# Multiplicative reasoning

If a child understands additive reasoning and the relationship between the whole and its parts, it is a fairly straightforward conceptual step to understand multiplicative reasoning.  Multiplicative reasoning should be modelled as repeated addition in the first instance.  Adding multiple equal parts (for example 5) might look like this:

5 + 5 + 5+ 5 is equal to twenty.  Children need to understand that multiplication allows for efficient repeated addition.  You  have your thing to be multiplied (5) and the multiplier (4): 5 +5 + 5 + 5 = 5 x 4.  Creating arrays and deliberately connecting repeated addition with multiplication makes for sound understanding.

How children work out the whole should not be taken for granted.  At first, children might count each item in the array.  Counting in multiples can be achieved by first skip counting.  Children might whisper the numbers while counting except for the last in each row, which is said out loud.  Then replace the whispering with counting in their heads and then simply saying the multiples.  Over time, given sufficient practice, children will internalise these times tables.

Commutativity is important here – the array used above shows 5 x 4 but rotated it shows 4 x 5.  Times tables taught systematically and with such conceptual support should be straightforward for children to learn comfortably before the end of year 4, especially when we consider it like this:

Of course, children need time to practise well and multiple representations help children to make connections.  Graham Fletcher’s blog post describes the use of pictorial representations on flash cards – an approach that is a great form of low stakes testing to support the learning of times tables.

This image supports the understanding of having a ‘thing to be multiplied’, a multiplier and a whole.  With practice, children will be able to subitise from glancing at the flash card, becoming fluent and accurate with times tables recall.

Some children will grasp all this quickly and can work at a greater depth while children that need more practice with the basics get it.  Still using the array, children can easily begin to think about distributivity simply by splitting the array into parts:

The part above the line is 5 x 2 and the part below the line is 5 x 2:

5 x 2 + 5 x 2 = 5 x 4.

There is lots of scope for systematic thinking about equivalence with a task like this.

Arrays are perhaps not the most efficient of representation so a progression is to get children to be able to represent multiplication in bar models.  First though, Numicon to work on the language of size of each part, number of equal parts and the whole:

Numicon is a great manipulative to represent multiple parts because of its clarity of the ‘size of each part’.  Multi-link cubes could work too, but children would need to organise the parts into different colours to differentiate between them:

Building worded statements using a manipulative will ensure children practise the language needed to internalise the concept of multiplicative reasoning.  Dropping in some of the  inverse relationship between multiplication and division could be useful here too.  Doing it systematically can also help keep times tables knowledge conceptual and not shallow:

Commutativity could be brought in again – showing that 3 groups of 4 is the same as 4 groups of 3 using manipulatives arranged with intent.  Alongside this, comparing the similarities and differences with the worded statements will get children to think with clarity about equivalence between two multiplicative expressions.

Bar models are a versatile representation that can be used to solve a wide range of problems later on, so getting children to sketch out multiplication and division statements using bars enables them to practice a versatile skill.  We should expect great accuracy in their drawings – they should be representing equal parts.  If children also represent the same expression on a number line beneath the bar model, we can encourage links between representations and lay the foundations for trickier calculations and problem solving as they progress through school.

# Before, then, now – modelling additive reasoning

One of the parts of the NCETM’s Calculation Guidance for Primary Schools is the ‘Before, Then, Now’ structure for contextualising maths problems for additive reasoning.  This is a very useful structure as by using it, children could develop deep understanding of mathematical problems, fluency of number and also language patterns and comprehension.

The first stage is to model telling the story.  We cannot take for granted that children, particularly vulnerable children in Key Stage 1, will know or can read the words ‘before’, ‘then’ and ‘now’.  Some work needs to be done to explain that this is the order in which events happened.  Using a toy bus, or failing that, an appropriate picture of a bus, we would talk through each part of the structure, moving the bus from left to right and modelling the story with small figures:

Before, there were four people on the bus. Then, three people got on the bus. Now there are seven people on the bus.

The child could then retell the story themselves, manipulating the people and the bus to show what is happening.  For the first few attempts, the child should get used to the structure but before long we should insist on them using full, accurate sentences, including the correct tense, when they are telling the story.

I have chosen a ten frame to represent the windows on the bus, which enables plenty of opportunity to talk about each stage of the problem in greater depth and to practise manipulating numbers.  For example, in the ‘Before’ stage, there were four people on the bus: if the child could manage it, it would be interesting to talk about the number of seats on the bus altogether and the number of empty seats.  By doing so, they are practising thinking about number facts to ten and building their fluency with recall of those facts.   The task could easily be adapted to use a five frame or a twenty frame.

The next stage could be to tell children a story and while they are listening, they model what is happening with the people and the bus.  After each stage, or once we have modelled the whole story, they could retell it themselves.  Of course, the adult would only tell the ‘Before’ and the ‘Then’ parts of the story as the child should be expected to finish the story having solved the problem.

When the child is more fluent with the language and they understand the structure of the problems, we can show them how it looks abstractly.  For the ‘Before’ part, the child would only record a number – how many on the bus.  For the ‘Then’ part, we would need to show the child how to record not only the number of people that got on or off the bus but the appropriate sign too – if three people got on they would write +3 and if two people got off they would write -2.  Finally, for the ‘Now’ part, they would need not only the number of people on the bus but the ‘is equal to’ sign before the number.  Cue lots of practise telling and listening to stories whilst modelling it and writing the calculation.

A more subtle level of abstraction might be to repeat the same problems but rather than the child modelling them using the bus and people, they could use another manipulative such as multi-link cubes or Numicon.  They could also draw a picture of each stage – multiple representations of the same problem provide the opportunity for deeper conceptual understanding.

The scaffolding that the structure and the multiple representations provide allows for some deeper thinking too.  In the problems described so far, the unknown has always been the ‘Now’ stage or the whole (as opposed to one of the parts). It is fairly straight forward to make the ‘Then’ stage unknown with a story like this:

Before, there were ten people were on the bus.

Then, some people got off the bus.

Now, seven people are on the bus.

This could be modelled by the teacher, who asks the child to look away at the ‘Then’ stage.  Starting with ten people on the bus and using a ten frame is a deliberate scaffold – deducing how many people got off the bus is a matter of looking at how many ‘empty seats’ are represented by the empty boxes on the ten frame in the ‘Now’ stage.  A progression is to not use a full bus in the ‘Before’ stage – it is another level of difficulty to keep that number in mind and calculate how many got on or off the bus.

Another progression is to make the ‘Before’ stage unknown.  The child will need a different strategy to those already explained in order to solve this kind of problem.  Then story would have to be started with: ‘Before, there were some people on the bus.’  Of course, the adult would not show the child this with the bus and toy people, but they would show the completed ‘Then’ stage: ‘Then, four people got on the bus.’  Finally, the adult would model moving the bus to the ‘Now’ stage and completing the story: ‘Now, there are eleven people on the bus.’  The child would have to keep in mind that four people had got on and now there are eleven, before working backwards.  They would have to be shown that if four had got on, then working out how the story started would mean four people getting off the bus.  They could be shown to run the story in reverse, ending up with seven people on the bus in the ‘Before’ stage.

This task has the potential to take children from a poor understanding of number facts, calculating and knowledge of problem structures to a much deeper understanding.  The familiar context can be used as a scaffold to build fluency and think hard about complex problems with varied unknowns.

# Penn Wood Professional Development – Language acquisition and reading comprehension

The York Reading for Meaning Project (Snowling, 2010) compared three interventions with a control to determine their effectiveness in developing reading comprehension. The interventions were led by teaching assistants and lasted for 20 weeks, each week comprising of three 30 minute sessions. The three interventions were:

• An oral language comprehension programme
• A text comprehension programme
• A combined oral and text comprehension programme

Their findings showed significant gains in reading comprehension scores for each intervention compared to the control group. Interestingly, the most effective intervention was not the text comprehension programme, but the oral language comprehension programme, which also resulted in greater gains in reading comprehension scores than the combined programme. These gains were still evident 11 months after the interventions ended.

With such an impact, it makes sense to attempt to turn this effective intervention by TAs into part of our day to day teaching. Perhaps we can adapt the programme to see even greater and longer lasting gains in language acquisition and reading comprehension if the ideas were embedded in our English lessons.

The simple view of reading identifies the importance of decoding and language comprehension in tandem to master reading for meaning. Hirsch would add to this the importance of domain knowledge, without which a reader would not make rapid connections between new and previously learned material. As such, explicitly teaching the general knowledge required to understand a text can support comprehension significantly. Of course, the challenge to this idea is that we can’t teach children the entirety of general knowledge. However, selecting great texts which reflect a variety of general knowledge schemas gives children the opportunity to develop key chunks of general knowledge on which further domain knowledge can be built through listening and reading.

Using great texts to teach language acquisition and reading comprehension is a perfect place to start. Once these texts have been selected, the first thing that teachers need to do is consider the following question:

Which words, phrases or concepts are children likely to find difficult to understand?

Jean Gross, in Time to Talk talks of three tiers of vocabulary. Tier 1 vocabulary includes words and concepts that children will come across first when they begin to communicate. Tier 2 vocabulary includes language that children will be able to understand the concept of and that is tricky yet functional. These words could be used in a number of contexts. Finally, tier 3 vocabulary includes language that is domain specific and only used in a small number of contexts.

When we’re looking at the bits of a text that children are likely to find difficult to understand, we’d need to be looking for those tier 2 words within a text. For children learning English as an additional language and for children in the early years, we’d also need to explicitly teach tier 1 vocabulary. Usually, these would be common nouns, verbs and concepts and these guidelines from Stories for Talking by Rebecca Bergmann are helpful when selecting them:

Nouns

• High frequency
• Functional
• Related to the story being studied
• Related by topic
• Feature around the classroom or school
• Easily supported with concrete objects

Verbs

• High frequency
• Functional
• Relate to the chosen nouns
• Easy to act out

Concepts

• High Frequency
• Functional
• Relate to the chosen nouns
• Most visually represented or repeated in the story
• Can be studied as a pair (big/little)
• Can be experienced practically around the classroom

Once that language has been identified, teachers can introduce it to children. By introducing it before children listen to or read a text, we can go some way to guarding against cognitive overload. Also, by increasing the number of interactions with this vocabulary, and by spacing those interactions, we increase the likelihood of long term retention of those ideas. Having said that, language is best learned in context so defining words for children will not suffice. The image below is an example of how language is introduced from the York Reading for Meaning Project programme materials:

This works well because the images provide contexts in which the word is used. The variety of images and contexts helps children to make connections between ideas. A slight amendment that includes the Talk for Writing approach would be to include the sentence from the text that the word is in.

Children will not internalise this language after one interaction with it. Children need to think hard about the meaning and application of the vocabulary over time if it is to be assimilated. The following question types come from Bringing Words to Life by Beck, McKeown and Kucan.

Where children have to differentiate between two scenarios, such as in the Example or non-example?’ question, the quality of the question comes from the two scenarios being minimally different and rooted in misconceptions about a word’s meaning.  With a set of questions like this for a number of focus words across a unit of work, children’s practice of thinking about and using language can be spaced over time in a variety  of contexts, giving children a great chance of adding permanently to their vocabulary.

Oral and text comprehension

By understanding the typical difficulties that struggling readers experience, we can plan to address those issues with some carefully panned practice. If we then consider the implications from the York Reading for Meaning Project, that the materials from both the oral and text comprehension can have such an impact on reading comprehension, then we can provide great lessons.

Developing Language Acquisition and Reading Comprehension at Penn Wood outlines those difficulties and what might be done. The York Reading for Meaning Project found that oral comprehension work is more effective than a text comprehension or a combined oral and text comprehension programme when measured using reading comprehension tests. All of the suggestions for addressing the profile of the struggling reader could be applied through reading a story but also through listening to one. Talk for Writing provides a great opportunity for this as children internalise and retell stories using text maps. Oral comprehension work can quite easily be introduced at the point of retelling. With the opportunity presented, the next step is to find ways of making oral language development work in the classroom on a day to day basis.

# 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