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Secondary Magazine - Issue 128: Building Bridges


Created on 26 November 2015 by ncetm_administrator
Updated on 16 December 2015 by ncetm_administrator

 

Secondary Magazine Issue 128'Under the Bridge' by Alan Levine (adapted), some rights resered
 

Building Bridges

There’s been a lot of debate recently about whether children should memorise the times tables (for example, here, here, here, here and here!). Irrespective of how primary teachers can best ensure that their pupils do know their tables (and some of the most strident objectors are clearly confusing learning and testing), it is undeniable that older pupils in KS3 and 4 are often severely hindered by their lack of knowledge of the multiplicative facts encapsulated in the times tables. Many teachers and teaching assistants with experience of working with pupils with low prior attainment will be reading this, and will know deep down that the root of so many of the barriers to these pupils’ progress in their class is a lack of fluency in times tables. It’s hard to think of a better example of a mathematical bridge between KS2 and 3 than times tables – and one that in far too many pupils makes the Tacoma Narrows bridge look like the epitome of sturdy stability in comparison!

Let’s not pretend that if only every pupil in KS3 knew that 7 × 8 = 56 then all would be well and no pupil would achieve less than a C grade. John Holt, in How Children Fail (an excellent book despite the title!), writes that “[p]ieces of information like 7 × 8 = 56 are not isolated facts. They are parts of the landscape, the territory of numbers, and that person knows them best who sees most clearly how they fit into the landscape and all the other parts of it”. Importantly, he stresses that knowing times tables is necessary but not sufficient:

“The child who has learned to say like a parrot, “7 times 8 is 56” knows nothing of its relation either to the real world or to the world of numbers. He has nothing but blind memory to help him. When memory fails, he is perfectly capable of saying that 7 × 5 = 23, or that 7 × 8 is smaller than 7 × 5, or larger than 7 × 10. Even when he knows 7 × 8, he may not know 8 × 7, he may say it is something quite different. And when he remembers 7 × 8, he cannot use it. Given a rectangle of 7cm × 8cm and asked how many 1cm2 pieces he would need to cover it, he will over and over again cover the rectangle with square pieces and laboriously count them up, never seeing any connection between his answer and the multiplication tables that he has memorize.”

Nonetheless, factual recall is a start. How liberating would it be if maths topics could be explored without the voyage of discovery getting stuck at base camp needing emergency rations of knowledge of and confidence in times tables? The list of topics that require confident, fluent times tables knowledge is very long indeed, and encompasses a very large proportion of the GCSE syllabus: everyday arithmetic (especially, division), operations with fractions and decimals, using and understanding percentages, multiples, factors, LCM, HCF, algebraic expansion, algebraic factorization, and so on. William Emeny’s beautiful network model of the GCSE curriculum makes it clear how important multiplicative confidence is.

So why not set aside the time to hit the times tables where they hurt, and ensure your pupils smash through “the wall”? You might need more time than you expect, and you’ll probably want to adopt a “little and often” approach. It seems like a daunting task doesn’t it? But when broken down sufficiently, it will be an excellent investment of class time.

What tables facts do they need to KNOW, at their fingertips? Only forty five:

(see Kangaroo Maths and Trinity Maths for aesthetically more pleasing representations!). This depends on the pupils understanding commutativity, but that’s a fundamental concept which they need to grasp irrespective of learning their times tables.

To begin with, let's assume that the pupils in your class know some of the tables in order to help advance to the more difficult ones. For example, if 8 × 7 is required, then perhaps they can recall 8 × 5 and add on two more 8s until reaching 56. However, this writer has observed Year 11 pupils who cannot count on fluently in 8s, let alone recall any of the 8 × table. Where does that leave them? Very stuck indeed! It would take an awful long time to use their fingers to count on one by one.

So let’s start at the beginning, with the 2 ×, 10 × and 5 × tables.

  • 2 × can be described as doubling, which gives them something familiar to rest on.
  • 10 × has the pattern that the digits shift one place to the left and the ‘ones’ digit becomes a 0. Don’t let your pupils say or think that they’re adding a zero – that causes havoc when later they face 2.3 × 10.
  • 5 × could be approached by linking to 10 ×. Perhaps focus on the even multiples of 5 first, showing how the number can be halved and then the half-number is multiplied by 10 (e.g. to find 6 × 5, note that half of 6 is 3 so 6 × 5 = 30). Then for odd multiples, use the even multiples as a starting point and add 5 (e.g. to find 7 × 5, first work out 6 × 5 as described above, then add 5).

All the time, to develop fluency, keep practising

  • commutativity (6 × 5 and then 5 × 6)
  • related division facts (if we know that 6 × 5 is 30, then we also know 30 ÷ 6 = 5 and 30 ÷ 5 = 6)
  • “empty box” questions with procedural variation (e.g. 7 × \bigcirc = 42 and \bigcirc × 9 = 45 as well as 4 × 7 = \bigcirc)

Evidence suggests that an effective sequence for teaching the tables is: 10 ×, 5 ×, 2 ×, 4 ×, 8 ×, 3 ×, 6 ×, 9 ×, 7 ×. So now the 4 × table: a quick way is the 'double double' rule (and divide by 4 can be reviewed at this point using the 'half half' rule), and similarly ‘double double double’ for the 8 × table. If pupils know some of the 8 × table and want to count on, help them to see that they can add 10 and subtract 2. This is a useful exercise for improving addition skills, and should give them the confidence to count on quickly and accurately.

Onwards!

  • Point out that 3 × can be thought of as ‘double then add again’: 3 × 7 = double 7 and another 7.
  • There is a pattern with 6 × which is well worth pointing out because it’s a check of accuracy: when multiplying 6 by an even number, the ‘ones’ digit of the answer matches the multiplying factor e.g. 4 × 6 is 24 and 8 × 6 is 48 (Note that this sounds more rhythmical sounded out as 6 × 4 is 24 and 6 × 8 is 48). For the odd multiples, pupils can work from the even multiples and add 6 (or, add 5 and then 1).
  • 9 × is popularly calculated using the finger method (yes, you know the one – loved by pupils, hated – unreasonably, it’s great for fluency – by many teachers). Counting on 9s should be done by counting on 10s and subtracting 1 each time; this is good practice for adding “near 10s” (e.g. adding 9, 19, 29 etc. to a number): well worth a review.

And finally …

  • 7 ×. But there’s only 7 × 7 left! So why not do a review of square numbers at the same time as 7 × 7 to reinforce 12 to 102?

After reinforcing each times table individually, overall fluency in recall will only be achieved by mixing up questions randomly and gradually building in more and more individual times tables to the mix: try the spreadsheets 60 Club and 100 Club.

Why not get a competition going to inspire more and more of your pupils to join the club!? Fluency in the times tables could be judged as 100 correct (for the 100 Club) in 2½ minutes or 60 correct (for the 60 Club) in 1½ minutes – but reserve the Gold medals for the pupils with “division fluency” (56 ÷ 7 = \bigcirc and 63 ÷ \bigcirc = 9 and \bigcirc ÷ 4 = 7) as well as “multiplication fluency” (8 × 9 = \bigcirc and 6 × \bigcirc = 30).

There is a myriad of websites out there to focus on times tables, both free and subscription, for example:

  • Woodlands Junior School’s Interactive Times Tables Games;
  • to display patterns within the times tables on the board use this from Maths-Resources.com - great for identifying the symmetry within the times tables, square numbers and just how few times tables may be causing issues;
  • if your school is willing to spend a small amount of money on a subscription, look no further than Times Tables Rock Stars, which really gets the enthusiasm going through avatars, games and competitions.

Let us know what you try and what you like: email info@ncetm.org.uk or share a picture on Twitter @NCETMsecondary.

Image credits
Page header by Alan Levine (adapted), some rights reserved

 

 

 
 
 
 
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Comments

 


03 January 2016 19:07
Very informative !
By Huma
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18 December 2015 09:12
Trinity Maths gets a mention in this article - the relevant graphics for this item can be viewed free on the author of Trinity Maths twitter account @MathsDunbar
By mattgeo4
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17 December 2015 10:19
As one who works with adult learners from Entry Level to GCSE, I can split every adult learner I have in to two groups - those that know their times tables and those that don't. For those that don't have such fluency, non-calculator tasks are difficult and time consuming and effectively these learners are discriminated against when working against the clock uin an exam.

I have to make it part of my introduction to GCSE to review knowledge of times tables and look at simple, repetitive methods of helping adults learn these useful facts.
By tony.jenkins
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16 December 2015 13:33
It's so exasperating to witness generations of pedagogues dreaming up novel ways to learn multiplications, when the answer has been under their noses for decades. Learning times-tables in a parrot fashion is looked down on as barbarian by those who think 'old is bad' and 'new is good'. However, the evidence of millions of lives proves that this method, which takes just a few weeks, ingrains the tables so firmly in the mind that they're remembered for the rest of those people's lives, with no need to revise them. Learning the times tables, and in that fashion, has served me well throughout life, and is the single most useful skill I learned at school.

Until the educators humble themselves and admit their new, contrived methods are inferior, I'll have to continue to watch children take half a minute of counting on their fingers to work out a multiplication they should know in a split-second.
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