Fluency in maths

Fluency in language

I can recite Latin verbs: the present tense of love is amo, amas, amat, amamus, amatis, amant. I recited them as I swam up and down the pool forty years ago: Amabo, amabis, amabit (breathe) amabimus, amabitis, amabunt (breathe). But if I were suddenly faced with an ancient Roman and had to express my affection, it would take a bit of thinking. I lack fluency in speaking Latin.

When we are fluent in a language, we can respond and converse without having to think too hard. The language comes naturally, and we do not use up space in our brain thinking about what word to use. Fluency comes from using the language in multiple settings, from trying things out, and failing and trying again. In order to learn a language we need to overcome the fear of looking stupid, and just give it a go. We also hope that the native speakers around us are kind.

Fluency in mathematical procedures

The idea of fluency applies also to mathematics. The National Council of Teachers of Mathematics provides this definition of procedural fluency:

“Procedural fluency is the ability to apply procedures accurately, efficiently, and flexibly; to transfer procedures to different problems and contexts; to build or modify procedures from other procedures; and to recognize when one strategy or procedure is more appropriate to apply than another.”

You can read more about their position here: Position Statement

Fluency and automaticity with basic facts can ease later maths learning. When learners are grappling the division algorithm, it can derail the process if they have to stop and look up the seven times table, for instance. Conversely, when learning to add fractions, a student who is automatic in their multiplication facts is at an advantage when thinking of the Lowest Common Multiple of two denominators.

My recent survey of high school teachers listed fluency with multiplication as one of the key skills they would like students to have when they arrive in Year 9.

Fluency in mathematics is a hot topic in mathematics teaching. There is universal agreement that fluency in applying basic mathematical facts enables later mathematical learning. There is considerable difference in how fluency is achieved.

Speed kills

For generations children have learned their tables by rote and been required to answer questions involving basic facts and multiplication quickly, often with some jeopardy (the ruler across the knuckles or public humiliation). Speed was taken as evidence of fluency. Most adults will be able to recall instances of having to recall facts under pressure in a classroom. For many this evokes memories of considerable emotional trauma. As a competent and keen mathematician, I still remember the grip of fear that I would be too slow. I was never quick with my tables at school, though I was fluent enough. I was fortunate as I felt good enough at mathematics for it not to put me off the subject, but this is not usually the case. I have written about maths trauma, which is related to maths anxiety. One of the traumatising traditions listed was “using speed to measure understanding”.

Jo Boaler is waging a war against time-pressure tests of basic facts. Her paper Fluency without Fear summarises research and outlines her concerns around some ways of developing fluency. She encourages the development of number sense through games, activities and number talks.

The dilemma

It is important to give learners opportunity to develop fluency in procedures and automaticity in number facts.

But

It is damaging to drill learners in maths at speed, as the effect of time pressure causes blockages in the very part of the brain that is needed for processing.

Solution

So how should we develop fluency? We need to use activities that apply the number facts in meaningful ways. We need games that use the mechanism of mathematics. Learners need distributed practice at the right stage of conceptual understanding. Conceptual understanding and procedural fluency need to proceed in tandem.

At younger levels there are Number talks and Choral counting that can be used to explore and develop number sense.

One way is number explorations. For example twenty-four is a multiple of 1, 2, 3, 4, 6, 8, 12 and 24. Time spent in the company of twenty-four will embed in learners the idea that it has multiple factors. Physical, pictorial and abstract representations will aid memory and provide hooks. Then later when they are factorising a quadratic involving 24, it should trigger memories of its factors.

Great book

My favourite resource in learning about procedural fluency is a webinar by Jennifer Bay-Williams, which you can see here: Research-based strategies that build procedural fluency.  It changed the way I think about teaching maths facts.

Jennifer Bay-Williams has since co-written the book: Math Fact Fluency.

One of the five fundamental principles espoused in the book is that “students need substantial and enjoyable practice.”

Substantial and enjoyable practice

The best form of substantial and enjoyable practice is games, games and more games. In this context I am not talking about computer or on-line games, but games between real live people. Games encourage flexible thinking and discussion. They are social and develop other skills. Mathematics underpins all games. Maths games can be played using minimal equipment, such as cards, dice and counters. These games can be competitive or co-operative, they can be individual, small group or between teams. They can pit the class against the teacher! Games can also be a home-school link. Teach the game at school and get the learner to play it at home or with their friends or at the after-school care. If it is fun, it will not seem like homework.

Some games are better than others and we have developed a checklist for evaluating games. You can read more about it on this post: Evaluating Mathematics Games. And there is a copy of the checklist available here: Checklist for Evaluating games

Here at Creative Maths we are committed to building a world of mathematicians. To help with this we have games and free resources that will help to develop fluency in multiplication and fractions. Hundreds of people have downloaded our free resource for developing fluency in multiplying by three. There is a low cost “Print and Play” file for all the multiplication facts available also.

We love to invent games as well as make videos. In the comments below or our Facebook page, let us know what particular skill or procedure you are wanting to develop and we will recommend a game, or invent one if we cannot find one.

If you have a game you find really works for your learners, let us know in the comments also or on our Facebook page.