Bar modelling – an introduction for parents

Over the last few years bar modelling has been increasingly used in our schools with all children from years 1 to 6. Bar Modeling is not new and is one of the heuristics, or problem solving tool which uses a practical method, used in the Singapore approach to teaching and learning mathematics. Associated in its current form with Singapore, Dr. Kho Tek Hung who worked for the Ministry of Education for over 25 years readily admits that he did not invent it, but rather refined and standardized the concept.Bar modelling was introduced in Singapore many years ago, to try to do something about the fact that Singapore’s children were poor at problem-solving, particularly word problems. They succeeded and problem solving is now at the heart of their curriculum! Their success is evidence based as Singapore is consistently one of the top performing countries in International comparison tests such as TIMSS and PISA ( Trends in Maths and Science Study for 15 year olds and PISA, Programme for International Students Assessment for 9-10 year olds and 14-15 year olds). In fact in the latest PISA tests Singapore was the top performing jurisdiction. Our Department for Education has researched high performing jurisdictions like Shanghai and Singapore, in their drive to improve standards of teaching and learning of mathematics with a focus on teaching for mastery in our schools. Here are the aims from our national curriculum for England, taken from the mathematics programmes of study. You will understand as you read this article how bar modelling is just one of the methods we can use to ensure that we are meeting the following aims

https://www.gov.uk/government/publications/national-curriculum-in-England-mathematics-programmes-of-study

Bar model drawing is a valuable resource to strengthen and deepen pupils’ understanding of mathematics, focusing on fluency and mastery. Bar modelling is used to develop pupils’ understanding of mathematical concepts, is a valuable problem solving tool and provides an important transition from the concrete to abstract concepts in key stages 1 and 2 and beyond.

Bar modelling supports children’s reasoning and gives children the opportunity to engage in mathematical conversations. The models will not do the maths for them but supports children in making connections, looking at what is the same and what is different and helps them to understand if they can do the question in a different way or to make up further questions of their own. The children transfer the maths they are doing onto the bars, the cubes or the pictorial representations. The models and pictures support their understanding and reasoning as they look at relationships. We bring the maths to the representations through reasoning. After time and practice. children eventually know which bar model to draw when they know which mathematical structures are being used in the question and whicih relationships are involved. Focusing on teaching the language of mathematics is very important, as this allows the children to access the maths within the problem.

When children focus on the question from an early stage it strengthens their understanding of the problem and helps guide their thinking and modelling.The hardest part of problem solving for children is what to do with the numbers. Models support their thinking, their reasoning and their problem solving

All children new to bar modelling need direct teaching and experience in the concrete – pictorial – abstract approach. Moving from the concrete to the pictorial to the abstract is called the CPA approach. This is not a one way linear journey and children will be moving back and forth at all stages of their mathematical career.

Concrete Pictorial Abstract

Children can start using this approach when they are in Foundation or Year 1 building models from concrete materials like cubes for example or drawing pictorial models. They will start with the discrete model which supports one to one relationships.

Theo has 4 red cars and 3 blue cars. How many cars has he got altogether?

Theo has got cars altogether.

This pictorial representation can then be made using multilink, Numicon, rods and transferred to abstract numerals.

4 + 3 = 7This is called a discrete model and is appropriate for children who are at the stage of understanding one to one relationships.

The bar model is just one approach to help children visualise the structure, make connections and solve the problem.

Here is an example of Sofia’s part - whole diagram supporting number bonds to 10 when she was in Foundation with Mr Poote. There had been a lot of conversation around making 10 and what this looked like with manipulatives and with abstract numerals. The children were making connections and asking questions of their own. You will notice that her three was the wrong way round and although it is ofcourse important that our

children know how to write the numerals, at this point it would have detracted from the wonderful conversations the children were having.

At any time, for differentiation children should be encouraged to find different ways to arrive at the correct answer and to discuss their findings. This strengthens their number sense and they learn from each other. Looking at Theo’s question what other questions could you ask?

It is generally agreed that there are four different types of models.

1. Discrete Model2. Part-Whole Model 3. Comparison Model 4. Change/Transformation or “Before and After’ model

1So, model drawing begins with discrete representation where one square represents each object in the problem. When children understand cardinality they know that the number they get to in the count represents the number of items in the group. This is irrespective of size of the items and order of the count. The cardinal value in Theo’s question is 7 because it gives the number of cars, regardless of arrangements, colour, size or the order in which the cars were counted.

When problem solving, children are encouraged to answer in full sentences. The sentence structure encourages the children to get a true calculation and not just to count and writing out the answer ensures that their answer is in the context of the problem. This approach encourages children to read the question and then write out the sentence with a space for their answer straight away. It is very important that the “thinking” is incorporated in with the doing and manipulating.

2.Following the use of discrete models, children move on to the continuous model or part whole model. In this model the lengths are not in proportion. In my experience with both children and adults, these models are less confusing when drawn on plain paper.

Take a look at this type of question, which can be supported with a part whole bar model.

Sofia has 7 sweets and Oscar has 5 sweets. How many sweets do they have altogether?

57

?

Sofia and Oscar have sweets altogether?

Consider what further questions you could ask, such as how many fewer sweets does Oscar have than Sofia? How do you know? Can you show me? How many sweets would Sofia have to give to Oscar so they have the same amount?

This model represents a quantitative relationship among three variables: whole, part1 and part2. Given the values of any two variables, we can find the value of the third one by addition or subtraction.

As mentioned previously, in this pictorial representation the bars do not have to be in proportion. The important thing is that the bar which is the longest represents the bigger number and for the person with fewer sweets, the bar is smaller. You will also notice the use of curly brackets. This notation is common practice. However curly brackets can be very difficult for our children to write so using rounded brackets is fine. The question mark is purposefully placed to indicate where the children should look and focus their attention, to find the answer to the given question.

3. Take a look at this question.

Nell has 12 books. Fenner has 10 books. How many more books does Nell have than Fenner?

Nell has more books than Fenner.

Since we are comparing the amounts, how many more Nell has is arguably much easier to see using a comparison model, when one bar is above the other.

Within the comparison models we have multiplicative models where the bars are of equal size. This supports ratio questions. These models help with understanding conservation, keeping things the same size and children’s knowledge and understanding of scaling and repeated addition.

For example;

Anya and Jordy share football cards in the ratio 2 : 3.If Anya has 12 cards, how many cards does Jordy have?How many cards do they have altogether?

Anya

Jordy

Nell

Fennerer

Once the size of one of the parts has been found then we know the size of Anya’s bar and the size of Jordy’s bar. Anya has 12 cards which is represented by 2 parts so we can work out that each part is 12 divided by 2 which is 6.So, Anya has 2 x 6 cards and Jordy has 3 x 6 cards.

Jordy has cardsJordy and Anya have cards altogether

4.

Here is an example of the change or transforming model used in before and after problems. These types of questions are generally trickier. Children may start by drawing one bar and then change or refine it to support further understanding of the problem. No models are incorrect so we would not want to see children crossing these out. They may not use some of them but that is fine, as the refinement in the models are supporting their reasoning and understanding of the problem.

The ratio of the number of Katie’s marbles to Emma’s marbles is 4 : 5. Katie is given 12 more marbles.

Katie now has twice as many marbles as Emma. How many marbles did Katie have to begin with?

Katie had marbles to begin with.

Before:

Katie

Emma

After:

Katie

Emma

To make Katie’s bar twice as long as Emma’s bar, 6 extra blocks are required. The extra 6 blocks represent 12 marbles so each one represents 12 divided by 6 which is 2. So, if each block represents 2 marbles then Katie had 4 x 2 at the start. Katie had 8 marbles to begin with.

Lev Vygotsky, a famous Soviet psychologist discusses that our development of reasoning is mediated by signs and symbols and that we learn through dialogue with others. The bar models support social interaction and mathematical communication which play a fundamental role in the development of cognition.

I have been very fortunate to work in Singapore with the MoE, NIE, teachers and children; also in Washington and Boston. I must thank Oxford University Press for these opportunities which were invaluable whilst researching my understanding of bar modelling and teaching for mastery and deep conceptual understanding.

I will follow up this piece with more questions for you to try if you want to. If you don’t want to try them, you can simply ignore the next instalment. If you do want to try, I suggest you enjoy them with a friend or friends over a cup of coffee.

Here is a taster question. This is a photo of the waiter serving a Singapore Sling. A word of caution, I have made up this recipe myself so I do not know how tasty or alcoholic it is. You may want to let us know

In my Singapore Sling there is twice as much soda water as lime juice and twice as much lime juice as cherry brandy.

There is three times as much gin as cherry brandy and twice as much cherry brandy as there is grenadine

If there is 60ml of lime juice, how large is the Singapore Sling?

If the bar tender adds one ice cube to every 25ml, how many ice cubes will be added?

Sue Lowndes. Professional development leader for mathematics at Oxford University Press