reflecting on timestables practice
We all know that students need to have a fluent knowledge of timestables. But what do we do when they don’t?
Many of the weakest students I have taught do not have a good knowledge of timestables. Ask them 4 x 7 and they’ll start writing 4, 8, 12 adding on four each time until they reach 28. This is a problem. So much within maths becomes overloading for these students’ working memory if they do not have basic multiplication facts secure in their long term memory.
In this post I will reflect on what I tried this school year to improve my students’ timetable knowledge, why that didn’t work and what I am going to do instead.
This is aimed at the very weakest students we have at our comprehensive secondary school. These are students who are on track for grades 1 and 2 at GCSE, the students I aimed my previous post on Pythagoras at.
Schön, Ebner and Kothmeier (2012) state that ‘… learning the multiplication table is a central subject of mathematics in primary school’, and similarly Caron (2010) explains ‘the need for every student to know their timestables well remains critically important’. With regard to the recall of timestable facts, Westwood (2003) says ‘to be competent in problem solving one needs to be able to easily draw on essential declarative knowledge’. This Cambridge Mathematics Espresso summarises research in the area well.
It is essential that our students know timestables fluently. Students who do not know timestables facts with automaticity must use their valuable working memory to either recall or compute them.
What specific timestables facts do students struggle with? The graphic above, produced from this study by the Guardian, shows this. This data is from 232 primary students’ in a single school and their responses to timestables questions within an app. The colour indicates the error rate, with more red meaning more incorrect responses. It shows 1,2, 5, 10 and 11 as being the easiest and the middle 7, 8, 9 region and parts of 12 as being the most difficult.
what i tried
“Practice is what they need” I thought, “so practice is what they will get”. Throughout this school year I would regularly use starters like the ones below with my lowest ability classes, about once a week or so.
Students would get either one or both of these grids and be asked to complete it. The order of the numbers in the second grid would change each time. On a good day this could keep the students occupied and on task for 15 minutes or more. Watching bottom set year 10 work silently on this for 15 minutes felt like I had achieved a miracle. “Aren’t they doing well” I thought each time, “aren’t they learning lots”.
But the problem is, honestly, I don’t think this actually helped. Engagement and learning are very different. The students were engaged, but I’m not convinced they were actually learning much.
If they were doing the first grid students would start by doing the timestables they already knew fluently like the 1s, 2s and 10s and therefore would spend the first 5 minutes writing without actually thinking. Then they’d do the timestables they didn’t know by counting up each time.
Watching how students tackled the second grid was fascinating. It would often go something like this (supposing they were working across). They probably knew 4 x 5 and 4 x 4 so they would just write them in. But then they’d get to 4 x 12 and not know it off hand. To work it out I would usually see them writing out the multiples of 4 by adding 4 each time: 4, 8, 12, 16 and so on (usually by counting up on their fingers). Then they’d count down their list to the 12th one: “ah, it’s 48”. With that they’d carry on; 4 x 1 well that’s easy, as is 11 and 3, and then they’d get to 4 x 6. They’d go back to their list and count down to the 6th one: 24, done! And so on.
The thing is, these students were doing that same process each time. Whenever they got to a timestable fact they didn’t know they’d write out the multiples and count down the list. That was what they were practising, that was what became permanent. It was a process that (mostly) worked for them, so they stuck with it. This practice became their permanent method.
Westwood (2003) mentions this exact procedure as being ‘unnecessarily time-consuming and heavily demanding of cognitive effort’. What’s more, the students didn’t magically learn and remember that 4 x 6 = 24 by writing the multiples out each time. They didn’t happen to notice patterns by chance. They didn’t get better at timestables this way.
Unscientifically, the students didn’t seem to get more accurate or quicker at this no matter how often we did it. I think that’s why, reflecting on this at the end of the school year, I realise it didn’t work. As a task to solely settle students it may have worked (but then, so would a colouring-in task), but as a task to improve their timestables knowledge it didn’t.
what i’m going to do now
Brendefur et al (2015) showed that when students learn timestables through strategy methods (such as physical representations, derived facts, discussing strategies with peers and creating method flashcards) they learn timestables facts several times more efficiently than students using drill methods (memorisation and rehearsal).
The problem with the starter I was using was that students were seeing timestables as independent facts. Their only way to find 4 x 12 was to list the multiples. They didn’t consider that this was 4 x 10 + 4 x 2, or 2 x 2 x 12 etc. Any of these facts would allow them to calculate 4 x 12 without writing the multiples. Moving forward I need to force students to work out answers this way, I need to ensure they are using the strategy methods mentioned in the above study.
Instead of the starter I used before, I will now use something like this. This is based on the ideas from Craig Barton’s book Reflect, Expect, Check, Explain. This task forces the conceptual understanding and prevents the inefficient listing procedure.
Here’s how it will work. Students start with 3 x 5 and write the answer in the box. Then they work downwards. They must write something in the Reflect column each time. They are comparing the 5 to the 10: what has changed? It’s doubled, so what do I do with my answer? Double it. Now 3 x 9. Well that’s one less 3 than 3 x 10, so what do I do to my previous answer? Once they’ve worked down the 3 column they then do the same thing for the 5 column, then the 6 etc. Half way down the 3 column their answers might look something like this:
It is not as immediate to students what they are supposed to do when presented with this sheet. I will have to introduce this task, make my expectations clear, explain to them why we are doing it this way and model examples. A list of multiples is henceforth officially banned.
A couple of points on this task:
- the order of the numbers across the top should be randomised each time, but it’s probably best not to start with 1, 2, 5 or 10 as students likely already know these timestables (so working downwards they won’t follow the process).
- the order of the numbers downwards should be changed each time. But this takes some thought. There should be connections, it’s about the relationships. Going 5, 11, 7, 1 isn’t that helpful.
- the variety in the downwards order ensures students become familiar with as many possible strategies – e.g. you can get to 10 by doubling 5, adding 1 to 9, 2 to 8 etc.
- the most important part is that students must be referring to the reflect column each time they go to answer a question.
- the very first question should be something easy that they will know (see Guardian graphic above). There’s no point starting with 8 x 7 if they don’t know what that is.
There is so much more to it though. That’s also where I went wrong this year. Just the grids alone aren’t enough.
The new reflect grids will help students understand the connections and relationships between multiples. Once established as a routine, that task will work well as a starter to settle students. But I will also ensure I explore representations like this:
Students need to be fluent with the idea that all of these are equal, and why they are equal. That means we need to work on it explicitly in class. “Give me a way to make 24. Give me another way. And another. Explain why those are the same… Now let’s do the same for 40”. Representations like the one below will also be used, ensuring that students understand what it is showing.
All of the work above is needed to improve students’ conceptual knowledge. But conceptual knowledge alone won’t automatically lead to students knowing 4 x 12 is 48 without having to work it out. Teaching students strategies is right, but at some point students need to transition to just knowing. ‘The importance of automaticity becomes apparent when it is absent…. conceptual understanding is necessary, but insufficient for mathematical proficiency’ (Wong and Evans 2007). My mistake this year was not embedding practice for automaticity.
An experiment by Knowles (2010) showed that using timed drill practice increased students’ performance in timestables tests compared to a control group. Woodward (2006) showed that using both strategy and time limited drills together was equally as effective at improving results as time limited drills alone, but that students taught using both methods were better at applying their knowledge to extended problems, like 40 x 60, and estimating answers.
Denham (2013) found that students were significantly quicker answering timestables after playing a computer game that incorporated timed drilled timestable practice, but that the inclusion of timetable strategy tips had no additional effect. As well as this, students had a positive feeling to this game and enjoyed playing it.
Evidently timed practice is important and has a place alongside strategy based methods. As a mental model, let’s consider each students’ knowledge of each timestable fact as being in one of two distinct groups:
- group 1 timestable facts they can recall instantly
- group 2 timetsable facts they cannot recall instantly
By instantly I mean within a second or two, without working anything out. In an ideal world we want students to have all facts in group 1 (of course with the understanding that facts won’t stay in group 1 forever unless practised). We want them to know that 4 x 12 is 48 without working it out.
Once students are comfortable completing the timestables grids with the reflect column and they have a good conceptual understanding of timestables and using representations etc I will then bring in timed online resources. The purpose of these is move students’ factual knowledge from group 2 to group 1 and to retain it in group 1. The conceptual practice will continue.
This, from Dr Frost Maths, is one possible tool. Students have to answer a timestable in a random order and they are timed as they do it. Of course, this requires the right classroom culture. Students are not competing against one another and it is not a competition. Instead they are trying to complete the timestable as quick as they personally can. Dr Frost Maths is great because it allows me as the class teacher to see how each student is progressing.
Another great tool is this game (flash required). It works well because it’s backwards – it gives an answer and asks students where that fits. The question on the screen is 28 (at the top), students have to click either 4 x 7 or 7 x 4 to place it in the correct space. A similar game, without requiring flash, can be found on Mathsframe.
Again, the timed element is essential here because we are testing students’ immediate recall, not their conceptual understanding. When I’ve used this site before students did only untimed practice and, like the basic multiplication grids, I’m not convinced it had an impact.
Both of these are a tool for me to assess students’ knowledge of timetable facts with regard to group 1 or group 2. At this point students should be comfortable knowing how to work out 6 x 8 say, but I want to assess whether they immediately know that it’s 48 without having to resort to a method like 5 x 8 + 1 x 8. Once it’s identified that a timestable fact is in group 2 and not group 1 it’s then a case of doing a lot of retrieval work and further specific timed practice, while also consistently emphasising and returning to the conceptual knowledge.
I do expect that students will take some time to get used to the reflect grids, it is a different structure and they have to engage with it for it to be useful. There is a need for clear explanations of the task, consistent expectations and routine building. Timed practice may prove challenging for the very weakest students so it is important to embed conceptual understanding first. Timed practice will require a strong and positive classroom culture. But I am hopeful that these approaches together will lead to significantly improved progress with both students’ conceptual understanding and their ability to immediately recall timestables facts.
It is essential that our students have a fluent understanding of timestables facts and that they can recall these with ease. Students must learn timestables through effective proven strategies, like derived facts, and not rely on methods with high cognitive demand, like listing multiples. Without teaching these strategies explicitly and without utilising timed drills we are inevitably overloading their working memories and limiting their chances of success.
The reflect timestables grid template can be downloaded here.
Contact me on twitter @drstonemaths.