Charlie Harber is the Deputy Lead Adviser for Primary Mathematics at Herts for Learning.  She has  researched the positive impact bar modelling has on pupils’ access to worded problems.

Recent analysis in many schools and discussion with subject leaders confirmed what many teachers have long suspected, that many children have the procedural skills but they  seemed to abandon all reasoning  when they need to apply them once they are embedded in a word/story problem. Many schools in the UK use RUCSAC to help the children, but have you considered why that doesn’t work? Is there a better way, one which just doesn’t prepare them to answer questions in tests, but also deepens operational understanding, exposes misunderstandings and develops reasoning – empowering the children to discuss the mathematics?

Simply put, yes I think there is a better way – bar modelling.

Why are word/story problems important? Why do we bother with them?

Word problems are not a means to test procedural understanding, they are more critical than that. It is important that children are able to identify the mathematics in a story problem as this supports them in translating their abstract calculations to genuinely real life situations, helping them clarify ‘Why am I learning this?’ and ‘When will I use this?’ When exploring word problems children should be engaging in creative thinking and exploring different solutions in a safe environment, learning the skills of team work, evaluation and persuasion; skills that business are looking for in their employees.

Are you able to work effectively in a team? That’s the skill employers most want when they are recruiting new college grads. The next most important skill: ability to make decisions and solve problems. Susan Adams ,  Forbes Staff

Additionally, they should be interweaving different aspects and domains of mathematics exploring across the whole of the maths curriculum. Moving mathematics from individual islands– this week we are learning ‘addition’ – to exploring the whole of the maths landscape, travelling different journeys with understanding.

So how do we support the children to access word problems?

A common solution is to use RUCSAC , a quick google search returned over 5,500 hits for RUCSAC display posters – that’s a lot of posters, all saying the same thing, I will admit they are pretty.

So how does RUCSAC work?

First READ the question – ok, let’s assume the children have managed that. But we do mean comprehend not just decode.

UNDERSTAND – how do we help the children understand? What does it mean to understand? How do we know if they understand? More importantly how do they know if they have correctly understood the problem? Find the key words? What are the key words? So what do the children do? – they underline any/all numbers and look for clue words or trigger language.

Let me be honest here – this doesn’t help, telling the children to hunt and find words isn’t fair, it all depends on the context of the word.

Many children switch off at this point, they don’t know what they are supposed to be answering and they give up. Who can blame them?

If they don’t understand what is happening, then we should not be moving them on to identify the calculation… they will apply whatever method you have been exploring most recently in class. They have no other choice, and if all of your word problems are a piecemeal approach they will be successful. Never having the opportunity to develop the critical and creative skills required but appearing on a surface level to be progressing well. Pages of correct solutions, never challenged and never debated.

 

Let’s reconsider the approach. If we can encourage the children to take the information one sentence at a time, then they can gradually build a picture of the information. This removes much of the anxiety surrounding word problems; we are no longer asking the children to provide us with THE ANSWER but for them to deal with information in a controlled way, a consistent structured approach which can grow with the information. As teachers we can then see where is the understanding breaking down

  • is it a calculation error?
  • Is it a misunderstanding of the language?
  • Is it a misconception?bar1

Let’s look at a question from this year’s KS2 SAT Paper 3 fraction problem, nationally 34% of children got this correct. Why so low? What is so challenging about this problem? Is it the numbers involved? No, they are low. Is the fraction referenced? Well its only three-quarters which is introduced in KS1. It is exploring whether children really understand fractions, and we don’t need complex mixed number fractions to do that.

  1. Lara has some money. What no start number? We are only told that she has some money. SOME money – how do we deal with that?
  2. She spends £1.25 on a drink. She spends £1.60 on a sandwich. There was no complex calculation required to see that in total she spends £2.85p.
  3. She has three-quarters of her money left. Three quarters has been written in WORDS, not in symbolic form. And this is what she has LEFT. What does this mean? Does £2.85 = ¾ ?
  4. How much money did Lara have to start with? How do we find this out? We have only been told that she has some money.

For many children there didn’t seem to be enough information in the problem to solve it also as the problem started with the unknown so they didn’t have an access point.

How would bar modelling approach this situation?

  1. Lara has some money

Ok. We haven’t got a number to start with so let’s represent it with a bar – it can be of any size but let’s make it big enough to put information on if we need to, then let’s label it so we don’t forget what it is.

bar2

  1. She spends £1.25 on a drink . She spends £1.60 on a sandwich

So we can add this information to the model? It must be less than how much she has got to start with – but we still have no idea how big the SOME is. We could label this ‘money spent’.  PLEASE REMEMBER THERE IS MORE THAN ONE WAY CHILDREN MIGHT ADD THIS INFORMATION.

bar3

  1. She has three-quarters of her money left

Now do we have some new information? Yes. How can we add it to the model? What does three quarters look like?

bar4

But she has three quarters left, where is that on the model? Can we label the other quarter? What does that represent? It must be what she spent.

We know how much money she spent so we can link the bars together.

bar5

From the model we can see that we need to solve £2.85 x 4.

bar6

Now how the children decide to tackle the £2.85 x 4 takes us into multi strategy discussion – there are number of different journeys possible, repeated addition, double and double again, times by 5 and remove one group, short multiplication and so on… interesting thoughtful discussions.

Bar modelling isn’t a magic bullet, it won’t overnight give your children a super power, but it does provide a consistent, logical and secure approach to EXPLORING contextualised mathematics. I have seen the confidence levels of children raise again and again, from refusing to start to getting stuck in and talking about the different models in the room. And don’t even get me started on the language/reasoning outcomes…

At HfL we believe so strongly in the power of the bar that we have developed a year by year WHOLE school progression from pre-operational learning to Year 6, addressing many of the learning statements and showing how you can use the bar not only to crack word problems but to challenge, support and to develop deep conceptual operational understanding (using the CPA approach) of maths.

Believe me, this is a bar worth visiting!

bar7
Sample page from HfL Progression in Bar Modelling document

 

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References

http://www.forbes.com/sites/susanadams/2013/10/11/the-10-skills-employers-most-want-in-20-something-employees/#10bd73f8752d