I think therefore I can? Modelling precise mathematical reasoning

    Published: 14 February 2019

    Picture the scene: you are a year 4 teacher exploring multiplication with your class. You’ve moved onto using formal methods for multiplying a 1-digit number by a 3-digit number. The children have used place value counters to explore why in 342 x 4, 4 tens multiplied by 4 is equal to 16 tens or 160. They have used scaffolding frames to help them to record their calculations and now they are rehearsing writing the formal written method of short multiplication in their books.

    Before you got to this point, you have of course, not only modelled using concrete resources but have used your flipchart to record the formal method. You’ve used different colours to show hundreds, tens and ones and any regrouping that you have had to do. On another model, willing volunteers have completed the calculation after discussing the procedure in pairs. Your models are pinned proudly to your working wall and the children are able to use these models, created together, to help them record their own work. It is all going swimmingly!

    Why is it then, that when you give the children the following question to answer that you are disappointed with the quality of their written response?

    Always, sometimes, never?

    When you multiply a 3-digit by a 1-digit number, if the digit in the hundreds place is more than 4, the product will be greater than 1000.

    ESSENTIALmaths Learning Sequence 24, destination question 8

    You have supplied all of the tools to be able to explore and respond successfully. Why then are the responses given unclear and use little mathematical vocabulary? Why have some children not even attempted to answer it? 

    What do we do to produce quality writing in English or in history or science? We provide the children with sentence openers; we spend time collecting relevant vocabulary; we provide writing frames if appropriate; we consider our audience and what style of writing would be best suited to this audience; we plan shared writing. We model. We model just as we would model a mathematical concept.

    And yet - are we modelling good quality responses to the sort of mathematical question written above? If we are not, are we then surprised that the children disappoint us? We are confident that they understand the concept – we wouldn’t give this question to a child who couldn’t yet multiply a 3-digit number by a 1-digit number - but yet we are disappointed by the lack of articulation in their answer.

    Are we actually providing the children with rich questions such as the one above, to allow them to truly reason and explain their thinking or do we fall into the trap of providing ‘prove it’ extension tasks? I have the great privilege to work with a number of teachers and look through many books and sometimes I come across work where the children are given a set of calculations to practice and then asked to prove or justify their answer. Surely the methods they have used provided enough proof or justification? That is not to say that prove it questions cannot still provide richness. Asking the children to respond with proof to the statement: ‘If I divide tenths by ten, I will get hundredths,’ requires pupils to make connections about the number system and explain what happens when dividing by 10. Just as we would make careful choices about the texts that we study in English, if we want to promote the quality of reasoning responses, then we need to give children the opportunity to grapple with rich questions.

    Consider this response to the initial question asked:

    This would be sometimes true.

    When you multiply 400 by 1 or by 2 then the product would be less than 1000 because 400 x 1 = 400 and 400 x 2 = 800. When you multiply 499 by 2 then answer would still be less than 1000 because 499 x 2 is 998.  

    But, when you multiply 400 by 3, then the answer will be greater than 1000 because 400 x 3 = 1200. I know this because 4 x 3 = 12, 40 x 3 = 120 and 400 x 3 = 1200. 400 x 5 would be 2000 and 400 x 6 would be 2400. I have noticed that any digit larger than 3 will give a product higher than 1000.

    Would we actual expect our 8 and 9-year-old pupils to respond like this?

    I would suggest that the answer would be no – unless of course we have modelled high quality responses and, just like with written pieces in English, given them the tools to be able to write in this way and the expectations that they can write in this way. After all, if children in reception can talk about graphemes and split diagraphs after being exposed to this language during phonics sessions, then why shouldn’t we expect our year 4 class to use the term ‘product’ appropriately?

    When we model mathematical concepts, we ensure that the vocabulary we use is precise and clear and we want the children to be mirroring this vocabulary when they are discussing their work. We want them to be using words such as multiply and product. We expect them to be able to use greater than and less than to discuss numbers.

    When exploring language in English, we display the vocabulary that we want the children to use on the working wall or a vocabulary board – but do we do the same in maths? Do we display sentence stems such as ‘I know this because…’ or ‘When you multiply … by… then’ or ‘I have noticed that’ or simply just the conjunction ‘because’. If the required vocabulary and sentence stems were displayed would our children use them in their writing more freely as they do in other written work?

    So, what could we do to help raise the expectations of written answers in our maths books? One strategy that we could use is to provide writing frames tailored to mathematics. This will enable children who have not previously been exposed to much mathematical language when writing a scaffold into the question. For example:

    This is …………. true.

    The product of 4... x …is less than 1000.

    The product of 4.. x … is greater than 1000.

    I know this because…

    I have noticed that…

    Frames such as this would not only help to structure the children’s responses but enable children who are less confident with their writing access to the same question as their peers. I would also suggest that if writing such as this was being encouraged in maths lessons, the quality of factual writing in other subjects would also benefit. 

    Just like in English, a guided writing group could be used to help scaffold the children’s response. This might be particularly useful to help your greater depth children with the clarity and precision of the language and examples that you wish them to give. 

    Most importantly, it is vital that when planning for the use of questions which expect a written response, that as teachers we spend some time writing our own response first. When planning modelled writing, I would always plan my model in advance to ensure that I included the sort of vocabulary that I wanted my pupils to use.

    Of course, there will be times when a more succinct answer will not only suffice, but will be the most efficient way of answering. Take question 20 from the 2017 reasoning paper 2:

    KS2 SATs question 2017

    Short and sweet, with no words at all, yet the mathematical symbols used explain exactly why Adam is correct.

    If we want our year 6 children to respond to explanations in this way, then just as we would teach them to make precise language choices in their narrative and factual writing, we should emulate this in our mathematics modelling.

    In this example, rather than encouraging a worded explanation, any unnecessary language has been removed yet it is evident that the reasoning has still been done successfully.

     

    If we know our expectations, and why shouldn’t we set them high, then we are more likely to provide the children with the relevant tools to write quality answers that are sure not to disappoint.


    References

    Department for Education (2017) Keystage 2 Tests: 2017 Test Materials. Available at: https://www.gov.uk/government/publications/key-stage-2-tests-2017-mathe… (Accessed: 19 Jan 2019).

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