Archive for the ‘Mathematical’ Category

Musings: Teaching Math With Examples by Michael Pershan

September 7, 2021

The book

Pershan, M. Teaching Math With Worked Examples. 2021. John Catt Educational

My reading viewpoint

My current role allows me to explore different ways in which mathematics can be taught to primary school children. Therefore, when I saw this book on Twitter, I felt it important to have a look and see what I thought. Teaching with examples is something that appears in a number of books including Craig Barton‘s. Of course I, like all teachers use examples in my teaching all of the time. This book focuses attention on using these effectively to teach new mathematical concepts.

I also have to thank the NCETM for being kind enough to interview me for their podcast, thus prompting this review!

Summary

Teaching Math with Examples does exactly what it says on the tin. The author is an experienced (American, hence the name) maths teacher who makes use of worked examples in his daily teaching. He has researched this approach extensively and recognises that teachers often do not have time to fully engage with educational research. In this book, he shares both an accessible introduction to the research and ideas around worked examples and also practical advice from his own classroom.

So what exactly is an example? As I said, most teachers will use examples constantly in maths lessons, so why do we need a book to learn about it? Well, Pershan argues that examples should be worked – that is, include the solution. While it is a little more nuanced than that, he argues that ‘if we value achieving mathematical understanding, we can see the studying of a solution for what it is: a core mathematical act.’ (p.12) So the book largely focuses on teaching using these worked, completed examples – sometimes correct, sometimes not, sometimes complete, sometimes not – and how using these can make us more effective.

This isn’t a new idea and I have read numerous books extolling the virtues of a worked example. However, what I really like about Teaching Math with Examples is the depth of understanding I have gained from this approach. Other books lack the detail and research. I also like a book that provides a resource and this is no exception – check out SERP’s Math by Example!

Teaching Math with Examples is easy to read and easy to see where it can be applied in the classroom. I like the layout with summaries ending each chapter, heaps of research distilled into easy-to-read bursts and a great final chapter which basically does my key takeaways for me! The only small drawback is since it is an American text, sometimes some of the explanation wasn’t clear to me, for example looking at equations. I am sure a more experienced mathematician would have no trouble, though, and it didn’t really detract from what I took out of the book.

My key takeaways

  • Just how much are my pupils thinking when I work through an example with them, and what can I do? Pershan acknowledges that worked examples are not some magic bean to understanding: ‘students learn when they think actively and deeply about a worked example’ (p.20). He suggests a process of Analyse – Explain – Apply when approaching a worked example and that made me think about how I show workings to pupils. One of the things I want to try out is focusing attention on a completed example.
  • Focusing on completed examples can help all pupils. Imagine a scenario where you flummox a class with a problem, wondering why they just don’t get it. Providing a worked example for pupils to engage with, followed by a similar example, is – according to Pershan and his research – likely to lead to more success and more learning. I also think this approach can be great for my higher attaining pupils – can they reason and explain the example? Can they articulate the need for each step?
  • Part- and whole-task practice may be something we miss in primary. Teaching Math with Examples dedicates Chapter 5 to this idea so I won’t go into too much detail. Essentially I read part-task practice as working on the small steps to answer a problem e.g. how to add, how to find an equivalent fraction, how to simplify. We do a lot of this in primary and many of us do it well. But what happens next? Do we ask them to do Step B while still thinking about Step A? Step C that relies on Step A? I’m not sure how often I explicitly help my pupils to connect the dots of part-practice.

I think you should read this book if…

… you teach secondary mathematics.

… you are a Maths Lead or someone with responsibility for curriculum / pedagogical decisions in teaching maths in primary

… you have read about worked examples and want a more detailed look.

Musings: Thinking deeply about primary mathematics by Kieran Mackle

February 8, 2021

The book

Mackle, K. Thinking Deeply about Primary Mathematics. 2020. John Catt Educational.

My reading viewpoint

I was lucky enough to be sent a copy of this book by Kieran himself. Since then, I have taken part in his wonderful podcast which I thoroughly recommend for all things primary: not just maths! I have also realised that my name is actually printed in his book – in fact, I get a whole paragraph. (Page 271 if you’d like to see the lovely things he wrote about me!) However, all of this has not impacted upon my honest review! This is also Kieran’s second book, the first of which I also reviewed, so needed to read this one.

Summary

Thinking Deeply about Primary Mathematics is written to do exactly what it suggests: support a new or early career teacher in considering the complexities and nuance associated with teaching mathematics in a primary classroom. With chapters including development of CPAL (concrete, pictorial, abstract, language), making use of stories and structures for planning a mathematics lesson, Kieran has selected some great key aspects that I would choose to develop with trainees that I work with. Images, clear explanations and walkthroughs all help make ‘thinking deeply’ something tangible and achievable.

Each chapter is set out into clear segments which are detailed at the beginning of the chapter, giving a clear roadmap and a way through the more challenging aspects, such as threshold concepts. Chapters are punctuated by thinking deeply tasks which readers are encouraged to do and engage with. If this wasn’t enough, Kieran has developed a website full of videos and additional support to guide readers through mathematics at primary.

While I do think that occasionally the writing can be a little heavy-going in terms of language used, on the whole Thinking Deeply is an easy text to read. While connections are made across the book, it is possible to dip in and out of chapters. Kieran’s dry sense of humour and anecdotes support enjoyment of reading and I found myself chuckling at various points.

It’s also worth noting that actually as an ‘expert’ in my field, I gained knowledge and insight from reading this book. While it is directed at more novice teachers, I think that the more experienced amongst us – particularly if we are in a position where we mentor others – could learn from these pages. Thinking Deeply is the first edu-book where I have simply highlighted an entire paragraph and written ‘THIS’ next to it. Underlined. If that doesn’t suggest I have learned something I don’t know what else will!

My key takeaways

1. We could all do with thinking deeply about the representations that we use in the classroom. Thinking Deeply dedicates a large chapter to CPAL representations, but prefaces this with the need to think deeply about the structures we use. Representations should highlight the underlying mathematical structure and focus attention on this. Kieran highlights some representations such as arrays that are vital to primary classroom practice, and made me think about bar models (specifically the way we represent subtraction as ‘take away’) more deeply.

2. Depth and challenge questions need careful thought: do the maths! This point is made much better in the book than here, but it really highlighted something to me, so worth a mention. We often take a challenge / depth task for granted: that is, someone tells us that it is a challenging task and we accept that, often because the person stating this is more expert. All teachers should really consider the task in terms of the maths but also in terms of why the task is challenging. Doing the maths is a key way to think deeply here.

3. A simple question allows access to maths. Amongst the many strategies shared, asking ‘what do you notice?’ stuck out for me simply because I don’t use it all that much. I ask similar questions around what pupils know, what is similar or different, but the simplicity of this question, worded in this way, allows access for all. As Kieran says, even if the pupils notice the most arbitrary, superficial aspects of the task, they have engaged with the mathematics’ (p.169).

4. Worked examples are (nearly) everything. A worked example isn’t something new to me, or most teachers. We make use of them regularly and many others (Craig Barton in particular) talk about them at length. However, I admit to not thinking deeply enough about my use of them and about their power. ‘The worked example is one of the most powerful tools at our disposal as teachers, particularly useful when utilised with disadvantaged/academically-disadvantaged pupils’ (p.192).

I think you should read this book if …

  • You are a trainee primary teacher
  • You are an early career primary teacher wanting to hone your mathematical teaching
  • You support / mentor / coach either of the above and want to think more deeply

Musings: Mathematical Tasks by Chris McGrane and Mark McCourt

October 16, 2020

The book

McGrane, Chris & McCourt, Mark. Mathematical Tasks: The bridge between teaching and learning. 2020 John Catt Educational.

My reading viewpoint

I chose to watch Chris speak at the last MathsConf event I attended, and so I felt it pertinent to get his new book. I love a good task, and I love the way in which I can be shown something and make further connections. I am also in a new role and part of this involves thinking about tasks that can be used in primary classrooms.

Summary

I am going to start by addressing the mathematics within the book. While some of the mathematical tasks used as examples were accessible to ‘primary maths’ me, I would say the majority involved concepts beyond my mathematical understanding. I know that some primary practitioners can find this daunting but I want to say first that with the exception of maybe one example, not understanding the mathematics did not hinder my understanding of what the intention of the task was, or what point the task was trying to illustrate.

This book has quickly become one of my favourite edu reads, simply because there is so much that I can take away and use straight away in class. In fact, I have planned out a whole sequence of learning on factors, multiples, primes and squares inspired by the book, to use with my Y5 class after half term. I am always a fan of a book that provides instant ‘I can do this in my classroom’ as well as further food for thought. This does both.

In essence, McGrane argues that tasks should be one of the fundamental foci when considering how to improve teaching and learning in the classroom. Through a readable blend of research, his own experiences and a series of interviews from some of the best mathematical minds, he sets out the importance of tasks, making explicit links with other fundamental foci including assessment and pedagogy. In order to develop pupils as mathematicians, we need to ensure they have a wide and varied diet of maths: tasks are an excellent way in which to do this. Obviously, he acknowledges that maths lessons will almost always contain tasks, but that we need to be more carefully considering the nature of these tasks, how they are presented to learners and what can be learned from them.

Chris establishes three main task types: procedural fluency, conceptual understanding and problem solving (p.103) although does recognise that this distinction can sometimes be arbitrary when classifying a task: instead we think more of a ‘primary aim’ (p.105). It is in the latter two thirds of the book that these task types are unpicked, explored and exemplified and this is the section of the book I feel everyone can get most value out of.

Drawing upon a range of approaches (some familiar to me as a primary practioniner, some less so) including spaced practice, retrieval, goal free problems and Increasingly Difficult Questions, Chris exemplifies each task type with numerous examples. What I particularly enjoyed was the reflection and analysis that followed: even if the mathematical concept of the task was beyond my reach I could still make sense of the purpose of the task and therefore learn from it. In fact, many of them I have identified as adaptable for my own classroom practice. I could spend an age summarising each area, but I have instead selected a few key ideas that were most interesting for me, below.

While I am sure that a number of the tasks included could be lifted or tweaked and used in the classroom, I very much like that this isn’t the point. I enjoyed reading about task development, pitfalls and how teachers can get pupils to think more deeply as opposed to completing more ‘mundane’ or routine tasks.

Chris’ writing style is easy to understand and clear, making it a joy to read. Trust me when I say that you will need highlighters / tabs / a notebook at the ready!

My key takeaways (well, a selection!)

1. Maths teachers should be aware of Teaching for Robust Understanding (TRU). Schoenfeld (2013) sets out some criteria for Powerful Mathematics Classrooms and although the ideas are ones I was aware of, I particularly like the joined-up thinking here. From noting that maths classrooms should be dynamic, with pupils being active learners to identifying the need to engage all pupils, I think this is a set of dimensions I could live by!

2. Procedural fluency tasks can be exciting. I am all for fluency, and pupils developing strategies but recognise that maths is so much more than this, and I need my pupils to see that too. McGrane brings this to life in a series of examples, including etudes, something I was unfamiliar with. If we are to truly meet the three aims of the primary National Curriculum, we should definitely be thinking about the ideas provided.

3. More, same, less tasks are awesome. I have in fact done similar tasks before without really understanding the power of them. McGrane argues that this task type develops conceptual understanding whilst allowing lots of opportunities for reasoning, and a focus on the key concept and not simply a procedure (p.168-169). The NCETM have a generic task template and, once pupils have been shown how the task works, the adaptations are endless. This is great for routine: pupils know what to do and what is expected, therefore they can focus on the maths and less on how to complete the task.

4. Task design is HARD but we are not alone. The thing about textbooks, the internet, books like this and Twitter is that myriad tasks exist. We can find one and use it, without much thought. However, is that task always ‘right’? Does it truly fit the purpose? Chris recognises that designing a task is hard, drawing on experience about the iterative, creative process. However, he also provides a number of suggestions for templates which could be a great starting point. Open middle tasks are a particular type I think are often under-used in the primary classroom.

I think you should read this book if…

  • You are interested in tasks, whether that be designing them or why the ones you use are designed the way they are.
  • You teach mathematics at any level and want to develop whole mathematicians.
  • You are developing a curriculum / scheme of work / resource bank and want to be inspired.

Musings: Reflect, Expect, Check, Explain by Craig Barton

March 7, 2020

The book

Barton, C. Reflect, expect, check, explain: Sequences and behaviour to enable mathematical thinking in the classroom. 2020. John Catt Educational.

My reading viewpoint

I was keen to read and review this book for two reasons:

1. I have read and reviewed Craig’s first book. I enjoyed it, it gave me a lot to think about and therefore wanted to read more.

2. A quote on the blurb struck me and colleagues as, well, interesting: ‘Some students think mathematically. They have the curiosity to notice relationships, the confidence to ask why, and the knowledge to understand the answer. They are the lucky ones.’ There is more on this in the summary, below, but research tells us that all human beings, even very young ones, have capacity to think in what is termed a mathematical way: often called innate learning powers or skills. So from the blurb along I wanted to know how this was going to play out.

As always, I read with a primary maths viewpoint.

Summary

The focus of Reflect, Expect, Check, Explain (from now abbreviated to RECE) is around mathematical thinking. Contrary to the above point in the blurb, Craig’s book sets out a series of activities, prompts and thoughts as his ‘attempt to find a structure to enable the majority of students, in the majority of classrooms, to think mathematically, for the majority of mathematical ideas’ (p.542).

As with all Craig’s writing, RECE is easy to read in the sense that it is clearly written. Key vocabulary and ideas are summarised and explained, images (including SO MANY FLOWCHARTS) support understanding of the ideas explored and how they fit with the bigger picture. It even made me laugh sometimes! I think it helps if you have read the first book, but maybe my summary will be enough. Certainly I think some pre-reading around cognitive load, variation and retrieval / memory is useful.

In terms of structure and layout, I also particularly loved the summary at the beginning of each chapter and the various points for the reader to stop and think about things.

RECE introduces some key ideas and activities which Craig has used himself to promote mathematical thinking. Throughout the activities, he makes use of a behavioural sequence forming the title of the book: by supplying students with lots of opportunities to reflect (what’s the same/different?), expect (predict), check (do the maths), explain (what is going on?) we promote mathematical thinking – opportunities to pattern seek, make conjectures, generalise and so on. These phrases and terms are much more nuanced than this. Craig allows this to happen in a number of ways but central to the book is ‘intelligent practice’ alongside purposeful practice, problem solving and other ideas more deeply demonstrated in the first book.

The examples and maths contained within the book are from secondary maths (as this is Craig’s domain!) with some being KS2 applicable. This made some examples in the book – which readers are encouraged to complete – tricky for me to do, and very tricky for me to see what was being explored through the patterns and so on. This isn’t a criticism, more a note for primary practitioners to be aware of.

To return to the point Craig makes on the back of the book … This summary does not fully do the book justice (nor do my key takeaways, below). It is a treasure trove of ideas: some you will like, some you will dislike, some you will want to use straight from the pages and some you will want to adapt. Craig himself makes this point regularly: RECE is an account of what he does, not a bible for what works in mathematical thinking. The point made in the blurb isn’t carried through in the text. What is carried through, I think, is the identification that we all see in the classroom – that actually some students appear to be able to think more mathematically than others. What RECE does is offer a series of ways to encourage all students to reach this level of mathematical thinking.

My key takeaways

1. Intelligent practice sequences are interesting, important, and I want to do more of them. While this blog post does not have enough words to fully articulate what Craig defines as intelligent practice, he makes an interesting point from Watson and Mason that if students only ever see ‘random’ sequences of questions they will not have opportunities to think mathematically (p.49)- to spot patterns and make predictions. Craig defines intelligent practice sequences as: ‘sequences of questions which enable students to gain practice in carrying out a mathematical method, whilst at the same time providing opportunities to think mathematically’ (p.58). RECE is littered with these and a much deeper explanation. In short, I really love this idea for exploring variation, boundary examples, non-examples and so on.

2. Practice is important, but it could be so much more. This is attached to takeaway 1 and 3. RECE makes the point that there is a place for practice and sometimes we need to check learning and understanding through a collection of problems. However, my own understanding of variation and pattern has been deepened through reading this book. I am more attuned to the idea that we could make much more out of these sequences, particularly through prompting pupils to reflect before each one. Making opportunities to discuss relationships, unusual examples and so on provides opportunities for students to get a more complete picture, and we should do more of it.

3. We need to think about learning as ‘episodes’ not lessons. I am sure Craig’s first book spoke about this, but what has really hit home for me in RECE is his consistent use of ‘learning episodes’ to describe the sequence of learning a particular concept. The learning episode is not tied to hour-long silos that are lessons; it does not have a set time scale for each element. Instead ‘it takes as long as it takes’ (p.229). Now of course with the external pressures on teachers this is not always something that we can achieve however I do think a shift is needed in the way we think about things.

4. We need to ensure we confront the ‘unusual’ and the ‘obvious’ when providing examples for pupils. Unusual examples are also called boundary examples and RECE makes the point that some teachers only present these as extension, making unusual examples ‘weird and unfamiliar’ because we don’t introduce them early, meaning students have and ‘incomplete’ and fragile picture (p.307). Personal case in point: when expanding brackets, RECE includes examples such as (2 + x) (x + 3). For the life of me I cannot remember seeing an example where the unknown did not come first in a bracket. And it threw me! Confronting the ‘obvious’ refers to examples which check understanding we may see as obvious. From the same example: question 1 is (x +2) (x +3) and question 2 is (x + 3) (x + 2). As Craig points out, introducing ‘obvious examples’ early on is a great way to check understanding.

5. Sequences allow us to bring back high value concepts. For Craig the high value concepts differ slightly from my ‘primary brain’ but for me, high value concepts at, say, Y6 primary include decimals, larger integers, number bonds, multiplication tables. Using sequences allows us to focus on the patterns and predictions and generalisations as well as returning to some key concepts that will crop up for pupils again and again.

6. It’s no good producing wonderfully well-thought out sequences if we don’t actually then think mathematically about them. Tied to all of this, I think, is teacher expertise and class discussion, particularly in the primary classroom. There’s a great sequence in RECE for example that, with a tiny bit of tweaking, I would happily give UKS2 pupils. Some might see the patterns, some might be surprised by answers. Some might get question 1 wrong and make incorrect assumptions about the rest. The role of the teacher here is key. Using reflect, expect, check, explain (or variations of) behaviours in class can turn a sequence of practice into a sequence really promoting mathematical thinking.

I think you should read this book if …

  • You are aware of variation theory / retrieval practice / mathematical thinking and want real examples of what this might look like in the classroom
  • You are a secondary maths practitioner, or a primary practitioner not put off by some more complex maths to think about

 

Musings: Maths on the back of an envelope by Rob Eastaway

February 15, 2020

The book

Eastaway, R. Maths on the back of an envelope: clever ways to (roughly) calculate anything. 2019. HarperCollins Publishers.

My reading viewpoint

I was lucky enough to attend a workshop run by Rob at Mathematics Mastery’s annual conference, where he shared some of the ideas found in the book. To add to this, he kindly donated a copy of his book to every attendee. As always, I read with a primary hat on but also with a love of maths!

Summary

This book does exactly what it says on the tin. Written in an easy style, straightforward to read, Rob takes the reader through a series of ideas and premises which allow you to explore how to roughly work out any mathematical problem. Why? Rob argues that in real life, we rarely need to know an exact number. Instead, we need to know a ballpark, a rough estimate. ‘The point of back-of-envelope maths is to help see the bigger picture behind numbers’ (p.6) and if you need convincing that this is the case then this book absolutely does that.

The book begins with some ‘tools of the trade’ – ideas on which the ability to roughly manipulate numbers in your head relies on (see takeaways, below). Then Rob explores some everyday mathematical situations where these could apply. This makes for fascinating reading, covering practical things such as conversions and statistics. Finally, Rob introduces some ‘Fermi problems’ to explore.

Overall, for someone who loves to manipulate numbers this was great food for thought and has given me some really great ideas for supporting estimation. If mental maths isn’t your ‘thing’ then the first two parts are still a great read to consider the applications for the classroom – both primary and secondary.

My key takeaways

1. If we want adults to be able to quickly consider the truth behind the numbers they see in everyday life, we need to support their recall of facts. In the ‘Tools of the trade’ section, Rob lays out a number of strategies and facts that, he argues, everyone should have an awareness of in order to be able to work with everyday maths. Alongside the standard number bonds, multiplication facts, powers of ten and percentages, Rob suggests we should all have an understanding of some key facts, including an idea of the world population, the UK population and so on. It made me think about how explicitly we are making pupils aware of these facts and how they can be used to support thinking about numbers they see in, for example, the press.

2. Estimation is key: teaching it, including it and thinking about it should be a regular part of maths teaching. What is interesting for me here is that often I taught estimation as a ‘thing you do before an actual calculation’. While that has benefit, after reading this book I am less sure how much we teach estimation for estimation’s sake. As Rob eloquently writes: ‘If we are trained to believe that every numerical question has a definitive, ‘right’ answer then we miss the fact that numbers in the real world are a lot fuzzier than pure maths might suggest.’ (p.3).

3. Miles and kilometres conversions are particularly important for UK children. While the conversion of miles to km is included in the KS2 NC, I wonder how much attention is paid to it in reality in the classroom. But as Rob points out, we are a ‘metric country’ which uses miles for our road signs, which still thinks about height in feet and inches. Go across to Europe and it’s kilometres; I even hired a car in Belfast once where the speedometer was in kilometres. With such a hybrid approach, we need to think carefully about how to support pupils to estimate conversions.

I think you should read this book if…

  • You have an interest in mental mathematics and find it fun to stretch your mind in this way.
  • You are a primary teacher or secondary maths teacher (or geography, science etc.) who wants to think more deeply about the role of estimation in your classroom.

Musings: A Compendium of Mathematical Methods by Jo Morgan

December 29, 2019

The book

Morgan, J. A Compendium of Mathematical Methods. 2019. John Catt Educational.

My reading viewpoint

I was lucky to receive a review copy of this from Jo. As a self-confessed maths geek I was keen to read and learn from this book. As always, I approach with a primary maths focus.

Summary

Jo’s book does exactly what it says on the tin: shares a huge selection of methods and strategies for a range of mathematical concepts and operations. Beginning with the essential four operations, links and connections are made across methods and more complex mathematical ideas. The methods range from those commonly taught across primary and secondary phases to methods found in other countries and other centuries. Excerpts from old textbooks adds to the intrigue.

Each chapter follows the same format, which made reading and grappling with the ideas more straightforward. After a useful vocabulary check, each method is exemplified using the same two examples throughout the chapter. Jo methodically works through each method, allowing for readers with less competence (like me with surds!) to develop their understanding and follow the process. Each method comes with a ‘Jo says’ box sharing her thoughts. As Jo herself states, these are not explicit preferences or criticisms but they do offer a real insight, for example by highlighting where a method would not work, is not intuitive or particularly efficient. The use of the same calculations and different strategies is brilliant as it allows for the reader to make connections between methods and across concepts.

As noted above, connections are made between different methods used in different chapters. For me personally this was a huge help, particularly when reading and thinking about concepts that are beyond my area of expertise such as dividing polynomials. The Compendium covers secondary mathematics however a number of chapters are incredibly interesting and useful for primary practitioners.

Overall, I found this book fascinating. I never knew there were so many methods and strategies and have learned a few myself (see below) as well as deepening my understanding of those I was aware of. Jo is clear and concise throughout, and I loved the ‘try it yourself’ options at the end of each chapter which allowed me to have a go with some of the methods I liked. While I read The Compendium cover to cover, it would be a great go-to ahead of teaching a particular concept and could easily be read by chapter.

My key takeaways

1. Exploring methods deepens subject knowledge. OK, this may be an obvious starting point. But for me, this was particularly the case for multiplication (see below). Through exploring and trying out a range of methods for different concepts, I have a more secure understanding of the way an operation works. This would be a great post-SATs exploration for Year 6. It also expanded my understanding. For example, I am aware of the constant difference model for subtraction, but I never thought to apply it to negative numbers i.e. 3 – (-4). My note simply says “This is AWESOME!”

2. Lattice method for multiplication is something I would teach. I have seen the Lattice method before, but have never fully understood it until now. Not only does Morgan provide two clear examples, she also explores why it works. It now may be my new favourite method.

3. I have learned new methods. I could include examples here, but I know without full explanation and exemplification they will be meaningless. Go buy the book and learn some for yourself!

I think you should buy this book if…

  • You have a procedural understanding of many concepts and want to deepen your understanding.
  • Like me, you love reading about different approaches and strategies.
  • You teach maths in UKS2 or secondary.

 

 

Musings: Teaching for Mastery, by Mark McCourt

July 15, 2019

The book

McCourt, M. Teaching for Mastery. 2019. John Catt Educational.

My reading viewpoint

I was lucky enough to hear Mark speak a couple of years ago on mathematics teaching in schools and, like many, was struck by his no-nonsense and, let’s be frank, sometimes a bit non-PG discussions. I have also attended a workshop run by him, and follow him on Twitter. While I do not always agree with his views I find him fascinating and thought provoking. It was only natural therefore that I was keen to read his book, a review copy of which was kindly provided to me by John Catt.

Summary

Trying to summarise the ideas within this book is like trying to hold 1000 Dienes blocks – some will inevitably slip through my fingers – so there may be some points that are important but fail to make my summary. I can only apologise (and suggest you read the book). This review is also a little sprawling: I have much I want to say!

Teaching for Mastery is a collection of ideas for a mastery approach to teaching. While applicable to all subjects and all teachers it does skew its examples and approaches to the teaching of mathematics.

Mark begins by setting out a view of mastery teaching, putting it against what he terms ‘conveyor belt teaching’ of schools cramming curriculum points into neatly assigned year groups. He acknowledges the fact that every child has capacity to learn, but that by attempting to, say, force an eight-year-old to learn what is deemed mathematics for age eight is not going to be successful if pre-requisite foundation knowledge is not in place. He puts a compelling argument forward for ensuring teachers take the starting points and existing schema for every child into account, building on this before introducing a new concept. However, while this is an ideal, I do think Mark acknowledges that in many schools now, and probably for a long time, the ‘conveyor belt’ will continue to exist. The majority of the book covers aspects of ‘teaching for mastery’ that can also be applied to those teaching a year-group based curriculum and, therefore, applicable to the majority of teachers in the UK and beyond. In this sense, whether you ascribe to the view of mastery teaching posited in the first 30-odd pages it does not matter – you can learn from it, and the rest of the book, and apply it to your own teaching.

With sections on cognitive science, variation and development of learning episodes focused on research into interleaving, spaced practice and other concepts, ‘Teaching for Mastery’ has applications beyond mathematics, and therefore the ideas can be considered by secondary teachers and applied by primary teachers to other subjects. However, with a large segment titled ‘A mathematical diversion’ I do believe the book will ‘speak’ to those of us who teach mathematics more. In all sections, Mark sets out what I perceive to be sound teaching pedagogy – whether your school is ‘mastery oriented’ or not, there is a lot to learn.

The term ‘learning episode’ and what it might look like is explored more comprehensively in Part V – Phasing learning but is in essence a recognition that learning a concept cannot take place in a single lesson, or in, say, the two-week unit in Year 3 on ‘Shape’. Again, this flexibility in approach to timings is discussed widely throughout ‘Teaching for Mastery’ and may feel uncomfortable with teachers used to teaching in timed units. However, I do think that these ideas and strategies put forward can influence schools teaching in this manner. After all, learning episodes can occur over years, and so, for me, it’s about the recognition of returning to elements and aspects, revisiting and building upon these.

McCourt’s voice comes through clearly – he is no nonsense, secure in his viewpoints (although with a caveat that they may change as understanding develops) and this makes the book easy to read. Words that are open to misinterpretation are established through development of a ‘common vocabulary’ – shared definitions which work throughout the book – and mathematical terms readers may be less familiar with are explained. Diagrams are used throughout to support understanding and illustrate points and I found these both effective and engaging.

While I personally believe that some of the ‘mastery school’ approaches laid out are unachievable for a teacher (requiring a shift in essentially Government reporting and monitoring systems) I do believe that elements can be reached by all teachers. Mark’s conclusion, that “a mastery model of schooling is a responsive cycle of teaching in which the teacher continually seeks to build knowledge of a discipline over time in a carefully scheduled, progressive journey” (p.317) is a model for teaching I believe every teacher can agree with and aspire to.

My key takeaways

1. “All pupils can learn all things, given the right time and appropriate conditions – and those conditions include impactful teaching.” (p.23) McCourt states this regularly throughout ‘Teaching for Mastery’ and I think this is a point that can be overlooked. But how do we achieve this? In current schooling systems, providing high-quality intervention can be challenging, and providing adequate time for building foundations when others are ready to move forward can feel like an insurmountable task. I am not sure that any suggestions Mark, or in fact anyone, can come up with can answer these questions in the current education climate. However, this point is worth mentioning, regularly reminding ourselves of, and considering how we can make it a reality in our schools.

2. Identifying pre-requisite knowledge is vital, and ensuring it is in place is key. Again, this point is one Mark recalls over and over again throughout the book. Subject knowledge of the teacher is key here – Mark argues that all teachers should have an understanding of the scope of mathematics to be learned at all ages, and a deep understanding of the connections between them. This can be a challenge, I feel, particularly but not limited to primary teachers where subject-specific knowledge a) may not be at the required depth and b) has to be deep for every subject taught. Within this, Mark also makes the key point that all pupils, at any age, will come to a novel concept with existing schema, and as teachers we need to tap into this and support them in making connections. He suggests that ‘ensuring a novel idea fits with the pupil’s current understanding’ (p.275) is key and, again, teacher subject knowledge plays a vital role in allowing this to happen.

3. Showing non-examples, even to novice learners (with caveats) is important in developing a deep understanding of the concept. Mark argues that a mastery teacher needs to plan experiences that highlight ‘not just what is the same but what is different’ (p.97). With novice learners, attention should be drawn clearly to the fact that these are incorrect, with the teacher identifying the flaws of the example (p.304).

4. Teachers need to do the maths. This is a point I cannot agree with more. McCourt suggests that while creating a task is worthwhile, engaging in the mathematics behind a created task will allow teachers to carefully consider the thinking behind it. This makes sense – doing the task will allow teachers to see potential pitfalls, misconceptions, directions to go and questions to ask.

5. Don’t lie to your pupils! OK, Mark puts this more eloquently but again this is a recurring theme, particularly in Part IV – A mathematical diversion. So often teachers (including myself) teach pupils incorrect mathematics e.g. you cannot take 7 from 3. The effects of this can be far-reaching, and so teachers need to find ways of explaining things correctly first time (p.114-119).

(I could go on. I took so much from this!)

I think you should read this book if…

  • You teach mathematics. Ever. To pupils of any age.
  • Note: You may also want to read this if you teach any subject.

 

Musings: Making every maths lesson count by Emma McCrea

May 15, 2019

The book

McCrea, EmmaMaking every maths lesson count: Six principles to support great maths teaching. 2019. Crown House Publishing

My reading viewpoint

I thoroughly enjoy reading different views on what makes great maths teaching, and I read a lot around the subject in order to support schools I work with, so this was an obvious choice. I was lucky enough to receive a review copy from Crown House and I am very grateful to them.

Summary

For those of you unfamiliar with the ‘Making every…’ series, allow me a moment. This series of books covers a huge range of subjects, these days (see my review on the primary teaching version here) and began with ‘Making every lesson count’. From my limited understanding (based on two books and the introduction to this one) each follows a framework of six principles, adapted and focused on the subject at hand. This means that reading this book felt familiar, which was pleasant!

The chapters are well laid out and clear, taking the reader through the six principles: Challenge, Explanation and Modelling (rolled together because, as McCrea points out, you cannot truly have one without the other in maths teaching) Practice, Questioning and Feedback. As I mentioned, these principles underpin good teaching and feature in all the ‘Making every…’ books, but they are specifically related to maths. Within each chapter, McCrea combines evidence-informed pedagogy (drawing on aspects including instruction, variation and cognitive load theory) with practical activities and suggestions, breaking each principle into a number of strategies (helpfully listed at the back of the book). While a higher proportion of the activities and maths problems provided are from secondary objectives, a number are primary and the strategies are applicable whichever phase you teach. Each chapter ends with thoughtful reflection questions to guide a teacher’s application of the ideas to their own setting.

Each chapter is laced with examples (pictures, tables, problems) and peppered with illustrations to break up the text. As noted earlier, the examples are often secondary in nature, but all, I feel, are within the grasp of any teacher in terms of accessibility. Mathematical concepts are also explained in a way teachers can understand even if subject knowledge of that concept is lacking. There are also some really great references to places to find out more, including sites to find specific types of activities. This is incredibly useful and they relate to a range of aspects.

My key takeaways

1. Challenge is important, but teachers should carefully consider how challenge is introduced and how it is developed. I, of course, am a huge advocate of challenge for all pupils in all lessons and understand the importance of aspects like desirable difficulties. McCrea delves deeper into this by introducing teachers to two ways of thinking about challenge – Depth of Knowledge (DoK) and FICT framework (familiarity, independence, complexity, technical demand) (p.21-28). These were new ways of thinking about challenge for me, although they encompass aspects I and most teachers will be familiar with. DoK has different levels to consider, and FICT considers the level of the four aspects in combination to increase challenge. This reiterates how challenge is not a case of ‘do it another way’ or ‘make the numbers bigger’ but needs to be a much more nuanced and considered part of planning and teaching.

2. Worked examples (and paired examples) are a powerful way to develop understanding. Worked examples also make an appearance in Craig Barton’s book, but McCrea stresses the use of paired examples – a modelled example, followed by a minimally different problem for pupils to solve. She argues that this supports the research of cognitive load and, by following this, allows pupils to learn. What also stood out to me was the concept of incorrect worked examples. Again, something that many teachers will use (spot the mistake, what went wrong and so on) but McCrea tightens ones thinking about these examples, stressing the importance on focusing on one error or misconception.

3. We need to think about fluency synthesis when setting tasks and problems. This is a big takeaway for me. ‘Fluency synthesis tasks require students to apply their knowledge to more challenging procedural problems’ (p.88) and McCrea goes on to explain that this involves recalling known knowledge. The example given sums up exactly what this means: while practising calculating the area of a triangle, pupils can develop fluency through the inclusion of fractions, decimals, metric/imperial measures and mixed units of measure. Furthermore, the procedure itself can be made more secure through inclusion of problems that are ‘boundary cases’ or non-standard, and those with too little or too much information. In this way, one task can secure a procedure as well as encourage pupils to retrieve known knowledge, which is a key aspect to making it ‘stick’.

I think you should read this book if…

  • You teach maths. Whatever the point of your journey in teaching the subject, I think you will learn something new.

Musings: Tackling Misconceptions in Primary Mathematics by Kieran Mackle

May 14, 2019

The book

Mackle, KTackling Misconceptions in Primary Mathematics: Preventing, identifying and addressing children’s errors. 2017. Routledge.

My reading viewpoint

This book was kindly donated to me by its author – someone I enjoy reading tweets from but was totally unaware he had written a book! If you aren’t following Kieran, do – he’s a wealth of mathematical knowledge. While I always enjoy a free book, I have a passion for mathematics education and always want to know more about misconceptions, so was eager to read.

Summary

To be honest, I found the book a little tough to read BUT not because of Kieran’s style or the content (more on that in a moment) but because the text is quite closely spaced. I do much of my reading on my commute, and the early start probably didn’t help matters. Having said this, clear chapters and subheadings, along with boxes demarcating helpful activity suggestions allowed the content to be broken up into manageable chunks.

The writing style is good, with Kieran’s dry sense of humour and absolute passion for the subject matter regularly coming to the fore. As mentioned, the book also includes lots of suggestions for activities covering a range of topics and phases, so you are bound to have something to takeaway and try.

This next part of my summary I am only saying because I have discussed it with Kieran, and it bears no real impact on the content of the book, I think, provided you understand it – I think the book is a little misnamed. The content is brilliant: it is essentially a deep-dive into some of the spikier objectives and wording in the National Curriculum, suggesting precision in teaching, language and activities: basically, if you follow Kieran’s advice you are well set up to avoid the potential pitfalls and misconceptions of the concept. However, what the book isn’t is a ‘this is the misconception, this is why it happens and this is how to avoid it’. It is much more than that – an exploration of ways in which to teach the nuances of the subject, in a way that means you won’t need such a list. Also, as Kieran pointed out to me, lists such as these exist. If you still want one of those, he or I can help with that.

So while it isn’t explicitly about stated misconceptions, the clever thing about the text is that it forces the reader to think really deeply about aspects of the curriculum one may gloss over – through thinking it unimportant, through a lack of subject knowledge or, let’s face it, because time is an issue. It walks through the importance of basic concepts, showing how ensuring these are right promotes success at later development and related concepts. Misconceptions are implied and ways to avoid, while not explicit, are well thought through.

My key takeaways

1. Language is important – get it right from the get-go. While not news to me, and referenced in a number of other books, this crops up too often in the book not to make it a key takeaway. As Kieran says, providing pupils with incorrect vocabulary ‘will only have a lasting effect on their concept development for years to come.’ (p.7). He makes some key language points teachers can overlook: a 2-D shapes drawer is not correct as the shapes within are 3-D and borrowing has no place in subtraction.

2. Number is weighted in the National Curriculum, and for good reason. Tackling Misconceptions includes a lot of writing on number, and its appearance in the NC. From counting to place value, Roman numerals to decimals, the NC is full of number and Kieran makes the point over and over again that the basics cannot be overlooked, and that it is worth taking time to get these right in every year group. He often cites the use of a counting stick, and I am completely with him in this – no classroom should be without one.

I think you should read this book if…

  • You find some of the nuance and description in the mathematics NC confusing or challenging, and want clarity on what it means, how it links to other aspects, and how you may go about teaching it to ensure precision
  • You are a trainee / new teacher grappling with the NC
  • You are a maths subject lead considering how the NC can be implemented in line with the new Ofsted framework – this has implementation written all over it.

Musings: Visible Maths by Peter Mattock

March 9, 2019

The book

Mattock, P.. Visible Maths: Using representations and structure to enhance mathematics teaching in schools. 2019. Crown House Publishing.

My reading viewpoint

As someone who works for a mathematics organisation which puts conceptual understanding at the heart of all that we do, reading about representations in mathematics was a natural step for me. Having been educated in the ‘it just does, learn it’ field of mathematics, I am always fascinated by representations which help me, and pupils, see the why of the mathematics and recognise the underlying structure.

I should also mention that I was given a review copy of this book by the lovely folks at Crown House Publishing. I am incredibly grateful to them for this (and they have a great selection of edu-books!).

Summary

The title of this book utterly sums up its purpose – to support teachers in making mathematics visible to learners. Visible Maths takes a series of representations – from counters and Cuisenaire to vectors on number lines – and, beginning with representation of number, leads the reader through the ways in which these representations can support the understanding of a huge number of concepts of mathematics, making connections between these concepts. It supports the idea of developing these representations from the beginning of primary to enable them to be built upon in secondary, helping pupils recognise the underlying mathematical structures. Visible Maths covers a huge number of aspects from positive and negative numbers, to irrational numbers and surds and is a great way to develop understanding of these as well as considering how representations can help support pupils’ understanding.

First off, I have to say the whole book is beautiful, visually. Colour illustrations are used throughout to explicitly show examples of these representations being used. These representations thread throughout the book, allowing the reader to flip back and forth and really see the connections and links between them. The illustrations really make the book, for me.

Similarly, Mattock’s writing style is great. Straightforward and clear, with some delightful maths tangents. Visible Maths is designed for both primary and secondary educators, meaning that some of the mathematics and terms may be unfamiliar to some teachers, but a clear glossary and concise explanations support understanding without it feeling patronising. It also assumes little in the use of representations – clearly explaining, for example, how negative numbers may be represented.

As you will see from my key takeaways (below) I found this book enthralling. As a primary educator, it helped me to fully appreciate the importance of deep conceptual understanding and using representations effectively to help pupils ‘see the maths’. As a blueprint for providing some consistency between primary and secondary, Visible Maths has serious power. It really helps with the ‘joined up thinking’ concept too – that, as a primary educator, I need to see where pupils are going in order to fully appreciate the importance of building those foundations.

My key takeaways

1. Pupils need different representations, and not all representations work for all maths. I am guilty of having my favourite representations (Cuisenaire, anyone?) and have been known to try to shoehorn the maths to fit the representation. What I love about Mattock’s writing is that he is open and honest about the strengths as well as the drawbacks and limitations of different representations. While he maintains the thread of repeating representations throughout, he acknowledges the limitations of the representations but crucially (see next point) makes connections between representations to help the transition.

2. Making explicit connections between representations is something teachers should know how to do and have secure knowledge about. As I have said, Mattock uses a range of representations – counters, bars, Cuisenaire, vectors on a number line – to represent different mathematical concepts. In doing so, and in the way in which the book is laid out, it forces you to think about the relationship between the representations and the way in which these can be introduced to make connections. As he says himself, there is a huge amount of writing on different representations, but this is the first time I have come across so many in one place.

3. Rounding ‘rules’ can be represented using counters. This blew my mind, if I am honest! While I have often used number lines for rounding, to deepen conceptual understanding, I am the first to admit I used ‘because you do’ when explaining why 5 (or 5 units e.g. 0.5, 5 tens) rounds ‘up’ to the next multiple of 10/100 etc. I have always been uncomfortable with depictions of hills or bus stops or any other explanation. On p.184, Mattock provides a diagram of rounding which made it so clear I goggled at its simplicity. Essentially, when considering rounding of 2.5, if we show all the possible tenths between 2 and 3 using place value counters, we can see we have ten possible options. Dividing these into two equal groups (the halfway point) actually places 2.5 in the group with 2.6 onwards, and therefore it should be treated the same. Take a look – he explains it much better than me!

4. Algebra genuinely is every teacher’s responsibility. Through linking representations, Visible Maths shows the connections between algebraic thinking and other mathematical concepts often deemed ‘simpler’. Mattock says, “If we see algebra as a generalisation of the relationships we observe in number, then this representation [bars] exemplifies the emergence of the generalisation – the idea that, regardless of value, the relationship between a single square and the orange bar remains the same.” This, for me, was a really powerful moment. In KS1 we should be promoting this idea, just hinting at it, so that when pupils encounter algebra it is not a ‘new’ concept.

5. Don’t wait until you need the representation to use it. One of the representations Mattock uses that was new to me was that of vectors (essentially arrows, but a bit more nuanced than that!) both with and without a number line. He suggests these are particularly useful for algebra however if a pupil has never seen this before they are going to find it challenging to apply to the concept (p.202). This is really important for any representation and teachers should again engage in ‘joined up thinking’ to ensure exposure to representations throughout school.

I think you should read this book if…

  • You teach maths. Seriously. If you are a teacher who, at any point, to pupils of any age, teaches them mathematics, then read this book. You may choose to skip over the more complex concepts if you don’t teach them, but I think as an explanation of different representations, why they should be used and when, you can’t go wrong with this book.
  • You are a Maths lead/HOD looking to get some consistency in representations.