Conception of the good

Today’s blog post outlines how I have broken down the teaching and planning of the initial (first-time) teaching of fractions to bottom set Y7. Last summer, my faculty and I worked on tailoring our own Mathematics Mastery Scheme of work for the upcoming cohort of year 7 pupils.

The teaching and planning of our year 7 curriculum is part of a five year plan to ensure that, as teachers, we are preparing our KS3 students with essential knowledge; that through extensive retrieval quizzes, interleaving and distributed practice of core concepts our pupils will be able to problem solve and think critically in KS4.

However, there are a few key points about critical thinking that must be brought to the forefront of this blog:

1) to think critically depends on having adequate content knowledge;

2) you can’t think critically about topics you know little about nor solve problems not understood well enough to recognise and execute the type of solutions they call for; and

3) when one is very familiar with a problem’s deep structure, the knowledge of how to solve it transfers well.

Essentially, I am trying to emphasise that content and factual knowledge are the bedrock of learning. I am going to demonstrate how I have taught a selection of lessons on the topic of fractions, where I fragmented different pieces of inflexible knowledge, linking them to what can be referred to as a knowledge framework. Kris Boulton clearly explained the idea of knowledge frameworks in his most recent blog – of which I highly recommend reading.

A Knowledge framework “is a web of connected knowledge.”

I taught each concept below as fragmented pieces of inflexible knowledge to slowly transition students to interlink each concept taught, in turn forming a bigger picture. Eventually they saw the connection between each idea. How does halving by partitioning link to the concept of halving by dividing by 2? How can we take our understanding that halving entails the splitting of an amount into two equal sized groups to calculating a quarter of 44? How can we calculate a quarter of 44 using our basic times table propositions?

I spent a significant amount of time ensuring that my students were able to identify that of half of all the multiples of 10, and that this was committed to memory; initially, teaching them the half value for each multiple of 10 to 100 using a double number line, as previously demonstrated by David Thomas at the La Salle Education Conference in March.

 Figure 1 – Double number line

Students were taught the link between 40 and its half value. We repeatedly practised this on mini-whiteboards throughout revision and testing. This was continuously practised for 2 – 3 lessons, followed by testing of this continuing less and less throughout the teaching of the unit. Here, students could successfully recall basic facts, e.g. that half of 70 is 35.

We moved onto calculating a half of 76 through partitioning (figure 1). Students were then halving two to three digit numbers by halving via partitioning (figure 2). Again, repetition was intense and then spaced out, with each test taking less time: students were now recalling factual knowledge; that the half of 70 is 35, in 3 – 5 seconds, but they were now also procedurally calculating the half of 76.

Figure 2 (left) – Student’s work on halving two digit even numbers via partitioning.

Figure 3 (right) – Student’s work on halving three digit even number via partitioning

As regular and spaced testing continued, students were starting to recall from memory that the half of 76 is 38; whereas the half of 46 is 23. If asked to explain, they could provide an accurate explanation, by demonstrating by portioning the two digit number.

Even though students were confident to half two and three digit numbers (specifically even numbers and not odd as of yet) I wanted to build upon their understanding of halving but in a different context. It went as follows:

Figure 4 (left) – Student’s work on multiplying fractions to determine a half of an even number

Figure 5 (right) – Student’s work on multiplying fractions to determine a half of an even number with division to go alongside

My rationale behind this was that it is absolutely irrelevant for students to explain to me right now why when we multiply fractions we multiply the numerator of both fractions to find the numerator of the result, and we multiply the denominator of both fractions to find the denominator of the result. What was valuable was that of students identifying that the result of 76/2 is equivalent to the familiar representation of 76 divided by 2 to achieve the result of 36. My students squealed with amazement – where the author does not exaggerate. They were dazzled that they could calculate half of 76 by partitioning; through the multiplication of fractions, as well as by dividing 76 by 2.

Here, I linked three fragmented and yet exactly the same pieces of factual knowledge in different contexts, where in terms of the procedures students followed them all to determine half of 76.  As three separate pieces of knowledge, they were inflexible because independently no student has completely mastered the concept of halving. Now, through building a knowledge framework where students have linked the three different representations of the same concept, their knowledge and understanding of halving has become flexible: it can now be accessed out of the context in which it was learned importantly it can now subsequently be applied to new contexts.

Figure 6 – An example of a knowledge framework used during planning.

After a selection of worked examples, students with their sheer enthusiasm and revelation powered through a carefully designed exercise of questions. I then ran through the concept of calculating a quarter of 76, and the students were recognised as benefiting from the following:

Figure 7 (left) – Student’s work on calculating a half or a quarter of a number (all multiples of 4 when calculating a quarter)

Figure 8 and 9 (middle and right) – Student’s work on calculating a quarter of multiples of 4 using long division as well as multiplying fraction.

1) Students have a clearer and wider understanding of halving through building upon their factual knowledge of halving but in different representations. As a result this will aid students’ memory retrieval when they come across the same or similar problem types;

2) Students are now seeing the multiplication of fractions in a context which will not be so separate and fragmented in year 8 or year 9, because they have been introduced to it in a different capacity in year 7 – different but relevant.

3) The connection and interplay between the different representations of the same factual knowledge of halving enables students to see new ways of thinking, inducing deployment of the right type of thinking at the right time. This is what we need to achieve as teachers in order to teach our students to think critically.

It is truly a combination of building a solid bank of content knowledge through distributive, spaced and tested practice, and by showing the connections of the different representations of the same piece of knowledge. These separate representations are inflexible singly but together create a fluidity that enables critical thinking.

Today, students smoothly could solve the following questions with only two worked examples being demonstrated to them by me. Their knowledge has become flexible and fluid. It is amazing!

Figure 10 (left) – Independent Task completed by students

Figure 11 (right) – Student’s work on finding a fraction of an amount

Today’s post links to my previous post concerning the importance of inflexible knowledge and why it is the foundation of expertise. As a result, it was argued that we should spend a significant amount of time teaching our students topics until the point of mastery.

What I mean by mastery is that students should be able to access key pieces of knowledge from memory, specifically their long term memory, instead of students attempting to complete different calculations in their working memory to solve a problem. For example, students should be able to automatically tell me that the square root of 9 is three, rather than calculating the square root of 9 in their working memory.

It is that time of the year where Year 11 students are frantically revising and, as teachers, our Y11 lessons are spent on consolidating prior knowledge, covering exam techniques and clarifying any existing misconceptions. I have been looking at a few AQA GCSE Maths papers, in order to get my head around how to prepare my students better in the years to come. One evening I decided to complete a past paper and here is what I realised:

We need to ensure that our teaching and learning of the GCSE is seen as a 5 year plan, where in KS3 we teach different concepts to the point of mastery – to the point of memory retrieval.  Inflexible knowledge is the starting point. Can students instantly recall basic times table calculations?  

Let’s look at this question, and my solution to it:

In this question here are the different pieces of knowledge I must have had access to in order solve the problem:

1) Calculating the lowest common multiple (LCM) of √2, √18 and 1;

2) Multiplying and simplifying surds where √2 x √9 = √18;

3) Adding and subtracting fractions of unlike denominators;

4) Square numbers and square root of numbers from √9 to √900;

5) Rationalising the denominator of a surds in a fraction to present the answer in the form of a√2 where a is an integer;

6) Recognising key mathematical words such as integer; and

7) Simplifying fractions where we divide 126 by 3 to obtain a numerator of 42 and a denominator of √2.

Evidently, there is a lot of thinking required to answer this question. There are many logical steps that need to be taken to progress from the problem to the solution. In this case, the difference between students who are experts and those who are novices is that the experts will be able to tell me that the LCM of √2, 18 and 1 is √18. As a result, we must multiply each fraction (numerator and denominator) by a common factor to achieve each fraction in the question to have a like denominator of √18.

The only thinking in working memory should be the logical steps to take from the problem to the solution. Not the calculations to each step – that should be an act of memory retrieval. How can this be achieved?

Only through extensive and distributive practice of finding the LCM of multiple numbers, finding equivalent fractions when adding and subtracting fractions with unlike denominators. This is the first step in the solution, which needs to be recognisable to all students. Automatising procedural knowledge of adding and subtracting fractions is necessary. It is one example of why inflexible knowledge is the bedrock of academic success. Similarly, only by identifying that √9 is equal to 3 can we then change the numerator of the first fraction. Again, inflexible knowledge such as square roots and square numbers represents another step towards the solution.

As teachers, we need to analyse exam papers and problem types, and look at our schemes of work as a five-year plan. How can we structure our teaching and learning to ensure that our students have the necessary background and procedural knowledge to attempt such problems? That these problem types will never be repeated? We cannot get students to memorise a model exam answer. This is not possible and neither is it recommended. We desire our students to see the connection between different concepts, however this is only achieved if students automatise different pieces of inflexible knowledge in the early years of KS3 and only then introduce problem types which connect different concepts.

My next post will outline the practical techniques I have been trialling within my lesson planning for my Y7 class. Watch this space.

Knowledge is important – particularly in maths. There are some concepts such as square numbers, where for example 2^2= 4 and 5^2= 25. Students are taught that 2 x 2 = 4 and 5 x 5 = 25. Through extensive repetition and practice, students are able to identify 2^2= 4 because they are no longer processing the calculation: instead they are retrieving this piece of knowledge from their long term memory. It should be an act of memory retrieval. Something I emphasised a lot I my previous blog.

Square numbers are an essential part of Pythagoras’ theorem, but without having any knowledge of Pythagoras’ theorem a student’s knowledge of square numbers is inflexible. Inflexible knowledge is “the unavoidable foundation of expertise, including that part of expertise that enables individuals to solve novel problems by applying existing knowledge to new situations.” (Willingham 2002:32)

I want to emphasise that inflexible knowledge is the foundation of learning. It is vital that we ensure our KS3 students build a bank of knowledge in their long term memory, even if that knowledge is inflexible, because as students start to accumulate knowledge over the years the connections and interplay between different pieces of knowledge allows for deeper learning to develop. This deeper learning in turn induces this bank of ostensibly inflexible knowledge to become flexible.

For example, square numbers should be drilled. Students should be able to recite 2^2= 4 and 5^2= 25 without thinking. They should not be processing that calculation; instead, they should retrieve it from long term memory. Now, squaring numbers can be extended to completing calculations such as 2^2 + 5^2 = 4 + 25 = 29. When introducing Pythagoras’ Theorem, students need to be taught that a^2  + b^2 = and that a and b refers to the two lengths which meet at a right angle of a right angled triangle: that the total of both squared lengths is the squared length of the hypotenuse.

Until students take on board this information they cannot proceed with Pythagoras’ Theorem. Once students learn the formula, the application of square numbers evolves into new knowledge; this is flexible knowledge, where “It can be accessed out of the context in which it was learned and applied in new contexts.” (Willingham 2002:32) In KS4, students can start solving questions like this with more ease given that different pieces of knowledge now connect to become flexible:

Inflexible knowledge consists essentially of facts, e.g. square numbers, the formula to calculate the area of a circle, and quadratic formula. If given the time to be learnt to the point of memory retrieval, we empower our students to apply such inflexible knowledge to a deeper instruction, e.g. Pythagoras Theorem, the area of a sector, and finding the solution of a quadratic expression. Surely we desire our students to be experts?

If so, we must begin with teaching knowledge – facts – only then will our students become experts and only then will “their store of knowledge…become larger and increasingly flexible, although not immediately.”(Willingham 2002:49)

These increasingly large stores of facts and examples are an important stepping stone to mastery. (Willingham 2002:49)

This second blog post discusses the pedagogy I observed from the brilliant Dani Quinn during my visit at DTA.

Firstly, what is pedagogy? Pedagogy is the art or science of teaching. It is the preparatory training or instruction of how best to teach.

I have been doing a significant amount of reading in the last few weeks of which I am going to refer, whilst discussing the pedagogy I observed.

Learning intention: To be able to calculate the area of a regular hexagon.

Dani started the lesson with a selection of prompt questions to induce students to recall important background knowledge, in this case the characteristics of an equilateral triangle. Following this, students were referred to recalling essential procedural knowledge of how to calculate the area of an equilateral triangle.

Background knowledge consists of essential core facts. Such facts do not change. A triangle is a shape which has three sides. An equilateral triangle has three sides of equal length and has three angles of equal size.

Procedural knowledge “is your knowledge of the mental procedures necessary to execute tasks…[it consists of] a list of what to combine and when.” [Willingham 2009:8]

Below are some of the prompt questions Dani asked, in order to check students’ background knowledge, as something which must be accurate in order to then carry on the lesson:

Figure 1: Equilateral triangle – highlighting essential background knowledge

“What facts do we know about an equilateral triangle?”

“What is the size of each angle in this equilateral triangle?”

“What is the name of this length?”

“What is the keyword which describes two lengths that meet at a right angle?”

“What angle does the perpendicular height meet the base length?”

“How can we calculate the perpendicular height of this equilateral triangle given the base height and the slant height?”

“Remind your partner why we need to use Pythagoras theorem in this case when trying to find the area of an equilateral triangle?

Here I was witnessing Dani induce students to recall essential knowledge. The thinking process students are engaging in here is essentially memory retrieval, where they are recalling learned facts. This is because “thinking well requires knowing facts.” [Willingham 2009:8] If students do not have the necessary background knowledge embedded in their long term memory then:

1. students cannot really consider or attempt the question which, in this case, is how to calculate the area of an equilateral triangle, and then to find the area of a regular hexagon; and
2. their working memories will be overloaded with not only previously learnt information but also novel information that they are interacting with for the first time.

Working memory is your conscious processing part of your mind. It is where you hold information that you are thinking about. Furthermore, it has limitations which your long term memory does not have. Your working memory can effectively process two or three new or to-be learned pieces of information. Whereas your long term memory has no limitations to the amount of information that it can store. Long term memory “is the vast storehouse in which you maintain your factual knowledge of the world…It lies quietly until it is needed, and then enters working memory, and so becomes conscious.” [Willingham 2009:7]

Figure 2. Aitkinson and Shiffrin (1968) Sensory memory-working memory-long term memory model

The prompt questioning students were engaging in was a memory retrieval activity where students were retrieving background knowledge of an equilateral triangle from their long term memory to their working memory. This act of recall not only improves your ability to retrieve information learned previously from your long term memory quickly, but it also means that each and every time you retrieve such background knowledge it will require less thinking – less conscious processing – because your memory is guiding your behaviour. Hence why such questioning is an act of memory retrieval rather than an act of thinking. Students are retrieving factual knowledge from their long term memory, something crucial to engage with for effective learning.

Students recalled basic background knowledge and were able to identify that an equilateral triangle is composed of two right angled triangles; that we divide the base length by two to get the base length of the right angle triangle, and given the slant height of the equilateral triangle being equal to the slant height of the right angle triangle, we can use Pythagoras theorem to calculate the perpendicular height. Dani did not teach any factual knowledge here; she did not need to because students had all this knowledge embedded in their long term memory. They were recalling it all, beautifully.

Figure 3. Calculating the perpendicular height using pythagoras theorem [This image was not used in the lesson – just one that I am using here]

Students guided Dani through a worked example on how to calculate the perpendicular height of an equilateral triangle using Pythagoras theorem. Here students were recalling procedural knowledge. They were recalling “extensive experiences stored in their long term memory in the form of concepts and procedures, known as mental schemas…they retrieve memories of past procedures and solutions,” of where they have calculated the perpendicular height using Pythagoras theorem. [Clark, Kirschner and Sweller 2012:9]

Procedural knowledge is knowledge around the logical steps needed to take to go from the problem to the solution. Extensive practice of solving several problem types allows students to embed such procedural knowledge into long term memory. Again this becomes an act of memory retrieval. Students here are retrieving procedural steps to calculate the perpendicular height from their long term memory to their working memory. Their working memory is not overloaded because they are conscious, processing the problem to already learned information, not to-be-learned information.

 So far:

1. Students had recalled background knowledge of their key characteristics of an equilateral triangle;
2. Students had recalled how to separate an equilateral triangle into two right angle triangles; and
3. Students had recalled how to calculate the perpendicular height using Pythagoras theorem.

Dani subsequently posed this question:

“Given that we know the perpendicular height of this equilateral triangle, and the base length, how can we calculate the area of this equilateral triangle?”

Students’ shot up in the air (popcorn technique), because they instantly recalled that to calculate the area of a triangle we multiply the base length and perpendicular height and divide by two. The structure of questioning and sequencing of prior learning carried students to the penultimate stage of the lesson. This is effective pedagogy: it takes into consideration the essential background knowledge and procedural knowledge which needs to be recalled from long term memory, and brought into working memory, in a manner which does not cognitively overload students. It allows students to identify the procedural sequencing of solving a problem type to calculate the area of a regular hexagon.

 The learning intention is to calculate the area of a regular hexagon and everything up to this point has been recall, because the search for the solution of the Learning Intention is where thinking will take place. It is at this point where students will learn new information. They have not been overloaded because the search for the solution is now only starting 35 minutes into the lesson. Students’ working memories are  not being overloaded with storing new information because everything that has been discussed thus far is all learned information. We are not storing new information inasmuch we are about to do so. This hammers home the key point that background knowledge and procedural knowledge is crucial for effective thinking. This also complements the propositions that students cannot begin considering the learning intention without the essential background knowledge discussed above.

Final stage of the lesson:

“Look at this hexagon, can I split it into a number of the same shape?” [Thinking time of 20 seconds]

“Here let me split it into half, and half from a different angle, and now from a different angle? [Here Dani split the hexagon into 6 equilateral triangles] How can I now calculate the area of a hexagon?”

Figure 4. Splitting a regular hexagon into six equilateral triangles

At this point, Richard and I were astonished with the sheer enthusiasm and excitement of students standing up ready to answer the question. At this stage of the lesson Dani introduced new learning. In students’ working memories they were learning new information: that to calculate the area of a regular hexagon we multiply the area of one equilateral triangle by 6.

The lesson was well sequenced. The foundation of academic success was in the recall of essential background knowledge and procedural knowledge. Students were recalling their prior learning, and through extensive practice in their time at DTA were they able to get to this end stage. As they practised a completion exercise on this topic, they were rehearsing new learned information and embedding it into their long term memories.

Background knowledge is essential.

I have heard raving reviews of Dixons Trinity Academy in Bradford and, as such, this Easter break I was adamant that I would book in a visit. There are several Dixon Academies, all of which are part of the Dixon’s City Academy Charitable Trust – something I learnt when the taxi driver dropped me off at Dixon’s City Academy instead of Dixon’s Trinity Academy.

Currently, Dixon’s Trinity Academy has three year groups. It’s core values are trust, fairness and hard work. The three key drivers are Mastery, Autonomy and Purpose:

1. Mastery: the urge to get better and better at something that matters.
2. Autonomy: the desire to direct our own lives.
3.  Purpose: the yearning to do what we do in the service of something larger than ourselves.      

Learning is put first. This is achieved through a partnership between parents, students and teachers. Together they are achieving one goal which is to ensure that the children of Trinity Dixons Academy are receiving the best learning experience

“We help our students to value learning by activating them as owners of their own learning.”

Luckily, Richard Deakin and myself walked in on time to be greeted by the wonderful Dani Quinn, Head of the Maths Department. In doing so, we witnessed her welcoming and greeting students passing by, and that they were wishing her a good morning in turn. Straight away, I saw that this was school a full of love.

Here, I outline my day: what I saw; what I loved.

Purposeful Start

The current Year 9 cohort silently and swiftly take their seats in the lecture hall, ready for their 10 minute maths task of the morning, as composed of a selection of revision questions. They are all fully equipped for the task. They are all eager and ready to start their school day n a purposeful manner. Ms Quinn’s instructions are remarkably clear, succinct and of the highest quality:

“Welcome Y9 thank you for entering in silence, raise your black desk, open your book, using your black pen to write down the first 15 prime numbers. Off you go.”

3 minutes later

“Thank you for completing the task in silence. Can you get your green pen. We are going to tick our work if it is correct, and put a cross next to our answer if incorrect. Tracking the speaker.”

Each student was engaged; focused; happy; and responsive to the clear instructions. Each and every minute – within those first 10 minutes of the day – I witnessed was completely purposeful.

“Mexican wave.

If you got all the prime numbers that you wrote completely correct then save your Mexican wave until the end.

If you correctly wrote the first 5 prime numbers correctly then Mexican wave.

If you correctly wrote the fist 10 prime numbers correctly then Mexican wave.

If you correctly wrote the first 15 prime numbers correctly then Mexican wave.”

Fairness

Students and staff transition around school in silence. Dani explained that we want to model what fairness is in all areas of the school. As a result, teachers transition in silence as well as pupils. Fairness is one of the three core values, as well as trust and hard work. These three values are implicitly and explicitly reiterated. Everything is rationalised to ensure that every action or behaviour is purposeful.

Whilst watching Dani teach, one pupil did not put her pen down when asked to, in a manner so as to track the teacher, ensuring all students were listening. Dani’s response was:

“Saira can you empty your hands please as it is not fair to make everybody wait for you. I know it is because you are really enthusiastic to get started so I appreciate your enthusiasm, but you will need your pen in just two minutes.”

I loved how she reiterated how important fairness is in the classroom setting. This student apologised. Dani thanked the student for her apology. There was no confrontation. There was no answering back. Students are aware of how to behave. They are aware of teacher’s expectations. It is incredibly simple.

Autonomy and Trust

There is autonomy for teachers as well as students.

Teachers have autonomy in how they are able to teach their lessons. There are a few whole school practices which all teachers abide to, such as having lollipops with students’ names on when questioning pupils. High quality and high pace questioning is the main form of quizzing pupils’ understanding of the main teaching point. This ensures that all students are engaged and listening to the teacher. There is no set lesson structure which teachers are told to follow; teachers are trusted professionally to deliver high quality lessons.

Students are given autonomy in different aspects of their school life. Students are given a choice of three different colour school jumpers and school polo shirts to wear. Student ambassadors in school conduct tours for visitors and they are not scripted. They are given autonomy in how they describe the school. Whilst on my tour, I asked myriad questions on the structure of the school, co-curricular activities and their feelings about the school etc. Both my wonderful guides were incredibly positive and informative.

Hard Work and Mastery 

After lunch I observed Dani with her Advisory (form group) where all students were taking part in DEAR (Drop Everything and Read). There is again autonomy in students either reading their own books or that their form tutor reads to them. Dani read a few pages and requested feedback on whether she was reading slower, in line with her Advisory giving her feedback the day before. She thanked them for her feedback and hoped to read slower today. In turn, students were able to come and read out to the class with Dani. Students were keen, enthusiastic and very eager.

Whilst a student read out to the class, Dani asked a selection of comprehension questions in order to ensure that students were both engaged and were fully understanding the text being read out. She would ask students to define words, or to provide synonyms of particular words. What does the work untameable mean? Why can you not tell when the monster comes if it is a dream or a reality? Students did not raise their hands but would popcorn. They would stand up to answer the question. This allowed the teacher to know which students knew the answer; as well as those who did not.

Here, DEAR was peaceful, enjoyable and purposeful. It also allowed students to master their understanding of the text. During DEAR, students were being quizzed on their comprehension skills, vocabulary, reasoning and listening skills.

Family Lunch 

I also experienced Family Lunch – something which I had previously experienced whilst visiting King Solomon Academy and Michaela Community School. Students are sitting in a set group where they are all provided a role such as serving the food, distributing plates, cutlery, pouring water for each person on the table etc. I was kindly welcomed to a table, with students from Dani’s Advisory. I started to pass a plate full of food down the table when a student said “No, Miss you start to eat first you’re our visitor.” I thanked Kashif who then responded by welcoming my thanks. The lovely group of students asked about my day, my school and took a genuine interest in what I had to say. Through such daily experiences of Family Lunch, the students evidenced confidence, manners, etiquette and excellent conversation and company. This was the highlight of my day.

If you haven’t yet visited Dixons Trinity Academy I would make that your next action for CPD. Part 1 outlined the ethos and philosophy behind the school. What I witnessed and what I appreciated. Next week, Part 2 will follow, outlining the amazing pedagogy I observed.

At 7:07 AM on Saturday 14th of March I jumped on a train from Manchester Piccadilly to Birmingham New Street with @Mr_Neil_Turner, @Linds_Bennett and @RichardDeakin for our third La Salle Education conference. After another manic week at school, it was definitely worth the trip. Each and every time Mark McCourt delivers a conference of the highest of quality, with useful workshops and at the same time he successfully gathers some of the most wonderful teachers up and down the country. This is my account of the day.

Workshop #1

David Thomas – 3 teaching techniques you need to know

I have seen David deliver a workshop at each La Salle conference and without fail he presents a different workshop each time and every time. I always walk away with a better understanding on assessment, a clearer insight behind the philosophy of Mathematics Mastery, and this time David discussed the use of effective bar modelling, the use of algebra tiles and the double number line. I have seen different workshops on bar modelling, but David’s interpretation was very interesting. I have seen bar modelling examples where the variables are all represented as different bars but in this case David promoted the use of a single bar. I am still to get my head around using a single bar for a selection of different variables but I am leaning more towards having each bar in a column representing one variable. I hope to pick his mind soon. Furthermore, his introduction of the double number line to introduce the topic of proportion was fantastic. What a simple yet very brilliant manipulative to use. You are simply building upon the times tables! He discussed how you can scaffold the use of the double number and then transition to more challenging mathematics by introducing the concept of recurring decimals. This something I am keen on trying in the classroom very soon. Thank you David! 

Workshop #2

Rob Wilne – The Hows and Whys of “how?” and “why?”

I had the coincidental pleasure of sitting next to Rob Wilne on a train journey to London a week before the conference. After a long chat on the train I was definitely set on seeing his workshop. Rob (@NCETMsecondary) who is the Director for Secondary at NCETM discussed his findings whilst visiting a selection of schools in Shanghai (Mark reiterates – Shanghai is not a country!) Rob discussed that reasoning is composed of three essential components – factual knowledge, procedural fluency and conceptual understanding. He then went onto explaining and analysing a selection of examples on how to develop conceptual understanding of number through the use of concrete-pictorial-abstract (CPA) manipulatives such as bar modelling to generate number sentences. The value of using CPA manipulatives is to develop student’s conceptual understanding of number, and then he moved onto showing examples of students reaching a stage of procedural fluency where they start reasoning about number because of a deeper understanding of number. The very important difference between the two ideas struck me.    

The deeper understanding of number serves the purpose of enabling students to reason using the CPA manipulatives shown. Whereas the skill of reasoning about number is strongly linked with procedural fluency.  

He then explained that some manipulatives (in this case Farmers’ Fields) is “good for the development of conceptual understanding, less so for procedural fluency: pupils must be supported to develop the standard algorithm from this.” Time is required on developing conceptual understanding using a concrete and then pictorial manipulative before moving onto the abstract. Rob emphasised that pupils require “years of experience and familiarity with a mathematical concept before they can reason with confidence and creativity in the domain around that content.” From a solid foundation of factual knowledge, and then a strong development of conceptual understanding of number are then students able to effectively reason about number and develop that procedural fluency that challenging problem solving tasks demand of our pupils.  Pupils must understand the structure of the manipulative used and identify the standard algorithm from it. For example, “I share some sultanas between Alice and Bob in the ratio 3:5. Alice gets fewer sultanas than Bob. How many grams of sultanas does Bob get?” In this example, Rob emphasised the importance of bar modelling to draw students’ attention to the additive structure of the problem where the difference in amount is highlighted by Alice’s bar being smaller than Bob’s by two equal sized boxes. Then it is crucial to spend time on enabling students to acquire deep conceptual understanding of the multiplicative structure of the problem in regards to the ratio element of it. From a deep understanding on using such concrete and pictorial manipulatives students are developing that conceptual understanding, but procedural fluency requires pupils’ to manipulate problems of increasing complexity with confidence. As a teacher who is intrigued and fascinated by curriculum design and pedagogy it just fuelled me with 1001 ideas. Thank you Rob!

Workshop #3

Sue Lowndes – Bar Modelling

In my opinion, this was my favourite workshop of the day. Sue Lowndes of Oxford University Press outlined how to effectively teach and plan using bar models to ensure effective student learning. Firstly, Sue explained the importance of the key steps students need to take when learning using bar models as the main pictorial manipulative. 

The key point which I haven’t heard before, and which stuck with me was in highlighting where the solution of the problem lied within the bar model using a curly bracket and a question mark. How obvious but something I completely did not think of! Yet it is essential because it narrows students’ focus on identifying where the solution of the problem lies.

She then outlined the different types of bar models and explained the key features behind each one: (1) Discrete bar models (2) Comparison bar models for addition and subtraction. (3) Multiplicative comparison models (4) Ratio models Sue then allowed all of us excited teachers to put pen to paper and attempt a variety of our own which proved to be rather challenging in the sense of following the instructions provided at the start. It requires a great deal of thought in terms of structuring how to determine the solution using the bar model, and then using the correct bar model for the problem given. She then showed us some classwork from a selection of students on using bar models to solve simultaneous equations. I am teaching that topic very soon and cannot wait to plan the unit using bar models. Now it is your turn to try a question that Sue asked us keen teachers to try? I shall reveal the answer next week. Overall, the conference was definitely the best that I had attended. This was not only due to the excellent workshops delivered by a selection of wonderful individuals, but also through the sheer organised manner of the event by Mark McCourt’s team. We were all in the great hall for the start of pi day. Not to mention the excellent cake and the great company. Yes, I did eat my body weight in cake that day. Next one is in Manchester, if you remember hearing a loud “YES!” during Mark’s announcement that was me. I hope to see you all at the next one.

Peak of my day was spending it with these wonderful people who inspire me to become the best teacher I can be:

Neil Turner, Richard Deakin, Lindsey Bennett, Kristopher Boulton, Bodil Isaksen, Bruno Reddy

and it was finally lovely to meet Caroline Hamilton, and see again, Dani Quinn and Paul Rowlandson.