You, meanwhile, would be free to work with individuals or a small group, and at the end of the lesson you would get the other children’s work handed in already marked. Brilliant!

What’s more, imagine if the marking in your children’s books showed clearly what each child did on their own unaided and what they did in collaboration with someone else, so you had a clear record of learning that you, the child and the parents could all look back on.

Welcome to the world of autonomous learning…

We have used the 333 process successfully with children in my own school from as young as 8. IT REALLY WORKS – the secret is all in the careful training. Here’s how to get started.

Step 1: Introduce CEDRIC (see separate post)

Step 2: Choose a task (eg worksheet or text book page) with a series of straightforward questions on your current topic. Prepare another task (eg a solo game or problem sheet) which quick finishers can work on individually whenever they are waiting.

Step 3 : Arrange the pupils in like-speed groups of 3 (or 4) (ie fast workers together, slow workers together.) Explain that you are going to teach them a team strategy for doing and marking their written work.

Step 4: Talk through the process as follows:

-follow the 3-step process: plan together, work apart, check together- this ensures that everyone does their own thinking and no-one simply relies on the team

-work in groups of about 3

-plan how many questions you are going to do (eg 3)

-work independently to complete the questions you have agreed on and let your team members know when you have finished (then do something else as you wait)

-when you are all finished, compare answers – if you all have the same answer to a question, assume it is correct – if you have different answers, discuss and support each other in working out who has made the error

-when you are agreed, use a coloured pen or pencil to mark your answers right or wrong and correct any wrong ones by writing the correct answer next to the wrong one in colour – do not rub out wrong answers

-agree how many questions you are going to do next and repeat the process.

Step 5: Set the children working, then circulate and make sure they are following the process correctly, paying particular attention to the following:

– each group has planned together which questions they are doing

-they are working independently without collaborating when answering the questions – they are letting each other know when they have finished

-they are marking their work in coloured pen or pencil

-they are respecting wrong answers by marking them wrong – not rubbing them out!

-they are having in-depth discussions to make sure everyone understands everything (this will require the most training – reinforce good practice when you find it )

-they are writing correct answers in colour after the cross

Steps 6, 7 & 8 etc: Train the children thoroughly for several lessons, then reap the rewards! It works!

PS There is a little bit more to getting this to work than can be explained in one article. If you would like to chat about any of these ideas in more depth, please get in touch.

I walked into a colleague’s classroom some months ago and saw this above the board:

“Mistakes are to be expected, respected, inspected and corrected.”

Brilliant! This, in a nutshell, was what we had been trying to teach our children for years – that it is not just OK to make mistakes, but mistakes are actually where the learning happens. So don’t rub them out! Instead, value them and talk about them.

I shared this mantra with a primary Year 4 class. Within a day, the class teacher had told me that the children had decided that there was a word missing. Detected! We have to detect our mistakes before we can respect and inspect them. So now we had: Mistakes are to be expected, detected, respected, inspected and corrected! 

We tried sharing this with other classes. But there were now too many words and no one could remember the correct order!

Then I had a brainwave. Was there a mnemonic? If you put M for ‘mistakes’ at the beginning you would get MEDRIC. Not very memorable…

Was there a word for mistakes which began with C, I wondered. Then we would have CEDRIC! The thesaurus revealed the answer – the only synonym that began with C: Confusion.

The more I thought about it, the more I realised that this was the perfect word to start our slogan…

Our maths working group met earlier this term, so we discussed CEDRIC. The group unanimously agreed that this was the one thing we should focus on sharing with our children this session. It linked beautifully with all the work we were doing on mindset and, if we could really get the message embedded across the whole school, we reckoned it could be quite transformative. 

Step 1 would be to design a CEDRIC character. This was delegated to our primary Year 6 pupils through a competition. Shortlisted entries were voted on by our primary Year 4s and 5s.

Here is our winning design, a combination of two pupil designs that have been combined by a graphic design colleague and branded with our school colours.  

Image copyright 2019 George Watson’s College, Edinburgh

Next step was to decide how we would get the CEDRIC message out to the whole school. We’ve decided to do an official launch through assemblies in the summer term. Meanwhile, several of us have been experimenting with our maths classes. 

Here’s how it has been working for me so far…

In any real learning situation, confusion is expected. If our brain is being stretched we will sometimes feel confused and we may make mistakes. In order to learn, we need help from others so our confusion can be detected. Peer marking is great for this. If we and others have different answers, there is an opportunity to investigate.

Don’t be afraid of making mistakes

With the classes I have worked with so far, we have used the 333 approach, where pupils work in threes and compare answers after every three questions using a three-step process: plan together, work apart, check together. Most pupils mastered this part of the process quickly. A few impatient ones needed reminding about team working.

The important next step is that our confusion or mistakes should be respected.  We shouldn’t be embarrassed by our mistakes or hide them or rub them out. Instead, we should get out our favourite colour and put a bright coloured cross next to each mistake. This, the children found more difficult.  Some were used to rubbing out mistakes and changing them – so some retraining was needed to get them to leave a mistake in place, or indeed even put it back again after they had rubbed it out.

The key was to continually remind the pupils (and staff) that the work you do on your own is done in pencil and anything you do with your team is in colour.  Suddenly you have a clear record in your book of what you did unaided and how you improved your understanding with others – great for personal reflection and for sharing with parents.

Having highlighted a mistake, it then needs to be inspected. At this point, we encouraged the pupils to get together with others in their team and talk, in depth, about the differences in their understanding. This is probably the most challenging step in the process – it requires a real change of gear from some children.  The objective is no longer to get through lots of questions, but instead to engage deeply with the thinking of each member of the team.

We encourage our children to use pictures and diagrams to explain their thinking to each other and to write things down. Individual whiteboards are great for this. We are also finding ourselves increasingly giving support with spoken vocabulary and sentences. Speaking maths is a really valuable skill, but, like any skill, it requires much practice.  

We are also encouraging the children to explore ideas from different angles until they are all in agreement with their understanding and are all confident that each of the others can explain their ideas clearly to someone else. It is early days yet, but I am confident that, with persistence, we can bring about real change.

Finally, when the confusion is sorted, mistakes can be corrected. But even now, we don’t rub out the earlier mistake. Instead, we keep it as evidence of our learning and write the correct answer next to it – again, using a bright, stand-out colour.

The result – if we do it well – is that we have truly learned from our confusion. And…we have jotters or learning logs where everything that we have learned and clarified is clearly highlighted – clear evidence of thinking, and great for sharing with parents. A teacher’s dream…

P.S. The idea of CEDRIC is not copyright. So feel free to use the CEDRIC slogan in your own school. However, the CEDRIC image shown above is ours. If you want a CEDRIC image to use in your school, why not run your own design competition?

Rob Porteous, deputy head learning and teaching, George Watson’s College, Edinburgh

In my last article, I shared what a fantastic resource metre sticks are for teaching early addition and subtraction skills. In this article I will highlight how they are also great for teaching fractions, decimal fractions and percentages in upper primary.

Halves and mixed numbers

Let’s assume your pupils already have a secure understanding of the idea that a fraction is part of a whole and are confident in using fraction notation. Now introduce the metre stick!

The whole stick is one metre (1m). The graduations show centimetres (cm). There are 100 centimetres in a metre. How many centimetres in two metres? How many in no metres? How long is half a metre? What about no halves? What about two halves? Is it possible to have three halves? Yes! If you use another metre stick! What about four or five halves? How many centimetres is that? New idea: 11⁄2m and 21⁄2m are called mixed numbers. Mixed numbers are made up of wholes and fractions.

Quarter metres & improper fractions

Once the children have grasped the idea of halves we can then move on to quarters. How can we divide the metre stick into four equal pieces? Where would 1⁄4 of a metre be? What about 0, 2, 3, 4 quarters of a metre?

Can you get five quarters? Yes! This would be 1 1⁄4 metres or 125cm. How about six quarter metres? We can write 5, 6 or 7 over 4. These are called improper fractions.

Decimal fractions and equivalences

To understand common and improper fractions well, children need to understand their relationship with decimal fractions. First explore tenths. How can we split the metre stick into ten equal parts? What fraction of the metre stick is each part? Point out that each coloured section is one tenth of the metre stick, or 10cm. How many centimetres in 4/10, 7/10, 9/10 of a metre?

We can write one tenth like this: 0∙1. We say ‘zero point one’. The zero shows there are no whole metres and the one shows we have one coloured section, or one tenth. How would you write two tenths as a decimal? How would you read that?

Where is 1⁄2m? How many tenths is that? How could we write 1⁄2m as a decimal fraction? How would you read it?

Ten tenths is one whole metre. Can you get 11 tenths? Yes! You need another metre stick! One whole metre and one tenth of a metre can be written as 1∙1. What do you think 1∙2 means? How about 1.7 etc.?

How could you divide the metre stick into five equal pieces? [see ‘Chopping up a Metre Stick’ for a more in depth practical investigation] What fraction of the metre stick is each piece? One fifth of a metre = 20cm = 0.2m. What would two fifths of a metre be? How about three fifths, four fifths, five fifths? Which is bigger, one fifth or one tenth? How do you know? We say 1 fifth is equivalent to 2 tenths. What other equivalences can you find? (2 fifths = 4 tenths, 3 fifths = 6 tenths etc.)

Add an element of challenge by introducing twentieths. Which is bigger: one twentieth or one fifth? How many times bigger / smaller is a fifth compared to a twentieth / tenth? What other equivalences can you find between fifths, tenths and twentieths?

Percentages

If one hundred percent (100%) equals one whole how would you show 1% on your metre stick? Suddenly all the easy equivalences between fractions, decimal fractions and percentages are open to us:

One quarter of a metre = 25cm = 25%. How would you write two quarters, three quarters as percentages?

1 fifth of a metre = 20cm =20%. How would you write two fifths, three fifths, four fifths as percentages?

1 twentieth of a metre = 5cm = 5%. Challenge the children to find all other equivalences from two twentieths = 10% up to nineteen twentieths = 95%

Extend these ideas further to include twenty-fifths and fiftieths.

Factor rainbows

Finally: if you have never used factor rainbows you have been missing out on something wonderful. [see ‘Multiplication Rectangles and Factor Rainbows’ for a more. In depth investigation of these] Here is the factor rainbow for 100.

Something to puzzle over: What do these numbers have to do with the denominators of the fractions that can be easily converted to percentages? Aha!

Author:
Rob Porteous
Deputy Head, Learning and Teaching, George Watson’s College, Edinburgh
Email: [email protected]
Creator of the Maths Investigations Website.

Two years ago we bought 100 metre sticks.  Last year we bought 100 more! Used together with base ten materials, metre sticks are a great resource for teaching key numeracy skills.

The ones we use have the multiples of ten and five written on them but have marks only (no numerals) for the numbers in-between.

These partially numbered metre sticks can be difficult to find. We sourced ours from the Ruler Company – www.rulerco.co.uk.

Number sequence 0 to 100

To familiarise children with the metre stick I like to play the ‘two hands, one piano’ game. This is a simple ‘show-me’ activity where one child ‘plays’ the numbers from 0 to 50, and the other plays the numbers from 50 to 100.  

Call out a range of numbers and challenge the children to find them as quickly as possible. I start with the multiples of ten, then extend this to include the multiples of five. Next, children progress to identifying the unmarked numbers. This requires them to count on or back, from the nearest 5 or 10, to find the given number’s position.

Adding and subtracting tens

The children can then move on to using the metre stick, along with base-ten rods, to add and subtract tens. Assuming you have an up-to-date set, each base-ten rod should measure exactly 10cm.  So, for example, if one end of the rod is placed on the stick at 15, adding on ten takes you to 25. Taking ten away returns you to 15.  

There is general agreement that addition and subtraction are best taught together. The metre stick shows beautifully how the two operations are related.  It is also effective in helping children understand why, when they add or subtract ten, only the tens digit changes.  

Pattern is a key element of mathematics.. Moving the tens rod along the stick from 42 to 52 to 62 etc. clearly shows a repeating pattern. By using these patterns, children quickly learn how to add or subtract 10, and multiples of ten, to or from any 2-digit number.  

For pupils in need of challenge, place two or three metre sticks end to end and explore calculations such as 123 + 70, 156 – 40 etc. 

Complements to 100

We will assume for what follows that children already have instant recall of addition facts to ten.  

Take two metre sticks, turn one round the other way and place them back to back.  Ask the children to observe and describe what they see.

By carefully placing tens rods along the appropriate part of each stick, it should be easy for children to see that, for example, 60 + 40 = 100 because 6 tens and 4 tens equals 10 tens.  

This activity can be extended to include the multiples of five, for example 35 + 65 = 100. Two lots of five unit cubes make up the middle ten.  

As before, the ‘in-between’ facts can then be explored, for example 64  + 36. The question as to why 36 (not 46) is the complement of 64 generates interesting discussion and is extremely useful when working out change from £1.  The metre stick becomes your ‘pound’ and the centimetres your ‘pennies’!

Differences

Difference is often taught as another word for ‘taking away’ but in actual fact these two ideas are quite different.

To find 100 – 70 we start at 100 and count down to 70. 

However, the difference between 70 and 100 is the distance on the stick between 70 and 100,

The relationship between addition and subtraction is again highlighted. The children come to understand that the difference between 70 and 100 is 30 because 70 + 30 = 100 and 100 – 30 = 70

Differences between the fives numbers is a good follow-up investigation, again using base-ten rods.

Subtracting 8 or 9

Subtracting ‘near tens’ is something children often find challenging.  Turning the metre stick into an elevator illustrates this beautifully!

Stand the stick on its end with zero on the table and 100 in the air.  The metre stick now represents a tall skyscraper with 100 floors.  A staircase runs from top to bottom and an express lift stops at every tenth floor.  So, if you are on floor 50 and want to get to floor 40 you can go down 10 floors in the lift.  

However, if you only want to go down 9 floors, the quickest thing to do is to get the lift down ten floors then walk back up one level using the stairs:  50 – 9 = 41  

The same method will allow you to subtract 19, 29 or 39, for example by zooming down 40 floors and walking back up 1. This idea can be extended to include subtracting 8, 18, 28…. by zooming down 10, 20 or 30 floors and climbing back up 2 levels.

Once children have grasped this idea they can explore what happens if they start on an ‘in-between’ number and subtract 8 or 9. They can also investigate the patterns that occur when they add near tens by zooming up and walking down.

Well I hope that has got your creative juices flowing!  Go and get those metre sticks ordered and get counting.  

P.S. The same metre sticks are also great for teaching fractions, decimal fractions and percentages. But that’s a story for another time ….

Author
Rob Porteous, Deputy Head Learning and Teaching, George Watson’s College, Edinburgh;
Creator of Maths Investigations.

All are offered to you, in the hope that they will transform your practice as they continue to do mine.

None of these ideas can ever be complete or finished. As a I work with them with different groups of children from year to year, they inevitably change and adapt. Please feel free to try them in your own context, adapt them further as you see fit, and then share your experiences here for others to learn from.

Hopefully this web site can then become a place for those of us who are trying new approaches with our children, to share ideas, share what works for us, and be further inspired by each other!

Rob Porteous, Edinburgh, 2019