'Logic, Codes, Puzzles' is a workshop being run by Robert J Smith at the Mathematics Teacher Network in Southampton (04/12), Northampton (05/12) and Leeds (06/12). This is a FREE event, some tickets still remaining.
Look at this paragraph.
What is vitally wrong with it? Actually, nothing in it is wrong. But you must admit it is a most unusual paragraph. Don’t just zip through too quickly. Look again - with caution! With luck you will spot what is particular about this paragraph and all words in it. Apart from it’s poor grammar.
Can you say what it is?
Tax your brains and try again. Don’t miss a word or symbol. It isn’t all that difficult.
Having looked at the above paragraph above, can you see what is wrong? Let me know what you think it might be by sending a tweet to @LaSalleEd and use the hashtag #MTN_Codes.
Next week, La Salle Education are running a series of Maths Teacher Network sessions across the country. This series of Network meetings include workshops from AQA in the guise of Roger Ray (@AQAMaths), Sian Thomas (Leeds) and Bernie Westacott (@berniewestacott) (Northampton and Southampton) from Oxford University Press. They will all no doubt put on fabulous sessions that you should definitely attend, but I wanted to tell you about my session as I will be looking at Logic, Codes and Puzzles. The Maths Teacher Network meetings are an opportunity to get together and talk and discuss Maths. Something that we don’t always get time to do.
Having looked at the above paragraph above, can you see what is wrong? Let me know what you think it might be by sending a tweet to @LaSalleEd and use the hashtag #MTN_Codes
I really don’t want to give too much away about the session as I want you to attend. So instead, I thought I would let you think about this (fairly simple) Atbash cipher.
By the way, for those that haven’t seen an Atbash cipher before, it is a particular type of monoalphabetic cipher formed by taking the alphabet (or abjad, syllabary, etc.) and mapping it to its reverse, so that the first letter becomes the last letter, the second letter becomes the second to last letter, and so on. For example, the Latin alphabet would work like this:
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
Z Y X W V U T S R Q P O N M L K J I H G F E D C B A
Due to the fact that there is only one way to perform this, the Atbash cipher provides no communications security. So the following should be easy to decode:
ZGGVMW GSV NZGSH GVZXSVI MVGDLIP
The following is the description for my session:
Logic, Codes, Puzzles
This session involves using maths to solve codes and puzzles. From simple addition and subtraction, to data handling and logical thinking, the session will show how we can use mathematical concepts and understanding to explore topics in greater depth. An opportunity to look at how all students might access a problem. What strategies can be used? Which are the most effective? And Why?
It will be my first opportunity to attend a Maths Teacher Network meeting but I am hoping that I will be able to organise and attend many more. (If your school can host such an event then please get in touch!!)
'Assessment and Feedback: Making it Work for Everyone Involved' is a workshop being run by Jemma Sherwood at the Head of Mathematics Conference in Birmingham (Aston University) on Friday 14th Decemeber 2018, some tickets still remaining.
Assessment and feedback are different, but related, ideas. Anything that involves seeing what pupils can or can’t do at a particular point in their learning journey constitutes assessment. How we use that information and what we tell pupils in response to it constitutes feedback.
This makes assessment and feedback two of the intrinsic parts of the teacher’s job. We assess and feed back to our pupils in lots of ways, some formally, many more informally. How many of the following do you employ?
Marking books with ticks and crosses
Marking books with written comments or targets
Obtaining pupil responses to written comments
Q&A with mini-whiteboards
Q&A with hands-up
Asking questions to the class
Talking with pupils about their work
Exit or entrance tickets
Longer topics tests
Setting GCSE papers
Online homework or assessments
From each of these activities we can gain an insight into what our pupils can imitate in the short-term, and what they are learning in the long-term, but each comes with varying degrees of success and requires varying degrees of effort. Our job, as Heads of Maths, is to find and promote those activities which are most useful (to teacher and pupil) and which require the least effort. Fortunately, we have a wealth of research and information to help us make these decisions.
In my session at the Heads of Maths conference, we will analyse these activities and more, while asking ourselves the following questions, considered from the viewpoint of pupils and teachers.
What positives (in the context of learning) does this activity bring?
What negatives (in the context of learning) does this activity bring?
How much time or effort does this activity take?
Is the gain worth the time or effort invested?
As a taster, let’s consider the idea of marking books. In their hugely influential work, Inside the Black Box (1998), Dylan Wiliam and Paul Black taught us that giving numerical marks in books alongside comments was pointless – the presence of the mark would nullify the effect of the comment. From that point, comments and target-setting in books became prevalent in schools, who tried to get teachers to be more detailed than just “6/10. Try harder.” The intention was honourable, the implementation diabolical. By 2015 we had teachers up and down the country writing comments, handing books back and asking their pupils to respond, taking the books back in, responding to their pupils’ comments, only to hand the books back and never have the page looked at again, apart from by SLT in book trawls, who smiled at the best practice they were seeing, which was clearly going to raise standards. Except it didn’t raise standards, it raised teacher dropout rates and levels of exhaustion.
This is a problem because, put simply, we have no evidence whatsoever that this kind of marking is effective. None. The EEF wrote a report in 2016 entitled, “A Marked Improvement: A Review of the Evidence on Written Marking” which concluded that there are not enough robust studies to assert the efficacy of written marking.
What a hugely important conclusion! All those hours invested are, quite probably, a huge misdirection of time. Schools are enforcing unevidenced practice because they see it done elsewhere and assume they must follow suit (driven, of course, by fear of the double-pronged stick of Ofsted and league tables.)
So, in response to our four analysis questions above we have something a bit like the following:
Assessment/feedback type: Written marking (comments/responses/targets)
Use as assessment: Teacher assesses pupils’ work when reading books.
Use in feedback: Teacher writes comments or targets, pupils read and possibly respond. Teacher may be able to plan next steps based on the activity, but this is dependent on how far after the lesson the books are marked.
- Sees what pupils can/can’t imitate during a lesson.
- Sees how well pupils present their work and their thought processes.
- Is given a target to improve and chance to consider the target.
- Doesn’t find this information out until books are marked, which can be weeks later.
- Cannot plan next steps if books aren’t marked immediately.
- Doesn’t learn anything about long-term retention of material.
- Target given too far after the event (they’ve forgotten what the lesson was about).
- Comment either so specific it doesn’t cover enough work in enough detail, or so generic it is useless.
30 books take approximately two hours.
Lesson time to respond to comments – this may be a positive or a negative use of time, depending on context.
No. If book marking were to inform planning, it would need to be immediate and after every lesson. This is impossible (5 hours of lessons a day would be accompanied by 10 hours of marking).
Probably not. There is some benefit in having to remember something from two weeks ago in order to respond to comments, but this can be achieved in better ways.
Join us at the Heads of Maths Conference in December where we can take the time to find some better ways of assessing our pupils, of gaining feedback for us, and of giving them feedback.
'Time to revisit…Teaching for Mastery
' is a workshop being run by Peter Mattock at #MathsConf17.
“Mastery”. Some people see it as the latest buzz-word to be shunned until we wait for the next “big thing”. For others it is central to teaching. For some it is a confusing term with no clear idea of what it actually means. And I can sympathise with all of these views…
The idea of “mastery” has been around for a long time. People much more knowledgeable have written about its provenance, its history and its progress to the modern day. Neither this blog nor my mathsconf session will be trying to reinforce or reinterpret any of this. I will not be attempting to explain the structure of a mastery curriculum (which is not exclusive to a mathematics curriculum). Better men than me have already done this, not the least of which is the LaSalle CEO Mark McCourt (if you haven’t read his blogs on mastery then you must). Saying that, it is important to understand certain aspects of its structure to understand where I hope my session fits in.
One of the central aspects of a mastery curriculum is teaching in a way that all pupils can access from their starting point, and then carefully assessing their understanding throughout the teaching process. A second is the use of correctives where the initial teaching isn’t successful – having different ways of approaching concepts when the first way falls short. The biggest aim of my session is to try and showcase some of the ways that teachers can approach this. Starting with what I see as important ideas to consider when thinking about structuring learning, I then aim to share practical examples of approaches that could be used either as part of the initial teaching or as a corrective approach. For those that know me, it won’t be surprising to hear that much (but not all) of this focuses on the use of representations to reveal the underlying structure of an idea (given that my book “Visible Maths” is entirely concerned with the use of representations and manipulatives to reveal underlying structure).
As an example, but not one I am using in the session, consider the “rule” that one negative number divided by another negative number results in a positive answer. Consider –15 ÷ –3:
One way of representing this is to use double sided counters; these usually appear with a yellow side (positive) and a red side (negative). Two different coloured counters can also work, and in fact to model this calculation we only need to consider negatives so a single colour of counter will suffice. The image above shows -15, and now we have to think about how we divide that by -3. One way of thinking about division is to think about creating groups, so a possible way of looking at this calculation is, “Start with -15 and create groups of -3.” These groups can be seen below:
When we think about division like this, the result of the division is “How many groups can we create?”. We can see that this process creates 5 groups, which means that -15 ÷ -3 = 5.
Often this “rule” is taught as an arbitrary rule, without any attempt to show where it comes from. In many classrooms, one could be forgiven if kids believed that the only reason this is a “rule” of maths is because teachers says so. But this rule is a necessary rule – if division works in the way we know it does then the answer to -15 ÷ -3 cannot be anything but 5. I finish my session with a discussion around other “rules” of maths, how appropriate representations can show where these rules actually come from, and also discuss how we can manage the transition from using representations/manipulatives to the abstract calculations. Hopefully I have whetted your appetite to hear more about teaching approaches that can support mastery in mathematics, and I look forward to seeing you (whether in my session or not) in Birmingham. Don’t forget to join us for the pre-drinks and networking the night before as well!
'Collaboration & Creativity with Technology in the Maths Classroom' is a workshop being run by Patrick McGrath at #MathsConf17.
Technology. It’s often perceived as a challenge for the maths classroom. We’ll check out what our peers are using, we’ll look at what EdTech companies are offering, but when we voice our concerns, we’ll often be met with the line “There’s an app for that” . The problem of course is that, sometimes, there isn’t.
Sure, we may all have access to amazing apps, programmes, web tools and resources that help explain or visualise key concepts, or help with repetitive practice, but the thing is, that's not the purpose of education technology. Its use should be grounded in teaching and learning - in providing context, in deepening the learning experience, and in providing ways for students to articulate their learning in ways never before possible.
In my role as EdTech Specialist at Texthelp, maths teachers are without doubt my favourite set of practitioners to work with - they’re passionate about learning and equally so of their subject, but the one thing I hear a lot when we talk technology? “There’s nothing in it for me”. When we poll a room, the number one piece of technology chosen is the good old Interactive Whiteboard. It’s a great tool and one not many of us could do without, but it’s not representative of the amazing learning opportunities we can provide to our pupils with the vast range of tech available to us as educators.
So, it’s time for a change. It’s time to move beyond the Interactive Whiteboard. In my upcoming session at MathsConf17, together we'll explore and discover 10 amazing ways to build technology into the maths classroom and to encourage pupils to use technology at home to enrich their learning.
We’ll be looking at collaboration and creativity and how, by using just a small selection of tools, we can enable these things and in doing so increase engagement - building towards a love of learning and of maths itself. Let’s see how we can truly build context around maths activities and understand the positive impact that providing multiple means of expression can have.
We won’t get bogged down in complex techspeak. We won’t be talking in acronyms and we won’t be talking about Google vs Microsoft vs Apple. Technology should be there to support regardless of your ‘platform’. So, we'll stick as always to teaching and learning and focus on creating meaningful outcomes for pupils.
We’ll also be ‘building bridges’ with a new and exciting tool from Texthelp called EquatIO®. Created by a maths teacher, it’s a tool actually grounded in teaching and learning. It helps pupils use maths in a digital environment, express maths concepts, explore graphing with Desmos integration and enables amazing opportunities for Assessment for Learning.
EquatIO helps students and teachers - and we provide it free to any teacher via https://text.help/e8VcrH - so grab yourself a copy, and join me for Collaboration & Creativity with Technology in the Maths Classroom at MathsConf17. I promise to send you away challenged but armed with a knowledge of the tools you can start with right away. Plus, if you’re lucky, you’ll be in with a chance to grab one of our amazing ‘I Love Maths T shirts’
Lot’s of learning, lot’s of resources, free software and maybe even a T Shirt - what’s not to like?!
Check out our video below to get a quick introduction to EquatIO:
You can read more about EquatIO at https://text.help/jxOlom and discover how Texthelp have helped over 15 million people around the world understand and be understood.
'How to Solve An Adfected Quadratic' is a workshop being run by Joanne Morgan at #MathsConf17.
Factorising a quadratic when the coefficient of x 2 doesn't equal one (a 'non-monic') is apparently one of the more challenging skills that our pupils learn at GCSE. I have seen many pupils struggle with it, even those who achieve a grade 8 or 9 at GCSE and go onto take maths at A level. Interestingly, it doesn't seem like it should be challenging at all. I think it's way more straightforward than some of the harder questions that come up at GCSE - so why do pupils struggle with it so much?
When I first became a teacher, I taught my pupils to factorise harder quadratics in the way I've always done it: by inspection. Simply write out two empty pairs of brackets and try some numbers - thinking logically about what those numbers could be - until your terms expand correctly. It's very quick once you get the hang of it. In my NQT year this method was met with frustration by my Year 11s. They wanted a more defined procedure. I looked online to see if I was missing something and discovered 'the grouping method' which I then showed them as an alternative. I felt that it was an unnecessarily convoluted method but they clearly preferred having a set of rules to follow rather than having to reason for themselves. It made me a bit sad.
This grouping method, particularly the last step where terms are gathered together, is a bit of a leap of faith for pupils who have never seen this kind of factorisation before. It kind of seems like magic.
Ten years on, I still prefer inspection (the 'guess and check' method) but I always teach my pupils the grouping method as an alternative. I know they'll see it elsewhere even if I don't teach it to them - in textbooks, revision guides and online videos. I find that only the strongest pupils favour inspection - most pupils choose to use the grouping method but very often forget it. Days before the GCSE exam I hear cries of 'what's that thing you do to factorise hard quadratics? Something to do with the middle term...?'. The steps in the grouping method are not intuitive, and as a result it's difficult to remember.
Given I had never heard of the grouping method before I started teaching, I was surprised to learn that it's actually an incredibly popular method. In fact, it appears to be the method that most maths teachers now use to teach non-monic factorisation.
How did I miss this method during my time at school? I did my maths GCSE in 90s. I have a couple of Bostock and Chandler textbooks from the 1990 - both only feature inspection, with no mention of grouping.
This probably explains why I'd not seen it before. It wasn't in fashion when I was at school.
Looking further back I was surprised to see that older textbooks do feature the grouping method. Here, in New Algebra for Schools (Durell, 1953), we see an example of the grouping method. Durell recommends this method for both monic and non-monic quadratics.
Note though that this follows on from extensive use of grouping elsewhere. By this point in the textbook pupils have had considerable experience of factorising expressions like p(a + b) + q(a+b) and ax - ay + bx - by. This is absolutely key. There is a clear progression here that I feel is often missing from modern teaching of factorisation. I'm not sure it make much sense for pupils to only use the grouping method for non-monic quadratics, having never seen it before.
Later, this chapter tells us that simple quadratic functions can often be factorised at sight without using the grouping method. It says "use the grouping method whenever you are not able to obtain the factors by inspection, quickly".
Another textbook from eight years later (The Essentials of School Algebra, Mayne, 1961) also features both the grouping and inspection methods. Again, earlier in the chapter there is a considerable amount of work on the skills and understanding required for the grouping method. Of inspection it says,
"After a little practice, the pupil will be able to reject mentally the impossible pairs of factors... With simple numbers it is slightly quicker than the method of grouping terms, but the grouping method is the method to rely on. It should always be used whenever the pupil is not able to obtain the factors quickly be inspection."
So it seems that the grouping method may have been popular in the mid-20th Century.
Looking even further back, in Elementary Algebra for Schools (Hall & Knight, 1885) we are told that we should factorise non-monics by inspection:
"The beginner will frequently find that is it not easy to select the proper factors at the first trial. Practice alone will enable him to detect at a glance whether any pair he has chosen will combine so as to give the correct coefficients of the expression to be resolved".
There's no mention of any alternative methods here.
I haven't yet read enough old textbooks to track the full history of the grouping method, but from what I've seen it has come and gone over the years, and sadly seems to have become detached from prerequisite skills along the way.
If teachers are teaching the grouping method to factorise non-monics and have not already taught pupils how to factorise expressions like p(a + b) + q(a + b) and ax - ay + bx - by then I think they may be trying to teach too many new skills in one go. It's certainly something to think about.
If you find this kind of thing interesting, come along to my workshop at MathsConf17. I start will by sharing some cool stuff I've seen in old maths textbooks, and then I will focus on quadratics. We won't look at factorising, but we will look at some methods for solving quadratic equations and how they've changed over the years. I really think that this kind of subject knowledge development helps to improve our classroom practice. I hope to see you there - and I will bring along some of my old textbooks in case you'd like to have a look for yourself.
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.