Nicola Randall is a Primary Mathematics Adviser for Herts for Learning where she has been working on closing the gaps projects with disadvantaged learners, working with schools to champion their needs. She has worked on behalf of the Virtual School.
Over the past few years, I have been working with schools across Hertfordshire to accelerate the mathematical progress of pupils in receipt of the pupil premium funding. The DfE research paper (November 2015) suggests that the 3 strategies which have the greatest impact on the attainment of disadvantaged pupils are: paired or small group additional teaching, improving feedback and one-to-one tuition. Whilst these are clearly helpful for closing gaps in understanding or knowledge, sometimes it’s the small tweaks to whole class practise that can make the biggest difference. Having worked alongside teachers, I have looked in great depth at many pupils’ work and together with teachers we have reflected on how the pupils conduct themselves during maths lessons. What is clear, is that each individual pupil comes with their own experiences, strengths and areas of difficulty yet interestingly, through this approach, I found there were some common barriers holding this vulnerable group back in mathematics.
What I will aim to capture in this article is 5 common difficulties and details of the tried and tested strategies that have contributed to overcoming these barriers. I do not proclaim to have found the snake oil or the magic wand, but hopefully, by sharing what has worked for some, might just help others. And the good news is that many of the strategies target several of the barriers at once.
5 Common barriers in mathematics
- Mathematical language
- Making connections in learning
- Solving multi-step worded problems
- Recall of facts
Barrier to learning: Mathematical Language
Top of the list for a good reason! This is the lesser-spotted barrier for many disadvantaged pupils. On the surface, this may not appear to be an issue (especially if English is not their second language) but children are very good at developing coping strategies, such as waiting until others get started, to confirm what they need to do or to look for patterns to be able to replicate. Consider the following question:
Tickets for a school play, of which there are 230 in total, cost £5 for adults and £1.50 for concessions. If 50 children and 120 adults bought a ticket each, how much money would the play make? How many tickets would be left un-sold?
There are two fundamental issues with accessing this problem. The first is the vocabulary. If pupils have not had the experience of attending events, such as the theatre, they may not know that a concession is a child. It requires them to identify that it’s a child’s ticket from the following sentence, where it is implied.
Secondly, the language structures in this sentence are quite advanced. There is an embedded clause with the phrase ‘of which’ providing crucial information using formal language in a way they may not be used to seeing.
On top of all that, there are several steps. According to Newmans Error Analysis (1977) ‘transformation’ and ‘process skills’ account for 70% of all errors in worded problems. The pupil is required to transform the written words into a process, identify and then apply the correct mathematical skills, which is the trickiest bit of all. The working memory involved in this worded problem is overwhelming for many pupils, let alone those whose language experiences are limited
Link to English lessons to provide some cross curricular examples when exploring elements such as embedded clauses. The same techniques used to explore sentence structure can be used for maths too. Try writing each sentence on a separate piece of paper. Explore: How does the order of sentences affect the overall question? Which bits are statements and which are questions? How can this information be used to find a starting point to solve the problem?
To build a wide range of mathematical vocabulary, be aware of the words that you will be using within the sequence and ensure they are being used regularly and consistently, alongside images to help reinforce their meaning.
Pre-teaching, if organised effectively, can be one of the most effective forms of differentiation in the teacher’s toolkit. By introducing key words before the session, pupils have the knowledge required to access the learning in class. In addition, the fact that the ‘catch-up’ happens before the session means that it is a much more positive experience for the pupil, which will help to build confidence. Explore mathematical vocabulary with pupils by linking it with other learning, hearing the way it is pronounced, drawing an image of it, consider ways of helping to remember the word, compare it to other similar words and finally putting it into a problem solving context so pupils can see its relevance.
Barrier to learning: Estimation
Many of the pupils I have worked with struggle to make suitable estimates. They do one of two things: either they pluck a random number out of the air which, if they’re lucky, might happen to be somewhere near the answer or they solve the question to find the actual answer and then write their ‘estimate’ afterwards.
The skill of estimation starts with having a good understanding of number magnitude – how ‘big’ a number is in relation to others. Things to watch out for are when pupils plot numbers on an empty number line without leaving appropriate spaces in between the numbers. They might also put 0 at the start of the line, regardless of what numbers they are trying to represent.
The second reason pupils are not particularly strong with estimating is that they do not always see the purpose – it’s not all about checking solutions. Pupils will need to use this skill to problem solve, apply into decimals and fractions and apply during calculation strategies.
Strategy – Do not assume that pupils will instinctively know how to estimate effectively: Teach the skill of estimation explicitly. Start by building 2 numbers, like 12 and 25, using a base 10 resource like Dienes and making comparisons. Place the 2 models next to each other to get a clear view of what is different and articulate this using a talking frame:
I can see…
This is similar to…because…
This is different to…because…
When I compare…I notice…
Now that the pupil has a point of reference, you can begin to ask them to estimate by visualising the base 10 resource. Ask questions such as 2 x 25 to begin with and ask them to estimate how big the answer might be, then build up to larger, more complex numbers.
In order for pupils to make an estimate, they must make a comparison in their mind. For example: How many balloons would it take to fill the empire state building?
There are several strategies that can be applied to estimate this answer. Pupils could visualise one floor, count the balloons on the surface then multiply to find the volume of one floor, before scaling up for all the floors in the Empire State Building. Pupils might estimate how many balloons fit in their classroom and then scale that up. Both of these strategies require pupils to compare with something they know.
An activity you can practise every day to help pupils estimate is simply by using a strip of paper or an empty number line and asking pupils to mark on where certain numbers are in relation to each other. The proportional representation is crucial, as this helps them assimilate the size of the number.
Barrier to learning: Making connections
This is raised as a barrier time after time, and not just for our disadvantaged pupils. Unfortunately, this is often the result of maths being taught discretely and in ‘weeks’ of learning, where cross domain links are only made in a lesson at the end of the week which include worded problems (the next barrier discussed in the article). Pupils often display the ability to recall facts, such as times tables, during a lesson on that particular topic but as soon as the context changes, such as applying this information into volume problems, pupils are unable to recall the required information to solve it.
In lessons, pupils have the opportunity to use a number line to calculate and often use this model to make sense of decimals and fractions too, but the connection to other scales is often not made explicit enough. An example of this is linking scales to time. The analogue clock face is a scale, even though it is circular and continuous. The challenge with time is that there are several different elements all being shown on the same scale: seconds, minutes and hours. This is very confusing to pupils who have not had much experience of reading an analogue clock. My advice is to teach reading clocks through scales and then incorporate the other elements into daily life at school – lunch time, school timetables, home time and so on.
Strategy – Try to avoid teaching areas of maths in isolation. Unpick the maths which underpins the strategy or skill in order to identify the bits that are transferable. For example, when we multiply and divide by 10, 100, 1000, this is only the beginning. This skill is underpinned by knowledge of place value – the base 10 nature of our place value system to be precise. Pupils need to explore the fact that each place means the number has multiplied by 10 each time and understand what this means as numbers increase in size. Considering a range of representations to allow pupils explore this will help them construct their own understanding – simply being told ‘the digits move to the left’ is not enough for them to retain the information.
Using repetitive questions which encourage pupils to make connections has also had a good impact on pupils with this as a barrier. Questions such as:
What do you notice? If I know, what else do I know? Once pupils are trained to respond effectively to this sort of question, they automatically start to look at the big picture in new learning. Another tried and tested strategy is to ask ‘what’s the same, what’s different’ questions, as this provides the opportunity to expose pupils to 2 problems with the same underpinning concept, but in different contexts.
Barrier to learning: Solving multistep worded problems
This is the barrier which is most frustrating for teachers because often worded problems do not involve complex calculations. The challenge for pupils is working out what the question is actually asking. As Charlie Harber explores in her blog on RUCSAC, we know that this approach is limited but teachers are desperate for something, anything that might help pupils access worded problems.
The first thing to clarify is what we are talking about when we say ‘worded problem’ because to some extent, isn’t every problem worded? When I talk about worded problems, I am referring to everyday questions such as:
Peggy buys 8.5m of wool, which costs £1.50 per metre. How much has she paid for her wool?
If pupils are able to read the question, they must try to apply their knowledge and understanding of length, money, decimals, place value and mental calculation strategies to solve it. This is why this barrier is often coupled with difficulty in ‘making connections’.
Strategies for this barrier link with some of the other ones I have already mentioned , such as asking pupils to consider the big picture by asking ‘what do you know’? But there is one strategy that I have found to work wonders: The bar model.
The bar model is a visual approach to solving worded problems, not a method. It doesn’t give you the answer but it does help you figure out which bit of the question is the answer. It relates to understanding number as parts of whole and helps pupils consider the size of the answer in relation to other parts of the question. E.g.
Notice that each metre is equal in size £1.50 x 8 and then half of the last metre which must therefore also be half of £1.50. The idea behind this model is that pupils draw what the question has already told them. For more information on how the bar model can be used across the school, check out our new bar modelling progression available here.
Another helpful strategy is to break the problem up. Get pupils exploring if the order of the sentences makes a difference to the overall question, or does it change it completely? If the problem is accompanied by an image, use just the image to open a discussion about what elements of maths might be involved, before slowly adding in the other bits of information and eventually introducing the question. By that point, pupils will already have a good idea as to how they can solve it.
Barrier to learning: Recall of facts
We know that pupils need to be able to recall facts swiftly to help them solve mathematical problems, but often memory and retention gets in the way. Professor Jo Boaler (2015) emphasises the negative impact that timed testing before pupils have truly learned the facts can have on pupils’ ability to retain and recall facts due to the pupil developing maths anxiety. Schools may want to question why they continue to test times-tables on a weekly basis in this way before the facts have had time to embed? Before they are tested (as we now know they will be in 2019) we need them to understand and apply facts to solve mathematical problems efficiently – which is true automaticity.
Strategies: CPA, the term coined by Bruner (1966), refers to the process of learning and acquiring mathematics through: Concrete, Pictorial and Abstract. Every time new learning is introduced, pupils need the time to explore using a concrete resource, such as counters and Cuisenaire for repeated addition and equal groups. Pupils are then encouraged to draw diagrams and pictures to construct what they understand the resource shows. The teacher’s job is then to refine the images into mathematical models, such as the array, which the pupil will use alongside the abstract form 5×8 until the image has been committed to long term memory. This results in a pupil knowing the fact 5×8=40 with deep conceptual understanding and being fluent enough to apply this to other contexts and problems. The difficulty is getting the balance right between exploration and teaching for conceptual understanding and building fluency through repetition and procedural learning.
Pupils in Y5 and 6 who still can’t recall all their times tables out of sequence suggests to me that they probably never will. If reciting tables in order and having weekly tests hasn’t work so far, it’s not suddenly going to and we can read in this recent blog about the history of times tables testing that it never really did!
Change your approach – consider the use of the Slavonic Abacus:
his image is a variation on the array and is organised into 5×5 squares. This is helpful for a pupil being taught to use what they know to solve unknown facts. Get the pupil to explore how a calculation such as 5 x 8 looks on the Slavonic abacus by covering up the other counters, leaving just 5 x 8.
Identify (5×5) + (3×5) = 5×8 and (5×10) – (5×2) so if pupils know their 2s, 5s and 10s, they should be able to work out the other tables. Allow pupils time to keep coming back to this strategy and drip-feed into everyday maths lessons. It will take time but once the pupil has developed the strategy, they will get quicker and quicker each time they apply it.
The second strategy I have personally found to have the greatest impact on learning times tables is to understand and use commutativity. If they do, it instantly halves the number of facts they need to learn.
Key strategies for learning times tables:
· Memory hooks (the 9s finger trick)
· Think 10, think 5 (using known facts)
· Pattern spotting (both within a times table and across times tables)
· Doubles and halves (and near doubles and halves)
· Applying the commutative law (actually understanding it, not only writing the ‘fact families’ as this is all procedural)
The final strategy I want to mention I have saved until last because it is the most important. Are you ready for the snake oil…?
Ultimately, achieving accelerated progress for disadvantaged pupils, or any pupils for that matter, is only possible when the teacher knows the pupil inside out, takes the time to understand their barriers and has a high expectation that they can achieve.
DfE (November 2015) ‘Supporting the attainment of disadvantaged pupils: articulating success and good practice’
Boaler, J (2015) Mathematical Mindsets: unleashing student’s potential through creative math, inspiring messages and innovative teaching’ Jossey Bass
Bruner, J. S. (1966) ‘Toward a theory of instruction’ Cambridge, Mass: Belkapp Press
Harber, C (2016) ‘RUCSAC pack your bags, lets hit the bar instead!’ Herts for Learning blogs
Newman, M. A. (1977a) ‘An analysis of sixth-grade pupils’ errors on written mathematical tasks’ In M. A. Clements & J. Foyster (Eds.), Research in mathematics education in Australia, 1977 ( Vol. 2, pp. 269-287). Melbourne: Swinburne College Press.