On teaching additive reasoning

The question was on the screen:

AR13

One year 6 child said: ‘The empty box is in the middle so you do the inverse.  You have to add the numbers together’.

This got me thinking about how children build on their early concepts of number to be able deal with problems like this, which I’ll call ‘empty box problems’.

AR1

The underlying pattern of additive reasoning is the relationships between the parts and the whole.   Getting children to think and talk about the whole and parts using concrete manipulatives early on should lay the foundations for them to internalise this underlying pattern.  Every time children think and talk about number bonds, they can be practising identifying the whole, breaking it into parts and then recombining to make the whole once more.

AR2

Alongside talking about the whole and parts, children should begin to generate worded statements whilst manipulating cubes or Numicon, for example.  At this point it is important to experiment with rearranging the words in the statement.  They should get to know that ‘four add two is equal to six’ and ‘six is equal to four add two’ are statements that are saying the same thing.  Some discussion around what is the same and what is different about these two statements would be worthwhile.

When children are then shown how this looks abstractly with numerals and the equals sign, this would hopefully go some way towards avoiding the misconception that the equals sign means that ‘the answer is next’.

AR4

In the examples used so far, the whole and each of the parts have been ‘known’.  Using the same manipulatives and language patterns, children can be introduced to unknowns.  It seems sensible to begin with giving children the parts and using the word ‘something’ to show that the whole is unknown, i.e., four add two is equal to something.  Some modelling alongside a clear explanation followed by plenty of practice should see children get used to the language patterns needed to think about the concept with clarity.  The next step is to show children the whole and one of the parts, using the word ‘something’ to replace the unknown part.  All of this talk and manipulation of objects is intended to support children to develop a concept of additive reasoning where they do not have the misconception that ‘inverse’ means ‘do the opposite’.

AR5

More sophisticated additive reasoning is the understanding of the inverse relationship between addition and subtraction.  Children need to fully understand that two or more parts can be equal to the whole.  From this, they need to internalise the underlying patterns: that Part + Part = Whole and that Whole – Part = Part.  From this, they should be able to work out the full range of calculations that represent one bar model.  Again, it is important to vary the placement of the = sign.

AR6

One more way to get children to think about the whole and the parts is to use bar models for calculation practice rather than simply writing a calculation for children to work out.  When done like this, children have to decide what calculation to do to work out the unknown.  Children often exhibit misconceptions such as ‘when you subtract, the biggest number goes first’.  These can be addressed using the underlying patterns; adding parts together makes the whole and, when you subtract, you always subtract from the whole.  When unknowns are introduced, they can be substituted into these basic patterns:

Part + Something = Whole           Part + □ = Whole              35 + □ = 72

Something + Part = Whole           □ + Part = Whole              □ + 35 = 72

Whole – Something = Part           Whole – □ = Part               72 – □ = 35

Something – Part = Part                □ – Part = Part                   □ – 35 = 37

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Knowing these patterns will help children to able to analyse problem types in order to decide on the calculation needed.  An additive reasoning bar model with one unknown generates both an addition statement and a subtraction statement.  Showing children empty box problems pictorially, they can talk through the calculations that can be read from the bar model, using the word ‘something’ to represent the unknown.  The next step is to show children abstract empty box problems and get them to map it onto a blank bar model.  They should be drawing on their knowledge that the whole is equal to the sum of the parts and that when you subtract, you always start with the whole.  Eventually, the hope is that the language alone should suffice to work out how to solve empty box problems, with children no longer needing the bars.

Which brings us back to that year 6 child.  Of course, children will develop misconceptions as they make sense of what is shown and explained to them.  By expecting them to think and talk about additive reasoning in the ways described above, it should go some way to building sound conceptual understanding.

Story Telling and Talk for Writing

Chris Smith’s session on storytelling during our day at the Story Museum included many ideas that could effectively complement our Talk for Writing work.  The suggestions in this post centre mainly on the imitation stage, although the Storytelling School approach of Tell, Deepen, Shared Writing and Independent Writing span the entire T4W process.  Specific details on each of these steps can be found in the Storytelling School Handbook: what follows is some thinking about how they can be applied in a T4W unit.

Imitation – Internalising the text

Hear it, map it, step it, speak it.

Children’s first interaction with the story is told by the teacher, not read aloud. Teachers should be prepared to tell the story with skill (see the Storytelling School Handbook for Teachers for a toolkit for great story telling).

Get children to map the story from memory.  In the first instance this can be a simply structured map depicting the main things that need to be remembered for the plot to work.  This should be a fairly short activity but it can be added to over time.  Children could have their text map in front of them as they listen to and practise the story during the imitation stage.  Before children use their text map to retell the story, they need to practise recalling the meanings of the symbols.  Try getting pairs to talk through their plans: ‘This is the bit where…’  There will still be a need for a teacher text map for the teacher to model retelling and for some children to use themselves.  It can be used later to support the planning of the innovation.

Stepping the story (details in the handbook) gets children to focus on key moments that trigger memories of the finer details for later retelling.  Each step could include words and actions to trigger those memories.  When children write at a later stage, each step could become a paragraph or a scene.  Consider here the planned repetition of tier 2 focus words to increase the number of interactions that children have with this vocabulary.  Stepping the story could focus on different things.  They could step the settings so that at each key moment in the story, children are thinking about the details of the setting.  They could step the characters so that at each key moment, they are thinking about bringing the characters to life.  They could step the dialogue, thinking hard about the interactions between characters.  Consider the intended effect for the story and the toolkit that will be later co-constructed.  By tweaking the ‘step it’ stage, foundations for great Reading as a Writer can be laid, preparing children for co-constructing the writers’ toolkit(s) later on.

Once children have internalised the main parts of the story through hearing, mapping and stepping, they practise retelling the story.  Paired retelling, taking turns, allows for flexibility which communal retelling may not provide.  The first person tells part of the story and then the other continues before passing the story back again.  This is not to say that communal retelling is not useful – many children, particularly those new to English, will need to hear and rehearse specific language patterns.  Some children may need to be encouraged to speak one idea at a time to get a feel for sentence demarcation which will support later writing accuracy.  Get children to build the sophistication of their story by magpieing phrases from others’ retelling.  One partner can tell the story while the other listens, text map in front of them, adding ideas that improve their version.  Something similar can be done with a group retelling performance: a group pass the telling of the story between them while the rest of the class listen, adding to their text maps.

Reading as a Reader – Deepening the storytelling

The sophistication of children’s storytelling will develop through the imitation stage and will be supported through established aspects of T4W such as shared reading, book talk etc.  The Storytelling Handbook includes other opportunities to deepen children’s understanding of the story.

Reading as a Writer

During this stage, we get the children to analyse how good writing is created.  Children need to have a sound knowledge of the underlying story structure but it is also useful to summarise the structure even more simply, for example, ‘Annoying thing won’t stop so… accept it.’  Boxing up the story helps children to analyse the writing section by section, which then contributes to planning the innovation.  Chris’ plot matrix (see the Handbook) may be an interesting alternative.

Whilst co-constructing the writers’ toolkit, it may be useful to return to ‘stepping the story’ with the intended effect in mind.  If the intention is to bring the character(s) to life, then thinking hard about the language use at each key moment that does this sets children up well to analyse other writing for how it could be done.

Non Fiction

A similar approach can be used with non-fiction writing.  Non-fiction arguably presents a different challenge in that the speaker / writer must have mastered the subject content as well as the appropriate language of non-fiction in order to effectively get their message across.  It is the subject content that must come first, though – children need something to think about before learning and using the language associated with non-fiction writing.

Imitation – Internalising the subject content

Just as the teacher would be expected to tell the story when working with fiction, so too with non –fiction.  The first interaction should be told, not read.  The teacher should tell the text in role in simple language so that children get to know the subject content quickly – informally tell children about the content from memory, making a connection just as for storytelling.

Get children to map the subject content, following the same structure as for fiction, including stepping it.  When children know the content well, the teacher can begin to change their retelling to make it more formal and to include the non-fiction language features necessary.  Children can then do the same throughout the Reading as a Reader stage, so that when they come to Read as a Writer, they would have already been talking some of the non-fiction language that we’d be asking them to analyse in other texts.

Great non-fiction writing relies on the writer knowing the subject content very well.  Sequencing some non-fiction work immediately after some fiction work and using the same subject content contributes effectively to writers’ subject knowledge.  For example, after a fiction unit on ‘Beating the monster’ using George and the Dragon, children could work on non-fiction, informing novices on the different types of dragon, arguing that dragons should not be held in captivity or instructing knights on how to defeat them.

10 Research Based Principles of Instruction for Teachers

Belmont Teach

I recently read an American Educator article from 2012 by Barak Rosenshine that set out 10 principles of instruction informed by research, with subsequent suggestions for implementing them in the classroom. It was also one of the articles cited in the “What makes great teaching? Review of the underpinning research” by Rob Coe et al and provided further elaboration on one of their six components of great teaching thought to have strong evidence of impact on student outcomes, i.e. quality of instruction.

Here’s my summary of the key messages from each of the 10 principles.

1: Begin with a short review of prior learning

Time-for-Review

Students in experimental classes where daily review was used had higher achievement scores. A 5-8 minute review of prior learning was said to strengthen connections between material learned and improve recall so that it became effortless and automatic, thus freeing up working memory.

Daily review could…

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Great teacher talk

Evidence into Practice

The ‘war’ on Teacher Talk Time was one of the more depressing developments in my teaching career. Obviously secondary students will rapidly get bored if they are subjected to lectures – no one is advocating that teachers should drone on and on for hours – but the obsession with cutting down teacher talk became dysfunctional under the Ofsted framework.

The profession became awash with strategies to prevent teachers from talking. Here’s an example from TeacherToolkit (a typically excellent resource bank) that I found dismaying at the time:

http://teachertoolkit.me/teachertalk/

“Teacher Talk can often be the root-cause of poor behaviour and debilitating progress during a lesson. … “Reducing Teacher-talk” will be my sole focus for 2012/13 in my own classroom, as well as observations of other colleagues and CPD training throughout the school year.”

Some of the Top Tips show the extent to which ‘teacher talk’ became taboo.

  • “Give the students a…

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#MathsCPDChat on times tables strategies

Pre-reading: Strategies for learning, remembering and understanding the times tables. Some additional thoughts for starting out teaching the times tables with year 2s onwards, prompted by a #MathsCPDChat These are the things I think are important for mastery of the tables (most of which, I suspect our primary colleagues are doing): 1. Begin with manipulatives…

http://mrreddy.com/blog/2015/01/mathscpdchat-on-times-tables-strategies/

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.

SVRHirsch reading

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:

Admire

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.

Aggressive

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.

Aggressive 2

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.

Contemporary educational ideas all my staff should know about

teacherhead

 

Key ideas from different sources. Key ideas from different sources.

As I look ahead to starting my new job at Highbury Grove,  I’m thinking about all the conversations we are going to have about learning.  To a large degree I want my teachers to be as up-to-date as possible within their own subject domains. They should know the latest OfSTED position ( eg with Moving English Forward or Mathematics: made to measure ) and be up to speed with exam specifications and assessment requirements.  Subject knowledge and subject-specific pedagogical knowledge are going to be key drivers of everything we do.

However, in order to fuel the collaborative effort of reaching the ambitious goals we have for the school, we’ll need to establish a shared conceptual language for talking about teaching across the school as well as within departments. Inevitably, different teachers will have engaged to different degrees with certain ideas depending on the books…

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