In the last blog The Tale of the Tail that Wags the Dog, I looked at how looking at the test developers’ framework can give us some insights into how well our curriculum is supporting learners to develop deeper and transferable maths skills. Some of the aspects that have really interested me are the expectations of language and the requirement that pupils transfer well between representations and can respond to questions in varied ways.
Is the language content of the tests difficult?
Well I believe yes and no. Often, I hear tell that the language is too complex. I wondered about that. Looking at the SATs papers I found the questions with the most accompanying language, such as the examples here.
Notice the layout, it’s basically lists of not very complex language. In fact a quick look at the tests didn’t result in any complex sentences at all.
The words are not tricky either. It’s actually quite accessible.
I put the reasoning paper text through a lexile calculator, which basically gives you an idea about how complex and dense the language is. It came out low. In fact the majority of pupils leaving KS1 will have accessed text of this difficulty.
The contexts in these examples are not unknown either though both are examples of mixed operation multi-step problems making them computationally more complex (see previous blog).
From the work we do in schools, our advisory team frequently meet pupils who are able to deal with the language and contexts of examples such as those above. What makes them more able to deal with these examples? The first indicator is that, where mathematical language is built up over time, with pupils expected to speak in full mathematical sentences instead of barking numbers, then they are less likely to baulk at a bunch of words and are more confident in translating them into mathematical operations. Furthermore, they are more likely to understand whether the number they produce is actually feasible in the context of the words. Secondly, if the diet for practice they are given is not just numbers, an operator and the equals sign then pupils are more likely to link them to words. Thirdly, those trigger words on that working wall can be misleading. For example the word decreases is often shown as a trigger word for subtraction but consider how misleading that would be in the following problem:
The temperature decreased by 6°c. It is now 14°c. What temperature was it originally?
And in the stickers question above, I see the word more in the addition section of some trigger word displays, whereas to find out how much more Jack pays than Ally we are finding a difference, two amounts are being compared. Bar modelling is a key model here in our experience, though truth be told, by the time pupils get to Year 6 without it, it can be hard to convince them they should use it. But like the build-up of language, if introduced over time it can be a really useful part of the toolkit. Below is an example of a handout from our first Year 3 routine problem solving sequence, building the understanding of operation. Included are solution sentences to encourage checking the answer makes sense in the context of the sentence.
To sum up, if pupils are not experienced at translating language into mathematics, the operational complexity is hidden from them.
Do the differing response expectations make it difficult?
Forget the clerical errors, we all make those in haste. But consider is the bulk of the practice you provide pupils limited in response to following a set procedure and placing an answer after an equals sign? Then yes the number of different ways pupils are asked to respond to the questions may present difficulty. Below I have listed all of the responses pupils have to deal with just for Reasoning Paper 3 2017.
- Missing number box simple
- Missing number boxes complex
- Simple number response
- Complex number response
- Rearranging digits into a calculation
- Circle on a time table
- Circle single example that has two required variables present
- Convert before circling examples that match given criteria
- Draw translation
- Demonstrate method
- Provide all possibilities
- Provide algebraic rule
Now some of this can and should be tackled by test prep, drawing pupils’ attention to these response types and discussing expectations for response. But where again, a build up over time of a wider range of expected responses is likely to lead to far more confidence in tackling the range in a test situation. Here we have a Y4 example of the range of question responses we might require of pupils once they can calculate with decimals to 2dp.
Are my pupils translating knowledge well?
The following question had us raising a collective eyebrow in the office. We could see that this would pose considerable challenge even for the best prepped pupil. Now whilst we are fully aware that the illustrations in the paper are no longer mere decoration for the most part, they can still provide a distraction for pupils. I return to thinking about the sleepy koala question – it didn’t matter that it was a cat and a koala, it could just as easily have been some other sleepy animal. The point of the question was to relate 18 to being 75% of 24. Just as in this question it really doesn’t matter that it’s a somersault other than it being the context for one and a half turns. The picture is a beauty but the maths is already stated in the words above. The picture confirms what a somersault is.
A graphic we regularly refer to when training is from the work of Jerome Bruner who states ‘It is not the type of representation that is indicative of the child’s understanding, but the ability to ‘translate’ between these models.’ (Bruner, 1966). This is part of what is being assessed, the ability of pupils to translate knowledge beyond the lesson and the representations already encountered.
So again, if what is present in the books is pupils unthinkingly repeating the teacher’s method 25 times, with no opportunity to translate and therefore deepen and secure understanding, then we cannot really expect them to translate it just for the purposes of a test. Mimicry goes so far, but innovating on knowledge must also be a part of the daily diet.
Consider how your provision is building up language (not triggers), representation and a wide range of responses over the year. For school leaders, what does that look like from Y1-6? These papers are called reasoning papers after all and this is not just about building up talk for example (although that is a useful tool) which I think is a current misapprehension I see in schools. There are many non-explicit opportunities where pupils are required to transfer their knowledge and understanding in the papers – this is also calling pupil reasoning though there may be no words involved. There are then, many lessons we can learn about our curriculum by looking at the assessments and the diet pupils need to be successful, not just at the ends of key stages but in mathematics generally.
Bruner, J. S. (1966). Toward a theory of instruction, Cambridge, Mass.: Belkapp Press.
STA, Reasoning Paper 3 (2017)