Keen to quickly find an answer, pupils will often skim-read the information given in a question, focusing mainly on the numbers provided.

They look for the first clues as to the correct operation to perform, often making errors or missing important steps. 

In contrast, an expert problem-solver may take longer processing and representing the information that is given in a question.

They can consider whether an answer is realistic. They may even have the capacity to course-correct if they make a mistake. 

Worded maths questions

The research paper ‘Removing opportunities to calculate improves students’ performance on subsequent word problems’ (Givvin, Moroz, Loftus & Stigler, 2019) has helped me to see how all children can learn to break down word questions.

If pupils are given all the information in a word question in full, they have a lot of data to process all at once, which can be a significant barrier.

Also, children often don’t think about the mathematical structure of the question as they quickly try to calculate the answer.  

But what if the information in a worded question was revealed more gradually? This limits the amount of detail that needs to be processed in any one moment.

Also, by making it impossible to calculate an answer straight away, children are forced to think about the structure of the question.

They can predict the extra information that they might need. Or given all the information, children can guess what the question could be.

We might not show numbers, so children can calculate different possible answers.

Suddenly, what was previously a closed question becomes an interesting, open task! 

Let’s consider an example ‘slow reveal’ prompt, where to begin with we’ve concealed parts of the question behind coloured bars, like this: 

Step 1

Pencils: 20p Rubbers: 15p 

Here, I can start by reading the information and asking, ‘What could the question be?’ and, ‘If that’s the question, what would the information be?’ 

The children might suggest it’s, ‘How much does it cost?’ or, ‘How much change does she get?’.

We can model some simple question-answer combinations as examples. 

Step 2

Pencils: 20p Rubbers: 15p 

How many rubbers can Jen afford? 

The question is revealed. To be able to answer this question, what information is needed? What could the missing information be? 

At this stage, we can recognise that we will need to know how much money Jen has.

We might ask a question like, ‘If the missing information is Jen has 50p, what would the answer be?’ 

Step 3

Pencils: 20p Rubbers: 15p 

Jen has £1. She buys ___ pencils and some rubbers. 

How many rubbers can Jen afford? 

Now we have all the information, but one number is still hidden. This allows us to calculate different possible answers. Some children might find all the possible answers!  

We could represent one example answer with a bar model: 

Jen could buy 4 pencils and 1 rubber 

Step 4

Pencils: 20p Rubbers: 15p 

Jen has £1. She buys 3 pencils and some rubbers. 

How many rubbers can Jen afford? 

Finally, we can reveal question in full. By now, all children have a much deeper understanding of the question. 

Division KS2

A classic division context for children in KS2 maths is to explore is eggs in egg boxes.

Typically, for a given number of eggs, we will ask, ‘How many egg boxes do we need?’.

Often, we want pupils to identify from the context what to do with any ‘remainder’ eggs.

Are they supposed to round up or round down? Here’s an example of how to explore this context.

At each stage, you can use counters to model the calculation if required. 

Step 1

The farmer packs ____ eggs into boxes. 

Each egg box can hold 6 eggs. 

Here, we can ask, ‘What could the question be?’ or, ‘What different questions could be asked?’. 

Step 2: 

The farmer packs ____ eggs into boxes. 

Each egg box can hold 6 eggs. 

How many egg boxes can we fill? 

Now that children know the question, they can give a possible answer. Often, they will choose to work in a number range that is comfortable for them, for example choosing 12 eggs. 

Step 3

The farmer packs ____ eggs into boxes. 

Each egg box can hold 6 eggs. 

How many egg boxes can we fill? 

Answer: 3 boxes 

This time, rather than giving the number of eggs, we could give children the answer and ask, ‘How many eggs could there be?’.

Now there are different possible answers. Can children find them? There could be 18 eggs. But there could also be as many as 23 eggs! 

Step 4

The farmer packs ____ eggs into boxes. 

Each egg box can hold 6 eggs. 

How many egg boxes do we need to hold all the eggs? 

Answer: 3 boxes 

For the final step, we make a slight change to the question. Now, if there are any remaining eggs, we round the answer up.

By making only a small change to the question, and by keeping all the other information the same, children can see the difference between the two questions.

You can change the challenge in the number range used as appropriate, and again the answer(s) can be modelled using counters. 

By revealing the information in word questions slowly, or by making small changes from one version of a question to the next, we help to slow down children’s thinking and create rich opportunities for discussion.

In using this technique, I have seen pupils experience less anxiety when working through word questions, as they have had more time to process the information.

It can also support children with English as an additional language (EAL) to identify the meaning of key terms.

Finally, it helps all pupils to understand and explain the process of breaking down a multi-step question.

I hope it’s a useful tool for you too! 

Gareth Metcalfe is director of I See Maths Ltd. Follow him on Twitter @gareth_metcalfe  

The post KS2 maths – Make word questions more approachable appeared first on Teachwire.

When I describe maths in this way, you probably think of your favourite problem-solving task, and rightly so. I love the buzz of the classroom while children grapple with an exciting logic puzzle or an open-ended Nrich task.

But what if we could take some of the traditionally more routine aspects of maths and open them up to more wide-ranging ways of thinking too? Even in addition and subtraction!

Adding and subtracting

A big emphasis is placed – quite correctly – on developing children’s mental fluency in addition and subtraction within 20. So, for example, we hope that learners develop a range of strategies for deriving the answer to 7+5, supported by practical experiences and visual representations, like 10-frames.

We encourage pupils to make a 10, seeing the 5 partitioned into 3 and 2, and want them to notice that 7+5 is 2 more than 5+5, by partitioning the 7 into 5 and 2.

We may also draw out that 7+5=6+6 by moving one counter from the left 10-frame onto the right 10-frame. Now we can use our doubles facts, too, in order to find the answer.

Similarly, let’s consider the different strategies that children can use for calculating 13 – 8. I like to show children 13 on two 10-frames. Then I ask them where they visualise the 8 being subtracted from: the left 10-frame, the right 10-frame or both 10-frames?

Subtracting from the left 10-frame encourages a ‘counting-on’ thought process. When subtracting from both 10-frames, pupils are encouraged to see the parts of 3 and 5 being subtracted. Again, we can promote different calculation strategies here.

Mental maths methods

By developing a range of efficient mental methods for addition and subtraction within 20, children will become genuinely fluent and we can encourage them to start choosing the best method for any given calculation.

How, then, can this ethos be continued when pupils progress to adding and subtracting two- and three-digit numbers? If we revert to working predominantly in the abstract, children often end up using written methods to perform every calculation.

But in coming back to the visual representations, and by celebrating the different methods that learners can use, we continue to develop these flexible, diverse calculation strategies.

For example, when representing 27+26 as quantities in 10-frames, maybe children will suggest moving 3 counters to change the calculation into 30+23.

Perhaps they will suggest adding the 10s then adding the 1s. Or maybe they will also recognise that they could calculate double 25 plus 3.

Again, all these strategies can be modelled and celebrated. From this foundation, as children progress into performing a calculation like 297+154, they can come to recognise that the answer will be equivalent to 300+151.

Think big

These big ideas can be explored further even as pupils practise calculation methods. Consider, for example, this sequence of three questions:

573+245=

543+275=

537+281=

Children are likely to calculate the answers using a column method. And if they answer the questions correctly, what will they discover? That all the answers are the same!

Why? Well, the only difference between question one and question two is the position of the 10s values within the addends. This doesn’t affect the sum. And we can explain the link between question two and question three by explaining that 537 is 6 less than 543, and 281 is 6 more than 275; therefore, the answer is the same!

By putting the questions in this sequence, as well as practising their calculation skills, children are exposed to these big mathematical principles and have more opportunities for reasoning mathematically.

The key idea that I focus on in subtraction is ‘constant difference’. So, for example, we might start by giving children sequences of questions like this…

9-6=

8-5=

7-4=

…and then using a visual representation to show that when the minuend (the number from which we’re subtracting) and subtrahend (the number to be subtracted) increase or decrease by the same amount, the difference between the two numbers remains the same.

This is a pattern that I will keep coming back to throughout lessons, and this idea can then be explored in a larger number range. For example, we can recognise that 41–18, 43–20 and 39–16 all have the same answer.

However, we might also see that some of the calculations are easier to perform than others!

Now, let’s consider another short sequence of questions that can draw children’s attention to these big ideas, as well as giving them the opportunity to practise their written calculation methods. Have a go at answering these three questions:

558–233=

564–239=

584–219=

The first question can be calculated relatively easily using a column method, as regrouping is not needed to find the answer. Question two does require regrouping, but having found the answer to question one, children might notice that the answer to this is actually the same.

Why? Because the minuend and the subtrahend have increased by the same amount. And for question three, pupils may expect the answer to be the same as question two if they recognise that the minuend and the subtrahend both change by 20. But, of course, the minuend increases by 20, whereas the subtrahend decreases by 20.

This means that the answer to question three will be 40 more than the answer to question two. Children might even notice this before they carry out the calculation, but they normally complete a written calculation to check their theory. This is such a rich opportunity for reasoning and pattern spotting.

Of course, for children to have sustained success they need these methods and approaches to be promoted consistently, year upon year, using visual representations and reasoning techniques.

It’s not a fast road, but it promotes rich and diverse thinking. By focusing on these big ideas and relationships, children are given the tools to approach mathematics with curiosity and flexibility, and we tap into a deeper expression of what it is to be a true mathematician. For me, these ideas have brought previously dry aspects of the maths curriculum to life!

Gareth Metcalfe is the director of I See Maths and author of the I See Reasoning and I See Problem-Solving eBooks. Follow him on Twitter @gareth_metcalfe

The post Primary school maths – use visual representations to make adding and subtracting more interesting appeared first on Teachwire.

When they notice a pattern and want to explore it further, or when they go off and create their own thought-provoking questions. These moments are precious and uplifting, encapsulating the joy of being a mathematician.

I used to look forward to the end of a sequence of lessons, once children had mastered the basics, when these moments were more likely to happen. Then the rich mathematical learning could really take place!

But what if children could experience more of these moments in day-to-day lessons? Could there be more space for reasoning whilst learning new concepts or practising calculations?

Mathematical reasoning

My eyes were opened to this new possibility by Craig Barton’s brilliant book, Reflect, Expect, Check, Explain, which shows the power of sequences of questions for stimulating mathematical reasoning. As my Pilates teacher says, ‘experience trumps everything.’

So let’s start by answering this little sequence of questions:

A. ¹⁄¹⁰ of 40 =

B. ¹⁄⁵ of 40 =

C. ²⁄⁵ of 40 =

D. ⁴⁄¹⁰ of 40 =

Which relationships did you notice?¹ Did you anticipate these relationships, or were any of them a surprise?

For example, the answer to question C and question D is 16. Why is the answer the same for both questions? That’s because ²⁄⁵ and ⁴⁄¹⁰ are equivalent.

But did you notice that before you calculated the answers? I didn’t!

On one level, this sequence of questions allows children to practise the skill of finding fractions of quantities. But there’s also scope for making predictions and explaining relationships.

To deepen, children can have a go at extending the sequence. For example, can they write two other fraction of quantity questions with an answer of 16? It’s not easy!

Find the pattern

I try to show children how to approach these sets of questions. I train them to look for the similarities and differences between the question they are answering and the one before.

I explicitly model the thinking behind making predictions (these predictions do not have to be correct). I show how we can explain the relationships that we find, sometimes using sentence stems to scaffold learning.

Then we explore how sequences can be extended to show our understanding of the mathematical patterns.

Time for another example. Have a go at this sequence:

A. 28 ÷ 7 = ___

B. 28 ÷ 7 = ___ + 1

C. 28 ÷ 7 = ___ ÷ 1

D. 28 ÷ 7 = ___ – 2

E. 28 ÷ 7 = ___ × 2

Once you have completed the sequence, check the answers² and think about the observations that children might make (or that you can lead them towards).

For example, would children expect the missing number in question B to be less than the missing number in question A? Or can we explain why questions A and C have the same answer? 

Then, by creating their own sequences of questions, children can show their understanding in a way that’s unique to them. You might even get some fantastic questions that you can use in your next lesson.

Keep it simple

The key principle in creating a sequence is to limit the number of differences between the questions. If only one thing changes from one question to the next, children can directly see the effect of each change.

This is like running a science experiment, where only one variable is adjusted. In contrast, if lots of things change from one question to the next, children will simply answer each question in isolation.

When I used to teach the order of operations to my Y6 class, I would teach the children the basic rules. For example, rather than carrying out calculations from left to right, we do the bracketed calculations first, then we do the division/multiplication, and finally we carry out the addition/subtraction.

I used to give the children a range of unrelated questions to work through. Now, I use sequences of questions like the one below. The questions, without being harder in themselves, take us a little deeper. Have a go!

A. 20 – 5 × 2 =

B. 20 – (5 × 2) =

C. (20 – 5) × 2 =

D. (5 – 2) × 20 =

E. 5 – 2 × 20 =

Before reading on, make sure you check your answers!³

For simplicity, I use the same numbers and operators for all the questions. Also, I try to minimise the difficulty of the calculations, as I want children’s limited attention to be directed towards spotting those key similarities and differences.

For example, the answers to questions A and B are the same because the brackets in question B do not affect the order of the operations. However, the position of the brackets in question C does change the answer. Why? Because the brackets here tell us to carry out the subtraction first.

I sometimes like to throw in a curveball too. Here, the answer to question E is a negative number. For this sequence, an effective extension task is to ask children to find the largest and the smallest possible answer using the same numbers and symbols. It takes a bit of thinking…⁴

It’s so exciting when children get used to tackling sequences of questions, especially when you have taught them how to attack the questions skilfully.

There is such a buzz as pupils make connections and explain their thinking. So much reasoning can be drawn out from so few questions! And I can use this technique early on in a series of lessons too, so long as children have the skills to carry out the calculations.

It is an approach that has given me so many great classroom moments. I hope it can be useful for you too!

¹Answers: 4, 8, 16, 16 ²Answers: 4, 3, 4, 6, 2 ³Answers: 10, 10, 30, 60, -35 ⁴Largest: 20 × 5 – 2 Smallest: 2 – (20 × 5)

Find Gareth on Twitter @gareth_metcalfe

The post Sequence maths – helping children spot patterns appeared first on Teachwire.