The difference between being a teacher with 20 years of experience and one with one year’s experience, repeated 20 times, is reflection on your practice. This is often achieved through partaking in professional development activities.
Membership of, or involvement with, a subject specific professional association such as ATM (The Association of Teachers of Mathematics) offers a variety of ways to do this.
The annual Easter conference offers three full days of workshops and events, as well as the opportunity to chat with others involved in mathematics education, to inspire you in your practice. One day CPD courses, with a member discount, give you an opportunity to explore an area in depth, led by an expert in that field. Our branch events, held around the country, give you the chance to listen to a national expert talking about their work or an opportunity to explore an idea with fellow professionals.
At Mathconf18 we have a bookstall with our most popular publications available to buy. These contain ideas for your classroom, together with some thoughts about how to use them and the pedagogy, or teaching strategy, that underpins the task. Trying a new approach in the classroom can be scary and we often include suggestions on how to manage its implementation with minimal risk.
In particular, ‘Assessment in the new National Curriculum’ gives prompts and questions to support learners getting the most from a task, both in terms of content and skills. Geoff Faux’s books, ‘Exploring area and fractions using square geoboards’ and ‘Exploring geometry with a 9-pin geoboard’ contain many rich tasks, as well as advice linked to how children learn mathematics. All of these contain ideas that you will want to reflect on and explore to develop your teaching.
Membership of ATM brings with it a subscription to our journal, Mathematics Teaching, with many articles from teachers exploring and reflecting on their practice. Reading any one of these will get your mind buzzing, whether you would like to use the ideas or not!
As a reflective practitioner you will not only see that a task or strategy worked well in the classroom but will also begin the journey of understanding why it worked well. We all (hopefully!) reflect on lessons that go badly in order to avoid that happening again, but it is as important to understand why things work when they do. It is not a matter of chance – even though it may feel like it sometimes!
You can see The Association of Teachers of Mathematics in the networking / Exhibitor slots during #MathsConf18 at the City Academy Bristol on Saturday March 9th 2019.
The use of technology must permeate the study of AS and A-level Mathematics and this is making a difference to how we teach and how students can solve problems. Now students can get answers in seconds to the kind of problems which would previously have required mathematical techniques like the quadratic formula, taking logs or integration.
This has brought changes to the way questions are asked in the examinations and also to the skills which a student needs to have. Finding the roots of a quadratic equation by hand wastes time and gains no extra credit so students should use their calculator. However, also they need to be able to interpret the answer. A calculator will give them the roots, not the factors for the quadratic.
So for students to do well, they should recognise when they can use a calculator, know how to use it and understand the information it is giving them. The following is a selection of key calculator skills which students need to have:
Solving equations for specific values
Integrating and differentiating for specific values
Finding roots of quadratic and cubic equations
Working with Matrices
Working with Complex numbers
Finding values of statistical distributions
At MathsConf18 in Bristol, I will be delivering a session on the Use of Calculators in the new A-levels. In this session I will focus on the use of a scientific calculator, particularly seeing how the best use of it could have made a real difference in last summer's A-level exams. Using actual student responses from last summer’s exams, I will show how to make the best of the calculator and help you to understand the thinking behind setting the assessment.
I hope to see you there and if you are then make sure you bring a gee-whiz kind of calculator with you!
You can see Dan Rogan speak about "Use of Calculators in the New A-levels" during #MathsConf18 at the City Academy Bristol on Saturday March 9th 2019.
Back in 2005, I began to write my Risps (short for Rich Starting Points) website, with the backing of the Gatsby Foundation. It took a year, and by the end of that time I’d posted forty pure investigative activities for A Level Maths. Since then, the site has gradually become popular, and that’s happened alongside a growing interest in using open tasks to teach mathematics at all levels.
A couple of years ago, I took what seemed to me to be the obvious next step;
How about a collection of risps for pure Further Maths A Level?
It could be argued that Further Maths sees a larger percentage of teacher exposition than elsewhere, and yet FM students are often well-suited to self-study and tackling problems under their own steam. Over the next two years, I wroteFurther Risps, and I’ve recently self-published that both as a hard copy book and as a pdf. I don’t stick to any particular syllabus, but any Further Maths teacher should find that the majority of the forty tasks here will be adaptable for their students’ situation.
I hope during my workshop to encourage people to try a task from the collection, and then enter into a discussion about how such material can sensibly be integrated into an A Level course. I also hope to offer some tips on how a teacher might write open tasks of their own.
You can see Jonny Griffiths speak about "Further Risps; open tasks for FM A Level" during #MathsConf18 at the City Academy Bristol on Saturday March 9th 2019.
Atomisation: Breaking down your teaching like you have never seen before…
Atomisation? Is this the new fad? Why is there a buzz around this word?
It’s because atomisation is… awesome!
It’s a term that all maths teachers can use to describe the first step of their planning process. Atomisation is the process where teachers break down a topic into its sub-tasks.
When you break down a topic into its sub-tasks you avoid two common pitfalls to the planning process. Firstly, it avoids re-teaching, and secondly, it avoids missing out sub-tasks to teach. Re-teaching is frustrating for every teacher because it means that teachers lose time to teach the remaining scheme of work. Additionally, when you miss out teaching certain aspects of a topic which really need to be taught explicitly then that’s equally frustrating. Why? Simply because teaching certain aspects of a concept within a sequence can determine how successful a pupil is able to develop a big picture understanding of the topic being taught. If we plan what we want to teach, and then sequence the order in which we want to teach before we start creating any resources, then we can avoid the second pitfall.
Atomisation also allows a teacher to develop a big picture understanding of the topic before they start teaching it. Teachers can see the starting point of what prior knowledge needs to be recapped for the topic to be taught successfully to then see the most complex application of the topic which will be taught. Most importantly, the sequence from the easiest to the most complex application is designed for the greatest percentage of pupils to learn the topic successfully on the first teaching attempt. This refers to all pupils, especially the weakest.
In this session, I shall discuss how I broke down the teaching and planning of the topic ‘Exterior and Interior Angles’.
Here is the breakdown of the topic:
Short Division Recap
Exterior Angles and Interior Angles
Exterior Angles of Regular Polygons
Finding the Exterior angle from a regular polygon
Use the exterior angle to identify a polygon
Interior Angles of Regular polygons
Finding the Interior angle from a regular polygon
Use the interior angle to identify a polygon
Finding Exterior Angles between combined regular polygons
Finding Angles within an Isosceles triangle between combined regular polygons
Algebraic: Exterior angles
Algebraic: Interior angles
You can see Naveen F Rizvi speak about "Atomisation: Breaking down your teaching like you have never seen before…" during #MathsConf18 at the City Academy Bristol on Saturday March 9th 2019.
In the latest of his “time to revisit” series, Peter Mattock focuses on division…
Division is one of 'those' concepts. Seemingly so simple, and yet pupils nonetheless can really struggle with division. In particular, pupils can struggle with the process of dividing one number by another or equally with identifying when a particular situation requires a division. In his "Time to revisit..." series, Pete has prepared a session that will hopefully help you think about your teaching of division.
Pete says. "Look at how we introduce the concept of division to support calculation strategies, but also to help pupils make sense of situations that require division to solve them. By examining different interpretations, supported by suitable representations and manipulatives, that are useful ways of being able to think about division."
In his session, Pete will take us on a journey through division. Starting with simple positive integer divisions, through division of negatives, divisions that result in fractions, divisions with larger numbers and finishing with division of both decimals and fractions. How we move pupils away from the manipulatives and imagery, and how the work with those can support moving to a purely numerical calculation.
And finally, by looking at practical situations involving division, and how, as teachers, we can use our deeper understanding of division to see why each of the contexts offered result in division problems.
Professor Emeritus in the department of education at the University of Oxford calls division 'The Dragon'. Those pupils who slay 'The Dragon' tend to go on to do well in mathematics; whilst those who don’t tend to struggle from that point on.
Ultimately, Pete's session is about supplying the weapons necessary to help as many as pupils as possible to slay this metaphorical 'dragon'.
You can see Peter Mattock speak about "Time to revisit... Division" during #MathsConf18 at the City Academy Bristol on Saturday March 9th 2019.
Most of the #MathsConf workshops I’ve done over the years have been directly about our qualifications but occasionally I get the opportunity to look more broadly at assessment and look at how things have changed over the years. In past workshops, I’ve focussed on problem solving and exam structures but this time I’m looking at how questions and papers have evolved over a sixty year period.
So my starting point will be the late 1950’s
By this time, ‘O’ levels were well established though most of the population took no qualifications at the age of 16. Those that did were expected to deal with imperial units and calculate with only log tables to help.
Next I’ll look at papers from the late 70’s and early 80’s
By this time, calculators were becoming available and the metric system was in use. Schools were changing with most becoming comprehensive and offering both ‘O’ level and CSE exams. For maths teachers, this period was just before the Cockcroft Report which influenced how we teach from 1982 onwards. For me, this was the period between experiencing the system as a student and becoming a teacher within it.
Finally, the late 90’s
GCSE has been the standard exam for most students for 10 years and the three tier system is well established. Coursework in maths is the norm but not yet compulsory and performance measures are becoming increasingly important in measuring schools and teachers. Personally, I was an experienced teacher and head of department having survived two Ofsted inspections which lasted a week each.
I’m interested in whether there are topics and question types that have remained pretty much the same through this lifetime of change and where the biggest changes have happened. I want to explore how the more demanding questions we set now compare to the rigours of ‘O’ level and how the breadth, and depth, of what we assess has changed. If I’m lucky enough to have an audience, I’ll challenge them to ‘guess the year’ when differences in wording and layout are taken away and we get to grips with the underlying mathematics.
If, like me, you love old questions, then come along or follow @AQAMaths on Twitter and look out for our weekly #AQAmathsarchives questions.
You can see Andrew Taylor speak about "A rummage through the archives" during #MathsConf18 at the City Academy Bristol on Saturday March 9th 2019.
'Not equal to the task? Alternative methods for teaching inequalities’
' is a blog preview of Sam Blatherwick's #MathsConf19 session/workshop being run at #MathsConf19.
Pedagogical approaches to inequalities can sometimes feel like a set of disparate methods that don’t seem to have much connection to one another, other than a shared set of symbols. It’s easy to resort to presenting inequalities as a set of tricks that work ‘like’ other topics, rather than demonstrating the many links within the topic itself.
This session will look at ways to bridge the gaps between various methods, from representing solution sets to solving quadratic inequalities. The workshop will explore strategies for building pupils’ conceptual understanding of inequalities by looking at how different approaches to teaching the building blocks can give alternative methods to solving the trickiest problems.
Here are some further thoughts from Sam:
The inequalities we see at school are the first small step on a long learning journey for our best mathematicians: look, for example, at proofs from A-level, or the validity of a binomial expansion, or the range and domain of a function. Or take this, from my first year undergraduate notes:
However, this is a topic that, at GCSE, seems a bit strange and unfamiliar. Sometimes techniques can be lumped together with algebra, sometimes with number, and the concept of things not being equal must seem bizarre to students after years spent studying equations. When looking at the various methods and techniques that need to be covered for GCSE inequalities, it’s easy to see a variety of separate methods, ‘tricks’ that work (“it’s like an equation but don’t divide by negatives”) and throwaway phrases (“bullies point at small things”, “crocodile eats large things”). The route through can seem muddy and impractical.
When considering the GCSE specification, you could see this as a starting point:
but then not even consider linking it to this:
In the past I would have said “well, just solve it like an equation”, which then changed (when students didn’t recognise this) to “well… just replace the inequality sign with an equals sign and then solve it like an equation”. None of this really feels satisfactory in hindsight, especially when the question had an unknown on both sides, bringing all sorts of potential pitfalls, or when it looked like this:
In which case, time for another method. Or make sure you don’t divide by a negative. Or try to remember to swap the sign when you do divide by the negative.
So what exercises and explanations can we use to promote a conceptual understanding?
I adapted the methods I used to build an understanding of inequalities. The session hopes to cover a range of question types, from the inequalities you see above to the quadratic inequalities and regions that we hope our best GCSE students will master.
You can see Sam Blatherwick speak about "Not equal to the task? Alternative methods for teaching inequalities" during #MathsConf19 at the Penistone Grammar School on Saturday 22nd June
If you work as a Maths teacher in a secondary school, chances are that your focus is primarily on
your Year 11 or Year 13 classes and giving them the best chance to make the progress they need to
obtain a good grade at the end of the year.
But what about our other year groups? In particular Key Stage 3. In 2015, Ofsted produced a document titled “Key Stage 3: the wasted years?” The findings include:
“In too many schools the quality of teaching and the rate of pupils’ progress and achievement was not good enough”.
“Inspectors reported concerns about Key Stage 3 in one in five of the routine inspections analysed, particularly in relation to the slow progress made in English and mathematics and the lack of challenge for the most able pupils.”
I have worked at my current secondary school since September 2012. During that time, until September 2016, we underwent a few different Schemes of Learning, but never got to a point where the pupils were “GCSE ready” at the start of Year 9. I along with other members of the Maths department just focused on making sure that our Year 11’s and 13’s perform, make progress and reach their Year 11 target.
How can Year 11’s reach their target grades if they have a poor experience of Mathematics at Key Stage 3? In response to this, my school decided to focus on a long term change and decided to radically transform the Key Stage 3 curriculum by hiring experienced primary school teachers, including James (@HowWeTeachIt), to use their knowledge and experience of KS2 mathematics to ensure that KS3 builds on the successes of students time in Primary while ensuring they continue the progress made in KS1 and 2 into KS3.
Initially, I had reservations, but as time went on, I became more convinced that what they do is the right approach for our Key Stage 3 pupils.
A new Scheme of Learning was introduced with an increased focus on three key areas within every topic, whether Number, Algebra, Ratio, Proportion and Rates of Change, Geometry or Handling Data. These were:
Fluency – varied and frequent practice with increasingly complex problems as lessons progress
Reasoning – Making conjectures, generalisations, justifying, arguing and proving methods using mathematical language
Problem solving – Applying maths to routine and non-routine problems, breaking down problems into bitesize chunks and persevering in finding solutions
This is the approach that is the norm in Primary practice and has been for a number of years, even before the new curriculum was introduced in 2014. The focus on ‘Using and Applying Mathematics’ with particular emphasis on Problem Solving, Communicating and Reasoning became enshrined in the three aims of the new primary (and secondary?) curriculum and increased the importance of exposing students to increasingly complex mathematical tasks that went beyond a simple requirement for students to do purely procedural mathematics.
The team decided, following inspiration by Steve from Kangaroo Maths, to change the three names to ‘Do It’ (Fluency), ‘Twist It’ (Reasoning) and ‘Deepen It’ (Problem Solving).
Our initial Do It planning had varied questions, with no links between questions, and no flexibility. After reflections from lessons, and reading books such as Craig Barton’s “How I Wish I’d Taught Maths”, we now include variation of questions where appropriate, better thought out questions, and more questions with flexibility of where pupils start with their work.
Examples of our work include:
Our Twist It planning initially were just worded problems, and fluency in words. Now, what we do include any of the following:
Which one doesn’t belong?
Spot the error
Examples of our work include:
Our Deepen It planning focused on goal specific problems and had heavy cognitive load. However, our working memories are limited, and it can be hard for pupils to know where to start. So, James and I used Craig Barton’s “goal free effect” method and introduce more “Tell me what you know” problems. Examples of our work include:
We want to keep training the pupils and ask them “What maths can they do rather than what they do see?”, and to build connections.
Most importantly for our approach is how we use these resources to work towards a Teaching for Mastery approach. In my workshop, we will look at the three areas in more detail, and we will also discuss about assessments, revision techniques and the use of exit and entry cards.
This is a work in progress, and by all means, we haven’t yet found the perfect lesson for all topics. But I hope that by attending #mathsconf18 and signing up to this workshop, you will be able to gain inspiration into planning good to outstanding lessons for all pupils, and not just on the examination groups. Maybe, just maybe, you can join us in this exciting adventure!
You can see Matthew Man speak about "How we teach it - The Mastery Way" during #MathsConf18 at the City Academy Bristol on Saturday March 9th 2019.
I’ve used Cuisenaire occasionally in my career. Much of the time it was as an aide in the teaching of fractions to younger secondary pupils. However, this fabulous resource has so much more potential. It can be used to introduce the very basics of arithmetic such as additive relationships, or extended into harder topics such as simultaneous equations, Pythagoras and equation of a straight line.
Allow me to share a reflection of “a learning episode”.
This evening my five-year-old son, who is as inquisitive as children of that age tend to be, spotted a small bag of Cuisenaire rods on my desk. He was immediately drawn to them. “What are those daddy? Can I see them?” The verb “to see”, for a five-year-old is not just an interaction of the eyes and brain. It is a tactile action, it involves touching the object and interacting with it in some way.
He poured all of the blocks over the table, gazing in my direction to ensure that this was OK. Immediately, he began to play with them. He built little patterns and began to group the rods. There is something about these little rods that is inherently enticing.
Mark McCourt had told me that young children will begin to behave mathematically with these blocks, given enough opportunity to play with them. I was stunned, when, after just a few minutes, my son said “Maybe after this I could do it by sizes”. The level of categorising went beyond the first level I’d expected him to consider; colour. Instead it was a mathematical idea. I let him play with them for a while. I was minding my own business, leaving him to it and not prompting him in any way.
All of a sudden, a loud announcement, coloured with the excitement and joy of a profound revelation: “Its colour is its size!” In that moment, these little rods had gone from being toy blocks to being something else. It’s impossible to make inferences about the connections he was making. However, what was to follow demonstrates, to me, that he was thinking hard.
“Orange is the biggest one!”
I’d resisted the urge to prompt or direct him until now, but I couldn’t help myself, I wanted to play too. Displaying a little bit of shock for his benefit I asked him “Is it really bigger than the blue?”
He was, correctly, adamant that it was. Having his conjecture challenged, he did what any mathematician would do – he sought out a proof! Carefully lining up the blue and orange he showed me that there was a gap. “Look – you can put a white one there”.
He’d just modelled a number bond to ten. While he can already “do” addition he hadn’t yet recognised that the calculations he does at school were synonymous with his demonstration with these little rods. I think that will come in time – after all, the pace of progress in his use of the rods is startlingly fast.
He continued to play freely with the rods. He made some domino trails. This is the beauty of this manipulative – there is fun to be had with it! A short while later I saw him looking at the purple and dark green. “This is four more taller than purple”. I was perplexed with this idea of four, as the green is only two blocks more than the purple. I chose not to judge, but instead try to understand his interpretation of the situation. I asked him to show me why.
He motioned with his finger four equal steps from the end of the green to the end of the purple. I suggest that there were two possible thought patterns here: the first is that there was some unit of measurement, known only to him, which was his point of reference. Alternatively, he hadn’t quite grasped the relative size of the white block to the others.
Maybe in asking why, I challenged him in a way that made him reconsider things. He presented me, absolutely delighted with himself, the following set up:
The mathematics is simply pouring out of this free play. These are exactly the sort of comparative models I watched Mark McCourt share with teachers yesterday!
The free play continued with “now I want to count them all”. This was going really well. He had counted past 50 when, all of a sudden, his twin sister appeared. He continued to count but her presence (she was asking me about the rods) put him off a little. He said he thought he’d counted properly, but wanted me to double check. His sister volunteered – she was keen to get involved too. Midway through counting I heard her brother say to her “you’ve missed out all of the fifties and sixties”. He had been listening intently. They decided to count them again together, this timing getting the correct total. I didn’t check the total for them. They have the knowledge between them to be sure of succeeding.
They began to discuss the orange rod. He told her how it was the biggest one. She replied, clearly insulted that he thought she hadn’t realised this “I know! Look – it’s two yellows”. She lined up the rods to show him her thinking. I hoped they’d follow this line of inquiry further, so offered a suggestion “how many white ones to get the orange?”. The guesses were wildly inaccurate. One thousand is the phrase they like to use for “lots of something”, so this was the figure they last mentioned. They each made their own models, slowly and deliberately placed the whites against the orange. This was a real test for their fine motor skills.
“The big one is the same as ten.” I noticed that neither of them said “ten whites”. Could it be that they had stumbled upon the standard numerical values of the rods? I was about to offer another prompt when my son asked me for a pencil, so he could measure it. They have done a little bit of measuring in school recently. Did the number ten resonate with him in some way as to remind him of this?
Before long the pencil was cast aside and a box was to be measured. This looks like a potentially intuitive introduction to the idea of perimeter. Yet more rich mathematical activity.
All of the above happened in less than 30 minutes. With no direct instruction from me a whole wealth of possible starting points for further exploration have been encountered. Cuisenaire is an incredibly powerful and versatile manipulative. The extent of how it can be used to support learning and teaching is vast. You can learn more about this by coming along to one of our Concrete, Pictorial, Abstract and Language CPD days.
Focusses on retention and independent learning.
All students know how they can make progress at all times.
Ensure students see that effort = success
All students are making progress from our regular low stakes assessments (feed forward). The feedback cycle is completed as feedforward is checked.
If feedforward indicates they are not making progress from the feedforward then we follow up from this.
Teachers know where student weaknesses lie and plan responsive teaching to work on these.
Teachers know the students who are really struggling overall and can put intervention in place to work on this.
Teachers know more information about disadvantaged students and give personalised intervention (for example through personalised questioning).
We aren’t waiting till the half termly assessment to realise that students haven’t understood something.
Ensure teachers have time to plan responsively and are able to personalise their lessons.
How it works
Students are given a pretest. This includes 5 topics.
2 are topics that have just been studied.
3 are topics that have been studied previously (sometimes in a previous year).
The questions will typically be of an A01 style – the idea here is to check they understand the topic and technique. There is a separate problem solving section at the end.
1 of these 3 topics will often be a prerequisite for a topic that is about to come up – particularly useful as this can guide planning for the new topic.
Topics are repeated over quizzes (generally with almost identical questions but different numbers – so students can use previous quizzes to revise from). Also, questions will be quite similar to the questions on online clips/quizzes be that from CorbettMaths or Hegarty Maths. This ensures students see that effort = success.
The pretest includes clip references to Hegarty Maths or Corbett Maths – allowing students to see videos as well as do questions and get immediate feedback.
Part of the students homework is to revise for these quizzes. Students are often given the choice of which ones to work on (e.g a typical HW would be complete 2 Hegarty clips from the pretest sheet).
This homework must be written in full in student’s books and must be marked as they go along.
Quizzes are given every 6-8 lessons (approximately every 2 weeks).
Quizzes are made up of the 5 topics discussed above and a separate problem solving section.
Each question has a clip reference next to it. This means after the quiz is complete students can do feed forward independently by watching the video written next to the question and do not need input from the teacher at this stage.
Students RAG rate each section to indicate their confidence.
Students peer assess the 5 topics from the assessment.
Teachers mark the problem solving section (typically 3 questions of 3-5 marks).
Teachers also complete an analysis sheet (explained later).
Quizzes and feed forward sheets are kept in cardboard folders which students bring to every lesson.
Books are not marked in a traditional way- they are checked by teachers in the lessons – instead this teacher time is used to mark problem solving section of quizzes, analyse quizzes and mark feed forward. Students complete Hegarty Maths work in their books and mark it themselves (this is checked in class by teachers). Worksheets set for HW are peer/self assessed.
A feed forward sheet is attached to the front of every quiz.
The lesson after the quiz is a ‘feed forward’ lesson. The laptops/iPads/computer lesson are booked for this lesson
Once teachers have marked the problem solving section (and checked the marking of other sections) they give back quizzes. No comments from teachers need to have been written (although teachers will sometimes choose to do this if they feel this will be beneficial) as the video clips clearly indicate where students can go to be shown how to do this type of question.
The good thing about this is that students can choose to do more than 1 clip from the quiz if they want to improve further. This is much more powerful than 1 comment which is likely to only be read once. If students wanted to they could feed forward on every single question they got wrong at home. Teaching them independent learning and resilience.
In the feed forward lesson students pick one question they got wrong and look at the clip number attached to this question. They fill in the feed forward sheet with notes, practice questions, corrections to the original quiz question and some hints/tips based on the topic. This takes a whole lesson.
The quizzes are taken back in. Teachers RAG the feed forward and record this in their mark book. They also write a short (typically 1 sentence) comment relating to the feed forward.
This ensures the feedback cycle is completed. By the end of the cycle
Students have worked on topics over 2 weeks in class and at home
They have been tested in exam conditions.
After their quiz is marked they have the chance to use video clips and their books to make progress on one particular topic in class (feed forward).
Teachers check that feed forward demonstrates that students have made progress. If not teachers ask students to redraft or intervene and give extra help where required.
Noting down the RAG rating for feed forward means teachers are aware if any students are repeatedly not making progress from the assessment cycle.
Analysis sheet and responsive teaching
When teachers initially mark the problem solving section of the quiz they analyse the marks on each section of the quiz (each topic)..
If students are a ‘concern’ on any topic (typically less than 50%) then their name is noted on a spreadsheet. In the topic section only students of concern are written down – if they’re doing well then that’s great – we don’t need to worry about them!
Teachers track percentages for quizzes over time – if students are dipping then need to look at why this is the case.
There is a section on the analysis sheet for disadvantaged students and students having interventions. Teachers write a short comment about each of these students. Teachers use this to personalise lessons for these students. People teaching additional intervention sessions can also use this to ensure the sessions are used effectively.
Teachers are expected to use this information to responsively teach. For example they may do a starter on a topic from the previous quiz and guide their questioning toward students who were previously a concern on this topic.
The RAG rating of feed forward (discussed previously) is also noted on this analysis sheet
This may sound complicated and may seem like teachers are doing rather a lot. So to clarify… over one cycle (approximately 2 weeks) teachers are expected to
Give out pretests at the start of the cycle
Set homework based on this pretest
Mark the problem solving section of the quiz (typically 3 questions)
Complete an analysis sheet where they note which students are a ‘concern’ using the scores for each topic. I would expect the analysis part to take approximately 1 minute per student. Additional detail for all disadvantaged students.
Teach responsively using this information (e.g. plan a starter based on 1 topic – focus questioning towards the students who are written as concerns for this topic, while others are getting on with something group together students of concern on a topic and reteach).
Respond to feed forward sheet with a RAG rating and short 1 sentence comment. Typically this comment will be something like “Excellent progress shown” for a green rating, “Ensure you make careful notes from the video” or a comment about a specific misconception if giving a red/amber rating.
This could be seen to be ‘triple marking’ – the assessments have been marked, students have made corrections/done additional work on the topic and then teachers are marking again.
However, we see this differently – we have put our comments at a different stage in the cycle than traditional – instead of immediately giving comments when students get something wrong, we give them the time to use videos to put this right and then write a short progress focussed comment – at the time when it’s needed the most. If the student has made effective progress – excellent – job done! If they haven’t – then teachers do something about it and they keep a record of this to ensure this doesn’t keep happening over time.
Would be really interested to hear people’s thoughts and how we might be able to improve our current process.
We’ve found we spend less time ‘marking’ but are much more aware of where our students misconceptions/weaknesses lie and we spend much more time planning to deal with this responsively and personalising our planning to meet the needs of our students.
Also, previously we would get to the end of a half term, do an assessment and then feel awful about how much some of our students had not retained. Now we know this much more quickly and can keep assessing them on topics to ensure this information is retained. There isn’t that horrible stomach dropping feeling at an end of term assessment!
You can see Cat Ashby speak about "Feedback NOT Marking in the Maths Classroom" during #MathsConf18 at the City Academy Bristol on Saturday March 9th 2019.
'Logic, Codes, Puzzles' is a workshop being run by Robert J Smith at the Mathematics Teacher Network in Southampton (04/12), Northampton (05/12) and Leeds (06/12). This is a FREE event, some tickets still remaining.
Look at this paragraph.
What is vitally wrong with it? Actually, nothing in it is wrong. But you must admit it is a most unusual paragraph. Don’t just zip through too quickly. Look again - with caution! With luck you will spot what is particular about this paragraph and all words in it. Apart from it’s poor grammar.
Can you say what it is?
Tax your brains and try again. Don’t miss a word or symbol. It isn’t all that difficult.
Having looked at the above paragraph above, can you see what is wrong? Let me know what you think it might be by sending a tweet to @LaSalleEd and use the hashtag #MTN_Codes.
Next week, La Salle Education are running a series of Maths Teacher Network sessions across the country. This series of Network meetings include workshops from AQA in the guise of Roger Ray (@AQAMaths), Sian Thomas (Leeds) and Bernie Westacott (@berniewestacott) (Northampton and Southampton) from Oxford University Press. They will all no doubt put on fabulous sessions that you should definitely attend, but I wanted to tell you about my session as I will be looking at Logic, Codes and Puzzles. The Maths Teacher Network meetings are an opportunity to get together and talk and discuss Maths. Something that we don’t always get time to do.
Having looked at the above paragraph above, can you see what is wrong? Let me know what you think it might be by sending a tweet to @LaSalleEd and use the hashtag #MTN_Codes
I really don’t want to give too much away about the session as I want you to attend. So instead, I thought I would let you think about this (fairly simple) Atbash cipher.
By the way, for those that haven’t seen an Atbash cipher before, it is a particular type of monoalphabetic cipher formed by taking the alphabet (or abjad, syllabary, etc.) and mapping it to its reverse, so that the first letter becomes the last letter, the second letter becomes the second to last letter, and so on. For example, the Latin alphabet would work like this:
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
Z Y X W V U T S R Q P O N M L K J I H G F E D C B A
Due to the fact that there is only one way to perform this, the Atbash cipher provides no communications security. So the following should be easy to decode:
ZGGVMW GSV NZGSH GVZXSVI MVGDLIP
The following is the description for my session:
Logic, Codes, Puzzles
This session involves using maths to solve codes and puzzles. From simple addition and subtraction, to data handling and logical thinking, the session will show how we can use mathematical concepts and understanding to explore topics in greater depth. An opportunity to look at how all students might access a problem. What strategies can be used? Which are the most effective? And Why?
It will be my first opportunity to attend a Maths Teacher Network meeting but I am hoping that I will be able to organise and attend many more. (If your school can host such an event then please get in touch!!)
'Assessment and Feedback: Making it Work for Everyone Involved' is a workshop being run by Jemma Sherwood at the Head of Mathematics Conference in Birmingham (Aston University) on Friday 14th Decemeber 2018, some tickets still remaining.
Assessment and feedback are different, but related, ideas. Anything that involves seeing what pupils can or can’t do at a particular point in their learning journey constitutes assessment. How we use that information and what we tell pupils in response to it constitutes feedback.
This makes assessment and feedback two of the intrinsic parts of the teacher’s job. We assess and feed back to our pupils in lots of ways, some formally, many more informally. How many of the following do you employ?
Marking books with ticks and crosses
Marking books with written comments or targets
Obtaining pupil responses to written comments
Q&A with mini-whiteboards
Q&A with hands-up
Asking questions to the class
Talking with pupils about their work
Exit or entrance tickets
Longer topics tests
Setting GCSE papers
Online homework or assessments
From each of these activities we can gain an insight into what our pupils can imitate in the short-term, and what they are learning in the long-term, but each comes with varying degrees of success and requires varying degrees of effort. Our job, as Heads of Maths, is to find and promote those activities which are most useful (to teacher and pupil) and which require the least effort. Fortunately, we have a wealth of research and information to help us make these decisions.
In my session at the Heads of Maths conference, we will analyse these activities and more, while asking ourselves the following questions, considered from the viewpoint of pupils and teachers.
What positives (in the context of learning) does this activity bring?
What negatives (in the context of learning) does this activity bring?
How much time or effort does this activity take?
Is the gain worth the time or effort invested?
As a taster, let’s consider the idea of marking books. In their hugely influential work, Inside the Black Box (1998), Dylan Wiliam and Paul Black taught us that giving numerical marks in books alongside comments was pointless – the presence of the mark would nullify the effect of the comment. From that point, comments and target-setting in books became prevalent in schools, who tried to get teachers to be more detailed than just “6/10. Try harder.” The intention was honourable, the implementation diabolical. By 2015 we had teachers up and down the country writing comments, handing books back and asking their pupils to respond, taking the books back in, responding to their pupils’ comments, only to hand the books back and never have the page looked at again, apart from by SLT in book trawls, who smiled at the best practice they were seeing, which was clearly going to raise standards. Except it didn’t raise standards, it raised teacher dropout rates and levels of exhaustion.
This is a problem because, put simply, we have no evidence whatsoever that this kind of marking is effective. None. The EEF wrote a report in 2016 entitled, “A Marked Improvement: A Review of the Evidence on Written Marking” which concluded that there are not enough robust studies to assert the efficacy of written marking.
What a hugely important conclusion! All those hours invested are, quite probably, a huge misdirection of time. Schools are enforcing unevidenced practice because they see it done elsewhere and assume they must follow suit (driven, of course, by fear of the double-pronged stick of Ofsted and league tables.)
So, in response to our four analysis questions above we have something a bit like the following:
Assessment/feedback type: Written marking (comments/responses/targets)
Use as assessment: Teacher assesses pupils’ work when reading books.
Use in feedback: Teacher writes comments or targets, pupils read and possibly respond. Teacher may be able to plan next steps based on the activity, but this is dependent on how far after the lesson the books are marked.
- Sees what pupils can/can’t imitate during a lesson.
- Sees how well pupils present their work and their thought processes.
- Is given a target to improve and chance to consider the target.
- Doesn’t find this information out until books are marked, which can be weeks later.
- Cannot plan next steps if books aren’t marked immediately.
- Doesn’t learn anything about long-term retention of material.
- Target given too far after the event (they’ve forgotten what the lesson was about).
- Comment either so specific it doesn’t cover enough work in enough detail, or so generic it is useless.
30 books take approximately two hours.
Lesson time to respond to comments – this may be a positive or a negative use of time, depending on context.
No. If book marking were to inform planning, it would need to be immediate and after every lesson. This is impossible (5 hours of lessons a day would be accompanied by 10 hours of marking).
Probably not. There is some benefit in having to remember something from two weeks ago in order to respond to comments, but this can be achieved in better ways.
Join us at the Heads of Maths Conference in December where we can take the time to find some better ways of assessing our pupils, of gaining feedback for us, and of giving them feedback.
'Time to revisit…Teaching for Mastery
' is a workshop being run by Peter Mattock at #MathsConf17.
“Mastery”. Some people see it as the latest buzz-word to be shunned until we wait for the next “big thing”. For others it is central to teaching. For some it is a confusing term with no clear idea of what it actually means. And I can sympathise with all of these views…
The idea of “mastery” has been around for a long time. People much more knowledgeable have written about its provenance, its history and its progress to the modern day. Neither this blog nor my mathsconf session will be trying to reinforce or reinterpret any of this. I will not be attempting to explain the structure of a mastery curriculum (which is not exclusive to a mathematics curriculum). Better men than me have already done this, not the least of which is the LaSalle CEO Mark McCourt (if you haven’t read his blogs on mastery then you must). Saying that, it is important to understand certain aspects of its structure to understand where I hope my session fits in.
One of the central aspects of a mastery curriculum is teaching in a way that all pupils can access from their starting point, and then carefully assessing their understanding throughout the teaching process. A second is the use of correctives where the initial teaching isn’t successful – having different ways of approaching concepts when the first way falls short. The biggest aim of my session is to try and showcase some of the ways that teachers can approach this. Starting with what I see as important ideas to consider when thinking about structuring learning, I then aim to share practical examples of approaches that could be used either as part of the initial teaching or as a corrective approach. For those that know me, it won’t be surprising to hear that much (but not all) of this focuses on the use of representations to reveal the underlying structure of an idea (given that my book “Visible Maths” is entirely concerned with the use of representations and manipulatives to reveal underlying structure).
As an example, but not one I am using in the session, consider the “rule” that one negative number divided by another negative number results in a positive answer. Consider –15 ÷ –3:
One way of representing this is to use double sided counters; these usually appear with a yellow side (positive) and a red side (negative). Two different coloured counters can also work, and in fact to model this calculation we only need to consider negatives so a single colour of counter will suffice. The image above shows -15, and now we have to think about how we divide that by -3. One way of thinking about division is to think about creating groups, so a possible way of looking at this calculation is, “Start with -15 and create groups of -3.” These groups can be seen below:
When we think about division like this, the result of the division is “How many groups can we create?”. We can see that this process creates 5 groups, which means that -15 ÷ -3 = 5.
Often this “rule” is taught as an arbitrary rule, without any attempt to show where it comes from. In many classrooms, one could be forgiven if kids believed that the only reason this is a “rule” of maths is because teachers says so. But this rule is a necessary rule – if division works in the way we know it does then the answer to -15 ÷ -3 cannot be anything but 5. I finish my session with a discussion around other “rules” of maths, how appropriate representations can show where these rules actually come from, and also discuss how we can manage the transition from using representations/manipulatives to the abstract calculations. Hopefully I have whetted your appetite to hear more about teaching approaches that can support mastery in mathematics, and I look forward to seeing you (whether in my session or not) in Birmingham. Don’t forget to join us for the pre-drinks and networking the night before as well!
'Collaboration & Creativity with Technology in the Maths Classroom' is a workshop being run by Patrick McGrath at #MathsConf17.
Technology. It’s often perceived as a challenge for the maths classroom. We’ll check out what our peers are using, we’ll look at what EdTech companies are offering, but when we voice our concerns, we’ll often be met with the line “There’s an app for that” . The problem of course is that, sometimes, there isn’t.
Sure, we may all have access to amazing apps, programmes, web tools and resources that help explain or visualise key concepts, or help with repetitive practice, but the thing is, that's not the purpose of education technology. Its use should be grounded in teaching and learning - in providing context, in deepening the learning experience, and in providing ways for students to articulate their learning in ways never before possible.
In my role as EdTech Specialist at Texthelp, maths teachers are without doubt my favourite set of practitioners to work with - they’re passionate about learning and equally so of their subject, but the one thing I hear a lot when we talk technology? “There’s nothing in it for me”. When we poll a room, the number one piece of technology chosen is the good old Interactive Whiteboard. It’s a great tool and one not many of us could do without, but it’s not representative of the amazing learning opportunities we can provide to our pupils with the vast range of tech available to us as educators.
So, it’s time for a change. It’s time to move beyond the Interactive Whiteboard. In my upcoming session at MathsConf17, together we'll explore and discover 10 amazing ways to build technology into the maths classroom and to encourage pupils to use technology at home to enrich their learning.
We’ll be looking at collaboration and creativity and how, by using just a small selection of tools, we can enable these things and in doing so increase engagement - building towards a love of learning and of maths itself. Let’s see how we can truly build context around maths activities and understand the positive impact that providing multiple means of expression can have.
We won’t get bogged down in complex techspeak. We won’t be talking in acronyms and we won’t be talking about Google vs Microsoft vs Apple. Technology should be there to support regardless of your ‘platform’. So, we'll stick as always to teaching and learning and focus on creating meaningful outcomes for pupils.
We’ll also be ‘building bridges’ with a new and exciting tool from Texthelp called EquatIO®. Created by a maths teacher, it’s a tool actually grounded in teaching and learning. It helps pupils use maths in a digital environment, express maths concepts, explore graphing with Desmos integration and enables amazing opportunities for Assessment for Learning.
EquatIO helps students and teachers - and we provide it free to any teacher via https://text.help/e8VcrH - so grab yourself a copy, and join me for Collaboration & Creativity with Technology in the Maths Classroom at MathsConf17. I promise to send you away challenged but armed with a knowledge of the tools you can start with right away. Plus, if you’re lucky, you’ll be in with a chance to grab one of our amazing ‘I Love Maths T shirts’
Lot’s of learning, lot’s of resources, free software and maybe even a T Shirt - what’s not to like?!
Check out our video below to get a quick introduction to EquatIO:
You can read more about EquatIO at https://text.help/jxOlom and discover how Texthelp have helped over 15 million people around the world understand and be understood.
'How to Solve An Adfected Quadratic' is a workshop being run by Joanne Morgan at #MathsConf17.
Factorising a quadratic when the coefficient of x 2 doesn't equal one (a 'non-monic') is apparently one of the more challenging skills that our pupils learn at GCSE. I have seen many pupils struggle with it, even those who achieve a grade 8 or 9 at GCSE and go onto take maths at A level. Interestingly, it doesn't seem like it should be challenging at all. I think it's way more straightforward than some of the harder questions that come up at GCSE - so why do pupils struggle with it so much?
When I first became a teacher, I taught my pupils to factorise harder quadratics in the way I've always done it: by inspection. Simply write out two empty pairs of brackets and try some numbers - thinking logically about what those numbers could be - until your terms expand correctly. It's very quick once you get the hang of it. In my NQT year this method was met with frustration by my Year 11s. They wanted a more defined procedure. I looked online to see if I was missing something and discovered 'the grouping method' which I then showed them as an alternative. I felt that it was an unnecessarily convoluted method but they clearly preferred having a set of rules to follow rather than having to reason for themselves. It made me a bit sad.
This grouping method, particularly the last step where terms are gathered together, is a bit of a leap of faith for pupils who have never seen this kind of factorisation before. It kind of seems like magic.
Ten years on, I still prefer inspection (the 'guess and check' method) but I always teach my pupils the grouping method as an alternative. I know they'll see it elsewhere even if I don't teach it to them - in textbooks, revision guides and online videos. I find that only the strongest pupils favour inspection - most pupils choose to use the grouping method but very often forget it. Days before the GCSE exam I hear cries of 'what's that thing you do to factorise hard quadratics? Something to do with the middle term...?'. The steps in the grouping method are not intuitive, and as a result it's difficult to remember.
Given I had never heard of the grouping method before I started teaching, I was surprised to learn that it's actually an incredibly popular method. In fact, it appears to be the method that most maths teachers now use to teach non-monic factorisation.
How did I miss this method during my time at school? I did my maths GCSE in 90s. I have a couple of Bostock and Chandler textbooks from the 1990 - both only feature inspection, with no mention of grouping.
This probably explains why I'd not seen it before. It wasn't in fashion when I was at school.
Looking further back I was surprised to see that older textbooks do feature the grouping method. Here, in New Algebra for Schools (Durell, 1953), we see an example of the grouping method. Durell recommends this method for both monic and non-monic quadratics.
Note though that this follows on from extensive use of grouping elsewhere. By this point in the textbook pupils have had considerable experience of factorising expressions like p(a + b) + q(a+b) and ax - ay + bx - by. This is absolutely key. There is a clear progression here that I feel is often missing from modern teaching of factorisation. I'm not sure it make much sense for pupils to only use the grouping method for non-monic quadratics, having never seen it before.
Later, this chapter tells us that simple quadratic functions can often be factorised at sight without using the grouping method. It says "use the grouping method whenever you are not able to obtain the factors by inspection, quickly".
Another textbook from eight years later (The Essentials of School Algebra, Mayne, 1961) also features both the grouping and inspection methods. Again, earlier in the chapter there is a considerable amount of work on the skills and understanding required for the grouping method. Of inspection it says,
"After a little practice, the pupil will be able to reject mentally the impossible pairs of factors... With simple numbers it is slightly quicker than the method of grouping terms, but the grouping method is the method to rely on. It should always be used whenever the pupil is not able to obtain the factors quickly be inspection."
So it seems that the grouping method may have been popular in the mid-20th Century.
Looking even further back, in Elementary Algebra for Schools (Hall & Knight, 1885) we are told that we should factorise non-monics by inspection:
"The beginner will frequently find that is it not easy to select the proper factors at the first trial. Practice alone will enable him to detect at a glance whether any pair he has chosen will combine so as to give the correct coefficients of the expression to be resolved".
There's no mention of any alternative methods here.
I haven't yet read enough old textbooks to track the full history of the grouping method, but from what I've seen it has come and gone over the years, and sadly seems to have become detached from prerequisite skills along the way.
If teachers are teaching the grouping method to factorise non-monics and have not already taught pupils how to factorise expressions like p(a + b) + q(a + b) and ax - ay + bx - by then I think they may be trying to teach too many new skills in one go. It's certainly something to think about.
If you find this kind of thing interesting, come along to my workshop at MathsConf17. I start will by sharing some cool stuff I've seen in old maths textbooks, and then I will focus on quadratics. We won't look at factorising, but we will look at some methods for solving quadratic equations and how they've changed over the years. I really think that this kind of subject knowledge development helps to improve our classroom practice. I hope to see you there - and I will bring along some of my old textbooks in case you'd like to have a look for yourself.