We all have stories like this:
Mick was giving us a quote for laying a patio. Having measured the rectilinear space he almost immediately told us that he would need so many 4x4s, 4x2s, and 2x2s and that would cost us so much, plus labour at so much and VAT at so much. I can’t remember if this was before or after he told us that he was no good at mathematics at school.
A group of mathematicians was visiting a church during a break in a conference and found a woman carefully reorganising the hassocks, which all bore geometric designs. They watched for a while and when they were convinced that there was some system behind the process they asked her about it. She explained the transformations and tessellations involved and asked them why they were interested. On hearing they were mathematicians, she announced that she was no good at mathematics, but her brother was a mathematician.
Mathematical reasoning is expected of everybody for GCSE, and yet I often hear teachers saying that some of their students ‘cannot do’ mathematical reasoning or problem-solving. In particular some ‘cannot do’ multistep reasoning. Of course most people can do multistep reasoning otherwise they would not be able to do shopping, plan and cook meals, organise their time, get from A to B, or follow sports — still less analyse football games and suggest alternative tactics. So it would be more correct to say that some students do not do this in mathematics than that they cannot.
In 1988 a few comprehensive schools pioneered an approach to teaching and assessing mathematics at GCSE using problem-solving approaches to teaching and assessment. We — the teachers involved — found that if problem-solving was expected and encouraged then that is what students did. However, that approach did not necessarily ensure that all students were developing mathematical reasoning, since problems could often be solved by ad hoc methods. That experience convinces me that all students can work with multi-step problem solving situations where ad hoc methods of solution are available. However, is there any evidence that all students are capable of mathematical reasoning where this means something more specific than solving problems?
Nicola Clarke sat through many Year 11 lessons designed for students who had expected and ended up attaining the lowest grades. She observed and recorded as much as she could of what went on when students were working together on standard curriculum topics and she found plenty of evidence of logical reasoning among students. Some of them seemed as quick as anyone to construct ‘if… then… because’ arguments based on mathematical facts. What made them slower in their work was when they lacked a fact or technique and had to start calculations from a very simple and error-prone base. This meant they could not put their reasoning into action, so could easily get bored, frustrated, or distracted, and the reasoning they were doing was not obvious to teachers because they could not put it into action.
I once spent a term in a KS4 maths class that was taught by a Special Needs Coordinator. There was hardly any conceptual input for the whole class, most of the teaching being a brief procedural introduction followed by a series of related worksheets. I therefore spent a lot of time watching how the students used the worksheets and often saw students working in ways that were creative.
- Students were supposed to be filling in a grid using multiplication and division facts involving 7 in sequential order. To speed up the work they were not doing multiplication or division but were working vertically using patterns in number sequences, such as adding 7 each time, writing successive integers for the third column, and hence avoiding the purpose of the worksheet. However, when asked to write 23 x 7, they were able to do it and some wrote out further non sequential calculations.
- Students were supposed to be plotting points using a list of pairs of coordinates and then join them up to complete a picture of an ostrich. I watched one student peruse the list and pick out adjacent coordinates which would be joined by specific sectors, for example, finding all the pairs of points that would be joined by the vector:
- Students had been calculating the outputs of flow diagrams for which the input and two or three operations were given. One student suggested finding the input if the output was given and this unleashed further creativity, such as giving the input and output and having to find appropriate operations.
- Students had to find missing coordinates for the vertices of standard geometrical shapes for which some vertices had not been defined. Several students extended this task to thinking about families of shapes and generalising where the vertices would have to be – can you recognise what shape it will be from studying the coordinates?
I observed these kinds of mathematical reasoning regularly and frequently, and often in situations where the students took the initiative to be creative in their work. By ‘mathematical reasoning’ I mean that I saw students:
- identify and use patterns
- look back on their processes and notice generalities
- generate examples that did more than what the teacher had offered, and even some counterexamples
- change representations when it was helpful, and manipulate the representations
- perform actions that showed they had observed some underlying mathematical structure
- use ‘if…then…because’ arguments
Now I admit that these were not in very challenging mathematical situations — completing the drawing of an ostrich on a dotted grid relates more to the KS2 curriculum than KS4 — but given that these aspects of mathematical reasoning were happening in the context of a diet of procedural worksheets, it made me think hard about expectations.
It seems to me that the first ingredient to enabling mathematical reasoning is to believe that anyone can reason. The second ingredient is to pose questions that call on specific kinds of reasoning that students show in other non-mathematical contexts. The third ingredient is to leave space for thinking and creativity, all within an environment that makes reasoning possible. So looking back at the kinds of reasoning I listed above, the environment should allow:
- useful patterns to be observed
- key generalities to be exemplified
- students to make their own examples
- more than one representation to be available
- tasks that are based in mathematical structures
- talk about ‘if…then…because’
I know nearly everyone can reason, the challenge is to ensure it is enabled and encouraged for everyone in mathematics lessons.
All the best,
Anne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications. Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent.