Fluency in Mathematics: Part 1October 4, 2014
I gave a talk on fluency in mathematics in March at Pedagoo London (my first public appearance) and again last weekend at the La Salle Education maths conference. This post is based on those talks and so, inevitably it is long enough to take several posts and revisits some old ground.
I started by asking the audience the following questions (yes I know, exciting start), and giving them a couple of minutes to work them out:
- Simplify 49/84
- Find √729
- Solve 3x+40=19
- Write √7/√175 as a decimal
At both talks there was somebody who took not much more than a minute and others who struggled. It is possible to answer every one of those questions in a few seconds if you have memorised the correct basic facts, such as times tables, and how negative numbers and surds work, and can recall them fluently. Both times though, I appeared to be the only person in the room to have memorised the first 27 square numbers.
As well as being useful for solving problems, fluency is now one of the major aims of the new maths National Curriculum, quoted below:
The national curriculum for mathematics aims to ensure that all pupils:
- become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.
- reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
- can solve problems by applying their mathematics to a variety of routine and non routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.
To have fluency as the first of three aims is a big change compared with the old national curriculum, the aims of which seemed to include everything else but fluency:
Learning and undertaking activities in mathematics contribute to achievement of the curriculum aims for all young people to become:
- successful learners who enjoy learning, make progress and achieve
- confident individuals who are able to live safe, healthy and fulfilling lives
- responsible citizens who make a positive contribution to society.
The importance of mathematics
Mathematical thinking is important for all members of a modern society as a habit of mind for its use in the workplace, business and finance; and for personal decision-making. Mathematics is fundamental to national prosperity in providing tools for understanding science, engineering, technology and economics. It is essential in public decision-making and for participation in the knowledge economy. Mathematics equips pupils with uniquely powerful ways to describe, analyse and change the world. It can stimulate moments of pleasure and wonder for all pupils when they solve a problem for the first time, discover a more elegant solution, or notice hidden connections. Pupils who are functional in mathematics and financially capable are able to think independently in applied and abstract ways, and can reason, solve problems and assess risk. Mathematics is a creative discipline. The language of mathematics is international. The subject transcends cultural boundaries and its importance is universally recognised. Mathematics has developed over time as a means of solving problems and also for its own sake.
Which to me, now, really sums up an era where you could have every variation of progressive education together in one document, while missing any mention of what is worthwhile knowledge.
One of the main reasons I think this has now changed, is a change in the understanding of how we think. One of the more popular diagrams in education today is this one from Dan Willingham’s book:
It’s a very simplified, but uncontroversial, model of how we think. We have a limited working memory, where conscious thought takes place, and a potentially unlimited long-term memory. To use working memory effectively, we draw on information that is already in long-term memory. To get things into long term memory we have to overcome the limitations of working memory. Having useful information in long-term memory, and being able to recall it without difficulty, makes thinking easier. In maths it is useful to be able to fluently recall a lot of knowledge, particularly basic number facts, rather than work everything out from first principles. The question at the start about simplifying 49/84 was an example of this, as fluency with the 7 times table makes the question trivial.
In order to build fluency we need to acquire knowledge and to learn to remember it without effort (automaticity).
And if that is our aim then, in maths, the best method is to tell kids what they need to know and set them lots of questions where they practise recalling it.
Continued in Part 2