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! 

On The Ambition School Leadership Blog: Looking Back On My Moleskine Moments

Last year I took part in the Teaching Leaders programme from Ambition School Leadership. In my latest blog post for them I reflect on a year well spent with the aid of my trusty Teaching Leaders Moleskine notebook.

Read the blog post here: https://www.ambitionschoolleadership.org.uk/blog/looking-back-my-moleskine-moments/

If you are interested in their primary middle leaders programme, click here: https://www.ambitionschoolleadership.org.uk/programmes/teaching-leaders-primary/

And for their secondary programme, click here: https://www.ambitionschoolleadership.org.uk/programmes/teaching-leaders-secondary/

For their other programmes, explore their website: https://www.ambitionschoolleadership.org.uk/programmes/

On The Third Space Learning Blog: Maths Tricks or Bad Habits? 5 Mathematical Misconceptions We Still Teach Pupils (And How To AvoidThem)

Whilst I'm sure I've been guilty of all of these 'tricks' during my time as a teacher, undertaking my role as maths lead and learning more about best practice has prompted me to become rather passionate about avoiding trick-based teaching in maths

It is also certain that the root of my desire to eradicate this kind of teaching which does little to support conceptual understanding can be found in my own school experience. I remember asking one question constantly in maths: "Yes, but why?". Teachers expected me to rote learn and regurgitate maths procedures but I struggled to remember them because I didn't understand them.

Whilst the list of tricks I've outlined in my latest blog post for Third Space Learning is by no means comprehensive, it will hopefully serve to provoke thought on this matter and will be a starting point for some who are not yet teaching so that children truly understand the maths:

https://thirdspacelearning.com/blog/maths-tricks-bad-habits-we-teach-pupils/

Giving the Gift of Reading: Activities That Promote Reading for Pleasure

After reading this great blog post by Rob Smith of the Literacy Shed I spent some time in reflection, questioning myself, my past practice and the blog posts I've written. I thought about the implications of the piece for the future, in my practice, in my advice to others and in my writing. Actually, I, without noticing initially, demonstrated my mastery of reading, and showed that I am a truly independent reader who automatically spends time responding to what I have read. Exactly what we want to develop in the children we teach, yes?

Amongst lots of other good stuff about reading for pleasure, Rob's blog post paraphrases part of a conversation he had with author Frank Cottrell-Boyce:

'"We make children pay for listening to us read, or reading a great book by making them do ‘stuff’ afterwards. We need space for just giving without the need for payback."

He later echoed this sentiment when he said, "A book given freely unlocks doors for children."'

And it was this, as well as a list of 'payback' activities (see below), that challenged my thinking the most.

Types of payback 

• Comprehension questions based upon the text that was read;
• Finding similar themes/devices/vocabulary in other texts;
• Writing a review based upon the text;
• Writing anything based upon that which has just been read;
• Restricting a child to certain books (band) until they have read enough books sufficiently well;
• Having a test which measures how well the child has been reading;
• Reporting how well – or not well – the child has been progressing in their reading;
• Depending on their responses the child receives extra reading practice to do even if they want to or not.

A note first on the list: the first four types of payback are quite different to the second four types. The first four are more focused on practise of reading skills whereas the second four have a strong assessment focus.

To begin properly then: a look at that term 'payback'. The word 'payback' suggests that when we require children to complete 'payback' activities the reading a child has just completed was not for them in the way that a gift would be for them. It asserts that if we expect them to respond with a 'payback' activity that the activity itself is not beneficial to them, but is for the benefit of the teacher. It also assumes that if teachers give children reading without 'payback' activities that the reading is somehow more for them. But surely children need to value 'payback' activities as for them too, as part of the gift?

Correctly designed 'payback' activities (more on this later as this is key) should develop essential reading skills which, when exercised, lead to better comprehension and as a result, increased enjoyment of the text. We can only really enjoy reading when we have a good understanding of what we've read. No-one who is not at the early stage of reading finds lasting pleasure in the act of decoding. If a 'payback' activity helps a child to learn or practise a skill, then the sense of fulfilment and achievement they experience doing that can too be part of the gift we teachers give. It's at this point that my thinking may have moved away from what Frank Cottrell-Boyce and Rob Smith meant by 'payback' activities - it is probable that I am now thinking about what the alternatives are to true 'payback' activities.

During a child's time in school we have the opportunity to give them more than they be able to give themselves. Yes, we need to do all we can to enthuse them to go home and read books, giving themselves the gift of time spent reading, and we should ensure that the activities we ask them to do in school don't put them off doing this, but when they are in school we have a chance to give them skills and understanding that they might not gain or develop just in the act of independently reading.

The journey to independence is not one of independence. Children need to be scaffolded and supported to reach ever higher levels of independence. You don't become a master swimmer by being shoved in at the deep end. By giving good 'payback' activities (we'll stop using that term from now on because we're talking about something else now) we support a child on their road to mastering the act of reading by becoming independent, just like I was after reading Rob's blog post. Achieving these levels of mastery and independence is what leads to the enjoyment of reading: children who know they will struggle to understand a book will not want to read it for pleasure. And if they are made to read it without having mastered the necessary skills to understand it, they won't experience the pleasure that a child with the relevant skills will.

It then becomes our job (teachers, parents, any adult who interacts and reads with children) to help children to see the value of the activities we guide them to do in response to their reading. This recognition may come implicitly as they see for themselves that the skills we are teaching them and allowing them to practise through the response activities we provide are helping them to understand, reflect more deeply and enjoy what they have read. We may have to spell it out a little more and explain how the response activities are for them, and that being taught skills and having a chance to practise them is part of the gift we give as educators.

The CLPE's Reading for Pleasure publication doesn't actually cite any type of response-less reading amongst its ten things that work when it comes to reading for pleasure (although their suggestion of a read aloud programme doesn't specifically require response). Many of the ten suggestions (numbers 5, 6, 7 and 8 specifically) suggest that some kind of response to a text leads to reading for pleasure - I assume that the responses they suggest link to and flow out of the read aloud programme they advocate.

As well as highlighting to children that response activities can be part of the gift we give, we have to ensure that what we are providing is a gift. This is where the 'correctly designed' part comes in.

A key factor to consider when designing response activities is that they improve interaction - with the text and with other readers. Response activities can be used to develop reading communities (as found in the OU research into reading for pleasure). The kinds of reading communities in the schools they studied didn't happen overnight, they developed: as adults realised there is a natural need to respond to reading they provided more opportunities to do so which eventually led to 'new and extended opportunities for interaction around texts'. We can create these communities, but only by giving children the opportunities and skills to do so - the response activities that we give should fit this criteria.

The OU findings also recognise that developing reading for pleasure in this way is complementary to other reading instruction. Without a teacher planning to develop reading communities and providing reading instruction (amongst other important factors) any time a child spends with a book runs the risk of becoming 'a routine procedure void of reader engagement and interaction'. In avoiding providing 'payback' activities we must ensure that we don't remove that which actually helps a child to receive the gift that books can give. The OU research suggests that without reading aloud and book talk in a social reading environment, time given over to just reading may not always be time well-spent. Click here to read more from the OU research on independent reading.

So, a key message is that response activities (not 'payback' activities), when focused correctly (on talk especially, which can be facilitated by the asking and answering of questions), are actually a part of the gift that we give to children. Allowing and encouraging children to value these activities, rather than seeing them as the price they have to pay for reading something is crucial to their success as readers and actually should lead them to greater engagement with, and enjoyment of, what they read. Even when response activities don't centre on talk, different forms of reading instruction (including other kinds of response activity such as the ones listed as 'payback' activities) can focus on allowing children to engage with and enjoy reading.

It's important that we don't prioritise response-less reading if we have not given children the tools they need to respond independently. Only when we have worked towards this, to paraphrase Frank Cottrell-Boyce, will a book given freely unlock doors for children. As we develop these skills with children through our instruction and the response activities we give, children will begin to read more independently in scenarios where they don't have to complete 'payback' activities - and this should be our goal. Of course, I definitely don't believe that children should never have the chance to read without having to respond in some way, but I also think that we should seize opportunities where we can to develop the skills they need to become successful and joyful readers who respond naturally just as I did when I first read Rob Smith's article.

My Guide To Bar Modelling

My guide to bar modelling, written for Third Space Learning, is now available to download. The guide includes information on different types of bar model, how to use them across the primary phase and in different areas of the maths curriculum.

The download also includes a PDF of PowerPoint slides which can be used for staff training purposes.

https://www.thirdspacelearning.com/resource-ultimate-guide-bar-modelling/

Book Review: 'The Great Paper Caper' by Oliver Jeffers

Crime novels aren't for children, are they? Are they? Well 'The Great Paper Caper' is. As usual Oliver Jeffers matches his quirky imagery with text bursting with wry, dry humour. Someone is chopping down the trees, but who? And why? The inhabitants of the wood set out to discover the answers to their questions with amusing and heartwarming results.

If ever you wanted to introduce small children to technical legal vocabulary then this book is for you. Alibi, investigation, examined, eyewitness report, evidence, culprit... they're all in the there. And not only is the terminology introduced, the book also provides a great place to begin learning about the process of solving crimes and the following legal processes.

The book throws up some excellent discussion points around justice (Did the culprit get what he deserved? Should criminals be allowed a second chance?) and motivation (Can crime ever be justified? Should we be empathetic towards criminals? How far should you go to be the winner?) making this a useful text to read to inspire debate in the classroom. It also has an environmental focus: teachers could look further at how nature is being damaged by humans, potentially by providing linked non-fiction texts which children would be more inspired to read because of the story's context.  

'The Great Paper Caper' is a great example of how picturebooks use images to do more than illustrate a text. The illustrations have to be read and interpreted too - without them the story would be incomplete as the text alone does not give all the details. In the context of the whole story there are plenty of opportunities for teachers to do reading comprehension style activities (particularly focusing on inference skills, it is all about finding clues, after all) using just the pictures.

This is another triumph for Oliver Jeffers, and one that has stood the test of time; it was first published in 2008. Here is a book which a child can enjoy alone, with an adult, or during a variety of different school lessons, and one that's sure to raise a few smiles as well as questions.

From The @TES Blog: Will Boys Be Boys?

Whilst I acknowledge biological differences between boys and girls, I also think we should judge them individually, rather than on their gender. Anything gender-specific that means they might underperform needs to be addressed, rather than pandered to. And, regardless of gender, all children should be subject to high expectations. Boys deserve to be expected to do well at school.

And when I refer to equal treatment I mean something along the lines of providing children with different opportunities to help them achieve the same outcomes. Some may refer to this as equity rather than equality.

Now, please read my piece for the TES: on how I think we need to treat boys in the classroom:

https://www.tes.com/news/school-news/breaking-views/boys-will-be-boys-wont-they-only-if-we-let-them-be