I often have teachers telling me that their pupils are limited in their mathematical success because they just can’t reason. In trying to help unpick the problem and work out how to respond I often ask two questions:
When are these pupils reasoning?
How have you taught them to reason?
This blog reflects on some of the things that can be a barrier to pupils developing reasoning skills and what teachers might consider in order to support all pupils in this endeavour.
One problem that I come across is that sometimes reasoning is considered as an extension activity, a challenge for the rapid graspers or a space only occupied by those “working at greater depth”. Certainly developing expectations of reasoning can be an excellent opportunity for pupils to think deeply about the maths they know, how to make connections and how to express this in different ways. However, I think it is worth reminding ourselves that there are 3 National Curriculum aims in mathematics: to become fluent in the fundamentals of maths; to reason mathematically and to problem solve (DfE, 2014)[i]. These 3 aims are for all children and will together, enable secure mathematical learning. Are there some children who are not getting access to frequent reasoning opportunities? If they do not have regular opportunities, then how can we expect them to be able to reason effectively?
Many schools I work with have been using ESSENTIALmaths resources to help embed reasoning within teaching sequences. This means that opportunities for regular reasoning are available, but ensuring that all pupils can access and achieve success in this reasoning is often something that teachers find difficult to manage within a whole class context.
Sometimes I think we can be guilty of expecting pupils to learn how to reason by osmosis. Either we give them some reasoning problems and they are able to answer these, so therefore their reasoning is sorted or we give them some reasoning problems and they can’t answer them so we stop giving them any as they are obviously just too tricky.
When do we make our thinking explicit? When do we tease out the skills involved in reasoning? When do we find a moment to reflect on useful strategies to support finding a starting point to a problem and when do we give time to evaluating the effectiveness of different approaches to the same problem?
Then sometimes we go the other way and try to do all of these things every time that we consider a mathematical problem. It is a bit like when we thought we had to write up the whole scientific enquiry each time we did one. Realising that if we focused on one specific element of the enquiry we could provide better focus, scaffolding and clarity of expectation, made the whole process more effective.
I know some teachers find it particularly hard to support some groups of pupils in their reasoning. When pupils are struggling with the mathematical concepts being presented, there can be a temptation to take pupils step-by-step through problems to lead them to the answer. Now, I would not deny the need to articulate and make explicit the reasoning that is being undertaken. But, if teachers only lead pupils to respond to this one problem, then the extent to which this can be applied would have to be questioned. It might be more effective to consider how pupils are being scaffolded to reasoning success throughout the learning sequence and what they are being asked to reason with. One example that often occurs to me is from ESSENTIALmaths sequence 3LS8. In this sequence, pupils’ learning is built from their understanding of place value and mental addition strategies towards making effective use of formal written addition. I often draw attention to the practice all pupils are expected to have within the early steps and the opportunities to model reasoning.
Formal written method with no regrouping (exchange)
What I want teachers to note is that here is an opportunity for all pupils to reason with what they have been taught. Having all had access to modelling with concrete and pictorial representations, pupils independently practice using the scaffold of the speaking frame and their choice of representation. The example practice questions have then been carefully chosen to ensure that pupils play close attention to what they have just been taught and importantly, to ensure that all pupils have the opportunity to reason with a missing number problem. This is the perfect opportunity to allow the teacher to model how pupils can approach this type of problem and provide practice, making links to what they already know before the missing number questions become more complex and involve the additional difficulty of regrouping. It is interesting when unpicking diagnostic assessments, that sometimes pupils are more than capable of the maths involved in a missing number calculation, but just don’t know where to start and how to reason through it. If our focus is on teaching reasoning to problem solve, then pupils need to be confident with the content.
So having identified the need for all pupils to have opportunities to reason and for teachers to model the skills needed carefully and systematically, I often return to how this can best be achieved within a class context.
I have concluded that if we want to make opportunities to discuss and develop reasoning, the answer is often a distraction. I may want to draw attention to the communication of what has been understood, the ability to find a starting point to something completely unfamiliar or the evaluation of an approach to working. In any of these cases, I do not want attention diverted to what the answer is and I will often do one of two things: provide the answer or answers or remove the numbers so that there is no answer to find…yet!
By doing this, I can have the conversation that I want to have, rather than being distracted by the “quick calculators” who want to give me their endpoint. I have found that a further benefit of removing the numbers is that the problem is open. Not only does this lend itself to low entry points so that everyone has something to offer, but it also allows me to differentiate challenge within the same reasoning context using “what ifs” and adding parameters. So all pupils can have access to a structured approach to reasoning whilst using the maths content that they can access. This has become one element of, what I consider, a whole class structured approach to reasoning that aims to leave pupils able to independently reason so that they can solve problems.
In developing a whole class approach, I will always consider first what I want my pupils to gain. Whether this be development of a problem solving skill or the ability to successfully tackle a reasoning question. For example, consider how reasoning can be developed starting with the visual below and some sentence stems.
I want pupils to look closely at the image. I want them to notice the scale and the labels and I want them to start to make sense of these. All children should have something to say about the image, but by talking and sharing what is seen, pupils slow down and are made to articulate - the internal monologue of interpretation is shared and pupils can hear each other making sense rather than racing to do something with what is in front of them.
At this point, I often provide some vocabulary to draw attention to where I know this is leading.
This enables me to check that pupils have the words they might need to express what is in front of them. But as the vocabulary is not exhaustive, I also ask pupils to think about other words that they might feel are appropriate to discuss. The challenge is to try and use each of the words or phrases in the context of the image and this generates further discussion and opportunities for clarification. I often find at this point that pupils are guessing what the question might be and usually it is far more elaborate than the one that is later revealed. That is no problem though as the act of constructing questions is a skill in itself and I always have takers for these questions – those who are intrigued by how they could be answered.
Sometimes this “getting into a problem” is the main focus of my whole class teaching and there is less focus on what happens next, but sometimes once the question is revealed, I want pupils talking about what they have now found to be important about the image. How can they annotate it? What is a distraction and can be removed? Can they see the structure of the problem now, or is it helpful to translate what is known into a different representation.
On other occasions, I might draw out the internal monologue of how to decide where to start and choose what to do next or I might focus on considering how my findings can be communicated precisely. The point is that in my structured approach to reasoning, I aim to scaffold all to be able to access the problem and I focus on articulating what is actually being done to solve it. Ultimately, I want pupils to be able to independently look at problems and draw on their problem-solving skills, but they are far more likely to be able to do this if they are clear what these are and they regularly use them.
In the upcoming ‘Reasoning to Problem Solve’ courses, we will be further exploring some of these ideas using practical examples and we will share the structured approach to reasoning.
In particular, we explore the key focuses for success. The first, as alluded to earlier, involves being explicit about the exact skill and ensuring this is the main driver of the learning (and teaching). The second, and often overlooked, are teaching strategies we can use to help pupils ‘get into a problem’. Unless pupils are ‘in deep’ they will not be able to develop their reasoning and problem solving skills. Aligned to this is the skill of the teacher to steer pupils’ attention. We often describe this as the art of helping pupils block out the white noise (distractions) to sharpen their focus. To ‘notice’.
Developing pupils’ mathematical representations are crucial. These can take many forms but are essentially tools pupils can use to mathematically model. They help pupils organise their thinking, analyse, sift information, test ideas and keep track as they proceed. And as with all skills, this mathematical notation can be refined. Once we are clear about purpose and focus, a whole myriad of similar problems are easier to design.
So if developing reasoning and problem solving are development focuses be sure to join us.