fluency, reasoning and problem solving

Maths in primary schools

Put the kettle on, and let’s talk about maths.

When I get home on a Friday night, members of my family (once they’ve eventually looked up from their respective screens of various dimensions and realised I’m back) will often ask, ‘Where have you been this week?’.

And I will normally reply, ‘I don’t remember.’

Not because the work in the schools or speaking at the conferences has been forgettable. It’s the geography that slips my mind. It’s ironic in a way, because geography is what I studied at university, alongside teaching. Ask me about Eastern European settlement patterns before the fall of the Berlin Wall, and I might vaguely remember something. And honestly, who would actually know, or indeed care, if I was correct? But ask me where I stayed last Wednesday, and it’s all a blur, often shaded in Premier Inn purple with hints of Holiday Inn green.

However, one linguistic feature does help me identify a location. And that feature is how school staff refer to drinks during the day. If I’m offered ‘a brew,’ I know I’m pretty near Focus’s offices in Oldham. You don’t hear that in Essex. Tea might be quintessentially British, but its nomenclature can be regionally diverse. (A school office manager once proudly made the point that the brew she’d made was proper Yorkshire tea but was slightly phased when I innocently enquired exactly where the plantations were located in relation to the surrounding moorland. The pedantic geographer in me couldn’t help it.)

And if you’re offered a ‘cuppa’ on arrival and Earl Grey is an option, it’s likely the majority of blue bars on the school’s context page of RAISEonline are on the left of the page.

The three strands of maths in the national curriculum: Fluency, reasoning and problem-solving

You may wonder what this has to do with maths in primary schools. It is, albeit very loosely, the fact that schools as organisations have their own lexicon of language. An aspect of this is related to the three strands of mathematics: fluency, reasoning, and problem-solving.


In the first year of the latest National Curriculum, I often saw a lot of arithmetic in pupils’ books. And I mean a lot of arithmetic in some schools. It sometimes took me back to my school days with pages and pages of column addition and subtraction. This mirrored the weighting in the programmes of study being very ‘fluency heavy’ in their demands and content. Both reasoning and problem-solving are mentioned, but not to such an extent or with such exactitude.

So, as schools got to grip with a new curriculum, this was reflected in the amount of calculation work in many settings, coupled with teachers realising that pupils needed to make up quite a lot of ground in many cases to meet the expectations of each year group. In schools, this work was referred to variously as calculation work, number, arithmetic, developing fluency, four rules, addition… Hence my comparison with the different names for drinks. But at least everyone knows what these mean, just like we know a cuppa is also a brew is also tea.

Reasoning and Problem Solving

However, with reasoning and problem-solving, this variation can be more (ahem) problematic. I have worked in schools where the two seem to be conjoined to form one long word, for example, with teachers saying, “Would you like to see my planning for reasoningandproblemsolving?” And the subsequent planning then shows no clear distinction between the two either.

In other settings, there is a clear collective understanding of the two as distinct strands of maths, but even then, teachers’ subject knowledge can be shaky. I’ve asked teachers and TAs to try an activity during relevant training sessions when I’ve given them a simple Venn diagram and asked them to write in one circle what problem-solving means and in the other circle what reasoning means and then explain any overlap in the intersection. It very quickly becomes obvious whether or not delegates have a clear understanding of the definition of the two strands. And while they do this, I can have a cup of tea/ a cuppa/ a brew, or even some good old Rosie Lee if I wish to reinforce stereotypes even further.

How effective is the teaching of reasoning in your setting?

Reasoning was identified as the weakest of the three strands in the first year of the latest curriculum. It is worth considering how effective the teaching of reasoning is in your setting now, especially in the light of the first set of statutory assessment outcomes in maths and any question-level analysis you may have carried out.

I was recently in a school where they link mathematical reasoning with the development of spoken language. Words and phrases are displayed on maths working walls to encourage mathematical discussion. The pupils’ progress is carefully planned, from explaining their maths into inductive reasoning (where the underlying mathematical argument may or may not be accurate yet is likely to have more coherence and completeness than the explaining stage) and finally into deductive reasoning (proving a watertight argument that is mathematically sound, often based on generalisations and underlying structure).


Problem-solving can be the first activity to introduce a unit of work, often with the teacher or TA modelling the problem-solving process in a similar way to how we have modelled writing for years. Pupils will then have opportunities to take a wide range of problems ‘apart’, learn the maths needed to solve the problem and then arrive at a solution or debate the merits of different solutions or different strategies to solve a problem.

This is a long way from the approach of the school where ‘reasoningandproblemsolving’ are interchangeable, only the preserve of ‘Problem Solving Friday’ after four days of column addition, and then based mainly on the traditional ‘If a train leaves a station at 5 o’clock, how many sweets does Susan have?' type of question.

So, where would your school sit between these two examples?

I believe it is vital our teachers and TAs have a clear understanding of what is meant by mathematical reasoning and what is meant by problem-solving. I’d suggest a staff meeting to sort it out, share best practice, and develop that vital consistency of approach and use of pedagogical language synonymous with effective schools.

And while you do this, everyone can have a drink. Mine’s white with no sugar please. Unless it is Friday evening and I am finally home, in which case it’s red.

Continue the Conversation

For more information on mathematics, fluency, reasoning and problem-solving, keep an eye on the Focus Education blog, or get in touch with the Focus Education office on 01457 821 818.


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