Claxton, G. The Future of Teaching and the Myths that Hold it Back. 2021. Routeledge.
My reading viewpoint
Claxton’s latest has caused quite a stir on Twitter, with a number of people I follow reading and commenting. I couldn’t resist taking a look myself. As always, I read from a primary teaching viewpoint.
Also, for reasons that will become clear, I feel I sit more on the ‘progressive’ side of the imaginary fence.
Summary
In short, this book is about the ‘traditional’ or ‘direct instruction’ views of teaching versus the ‘progressive’ or ‘discovery learning’ views. It’s more than that, of course, but at its heart it is Claxton arguing that the ‘direct instruction’ collective of teachers and experts are incorrect about the way in which children should be taught and in fact we all need to be more ‘progressive’ and more open. For someone who sides themselves with the more progressive views, you may be thinking that I therefore loved every minute and want to put this book into the hands of every teacher going.
Alas, no.
The biggest issue I have with The Future of Teaching is the polarising and frankly very personal attacks made on what Claxton calls the DIKR group (direct instruction in a knowledge-rich curriculum). He tells the reader that this particular group (naming names and schools) believe completely that direct instruction and facts are the only way to get children to learn well. He goes on to suggest that this is wrong, and that there are many better ways to get children to learn and grow as human beings. The thing is, this isn’t really something I disagree with – although I don’t believe Claxton had to be so personal. However, and this is a big however … I have never met a teacher or visited a school who actually uses the DIKR teaching Claxton describes.
Without doubt, I don’t have Claxton’s range of experience. But … I have visited probably over 100 schools in my previous role, and I work in a Trust which is described as more ‘traditional’. Yes, I have seen teachers teach in a ‘sage on the stage’ instructional style. Yes, I have seen Knowledge Organisers. Yes, I have met teachers who believe pupils need to know a lot of facts and knowledge, and feel rote learning is a great way to teach this. But I have never met a teacher who has such a narrow view as defined by Claxton. Interestingly, he is quick to identify that the narrow view of what he believes traditional teachers think of as a progressive classroom (think extreme discovery) doesn’t exist, but I am not sure I believe his view of a traditional classroom exists, either.
So my thoughts on The Future of Teaching have been somewhat shaped by my irritation of the personal attacks and by the frankly impossible reductivist definition of ‘traditional’ teaching. But what does Claxton say should be done instead? Claxton talks about the need for pupils to develop in lots of aspects: ‘to be fully human, in my book, is to score well on knowledge and reason and imagination and empathy and literacy and and and…’ (p.7). Early on, he identifies a learning river which recognises that learning happens in different ways and that aspects further under the surface (such as attitudes and dispositions) are harder to see and judge. This reminded me of Sherrington and the rainforest, and is sort of similar to how I feel about teaching.
In his discussion of knowledge (Chapter 3) Claxton notes that we need to think carefully about powerful knowledge, or what we consider to be ‘the best’ knowledge. He says that ‘powerful knowledge is only functionally powerful if it empowers real people, living real lives, to achieve things that are of value to them or their communities’ (p.46). He suggests that we as teachers need to be critical of what is being taught and acknowledges that a curriculum that funnels pupils towards academia is flawed. I agree with this and I think it’s an interesting point to consider. What knowledge do and should we be teaching our pupils? To be fair, while Claxton is quick to criticise he is less quick to offer an answer.
The Future of Teaching dismisses the theories around memory, and cognitive load theory (CLT) in particular. In fact, he states that the forgetting curve and theories ‘do not apply’ except when we need to learn nonsense (p.99). I like that I have been made to think more critically about CLT – not that it will change my mind about how I apply ideas from this in the classroom and in my teaching.
He goes on to say that ‘working memory turns out to be not a basic feature of cognitive architecture but an emergency strategy that only needs to be deployed under very specific conditions’ (p.107) and suggests that teachers don’t actually need to consider the confines of working memory as often as we do. This, in my opinion, is one of many strange takes from Claxton and I actually think he contradicts himself. On the previous page he identifies ‘cognitive conditions’ where working memory seems to matter and these include information coming too fast, several tasks or actions competing for your time and environmental / psychological distractions. For me, much of this can happen every day in the classroom – surely then as teachers we NEED to consider working memory because these ‘very specific’ conditions occur constantly in a school?
I don’t think that Claxton spends a huge amount of time actually suggesting what the future of teaching should look like, nor does he offer much in the way of an alternative. He does spend time discussing the ‘4E cognition’ idea (p.112) which embodies more of the way I think about teaching. This recognises that teaching needs to be receptive to the learner and the environment and that knowledge and learning are tangled webs, amongst other things. But I think that the amount of time given over to exploration of these ideas is overshadowed by the constant ‘DIKR is terrible’ narrative.
Similarly, he mentions a number of primary schools who take a more progressive stance and outlines their approaches, but does very little to flesh these out. Instead, he chooses to extoll some of the virtues of Michaela (from p.159) despite having some unkind things to say about its approaches and its Headteacher earlier. To be fair to Claxton, I found reading his take on Michaela interesting, but again … Lacking.
Chapter 10 is a summary of the book and Claxton’s ideas – cue more ‘DIKR is terrible’. However, if you do want to read the book I’d start there. Read that chapter and then, if you want to read the rest, you can.
In conclusion, the Future of Teaching is, without doubt, going to irritate. It irriated me, and I sort of agree with Claxton’s stance on teaching. It IS an interesting read, and it DID give me some things to think about. However, it seems so wrapped up in its need to slam a type of teaching I don’t think exists, and name and shame a handful of people personally (which I don’t think is necessary) that it’s hard for me to think about his points without feeling somewhat disappointed.
]]>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
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.
]]>Sharma, Lekha. Curriculum to Classroom: A handbook to promote thinking around primary curriculum design and delivery. 2020. John Catt Educational
My reading viewpoint
In my current role I am working on curriculum design for mathematics, and so have been reading books related to curriculum design. As always, I read with more of a focus on primary mathematics.
Summary
The first thing to note about Curriculum to Classroom is that it is short. At less than 100 pages, A5 sized, it is a quick read. Predominantly this has come about due to the conciseness of the writing: not a word is wasted. The writing style is clear and to the point and while it does assume some knowledge in aspects such as how pupils learn, it incorporates enough explanation to stop you reaching for Google.
Curriculum to Classroom does exactly what the title suggests – it is a roadmap for thinking about designing a primary curriculum, with a focus on what that looks like in the classroom. Each chapter includes short, medium and long term considerations and many include reflective questions to prompt the reader to think carefully about curriculum design for their school. Sharma emphasises the importance of community and individuality at every turn, sharing research, expertise and her own understanding all under the caveat of what worked for her and colleagues may not work for you. The focus is on reflection.
Chapters are well laid out with visuals to support understanding of key ideas. You could dip in and out but as I mentioned, with such a short book I would advise reading cover to cover as the journey is set out much more clearly in this way.
My key takeaways
I think you should read this book if …
… you are at the start of a curriculum journey and in a position where development is down to you. I will be honest and say that I am further down the road in terms of design than perhaps the intended reader of this handbook and, as a result, didn’t learn a great deal, although it DID make me reflect and I will continue to use the reflective aspects in development.
]]>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 …
Lemov, D. Teach Like A Champion 2.0. 2015. Jossey-Bass
My reading viewpoint
Since beginning my new role in September, I have learned that the school I primarily work in follows the principles of TLAC. Like most teachers, I have heard of and even do many of the principles set out in this book however I have never read the text myself. Therefore, I decided to fully get to grips with the ideas and understanding behind the techniques that I have heard a lot about in order to be able to make use of them and understand how they are used in the school.
Summary
I think it is impossible to be a teacher and have never heard of Lemov and the principles that are found in Teach Like a Champion. The book comprises 62 techniques that, according to the front cover, ‘put students on the path to college’. From checking for understanding, to behaviour, to routines, every aspect of daily teaching is covered in this book. While it is impossible for every teacher to use every technique (I think), Lemov suggests that the tools here allow teachers to make independent decisions about how and when to use the techniques. It also suggests that mastering these techniques allows teachers to do their jobs: teach.
I think what’s interesting about the quote on the cover is that I am not sure that I fully agree that these techniques will do what they say on the tin. Interestingly, I have seen these techniques used with no understanding or questioning about the effectiveness of them for the pupils in the classroom. What TLAC can do however is provide a teacher with understanding of where these techniques have come from, and also allow them to build a personalised selection to use in their classroom. While I don’t know that following these will lead to such a path, I do feel that employing aspects such as routines, expectations to listen and respond and high expectations will allow for more effective teaching.
The 62 techniques feel overwhelming when viewed as a whole however when broken down and thought about in smaller segments, you begin to realise how many are a staple part of many teachers’ toolkits. There are also pretty summaries available. Some techniques I will share in more detail later on, but they include ideas around checking for understanding, having high expectations of all students, insuring whole class buy in and participation and ways of ensuring pupils are doing more work than the teacher. I understand that not everybody sees the need for such a toolkit however having observed so many lessons I do feel that having a text such as this is a great starting point for many teachers.
The book itself is long, and probably not meant to be read cover to cover in the way that I have done. In fact, it should be recognised that I did not read every single word. Some of the descriptions of the techniques were very familiar to me, enabling me to skip through and consider how they could be applied in my classroom.
The way the book is laid out allows for teachers to dip in and out, considering which aspects are most important for their classroom at the time of reading. Each chapter is formulaic in the sense that the specific techniques are easily identified through the layout, and each is further exemplified through descriptions, video content and suggestions that link to the website. This means that the book itself is easy to navigate and teachers are able to easily consider which techniques to look at and which may make most impact in the classroom.
Overall, as a more experienced teacher who has seen hundreds of lessons in classrooms across the country, many of the techniques in the book are familiar to me as ‘just good teaching’, or around the ideas of ‘quality first teaching’. Some of the techniques are are are, in my mind, blindingly obvious. However, this year I am lucky to work with two fantastic trainee teachers and can see the benefits of making these techniques explicit to those who may not have as much experience as me. Similarly, I have benefited from rereading some of the techniques that I feel are more obvious in order to develop a deeper understanding of how they might work in the classroom.
It is also interesting to read about techniques that I have only read about elsewhere in a negative light such as SLANT (an acronym for specifying a certain way of sitting) and see and understand some of the theory behind it. I am not saying that reading this book will change your mind on those techniques that you feel will not work for you or your classroom but it has helped me understand why such things may be useful for some teachers. As my friend and colleague says, everything works somewhere!
My key takeaways
I think you should read this book if…
Roiha, A & Polso, J. How to succeed in differentiation. 2020. John Catt Educational
Summary
I’m going to be honest and say that I really struggled with this book and it has taken me a long time and several attempts to read and review. One of the main sticking point for me was the fact that this book has been translated, meaning that some of the nuance has (in my opinion) been lost. For example, the book sometimes refers to young children as small students or similar and this therefore makes for a confusing read. Equally, some of the sentence structures are not quite what you’d expect in English.
The other frustration of this book is that it is based on the Finnish national curriculum and this, as shown in the book, is written and designed for teachers. Anyone working with the English national curriculum will know that our curriculum is simply not the same and that national tests and a large content base makes teaching challenging. While of course it is not the authors’ fault that they have written a book based on the different curriculum, it does make the title quite difficult to achieve!
One of the aspects of the book that I really liked was its long discussion of what differentiation actually means, and I think that this is important for all teachers to have a deep understanding of. They discuss differentiation in terms of reactive and proactive and highlight the importance of constant reflection and monitoring. For a similar discussion, it is worth looking at Cowley’s book.
Unfortunately, while there are some gems to be taken from this book that I will share later, I feel that as a whole there is too much focus on being able to change lessons, materials, instructions and outcomes for individual pupils. While of course this is the dream, and we would love to be able to differentiate for every pupil, in a class of 30 with one adult it is simply not possible to meet individual needs. Even when you realise that it is in fact possible to loosely group pupils together, every lesson may require five or six levels of differentiation which can be a monumental challenge. I feel that while of course the individual need is hugely important, the book is a little unrealistic.
Similarly, while differentiation for higher attainers is mentioned, there is very much a focus on lower attaining pupils and differentiating for them. Whether this is due to the translation or not is unclear, but there seems to be a focus on giving lower attaining pupils less access to deeper understanding of a concept: the idea of surface level understanding crops up a fair few times. With my maths hat on I feel this may lead to or be interpreted as lower attaining pupils not having access to reasoning and problem solving opportunities and for me this is not what differentiation is about.
Overall, the book is not challenging to read despite the sometimes confusing parsing where the translation has not quite reflected the nuance of the original. Each chapter begins with profiles of four students, all of whom teachers are likely to have encountered, which means that you are able to consider how the content could be applied to specific children in your class. I also like the fact that towards the end of the book, specific curriculum areas are covered. I have course only read the maths section but other areas are included which may be of use if you are looking for specific advice. The book is called ‘how to succeed in differentiation’ and I am not sure that through reading this book anyone will feel that they could succeed in an English classroom.
My key takeaways
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…
Shah, Paarul et al. The Oxbridge Formula. 2020. STEP Maths Publishing.
My reading viewpoint
I was sent this book to read and review via Twitter. As someone who (many moons ago) was interviewed at a Cambridge college, and someone with a passion for maths (which, along with other STEM subjects, is the focus of this book) I was intrigued.
Summary
The Oxbridge Formula does exactly what it says on the tin: provides a formula (of sorts) and a lot of additional reading and support aimed at preparing a young person to apply and secure a place at Oxbridge.
The introduction gets quickly to the crux of the matter and sets out the need for ability, passion and potential, all of which are further defined and referred to throughout the text. As an adult who has been through university, these traits seem obvious however I recognise that for a potential student these may not be! Throughout the book, references are made to how you can let these traits shine in order to improve your chances of securing a place.
It is also worth noting that the two main contributors are women. This is a huge plus for women in STEM and encouraging more female students to take their learning further.
The book is set into formulaic and clear chapters covering a variety of STEM degrees, meaning that the interested party can focus their attention on the relevant information. Each chapter covers the same content. I will be honest: I read the maths section in full and skimmed the rest! But from application and personal statement, to relevant tests and interviews, all the information you could need to support preparation is there. There are examples of personal statements (complete with annotations), sample interview questions and a script and overviews of the various courses and combinations so that you are able to focus in on what might work for you. I felt the interview questions and samples were particularly useful here. In the maths section, they provide opportunities for potential students to practise and think about the problem.
Shah and the other contributors (including a number of quotes from Oxbridge students) are ‘at the chalk face’ of Oxbridge and this shows in their wealth of knowledge and understanding. The book is also written in easy-to-access tone and style – as Birbalsingh says in the foreword: ‘You will feel as if someone is sitting next to you chatting away’. This tone and style serves (to a point) to put a potential candidate at ease and allows information to be assimilated pretty quickly. This means that you are able to absorb the key points and easily consider areas you will need to develop in order to apply and hopefully secure a place.
But …
From here on in, I am unsure if the rest of this is down to Oxbridge and its reputation around economically disadvantaged students, or my own reservations, or the book itself. The Oxbridge Formula regularly mentions the need for prospective students to show ‘super curricular’ activities along with reading and work experience (xvii). I understand this: the need to show passion and potential is a key aspect of the formula and I would expect any student studying maths at degree level to have these traits. I also understand that these can only be shown by more than ‘doing an A Level’.
However if, like many students, you are from a deprived socioeconomic background and/or lack an enthused and dedicated teacher / parent / carer, I can imagine reading this repeatedly would make you question your ability to achieve a place at Oxbridge. Shah et al have a huge number of great book suggestions, websites / videos / podcasts to make use of and suggest summer projects and lectures as ways to boost this for the application. These are brilliant – the maths books are some I am familiar with. But if you are from a household that lacks finances for books, in an area where your local library is closed, don’t have access to your own laptop, have to work over the summer in order to support your family… One or any combination of these, from reading the book you may well come to the conclusion that Oxbridge is unattainable.
I fully understand that students looking to apply to Oxbridge have intrinsic motivation to get there and the ‘where there’s a will there’s a way’ but, let’s face it, attending conferences, lectures, buying books or subscribing to the Oxbridge Academy’s wealth of support offered is something some prospective students will not be able to achieve.
Ultimately, it cannot be denied that The Oxbridge Formula is filling a gap in the market. There will always be applicants for Oxbridge and there is not enough support for students particularly in STEM subjects. I am all for anything that promotes further education in STEM and if you are a student who wants to get in – and has the means and/or will to do so – then I fully believe this book will support you in your effort to secure a place.
I think you should read this book if…
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 …
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