Making it Stick

Up until very recently, there was a big difference between learning and performance in the maths classrooms of the academy I work in. In this blog, I’ll explain the journey our school has been on to make learning in maths stick. There are no original, flashy ideas in here – they are all synthesised from educational research and far more experienced teachers than I, and I hope I have correctly referenced and credited those who have influenced my thinking.

The problem:

Our children’s maths books were great. They showed challenging tasks being successfully completed by most children. There was a mix of fluency tasks, problem solving and reasoning. Yet test results – not just end of key-stage SATs, but just about ANY tests we administered across KS1 and KS2 – showed very little learning was ‘sticking’. The children were performing, and not learning.

Concepts that should have been built on year-on-year had to be started from scratch each September. Take finding fractions of numbers as an example. The foundations are laid in EYFS by finding half of a number. By Year 2, children should be confident with the standard representation of a fraction and should be able to find non-unit fractions of number, such as two thirds of 27 or three quarters of 48. It is a skill that was covered (note – ‘covered’ is a deliberate choice of verb!) year-on-year up to Year 6. Yet every teacher across KS2 found that every time they revisited the concept it was met with a sea of blank faces, including the old favourite, “we’ve never done this”!

“What don’t they get?” went the typical staff room conversation. “It’s just divide by the bottom, times by the top!”

There we have exactly what they “don’t get”. Year on year, the children were told to find a fraction of a number by dividing by the ‘bottom number’ and then ‘timesing’ (another questionable word) by the ‘top number’. Having been given this shortcut, the children successfully worked through a page of calculations, and moved on. By the time the books were away at the end of the lesson, many children had already started to forget how to find a fraction of a number. But performance in books was very good. Was any actual learning taking place? No. No it was not. We had teaching that was for performance, and not understanding.

First try:

Around this time I switched from English lead to maths lead, and an Exec Head came in to work with me to develop teaching and learning. I was told teaching needed to be more memorable. I agreed. But it turns out we massively disagreed on what ‘memorable’ meant. I was encouraged (read: told) to ensure maths was more ‘fun’. Throw away those text books! Do your column additions in chalk on the playground! How can you make prime numbers memorable? Go apple bobbing for prime numbers! Yes, that one was genuine! We also had to link maths to the wider curriculum. I once spent a lesson with the mathematical aim of teaching children to be able to add large numbers including decimals by getting them to add up the cost of a round the world trip, using iPads. One hour of messing about on Sky Scanner later, the children had a thoroughly enjoyable time but hadn’t completed more than 3 column addition calculations between them. They were engaged; tremendously so. But still not learning.

After a term or so of this we realised – in summary - children remembered the fun activity of apple bobbing but didn’t retain that 17 was a prime number. Clare Sealy writes about the difference between memory and memories in her excellent blog: https://primarytimery.com/2017/09/16/memory-not-memories-teaching-for-long-term-learning/

After joining a MAT we decided to abandon the ‘engaging’ approach to maths and we decided to focus, rightly, on maths in its own right. This was the beginning of the change in culture from performance to learning. It took a long time to address that. Why? Because it’s more difficult. And it can be slower. And the books don’t always have as many ticks in. It was a complete change in philosophy, moving away from tricks and shortcuts to images, models, representations, connections and understanding. A sea-change in the approach to the teaching of concepts such as fractions would be a whole other blog, but it essentially embraced the idea of using concrete and pictorial representations before moving into the abstract. Rather than the best way to teach individual mathematical concepts, this blog will focus on the four significant changes in our approach to curriculum and pedagogy that helped make the learning actually ‘stick’.

Coherence:

We introduced the idea of ‘coherence’ - this is one of the NCETM’s ‘five big ideas’ for teaching mastery. The idea, in a nut-shell, is that “small steps are easier to take”. Rather than rushing in to a concept, we now spend time breaking down the small steps required for deep understanding and introduce them one at a time. For example, in KS1 before going straight into the concept of a half, we spend a lesson on understanding the concept of wholes, and parts of a whole. Then we spend a lesson looking at what equal parts are, before even introducing the term ‘half’. This means that when we come to define a half, the children truly understand the idea of something being split into two equal parts.

Coherence for us also meant that each lesson in a unit built on the lesson that came prior. This way, through the natural course of teaching, Friday’s lesson revised Thursday’s concepts, which revised Wednesday’s concepts, and so on.

The NCETM call this ‘coherence’, but these concepts are not new or unique to an approach advocated by the NCETM. They are grounded in research and are part of Barak Rosenshine’s 17 ‘Principles of Effective Instruction’ (Rosenshine, 2012). The two relevant to coherence are:

• ·      Present new material in small steps with student practice after each step
• ·      Limit the amount of material students receive at one time

Review:

Linked to the idea of coherence, and a further one of Rosenshine’s principles is to “begin each lesson with a short review of previous learning”. We introduced this daily review as a ‘Do Now Activity’, a strategy taken from Doug Lemov’s Teach Like a Champion 2.0 (2013). The children begin each lesson with an activity based on yesterday’s learning (which will be essential to today’s new learning). The task must be something that the children can do themselves; this frees the teacher up to deliver precision teaching to those they know didn’t quite master yesterday’s work, thus making sure they can catch up in the coming lesson.

Towards the end of this year, I found a tweet from headteacher @ShayGibbons1 who, rather than just reviewing yesterday’s learning, uses four short tasks based on learning from yesterday, last week, last block and last half term. I began trialling this with my Year 2 children and it’s something we will be rolling out next year to make the review part of our lessons even more rigorous. This also links strongly to the next concept in the blog – interleaving.

An example of a Year 2 ‘Do Now Activity’ for review

There are countless other ways to review prior learning; these are just the main ones that we have used. The principle remains the same – regular, low-stakes quizzing improves memory. Other great ideas are the White Rose diagnostic questions, creating a Kahoot quiz and using past SAT questions, to name but a few. Aidan Severs (aka That Boy Can Teach) has also written a great blog on ‘No Quiz’ retrieval practice which is full of great ideas: http://www.thatboycanteach.co.uk/2018/06/no-quiz-retrieval-practice-techniques.html

Interleaving:

We still had a curriculum that put things ‘in their box’. So, for example, Year 1 might do two weeks on shape in the summer term. Here, the whole of the shape curriculum would be covered – very well, and in great depth – but then it was never revisited. This had to change, so we looked to create links across the curriculum while still having a blocked approach. A simple example would be in Year 4, during the fortnight block on statistics, ensuring that some of the graphs chosen included negative numbers thus reinforcing learning from the previously studied place value block.

Our work on problem solving also needed adapting if we were going to make learning stick. I had seen many children sleepwalking their way through ‘problem solving’ (term used lightly here), because at the top of their worksheet was the title ‘multiplication problem solving’. So rather than thinking about what the underlying mathematical structure of the questions, children were just picking out the two numbers from the question and multiplying them. Again - performance in books was very good. Any actual learning taking place? Still, no.

We used Craig Barton’s (2018) idea of ‘Same Surface Different Deep’ (SSDD) problems to improve problem solving. Craig writs about them far better and in more detail than I can about them in his book and on his website – https://ssddproblems.com/  – but the theory behind this technique is that finding the correct solution to most mathematical problems involves two steps: identify the strategy needed to solve the problem, and then successfully carry out that strategy (Roher, Dedrick and Burgess, 2014). By giving children lots of different problems with different surface features (names of characters, settings, objects, etc.) but the same underlying deep structure (e.g. calculating perimeter) then we are denying children the opportunity to identify the strategy needed to solve the problem.

An example of SSDD problems we’ve used in KS1

Craig Barton’s SSDD Problems tend to have one stimulus and four different questions around the outside, like the example given. But this isn’t the only way to build interleaving into the curriculum. It can be as simple as after a lesson on multiplication, rather than giving children 12 multiplication word problems, swap some so that they are based on other concepts studied, such as addition of subtraction. Or weave in other areas of the curriculum previously studied, such as the KS1 problem, “If I have three octagons in my bag, how many sides are in the bag”?

A great example came from the White Rose scheme of learning for Year 1, where during the positional language block, children had to use words such as ‘below’ and ‘next to’ to describe the position of a 50p coin relative to four other, different coins. This built on and reviewed their measurement work, as one of the Year 1 objectives is to recognise the value of the different UK coins. Interestingly, if I had given the children this task before teaching the measurement block, this task would have very likely caused cognitive overload, as the children wouldn’t be fluent in recognising the values of the coins which would detract from the actual learning, which was using positional language. Sequencing is very important!

Fluency

It is obvious to anyone who has taught in Upper KS2 that children who know their multiplication facts do better at long multiplication. This seems fairly intuitive. But I never really understood the science of why this was. Then one day I read about cognitive load theory, and this explained it. Craig Barton has written extensively about how cognitive load theory can be applied to the learning of mathematics here: http://mrbartonmaths.com/teachers/research/load.html

Children working on long multiplication - for example - who don’t know their times tables will have to use their working memory to calculate each individual multiplication, rather than pulling it from long term memory, where it would be stored if it was known ‘by heart’. The cognitive load this places on working memory means that there is less capacity for students to think about the structure and process of long multiplication. Less time thinking about long multiplication means that it is less likely to be learned; after all, “memory is the residue of thought” (Willingham, 2009).

To combat this, we introduced a rigorous system for learning number facts. There are many out there – or you could just make your own – but we went for ‘Rapid Recall’ from Focus Education (slightly tweaked to match the 2014 NC). But fluency is more than just learning number facts by heart. Russell (2000) defines fluency with numbers as having three elements: efficiency, accuracy and flexibility. To help make learning stick, as a school we’re now focussing on flexibility, defined by Russell as “the knowledge of more than one approach to solving a particular kind of problem.”

Quite often as a school we would teach children one way to solve a problem, in the (misguided) belief that introducing the children to more than one would be confusing. But this doesn’t lead to true fluency. This year, looking at returned SAT scripts from the arithmetic paper we can see children using inefficient strategies. Hardly anyone spotted that 0.5 x 28 was just finding half of 28; most children did a long multiplication. My colleagues and I also had an interesting debate about the question 28% of 650. All three of us would have solved it a different way: I would have found 25% (a quarter) and found 1% by dividing by 100 and added 3 1%’s to 25%. My first colleague would have found 10% and 1%, doubled the 10%, multiplied the 1% by 5 and added the two totals. The other colleague would have found 30% and subtracted 2%. All are efficient and credible. We decided that we needed to give the children more tasks where they must find multiple solutions to problems and critique the approaches for efficiency. This is our next challenge when it comes to making learning stick.

Summary:

After a long journey, we are finding that children are now retaining what has been taught, even after longer periods of time. The four principles of coherence, review, interleaving and fluency have made that difference for us. We are not at the end of our journey; attainment at the end of KS2 has risen from 40% of children achieving ARE in 2016 to 71% in 2018 – still below national but just by 5% now. And we are getting more children reaching this expected standard than we were getting achieving Level 4 in 2015 and earlier. I hope this blog has been of some use to you, and I’d like to thank the authors of the blogs, journals, websites and books mentioned for their ideas which helped to transform our maths teaching.

References:

Lemov, D. (2013). Teach like a champion 2.0. San Francisco, Calif.: Jossey-Bass.

Rohrer, D., Dedrick, R. and Burgess, K. (2014). The benefit of interleaved mathematics practice is not limited to superficially similar kinds of problems. Psychonomic Bulletin & Review, 21(5), pp.1323-1330.

Rosenhine, B. (2012). Principles of Instruction: Research-Based Strategies That All Teachers Should Know. American Educator, Spring 2012 pp.12-19,39

Russell, S. (2000). Developing Computational Fluency with Whole Numbers. Teaching Children Mathematics, 7(3), pp.154-158

Willingham, D. (2009). Why don't students like school. Hoboken, N.J.: Wiley.