La Salle’s platform includes everything needed to teach, learn and assess mathematics from primary to A-Level, both in-school and remotely.
London, United Kingdom: La Salle has successfully completed a new round of funding of just under £1M. La Salle’s existing shareholders supported the round and were joined by new investors who see a substantial opportunity to address a new, direct-to-consumer market by utilising La Salle’s digital technologies.
La Salle’s existing product, Complete Mathematics, is already well used by schools in the UK and overseas, supporting teachers to plan and deliver highly impactful lessons on a platform underpinned by the world’s most comprehensive maths curriculum. Building on its existing technology, La Salle has now developed a new, complementary offering in response to the post-COVID demand from parents and families seeking high-quality, comprehensive maths tuition online. This new ‘digital tutor’ will bring about all the benefits of a real-life tutor but at a fraction of the cost to schools and families.
“I am delighted that current and new investors are supporting the business to launch our new ‘digital tutor’ product” says Mark McCourt, CEO at La Salle. “We know that around 25% of pupils in the UK already benefit from private tuition, but those benefits are not accessible to the rest of the population due to the high cost of a real life tutor. This new funding will support the rollout of our ‘digital tutor’, thereby levelling the playing field and bringing maths tuition to all families regardless of income level. It’s a hugely exciting initiative that will really help pupils from all backgrounds and from anywhere in the world be successful in learning mathematics.”
This new round of funding will be used to grow the customer base in the UK, where catch-up funding has been made available for schools to invest in high quality tuition aids and internationally, bringing universal access to high quality mathematics education and support.
In 2021, La Salle will:
Launch the world’s first fully comprehensive ‘digital tutor’ for mathematics, covering all areas of the curriculum for all age and ability levels.
Provide ‘summer school’ access for pupils free of charge, with the intent of ensuring low income families are equally well-equipped for a successful academic year.
Meet international demand for individually tailored maths tuition that removes the need for a conventional tutor - with hyper-personalised digital tuition delivered at a truly affordable price.
About La Salle:
La Salle Education was founded in 2013 to support teachers of mathematics. The community of teachers using its services has grown rapidly, with thousands of teachers from around the world regularly attending La Salle events and receiving professional development from the company. La Salle supports teachers effectively and efficiently through its online teaching, learning, assessment and monitoring platform, Complete Mathematics. In 2020, La Salle launched trials of its ‘digital tutor’ product, attracting an immediate user base. The company is now taking the next step in helping pupils by using its intelligent technologies and comprehensive content to offer online tuition at a genuinely affordable price.
Very significant challenges lie ahead for schools. With the closure of schools across the country, pupils are working hard at home with the incredible support and provision put in place by their teachers and schools in a remarkably short time under incredibly challenging circumstances. As has ever been the case, when real challenge presents itself, teachers rise to it and go to great lengths to ensure their pupils have the best possible chances.
Although schools continue to do an amazing job, there is the very real risk that, during these times of school closures, the gap between the most disadvantaged pupils in our society and those who are most advantaged will widen even further, with those families with the means putting in place private tuition to ensure continuity.
We know that approximately 25% of all pupils in the UK were already benefiting from private tuition beyond school, giving them significant advantage over those pupils for whom this was not possible. The average spend on mathematics tuition in the UK is around £1000 per year. This is out of reach of many families.
Now we are seeing the differences in opportunity become even more extreme. Away from school, it is difficult for teachers to intervene and lift up those most disadvantaged pupils in the way they routinely do when the pupils are on site.
For many pupils, particularly those in in the most disadvantaged circumstances, the coming months could represent significant lost opportunity.
When pupils return to school, teachers face an unprecedented challenge: to provide a schooling of such exceptional quality that all pupils are accelerated to (and hopefully beyond) a point in their learning as though no interruption to their education had happened. Put simply, teachers face the challenge of providing the most important academic year in generations.
This will be a tremendously difficult task and schools must turn their focus to it now.
It is understandably tempting for teachers to focus on the immediate issue of providing pupils with access to learning materials during the school closure. It is absolutely right that this happens, but we must not take our eyes off the bigger challenge of preparing for an exceptional year to come.
This will require a remarkable quality of curriculum planning, resourcing and monitoring. That is why my focus now is on supporting schools to ensure that staff are well trained and prepared, that the curriculum is coherent and of the highest quality, that resourcing is in place and tracking pupil progress is automated such that all teachers, when finally faced with the return of pupils, are able to focus 100% on pedagogic decisions and working intensively with individual pupils.
We are already the UK’s largest provider of mathematics teacher professional learning, with thousands of teachers in our network and the most extensive programme of CPD across the country. Now we are going even further to help teachers to boost their subject specific content knowledge and subject specific pedagogical knowledge.
Last week, we launched a comprehensive programme of online CPD sessions for maths teachers. We are also working intensively with our member schools to support them with detailed curriculum planning – in the coming year, the curriculum needs to achieve something amazing, so we are working with our members to ensure that, when schools return, everything is in place for a hugely successful year.
For new school members, our focus is on ensuring they are fully trained and equipped to make the most of our curriculum and platform.
The adoption and successful deployment of a serious educational technology requires rigorous, dedicated teacher development and planning. We do not throw technology at teachers and pupils for the sake of throwing technology at them – this always does more harm than good. The deployment of educational resources requires strategic planning and critical evaluation of approaches. Without this step, there is every chance that pupils will actually regress rather than improve. This is why no school is allowed to join Complete Mathematics without also agreeing to receive the appropriate training (at no cost, of course). We are interested in our work having real impact; this is far more important to us than trying to do a ‘land grab’ of a schools market at a time when schools are having all sort of opportunistic offers presented to them. Our work is carefully considered, strategic and sustained. Complete Mathematics schools and colleges are fully supported by our expert team at all times.
A recent report showed that for every £1 spend on education technology, just 4 pence is spent on the relevant CPD. This is why almost all ‘edtech’ fails. We all know that schools picking up and deploying products without taking the professional development needs of teachers seriously are simply contributing to making things worse. Because we are a well-established organisation with expert mathematics education staff, we are able to ensure our members are fully supported to deliver impactful mathematics lessons and increased pupil outcomes.
We do things differently. We take the longer view.
We are supporting schools and colleges to use the time now unexpectedly made available to them to thoroughly prepare for the most successful school year ever. This means helping current and new Complete Mathematics members direct their efforts into the forthcoming academic year. Of course our members are using Complete Mathematics to help their immediate work, with some pupils learning at home, but we are also determined to significantly strengthen the planning and preparation for accelerating the learning of all pupils once they return in the new academic year.
With teachers’ energy going in to preparing for the most important academic year in generations, we are also going further in supporting pupils and taking more and more workload away from teachers.
All Complete Mathematics pupils have a login for our extensive platform, where they can follow lessons set by their teachers or engage with independent learning. But with the risk that pupils in the most disadvantaged circumstances will not be able to access the same additional tuition support that their more advantaged peers can, we are now putting in place a new form of provision: private tuition for all.
Our expert mathematics team has devised and planned a series of ‘Preparing for success…’ courses. These courses are available to Complete Mathematics pupils for free. Each will be a series formally taught sessions forming a single course. The lessons will be delivered by expert, qualified teachers. For those pupils who are unable to attend a session or who just want to revise further, the recordings of the courses will be available in the Complete Mathematics platform for all to view at a time that suits.
Complete Mathematics subscription is just £950 for a school or college, giving all teachers and pupils full access to the most comprehensive online learning platform for mathematics. We now also offer a full course of CPD for teachers and expert pupil tuition for key courses.
And we want to go even further still. We recognise that there are many families who would like to access the benefits of these online courses, so we are making them available to non-members too. This is an ideal use of pupil premium or PEF funding for those schools that wish to enrol specific pupils.
We refuse to create provision that is beyond the reach of families. So, rather than the typical £20-50 per session fee that parents are often asked to pay for online tuition, we are making each full course available for just £30. That’s over 30 sessions, spread across the next couple of months, for just £30. This super low-cost tuition is designed to be accessible to all at under £1 per lesson.
In the coming weeks and months, we will work with our schools and colleges to:
Provide a comprehensive programme of online CPD for teachers
Provide extensive support and training on how to use our platform effectively and with impact on pupil outcomes
Fanatically support you in preparing for the new academic year with planning, resourcing, assessing and monitoring help
Ensure that all pupils who are now working from home can access online learning – both that set by their teachers and automated independent learning materials
Ensure the most disadvantaged pupils are not left behind and can access free or super low-cost private tuition
The Covid-19 virus has disrupted all of our lives. Our job now is to ensure that all pupils, regardless of background, can return to the most impactful and amazing academic year ever.
Mastery is a commonplace word now in mathematics education, and social media is awash with 'mastery lessons', 'mastery resources', and 'mastery curricula' - is this mastery? What do we mean by mastery? Is it a teaching style? Is it a curriculum design method? Is it an intervention strategy? The answer is, always, firmly, no!
When Benjamin Bloom and John B Carroll were squirrelled away codifying what Carleton Washburne (and others) had mapped out, they certainly did not intend for mastery to be distilled down into lessons, pedagogy or even how a curriculum should be written. Mastery in its purest sense is a way of schooling, a way of ensuring every child can succeed given the right conditions.
Using Carrol’s model of school learning we can formulate the degree of learning into 5 areas: perseverance, opportunity to learn, learning rate, quality of instruction and ability to understand. However, we strive to ensure that the function is equal to one; where an equal amount of attention is dedicated to each of these key principles. In this blog, we shall focus on ‘quality of instruction’ and all that is encompassed by that. It must be noted at this point that learning rate is often what we define as ‘ability’ and it must be clear that ability is only a measure of learning rate. For example, we can have pupils who are ‘low ability’ but in the same regard, ‘high attainment’, i.e. their pace of learning is fairly deliberate but can, and will achieve well - I very much put myself in this category!
So when I refer to ‘mastery’, I mean Bloom’s mastery. Mark McCourt describes teaching for mastery in his latest book, ‘Teaching for mastery’ and Chris McGrane has outlined how utilising the Complete Maths platform to teach for mastery and I plan to complement this with my experience of implementing mastery in a comprehensive secondary school.
Mastery hinges on responsive teaching and not only after summative assessments, but in the moment responsive teaching. Interventions must be made as soon as they are needed and the goal of mastery is to scale the one-to-one tutoring model of teaching to one-to-many. To explain my take on implementing mastery, I’ll exemplify what a learning episode might look like and how we make the mastery cycle work
In this episode the phasing model of learning is used, i.e. teach, do, practise and behave. Mark McCourt provides a sensible proportioning of content as follows:
Teaching and doing are blended with the opportunity to do purposeful practice also. The final phase of ‘behaving’ is somewhat the most challenging, but most important - this is what develops the mathematician and proves the understanding is deep and connected. I plan to explore this phasing model in more detail in upcoming conference presentations or at our public CPD events. Furthermore, to keep this blog somewhat succinct, I have omitted some of the detail but hopefully left enough for it to be comprehensive.
The learning episode is on directed number arithmetic. The basis of learning directed number arithmetic is best modelled using algebra tiles (usually with a visualiser) and allowing pupils to use the concrete materials to help conceptualise the ideas. I believe that at the early acquisition stage of learning (when knowledge is inflexible), example-problem pairs are a powerful tool and a plethora of research on this support their utility.
Above is one example-problem pair using algebra tiles. It must be noted at this point, that much of this is usually done with pen, paper and algebra tiles under the visualiser. Prior to this, a lengthy introduction to the idea of ‘zero pairs’ has taken place and we have explored this profound idea with pupils. Building on this example, the class and I would explore lots of different addition calculations and encouraging the pupils at every opportunity to create their own questions. There are two parts to self-generation: one, it alerts me immediately to lack of understanding for those who cannot do it and two, provides an insight to the depth of understanding at this point - this is far superior to just providing or asking questions in my opinion. You will see in the example above I like to include ‘Make another with the same answer’ box pop up when pupils are working on the problem question; this allows pupils to maximise their ‘up-time’ in lessons and creates space in the lesson to allow me to get between the desks. Once happy with addition then next comes the dreaded subtraction! Not anymore! We strive to teach children proper mathematics and in particular, proper arithmetic. No silly rules or sayings or tables to spot what signs they have in comparison to the magnitude of the number - just simple arithmetic.
In the example above we use the additive inverse property as a basis for what we usually refer to as subtraction; pupils love the conversation I have at this point to hook them in about ‘take-away’ and subtraction not being a ‘real thing’. We are proper mathematicians now and we only work with two operations: addition and multiplication. Again, like with addition, example-problem pairs are utilised and ‘make me another’ prompts but before I move to more demanding questions I like to check for understanding or more replication at this point and offer something like this:
A fairly standard set of questions, but maybe not as many as you would have hoped for? Before we delve into why that is, take note of the ‘think’ and ‘do’ prompts: here I offer some undoing style questions and a type of non-example to provide opportunities for pupils to turn inflexible knowledge into more flexible, usable knowledge. Furthermore, maximising ‘up-time’ in lessons is crucial and this ‘buys’ me time as a teacher to meet the needs of everyone in the room. Returning to my first point, the need for endless exercises on a specific skill is simply not required. Multiple studies by Rohrer and Taylor have shown that for over-learning to occur, pupils only need to answer two questions correctly and the skill is over-learned. However, it is noted that educators should probably err on the side of caution and offer more than two. In a simple experiment between two groups, where one group answered 3 questions and the other 9, the differences in far transfer are negligible and even more worrying, the drop in accuracy after only one week (both groups were observed to have a mean of 94% accuracy at the initial teaching stage) is mind-blowing:
Do we waste too much time doing repetitive questions? How can we properly practice mathematics?
Would a task like this help pupils practise directed number arithmetic and also draw upon reasoning skills and ultimately aid the transition from inflexible knowledge to flexible knowledge? Based on my experience, this does! With the practice phase, we also want to interleave previously taught ideas to take advantage of retrieval and method selection. Again, Rohrer and Taylor provide us with another study indicating the benefits to far transfer in ‘shuffling questions’ when practising mathematics.
We can see those who block practice (i.e. practise what has been taught explicitly) perform well initially but over time do not learn as well. Those who work on a mix of questions from previously taught content, perform far better over time. Once again in this study, the questions were very procedural and we can see accuracy drops significantly over time - this begs the question: what else can we do? Although I must note at this point, interleaving in its truest sense, is something as Maths teachers we already know and have always done but maybe we need to ensure opportunities to do so are embedded in our curriculum scheduling.
In my school, we utilised the custom diagnostic quizzes you can construct in Complete Maths for this very purpose. We creating a quiz, worksheet or task that is interleaved we need to consider what has been taught before and look for opportunities to call upon method selection, i.e. questions that might appear to look the same but have very different structures underneath. Also, we can include questions where on the surface they look nothing alike but when you drill down they are very similar. Interleaving is not using something like perimeter as a vehicle and changing the sides to decimals, fractions or algebraic terms - interweaving is a better description for this type of intention.
Moving to the behave phase of learning we want to draw upon previously learnt ideas and build connections to the current idea. We need to take care that when the problem-solving demand is high, the level of the mathematics needs to be fairly trivial (dependant on the pupils whom you have in front of you of course) to provide easy access but a high ceiling of mathematical opportunity. We need to consider maturation and it is suggested that 2 years is around the typical period, however in this case of directed number arithmetic a suitable behaving task is the classic always, sometimes and never true type of task. Pupils are handed statements about negative numbers, e.g.
“Two negatives make a positive”.
“A positive number is more than a negative number”
“Adding a positive number to a negative number will make a positive answer”
They must put each statement into the always, sometimes, and never true category, but also provide evidence of their decision. Using only prose statements pulls the maths out of the pupils' heads and getting between the desks to challenge them on their decisions (or sometimes settle debates) is a fruitful way in which to gather intelligence on your pupils’ learning.
However, we always need more when it comes to information on learning and in my opinion, building in as many different opportunities as possible to do so helps teachers make well-informed decisions and become more responsive. Back the mastery cycle! Our curriculum was designed in the following way:
Each block has three parts: number, algebra and another strand. Number and algebra are hugely important strands of mathematics and having opportunities to generalise was important to me when sequencing topics. Each block required mastery to be met or future topics would provide problematic. Teachers have complete autonomy to work across the block left to right, right to left or to be honest whatever is best for the pupils in front of them. Professional autonomy is the foundation upon the re-design of our curriculum and I was keen to foster the ethos of doing what is best for the pupils. We grouped the pupils by prior attainment, although not solely based on primary information and we ran diagnostic quizzes to ensure we had accurate data to determine where each child should start on their journey to learning mathematics at secondary school. In the interests of brevity, I go into much more detail on this in my conference presentations, but I must stress that the groupings were very flexible and I would regularly move pupils from class to class based on the judgement of the teacher. I used a model whereby once grouped, the teacher would keep this class for a minimum of four years. Relationships are critical in schools and even more so in secondary; short 50 minute periods are a significant change for pupils and in some cases, we don't see them each day.
Circling back to assessment: we offered diagnostic quizzes after each strand and a summative assessment after the entire block where we would look at quantitative data to check for mastery. Strand assessments were invariably multiple choice and the block assessment would more of an extended response. I had to consider workload of marking and the workload of pupils, continually doing assessments, however, teachers run experiments and gather data every period of maths, hence their judgement was of paramount importance to making the mastery cycle work.
Correctives are integral to making mastery work and often it is the barrier most schools face when attempting a global implementation. Something I considered, which is often never done in education was scenario-based practice - for the teachers! I created a scenario and we discussed how logistically we could make it work:
Teachers were faced with an example breakdown of quiz scores and then I indicated something that might be controversial but a very real problem: the concentration levels (and at times behaviour) during the five periods this particular year group attended maths:
I offered the following paragraph:
Think carefully about how you can correct learning before progressing. The topic of integers is an integral foundation upon which we build, hence careful thought and planning must be prioritised. We aim for pupils to sit the block assessment at a state of readiness, i.e. we are fairly certain they should all demonstrate mastery in the essential skills section. The earlier we can make interventions (or correctives) is of paramount importance to a successful implementation of the mastery cycle.
What is your course of action?
Scenario-based practice is a powerful tool and used frequently in other walks of like, for example, special operators in the military often ‘rehearse’ the scenario they face to ‘iron out’ the kinks in their procedures and delivery - why not in maths teaching?
So how can we manage correctives and what did we, as a department, collectively decide as to the best course of action? Let’s take the following example of quiz data:
Some pupils have not grasped some of the key ideas in each of the three strands: number, algebra and integers and in my conference presentations, I go into more detail on what these assessments look like. Based on the data, it is clear that we need to run correctives to ensure every pupil is ready for the next block of work; remember, we do not want pupils beginning collecting like terms if they cannot work with integers. I am holding myself accountable and building this culture in the room. Often I would explain to pupils that clearly my lessons had not been impactful and I need to work with specific groups to ensure I correct this - I want you all to be successful and let’s fix this together. How can we organise this?
Above exemplifies two models of how correctives can operate in a ‘real’ classroom. Model one offers those who are secure with integers are offered enrichment of the topics contained within the numbers strand and allows the teacher to interview and work closely with those pupils who have shown remediation is required. Model two offers those who are secure on all areas, enrichment of all three strands and the remaining pupils who need remediation across the block are grouped to allow the teacher to design a plan for them. It must be noted that interventions in the moment, throughout the teaching of this block of work, are critical to making this model of correctives manageable - if you wait until the end of the block before diagnosing problems I would argue that corrective teaching will be very much impossible and probably too late.
I hope that this blog makes some sense but as you can gather, just like teaching and learning, mastery is a complex process and distilling it into a blog is not easy! If you would like to know the finer details of how all of this worked for us at St Andrew’s then look out for my presentations at the PT Conference on Friday 13 March and also the next MathsConf (22) in Manchester on Saturday 14 March.
We regularly run our hugely popular Mastery in Mathematics CPD course across the UK. Discover upcoming dates and book your tickets here.
When entering Raw Data: Option to show the data as a Dynamic Dot Plot
Dynamic points in a dataset can be moved around holding CTRL and drag or HOLD and drag
Points can also be added in Point Mode (max 1000 points). Such points when selected (using marquee select) can add to an existing dataset or can create a new dataset. Subsequent dependent objects will adjust accordingly
A listed Raw Data set can be deselected to allow a new set of data points to be added.
Individually selected dynamic data point: coordinates given in status bar.
Box and whisker plots
Outliers: Data points further out than 1.5 x IQR are shown as potential outliners. The Box Plot extreme verticals are the lowest and highest data that can be considered not to be outliers.
Can plot multiple box plots. Must be entered one at a time, but individually labelled. Autoscaling optimised, and each on has a LABEL based on the individual data column header. Label can be edited using the right-click option with the parent box plot selected.
New Statistics Object: Q-Q Normal Plot
A test for normality used in Core Maths on a raw data set. So with a raw data set entered, right click option “Q-Q Normal Plot”:
Data is first sorted in ascending order and numbered 1 to n
The mean and SD of the dataset are calculated
The dataset is formed of three columns: Data, Number i (i... n), Quantile Normal
Normal Quantiles are plotted against the data on equal scales. The line y = x is drawn dotted
Select the Q-Q plot and right click > 'Table of Statistics' to show 3 columns in the Results Box.
Minor fix to the Statistics page: Normal areas plotted nicely
Objects can now be arranged in ORDER
Can add image
Variance now uses /(n-1) form throughout
Vertical scaling and its label are determined by the first object plotted
Mastery learning is a well-defined approach to schooling. It originated in the work of Carelton Washburne and was later developed by John B. Carroll and Benjamin Bloom. Mastery is a model of schooling which has, at its heart the belief that every pupil can learn the school curriculum.
Thomas Guskey further developed the work of Washburne, Carroll and Bloom and codified the core elements of mastery learning:
Diagnostic Pre-Assessment with Pre-Teaching
High-Quality, Group-Based Initial Instruction
Progress Monitoring Through Regular Formative Assessments
High-Quality Corrective Instruction
Second, Parallel Formative Assessments
Enrichment or Extension Activities
Mark McCourt, the UK’s leading expert on mastery learning and CEO of La Salle education has gathered these core elements into the following mastery learning cycle.
Implementing a Mastery Curriculum
Moving to a mastery curriculum is not a trivial process. Succeeding with implementing a mastery curriculum depends upon the following:
A rigorously designed curriculum – this is essential such that the development of the key skills, ideas and relationships are coherently planned for.
A well-resourced curriculum – with quality resources, teaching notes and assessment tools.
Expert teachers – teachers who have the understanding of pedagogy, didactics and reactive assessment processes.
Complete Mathematics subscriptions can support the implementation of a mastery curriculum in your school by addressing all three of these concerns.
A rigorously designed curriculum
Our curriculum is not based upon that of any one nation. Instead it has been developed with the best of mathematics education research and professional knowledge in mind. From the early learning of number through to advanced level ideas such as calculus, the entire curriculum has been planned out. It is broken down into units, each of which is made up of various key mathematical ideas. Each idea contains 'granules', with each granule representing one step towards the mathematical idea that is being learnt.
There are 204 units, containing 320 big mathematical ideas. These ideas appear many times in different units over the years as the curriculum deliberately spirals and pupil understanding strengthens. Each idea is scaffolded carefully as a journey to building the idea; these carefully sequenced steps gradually build up each idea. There are over 1600 steps in Complete Mathematics.
A well-resourced curriculum
The Complete Mathematics platform contains all of the resources you need to plan and implement a mastery curriculum in your school. It includes tools for planning, assessing and reporting. Every granule has a full set of supporting materials and teaching aides for use when planning as well as content to use with your pupils.
Subscription to our Teacher CPD College entitles each teacher to unlimited access to our CPD courses, accessible anytime, anywhere. These high quality CPD courses develop teachers practical and theoretical understanding of how to teach mathematics effectively – be it in the primary, secondary of post-16 sector. Teachers who subscribe to the CPD College have exposure to a vast array of professional learning.
Using Complete Mathematics to Implement the Mastery Learning Cycle
For every single granular step of the Complete Mathematics Curriculum there is a wealth of material to support your teaching.
You can view the pre-requisite knowledge for this granule and view the subsequent ideas which depend upon fluency and understanding with this idea. There are extensive pedagogical notes and lists of common misconceptions for every granule. These notes are drawn from both the literature on effective teaching of this idea and from the experience of the expert team at Complete Mathematics.
Importantly there are example questions for each of the granules. These are split into typical, probing or hard question types. These indicate the sorts of questions that pupils, who have understanding of the idea should be able to answer correctly.
For each of the granules there is a resources and tutorials section. This includes resources associated to this granule which have either been created by the team at Complete Mathematics or uploaded by our members. These resources can help to simplify the planning process for teaching each granule. The tutorials section contains links to videos which explain the ideas and skills related to each granule.
Also, for each granule there is a summary of 'Key Learning Points', 'I Can...' statements for pupil self-evaluation and a breakdown of key vocabulary. It is important that pupils are exposed to the correct terminology and can become fluent in their description of mathematics.
With this array of supporting materials available at every step within a unit, a teacher can spend less time searching for guidance or content and more time perfecting their methods of teaching this piece of maths. If corrective teaching is needed, a teacher can dip back in the curriculum via the listed pre-requisites and find the right support and content immediately. Additionally, with the more advanced materials found in the example questions and resources sections of a granule, there is great scope for topic enrichment and enhancement activities.
At the start of a new episode of learning we need to ensure that we are teaching pupils the correct mathematics. We know that pupils must be secure in the prerequisite ideas or else subsequent learning will not be possible. There is no such thing as a weaker pupil, only a pupil to whom we are teaching the wrong mathematics.
The assessment creation tool on the platform generates a pre-teaching diagnostic for the current topic. Pupils can do this on their own devices or you can generate a PDF and print this out.
The platform will generate a question by question analysis for this assessment to let you plan next steps for your class. For those pupils who need some re-teaching of this prerequisite work you can view the granules on the platform to plan for this and to find appropriate tasks and exemplar questions etc. For pupils who have demonstrated fluency the hard questions on the prerequisite granule can be used to offer challenge and further depth.
Similarly, as your class are working through the curriculum it is possible to generate quizzes, to formatively assess their learning. A powerful assessment tool which is built into the platform allows us to assess what pupils have been working on in class over a specific time period. This means we can think about retention and the long-term durability of the learning, rather than just instantaneous performance at the time of teaching. This also allows us to utilise key aspects of cognitive science such as the testing effect and interleaving effect to enhance learning.
Continuing round the Mastery Learning Cycle — following the completion of our teaching for each part of a particular Unit and having identified through our regular formative quizzes that no-one in the class requires further remediation, we use the Custom Diagnostic Test creator on the CM platform to create a Summative test on the whole of the Unit.
If the assessment markbook identifies further required remediation, we proceed to some corrective teaching, once more using the available support materials, resources and pedagogical advice found in each granular objective. If however everyone in the class scores appropriately on the summative assessment, then we move forwards onto the planning of the next Unit, and restarting the cycle, with the knowledge that we a building on strong foundations of understanding across the class.
The Complete Mathematics platform continues to evolve to help teachers implement a mastery approach. If you are interested in learning more about the work we're doing, or how you and your institution could benefit from Complete Mathematics subscription then do please get in touch. Our school support team would be happy to arrange a complimentary visit, webinar or phone call to discuss this — book your free demo.
Part of our Teacher CPD College is our hugely popular Mastery Learning course. This is also available as a stand-alone course, and you can find out more details here.
Just over a month ago we released a blog describing our work on a dynamic worksheet generation tool. This was our first experiment in dynamically generated content. Today we announce our next experiment, dynamically generated example questions. Below are the first few questions we have created, available to all for a short time.
Try it for yourself - simply click the DYNAMIC button to create a new version of the question. Worked solutions can be viewed by opening the section beneath the question. The questions can also be viewed in full screen by clicking the expand icon.
Complete Mathematics members will be familiar with the example questions that appear on our online platform, but for non-members here is a summary of what they are, and what they are for.
Example questions, along with a wealth of other support materials, are available to a teacher once they have chosen the particular objective(s) for their lesson from the Complete Mathematics curriculum. The example questions come in three varieties: ‘Typical’, ‘Hard’ and ‘Probing’. ‘Typical’ questions are designed to offer some guidance to the teacher on the sort of questions their pupils should be expected to answer. ‘Hard’ questions build on the ‘Typical’ questions and provide more of a challenge to the pupils. Finally ‘Probing’ questions provide the teacher with questions that may challenge their pupil’s preconceptions and are intended to broaden the pupil’s knowledge.
We have found that, as well as using these questions for guidance and to inform their planning, teachers are also using them in front of the class as a teacher led activity. It is for this use case that making these example questions dynamic is an especially attractive proposition. Teachers will be able to walk through a question with the class, with help from the worked solution, then click ‘Regenerate’, and immediately have another version for the pupils to attempt on their own.
Further to this, example questions are also used on the pupil side of Complete Mathematics after the completion of a quiz. Pupils are provided with a page to analyse their performance, as part of this process they are shown similar questions to the quiz questions they have answered (particularly useful for incorrectly answered questions). This provides another brilliant opportunity to put dynamic example questions to use, with pupils able to generate as many further practice questions (and worked solutions) as they want. This gives pupils an even greater opportunity to independently fill the gaps in their learning.
The goal of these experiments on dynamically generated content is to lay the groundwork for making Complete Mathematics an even closer mapping of the mastery cycle. If we can use computing power to make the loops of the cycle more efficient it frees up teachers to spend more time planning and teaching brilliant, insightful lessons. But it is no good when creating a quiz or worksheet if the generation of questions is not intelligent or when generating example questions we don’t also provide responsive worked solutions. Our goal is to produce content that is as good as the teacher would have produced if they had the time (or pretty close to it!).
So what’s next? As well as continuing to populate Complete Mathematics with more dynamic example questions and more dynamic worksheets we will be adding new functionality to allow users to input (and check) their answers. This will make these questions much more powerful, whether they are used in a worksheet for homework, at the front of the class, or in a quiz.
If you are not a Complete Mathematics member, and you want to find out more, you can sign up for a free school visit or online demonstration here.
Today we launch a new feature to be added to Complete Mathematics: our Dynamic Worksheet Generation tool. This initial launch is focused on the creation of worksheets for the factorisation of quadratics. We have created a tool which captures some of the ‘teacher logic’ inherent in worksheet design and have incorporated intelligent variation and sequencing in the questions generated.
The worksheet generator can produce worksheets with questions which come in a sensible teaching order for this topic. Intelligent variation sequences, are periodically included at appropriate places in the worksheets. These sequences of questions help pupils to attend to the key variants and invariants. One such example is shown:
x2 + 14x + 24 = (x + 12)(x + 2)
x2 + 11x + 24 = (x + 8)(x + 3)
x2 – 11x + 24 = (x – 8)(x – 3)
x2 – 14x + 24 = (x – 2)(x – 12)
x2 + 10x – 24 = (x – 2)(x + 12)
x2 – 10x – 24 = (x + 2)(x – 12)
x2 + 5x – 24 = (x – 3)(x + 8)
x2 – 5x – 24 = (x + 3)(x – 8)
Teachers have the freedom to toggle on and off which variations of quadratics to include on the worksheet. This is particularly useful for the implementation of the mastery cycle. Based upon formative assessment a teacher can identify where individual pupils need to focus their attention. The teacher can then create a worksheet specifically for this pupil in moments for them to remediate and practise on these sub-skills.
At La Salle we talk about a learning episode as opposed to lessons. Learning and teaching can’t be neatly contained in sixty-minute slots, instead a learning episode spans a series of lessons and may include a number of related learning objectives. We consider a learning episode in the context of the following phases:
During the Teach phase, the idea is entirely novel to pupils. It is just beyond their current knowledge and understanding. The teacher will instruct the pupils, tell them key facts, pass on knowledge, show and describe, use metaphor and model, all in order to bring about connections in the pupil’s current schema so that they can ‘meaning make’. This phase is often described as explicit teaching. It is a crucial phase – after all, the teacher knows things and the pupil does not; so tell them!
The end of the 'Teach' phase does not result in learning. It is merely the first step. At this stage the new knowledge is ‘inflexible’, and it is our job as teachers to bring meaning and understanding to the knowledge so that it becomes ‘flexible’ (more on inflexible and flexible knowledge later).
We now ask pupils to Do. At this stage, they do not yet know or understand the new idea, they are replicating what the teacher has told or shown them. The 'Do' phase has two important purposes. Firstly, the teacher is able to observe whether or not the pupils have made meaning of the model, example, metaphor or information they have been given or shown. The teacher can see and act; are the pupils able to replicate what I have demonstrated? If not, the teacher can change their model, example or explanation, perhaps making stronger and more explicit connections to previous knowledge and understanding. The second reason for the 'Do' phase is to give pupils a sense that the idea or task is surmountable – that they, quite literally, can do what they are being asked. Well structured 'Teach' and 'Do' builds pupils’ confidence and shows them there is nothing to be afraid of, the new idea is within their reach.
The Complete Mathematics platform supports teachers in delivering the 'Teach' and 'Do' phases by including pedagogical notes, typical misconceptions, key vocab, and example questions. Additionally, on our CPD days we explore evidence-based techniques to enhance the effectiveness of these phases.
Once both teacher and pupil are clear that the pupil is able to ‘Do’ – that is to say, they can perform – the teacher now segues the pupil to the Practise phase
During 'Practise', we wish to move beyond simply performing. We want the pupil to gain a confidence in working with the new idea, to see its underlying relationships and to assimilate the new idea into their schema of knowledge. In order to achieve these more meaningful goals, the pupil needs to be able to attend to a higher level. In other words, as described earlier, the pupil needs to have achieved fluency at the performing level first, so that they may attend to connections, relationships and a deeper conceptual appreciation.
So, we shall define the point at which the pupil moves from 'Do' to 'Practise' as the point at which they achieve fluency. This is where the dynamically generated worsksheets can be particularly useful. They give pupils ample opportunity to build procedural fluency but also develop conceptual understanding through the built-in intelligent variation as described above.
The final phase, Behave, is the most important phase. This is the phase that brings about understanding. At this stage, teachers create opportunities for pupils to behave mathematically. Complete Mathematics has tasks which support this phase, however, the pedagogical actions are perhaps the most important factor in this phase. You can learn more about this on our CPD days: Deeper Understanding in Mathematics, Embedding Problem Solving in the Mathematics Classroom, Designing Mathematical Tasks and Curriculum Design.
This worksheet generator is our first experiment in this area and we would love to hear any feedback you have either by commenting below or talking to us on twitter. There are a multitude of other ideas we want to explore. For instance, allowing teachers to implement their own variation logic in these worksheets and allowing pupils to answer the questions online. Also, in addition to these further experiments on dynamic worksheets, we are working on dynamic versions of the example questions found on every objective in the Complete Mathematics platform.
Complete Mathematics members will be able to see all these developments and more appearing on the platform in the coming weeks and months. In fact members can use this worksheet generator right now in the 'Factorising Quadratic Expressions' objective in unit 10.5. If you don't want to miss out on these developments, or you want to take advantage of free attendence to the CPD courses mentioned above, you can find out more about becoming a Complete Mathematics member here.
I’ve used Cuisenaire occasionally in my career. Much of the time it was as an aide in the teaching of fractions to younger secondary pupils. However, this fabulous resource has so much more potential. It can be used to introduce the very basics of arithmetic such as additive relationships, or extended into harder topics such as simultaneous equations, Pythagoras and equation of a straight line.
Allow me to share a reflection of “a learning episode”.
This evening my five-year-old son, who is as inquisitive as children of that age tend to be, spotted a small bag of Cuisenaire rods on my desk. He was immediately drawn to them. “What are those daddy? Can I see them?” The verb “to see”, for a five-year-old is not just an interaction of the eyes and brain. It is a tactile action, it involves touching the object and interacting with it in some way.
He poured all of the blocks over the table, gazing in my direction to ensure that this was OK. Immediately, he began to play with them. He built little patterns and began to group the rods. There is something about these little rods that is inherently enticing.
Mark McCourt had told me that young children will begin to behave mathematically with these blocks, given enough opportunity to play with them. I was stunned, when, after just a few minutes, my son said “Maybe after this I could do it by sizes”. The level of categorising went beyond the first level I’d expected him to consider; colour. Instead it was a mathematical idea. I let him play with them for a while. I was minding my own business, leaving him to it and not prompting him in any way.
All of a sudden, a loud announcement, coloured with the excitement and joy of a profound revelation: “Its colour is its size!” In that moment, these little rods had gone from being toy blocks to being something else. It’s impossible to make inferences about the connections he was making. However, what was to follow demonstrates, to me, that he was thinking hard.
“Orange is the biggest one!”
I’d resisted the urge to prompt or direct him until now, but I couldn’t help myself, I wanted to play too. Displaying a little bit of shock for his benefit I asked him “Is it really bigger than the blue?”
He was, correctly, adamant that it was. Having his conjecture challenged, he did what any mathematician would do – he sought out a proof! Carefully lining up the blue and orange he showed me that there was a gap. “Look – you can put a white one there”.
He’d just modelled a number bond to ten. While he can already “do” addition he hadn’t yet recognised that the calculations he does at school were synonymous with his demonstration with these little rods. I think that will come in time – after all, the pace of progress in his use of the rods is startlingly fast.
He continued to play freely with the rods. He made some domino trails. This is the beauty of this manipulative – there is fun to be had with it! A short while later I saw him looking at the purple and dark green. “This is four more taller than purple”. I was perplexed with this idea of four, as the green is only two blocks more than the purple. I chose not to judge, but instead try to understand his interpretation of the situation. I asked him to show me why.
He motioned with his finger four equal steps from the end of the green to the end of the purple. I suggest that there were two possible thought patterns here: the first is that there was some unit of measurement, known only to him, which was his point of reference. Alternatively, he hadn’t quite grasped the relative size of the white block to the others.
Maybe in asking why, I challenged him in a way that made him reconsider things. He presented me, absolutely delighted with himself, the following set up:
The mathematics is simply pouring out of this free play. These are exactly the sort of comparative models I watched Mark McCourt share with teachers yesterday!
The free play continued with “now I want to count them all”. This was going really well. He had counted past 50 when, all of a sudden, his twin sister appeared. He continued to count but her presence (she was asking me about the rods) put him off a little. He said he thought he’d counted properly, but wanted me to double check. His sister volunteered – she was keen to get involved too. Midway through counting I heard her brother say to her “you’ve missed out all of the fifties and sixties”. He had been listening intently. They decided to count them again together, this timing getting the correct total. I didn’t check the total for them. They have the knowledge between them to be sure of succeeding.
They began to discuss the orange rod. He told her how it was the biggest one. She replied, clearly insulted that he thought she hadn’t realised this “I know! Look – it’s two yellows”. She lined up the rods to show him her thinking. I hoped they’d follow this line of inquiry further, so offered a suggestion “how many white ones to get the orange?”. The guesses were wildly inaccurate. One thousand is the phrase they like to use for “lots of something”, so this was the figure they last mentioned. They each made their own models, slowly and deliberately placed the whites against the orange. This was a real test for their fine motor skills.
“The big one is the same as ten.” I noticed that neither of them said “ten whites”. Could it be that they had stumbled upon the standard numerical values of the rods? I was about to offer another prompt when my son asked me for a pencil, so he could measure it. They have done a little bit of measuring in school recently. Did the number ten resonate with him in some way as to remind him of this?
Before long the pencil was cast aside and a box was to be measured. This looks like a potentially intuitive introduction to the idea of perimeter. Yet more rich mathematical activity.
All of the above happened in less than 30 minutes. With no direct instruction from me a whole wealth of possible starting points for further exploration have been encountered. Cuisenaire is an incredibly powerful and versatile manipulative. The extent of how it can be used to support learning and teaching is vast. You can learn more about this by coming along to one of our Concrete, Pictorial, Abstract and Language CPD days.
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.
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.