As a conclusion to Maths Week 2011 and as an aside to this year’s Tables Tips Twitter Project, I am resurrecting an article about learning tables off by heart. This post originally appeared on Anseo.net back in January 2008. Almost 4 years later, I wonder if opinions about learning tables has changed. Read on and see what you think!

A few days ago on Education Posts, I proposed that learning tables off by heart is rubbish. My proposal was met with general disagreement and in some cases, complete contempt! I decided to prove my point by asking teachers to learn a few sentences off by heart just like children are asked to learn number sentences off by heart.

They didn’t know the significance of them, exactly like a child doesn’t know the significance of learning tables when told to do so. So here’s the sentences. If you want to take part in the challenge, don’t read on after the sentences until you’ve tried learning them off by heart.

Fred Davidson lives in Aaron Zion Avenue

Greg Fredson lives in Aaron Clare Avenue

Isaac Davidson lives in Aaron Clare Avenue

Fred Davidson works in the Bill Davidson Building

Greg Fredson works in the David Bill Building

Isaac Davidson works in the Clare Fredericks Building

It was interesting to see how many people accepted the challenge. There were 32 responses to my challenge but only 7 actually took it on. The others used the thread to give their opinion on my hypothesis. There was a strong sway of disagreement with me (57%) and only about 22% agreed.

The general views of those who disagreed were:

- There is no other/better way to teach tables
- The sentences have no relevance to tables
- Learning tables off by heart did me no harm
- Call me old fashioned but…
- We shouldn’t spoonfeed children

Of those who attempted the challenge, only one out of seven considered it easy. The rest who found it difficult gave the following responses:

- I got muddled / scrambled / confused
- My head got exhausted
- I had no motivation / interest to learn them
- I expected them to be easy to learn but they weren’t
- They weren’t important to me
- It was frustrating
- I’m too busy

It was also interesting to see how some teachers recorded ways they tried to learn the sentences. Some said they tried looking for patterns in the sentences. Others made a story up (e.g. similar surnames became part of a family) and finally others grouped similar names together to try and find patterns, etc.

Next, before the reveal, I’d like to compare how learning tables is very similar to having to learn those sentences off by heart.

- When you give children tables, many of them will experience feelings like those expressed by respondents, being muddled, confused, seeing no point, no motivation, etc.
- Perhaps the majority of teachers are good at learning things off by heart, due to the Leaving Cert relying heavily on this skill. To become a teacher, you have to score very high points in the exams. However, perhaps most people are not good at learning things off by heart.
- The six sentences below have complete relevance to learning tables. In fact, the sentences below represent 6 number facts. They seemed meaningless to many of you because they have appeared in a new way. A child comes across tables initially as something new too.
- Every word below is also a real word and you can explain what every word means. Putting them together, however, they don’t seem to have any pattern. Likewise, every number in a tables fact is also known to a child but put them together and they don’t seem to make sense or be very interesting. The reason some of you weren’t bothered learning the sentences off is the same reason a child wouldn’t either.

Here’s how the sentences below are, in fact, tables facts.

- Any of the names represent a number. E.g. Aaron=1, Bill=2, Clare=3 and so on. Z words represent zero.
- Lives = Plus
- Works = Multiplied by
- Avenue = Equals
- Building = Is

So to translate:

Fred Davidson lives in Aaron Zion Avenue (6 + 4 = 10)

Greg Fredson lives in Aaron Clare Avenue (7 + 6 = 13)

Isaac Davidson lives in Aaron Clare Avenue (9 + 4 = 13)

Fred Davidson works in the Bill Davidson Building (6 x 4 = 24)

Greg Fredson works in the David Bill Building (7 x 6 = 42)

Isaac Davidson works in the Clare Fredericks Building (9 x 4 = 36)

As far as I would be concerned, all the responses in this discussion represented exactly how a child feels when given a list of tables to learn. Just because I represented my tables in words doesn’t mean they should have been any more difficult.

The people who looked for patterns, made relations and grouped similar sentences were all using strategies to help them learn. If you can give your pupils strategies like that, e.g. David Fredson also lives in Aaron Zion Avenue (commutative property), you’ll see the benefits straightaway.

My conclusion to this is that we need to accept that teaching strategies is far more beneficial than simply learning tables off by heart. No doubt, some will still disagree and I’d be interested to see your reasons, to which I’d be glad to respond.

**Last Update: ** March 31, 2019

People responded to my findings asking for an alternative – so here it is:

I have proposed different methods of teaching the tables in previous posts. For whatever reason, the majority of people seem to ignore them. It’s probably because it’s harder to plan for or that it’s less effort to simply tell children to learn a list of meaningless (to them) numerical garble for homework and check it the next day. Anyway, I’ll try again.

Firstly, what most people found from my sentences was there were patterns to be seen. That’s a great place to work from. For example, adding 10 to any number, you add a one to the ten’s side. Or Multiply by 10 then add a zero. The finger trick with the 9X tables provides an interesting pattern. 2,4,6,8,10x tables always have an even answer. 5x are half the 10x tables. There are so many patterns it’s unreal.

If children are learning their tables by rote, they have to learn 121 facts. (in this curriculum you only learn up to 10 times tables btw)

Learning by strategies, you always start with what children know. For addition, generally they know 0, 1, 2. Build on this knowledge. I’d go to 10+ after this, then 9 – (9 is 1 less than adding 10). I’d try doubles, then near doubles, numbers that add to 10. Also I always show the commutative property. So if they are learning 10 + 3, they will also learn 3 + 10. This process takes more teaching time but the results have shown it to be more beneficial.

With multiplication, again kids know 0,1,2 (if they don’t 0xa=0, 1xa=a, 2xa=double a). With 10x add a zero to the number, 9x-finger trick, 4x double the 2x, 3x = 2x then add x, 5x half the 10x, and you are left with 6,7,8x tables. With the commutative property, you are only left with 5 facts to learn and here’s one more strategy to help – 56=7×8. So the only things left to learn are 6×6, 6×7, 7×7,8×8. Children seem to be pretty good at squaring numbers so that cuts out three more. So now, they must learn one table off – the meaning of life 42=6×7. They then associate the meaning of life (thanks Douglas Adams) with this table. If they are in 6th class – they may even get to read the chapter of the Hitchiker’s Guide.

I have deliberately annotated this in algebraic form as I’m sure you can understand it. Children of course will not know what those “a’s” and “x’s” and “y’s” are but generally I use sweets or Euros.

So…to conclude, by using strategies such as pattern-finding, association and the commutative property, children don’t have to learn anything off by heart. It takes longer but it works.

If anyone has any other tips, I’d appreciate it, because I’m always learning. Only today, someone told me of a couple of tricks such as visualising a chessboard (64 squares = 8×8) or the San Fransisco 49ers can be remembered by having two number sevens scrumming for the ball (7×7=49 – American I know) If you’re in Dublin, maybe the number of the bus near your school into town could help, e.g. the 15 bus (for 3×5)

If rote-learning does work (and the majority of teachers are obviously still teaching this from reading these posts), why are secondary teachers giving out that children don’t know their tables coming into secondary school? I have yet to see convincing evidence that rote-learning is the best way of learning tables. If anyone can find some research, I’d be interested to see it.

Learning Tables

I really haven’t the time to be doing this but I feel compelled to bring clarity to the topic:

Teachers should have a detailed, solid, confident, forensic knowledge of the definitive evidenced based approach to teaching tables. The question is: have 100% of them got this level of knowledge and understanding? Are they sure in their current approach?

Learning by rote or learning underpinned by understanding? Of course the latter is preferable.

Learning mathematical concepts must be grounded in hands on work with manipulatives, so that children can move on to replacing actual present objects with visualisations – moving from concrete to abstract.

Here’s how I teach a set of tables facts: e.g. 6 Times Tables for the first time using various coloured counters. Same method repeated for all sets 2 to 10.

Ask the pupils to put out 6 green counters in a row on their table. On the whiteboard you draw 6 green counters and beside it write 1 x 6 = 6 and translate the equation verbally as ‘we have one group of six, which equals six altogether.

Now ask the pupils to put out 6 red counters underneath the 6 green counters and you draw six red counters underneath the six green and beside it write 2 x 6 = 12 and then translate it as ‘ we have 2 groups of six counters which give us twelve counters altogether. The x stands for ‘groups of’ – no more mystery! Stop using ‘times’ – use ‘groups’ when translating the ‘x’ – it unwraps the understanding and can be the key that unlocks the concept! Amazing ……. just shows how we need to be careful and precise in our use of language.

The principle of multiplication is now obvious: we are dealing with equal groups i.e. 2 groups of 6 or 3 groups of 6 etc. and we can see that 3 x 6 = 18 or simply: 3 x 6 = 6+6+6 : arrive at it by repeated addition.

Continue the pupils building groups of 6 and teacher drawing the groups and recording the facts on the whiteboard, until you have reached and recorded to 10 x 6 = 60 .

Now allow the pupils to draw the sets of 6 into their copies and record the facts.

HERES THE THING:

When my pupils are at home tonight learning the ‘groups of 6’ facts, they can visualise the sum i.e. when asked ‘what is 3 groups of 6’? – my students are visualising 6 green, 6 red and 6 blue and seeing 18 in the visualisation. They can see the constructs that deliver the total of 18.

So – they are not learning off the facts but seeing the facts in the minds eye.

The concept is laid down and is solid.

The process of teaching a set of tables: manipulatives handled on the table …. represented pictorially on the board with the abstract equation recorded beside it.

This works. Try it out with your pupils because if the concept that underpins multiplication facts is not taught well – then it can pull down many other areas of mathematics such as fractions, decimals, percentages, number computation, problem solving etc.

TEST: ask a pupil what is 5 x 6? O.K. – he/she says 30. Great. Now ask them to draw what it looks like! This will quickly reveal if the answer was delivered through rote learning or derived from real understanding of the concept i.e. he/she should be drawing a collection of objects divided into 5 groups with 6 in each.

Teaching Multiplication well is a teacher obligation!

Just a couple of thoughts …. needless to say – you could write a book on the subject.

Martin Pender @ Rainbow Education

True words of wisdom. There is no point knowing that 5×6 is 30 if it means nothing in the real world. I’m often asked for an alternative and I freely give it. However old school will never listen as long as there are enough of them!