Dynamic Example Questions from Complete Mathematics

Written by Tom Valsler Monday, 20 May 2019

Just over a month ago we released a blog describing our work on a dynamic worksheet generation tool. This was our first experiment in dynamically generated content. Today we announce our next experiment, dynamically generated example questions. Below are the first few questions we have created, available to all for a short time.

Try it for yourself - simply click the DYNAMIC button to create a new version of the question. Worked solutions can be viewed by opening the section beneath the question. The questions can also be viewed in full screen by clicking the expand icon.

Complete Mathematics members will be familiar with the example questions that appear on our online platform, but for non-members here is a summary of what they are, and what they are for.

Example questions, along with a wealth of other support materials, are available to a teacher once they have chosen the particular objective(s) for their lesson from the Complete Mathematics curriculum. The example questions come in three varieties: ‘Typical’, ‘Hard’ and ‘Probing’. ‘Typical’ questions are designed to offer some guidance to the teacher on the sort of questions their pupils should be expected to answer. ‘Hard’ questions build on the ‘Typical’ questions and provide more of a challenge to the pupils. Finally ‘Probing’ questions provide the teacher with questions that may challenge their pupil’s preconceptions and are intended to broaden the pupil’s knowledge.

We have found that, as well as using these questions for guidance and to inform their planning, teachers are also using them in front of the class as a teacher led activity. It is for this use case that making these example questions dynamic is an especially attractive proposition. Teachers will be able to walk through a question with the class, with help from the worked solution, then click ‘Regenerate’, and immediately have another version for the pupils to attempt on their own.

Further to this, example questions are also used on the pupil side of Complete Mathematics after the completion of a quiz. Pupils are provided with a page to analyse their performance, as part of this process they are shown similar questions to the quiz questions they have answered (particularly useful for incorrectly answered questions). This provides another brilliant opportunity to put dynamic example questions to use, with pupils able to generate as many further practice questions (and worked solutions) as they want. This gives pupils an even greater opportunity to independently fill the gaps in their learning.

The goal of these experiments on dynamically generated content is to lay the groundwork for making Complete Mathematics an even closer mapping of the mastery cycle. If we can use computing power to make the loops of the cycle more efficient it frees up teachers to spend more time planning and teaching brilliant, insightful lessons. But it is no good when creating a quiz or worksheet if the generation of questions is not intelligent or when generating example questions we don’t also provide responsive worked solutions. Our goal is to produce content that is as good as the teacher would have produced if they had the time (or pretty close to it!).

So what’s next? As well as continuing to populate Complete Mathematics with more dynamic example questions and more dynamic worksheets we will be adding new functionality to allow users to input (and check) their answers. This will make these questions much more powerful, whether they are used in a worksheet for homework, at the front of the class, or in a quiz.

Dynamic Worksheet Generation from Complete Mathematics

Written by Chris McGrane Tuesday, 02 April 2019

Today we launch a new feature to be added to Complete Mathematics: our Dynamic Worksheet Generation tool. This initial launch is focused on the creation of worksheets for the factorisation of quadratics. We have created a tool which captures some of the ‘teacher logic’ inherent in worksheet design and have incorporated intelligent variation and sequencing in the questions generated.

The worksheet generator can produce worksheets with questions which come in a sensible teaching order for this topic. Intelligent variation sequences, are periodically included at appropriate places in the worksheets. These sequences of questions help pupils to attend to the key variants and invariants. One such example is shown:

x2 + 14x + 24 = (x + 12)(x + 2)

x2 + 11x + 24 = (x + 8)(x + 3)

x2 – 11x + 24 = (x – 8)(x – 3)

x2 – 14x + 24 = (x – 2)(x – 12)

x2 + 10x – 24 = (x – 2)(x + 12)

x2 – 10x – 24 = (x + 2)(x – 12)

x2 + 5x – 24 = (x – 3)(x + 8)

x2 – 5x – 24 = (x + 3)(x – 8)

Teachers have the freedom to toggle on and off which variations of quadratics to include on the worksheet. This is particularly useful for the implementation of the mastery cycle. Based upon formative assessment a teacher can identify where individual pupils need to focus their attention. The teacher can then create a worksheet specifically for this pupil in moments for them to remediate and practise on these sub-skills.

At La Salle we talk about a learning episode as opposed to lessons. Learning and teaching can’t be neatly contained in sixty-minute slots, instead a learning episode spans a series of lessons and may include a number of related learning objectives. We consider a learning episode in the context of the following phases:

Teach
Do
Practise
Behave

During the Teach phase, the idea is entirely novel to pupils. It is just beyond their current knowledge and understanding.  The teacher will instruct the pupils, tell them key facts, pass on knowledge, show and describe, use metaphor and model, all in order to bring about connections in the pupil’s current schema so that they can ‘meaning make’.  This phase is often described as explicit teaching.  It is a crucial phase – after all, the teacher knows things and the pupil does not; so tell them!

The end of the 'Teach' phase does not result in learning.  It is merely the first step.  At this stage the new knowledge is ‘inflexible’, and it is our job as teachers to bring meaning and understanding to the knowledge so that it becomes ‘flexible’ (more on inflexible and flexible knowledge later).

We now ask pupils to Do.  At this stage, they do not yet know or understand the new idea, they are replicating what the teacher has told or shown them.  The 'Do' phase has two important purposes.  Firstly, the teacher is able to observe whether or not the pupils have made meaning of the model, example, metaphor or information they have been given or shown.  The teacher can see and act; are the pupils able to replicate what I have demonstrated?  If not, the teacher can change their model, example or explanation, perhaps making stronger and more explicit connections to previous knowledge and understanding.  The second reason for the 'Do' phase is to give pupils a sense that the idea or task is surmountable – that they, quite literally, can do what they are being asked.  Well structured 'Teach' and 'Do' builds pupils’ confidence and shows them there is nothing to be afraid of, the new idea is within their reach.

The Complete Mathematics platform supports teachers in delivering the 'Teach' and 'Do' phases by including pedagogical notes, typical misconceptions, key vocab, and example questions. Additionally, on our CPD days we explore evidence-based techniques to enhance the effectiveness of these phases.

Once both teacher and pupil are clear that the pupil is able to ‘Do’ – that is to say, they can perform – the teacher now segues the pupil to the Practise phase

During 'Practise', we wish to move beyond simply performing.  We want the pupil to gain a confidence in working with the new idea, to see its underlying relationships and to assimilate the new idea into their schema of knowledge.  In order to achieve these more meaningful goals, the pupil needs to be able to attend to a higher level.  In other words, as described earlier, the pupil needs to have achieved fluency at the performing level first, so that they may attend to connections, relationships and a deeper conceptual appreciation.

So, we shall define the point at which the pupil moves from 'Do' to 'Practise' as the point at which they achieve fluency. This is where the dynamically generated worsksheets can be particularly useful. They give pupils ample opportunity to build procedural fluency but also develop conceptual understanding through the built-in intelligent variation as described above.

The final phase, Behave, is the most important phase.  This is the phase that brings about understanding. At this stage, teachers create opportunities for pupils to behave mathematically. Complete Mathematics has tasks which support this phase, however, the pedagogical actions are perhaps the most important factor in this phase. You can learn more about this on our CPD days: Deeper Understanding in Mathematics, Embedding Problem Solving in the Mathematics Classroom, Designing Mathematical Tasks and Curriculum Design.

This worksheet generator is our first experiment in this area and we would love to hear any feedback you have either by commenting below or talking to us on twitter. There are a multitude of other ideas we want to explore. For instance, allowing teachers to implement their own variation logic in these worksheets and allowing pupils to answer the questions online. Also, in addition to these further experiments on dynamic worksheets, we are working on dynamic versions of the example questions found on every objective in the Complete Mathematics platform.

Complete Mathematics members will be able to see all these developments and more appearing on the platform in the coming weeks and months. In fact members can use this worksheet generator right now in the 'Factorising Quadratic Expressions' objective in unit 10.5. If you don't want to miss out on these developments, or you want to take advantage of free attendence to the CPD courses mentioned above, you can find out more about becoming a Complete Mathematics member here.

OCR MathsConf18

Written by Steven Walker Monday, 04 March 2019

Edited and compiled by Robert J Smith @RJS2212

The OCR (Oxford Cambridge and RSA Examinations) are exhibiting @OCR_Maths at #MathsConf18 on Saturday 9th March 2019, tickets still available.


The recent qualification reforms have brought with them an increased emphasis on problem solving. In this blog we take a look at incorporating puzzles into revision programmes to help build students’ problem solving skills.

What is Problem Solving?

In GCSE and A Level, problem solving questions will often require students to interpret given information and decide on the techniques needed to obtain an answer.

For example, at first glance this circle puzzle might appear to be beyond GCSE (as equations of circles are limited to circles centred around the origin in GCSE), but could be solved by bringing together coordinates, constructions and Pythagoras’ Theorem.

How to use Problem Solving questions

Problem solving needs to be incorporated throughout the academic year, but in the run up to the exam season puzzles can make good revision tasks. These help students make links across the curriculum, help identify technique weaknesses and prompt discussions.

At first, many will see this additional grid as having 3 unknowns, but on closer investigation a pair of equations can be seen with only 2 unknowns, which can then be solved before going on to complete the puzzle.

Creating discussion

Maths is often perceived as a quiet subject. Puzzles encourage discussion and the process of sharing ideas can help consolidate knowledge.

OCR Support

Each of our Check In tests (GCSE (9-1), FSMQ, GCE Maths, GCE Further Maths) include an extension problem and our practice exam papers can be found on Interchange.

If you have questions then submit your comments below. You can also sign up to subject updates to receive up-to-date email information about resources and support and follow us on Twitter @OCR_Maths


You can see OCR (Oxford Cambridge and RSA Examinations) in the networking / Exhibitor slots during #MathsConf18 at the City Academy Bristol on Saturday March 9th 2019.

Don't forget in March we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

Daydream Education MathsConf18

Thursday, 28 February 2019

Edited and compiled by Robert J Smith @RJS2212

Daydream Education are exhibiting @DaydreamEdu at #MathsConf18 on Saturday 9th March 2019, tickets still available.


Daydream Education

Using Short Feedback Loops to Improve Student Knowledge Retention and Recall

Getting teenagers to revise effectively has always been difficult. Though, with so many entertainment options at their disposal nowadays, it can sometimes seem like a losing battle. Thanks to recent technological advancements, not only does the average teenager have a shorter attention span but they also have endless distractions. All these factors combined means it can be harder to get them to focus on written information in the first place, and then even harder to convince them to return to that information to test their knowledge.

Thankfully, Daydream Education has come up with a double-pronged solution combining strong physical resources with short feedback loops to improve students’ knowledge retention.

According to Wired magazine a feedback loop is “a profoundly effective tool for changing behaviour [in which you] provide people with information about their actions in real time, then give them an opportunity to change those actions, pushing them toward better behaviours.”

The concept is simple. The shorter the feedback loops, the more often learners correct their behaviour and the faster they learn.

The first prong of the Pocket Poster revision guides’ solution is their high-quality printed format. They are small in size but huge in content, fitting the whole curriculum contained in normal revision guides into a travel-ready, more accessible package.

Designed in full-colour, strong paper and filled with engaging images and annotated diagrams, they are tailored to contain only the most essential information needed to master each school subject. At the same time, they are substantial enough to engage the brain’s sensory memory system.

As people tend to retain more information from physical paper texts, this first port-of-call, according to a study conducted by Scientific American magazine, provides the learner with a strong yet accessible revision foundation.

The second prong of their solution to revision comes in the form of the extra digital content Pocket Posters provide. Each book contains a code that learners can use to access a digital version of the book on a computer, tablet or phone – complete with a range of over 1,100 quiz questions to test their own knowledge.

Daydream Education

Questions are categorised into fluency, reasoning and problem solving to ensure students are being tested in all types of questions.

Not only do these quizzes correct students so that they can find gaps in their own knowledge and improve their academic performance, but they can also help teachers track students’ progress.

The Pocket Posters come with a teacher portal so educators can see their pupils’ progress and discover where there are common gaps in knowledge and, therefore, improve their own teaching practices. A similar tactic has recently been taken up by professors at Harvard University to improve their lesson plans.

Such a revolutionary concept isn’t just a way to improve exam performance, either. It can also improve attitudes towards learning itself. A 2001 study conducted by Stanford University’s Albert Bandura concluded that “people are proactive, aspiring organisms” who crave information about themselves in order to improve. The short feedback loops incorporated into Pocket Posters tap into this natural instinct, turning revision into an interactive self-improvement game.

Ultimately, Daydream Education’s Pocket Posters provide the best of both worlds – a strong, engaging print product and interactive digital content, all for a small price for schools on a budget.

Daydream Education

To check out samples of the digital content available to Pocket Poster users, check out Daydream Education’s app page by following this link: https://apps.daydreameducation.com/

Modules are available from Key Stage 2 Maths all the way up to Higher GCSE Maths, plus a range of other subjects. And it’s all free!

To view the full range of Maths books, click here.


You can see The Mathematical Association in the networking / Exhibitor slots during #MathsConf18 at the City Academy Bristol on Saturday March 9th 2019.

Don't forget in March we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

MA MathsConf18

Written by Mathematical Association Thursday, 28 February 2019

Edited and compiled by Robert J Smith @RJS2212

The Mathematical Association are exhibiting @Mathematical_A at #MathsConf18 on Saturday 9th March 2019, tickets still available.


MA Banner

The MA was the first teachers' subject association to be formed in England, in 1871, as the 'Association for the Improvement of Geometrical Teaching', the original catalyst being the need to develop and to lobby for alternatives to the then standard treatment of geometry. The Association's history is located within a broad context of changes in the educational system, developments in educational and mathematical thought, the growth of professionalism, and wider social, political and economic forces which influence the curriculum. MA reports and publications became standard references in the growing international interest in mathematics education throughout the 20th century.

Primary

The MA offers a variety of support for primary teachers and primary schools. We have a dedicated primary journal, offering a mixture of theory and practice, as well as ideas for instant classroom use. If you haven’t yet explored our Primary Mathematics Challenge, don’t miss it. It provides a fabulous resource for extending able children and developing problem solving skills with pupils in Y5 and Y6.

Mathematical Assocaition Image

Professional Development

A principal objective for the MA is "to promote and support the professional development of teachers", and so its Professional Development Committee organises a range of PD events for teachers of mathematics. The MA is an NCETM accredited provider of mathematics CPD courses and events.

Secondary

Four of the MA's journals (with issues between 3 and 5 times a year) are ideal for the 11-16 age range: 'Mathematics in Schools', 'Equals', 'SYMmetryplus' and 'Mathematical Pie'. The MA also publishes books aimed at secondary teachers. The 11-16 subcommittee monitors developments, responds to consultations (e.g. on the revised Key Stage 3, Functional skills, twin GCSEs) and produces resources.

Mathematical Assocaition Image

You can find out more about joining The Mathematical Association on the MA website Click here to #JoinTheMA


You can see The Mathematical Association in the networking / Exhibitor slots during #MathsConf18 at the City Academy Bristol on Saturday March 9th 2019.

Don't forget in March we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

ATM MathsConf18

Written by Heather Davis Thursday, 28 February 2019

Edited and compiled by Robert J Smith @RJS2212

The Association of Teachers of Mathematics are exhibiting @ATMMathematics at #MathsConf18 on Saturday 9th March 2019, tickets still available.


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.

Don't forget in March we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

Use of Calculators in the New A-levels

Written by Dan Rogan Monday, 25 February 2019

Edited and compiled by Robert J Smith @RJS2212

Use of Calculators in the New A-levels is a workshop being run by Dan Rogan @AQAMaths at #MathsConf18 on Saturday 9th March 2019, tickets still available.


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:

  • Iteration
  • 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.

Don't forget in March we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

Further RISPs

Written by Jonny Griffiths Tuesday, 19 February 2019

Edited and compiled by Robert J Smith @RJS2212

Further Risps; open tasks for FM A Level is a workshop being run by Jonny Griffiths @maxhikorski at #MathsConf18 on Saturday 9th March 2019, tickets still available.


Further Risps; open tasks for FM A Level

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.

Don't forget in March we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

Atomisation: Breaking down your teaching like you have never seen before… 

Tuesday, 19 February 2019

Edited and compiled by Robert J Smith @RJS2212

Atomisation: Breaking down your teaching like you have never seen before… is a workshop being run by Naveen F Rivzi @NaveenFRivzi at #MathsConf18 on Saturday 9th March 2019, tickets still available.


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
    • Isosceles Trapezium
  • Algebraic: Exterior angles
  • Algebraic: Interior angles
  • Irregular polygons

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.

Don't forget in March we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

Time to revisit - Division

Written by Peter Mattock Thursday, 14 February 2019

Edited and compiled by Robert J Smith @RJS2212

'Time to revisit... Division' is a workshop being run by Peter Mattock @MrMattock at #MathsConf18 on Saturday 9th March 2019, tickets still available.


Time to revisit…Division

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.

Don't forget in March we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

A rummage through the archives

Written by Andrew Taylor Thursday, 07 February 2019

Edited and compiled by Robert J Smith @RJS2212

'A rummage through the archives' is a workshop being run by Andrew Taylor @AQAMaths at #MathsConf18 on Saturday 9th March 2019, tickets still available.


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.

Don't forget in March we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

How We Teach It – The Mastery Way

Written by Matthew Man Thursday, 31 January 2019

Edited and compiled by Robert J Smith @RJS2212

'How we teach - The Mastery Way' is a workshop being run by Matthew Man @mr_man_maths at #MathsConf18 on Saturday 9th March 2019, tickets still available.


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:

  • item 1
  • Fluency – varied and frequent practice with increasingly complex problems as lessons progress
  • item 2
  • Reasoning – Making conjectures, generalisations, justifying, arguing and proving methods using mathematical language
  • item 3
  • 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:

  • item 1
  • Which one doesn’t belong?
  • item 2
  • Spot the error
  • item 3
  • Multiple methods
  • item 4
  • Multiple solutions
  • item 5
  • Comparisons (structure)
  • item 6
  • Working backwards

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!

Thank you.


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.

Don't forget in March we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

A five-year-old’s first engagement with Cuisenaire: Joyful Learning

Written by Chris McGrane Thursday, 31 January 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.

An initial arrangement

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.

Some organisation

“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:

Comparative thinking

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?

The first attempt at measuring
A further attempt

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.

Measuring round the box

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.

Maths Teaching and Assessment

Written by Catherine Ashby Friday, 01 February 2019

Edited and compiled by Robert J Smith @RJS2212

'Feedback NOT Marking in the Maths classroom' is a workshop being run by Cat(herine) Ashby @CMATHS3 at #MathsConf18 on Saturday 9th March 2019, tickets still available.


Principles of new assessments

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.

Old Cycle
New Cycle

How it works

Pretests

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

Year 8 Quiz 1
Year 10 Quiz 4

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. Feed forward

A feed forward sheet is attached to the front of every quiz.

FeedForward
Example 1
Example 2
Example 3
Example 4

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.

Complete Feed Forward Quiz

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

RAG analysis

Teacher responsibility

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.

Don't forget in March we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

Logic, Codes, Puzzles

Written by Robert Smith Thursday, 29 November 2018

'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!!)

ALL RIGHTS RESERVED
COPYRIGHT 2019
LA SALLE EDUCATION