# Adaptive reason: a vital ingredient in the recipe for primary maths

Iwanted to conduct some research with gifted and talented maths pupils in my primary school, but I struggled to define what exactly made a pupil gifted in maths. I was sure the answer had to be more than just good test results. I knew that there was a certain something that was there in great maths moments and not others. I just couldn’t pinpoint it.

So I turned to the model of mathematical proficiency, developed by Kilpatrick et al in the US in 2001. It was the result of a long research and consultation process and produced a rather concise diagram of what being ‘good’ at maths might mean.

The five intertwining strands of mathematical proficiency are identified as conceptual understanding, procedural fluency, strategic competence, adaptive reasoning and productive disposition. They are modelled as being interwoven and interdependent: no one strand is more important or greater than another. How true to this model are we in the classroom? And in looking at it in detail, might I find my missing skill so crucial to those advanced mathematic moments?

Procedural fluency is the easiest to spot - this is our old-fashioned arithmetic practice. Can we take a set of numbers, manipulate them to apply a basic calculation and generate a correct answer? This is the daily bread-and-butter of any classroom at any age range.

Alongside this, pupils are taught conceptual understanding - the land of place value and number patterns - as well as strategic competence, which includes diligently highlighting the numbers in a word problem and memorising all the possible synonyms for addition and subtraction.

Even productive disposition - the inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy, according to Kilpatrick - is now being given some attention as Carol Dweck’s growth mindset becomes the focus of staff training days.

## Childrens’ capacity for reasoning

So where does that leave us with adaptive reasoning and is it the elusive thing I have been looking for in pupils?

Defined as “capacity for logical thought, reflection, explanation and justification” by Kilpatrick and his team, it is the skill of being able to prove that your answer will always be true. This skill appears in the maths curriculum only at the very top end of our schooling and so is currently reserved only for GCSE and A-level students.

Yet, it forms one strand for mathematical proficiency, suggesting that without this skill, the remaining four strands are weakened.

Explaining and justifying your answers shows a deeper level of understanding; a dexterity in the manipulation of the problem and demands a perseverance and commitment to the challenge faced.

A number of researchers have argued that young children are not able to provide any sort of mathematical reasoning, but I disagree. A flawless, detailed mathematical proof that takes into account every possible exception is definitely an advanced reasoning skill that requires experience and a wealth of knowledge usually only attained towards the end of a child’s school life. However, proof is at the top of the reasoning ladder and younger children should be encouraged to engage with more basic reasoning skills - such as justification and explanation - from the beginning of their mathematical careers.

At this early stage, a successful justification can only be tested by the classroom community. Can their peers spot any problems or mistakes? If not, then the reasoning is accepted as true.

To do this, reasoning has to be taught. And this is where the challenge starts. Teaching reasoning can only be accomplished through modelling, allowing pupils to engage with the vocabulary needed to form explanations and critique those of others.

This can be taught in the following way:

- The teacher must pose a problem and allow pupils to attempt to solve it in pairs or small groups using their existing knowledge.
- After a period of time (half a lesson or as the plenary to the first lesson of a two-lesson series), groups present their existing answers to the rest of the class.
- At this point, the task becomes collaborative - can any of the other pupils spot any problems (we call these “bugs” - cross-curricular link to computer science) in the explanation? If so, how can we improve them?
- Together, an improved approach is designed, with support from the teacher to model vocabulary, phrasing or any notation that may be useful.
- During the second part of the lesson, pupils re-attempt the same problem and apply the new knowledge to form a new justification for their answer, presenting their final solutions at the end.

The collaborative nature of this activity is vital to generate the vocabulary at a young age. Without it, primary pupils will struggle attempting to notate their thoughts in a way that will be understandable to others.

It also allows them to rehearse their explanations in a smaller, safer setting before being asked to share these with the class. By using a variety of open-ended tasks, even Reception children can explain their thinking and justify their reasoning in a way that is acceptable to their classmates.

## Spotting the skills

The next challenge is assessing this skill: how can we judge the development of reasoning in a child?

The short answer is that currently we can’t. The long answer is that there is some current research happening in this area, but it is not efficient enough to be effective in a classroom setting. Our summative written tests certainly aren’t assessing this skill in isolation at the moment.

However, we can observe elements of this skill in written and verbal explanations of why they are doing what they’re doing but it takes some practice to find what you’re looking for.

It can happen naturally. I have taught a number of students who have excellent reasoning skills but do not perform well in tests; it is the missing ingredient in my gifted mathematician cake. But it can be taught to those children who do not develop this skill naturally and allows us to move away from a model of an innate ability within a gifted child.

*Agata Wygnanska is a Year 5 teacher and maths and religious education coordinator at St Mary’s Junior School in Cambridge*

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