Monday, 4 January 2016

Times Tables: What is Knowing?

@tombennett71: There should be nothing controversial about a mainstream expectation for children to know times tables and we'll look daft if we dispute it.

And I agree. Apart, perhaps, from the part about 'know'. What does 'know' mean? Merriam-Webster defines 'know' thus:

  • to have (information of some kind) in your mind
  • to understand (something) 
  • to have a clear and complete idea of (something)
  • to have learned (something, as a skill or a language)
If a child, when they are tested on their tables in 2017, can choose their own version of 'know' then I definitely agree. When you've wiped away your tears of laughter after watching Nicky Morgan avoid answering 11x12, read what she said: "We are introducing a new check to ensure all pupils know their times tables by age 11." She says 'know'. The 'by heart', 'by rote', 'by memory' rhetoric has been added by the papers who gleefully reported the news, glad at the chance to stick another knife in the back of the profession. So, theoretically children don't have to know their tables by heart.

The reason why this issue resonates with me, and with many others, is that as a child, despite my dad's best efforts, I found it impossible to learn my tables by heart. And I still don't know them all today. What I can do is work out multiplication problems very speedily using Merriam-Webster's second, third and fourth definitions. I understand what happens when you multiply one number by another so I can solve a problem. I have a clear and complete idea of how timetables link to other areas of maths. And I have learned lots of methods (you might say skills) to help me to work times tables questions out before anyone realises I haven't memorised them.

The beginning of my journey out of times-tables-embarrassment-land was when I realised that my dad, at random moments during the day, in an attempt to keep the practice up, would only ever ask me what 6x6 was. So I learnt 6x6 (it's 36 - see, told you I'd learnt it). I soon realised that if I knew that then I could work out 6x7 really quickly.

The next step of my journey was my realisation, in secondary school, that if a teacher tried to get me to learn a method without explaining how and why it was working, then I wouldn't be able to do it. I had to understand the mechanics of the mathematical process in order to be able to solve problems. As my teacher took the time to model processes in a way that I understood them, I began to improve in maths. I started to enjoy it too. In fact, I started to think mathematically, could problem solve, reason and I sure was getting fluent. Recognise those three terms? Yes, the aims of the National Curriculum. If I had only learnt by heart the formula for finding the area of a triangle without understanding why it worked then I'd have been far less fluent and would not have been able to problem solve or reason. So why are so many teachers hellbent on getting kids to memorise stuff like times tables?

OK, if a child can memorise them then great, but teachers beware, I truly believe there are kids out there in year 5 right now who will be better supported this year if you teach them some tricks and tips so that instead of rapid recall, they can do rapid work out of tables. Take it from someone who knows.

Here are a few tips and tricks for how you can help those children once you've identified who they are (probably by giving them one of the hundreds of times table check practise tests that will appear online by the time the month is through):
  • Find out which tables they have learnt by heart - the majority of children will definitely have 2s, 5s and 10s.
  • Assuming children know 1s, 2s, 5s and 10s they already have good reference points for other tables. 3s and 4s could be taught using manipulatives such as Numicon shapes or cubes (or you can get really creative - Ikea's dogs' bums coat hooks are fun for 3s) to reinforce what is happening when multiplying 3 and 4.
  • When learning 4s and 8s make links back to 2 times tables. Lots of simple investigation opportunities here too i.e. Which times tables does the number 16 appear in? If the kids can make these connections themselves they will be more likely to learn skills that they can apply in a test situation.
  • Similarly link 3s and 6s together. Later they can be linked to 9s and 12s.
  • Teach 9s using the finger trick. Make sure children have identified the pattern in the answers: the digits in the answers add to 9 - do investigation so that they find this out for themselves.
  • Teach 11s by looking at the pattern in the answers. 10x11, 11x11 and 12x11 might be a bit more difficult so these might need to be learnt by heart - reducing the number of answers that need to be learned by heart is still helpful.
  • This might sound totally ridiculous... OK, it absolutely will, but Weetabix taught me how to work out my 12 times tables quickly. I know the pack sizes.  Each tube inside a box contains 12 Weetabix. You can get boxes of 12, 24, 48, 72. Help the kids tap into outlandish methods like this - maths in real life will be a saviour to many. So many kids need to know why maths is important and relevant to them SO THAT they can begin to understand it.
  • Your children probably are capable of retaining a few facts. I could always remember 6x6 which inspired me to learn my square numbers. Mathematically square numbers are interesting and are more likely to stick in the head (nice links to actual squares in geometry too as a model). Once you've learned square numbers the world is your oyster, especially if you know the square of 6, 7, 8, 9, 11 and 12. You can use those as a reference point and quickly add or take from them.
  • Many children will be able to get a feel for which numbers sound like correct answers and which don't. Some work on what a prime number is might help - children will learn to avoid 17 as an answer in a tables test because it doesn't sound right. They will begin to know that 56 and 42 do appear somewhere - this gives them a reference point to check their answers by. 
  • I use the idea of The Hard Tables. This reduces the number of times tables that children have to really worry about in year 5 and 6. The Hard Tables (even the name shows a child who struggles to memorise them that you understand their plight) are basically any problem (beginning at 6x6) above the square roots of square numbers i.e. 6x6, 7x6, 8x6, 9x6 and 12x6 (most children will know 10x6 and 11x6)
  • Give the children tests where you model a thought process e.g. "The question is 8x7. So think of your square numbers: 7x7 is 49 so you need another lot of 7. 49 add 7 is... is your answer one of those numbers that we know is an answer in the times tables? Does it sound right?"
Put me in front of the nation and ask me a times table question and I'll answer it right away. Not because I know everything up to 12x12 by heart but because I will THINK MATHEMATICALLY about the answer. I will demonstrate fluency as I link areas of my mathematical understanding together. I will demonstrate, invisibly, my ability to problem solve and reason. I will demonstrate that I 'know' my times tables without actually ever having memorised them all. I will be using one or two of the above strategies to get to answer. It will take me a fraction of time longer than someone who has memorised the answer, but out of the two of us, I'll be the one demonstrating better mathematical thinking.

Photo Credit: WoofBC via Compfight cc

3 comments:

  1. Great post and thought provoking.

    I am 58 years old and learned my times tables off by heart in primary school in the uk. I used to practice them for the tests while laying in bed before going to sleep.

    I dont use any fancy tricks although I do know most of those listed above.

    I save tricks for difficult stuff.

    I just remember times tables which I started to learn by heart about 54 years ago by repetition.

    I dont think there is any harm learning them by rote and I think most people can do this if they really try and are motivated to do so.

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  2. A very insightful post.

    Like you, as a child I had trouble memorizing the multiplication tables, and, like you, I devised ways around my difficulties. I eventually became a mathematician, and I “know” arithmetic and am “fluent” in it. At least that is the case if one accepts that words like “knowing” and ”fluency” have meanings that extend beyond pure rote recall.

    In my country (Canada), there are many people who are concerned with the state of mathematics education. and they constantly push for a redesign of the elementary mathematics curriculum. They are fond of using these words, but, alas, it is not clear that they are willing to permit such a broad interpretation as you and I would like.

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    Replies
    1. Thanks for your comments (and sorry for the tardy reply!). I'm afraid there are many out there who look down on us and say that we just haven't tried hard enough because we didn't rote learn.

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