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Posted on March 6, 2016 by Jonathan Porter

The Golden Mean Part 2: Sample and Domain

In my last blog I spoke about the ‘Golden Mean’. This is the Aristotelian principle that what is virtuous is always between two states: one of absolute excess, the other of absolute deficiency. I suggested that this could be quite a helpful way of thinking of teaching and curriculum design. I said that over the next few posts I would try and explain how this insight has informed my thinking on teaching and assessment in KS3 history. Here are three questions that came to mind:

  1. How do I strike the right balance between the sample (the end of unit test) and the domain (in this case, the period of history in question)?
  2. How do I help my pupils remember what they need to remember for the assessment, as well as helping them to remember what I think is just important for them to remember?
  3. And how do I guide them towards an answer without answering for them?

Sample and domain

 In my last blog, I explained why I think the sample/domain issue is one that particularly afflicts the teaching of history at KS3. The more our assessment system leverages the assessment, the more tempting it is for the teacher to narrow the domain and teach to the test. This is particularly so in schools like my own where pupils, even in Year 7, do all assessments without notes and from memory. The alternative, however, is arguably just as bad: unpredictable tests sprung on new arrivals to the school, which help us to norm-reference one pupil against another, but potentially crush a child’s confidence in the process.

For this reason, I am in favour of giving younger pupils the question in advance. But only if the question itself avoids the sort of ‘narrowing’ I’ve described. An example of this sort of narrowing might be a very specific question about the significance of the Magna Carta or the ‘greatness’ of Alfred the Great rather than a broader question that aims to assess whether pupils have retained a good knowledge and understanding of the period in question. Alfred the Great is perfectly interesting, but I don’t want my pupils’ appreciation of Anglo-Saxon history totally dictated by what could become both a semantic and substantive discussion of his ‘greatness’.

In part, as was put to me, this is just the age-old question of breadth vs. depth. In the case of history, we often choose to sacrifice breadth over depth. With a limited amount of time, we cannot possibly cover everything that happened in, say, Anglo-Saxon or medieval England. We also want to ‘scale-switch’ and to help our pupils see continuity and change in the short, medium and long-term. I think this is true.

But I think what we’re also seeing is some of the effects of the ‘skills-centred’ attitude to the teaching of history. That is ‘We don’t need to worry about teaching a broad sweep of Anglo-Saxon history because it is learning to think about causality/significance/continuity/change that is most important.’ I think this is a mistake. Why? Because I think KS3 gives us a great opportunity to put in place the broad domain knowledge that we need for long-term expertise. In the long run, enquiry questions that are too narrow have the potential to erode the domain at exactly the point in a child’s development when, in my view, the domain should be most broad. With that in mind…

All enquiry questions are equal, but some are more equal than others.

 I think we can find a Golden Mean between the sample and the domain – between breadth and depth. I think we can have enquiry questions that ask our pupils to consider second-order concepts while also giving them the best possible opportunity to learn – and remember – as much as they can of the period in question.

Here is the question that we settled on for our unit of work on medieval England:

‘What was the most significant challenge to the King’s power in medieval England?’

 I expect a good response to include a paragraph on each of the three challengers (the Church, the nobles and the peasants) and top answers to include a fourth paragraph in conclusion. The pupils have 45-50 minutes and all pupils write from memory without notes or sentence starters. We try and achieve this through regular self-quizzing of both the upcoming ‘sample’ (What was the most significant challenge to the King’s power in medieval England?) and the broader domain (the chronology of medieval England). You can see that I’ve tried to place an equal emphasis on this on the pupils’ knowledge grid, which they use in weekly self-quizzing as well as in class.

Slide1

My hope is that this strikes a ‘golden’ balance between the sample (right-hand side) and the domain (left-hand side). The pupils learn about, and have to remember, important events in medieval English history (The Battle of Hastings, the Anarchy and the Battle of Agincourt), which are not needed to do well in the end-of-unit assessment. At the same time, I hope I’ve also supported all pupils to learn the specific information that they’ll need to do well in the assessment.

Next time, I’ll write a little bit more about how we use these grids, low-stakes testing and songs to try and help pupils automate people, dates and concepts so that their working memory is focussed on the most challenging parts of the assessment.

Posted on March 6, 2016 by Naveen Rizvi

Minimally different examples

School Visited: Uncommon Collegiate Charter High School

Class Observed: Algebra 1 Grade 9 Statistics (UK equivalent Year 10)

Theme of post: Pedagogy and Curriculum

This is the first blog post in a series complementing a workshop I delivered at Mathsconf6 entitled “Mathematics Pedagogy at USA Charter Schools”. Here I spoke of my experiences whilst visiting a selection of Uncommon and North Star academy schools in New York and New Jersey.

I discussed how the common denominator in each maths lesson I observed was the level of academic rigour in the teaching delivered to pupils. I am still trying to find a way to define this term, because it is overused and poorly defined. However, I do believe I may be able identify the evident features of academic rigour in the structure and delivery of each teacher’s pedagogy and instruction:

Academic rigour featurse

In my workshop, I discussed four examples of academic rigour that different teachers up and down the country can implement with minimal effort and maximum impact. Each blog post will discuss one example at a time. Today, this blog will address using minimally different examples during instruction.

Agenda

Minimally different examples are examples to explain different concepts where the difference in the algorithm between one concept and the other can be distinguished as one additional step. This was one example I witnessed in one of the classrooms I was in.

 

Minimally different examples 1

The teacher presented the first expression to factorise, where the coefficient ofa in the quadratic expression is 1. This was a recap exercise where she stated the values for the coefficients, and then she determined the two values of which the product gives us the value of c, and the sum gives us the value of b. She wrote the expression in its factorised form.

She then introduced the next expression to factorise. She stated that it looks visually very different to the first one, but that there is only one difference between the algorithm to factorise the first and second expression. She defined the minimal difference between the two as one additional step – which was at the start of the algorithm. She asked her pupils, “What is the greatest common factor (GCF) of all the terms in the expression?” Pupils spotted the GCF to be 3p, which she then factorised. She said “look, we now have an expression in the parenthesis which looks similar to the first expression we factorised.” The minimal difference between factorising the first and second expression was classified as one additional step. Traditionally, in my training, I would have seen the algorithms to factorise both the expressions as isolated and fragmented. Instead, they are connected; pupils were empowered to factorise more complex expressions from prior knowledge of the algorithm to factorise the first expression. Minimally different examples are a good strategy and example of academic rigour in the classroom because they help pupils to understand knowledge and concepts which are initially complex and ambiguous. Here, the difference between the two examples is classified and determined as one additional step in the algorithm. The possibilities are endless.

I plan to show different examples of how I have used the idea of minimally different examples in my teaching, sometime this week.

Posted on March 5, 2016 by Jo Facer

Reading Reconsidered

Teach Like a Champion,’ by Doug Lemov, changed my teaching profoundly: it was the most practical and helpful piece of writing I had ever encountered, and transformed my classroom practice, giving me specific aspects to hone and improve.

When I heard that Lemov had been an English teacher, it didn’t surprise me – in particular, in TLAC 2.0 there are several techniques which are especially useful for the English teacher. When I heard he was co-authoring a book on reading, I had very high hopes. ‘Reading Reconsidered,’ written by Lemov with Erica Woolway and Colleen Driggs, does not disappoint. With a nod to the poetic importance of literature (‘this book is about the enduring power of reading to shape and develop minds’), again, we have a manual for practice; specific things that teachers can do, day in, day out, to read effectively with pupils.

‘Reading Reconsidered’’s opening gambit is that text selection is key: in pages referencing Hirsch and Arnold, canon and cultural capital, the writers note: ‘part of the value of reading is to be able to read and talk about important books that almost everyone else has read.’ The great conversation of literature, intertextuality, ‘works only when pupils have read some texts in common.’ The writers extol the value of a common reading curriculum for all pupils, and warns us to select our texts carefully, noting: ‘a typical pupil might read and intentionally study forty or fifty books in English classes’ over their time in school and ‘these few books form the foundation of their knowledge of how literature works.’ If only there were as many as fifty texts! Such a sentence puts the demands of the new GCSE English Literature, with its four texts (I include the poetry cluster as a ‘text’) over two years into frightening perspective. To those who argue that canonical texts are unreadable by youngsters, the authors respond: we just need to get better at teaching them.

The writers go on to isolate the ‘five plagues of the beginning reader’, looking at five challenges all readers face in encountering tricky texts, and how we can overcome these in our everyday practice. One example is using ‘pre-complex texts’ to prime pupils for the canon, such as children’s classics like ‘The Secret Garden’ which use challenging syntax but have child-friendly story arcs.

The chapters on close reading are a must-read primer for all English teachers, going meticulously through how we should read closely in class, supplemented with specific questions designed to unlock meaning in complex passages. One key take-away for me was: teachers! Prepare to close-read! Annotate your text! It sounds blindingly obvious, but I know I’ve been guilty of sauntering into class, blank copy in hand, hoping for the best. Yet what more important preparation can there be for a lesson than our own annotation?

The most revolutionary chapter for me in ‘Reading Reconsidered’ was that on non-fiction. It made me recognise how vital it is for pupils to read non-fiction alongside fiction to assist with their comprehension and to enable really excellent analysis: ‘reading secondary nonfiction texts in combination with a primary text increases the absorption rate of pupils reading that text’; ‘when texts are paired, the absorption rate of both texts goes up.’ Overall: don’t teach non-fiction as a separate unit, but rather interweave non-fiction texts into your teaching of literature, either with short, contextual glosses or in-depth historical study of the time period in question to deepen analysis.

Though reading is this book’s chief subject, the authors do not neglect writing: ‘we are suggesting that pupils [should] write with more frequency and consistency as part of their daily work of responding to texts’. They recommend intervening at the point of writing to help pupils improve (no mention of lengthy, burdensome and delayed marking), explaining: ‘great teaching begins at the moment learning breaks down.’ ‘Writing,’ in this guide, also encompasses annotation, and again there is detailed advice for modelling these, with the goal of pupils eventually annotating autonomously.

Again, though the goal is for pupils to read independently, we need to be aware that if they do this poorly they are ‘inscribing errors’ (and of course we know from Lemov himself that ‘practice makes permanent’). It is vital that pupils read aloud, as well as listening to great reading being modelled for them. In considering ‘accountable independent reading,’ the writers give such guidance as using short sections with a specified focus, or scaffolding pupil comprehension by using questions.

Although the focus of the books is practical, with advice to be found on the specifics of vocabulary instruction and the dynamics of a classroom discussion, the underpinning voice here is one deeply concerned with children loving reading and doing it effectively. The voice of the parent in each writer is heard most clearly in the book’s dedication: ‘to our kids, with whom we have 16,000 more nights to read – not nearly enough.’ Foundational to this book is a personal and deep love of reading, for all the right reasons.

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Posted on February 28, 2016 by Naveen Rizvi

Act the anger, feel the warmth

I always believed that I had high academic and behaviour expectations of my pupils whilst working at my placement school in South Manchester. I was known to be strict at times because of my high expectations. I would give sanctions out to pupils who would not be giving me their 100% attention. I would insist my pupils SLANT whilst I was teaching to avoid fiddling. I would teach from the board and go through worked examples. My pupils knew how I wanted them to behave in my classroom. My pupils knew how I would behave if they were compliant or defiant. However, when I arrived at Michaela, my expectations were seen as low, and they were, and here I explain why.

At Michaela, we have 14 teachers and 6 teaching fellows where 9 members of staff are founding staff. I was observing a founding member of staff guiding and monitoring pupils to ensure that they were transitioning around school corridors in silence. I saw founding members ensuring that all pupils during assembly and class were slanting, if they weren’t then a teacher would give non verbal cues where a teacher would slant themselves and pupils would follow. I saw members of staff giving pupils demerits for not tracking them whilst they were speaking. At the same time, I saw founding members hysterically laughing with pupils during break time and lunch time when staff made witty comments about Mr Smith having luscious long locks of hair sarcastically (…Mr Smith has some hair). Pupils would be playing basketball or ping pong with teachers during lunch time. I would have pupils talking to me in the morning to greet me “Good morning Miss Rizvi, how was your evening?”

Now I look at pupils around the school and think now I know why Michaela pupils were so kind and polite. Let’s remember pupils did not come to secondary school as polite, kind or obedient as they currently are now. Pupils were trained and moulded to embody Michaela values. During our initial behaviour bootcamp, teachers taught pupils how to sit up straight in class. We taught pupils how to address a teacher. We would over exaggerate our behaviour when pupils made mistakes. For example,

“Excuse me Mr Rubbie, how dare you walk past me and not respond ‘Good Morning’ when I have wished you a pleasant morning. I am always polite to you so I expect you to always be polite to me,”

and “Zakye I am incredibly disappointed in your disingenuous apology and in you rolling your eyes when I was speaking to you! I am utterly horrified over how disrespectful you are to your teacher who works so hard for you.”

We would overexaggerate when pupils demonstrated exemplary behaviour:

“I am so proud of you Yasmine for scoring 100% in your science quiz this week, I would like to give an appreciation to Yasmine for her excellent self quizzing which resulted in her scoring 100%. We all want to be as successful as Yasmine. 1, 2 *two claps*,”

Olivia Dyer, Head of Science and Founding member of staff, said to whilst giving me feedback that “we act the anger, and feel the warmth.” Being angry and negative is emotionally draining for a teacher so we act when we over exaggerate and over justify why talking in the corridors is unacceptable.  Furthermore, we are wholeheartedly loving when a child is successful and we celebrate it to make that child feel valued, and to have surrounding pupils identify the kind of behaviour they need to demonstrate to be identified as an exemplary Michaela pupil.

This gives pupils clarity on how to behave and how not to behave. We do not give leeway to different extents of behaviour. If a pupil is doing something that they are not supposed to do then we would pick them up on it, no matter how subtle. At my previous school, I would give demerits because pupils would not be tracking me whilst I was teaching. If a pupil was rude even in the most subtle way such as smirking or curling their hands I would go ballistic. Surrounding teachers thought I was crazy for it.

At Michaela, teachers have sky-high expectations of pupils and our behaviour management is consistent because we all use the same language for rewarding and sanctioning pupils.

At my previous school, there was minimal consistency in rewarding and sanctioning pupils. We did not all have the same dialogue for sanctioning or rewarding students. If I ever pulled a child aside in the corridor for being too loud or demonstrating inappropriate and unprofessional behaviour then I would hear the response “Well [insert a member of SLT’s name here] did not say anything down the corridor when I was behaving this way.” It was exhausting. I would be the teacher who was crazy or too strict because of the high expectations I did have of the children. I would be unapologetic for giving a child a detention for not having a pen with them. I had one pupil who always forgot to bring in their exam prep folder, so for two weeks I proceeded to call the pupil’s house every morning at 7:15am to ensure that their daughter would bring their folder to school so they could be more prepared for my lesson.

At Michaela, all teacher’s have unapologetic and uncompromising high expectations of each and every pupil. Even more so our expectations are even higher for the pupils with the most tragic circumstances. We cannot change a child’s tragic circumstances at home but we can control their learning circumstances at school. If anything, it is more important to have high expectations for pupils who have those tragic circumstances so they have all the potential to escape their situation using education as the engine for such mobility. I know that the sanction I give to a student for a particular issue in science would also be sanctioned three floors downstairs in MFL by another member of staff. We are incredibly consistent. And students know that too…isn’t that the dream in any school? Students do not push their limits with us.

For example, we expect all pupils to bring a pen to school, and if they do not have one then we provide them with the opportunity to rectify the issue in a way which does not impact others around them. We have a stationary shop open between 7:30 – 7:50 am just before schools starts. The onus is on the children to meet our high expectations and we are explicit with what our expectations are. We are all consistent with the dialogue we have around school when we interact with children which contributes to a consistent and strong school ethos. As teachers we have bought into what is considered as Michaela standards. We have children bought into Michaela standards too.

There are many reasons behind the success of Michaela Community School and one of them is the uncompromising high expectations that all teachers have of each and every pupil our school serves. Children respect our standards because they understand that we want the best for them. Furthermore, to be the best pupil then pupils need to meet those high standards. My standards were considered high at my placement school where teachers were inconsistent with the dialogue they used and the reasons for sanctioning pupils or rewarding pupils. At Michaela, I realised my standards were not high enough. I had to raise my expectations, and I still am, but now I could not imagine reducing them. I would be failing myself as a teacher and the pupils that I teach. High expectations are underprioritised at majority of schools where at Michaela Community School it is part of the golden triangle of success.

We know that what we are doing having our sky-high expectations is right when a pupil who I have given detention to, which I have accidently forgotten to log, goes to detention and informs Mr Miernik that he has detention and that he is here to serve it even though Mr Miernik has informed him that he hasn’t got one. Or when I get an appreciation during Family lunch “I would like to thank Ms Rizvi for coming into school today and teaching us even though she isn’t feeling 100%” or when I get a postcard from a pupil expressing their gratitude.

Our Michaela high expectations in sanctioning and rewarding pupils comes from a place where we want our pupils to be the best possible student and human being. If you want to see us Michaela teachers and pupils in action then do come and visit us during a school day. Our doors are open to all. Come and have lunch with us.

Posted on February 27, 2016 by Olivia Dyer

Selling Science

Astronomy is my hard sell of physical science. Think of astronomy as the confectionary placed at the point of purchase in a well-known chain of high street stationary shops. How did the checkout boy, Alex, know I wanted a super sized bag of sour sweets at 10 am on a Saturday morning? I want the children to LOVE physics. I want them thinking, “How did Miss Dyer know I loved physics so much?!”. Which is why their introduction to secondary school physical science is astronomy. This is an eight-week unit designed with the number one purpose to blow their minds. The reason I chose to teach them about astronomy before electronics or mechanics is because it is truly fascinating. When I speak to pupils from other schools or my grown up friends about their experience of ‘physics’ at school, they tell me that they are or were taught so badly that they gave up, thinking that physics was not for them. If not for them, then who is physics for?

Caroline Herschel is the first scientific enquirer that pupils are introduced to in this unit. She was a pockmarked, four-foot three-inch woman whose family assumed that she would never marry and felt it was best for her to train to be a housemaid due to a childhood bout of typhus. In fact, Caroline Herschel beat the odds to receive many honours for her scientific achievements. Together with Mary Somerville, she was first woman to receive honorary membership of the Royal Society in 1835. Children are designed to leave the lesson where they learn about Caroline Herschel thinking, “Yes, physics is definitely for me”; “If Caroline Herschel can do it, why can’t I?”; “Nebulae are amazing!”. The astronomy unit taught at Michaela goes into far more depth than any other astronomy unit that I have ever taught at any other secondary school.

In this post, I have included an excerpt of the astronomy textbook that I have written and teach from, to give an idea of the content. The devil is in the detail. You cannot expect a pupil to love astronomy merely by learning the order of the eight planets and about the phases of the Moon. No, let’s not patronize the children that we are expected to teach. At Michaela, pupils learn about different models of the Solar System and references can be made to Ptolemy and his view of the Universe, thanks to Jonathan Porter’s history curriculum that includes Ancient Greece. Pupils learn that although Georges Lemaître first conceived the Big Bang theory, that the phrase ‘big bang’ was coined, ironically, by Fred Hoyle – an astronomer who disagreed with this theory. It is the links that can be made between science and other disciplines and the small pieces of information that are not usually found in science curricula, that children really love.

A recent visitor to Michaela asked a group of five year eight pupils what their career aspirations were. The responses were: experimental physicist, astronomer, ambivalent (!), mathematician and pilot. A year after being taught astronomy, some pupils actively want to pursue a career in the physical sciences. Using astronomy to sell the physical sciences? I say give it a go.

Posted on February 27, 2016 by Jo Facer

A guide to this blog

I’ve worked in education since 2010, as an English teacher, Head of Department and Assistant Head in four schools. I’m currently Head of English at Michaela Community School. I write about curriculum, teaching, leadership, English and reading. You can read about what education means to me and why I do what I do here.

Curriculum

Teaching

Leadership 

English

Reading

Posted on February 21, 2016 by Jonathan Porter

The Golden Mean: Aristotle and KS3 History

Aristotle

One of the most attractive principles in Aristotle’s philosophy is the Doctrine of the Mean. Aristotle says that what is virtuous is always between two states: one of absolute excess and the other of absolute deficiency. A good soldier, for example, is neither totally cowardly nor totally rash: he uses his reason to find a mid-point between the two – the ‘Golden Mean’. I think this is something that the best teachers do very well: they aim to be strict, without being austere, and kind, without being soft.

I think it’s also a principle that can very well be applied to curriculum design. One of the biggest challenges I’ve faced this year is striking the right balance between the sample and the domain. Greg Ashman and Daisy Christodoulou have written persuasively about how bad assessment (and notice I do not say ‘all’ assessment) leads to a situation where the sample BECOMES the domain. Greg has produced a really helpful diagram, which I hope he won’t mind me reproducing here:

Greg Ashman pic

But it is highly problematic, particularly for those who believe that the expert performance – in the long run – depends on wide domain knowledge. Over a long period of time, teaching to the test in this way erodes the domain so that our pupils only learn (and remember) the red dots (the sample), rather than blue shaded area (the domain). Clearly, this affects all subjects, but I think its impact has been keenly felt in the teaching of history.

History teachers are used to creating enquiry questions that act as lenses through which they view a period of history: What was the significance of the Magna Carta? What were the causes of the French Revolution? To what extent was Alfred the Great ‘great’? These questions are designed to assess the pupils’ understanding of significance, causality, continuity and change. They encourage the pupils to see history vertically as well as horizontally; because it is only through these sorts of questions that pupils can learn about the nature of ‘power’, ‘democracy’ and ‘tyranny’, and how these initially inflexible concepts bend and flex over time. Without them, history would just be ‘one damn thing after another’.

However, as important as I think enquiry questions are, I do think they present historians with a significant challenge, particularly at KS3. The reason for this is because, in effect, enquiry questions at KS3 often become the sample. In order to help our pupils write complex end-of-unit essays we provide our pupils, and our teachers, with the enquiry question weeks in advance. And, as such, the enquiry question will always inflect and distort the domain. If my enquiry question were ‘To what extent was Alfred the Great ‘great’?’ huge amounts of my teaching time will be devoted to the question of Alfred’s success as a king, rather than other significant features of Anglo-Saxon history. If my enquiry question were ‘How significant was the Magna Carta?’ much of my teaching, particularly toward the assessment, could centre on quite a narrow discussion of the events pertaining to that document, rather than other significant events in the period – the Peasants’ Revolt, the Anarchy or the story of Thomas Becket.

I want my pupils to learn as much as possible about medieval England, not because I want them to do well in their assessment (I do), but because I believe that, in the long run, their success in history will partly depend on broad domain knowledge. But also because learning about these events, and remembering them for years to come, has educational worth over and above my end of unit assessment. To defer to Aristotle again, I believe that such knowledge, combined with virtue, leads them to eudemonia – a state of human flourishing.

Over the next few blog posts, I’m going to write about how I’ve tried to find the ‘Golden Mean’ when it comes to curriculum design. How do I help my pupils to remember what I’ve taught them for years to come? How can I guide them to write complex answers without answering for them? And how do I strike the right balance between the sample and the domain?

Posted on February 20, 2016 by Lia Martin

What Happens Next, Miss?

Teaching English and thoughts on the future of education.

The success of ‘eating together’ has long been documented. Countless studies claim that children who eat with their families are more likely to have healthy relationships, achieve academically and maintain psychological stability and wellbeing.

Coined ‘family lunch’, at Michaela we do lunchtime differently. Gone are the long lines of children counting pennies for a burger. Gone are teenage clans claiming plots in the lunch hall.

We seat pupils in sixes, randomly shuffled, with a teacher or member of support staff at the head of each table. All six have individual roles (serving the food, clearing up, fetching dessert and so on) creating a joyous demonstration of working together to reap the benefits of a cooked meal.

Pupils are given vegetarian fare so that any cultural group can sit together and enjoy the same food. We have a topic of conversation every day that, led by the head of the table, is discussed over lunch. Not only do these topics provide a platform to explain why we do the things we do (see my post on narrating the why), but it often gives us English teachers an excellent opportunity to speak about the joys of reading.

The last minutes of ‘family lunch’ are set aside for appreciations, during which pupils and teachers will volunteer themselves to show gratitude to someone. We hear anything from, ‘I would like to give an appreciation to Mrs X for helping me to become more confident when reading aloud’ to, ‘I’d like to thank my mum for teaching me how to iron my shirts.’ All we ask is that appreciations are specific and that they have something to be grateful for every day.

In this ‘Whatsapp age’, it’s hard to monitor how much time our young people spend communicating without distractions outside of school. Having daily time set aside to talk, eat and be grateful together is precious. It strengthens our relationships and allows us to have meaningful dialogues with pupils outside of the classroom. It’s a sight to behold and, without a doubt, the very best part of our day.

Posted on February 21, 2016 by Naveen Rizvi

Efficiency in Simplifying Surds

Today, I had a random flashback to a formal observation I had at my placement school. I remember having a discussion with the observer, a maths teacher, where we were debating over one particular point. I was teaching my second lesson on simplifying surds: from √a  to the form k√b  where k is an integer. Given that this was the first time my 9X2 set were learning surds, I went for a very explicit selection of worked examples, structuring one type of method to simplify a surd to k√b form. In my lesson I was structuring the teaching as below, where I wanted pupils to identify the highest common square number, which is also a factor of the ‘a’ in surd √a.

In the lesson preceding this lesson, and in the weeks throughout the year, I would have pupils complete a recall activity frequently in which they had to identify the square number of the first 15 integers, and also square root the first 15 square numbers. The point was to ensure that pupils could recall these facts from memory, and develop automaticity in doing so, to the extent that there wasn’t an act of mental processing.

Picture1aPicture1b

Figure 1 – Worked Examples for simplifying surds effciently

I would ask pupils to select a factor which can divide 72 or 160 and which is also the biggest square number that can divide 72 or 60. They would then identify the square number and then write the root number of the square number below and continue on with the multiplication to simplify the surd to  k√b form. This was done because it is the most efficient and accurate method to simplify surds. If children are taught one accurate method to simplify surds at the start then they will get the correct solution and feel successful. If children are taught one method to get the solution and then you explore the different routes to get the same solution afterwards then you are building on their existing understanding of how to simplify surds.

The observer was suggesting that it would have been more beneficial for pupils to not have been taught one technique but to have explored instead a myriad of techniques. However, I think that what he was suggesting is only beneficial after they have first understood how to simplify surds in one accurate and efficient method. His suggested method could potentially cause several misconceptions insofar as you will have 30 pupils listing 6 possible attempts consisting of 12 different factors of 72. The lack of guidance and structure can lead to misinterpretations which leads to misconceptions.

In the early stages of learning such an abstract concept it is best to provide one accurate, structured and efficient worked example for students to replicate with different problems and in order for them to consolidate their understanding of how to simplify surds. This is, effectively, pattern spotting. Only then you can start to explore the different routes to the same solution without risking many misconceptions developing, as opposed to the converse.

Picture1c

Picture1d

Figure 2 + 3 – Comparing simplifying the square root of 72 using one algorithm, where we always select the highest common factor to be the highest square number that can divide 72, to the other multiplication sums when simplifying the square root of 72.

As a maths teacher, and even when I was as pupil myself, I knew the most efficient way to simplify a surd such as √60 was to have a multiplication sum with a factor, which was also the highest square number, because this would result in an integer multiplied by a surd. I knew this because my teacher explicitly told me. Later on we explored different routes to get to the same solution. Since I knew the answer for simplifying  √60 then when I got the same answer through different routes I felt successful and reassured. Why? Simply because I knew one concrete and accurate method to get the simplified solution for the problem. Teaching pupils one accurate method to solve a problem allows pupils to feel successful, and it further empowers them when exploring how to solve the same problem through different routes.

At Michaela, we spend a significant amount of time discussing our worked examples; whether DaniBodil or I have made the section of the textbook which is being taught that half-term, we discuss what is the best strategy to solve problems where pupils are adding and subtracting algebraic fractions and where the denominators are integers. What is the best worked example to solve problems where pupils are to substitute a positive integer into an algebraic term or expression? We outline it very clearly in our textbook, and we organise three or more worked examples where we interleave fractions, GEMS, decimals etc., but the cognitive process which pupils go through is similar in all three worked examples. This is because we want pupils to look at a problem and be able to identify each step between the problem and solution. How do we do this? We explicitly state it: step 1, identify the lowest common denominator; step 2, form the equivalent fraction by multiplying the numerators by the common factor; step 3, add the fractions with like denominators etc.

Picture1e

Figure 4 – Example of worked example made by Dani Quinn.

Our pupils are taught explicitly and we demonstrate clearly using our visualisers one concrete, accurate and efficient algorithm for the problem in order to get the solution. The different routes to the same solution of the same problem are explored later on once we know that all kids in the room are proficient at solving a selection of well-sequenced and crafted problem types with the one method we taught them initially. Our pupils are mathematically proficient; they love to learn and they feel this way because they feel successful knowing one accurate method between the problem and solution as to how to add and subtract fractions with integers as the denominator, or variables as the denominator, or expressions as the denominator. They then feel empowered when they can get the same answer through different methods.

And so, I respect that our opinions differed but I am sticking with the way I delivered the initial teaching of simplifying surds. It was the second lesson of this topic and despite not 100% of pupils were getting the right answer on their mini-whiteboards – where they were at the fourth lesson.

Posted on February 19, 2016 by Jo Facer

The Means and the Ends

In the past, I have confused the means and the ends.

In my first year of teaching, I thought back to my most recent experience of school: A-level English. Looking at the oldest class I taught, year 10 set 5, I thought there could be no better path than the one teachers older and wiser than I had taken me on. At ages 17 and 18, I had written an essay a week.

So, I decided to set my year 10 set 5 an essay a week.

Obviously, this was doomed to failure. My poor struggling year 10s, so far behind in literacy, failed so utterly in this first homework I lost their trust entirely. It took a very long time to build it back up.

I had confused the means and the ends. Of course I wanted year 10 to write beautifully crafted, intelligent essays. But I hadn’t considered that the way to get someone to write a great essay is not to just write a lot of essays.

I see this a lot in unit planning, especially at KS4. We’ve become awfully good at drilling to the exam. But two years is a very long time to drill to the exam. We have two years toteach, with perhaps two weeks (or, if desperate, months) to drill exam practice. Too many KS4 units on English language, for example, teach using unlinked, decontextualised texts, like random novel openings or random excerpts from unlinked news articles. Although this is the format students will encounter in their eventual exam, it is surely a wasted opportunity to only teach disparate content in the ‘teaching’ stage. Of course in the end, we want students to be able to write about decontextualised pieces of writing, but in the run-up it is surely much more effective to lead students through a well-designed scheme, for example short stories, or articles linked by a common theme like feminism, or social justice – schemes that will allow students to practice key exam skills, but also learn something.

The means don’t have to look like the ends. In fact, they rarely do.

When successful adults turn around and say: ‘I didn’t enjoy school. I want our children to have a more fun experience than I had,’ they are confusing the means and the ends. Weall want children to have fun; or rather, fulfilling, happy lives. But you don’t get to those ends by making school all about having fun. Many adults have succeeded because of schooldays filled with hard, hard work, not fun and games. We can have fun and gamesnow, because of that hard work.

In Education is Upside Down, Eric Kalenze writes about ‘engagement first’ teaching. This is the paradigm in which I was taught to teach. It was only after too many years of seeing my poorest students make insufficient leaps in their education that I realised my error. We can’t put fun first; we can’t even put exams first.

We have to put learning first, and the means do not often look like the ends.