual information to problems for them to solve. We're still not yet throwing random word problems at them though. Instead, teachers should choose a problem (Rising Stars Assessment Bank is a great place to look for these) and model how the problem works in its entirety without having any unknowns:
Having children able to solve the problem on the left is the end goal, not the starting point. As teachers, we must first model and explain the whole story of the problem, linking back to the structures that they learned previously. The teacher can then repeat the problem with different numbers and deliberately omit some information to change what is unknown, modelling how to represent the problem pictorially.
Children need to be familiar with the context to reduce cognitive load. Familiarity with the context as well as sound knowledge of the underlying structures is a solid scaffold to get all children thinking mathematically. It is only when children know a problem context inside out, coping with varied unknowns that I would introduce a problem in a different context, then another. This way, children get the variation needed to encourage transfer of learning whilst having the appropriate scaffold to be successful.
Thanks to Head Teacher, Nick Hart, for this blog post on Teaching Ratio. You could receive blogs like this directly to your inbox! Just log in to your My Rising Stars Account and select MATHS as your contact preference. Not got a My Rising Stars account? Register for free.