# Maths Marking and Feedback Decisions

This post is a record of the CPD session from Spring 2014 where we worked on our maths feedback.

These are the principles behind the decisions that we make about the type of feedback that we give. Marking should not be a time consuming chore so any decision that we make about how we give feedback needs to consider the impact for the time and effort that we invest.

Bright Spots

In this example from Year 1, the teacher, seeing that the child was successful with the given task, has written a more challenging question where the position of the empty box has changed. Any written work would have been a waste, as the child will have found reading and understanding difficult. Clearly, there has been some communication between child and teacher to explain the twist – that an inverse operation is needed.

In this example from Year 3, you can clearly see that the teacher’s explanation and modelling has led to the child understanding the calculation strategy well. A simply written, short question here probes understanding further and encourages links to be made.

This Year 4 example shows a couple of strategies. First, the teacher has reminded the child of how to approach the problem, which resulted in the child able to correct their initial mistake. Second, the teacher asked the child to clarify the calculation needed, which led to the child being able to sort the information in the problem and then solve it.

Here in Year 5, the child originally made mistake. The teacher helped the child to focus on the important bit of the problem, which enabled them to successfully correct the mistake.

In this Year 6 example, the child identified the calculation needed but made a mistake calculating. The teacher, though, knew that the child may not have understood the nature of the problem so the bar model was drawn to help the teacher explain the problem. Also, the teacher clearly intervened in terms of prompting a calculation method that enabled the child to correct the original mistake.

Mark the process or mark the answer?

This photo is from a child’s book in Year 2. The child has calculated accurately but the strategy that they used was particularly inefficient. In this case, that inefficiency definitely needs to be the focus of the feedback. This is important because we show what we value by doing this – that understanding is more important that simply getting questions right.

They got everything right!

Getting everything right can mean a number of things. It could mean that there was a lack of challenge; that it was pitched too low. Of course, there is also the case the child couldn’t do it before, had a clear explanation and understood it quickly. Knowing the situation determines the feedback. We also need to acknowledge another situation where children get it all right. Over learning something until they can do it with minimal thinking is an important part of mastery. In the example above, the child had already been shown how to divide using short division, but the purpose of the practice was to stave off forgetting. It was a situation where the teacher should expect that there’d be very few mistakes. There was only a few questions and this would have been repeated, spaced out over time to aid the transfer to long term memory.

The feedback still needs to be considered carefully though. There are a couple of choices. The feedback could focus on pushing further, perhaps introducing trickier numbers. Alternatively, the feedback could centre on the expectation that this is remembered, that the child should practise at home and that in a week or so, they can be as successful.

Making decisions when marking

So to aid the decisions that are necessary when considering the appropriate feedback, these decision trees are provided. One is for basic practice tasks, and one is for problem solving tasks.

What followed was some deliberate practice of this decision making. Teachers worked in year teams to decide on appropriate feedback, using some unmarked work that they brought with them.

To continue to ensure that our feedback in maths books is effective, it is important that we discuss and question the possibilities, so that those decisions can be made with increasing efficiency.

http://reflectionsofmyteaching.blogspot.co.uk/2013/10/can-i-be-that-little-bit-better.html?m=1

http://reflectionsofmyteaching.blogspot.co.uk/2013/11/can-i-be-that-little-bit-better-at.html?m=1

http://youtu.be/ag38OBjuMrQ

http://wp.me/p2uRcx-V9

http://feedbackasateachingstrategy.weebly.com/

http://www.learningspy.co.uk/featured/reducing-feedback-might-increase-learning/

http://meridianvale.wordpress.com/2014/01/25/what-if-feedback-wasnt-all-it-was-cracked-up-to-be/

# Growing Great Teachers – Which research group?

Penn Wood – Growing Great Teachers

Key Principles:

• Working on the ‘bright spots’ – building on existing strengths.
• The Pareto principle – 20% of teaching strategies yield 80% of the value.
• Deliberate practice – Focused, intentional practice supported by high quality feedback.
• Action research – experimenting with strategies to find out what works.
• Better never stops. All teachers need to improve, not because we are not good enough, but because we can be even better.

Teachers will come ready to think of a teaching sequence which has gone well.  Through discussion with year colleagues, using the coaching questions below, teachers will identify aspects of the teaching sequence that were good or better.  This will also provide some practice for teachers when coaching later in the action research cycle.

• Tell me about a time when behaviour for learning was great?  What did you do that supported them to do this?
• Tell me about a time when you could immediately respond to what a child said or their work with quality feedback?
• Tell me about the outcomes for different groups of children?  How did you meet their needs?
• Tell me about a time when you saw a real improvement in reading fluency /understanding?

These questions will help teachers to focus in on an area of strength that will then become the focus of a term’s CPD.   Teachers will develop on an aspect of good or better teaching in research groups, led by senior or middle leaders.  The groups are as follows:

• What are the most effective strategies for improving fluency and understanding?
• How can we create a positive climate for learning to read for pleasure and widely across the curriculum?

Modelling and explanations

• What are the most effective ways of authoritatively imparting knowledge?
• In what ways can our explanations develop children’s resilience and thirst for knowledge?

Meeting the needs of different groups of children

• How can we ensure that teaching strategies, support and intervention match individual needs accurately?
• How can we differentiate tasks so that more children attain the higher levels in national assessments?

Feedback and questioning

• How can we anticipate misconceptions, check for understanding and intervene to make a notable impact on learning?
• How can we use feedback and questioning to ensure that more pupils attain higher levels in national assessments?

EYFS

• What are the most effective strategies to secure the early acquisition of language?
• How do we increase the proportion of children meeting and exceeding national expectations?

Within these research groups, teachers will further discuss what worked well for them in their successful teaching sequences, with the aim of creating a toolkit for possible strategies.  This will be supported by short videos (available by logging into the school account), blog posts and books.

http://www.learningpt.org/pdfs/literacy/components.pdf

Modelling and explanations

http://www.education-consumers.org/CT_111811.pdf

http://wp.me/p3UXMS-2I

http://www.aft.org/pdfs/americaneducator/spring2012/Rosenshine.pdf

Meeting the needs of different groups of children

http://www.learningspy.co.uk/featured/deliberately-difficult-focussing-on-learning-rather-than-progress/

http://bit.ly/1iiwu1B

http://bjorklab.psych.ucla.edu/research.html

http://ow.ly/o8Anb

http://learninglab.psych.purdue.edu/publications/

Feedback and questioning

http://wp.me/p2uRcx-VJ

www.huntingenglish.com/2013/12/26/disciplined-discussion-easy-abc

wp.me/p43kJZ-4U

http://reflectionsofmyteaching.blogspot.co.uk/2013/10/can-i-be-that-little-bit-better.html?m=1

http://reflectionsofmyteaching.blogspot.co.uk/2013/11/can-i-be-that-little-bit-better-at.html?m=1

http://youtu.be/ag38OBjuMrQ

http://wp.me/p2uRcx-V9

http://feedbackasateachingstrategy.weebly.com/

http://www.learningspy.co.uk/featured/reducing-feedback-might-increase-learning/

http://meridianvale.wordpress.com/2014/01/25/what-if-feedback-wasnt-all-it-was-cracked-up-to-be/

The final part of the session will be for each teacher to settle on one strategy that they will experiment with in their classrooms over the next few weeks.  The research leader will ensure that each teacher in their working group leaves with a plan in place.

# Hassan’s internal number line

Hassan is a wonderful boy. He’s polite and has a great group of friends. But Hassan started Year 6 working significantly below his peers. His school history is of sustained underachievement with very little progress. He did not have an internalised number line with which to think about numbers, to the point where he could not reliably say which number out of two was biggest.

This post is an account of an intervention carried out by Amy Coyne. It is one of the most successful interventions I have seen and has resulted in vast improvements in Hassan’s ability to think about numbers. Here’s what happened:

These number cards were prepared: 53, 67, 54, 35, 76, 45

Two of the cards were presented to Hassan and, with the use of Numicon or dienes blocks or arrow cards, Amy modelled explaining which was the bigger number. Hassan picked this up fairly quickly, but to help him to retain this procedural knowledge, it was repeated little and often over the course of a few days.

A third card was added and Amy again modelled, using appropriate concrete equipment, how to order them. When he could consistently order three numbers, a further card was added until he could deal with ordering six cards. Using those six cards, Amy made seven different sequences:

53, 67, 54, 35, 76, 45

67, 35, 76, 54, 53, 45

76, 35, 67, 53, 45, 54

45, 53, 35, 76, 67, 54

54, 45, 76, 53, 35, 67

53, 54, 35, 76, 45, 67

45, 76, 54, 67, 53, 35

The cards were presented to Hassan in these orders, one set at a time, and Hassan was asked to order them. At first, with this slight change in task, he would place the numbers in fairly random order for each sequence. After completing each sequence, Amy ordered them with him, using concrete models when necessary. When Hassan was asked to read out the order, if he was incorrect, he often didn’t notice. However, when the sequence was read aloud to him, he could hear the error and would correct it.

This was repeated over several days for short periods of time. Sometimes this was in maths lessons and at other times it was not. The seven different sequences would be laid out in a straight line and he would pull the cards out and order them. As the days progressed, he could very quickly pick out card 35 and put it furthest left and also card 76 and put that furthest right – the smallest and greatest numbers. However the other cards in between were never placed consistently in order.

After a week of doing these sequences once or twice a day, he could order every sequence in the correct order. A new set of numbers was introduced: 12, 27, 45, 54, 59, 72

Hassan was very good at picking out the biggest and smallest numbers. The numbers in between were still more difficult for him. Amy modelled looking at the tens and units columns and this prompted him to order them correctly.

The cards were then mixed up, with more numbers being added one at a time to see if he could order them again. His confidence was growing and once he was happy with the order he had put them in, he was asked to read the numbers out to see if he could spot any errors himself. He often did and corrected them without Amy needing to intervene.

After a few days starting with six or more cards, he could reliably order them correctly every time. Next, some three digit numbers were added to make twelve cards overall. He was quickly able to deal with this progression. He was then given cards with multiples of ten to see if he could slot them into the correct places. He struggled a little with cards 10 and 20 but he placed multiples of ten more than twenty in the right places every time. If he needed to, he referred to a tape measure to check.

Once he was confident in ordering these numbers and could do it correctly every time, two numbers from the sequence were chosen. He was asked: ‘What are the smallest and biggest numbers that could go in the gap?’ This proved to be quite tricky for him and he would often say the number before the smallest card. This took him around a minute to process each time, and many answers were guesses. Amy modelled looking at a tape measure to find the two numbers (53 & 59). He then could see, using the tape, which numbers would come after 53 and before 59, and therefore the biggest and smallest that could go in the gap.

From here, Hassan is now working on adding and subtracting one digit numbers and multiples of ten from numbers in his card sets, with increasing success. Soon, the goal is that he can add and subtract any two digit number from any other.

Why this intervention worked, when other have failed

Spacing and interleaving

Regular short sessions, interspersed with other topics in maths lessons, with varied lengths of time in between those sessions has given Hassan time to internalise patterns of numbers and procedural knowledge for dealing with them.

Building knowledge of the number system

The more he practised recalling facts about numbers and procedures for how to think about them, the more successful he became. Each nugget of internalised knowledge enabled further memory development until he had internalised the basic number system.

Deliberate practice to mastery

The moment that Hassan understood and was successful did not signal the end of the intervention. It will continue until he never makes a mistake, even when tasks are altered.

Any new concepts were introduced alongside concepts that Hassan was familiar with.

Detailed dialogue between teacher and teaching assistant

Using video and observing ‘live’, the Amy and I talked about the nuances of the decisions that Hassan made to tweak tasks and feedback. This precise tailoring resulted in explanations, tasks and feedback which were accurately matched to Hassan’s needs.

This intervention was put in place because Hassan was working significantly below his peers at number. It was clear that he had not internalised a number line at the beginning of the year, but this shows that he now has. He will need more practice to cement his understanding but the progress that he has made has been good. We did not work on this with him for a week before the Christmas holiday. He had just over two weeks off school over Christmas and when he returned to school after the holiday, he could still deal with the number tasks accurately. Next, we are looking to see if the results are replicable with other children.

Details about the child have been changed to preserve anonymity.

# What makes great teaching?

Great teaching combines effective instruction with continuous improvement.

If you can dream—and not make dreams your master;

If you can think—and not make thoughts your aim;

If you can meet with Triumph and Disaster

And treat those two impostors just the same;

If you can bear to hear the words you’ve spoken

Twisted by knaves to make a trap for fools,

Or watch the things you gave your life to, broken,

And stoop and build ’em up with worn-out tools…

Rudyard Kipling, 1892

“English Teacher Joe Kirby has taken me to task in his blog ‘Pragmatic Education’… I take particularly seriously the concerns teachers such as Joe Kirby have about the teaching practices which our current examination system encourages.
He, and many others, are deeply worried about what he calls, ‘the enacted school curriculum: what actually gets taught in classrooms.’… Kirby’s challenge to us in government is clear.”

Secretary of…

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