Dr Flavia Belham is the Chief Scientist at Seneca Learning who are exhibiting at #MathsConf17.
Maths is a fascinating but complex subject to learn. Students need to understand the concepts but should also acquire procedural fluency when solving problems. What is the best way to do that?
Dr Doug Rohrer has conducted important research on effective learning strategies focused on Mathematical content. He has done experiments both in the lab and also in the classroom. One of his main findings taps into the idea of interleaving.
Routine 1 is blocked, whereas routine 2 is interleaved. Research in the field of cognitive sciences has consistently shown that routine 2 leads to deeper learning and stronger memory.
It is easy to understand this, if you use a gym analogy. Imagine you are going to take part in a fitness competition in 3 months. Would you train only your arm muscles for the first month, only the leg muscles for the second month, and do only treadmill exercises for the last month? Or would you spread out the three types of exercise so that they are frequent but mixed?
How does interleaving improves learning of Maths?
Interleaving improves learning. How does it actually work and how can Maths students benefit from it? There are two main reasons. The first one is that it helps students to differentiate between two concepts. It is easier to understand the difference between an elk and a moose if you see them side by side than one after the other. The same happens with other concepts.
The second reason why interleaving improves learning of Maths is that it helps students to figure out the right strategy or formula on the basis of the problem itself. For example, imagine that students are learning how to calculate the volume of different shapes. Usually, they would learn the formula for a cylinder and apply it to several problems involving cylinders. Then, they would lean the formula for a sphere and apply it several times to problems involving spheres. This is just like the blocked routine 1 from before.
The problem with this routine is that, even before they read the question, students already
know which formula to use! That’s simply because they know that block of problems will be
of the same kind and will require the same strategy.
If, however, you present them with a set of questions that can be about spheres or about
cylinders in a random order (like in routine 2), students will need to understand how to figure
out the right formula based on the problem and nothing else. Doing this will massively help
them on cumulative examinations, as questions can be about any topic. That way,
interleaved practice not only boosts learning, but also prepares students for future
Seneca is a free online homework and revision platform that helps students learn 2x faster
by using strategies from cognitive science - including interleaving. Our Maths content is 80%
questions and problem solving. More than that, the order of questions is pseudo-randomised
so that students need to think about the problem and figure out the right strategy on their
own. This leads to deeper learning, stronger memory, and better exam results.
To help with procedural fluency, a large part of our Maths courses is made up of worked
examples, in which students answer bit after bit and receive immediate feedback for each.
The content is written by senior educators and is exam board specific too.
Here are some of our Maths courses. Try them out and create a teacher account. You will be
able to link your account to your students’ and monitor their progress on the platform. All
'The use of Lego and video as part of teaching, learning and assessment' is a workshop being run by Martin McCutcheon at #MathsConf17.
This workshop looks at how to support all abilities in accessing and enjoying mathematics through the use of Lego. Lego demonstrations can help give clear explanations, provide fun and creative classwork, and allow teachers to tailor the task so that all can achieve success.
Right from pre-school and early years, play and fun is a way of learning. The use of Lego allows most students to access a toy they already know and enjoy. The colour and creations can inspire further experimentation and involvement in the tasks: they build it, see it, hold it. (Of course we have to watch out they don't just want to build a castle or spaceship!). Attendees of this workshop will get a chance to play and create with Lego right from the start in the workshop and create demonstrations of mathematical concepts and objects, from fractions to tessellations, symmetry to expansion of brackets or anything else that comes to mind!
I will discuss how I have been using Lego in class with a few slides of examples of topics used in class, as listed below, with ideas to explore:
Area and Perimeter
Volunteers will have an opportunity to show to the workshop what they have created on the suggested maths topics or any they want to add:
Nth term (eg dog diagram, windmill)
Plans and elevations
Factors and multiples
Attendees will then watch a few YouTube clips of students work in class, from fractions and symmetry to algebra, and can discuss what works in class and what might need improvement.
Students’ enjoyment of playing with the Lego should enhance their learning, Lego demonstrations should add to the explanations and teaching, and the ability to see what students create a way to assess the learning with the teacher able to differentiate and adjust the criteria to meet the needs of the students.
Following that we will learn how to make videos in class and post these on YouTube online, including talking about safeguarding. Some detail about equipment needed and editing will be given, and ideas for post-production will be discussed. And finally we will do some live uploads of attendees own Lego creations of mathematical concepts at the workshop.
Two chance events led me to this position, one I happened to find a box of Lego in my new classroom and thought why not try use it? And secondly, a tweet from a Jo Morgan said that the Maths conference was open for proposals, and I thought, why not?
Early in my teacher training and NQT years I was advised to keep experimenting to make sure I didn't get stuck in a rut and settle in my teaching ways as I got further into my career. So this led to several years of my worrying I wasn't doing interesting enough lessons with interesting enough activities! And then years later I found the Lego box. : )
The use of video has its inspiration long before I became a teacher or YouTube even existed, in the 1990s I used to watch the Learning Channel in South Africa, a TV show where questions were answered on TV from callers to the show, then while training we used recording software on iPads, finally I got a good smartphone and could start this channel.
At the end of the workshop I will briefly talk about how other subjects can use this format, from English to Science and also explore how else we can include fun and play from childhood toys, and look at other things like the maths in football or flags.
There will be opportunity throughout the workshop for attendees to ask questions, make suggestions and explore their own ideas in Lego and video. And lastly all attendees will get a MartinMaths sticker as thanks for attending!
'An Introduction to Cognitive Load Theory' is a workshop being run by Michael Allan at #MathsConf16.
In 2017, Dylan Wiliam tweeted: “I’ve come to the conclusion Sweller’s Cognitive Load Theory is the single most important thing for teachers to know http://bit.ly/2kouLOq “ (see here for original tweet).
In very simple terms, Cognitive Load Theory is about considering the limitations of pupils’ working memory at the point of initial instruction.
I decided to offer to run a workshop about Cognitive Load Theory at MathsConf16 at High School of Glasgow on 6 October 2018, and this blog post will be a summary of my presentation.
As well as Dylan William, Greg Ashman, Craig Barton and John Sweller, I have also read some of the work of Daisy Christodoulou and the paper by Kirschner, Sweller and Clark titled “Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential and Inquiry-Based Teaching”. Another great summary of Cognitive Load Theory can be found at this link.
Take a moment to answer this question before you read on:
What are some of the things that you know?
You know a lot of stuff. Some of it is important – like date of birth, phone numbers, passwords, pin codes. Some of it is not important – like the lyrics to Aga Do. Some of it is long lasting and easy to retrieve. Some of it is to do with what is happening right now – the brightness and temperature in the room you are sitting in. Some of it is to do with what happened tens of years ago and you probably can’t remember it right now. But it’s in there… What was the name of the teacher you had in Primary 1?
You know how to write but is that the same as knowing how to speak?
You know how to multiply numbers but is that the same as knowing how to count?
Is knowing that things fall towards the ground when they are dropped the same as knowing the formulae for potential energy and kinetic energy?
David Geary (2007) talks about two different types of knowledge: Biologically Primary and Biologically Secondary. Biologically Primary Knowledge includes things like being able to speak your native language, being able to read people’s body language and being able to make sense of how things interact in our physical environment. Biologically Secondary Knowledge concerns everything that has to be learned through effort. Learning a new language, knowing your times tables and being able to tie shoelaces are examples of biologically secondary knowledge. In fact, pretty much everything we teach in our classes in school can be described as biologically secondary.
To understand cognitive load, we must define what we mean by novices and experts and consider how they differ when learning new material. Novices are people who have a very limited experience in a particular domain. Experts are extremely knowledgeable in a particular domain. Novices and Experts think and learn differently. The differences are discussed further in this post by David Didau.
When we learn new material, our working memories are stretched significantly. Everything we think about contributes to working memory. It is thought that our working memories is limited to (7±2) items. There isn’t really an agreed consensus on the number of items that can be held in working memory at any one time, and it depends on many factors such as how complicated the items are and what we are required to do with them once they are in our working memories.
The main points here are:
our working memories are limited
everything we must think about uses up space in working memory
learning is defined as a change in long term memory (Kirschner, Sweller and Clark)
learning requires effort in working memory
There is no known limit to our long term memories. In the long term memory, information is organised in schemas.
You have schemas for everything. And you can have unlimited schemas (as far as we know). They can be vast or they can be simple. My schema for solving a Rubik’s cube is, like most of yours I am sure, vast and complex. But my schema for crochet patterns is very small – there are, I am assured, lots of different abbreviations used for different stitches, and these vary depending on the country where the pattern originated from etc.
A person with a highly developed schema for, say, solving simple problems involving differentiation (i.e. just finding the derivative of lots of functions) will have a more success learning how to find the stationary points of a function or the equation of a tangent to a function than a person who does not have that schema as well developed. A really good way to develop schemas is through practice of the component parts.
How can we tell if a student is a novice or an expert? We need to use formative assessment and perhaps diagnostic assessment before the first lesson in a particular topic. Note that a student who is an expert in one domain may not be an expert in another.
There are three main types of cognitive load:
Extraneous Load: caused by inappropriate instructional designs that ignore working memory limits and fail to focus working resources on schema construction or automation. This type of load is mostly environmental and always unhelpful for learning. This may include noise, unhelpful or unnecessary pictures/graphics/animations and poorly structured learning activities.
Intrinsic Load: caused by the natural complexity and structure of the material that must be processed. Necessary for learning – it is what makes it worth learning. Some things are harder to learn than others, based on their complexity and the prior knowledge of the learner. Learning capital cities is pretty easy – I tell you that Paris is the capital of France, you understand what I mean (as long as you know that France is a country and you have an idea of what Capital means) but if I tell you that the area under the curve sinx from 0 to pi/2 is 1 square unit you need to know quite a few things in order to understand it. The intrinsic load depends on two main factors – the complexity of the material and how knowledgeable you already are in that specific domain.
Germane Load: caused by effortful learning, resulting in schema construction and automation. This is the effort required to actually learn material (if our definition of learning is “a change in long term memory”).
As teachers (or “instructional designers”) we need to ensure we do the following:
Minimise extraneous load – consider the environment and anything you make students think about that isn’t to do with the new learning.
Minimise intrinsic load – break down the problem for novices. Present small parts at a time before approaching a whole problem that requires several new steps.
Maximise germane load – by reducing extraneous load and making the intrinsic load more manageable for learners, schema construction is much easier.
The Phonological Loop
The part of the working memory that processes written and spoken material is called the Phonological Loop. When you read something, you generate a sound in your head. When you listen to someone speak this is also processed as a sound. If you are trying to read something while someone is speaking, you get cognitively overloaded straight away. As teachers, we should avoid things like reading out slides or, even worse, talking about slides that have text on them while the students are reading the slides. For novices who are not familiar with the content, this will cause them excessive cognitive load. More on this when we get to the modality effect.
Cognitive Load Effects
I will mention 6 cognitive load effects briefly, and give some examples of each one.
The Worked Example Effect
At the point of initial instruction, novices benefit from seeing worked examples. An effective strategy is to present a worked example to the class (you can use questioning about the parts that they can already do – this isn’t necessarily chalk and talk) followed by the class completing a very similar problem for themselves. When I do this, my classes don’t copy the worked example, but they do write their solutions to the problems they will try into their notes. When we discuss the problem as a class and go over the correct solution (or a correct solution) they then have the chance to change their answers. The worked example should allow all pupils who are paying attention the chance to get the problem correct without too much of a demand on their working memories. This allows them to see the ways that the parts of the example interact and allows easier formation of schemas. Some examples of worked examples are given below:
Questioning and discussion of steps is what makes this effective. Cannot just be pupils following the same steps without using their brains.
The Expertise Reversal Effect
It has been shown that worked examples are more useful for novices than they are for experts. As expertise grows through experience, worked examples are no longer needed, and in fact can cause unnecessary cognitive load (extrinsic) for experts. Instead of presenting experts in a particular domain with worked examples, it is more beneficial to have them solving problems. Learning through problems is only possible when a strong foundation of knowledge has been built up by the student.
The Redundancy Effect
Any information that is additional to the problem is redundant information. For example, when students are solving geometry problems, an annotated diagram alongside text that tells you the lengths of the sides and the sizes of the angles (which are already marked on the diagram). In this case one of these sources of information is redundant, since the problem could be fully understood with just one of them.
Here is an example:
We can cope with this as experts, because we look at this question and instantly think “Pythagoras!” but remember that novices do not work in the same way. A novice needs to process everything in the problem.
Other sources of redundant information include teachers reading out slides and drawings/images on slides and worksheets that have little to do with the problem. At the point of initial instruction, these additional things are not helpful for learning, and so they should be avoided.
Some teachers tell me that the reason they read out slides is that they do not trust their pupils to read the slides for themselves. A simple fix in this case is to simply put a picture on the slide that represents the idea being discussed and to simply say the things that would have been text on the slide.
The Split Attention Effect
This occurs when two or more sources of information must be integrated in order to make sense of the whole problem or idea. This can easily be eliminated by integrating the two sources. This differs from the redundancy effect in that both pieces of information must be thought of together to make sense of the whole.
Here is an example from a Higher Maths past paper:
A simple fix:
The equations could easily be added to the diagram, thus removing the need to interpret two sources of information to make sense of the whole.
The Modality Effect
This concerns the way that new information is presented, whether it be auditory, written (which is also auditory by the time it is processed) or visual. We can cope with listening to speech and seeing something in a diagram at the same time without impacting on cognitive load. This is better than integrating text and a diagram. Have you ever been on a museum tour with a headphone set? This is effective because it is easier than reading text then looking at things. Yes, it’s saving us from having to read – effort – but also it cuts down on reading (with eyes) and seeing the exhibits (with eyes).
What we can’t do is listen to something while listening to something else. We can’t read something (which uses visual channel and auditory channel) and listen to someone speaking.
A diagram for a question (or to demonstrate a relationship) that would normally have text alongside it can be replaced with just the diagram and the teacher narrating over the top. If you have pupils who need the written form too (not all of them will) then you can give them a written copy, but it will be better for everyone else if they hear the question and see the diagram rather than having the text, which you will probably redundantly read out, and the diagram too – you get the split attention effect if they have to read about the diagram while looking at the diagram.
The Goal Free Effect
This effect concerns the idea of “problem solving search”. When novices are presented with a problem such as the one on the left in the diagram below, they tend to think of the whole problem in one go and suffer cognitive overload as a result.
Taking the specific goal out of the problem and re-framing it as is shown on the right eliminates problem solving search so that the novice learner may use any angle facts they know to fill in as many angles as they can. When the problem is framed this way, novices are able to make sense of the individual steps they take, and this allows them to assimilate long term memories of angle facts.
The idea that novices can learn new knowledge through discovery learning is flawed due to what we know from Cognitive Load Theory. Kirschner, Sweller and Clark (2006) state that “The goal of instruction…is to give learners specific guidance about how to cognitively manipulate information in ways that are consistent with a learning goal, and store the result in long-term memory”. Discovery Learning does not easily facilitate this. I used to attempt to teach Pythagoras’ Theorem through a discovery task. The class would investigate the relationship by matching around 15 squares to the correct 5 triangles by finding the sides that matched. No relationship yet discovered. They then had to measure the lengths of the sides of each square and work out the areas of each square.
Only a small number of pupils in the class managed to calculate the correct areas, and nobody noticed that the two small squares had a combined area that was equal to that of the large square. So I reluctantly told them that this relationship would exist. “It doesn’t work on mine! 3.1 squared plus 3.9 squared doesn’t make 5.2 squared”. If only they could measure accurately. This type of discovery investigation task looks lovely – I was observed by a depute head teacher doing it with a second year class. His comments were “You could just feel the learning in the room – they are so engaged”. No you couldn’t and their engagement was with glue sticks and scissors. They only learned Pythagoras’ Theorem in the last few minutes when I explained it quickly before the bell. They still were not convinced that it works because for their squares and triangles it didn’t work. It was a discovery learning failure.
I now start the Pythagoras’ Theorem topic by telling them that the two small squares have the same total area as the large square and I demonstrate it with a few Pythagorean Triples (3, 4, 5), (5, 12, 13). We sketch a diagram of a right-angled triangle with three squares every time we answer a question. The success rate is much higher and they feel like they are doing pretty advanced maths. The paper by Kirschner, Sweller and Clark in the references list is well worth a read for more on this, as is listening to Greg Ashman and Daisy Christodoulou on the @mrbartonmaths podcast.
This is a great way for pupils to apply what they have already learned in different and unfamiliar contexts. The trouble is, often interdisciplinary learning attempts to teach new content through interdisciplinary learning projects. It is not fair on novices to expect them to synthesise new material at the point of initial instruction. I’m not saying that Interdisciplinary Learning is a bad idea. What I am saying is that, when designing learning experiences, we need to be mindful of the fact that we are experts and that our pupils are novices.
Classroom displays often contribute to the extraneous load we impose on our learners, particularly when the displays are engaging. With this in mind, I have removed as much clutter as I could from the walls in my classroom. All of my displays are now on the back wall (my pupils sit in rows, facing the front). The only things worth looking at on the wall at the front of my classroom are the two whiteboards. Examples of pupils’ work are shown using the visualiser and do not become wallpaper on my walls. The walls at the side are plain, with the exception of the fire evacuation instructions. Perhaps you’re not ready to give up your classroom displays, but please consider what they add to the learning in your classroom. If it’s formulas for pupils to use, are you happy that they don’t need to commit these to their long term memories, and instead just rely on them being on the wall?
If you only remember three things from this blog post:
Novices and experts learn differently
Working memory is limited
Effects: Worked Examples, Redundancy, Split-Attention, Modality, Goal-Free
Barton, C. (2017) ‘Greg Ashman – Cognitive Load Theory and Direct Instruction vs Inquiry Based Learning‘, Mr Barton Maths Podcast.
Barton, C. (2017) ‘Daisy Christodoulou – Assessment, Multiple Choice Questions, 7 Myths about Education‘, Mr Barton Maths Podcast.
Barton, C. (2018). How I Wish I’d Taught Maths. John Catt Educational Ltd. Woodbridge.
Christodoulou, D. (2014) Seven Myths About Education. Routledge. Oxon.
At La Salle Education, we believe that pupils benefit enormously from having a deep understanding of multiplication and division facts, which can later be efficiently recalled for use in more complex problems.
A secure knowledge of times tables facts makes pupils able to engage in interesting mathematical problems without having to worry about working out basic facts first – these facts are part of the underlying mathematical grammar that pupils call upon to engage with mathematics throughout their learning and application of the subject.
But mathematics is not simply a list of facts to be remembered. At La Salle, we are interested in the interconnectedness of mathematical ideas. Most times tables practice is focused on simple rote learning and memorisation of the facts. This misses opportunities to build deeper understanding of multiplication and division and results in a superficial ability to simply regurgitate numbers. Our times tables app draws on variation theory to give multiple representations of multiplication facts, which builds more meaningful connections in pupils’ minds and gives a greater chance of the facts becoming embedded in the long-term memory.
Through a variety of representations and metaphors, the Complete Mathematics Times Tables app gives pupils a better chance to ‘meaning make’ than traditional times tables apps.
Representations and metaphors
The Complete Mathematics Times Tables app deliberately intertwines a variety of ways of looking at and thinking about multiplication and division (and their connections to addition). The app includes standard recall prompts
but also makes connections to multiplication grids
and introduces pupils to arrays
The app also includes a pinboard manipulative, which not only connects the tables facts to multiplication grids, but also draws on the metaphor of multiplication and division as a view of area
Why no timer?
Becoming mathematically literate is not a competitive sport, it is a fundamental basic right for all. Although we want all pupils to be able to quickly recall times tables facts and be able to work efficiently with a wide range of problems that draw on these facts, we believe that – at the point of learning and embedding – it is far more important to carefully consider the problems and metaphors and to build a deeper understanding through meaningful practice.
The Complete Mathematics Times Tables app is ideal for use in the mathematics classroom, at home, on the bus or… well… anywhere! Pupils can use the app on any device with a web browser.
With daily use, pupils will achieve a very secure knowledge of times tables facts. More than this though: unlike traditional times tables apps, which focus purely on the list of facts, using the Complete Mathematics Times Tables app daily, pupils will acquire a deep understanding of why the facts are true.
The times tables app could be used during tutor time, with pupils setting the quiz at 50 questions and recording each day how they are improving and which multiplication facts they need to continue to work on. Just 10 minutes per day for all pupils will help to drive up pupils’ mathematical literacy across the school.
So, why not try the app today with your pupils and start a journey towards truly meaningful understanding of times tables rather than just fast regurgitation of meaningless numbers.
Today, the Education Endowment Foundation has released its much anticipated report, "Improving Mathematics in Key Stages Two and Three"
La Salle Education welcomes the report and all of its recommendations, which we believe describes long established good practice in mathematics teaching. The report fully supports our mastery approach and backs up the model we use in the Complete Mathematics platform and CPD programmes.
Recommendation 1: Use Assessment to Build on Pupils' Existing Knowlege and Understanding
Complete Mathematics: contains extensive assessment and monitoring features, which are uniquely tied to what has been taught and future planning, giving teachers immediate insight into gaps in learning and quick and easy ways to adapt planning to account for such gaps. Our granular assessments also allow teachers to give targeted and contextualised feedback. Complete Mathematics also contains guidance on common misconceptions that can arise, meaning teachers are able to plan lessons that address such misconceptions
Recommendation 2: Use Manipulatives and Representations
Complete Mathematics: All Members have regular access to CPD on concrete, pictorial and abstract approaches to teaching mathematics, which includes extensive training on the use of manipualtives across the age and ability range. The Complete Mathematics platform also contains a suite of digital manipulatives for teachers and pupils to use when exploring mathematical concepts. Guidance is provided on the importance of seeing manipulatives as a scaffold, which is gradually removed to leaves all pupils with the ability to use quick and efficient abstract and symbolic methods.
Recommendation 3: Teach Pupils Strategies for Solving Problems
Complete Mathematics: contains extensive guidance on problem solving for all concepts in maths. Members also have regular access to our CPD events, including the popular Mastery in Mathematics day, which include deep exploration of strategies and dispositions for solving problems, reasoning and analysing. Our work on variation theory also includes guidance on understanding and being able to select from a variety of approaches. The Complete Mathematics platform includes thousands or problem solving tasks.
Recommendation 4: Enable Pupils to Develop a Rich Network of Mathematical Knowledge
Complete Mathematics: contains the whole of mathematics, with every single idea and concept from early years through to the end of A Level. The map through mathematics is presented to all pupils in their platform, giving them the ability to explore all maths and the detailed connections that exist. Our team spent many years creating the detailed map of mathematical ideas and the interconnectedness between them. All members have access to this map and can therefore plan schemes based on careful progression and connectedness. The platform contains extensive guidance for both teachers and pupils on every concept, including the underpinning knowledge and skills required.
Recommendation 5: Develop Pupils' Independence and Motivation
Complete Mathematics: members have access to regular CPD throughout the school year, including much about promoting thinking skills and developing metacognition. The platform contains an independent, adaptive learning system for pupils, which allows them to take ownership of their learning - pupils can pursue areas of mathematics independently, based on assessment and quiz data. We see large numbers of pupils taking quizzes on the Complete Mathematics platform and then choosing to do further study and solve further problems until they have better understood the ideas.
Recommendation 6: Use Tasks and Resources to Challenge and Support Pupils' Mathematics
Complete Mathematics: members have access to the UKs most extensive mathematics teaching and learning platform and the UKs largest network of maths teachers. The platform contains hundreds of thousands of questions, problems, activities and tasks. We believe, as the EEF does, that these resources are just tools, which must be use appropriately in order to be effective. This is why every single resource is also supported by pedagogical advice. The community of teachers also share their thoughts on the resources and how to use them for impact. All resources are tied to quizzes, which can quickly identify pupils' strengths and weaknesses and help teachers plan to overcome misconceptions. Complete Mathematics members have access to regular CPD exploring conceptual and procedural knowledge and how to use stories to build understanding.
Recommendation 7: Use Structured Interventions to Provide Additional Support
Complete Mathematics: platform contains extensive assessments with linked analytics, allowing teachers to target support and plan for early intervention. This means interventions can be explicit - teachers have the information they need to know at the granular level what mathematics is holding the pupil back and are then provided with comprehensive support in terms of pedagogical advice and resourcing to be able to address the specific issues. Furthermore, the platform allows for 'self-intervention' through its pupil interface, where pupils can explore mathematical ideas further based on the platform analytics of their understanding
Recommendation 8: Support Pupils to Make a Successful Transition Between Primary and Secondary School
Complete Mathematics: platform contains the pupil "Learning Diary", which records every interaction a pupil has with the system - all the work they do, all the questions the answer, all assessments and quizzes and associated analytics. This profile of the pupil grows with them. As the move from class to class, year to year, and primary to secondary, all of their data and information travels with them. This means that teachers meeting new Year 7 pupils can begin with a deep understanding of their mathematical backgrounds. Furthermore, the Complete Mathematics platform contains comprehensive diagnostic capabilities, meaning teachers can quickly identify strengths and weaknesses of new cohorts. Because Complete Mathematics is entirely integrated, these diagnostics can then be easily used to inform planning and the building of schemes for individuals, classes or entire year groups. The diagnostic information can also be used to identify the most appropriate pupil groupings.
The EEF report is a very welcome addition to the mathematics education canon. We wholeheartedly endorse the report and its recommendations and are proud to have already been doing all of the suggested approaches contained in the report.
At La Salle, we are determined to ensure our work truly reflects the needs of real classroom teachers. To achieve this, we work closely with schools across England. We are now recruiting additional Research Schools. Please read on for information on what being a Complete Mathematics Research School entails and how to apply.
Complete Mathematics is already the most extensive support platform for maths teaching and learning, but we are committed to keep growing, improving and making the system more and more useful, so that every maths teacher can benefit.
To help us make the right decisions, we have a number of Complete Mathematics Research Schools across the country, who we work closely with. We are now seeking to recruit 30 new secondary school partners this Autumn and then primary schools and FE colleges in the Spring term.
To apply, you must be a Head of Maths or the mathematics coordinator in a school or college in England.
WHAT’S IN IT FOR YOUR SCHOOL?
Completely free access for all staff and students to Complete Mathematics
Free on-site training for you and your team
Free tickets to all of our #MathsConf conferences for all of your maths team
Reduced fees on our national programmes of CPD
A Complete Mathematics Research School badge to use on your website and communications
Combined, this package of resource and support is worth tens of thousands of pounds!
WHAT’S THE CATCH?
There is, of course, a catch.
We are sincerely looking for Heads of Maths or Maths Coordinators who want to work together with us. You are the experts, you know what is going on in the classroom. We can only make the right product for you with your help. So, we are asking for your input and advice.
WHAT DOES BEING A RESEARCH SCHOOL REQUIRE ME TO DO?
There is no set format for our research schools, with teachers contributing in different ways, but being a research school might involve some or all of the following:
Having visits from one of our team
Running a workshop at a MathsConf
Running a TeachMeet (we will pay for refreshments and provide PR and a slot)
Making introductions to your feeder primary schools
Featuring in a case study or blog
Running a CPD event in your region (we will do all the PR and sign up delegates)
In addition, we ask all of our Research Schools to really throw themselves into Complete Mathematics. So, we do require you to get your entire maths team on board in using the system fully (we will give you all the training and support you need).
If this opportunity is something you are interested in and can commit to becoming a Research School, then we would love to see hear from you.
There is no doubt that Complete Mathematics is at its most powerful and effective when it is used across an entire school or college, group of schools (such as an academy chain) or used to make transition more meaningful when a secondary school and its feeder primaries are connected through the system. However, we recognise that there are many maths teachers working in schools and colleges who wish to engage with Complete Mathematics without the support of their institution. We believe that everyone should be able to join our network of maths teachers in making maths education better, that’s why we have now introduced ‘Individual Membership’.
Individual Membership gives you all the features and functionalities of the Complete Mathematics online environment as well as giving you full and free access to the national conferences, which we run three times per year (non-members pay a fee to attend). You will also benefit from significantly reduced course fees at any of our CPD courses.
We want to make it as easy as possible to join, so Individual Membership can be purchased using our easy monthly direct debit. We offer a special price for individuals of just £20 per month (inclusive of VAT).
Haven’t made a decision about Complete Mathematics yet? Why not book a free 1-to-1 webinar with one of our team. You will get a personal tour of the Complete Mathematics system and can ask any other questions you might have. Call today to arrange the best time for you.
We all know that the very best position for a school to be in is to have each and every maths lesson delivered by a specialist mathematics teacher. We share that aim and aspiration, but the reality is that many schools across the country are dealing with the impact of a national recruitment crisis. There simply is not enough maths teachers to fill the roles.
Head teachers are then faced with tough decisions about how to staff the provision of maths. In many cases, long term supply, the use of HLTAs or other non-qualified staff, or internal day to day cover by colleagues is the only option. These staff strive to provide the best possible learning experience for their students and heads of maths work hard to support them. But what if there was another solution? What if those colleagues standing in for a maths teacher were also able to deliver effective maths lessons, while at the same easing the crushing burden on the head of maths?
La Salle Education specialises in improving mathematics education in schools and colleges in England.
Using our extensive platform, Complete Mathematics, teachers are able to access teaching, learning and assessment resources and support covering the entire age and ability range. Many teachers use the system to deliver their maths curriculum.
La Salle is also able to offer schools a unique solution to a maths specialist shortage. Using the Complete Mathematics platform and working alongside your HLTA, cover manager or supply teacher, a La Salle mathematics expert will plan and monitor every lesson, giving extensive support to the temporary staff member to ensure that they are delivering impactful lessons that get the most out of your students. In addition, your Complete Mathematics Mentor will set regular, meaningful homework for every child and monitor their progress, providing frequent reporting to the head of maths.
The process is simple and flexible so that head teachers are able to continue their search for a specialist teacher, safe in the knowledge that the temporary solution is as effective as possible. A La Salle Mentor will visit your school, meet with the head of maths to learn about schemes of work and the current attainment of the students. Where possible, the Complete Mathematics Mentor will also meet face-to-face with the member of staff who will be delivering the lessons. Then, through the Complete Mathematics platform, the Mentor will plan every lesson for each class involved. Students will also have access to an online environment where they can see their maths lessons and collect and submit their homework. During the period of mentorship, the Mentor will discuss progress regularly with the HLTA, cover manager or supply teacher, engaging them with co-planning and exploring effective approaches.
We understand that head teachers need flexibility, so contracting a La Salle Mentor is made easy with a simple month-to-month commitment. We don’t tie you in and will even do all that we can to help you find a full-time specialist teacher to fill your post as quickly as possible.
Only La Salle has the ability to offer such a comprehensive service to schools. Complete Mathematics covers every single lesson from Year 1 to Year 11, so no matter what the ability range of the classes involved, we have it covered.
Of course, nothing beats having a specialist teacher, but in the meantime why shouldn’t your students receive the most effective lessons possible? For more information about the programme, please visit the Solving Maths Teacher Shortages page.
After 20 years teaching there are days when things seem not to be going well. My year 11 ought to be at their best, working hard to squeeze the most out of every last precious minute, but instead I find too many of them happy to chat and achieve very little. In the department we seem to be struggling to be aiming for anything more than a narrow focus on exam success, as a school the government’s cuts for sixth form funding are starting to bite and as a country education seems to be redefined almost daily as politician’s battle for the voters’ attention.
And then I spend an hour with my year 7 group, a small group of students who find maths a huge struggle, whose number sense isn’t complete, who mostly need to count on their fingers and who usually can’t remember very much from one day to the next. But they are so charming and so polite (most of the time) and keen to learn and although terrified of tests they do arrive for each lesson with a positive outlook and a cheerful nature. And we tackle probability and they make good contributions and ask good questions and a good hour is had by all. And our good hour isn’t because I’ve made good use of different learning styles, and I’ve not had to address thinking skills and they are happy to learn and discuss without any hint of “when are we going to need this”. An hour spent explaining and practicing and encouraging seems to have worked again.
And then I remember those in the year 11 class who are constantly asking good questions and have made good progress – those who were a struggling and nervous grade C in their December mock some are now a much more confident grade B with hopes (on a good day) of achieving a grade A in the summer. And then I’m grabbed by a conversation on Twitter and the teaching matters again. And then I’m planning some lessons with Complete Mathematics and are overjoyed to see resources that others have shared so that I’m not planning from scratch again. And the funding and the politicians? I remain hopeful that wiser heads than mine will triumph and sensible thinking will prevail and the outlook isn’t so bleak after all.
Over the last couple of years, the team at La Salle Education has been thinking about assessment without levels.
National Curriculum Levels served a purpose in their original guise – a way of suggesting that mathematics is an interconnected subject, where the learning of a concept rests firmly on the foundations of earlier concepts, which must be secure before moving on. Levels, in a very broad sense, drew a picture of what this progression might look like and suggested a pathway through mathematics as a child learned more and more.
There is nothing at all wrong with this idea. In mathematics in particular, not only is it a correct assertion, but it is a very useful one too. Ensuring that a student has secured underlying concepts before trying to build on top of them is the best way of giving them the chance to really learn mathematics and be successful.
Levels, in their original format, also did a good job at showing interconnections across strands of mathematics – showing that, say, some statistics should not be encountered before the underlying number work.
The abolition of National Curriculum Levels was due to a multitude of reasons, many of them ideological and many of them ignorant of the differences in subject areas. But there were also several good reasons for their scrapping. National Curriculum Levels, as with any metric in a high-stakes system, was an idea almost doomed to abuse at its inception. In no time at all, because of a desperate need to ‘measure progress’, the levels were subdivided and misinterpreted. Suddenly, the once broad and general pathway through mathematics became an ill-informed and utterly ridiculous statement of mathematics learning that simply ignored the way in which mathematics learning happens. The notion that a child can be a Level 4b in mathematics is nonsensical. What on earth does it mean?
This is a far cry from what Level 4 was supposed to mean – a broad statement of an approximate place on a journey of learning mathematics.
Then, with such predictability, we started to see the granularity become more and more extreme. The idea that a single test could tell you the level of a child or, worse still, that a single lesson with perhaps just one activity could brand a child with an extremely specific level. Even a 20 minute lesson observation was starting to demand an assessment of each child’s level.
One other issue with National Curriculum Levels is that they had become a fundamentally dishonest measures. With a fairly obscure language, it became easy for parents to misinterpret the true meaning of the metrics. A parent told that their child has achieved Level 3, when last year their child was just Level 2, might well be pleased and feel that all is well. But what if that child is 14 years old? Levels all too easily disguise low attainment because they can be used to exclusively highlight progress.
At La Salle, a group of around 30 people have been constructing an assessment system with two overarching aims. Firstly, the metric should be honest and show not only progress but attainment also. It should give a clear indication of a trajectory so that everyone (teacher, student and parent) can play a part in intervention as soon as it is required. Secondly, the system should reinstate the correct intentions of the Levels system – that is to say, the system should give clear guidance of a pathway through mathematics, where concepts underpin each other and the journey through an interconnected mathematics curriculum is one that gives the best possible chance of success.
We have experimented with the ideas over a couple of years. The team is made up of a wide variety of people, from those of us who created things like the NCETM self-evaluation tool or were part of creating national programmes of study, or members of the team who are currently working in schools of various types and circumstances, to colleagues who write assessment materials for awarding bodies. We have debated long and hard, tried various ideas and consulted with lots of teachers (and continue to do so).
The emerging result is MathsAge.
The early version of this system is already incorporated into Complete Mathematics and, as with every aspect of our work, we will continue to research and refine. You can imagine, as more and more students interact with the system and more and more teachers give feedback, the metric can be honed and continually made more rigorous.
So what did we do? Initially, our work centered around curriculum design and then learning design rather than assessment. All too often, assessment is allowed to drive curriculum and learning design – this is something we have no interest in. Curriculum and learning come first. This meant spending years creating a journey through school age mathematics from counting to calculus. There were many iterations of this curriculum. The result – a progressive journey through mathematics based on the fundamental principle of securing concepts before building on top of them – is, of course, not the only pathway. There are many areas of school age mathematics that are axiomatic, which means that there are many entry points to starting particular strands of the journey. Nevertheless, we wanted to put in place a journey that does work – not a unique journey – but one that will work if followed.
This curriculum design then led to the most extensive piece of work: learning design. We have been working on this for a while, with a very large team, and will continue to work on it for years to come (forever in fact), with every single teacher in Complete Mathematics also able to add to the design and debate.
Combined, these two areas of our work produced a framework - not unlike a neural network when we draw it out on paper! This gives us a way of identifying key signposts along the journey and dividing the journey in a practical and sensible way.
The obvious response is to work with what works for schools. So we took the full journey and divided it into 11 very broad steps, going from counting to early calculus. The idea being that any student who successfully passes through the 11 steps will be able to achieve a Grade 9 at GCSE.
These steps contain broad statements of attainment, showing the areas that must be secure before moving on. It does not mean that the mathematical topic will never be encountered again, but simply that there are stages in strands / topics / concepts that need to be addressed at certain points along the journey.
Why did we choose age rather than stage? It would be easy, of course, to use the 11 steps to say ‘this is what stage you have reached on the journey’, but we wanted the metric to be really honest, particularly for parents. So each stage is related to an age and that is what we communicate back.
It is nonsense to say a child is, for instance, a low age 12 because areas in the broad step will mature in different ways. So we make no judgment about this sort of granular level. Instead, and this is where the assessment design phase really kicked in, we are more interested in the ‘strength’ of the measure. Any interaction might result in a judgment being made, but how robust is that judgment and how reliably can a teacher alter their planning? That is our focus. So the MathsAge comes with a ‘strength’ measure. This will become more and more accurate as the child provides more and more evidence.
Unlike most systems available for maths education, Complete Mathematics is not a static system, which means that all of this work will continue to evolve based on real users’ interactions and real teachers’ observations and feedback. This means that the trajectory measure will become ever more accurate. In the meantime, the trajectory (the system is predicting what final GCSE grade a child of any age will achieve at the end of school) is underpinned by the statistics collected at national level over the last couple of decades, which give fairly accurate probability distributions of where a child will progress to given their current attainment. We have used these probabilities as the starting point, which can then be tested over time as real children progress through the age groups.
MathsAge is therefore a way of allowing everyone involved – teachers, pupils and parents – to have a clear picture of where the pupil is right now and where the pupil is heading.
Because the Complete Mathematics curriculum was the starting point, knowing a pupil’s MathsAge also allows us to know precisely which concepts are holding them back. This will give schools the ability to change their planning or to provide meaningful intervention, rather than generic booster sessions.
What a year it has been so far. With thousands of teachers getting involved in Complete Mathematics through our ‘Research Schools’ project, the National Mathematics Teacher Conferences and our regional CPD roadshows, ITE and TeachFirst students learning their craft with the support of collaborating teachers, and those schools that have adopted the system in school to help raise standards in maths, we have been amazed at the way the community has engaged with La Salle.
Summer term is now kicking off and we are determined that Complete Mathematics will continue to grow and reflect the views of our community of thousands of teachers. Over the last few months, teachers have been telling us what will really help them in the summer term. We have a standing joke in the La Salle office: Complete Mathematics, it will never be complete! You see, Complete Mathematics is not like other systems – it has always been our intention for the project to grow and grow and grow, reflecting the real needs of those actually teaching in classrooms day-to-day.
So we are proud to announce that during this next half term, we will add the following new features to Complete Mathematics:
1. Core Maths
With the school leaving age now increasing and all students expected to continue with the study of mathematics up to the age of 18, many schools and colleges will be faced with the enormous challenge of creating a new programme of study for those students not following an A Level course. For many, this will mean putting together a pathway and resources for the new Core Maths course. Thousands of teachers will be faced with the same problem of creating a robust structure and scheme of work. That’s why La Salle will add Core Maths into Complete Mathematics at the end of May. This will give schools access to a fully resourced scheme (and assessments for tracking!), which can be deployed to their students. Of course, being in Complete Mathematics, the course will not just be a scheme of work and some content – as with all other areas, Core Maths will be fully supported with pedagogical notes, misconceptions, exemplar questions and a host of other support materials. Given that many schools and colleges will be faced with the challenge of delivering Core Maths without additional specialist staffing, we hope that these support materials will enable all teachers to deliver the highest quality lessons for those post-16 students taking up the course.
2. New GCSE Mathematics Courses
We have already put together a comprehensive scheme for the new key stage 4 national curriculum (you can use it right now in Complete Mathematics), but will now be going further to create specific schemes for AQA, Edexcel and OCR GCSE mathematics. Rather than having to build a scheme, you will be able to simply select the level (higher or foundation) and the awarding body for an immediate, fully resourced and supported scheme of work and online learning and assessment materials.
3. A Level Maths
Complete Mathematics already contains a fully exemplified scheme of work for KS1, KS2, KS3 and KS4. Later in the term, we’ll also be adding schemes for A Level maths. Another great addition to the system, which will allow you to continue to use Complete Mathematics with your older students.
4. Scheme of Work publisher
Already, hundreds of teachers have built schemes of work in Complete Mathematics, using our quick and easy SoW Builder. We’ve been so impressed by the quality of curriculum design and thought that has gone into these that we are soon introducing a SoW Publisher – this means you will be able to send your scheme to other people in your school, colleagues you have added as ‘friends’ or the entire Complete Mathematics community!
5. Super-Customisable Lesson Planner
You have been telling us that you want even more flexibility in your lesson planner, so that is what we are working on now. Thanks to suggestions from teachers across the country, the Super-Customisable Lesson Planner will give you all the freedom you need to chop and change, add in additional concepts, repeat topics, or simply ‘go off on one’ mid-lesson!
6. Even more La Salle content for KS1
The primary specialist members of our team have been busying themselves with creating ever more content and support materials for KS1. These materials are already appearing online and will continue to grow in the coming weeks. All ‘new curriculum’ compliant of course!
7. Community Content
Perhaps the most rewarding aspect of Complete Mathematics to date has been to see the hundreds of resources and lesson notes being added by teachers across the country – particularly those ones marked ‘public’. It never ceases to cheer us up when we see how keen teachers are to support each other. Complete Mathematics is a very different environment to other maths education sites – everything that makes it onto the system goes through a curating and QA process, which means, unlike some sites, you can be assured that the materials you use are of the highest quality. During the summer term, we’ll be looking out for the most interesting and effective materials added by teachers – there will be a prize for the best!
If you haven’t yet heard about the Mathematics Teacher Network, summer term is the time to get involved. Oxford University Press, AQA and La Salle Education have joined forces to bring high quality CPD to maths teachers across the country entirely for free. See our MTN page for details.
During this half term, we will be preparing for the National Mathematics Teacher Conference. This will be the fourth conference and will take place on Saturday 20 June in Manchester. The venue is stunning and vast – but it needs to be, the conference is open to 1000 teachers! Our conferences have fast become the largest gatherings for maths teachers from primary and secondary schools. If you haven’t joined an event yet, be sure to come along. You will have the chance to collaborate and learn from teachers from across England.
So, a very busy half term on the way, but Complete Mathematics will never stop growing. In the second half of the summer term we have even more exciting new developments to tell you about. But, more importantly, we also want to hear from you – your feedback and suggestions are what lead to our developments. Please do get in touch if there is something we can do for you and your school. Help the wish list grow and influence the Complete Mathematics of tomorrow!
Have a great half term. Enjoy the sunshine and warming weather. And keep doing the most amazing job on earth: being a teacher.
A new collaboration will engage SSAT schools with Complete Mathematics, a virtual teaching and learning environment created by La Salle Education. SSAT schools will contribute to the further research and development of Complete Mathematics to ensure that the tool offers teachers of mathematics effective support. SSAT member schools that take on the use of Complete Mathematics will also have access to high-quality, subject-specific professional development delivered by La Salle Education as part of its National Mathematics Support Programme.
“With the changes in the National Curriculum and the higher expectation of pupils’ achievements in mathematics, it is vital that our member schools have access to up-to-date resources that genuinely enhance teaching and learning,” explained Anne-Marie Duguid, Head of Teaching and Learning at SSAT. “Complete Mathematics holds pedagogy at its heart and places the teacher at the core of its activity, using technology to support rather than replace their role, which we believe will help deliver excellence for our member schools.”
Mark McCourt, Chief Executive of La Salle Education said: “SSAT is focussed on improving the quality of teaching and learning and Complete Mathematics can be a valuable partner in achieving this. A core element of Complete Mathematics is continual assessment, giving teachers detailed information on each pupil’s progress, enabling them to develop personalised learning using the activities and tools within Complete Mathematics.”
Notes to Editors
La Salle Education and Complete Mathematics
La Salle Education is run by mathematics educators with the aim of supporting all mathematics teachers with high-quality resources and continuing professional development. In April 2014, La Salle Education launched Complete Mathematics, an online environment that caters for every aspect of mathematics education for the 5-16 age range. With the teacher at its core, Complete Mathematics enables all aspects of teaching, learning and assessment. As an online tool, Complete Mathematics is updated regularly to support developments in the curriculum and teaching and learning approaches.
SSAT, a membership organization with more than 2,000 schools as members in England, offers schools a range of products and services designed to improve the quality of teaching and learning and to raise achievement in schools.
The National Mathematics Support Programme (NMSP) is a series of seminars and conferences taking place across England to offer teachers high-quality continuing professional development (CPD). Created by La Salle Education, NMSP events are free to attend and are open to primary and secondary mathematics teachers and headteachers.
The National Mathematics Teacher Conference takes place on Saturday 14th June. It is the UKs largest single day event for mathematics teachers and is entirely free to attend.
The event is organised by La Salle Education as a means of ensuring that all teachers of mathematics, across primary and secondary schools, have access to high quality professional development as we move towards a new school year that sees the introduction of a new national curriculum and the removal of the longstanding system of curriculum levels.
The event is kindly sponsored by AQA.
The day consists of a variety of talks, workshops, networking, resource sharing and a panel debate.
With over 220 delegates attending, the vast majority being current practitioners, La Salle Education and AQA are proud to host what should prove to be an exciting, interesting and enjoyable day.
9:30 - 10:00
Arrival and Networking / Morning Refreshmets
Maths Magic with Dr Maths
10:00 - 10:20
Welcome and Introduction
Mark McCourt, Chief Executive, La Salle Education
10:20 - 10:30
Welcome from the Event Sponsor, AQA
10:30 - 11:00
Keynote Speaker, Dr Vanessa Pittard, Assistant Director, Curriculum and Standards, Department for Education
11:00 - 12:00
A: The New GCSE - Andrew Taylor
B: Images, Language, Symbols, Concrete - Mark McCourt
C: The New Mathematics Curriculum in Primary Schools
D: Why do we Teach Maths? - Richard Perring
12:00 - 12:30
The Idea Exchange
12:30 - 13:00
Lunch and Networking
Magic with Dr Maths
13:00 - 14:00
Afternoon Workshop 1
E: Assessing without Levels - David Thomas and Alan Gothard
F: Leading an Outstanding Maths Department - James Gatrell
G: Effective Use of Diagnostic Questions - Craig Barton
H: Developing a Framework for Fluency - David Cook and Rachel Rayner
I: Taking Maths Beyond the Classroom - Steve Humble
14:00 - 15:00
Afternoon Workshop 2
J: Blindingly Obvious... - Bruno Reddy
K: Making Sense of the New Curriculum - Tony Gardiner
L: Outstanding Maths Lessons Using Web - Douglas Butler
M: An Outline of PBL - Julia Smith
N: The Next Questions in Maths - Luke Graham
15:00 - 15:30
The panel will take questions from the audience and debate current issues.
La Salle Education is proud to announce the launch of the National Mathematics Support Programme (NMSP).
With the upcoming changes to the new national curriculum and the removal of national curriculum levels, the NMSP is a nationwide series of support seminars aimed at helping primary and secondary schools to prepare for the new school year.
NMSP events are taking place in 9 locations across England, with the vast majority of schools being within 60 miles of an event.
Events are entirely free to attend.
Places at events are limited - please use our online booking to sign up.
Regional Seminars to Support the New Mathematics Curriculum in Primary and Secondary Schools
The National Mathematics Support Programme is a series of seminars taking place across England to help ensure that all teachers of mathematics have access to the information they need in order to prepare for the introduction of the new national curriculum and the removal of the system of NC Levels.
At a time of unprecedented change, it is important that all schools have access to high quality CPD and information. At this seminar we will address three key issues facing mathematics teachers:
The New National Curriculum - What is different and what practical steps can schools take to successfully prepare for and implement the changes.
Assessment without Levels - What frameworks can be used to assess pupils' progress and how do teachers accurately measure understanding and mastery.
Outstanding Lessons - What practical steps can we take in our classrooms to ensure pupils make outstanding progress and have a thorough understanding of mathematics from which to build upon.
NMSP events are FREE to attend and open to mathematics co-ordinators and Headteachers in primary schools, heads of mathematics and senior leaders in secondary schools. Local Authority advisors may also attend.