Regarding times tables tests: a plague on both your houses!
There’s a well known cartoon that claims to illustrate the difference between equality and equity. And a newer version of that cartoon that simply removes the barrier that is causing the problem.
To some extent, this sums up my attitude towards times tables tests and the arguments between those who think that times tables should be tested often, using timing and speed as an indicator of success and those who believe that times tables tests are tantamount to child abuse, leading to increased mental health problems and of no practical use at all.
My position, probably like many, is somewhere in between these two extremes, but the debate is so often polarised between the two that other debate is either stifled or just doesn’t get any time, squeezed out in all the polemic.
So, what do I believe?
- Multiplication tables are really useful. Now, I am talking here about tables like the below, tables with a rich mathematical history and an amazing source of rich mathematical activity. Of course, there are modern versions!
(By Unknown — Popular Science Monthly Volume 26, Public Domain, https://commons.wikimedia.org/w/index.php?curid=11410038)
2) Fluent recall of multiplication facts is really useful for school study and beyond. Some element of timing is useful for reasons I go into below, but whether a question is done in 1.1 seconds or 1.23 seconds… not really interesting to me. Rather more interesting is whether the child knows 3 x 4, then they also easily know what 3 x 40 is, or 3 x 0.4. Or 12 ÷ 4 or indeed that 4 x 3 has the same result as 3 x 4.
3) Which now brings me to ‘times tables’. Here I am referring to the following format — and sorry, I’m not that bothered about multiplicand and multiplier here. Commutativity rules ok! Nice colours, fonts, etc. But all the same.
4) What’s the problem? I am a child of the 60’s. So my times tables learning experience was very traditional. Chanting in groups, tests, the lot. And do you know, I don’t think I was really secure with all the tables facts until well into my late teens/twenties. As for realising that if 3 x 4 = 12 then 3 x 40 = 120… no way! Now, I am not proud of this and am not one of those who say “I didn’t know my tables and look at me now”, as I do realise that knowing these facts are useful.
So, as I have taught mathematics and times tables over the years, I see children still struggling, I remember struggling. I see times tables being magnified out of all importance and now in England, we have the Government mandated centrally controlled times tables tests on the cards. And times tables continue to be ‘difficult’
5) What’s the problem? For me, the following.
The very format of the tables mean that learners are not trying to remember a number relationship, but a sum. With the equals sign (that then often gets remembered as meaning ‘the answer is’ as opposed to showing equality. Something that is known to cause issues later on) and then repeated with digits changing.
So the learner is not remembering a relationship, but trying to remember the place of that ‘sum’ in a list of other sums. In array of such lists. I am sure that this MUST add to the cognitive load and adds to the difficulty of remembering times tables. In my opinion it’s a great example of the ‘split-attention effect’ referred to in dual coding theory.
Then the learner starts to use other strategies to remember the list NOT the relationships. Chanting and repeating 2,4,6,8,… for example. Which again is not learning the tables, but learning lists and sequences. How many times do we see students mentally chanting? And as for those videos on twitter showing a class chanting tables, we can all see the ‘mumblers’, half a beat behind the pupils at the front. I know, because that’s what I used to do.
So, we add difficulty, encourage strategies that actually do not contribute to learning the multiplicative relationships and then once that’s all done…
We give them a calculator.
Yes, we do say “oh, but it will not be used for simple calculations”, or “but calculators are so useful” and point to relevant research.
But for almost everyone, we have such a gut feeling that they restrict our skills of arithmetic — and actually do not care! We just want an answer.
Again to use a personal analogy. My first calculator, a Casio FX39 back in 1978, was chosen because it was one of the first that had a fractions button and I hated fractions! My chickens came home to roost at A level once I had to do algebraic fractions though!
6) What to do?
The image at the top showed that a great way to cope with something causing a problem is to remove the problem. So many strategies and arguments about times tables are because, in my opinion, we just try to do more of what has always been done, but in a ‘better’ way.
Why not just remove the problem?
I thought about tables for many years and came up with a format for tables that removes the list of sums format. It still has a timed element, as I think that’s useful. But if asked for 6 x 7, then if the learner thinks in their mind of 7 x 6, that’s fine! but in the usual ‘list format’, that is a different question. The these are also easily extended to take care of 6 x 0.7 and also division facts. A completed test looks like the below image and all details and a variety of tests can be found free of charge on the teacherspayteachers website.
These tests are very low stakes, ‘personalisable’ and still get the multiplicative relationships remembered. You could have every student in a room doing a 4 minute test and EACH ONE doing a different test. You can mark these yourself… or do some peer marking (probably preferable). I have not made any rules on this. Especially as I have extended the tests. For easier versions, marking them myself was quick and easy.
The second thing that I feel will really make a difference is the use of the QAMA calculator. I have NO idea (well, I may have… because more thought is needed not just ‘getting the answer’) why teachers have not jumped on this already, as it has the potential to help improve skills, including times tables, across the world. Indeed, many eminent mathematicians, academics and mathematics educators love what it does. Just this year, Bob and Elizabeth Bjork from Stanford were discussing it on a mathematics education podcast with Craig Barton.
What the QAMA does is ask users to enter a reasonable estimate. So for 3.2 x 4.9, you could enter 15 and then get the precise result.
How does this work with multiplication facts? The ‘usual’ times tables?
The ONLY reasonable estimate for these is the right answer! So the user, on entering 7 x 6, would have to enter 42 themselves as the estimate. This functionality does two things. Firstly, the user never has the calculator replace their own though processes on multiplying numbers. Secondly, in a world where feedback seems to be one of the words of the decade, the user gets immediate feedback on their answer. If they are unsure, they can ‘fail quietly’ and redo the estimate. This can ONLY help students reinforce their memory of the tables.
There is a lot more the QAMA can do using estimation, but it is just the times tables that we are discussing here.
So, to get back to the title of this piece. A plague on both your houses.
Often, the antagonists in an argument, be it times tables tests or the use/non-use of calculators, are so entrenched in their positions that they do not consider another way — making a change that removes the problem.
We do, as teachers hold ancient grudges, considering only our preferred solution to a problem without identifying what might be the source of the problem and removing that.
Continuing to do so? Well the difficulties with times tables, with calculators will continue — to the detriment of all learners.
