Teaching Mathematical Problem Solving: What The Research Says

Recently the EEF published their guidance report for KS2 and KS 3 maths. It gives 8 recommendations for improving the teaching of mathematics:

In this blog post for Bradford Research School I focus in on problem solving but touch on the use of manipulatives, developing a network of mathematical knowledge and other areas of the guidance. In the article I outline a maths lesson which follows much of the advice given in the guidance (the cube trees at the centre of the lesson):

https://researchschool.org.uk/bradford/news/teaching-mathematical-problem-solving-what-the-research-says/

On The @TES Blog: Idealistic Leaders vs. Realistic Teachers

https://www.tes.com/news/school-news/breaking-views/fix-workload-put-ministers-school-leaders-policy-wonks-consultants

"Teachers must…", "Teachers need to…", "Teachers should…"

These are potentially my most used phrases when writing articles on education. Occasionally other groups will be on the receiving end of my strongly worded ‘advice’, but usually it’s teachers because teaching is what I know.

Recently, I have been pulled up on my use of these phrases – turns out teachers don’t like being told what to do. Now there’s a surprise.

My sharing comes from a desire to help others, never from a position of wanting to overburden and bludgeon teachers who are already striving to do their best. But I can see how it comes across sometimes and it got me thinking...

Click here to read more over on the TES blog

Why You Might Be Getting Feedback Wrong! (Part 2)

At the launch of Bradford Research School I held a workshop entitled Why You Might Be Getting Feedback Wrong! The workshop is now available in blog form over on the Bradford Research School blog. It comes in two parts:

Part 1 addresses 5 myths:

• Myth 1: marking is evidence-based
• Myth 2: feedback = marking
• Myth 3: marking is time consuming
• Myth 4: Ofsted require a particular kind of marking
• Myth 5: children need to know what level/grade their work is

Part 1: https://bradford.researchschool.org.uk/2017/12/08/why-you-might-be-getting-feedback-wrong-part-1/

Part 2 addresses 4 more myths:

• Myth 6: marking needs to be done in great detail
• Myth 7: marking will have an impact on progress
• Myth 8: all errors are equal
• Myth 9: triple/dialogic marking is best

Part 2: https://bradford.researchschool.org.uk/2017/12/15/why-you-might-be-getting-feedback-wrong-part-2/

I have also uploaded the PowerPoint I used in the workshop for anyone to use in their own training or discussions with SLT: https://www.tes.com/teaching-resource/why-you-might-be-getting-feedback-wrong-powerpoint-for-staff-training-11795794

Scaffolding Structures for Reading Comprehension Skills

This is a very different blog post to the sort I normally write; it represents some very hypothetical thinking and the purpose of writing it is to open it up to discussion. My hypothesis is that the reading skills outlined in the English Reading Test Framework for KS2 (and KS1) might be best taught in a particular order. I also hypothesise that when teaching particular skills (represented as being higher up the model pictured) teachers can guide children through how to use other skills (lower down the model) to arrive at a better ability to practice and use the skills that are higher up the model. First of all, here's the model I've put together to which I refer:

Skills (taken from English Reading Test Framework for KS2) are listed in the order that they might best be taught. This suggested order is based on the idea that some reading skills might be required prior to developing others. The most basic skills are towards the bottom.

The inclusion of 2d (inference) may depend on the text type. For example, in many non-fiction texts there is no requirement to infer information, only to retrieve it. In these cases the 2d (inference) step/building block can be skipped.

The only reading skill from the test framework which isn’t included here is 2h (make comparisons within the text). It is possible that texts can be compared at many different levels, for example, the vocabulary used can be compared (2a), summaries of plot can be compared (2c) or structure of the text can be compared (2f). The skill of making comparisons (2h) could be seen as a ‘floating’ skill – one which could be applied in different ways alongside other reading skills.

All of the following symbols and colours refer to the Reading Roles, a system I designed to make the different skills memorable for children and teachers. Read more about the Reading Roles here: http://thatboycanteach.blogspot.co.uk/2016/12/reading-roles-cognitive-domains-made.html

2d – Inference (this one is lots of detail here: http://thatboycanteach.blogspot.co.uk/2016/10/scaffolding-inference-trialling.html)

In order for children to begin to make inferences they need to at least be able to retrieve information in the text, and before this they need to be able to understand what the words mean.

(2a)       give/explain the meaning of words in context

(2b)       retrieve and record information/identify key details from fiction and non-fiction

(2d)       make inferences from the text/explain and justify inferences with evidence from the text

2f and 2g are very interlinked as they are both about meaning – one with a focus on word and phrase choice, and one with a focus on content choice. It is possible that 2g and 2f should precede 2d in the teaching sequence but if making inferences is one way in which we take information from a text, then arguably we need that information to make meaning; we can then go on to identify and explain how that meaning is enhanced through word choice and how the content included contributes to the meaning. The fact that these skills are not included in the KS1 test framework might suggest that this is correct, and that these are more advanced skills than making inferences.

2g – Author’s purpose

(2a)       give/explain the meaning of words in context

(2b)       retrieve and record information/identify key details from fiction and non-fiction

(2d)       make inferences from the text/explain and justify inferences with evidence from the text

(2g)       identify/explain how meaning is enhanced through choice of words and phrases

2f – Language structure and choice

(2a)       give/explain the meaning of words in context

(2b)       retrieve and record information/identify key details from fiction and non-fiction

(2d)       make inferences from the text/explain and justify inferences with evidence from the text

(2g)       identify/explain how meaning is enhanced through choice of words and phrases

(2f)        identify/explain how information/narrative content is related and contributes to meaning as a whole

Once children understand word meanings, can find and infer information, explain how language has been used to communicate meaning and, as a result, can understand the meaning of a whole piece of text, then they can begin to summarise the text, or make predictions based on their understanding. It might not be necessary to summarise a text before making a prediction, and the ability to summarise a text should not rely on the ability to make predictions based on it. These two skills are both included in the KS1 test framework, but children at this stage summarise and make predictions based only on word meaning, information retrieval and inference (missing out 2f and 2g) – summaries and predictions at this stage might be at a simpler level. It is probably true that in KS2 similar summaries and predictions could be made, without paying heed to 2g and 2f.

2c - Summarising

(2a)       give/explain the meaning of words in context

(2b)       retrieve and record information/identify key details from fiction and non-fiction

(2d)       make inferences from the text/explain and justify inferences with evidence from the text

(2g)       identify/explain how meaning is enhanced through choice of words and phrases

(2f)        identify/explain how information/narrative content is related and contributes to meaning as a whole

(2c)       summarise main ideas from more than one paragraph

2e – Predicting

(2a)       give/explain the meaning of words in context

(2b)       retrieve and record information/identify key details from fiction and non-fiction

(2d)       make inferences from the text/explain and justify inferences with evidence from the text

(2g)       identify/explain how meaning is enhanced through choice of words and phrases

(2f)        identify/explain how information/narrative content is related and contributes to meaning as a whole

(2e)       predict what might happen from details stated and implied

The model suggests that an understanding of word meaning is core to all reading – this model assumes that children already have the skills of decoding, sight recognition and phonological awareness. The model only includes reading skills outlined by the test framework and does not include factors such as the necessity of activating prior background and literacy knowledge when reading.

The model also suggests that there is a hierarchy of reading skills and that children might benefit from having some reading skills taught before others.

It also suggests that when requiring a child to work on a skill which is ‘higher up’ the model that they work through a sequence of skills usage in order to initially scaffold their ability to exercise the ‘higher’ skill. For example, if requiring a child to summarise a passage, they might first answer questions about the vocabulary used, the information contained within (given both literally and inferentially) and what the authors purpose was with regards to structure and language choices.

This model focuses on the following strands of Scarborough’s reading rope: vocabulary, verbal reasoning and language structures:

I hope I have made my thinking clear in this blog post and I would really appreciate any thoughts about what I have proposed. If you can back any of your comments either with research or with case studies from experience then even better!

If you would like Aidan to work with you on developing reading at your school, please visit his website at https://www.aidansevers.com/services and get in touch via the contact details that can be found there.

My Year As A Teaching Leaders Fellow

I use my blog as a kind of scrapbook, a central place to keep a record of things I've written that have ended up elsewhere in other publications and on other websites. This time I wanted to preserve these photographs from my graduation from the Teaching Leaders programme that I took part in during the '16/'17 academic year.

I was pleased to graduate with commendation and to also have won the Ann Brougham values award for the primary North cohort. I was peer nominated (thank you, whoever you are!) for the award which was created in memory of the first Lead Coach on Teaching Leaders, and is presented to a Fellow who has remained true to their values, supported their peers on the programme and displayed an unrelenting commitment to the Ambition School Leadership mission. The award goes to the Fellow that has demonstrated ASL's core organisational values of mastery, grit, empowerment, teamwork and integrity to an outstanding level throughout the programme. For the prize I was able to choose a book; my choice was Patrick Lencioni's 'The Five Dysfunctions of a Team', and there is significance in that choice.

Patrick Lencioni's 'The Advantage' really helped me to wrestle with gaining a clarity of vision way back in November of last year at the beginning of my Teaching Leaders journey; I blogged about it in a post somewhat bizarrely entitled 'Dogs & Sledges: Harnessing Action To Clarity Of Vision':

http://thatboycanteach.blogspot.co.uk/2016/11/dogs-sledges-harnessing-action-to.html

Patrick Lencioni recommends that once that first question 'Why do we exist?' has been answered leaders can then go on to ask themselves 'How do we behave?', 'What do we do?', 'How will we succeed?', 'What is most important, right now?' and 'Who must do what?'. In schools, we often have deeply entrenched answers to these questions and we carry on in those ways regardless of whether we know our 'why' or not.

In that blog post I also referenced many of the speakers from the Teaching Leaders residential: 

I was challenged by some inspirational leaders to ensure that I was clear in my vision. Steve Radcliffe, coach to powerful and influential figures the world over encouraged me first to think of the future before engaging others in that vision of the future. Andy Buck told me to focus on one thing in order to gain clarity. Baroness Sue Campbell reiterated the need to be clear on where we are going, asking me to consider if everyone gets my vision and wants to follow me. She also caused me to consider whether my targets were good enough and whether or not I knew what great looked like. James Toop discussed creating culture - my key piece of learning from that session: 'Be clear on what my vision is', I wrote in my Moleskine - I knew that without a clear vision I would struggle to create a culture within my own team. Sir David Carter issued a performance challenge, the first point of which was to 'de-clutter'...

Since then I have written three blog posts for the Ambition School Leadership blog, each reflecting on an aspect of my leadership journey:

My Ambition Isn't Just About Me: 
http://thatboycanteach.blogspot.co.uk/2016/11/my-ambition-isnt-just-about-me.html

The education system has its challenges but I see potential in a system whose workforce are positive and optimistic about how they can influence those within their sphere. Imagine the impact that could be had if every leader in every school saw the potential in being solution-orientated, finding innovative ways to make the system work for the schools we work in. It is my ambition to ensure that this is always done, for the benefit of the learners, at the schools I work in.

Leadership Lessons: Letting Go And Letting Them: 
http://thatboycanteach.blogspot.co.uk/2017/03/how-watership-down-and-unique.html

I realised that, as a leader, I often attempted to do it all, even when there were others in my team who were better for the job. Furthermore, the experiential brought it home to me that I actually felt threatened by those who were better at something than I was. I found that I harboured feelings of resentment towards those I was supposed to be leading and my negative feelings were not conducive to good leadership and teamwork. 

Looking Back On My Moleskin Moments:
http://thatboycanteach.blogspot.co.uk/2017/11/on-ambition-school-leadership-blog.html

Although so much of what I learned last year on Teaching Leaders is now internalised and has become a natural part of how I function as a leader, it’s good to know that whenever I need a reminder my moleskine is there, immortalising the wisdom of a year so well spent honing my leadership skills.

Why You Might Be Getting Feedback Wrong! (Part 1)

At the launch of Bradford Research School I held a workshop entitled Why You Might Be Getting Feedback Wrong! It's a slightly clickbaity title, and I wouldn't have been surprised if no-one came to it, after all, who wants to be told they're doing something wrong?

Anyway, people did come and I've begun to write up what I presented in the workshop. The information I was presenting all came from the EEF's review of the evidence on marking 'A Marked Improvement' - my workshop was an attempt to summarise their findings into a 20 minute bite-size chunk.

Read part 1 here: https://bradford.researchschool.org.uk/2017/12/08/why-you-might-be-getting-feedback-wrong-part-1/

Mathematical Misconceptions And Teaching Tricks: What The Research Says

Imagine a factory. Think of the vast machines clanking away. Think of the whirring, the turning, the raw materials becoming a finished product. Beneath those metallic exteriors cogs, cams, belts and levers are working together to effect that change. But all but the most initiated don't really understand how the machines do what they do, they just know that if they put the right parts in at one end, the machine will produce the desired item.

And this is how many children feel about maths. They know that putting some numbers into a calculation will give the desired answer, but they don't really have a clue what goes on inside the 'machine' of that procedure. This is all well and good until that child has to apply this learning - having no understanding of the mechanics of mathematics makes it very difficult to use procedures in context.

In my blog post for Third Space Learning entitled 'Maths Tricks or Bad Habits? 5 Bad Habits in Maths We're Still Teaching Our Pupils' I make several suggestions for how to use visual representations to teach good conceptual understanding of some tricky aspects of the maths curriculum, such as the ones below:

The recent EEF guidance document on improving maths in KS2 and KS3 backs up the importance of modelling good conceptual understanding in maths lessons, rather than relying on tricks that work but don't help children to have an understanding of the 'why' and the 'how':

Recommendation 4: Enable pupils to develop a rich network of mathematical knowledge 

"Pupils are able to apply procedures most effectively when they understand how the procedures work and in what circumstances they are useful. Fluent recall of a procedure is important, but teachers should ensure that appropriate time is spent on developing understanding. One reason for encouraging understanding is to enable pupils to reconstruct steps in a procedure that they may have forgotten. The recommendations in this guidance on visual representations, misconceptions, and setting problems in real-world contexts are useful here."

In order to teach maths well, and in order for children to succeed in maths, teachers need to make sure children understand what is going on when they carry out a mathematical procedure. A great way of developing this understanding is using manipulatives and representations:

Recommendation 2: Use manipulatives and representations 

"Manipulatives and representations can be powerful tools for supporting pupils to engage with mathematical ideas. However, manipulatives and representations are just tools: how they are used is important. They need to be used purposefully and appropriately in order to have an impact. Teachers should ensure that there is a clear rationale for using a particular manipulative or representation to teach a specific mathematical concept. The aim is to use manipulatives and representations to reveal mathematical structures and enable pupils to understand and use mathematics independently.

Teachers should: Enable pupils to understand the links between the manipulatives and the mathematical ideas they represent. This requires teachers to encourage pupils to link the materials (and the actions performed on or with them) to the mathematics of the situation, to appreciate the limitations of concrete materials, and to develop related mathematical images, representations and symbols."

As I wrote in the guide to Bar Modelling that I produced for Third Space Learning (click to download for free):

If we don't do this, we run the risk of allowing children to proceed in their mathematical education with misconceptions:

Recommendation 1: Use assessment to build on pupils’ existing knowledge and understanding 

"A misconception is an understanding that leads to a ‘systematic pattern of errors’. Often misconceptions are formed when knowledge has been applied outside of the context in which it is useful. For example, the ‘multiplication makes bigger, division makes smaller’ conception applies to positive, whole numbers greater than 1. However, when subsequent mathematical concepts appear (for example, numbers less than or equal to 1), this conception, extended beyond its useful context, becomes a misconception. 

It is important that misconceptions are uncovered and addressed rather than side-stepped or ignored. Pupils will often defend their misconceptions, especially if they are based on sound, albeit limited, ideas. In this situation, teachers could think about how a misconception might have arisen and explore with pupils the ‘partial truth’ that it is built on and the circumstances where it no longer applies. Counterexamples can be effective in challenging pupils’ belief in a misconception. However, pupils may need time and teacher support to develop richer and more robust conceptions."

When we do teach children using appropriate models and images so that they understand the mathematical concepts behind the procedures (or the 'tricks'), we provide children with something that they can actually look at and explain. Explaining something that is concrete is easier than explaining an abstract concept.

In the bar modelling guide (click to download for free) I pointed out that:

By developing children's skills to represent and explain their understanding using a model, we develop their independence and motivation:

Recommendation 5: Develop pupils’ independence and motivation

"Teachers can provide regular opportunities for pupils to develop independent metacognition through:

• encouraging self-explanation—pupils explaining to themselves how they planned, monitored, and evaluated their completion of a task; and

• encouraging pupils to explain their metacognitive thinking to the teacher and other pupils."

Next time you plan a maths lesson question how you will ensure that children have a good conceptual understanding of the content you teach. Often, concrete or pictorial representations will be the best way to show children the inner-workings of the concepts you cover. Following Psychologist Jerome Bruner's research-based CPA (Concrete - Pictorial - Abstract) approach means that children (and adults) are more likely to understand what is going on inside the maths machine as calculations and processes take place.

Further Reading and Resources:

• These AET Mathematics plans give ideas for how to represent every single learning objective in the national curriculum from year 1 to year 11.
• My article for Third Space Learning: Maths Tricks or Bad Habits? 5 Bad Habits in Maths We're Still Teaching Our Pupils
• Articles by Mike Askew: KS1 and KS2 Maths – using visual models and Models in Mind
• Mr Bee's article: Fractions? I Love Fractions!