Learning the Times Tables: teachers collaborate to find the most effective strategies
by Kate Frood, head teacher at Eleanor Palmer Primary School, London
In September 2016, a group of teachers from ten Camden primaries (supported by London Central and NW Maths Hub) formed a Teacher Research Group to look at the learning of times tables. Our agreed goal was teaching children recall of times tables, to the level of automaticity. We perceived that if children can commit key facts to long term memory, then working memory is freed to apply to deeper and more complex learning (this observation is borne out by cognitive science research – see Is It True That Some People Just Can’t Do Math? by the cognitive scientist Daniel Willingham, for example). However, we also shared an instinct that recall alone was not enough! We all know the child who can count in fives but cannot answer 7 × 5 at speed, or the child who can recall 5 × 6 but who cannot apply this to
? × 6 = 300. We shared concerns that the new Key Stage 2 arithmetic paper is leading to too much rote practice and ‘tricks’, encouraging children to use a taught method irrespective of the numbers involved (remember Q3 from last year’s arithmetic paper “326 ÷ 1”?).
Broadly we have four main conclusions to share:
- Teachers need a broad understanding of what fluency is to embed deep and permanent learning of key multiplication facts
- There are routines and strategies that help children to memorise key facts
- Clear messages that ‘remembering stuff’ is important, must be embedded in school culture
- Reasoning should be developed from the outset, alongside the expectation of fluent recall – a suite of simple routines and activities are needed to help with this.
What is fluency?
In Developing Computational Fluency with Whole Numbers in the Elementary Grades (2000), Susan Russell defined it as “a well-built mathematical foundation with three parts”, as follows:
- Understanding the meaning of operations and their relationships to each other e.g. inverse, multiplication as repeated addition
- Knowing facts and how they relate to each other (“if we know this, what else do we know?” 4 × 5 = 20, so I know 4 × 50 etc)
- A thorough understanding of the base 10 number system, how numbers are structured in this system and how they behave in different ways in different operations eg 24 + 10 = 34 and 24 × 10 = 240.
Real fluency combines a deep conceptual understanding with an ability to recall accurately and rapidly. It is not just repeating back the fact. It is about flexibility - efficiency - accuracy (Russell 2000). As a group, we wanted to emphasise the importance of this broader understanding whilst being uncompromising on the need to know stuff ‘off by heart’.
We concluded that fluency is about:
How can we better support children in learning times tables?
- Building conceptual understanding (of the facts we want them to learn)
- Building relational understanding (e.g. seeing the link between 4 + 5 and 40 + 50)
- Building an understanding of the structure of operations (e.g. 3 × 10 is the same as 10 + 10 + 10).
Back in our classrooms we found that:
- The counting stick (and a colour-coded card set of each times table from 2-12) is a core resource and should be on hand at the front of the class at all times. It reinforces the patterns, makes visual the structure of repeated addition and enables connections to facts outside those being memorized, such as 30 × 4
- Chanting using the stick should happen on a daily basis. We found having a class ‘table of the week’ created better outcomes than juggling 30 individual programmes. We loved this ATM video with Jill Mansergh modelling strategies for learning tables
- The rich patterns of multiplication should be exploited from colouring on the hundred square to exploring patterns created by looking at the ones digits within each times table
(click to enlarge)
- The principle of ‘start with what you know – build on what you know’ is important. Reference to ‘super-size’ (30 × 4, 700 × 8) and super-skinny (0.6 × 50), in Years 4, 5 and 6, should be introduced alongside each times table. This can serve to extend those who already know the core facts and makes key links to place value
- Weekly tables test, re-framed as ‘quizzes’ and thus less intimidating, should be a routine from Year 2
- Multiplication square visuals in clear and accessible places, really support children and could be left up during early tables quizzes
- Quizzes can be structured in a way that supports conceptual understanding and relational understanding. We constructed this quiz which had great results! We observed the less secure children referring back to ‘Round 1’ in order to answer Round 3. Not cheating – success!
Remembering stuff matters!
To support each of our schools and to deliver the core message that memorisation matters, we devised an inter-school ‘Spring Slam’ competition. All our Key Stage 2 classes took part in four rounds leading to a grand finale with the top child from each year group in each school attending and ‘best in borough’ crowns awarded.
Critical to the conception of this competition was that every child gave it a go and the submitted score (by set deadlines, to me as lead) was the class average score. A lower-attaining child immediately improved by seven marks in Round 2 by understanding what happens when you multiply by zero, contributing to an increase in the class average. To incentivise the higher-attainers, there was a bonus of five points for a 50/50 score, highlighting those who didn’t check! League tables were hotly anticipated and enabled us to benchmark our own children. Every class made significant progress over the competition, from 2-25% improvement.
Here is a sample quiz from Round 1 (Y5 & 6).
The key learning as we constructed and reviewed each round was the shift to assessing deep understanding and secure fluency. For Years 5 and 6 (who were scoring high class averages based on recall alone in Rounds 1 and 2), we re-designed next rounds to include: empty boxes, division, super-size and super-skinny questions. Here is a sample quiz from Round 4 (Y5 & 6).
Average scores plummeted by 8-10 points. There was a clear next step for many of our classes!
Routines to develop real fluency
Perhaps our greatest learning was the need to develop reasoning alongside fluency. We remembered, devised and tested many simple whole class routines that develop this deeper understanding. All centred around pupil talk and about the properties of numbers such as prime, factor, multiple, square, odd, even. We found that children were really empowered by being given a language to talk about numbers and they loved the activities. Here is an example of one such activity:
ALL, SOME, NONE
Write three or four random numbers on the board. Pupils talk with a partner, using a whiteboard and pen if that helps, to construct three sentences about those numbers, using the words ‘all, some, none?’
E.g. 2, 7, 11
“All of the numbers are prime”
“Some of the numbers are odd”
“None of the numbers are factors of 15”
All these statements lead into some lovely teacher-pupil dialogue.
We have set ourselves the goal of bringing all our learning together by the end of this school year. If you are interested in a copy of our final book, including templates for the Spring Slam Rounds and the structured quizzes, register your interest by emailing firstname.lastname@example.org.
Kate says “'Digging Deeper' articles from Issues 88 and 91 of the NCETM Primary Magazine inspired and guided us”
Those interested in reading a summary of the research regarding learning of times tables should visit Cambridge Mathematics' new Espresso site – expertly filtered research aimed at maths teachers. Specifically, Issues in learning and assessing times tables.