Mathematical Maturation

Written by Mark McCourt Thursday, 18 June 2020

As a Complete Mathematics subscriber, you will be familiar with seeing several different metrics for communicating pupil attainment within the platform, whether it be our own MathsAge, GCSE grades or National 5.

But what lies behind these metrics and how are they assigned?

At La Salle, we are interested in the journey that a pupil takes in learning mathematics from counting through to calculus. As pupils learn more and their schema of knowledge develops, they become more and more ‘mathematically mature’.

Taking a view of the curriculum in terms of mathematical maturation is incredibly important if we are to provide pupils with a truly meaningful, interconnected view of mathematics.

As pupils mature mathematically, they move through phases, or levels, of typical dispositions, behaviours, knowledge and understanding.

Pupils are growing mathematically. And hopefully heading to becoming young mathematicians themselves, who will be inspired to continue to study, use and love mathematics well beyond leaving school.

Behind every element of Complete Mathematics is a sense of this ‘mathematical maturation’, which we communicate by considering the cognitive Demand Criteria Level (DCL).

We thought you might like to know a little bit more about DCL and to see the descriptions of the levels that drive our metrics.

The Complete Mathematics DCL range from Level 0 to Level 22.

Here is every DCL in detail, followed by a discussion of how these levels map to attainment grades in the platform.

Complete Mathematics – Demand Criteria

The demand criteria are broad descriptions of mathematical maturation from the point of no mathematical education through to becoming a mathematician. It is not age related. It is not intended to be treated as a strict ladder of progression. Rather, the levels within the demand criteria aim to give a general sense of the capabilities and dispositions of a person learning mathematics as they reach stages of mathematical maturity.

Level 0

A pupil operating at DCL0 is assumed to have biologically primary mathematical senses, including cardinal and ordinal numerosity up to three. Pupils can distinguish between simple objects arranged in order of size and colour.

Level 1

A pupil operating at DCL1 is beginning to learn about mathematics beyond the intuitive and biologically primary mathematics that they have encountered and have a sense of from early childhood. The mathematics at DCL1 requires explicit teaching, particularly to move beyond natural numerosity. Pupils will develop number sense beyond first, second, third and one, two, three. They can order, compare and perform arithmetic within 20, though may require the support of concrete materials to do so. Pupils can count forwards and backwards within 100. They have an emerging sense of fractionness through a simplistic understanding of ‘half’ and ‘quarter’. Pupils can make simple statements about the relative position of an object and can name simple 2D shapes. They have an emerging sense of length, height, weight and capacity and can order objects based on the properties where the values are simple. They are beginning to tell the time and handle money with simple denominations.

Level 2

A pupil operating at DCL2 is developing key foundational knowledge in mathematics. They can order and compare within 100 using appropriate symbolism and count forwards and backwards in steps of 1, 2, 3, 5 and 10. Pupils are beginning to realise that not all calculations should give exact responses, rather that it is sometimes more appropriate to provide and estimate. They can work with arithmetic within 100 and are particularly confident when the calculations are in the context of money. Pupils are beginning to appreciate an inverse relationship between addition and subtraction / multiplication and division, as well as some simple constraints that apply to the operators. Pupils are developing a stronger sense of fractionness through imagery and objects. When describing simple shapes, pupils can make statements about symmetry and other basic properties. Pupils can gather information about length, weight, capacity, mass and temperature by using appropriate apparatus with simple scales. Pupils can represent information in a small number of very simple formats, including tables, tally charts, bar charts and pictograms, where both the values and scales are straightforward and discrete.

Level 3

A pupil operating at DCL3 is formalising key foundational techniques. They can work with addition, subtraction, multiplication and division with 3-digit numerals in a variety of problems, checking their answers where necessary and using quick mental recall of multiplication facts related to 1, 2, 3, 5, 4, 8 and 10 times tables. They understand the place value of digits within numerals to 1000. Pupils can identify, represent and solve simple problems with non-unit fractions where the denominator is small. They appreciate the meaning of the denominator as indicating how many equal parts a quantity or object is being split into. Pupils use appropriate units when working with money, length, weight, capacity and mass. They can describe properties of simple shapes, even when the orientation is changed, including perimeter and basic geometrical properties relating to angles, including identifying perpendicular and parallel sides. They accurately use the language of acute, obtuse and right angled. Pupils can calculate using simple time intervals.

Level 4

A pupil operating at DCL4 is becoming fluent in key foundational knowledge and techniques. Pupils are comfortable in communicating with numbers to at least 1000000. They quickly recall multiplication facts up to 12x12 and use their knowledge of factor pairs in working confidently with arithmetic within 10000, for which they use formal written algorithms and can solve problems in a variety of contexts. Pupils can confidently order and compare numbers and have a good understanding of place value. When counting backwards, they can bridge across zero into negative numbers. They can add and subtract with fractions and recognise and understand the decimal equivalence of simple fractions. Pupils can round numbers, including answers, to the nearest 10, 100 or 1000, or, when working with decimals, to the nearest whole number or to one decimal place. Pupils can convert between time formats between different units of measure, such as metres to kilometres. They can calculate perimeter and area of rectilinear shapes, though they may still rely on the use of counting squares to find area. Pupils communicate information more clearly, adding bar charts and time graphs to their repertoire. A DCL4 pupil is beginning to be able to hold mathematical conversations when solving problems, using correct terminology and appropriate formal algorithms.

Level 5

A pupil operating at DCL5 is becoming increasingly confident in using foundational knowledge and techniques to work in a range of situations. They have extended their mathematical literacy to include working with numerals to 3 decimal places, recognising and working with common multiples or factors, and using both square and cube numbers. Pupils understand what it means for a number to be primar and can establish whether a number up to 100 is prime. Their mathematical communication is largely through efficient formal methods. Pupils now use rounding as a method for checking calculations and can convert confidently between measures, recognising some common equivalences. Pupils use of fractions in problems is increasingly confident, including situations where they must work with improper fractions or identify equivalent fractions. Using their foundational knowledge, pupils are expanding their geometrical repertoire. They use standard units when work with measures and can find perimeters of composite shapes. They are working with polygons, volumes, nets and 2D isometric representations of 3D shapes. Pupils can describe the effect that straightforward reflection or translation has on simple shapes. They can state and describe simple angle facts.

Level 6

A pupil operating at DCL6 is confident with arithmetic, including with decimals, has sound mental calculations skills, can round numbers to a required degree of accuracy, and works comfortable with problems involving a mix of fractions, decimals and percentages, including when it is necessary to convert between those forms. They can do this because their foundational knowledge is embedded and well-rehearsed. Pupils are building on their foundational number knowledge to understand the general case. They are beginning to appreciate pattern and can work with linear number sequences as their sense of algebra starts to emerge. Pupils understand how points can be expressed on a plane using a coordinate system. When working with shapes, including triangles and parallelograms, pupils use standard units, can state simple angle facts and use formulas for finding area. In 3D, they use standard units for volume and formulas for finding volumes. A DCL6 pupil is on the cusp of becoming mathematically functional.

Level 7

A pupil operating at DCL7 can be considered mathematically functional. They can access and understand key mathematical information commonly encountered in day-to-day life. In addition to their already established foundational knowledge, pupils use negative numbers in context, can solve percentage problems and are able to use estimation as a method for attacking problems. Pupils can list combinations of two variables and can express missing number problems algebraically. Pupils can work with simple formulae. Their geometry repertoire has been extended to include translating shapes on a coordinate grid and working with simple scale factors. They are familiar with key parts of circles and how to name and label them. When working with data, pupils use pie charts and line graphs. They can find the mean average of a set of data and understand its meaning.

Level 8

A pupil operating at DCL8 is beginning to move beyond a simple functional use of mathematics, seeing more relationships between areas of mathematics and understanding more about its applications. Pupils can express one quantity as a fraction of another, work with roots in addition to powers, can round numbers to a required number of significant figures and understand the rules relating to the order of operations. Pupils are increasingly sophisticated in their appropriate use of calculators and other technologies to enhance their mathematical work. They routinely convert between standard units in a range of problems. Pupils use of ratio is becoming more useful, particularly now that they understand the purpose of reducing a ratio to its simplest form. When examining generality, pupils understand the meaning of simple expressions and know when they are used in equations. Their use of algebraic notation is consistent and appropriate to the simple problems that they work on, including problems involving substitution, generating sequences, simplifying expressions, collecting like terms and multiplying a single term over a bracket. The linear equations they solve are confined to those in one variable. Pupils use formulae when solving volume problems. They understand properties of parallel and perpendicular lines and a range of other properties of 2D shapes. Their use of coordinates is accurate in all four quadrants and they use this knowledge to plot linear graphs, understanding the meaning of gradient. Pupils can identify congruency in triangles and can interpret scale drawings. When working with data, pupils consistently make appropriate choices for best representations including frequency tables, bar charts, pie charts, and pictograms. Pupils understand the probability scale.

Level 9

A pupil operating at DCL9 has well established formal written methods for working with arithmetic, which they do with confidence and accuracy. Building on their appreciation of proportion, they can solve a range of problems involving ratio. Their use of percentages in solving a wide variety of problems continues to expand. Pupils can express numbers as multiples of primes by decomposition. Given a straightforward linear sequence, pupils can determine an expression for the nth term in the sequence. Pupils solve linear equations, including those that first require rearrangement or factorisation. Pupils use their established understanding of gradients and their use of graphs of linear functions to work confidently with conversion graphs. Pupils understand and can identify alternate and corresponding angles. They can construct bisectors of lines and angles and can interpret and construct loci. Pupils work with scale drawings in a range of problems and can construct enlargements. Pupils understand simple sets and unions and can express these in a range of ways including Carroll and Venn diagrams. Pupils established use of area is extended further to include composite shapes with circular parts and finding surface area of prisms or cylinders. When working with data, pupils can carry out a statistical project, using a range of representations including scatter graphs and the three averages.

Level 10

A pupil operating at DCL10 can be considered mathematically literate. Their mathematical knowledge and skills are at a level suitable for the majority of generalist jobs and can transfer across a range of workplace requirements. They have strong arithmetical skills, including arithmetic with mixed numbers and negatives, they can work with percentage change problems and understand direct proportion. Pupils can work with, and solve problems involving, a range of compound measures. Pupils can expand binomials, change the subject of a formula and solve equations with an unknown on both sides, including those with fractional expressions. Pupils knowledge of equations is built further by their new understanding of simultaneous equations, which they can solve in cases where both equations are linear. Pupils can plot and interpret graphs of quadratic and cubic functions. They have a good understanding of angle sum properties and fully appreciate congruence. At DCL10, pupils first embark on the formal study of trigonometry, starting with an understanding of Pythagoras’ Theorem. When working with data, pupils can use sample spaces, stem and leaf diagrams and frequency polygons.

Level 11

A pupil operating at DCL11 is moving beyond the mathematical knowledge required for a general level of mathematical literacy. Their representation of number now includes standard index and they can work with inverse proportion. Pupils can plot and interpret graphs of exponential and reciprocal functions. They are able to generate and find general terms in geometric sequences. Pupils can model real life situations using formulae and graphs. Their knowledge of simultaneous equations now includes situations where one equation is non-linear. Pupils formal study of trigonometry continues with an appreciation of the sine, cosine and tangent functions. Pupils can better describe data and its limitations through the use of interquartile range and box and whisker diagrams.

Level 12

A pupil operating at DCL12 can draw on a range of sophisticated mathematical techniques. Their understanding of the connections and equivalences across fractions, decimals, percentages and ratio is matured. They can use equivalent ratios, find reverse percentage change and compound interest or other repeated percentage change measures. Pupils representations of number are enhanced by effective use of index laws and they can perform calculations where numbers are in standard index form. Pupils further appreciate the impracticality of finding precise solutions and have added trial and improvement as a numerical method to their repertoire. Pupils work confidently with sequences involving triangular, square and cube numbers. They can solve equations involving direct and inverse proportion. Pupils can find the equation of straight lines and they use function notation appropriately. Working with non-linear expressions, pupils can solve quadratic equations through a variety of methods including the use of factorising and completing the square. Pupils have more formal approaches to defining turn and position, including the accurate use of bearings. They know the sine and cosine rules and can find missing lengths and angles using Pythagoras’ Theorem and trigonometry. They have committed to memory special trigonometric angles in simple cases. Pupils can use trigonometry to find the area of triangles. They are confident in using scale factors. When working with data and carrying out statistical projects, pupils adopt systematic listing strategies. They understand the probability of mutually exclusive events. A DCL12 pupil should be encouraged to study mathematics at a higher level in adult life.

Level 13

A pupil operating at DCL13 works fluently with number. They can perform calculations with surds, can find upper and lower bounds and use recursive formulae. Their confidence in working algebraically is enhanced by their new ability to work with algebraic fractions and transform functions. Pupils understand similarity and can identify a range of properties of similar shapes, using these to perform calculations. Presented with information about shapes in given formulae, pupils can undertake a process of dimensional analysis enabling them to state the type of property being expressed. Pupils are familiar with and can sketch graphs of sine, cosine and tangent functions. Pupils know a range of circle theorems and can use these in formal geometrical proofs. Pupils understand conditional probability and can accurately produce a tree diagram to describe and experiment and calculate probabilities of theoretical events.

Level 14

A pupil operating at DCL14 is an emerging mathematician. They are able to work with mathematics eloquently, using sophisticated mathematical terminology and conventions. Their use of number is matured, enabling them to choose appropriate levels of accuracy with which to communicate solutions; choosing to express values in significant figures, work with expressions involving quantities in ranges of lower and upper bounds, and reliably work in surd form. At DCL14, pupils can express generality correctly in a wide range of scenarios, including solving simultaneous equations graphically and algebraically. They can find roots of equations by transforming expressions and can work with simple algebraic fractions. Pupils’ understanding of geometry is well developed. They can work accurately with trigonometric functions to solve problems in two-dimensions. Pupils can find volumes and surface areas of frustums and spheres and are able to define and interpret simple vectors. At DCL14, pupils work with difficult probability questions, including ‘without replacement’ problems. A DCL14 pupil would be expected to pursue further study of mathematics to an advanced and higher level.

Level 15

A pupil operating at DCL15 is a young mathematician who will be able to specialise in a mathematical field at higher education and in their career. Their well developed number work allows them to tackle growth and decay problems. They can find the general term in quadratic sequences and solve both linear and quadratic inequalities. They are increasingly interested in and capable of producing mathematical proofs. Their understanding of geometry is expanded by the addition of new skills in using negative scale factors, describing combinations of transformations, working reliably with plans and elevations, and performing calculations involving arcs and sectors. When using data, pupils understand the meaning of and can produce cumulative frequency diagrams. They are confident in working with grouped data and can produce and interpret histograms, including those involving unequal grouping.

Level 16

A pupil operating at DCL16 is becoming increasingly specialised and sophisticated in their use of mathematics. In particular, they are studying pre-calculus in a meaningful way and are making more articulate and accurate use of formal mathematical argument in proofs. In pre-calculus, pupils appreciate characteristics of rates of change and can describe the meaning of gradients at points on curves and, in simple cases, calculate these gradients. Pupils understanding and use of vectors is further developed to a point where they can use combinations of vectors in geometrical proofs. Pupils can describe and plot coordinates in 3D. When undertaking statistical projects, pupils understand implications for sampling populations. In their analysis, they are able to discard outlying data by considering central tendency and measures of spread. Pupils understand correlation and can add a line of best fit to data, using it to produce commentary about the trends and make predictions.

Level 17

A pupil operating at DCL17 can use mathematics reliably in a wide variety of situations, particularly in describing the real world. They understand the laws of indices for all rational exponents and are able to rationalise denominators. Pupils understand and can use force, weight, displacement, speed velocity and acceleration. In solving non-linear problems, pupils can use the factor theorem and their knowledge of the discriminant of a quadratic function. They can represent linear and quadratic inequalities graphically and can describe asymptotes. Pupils use intersection points of graphs to solve equations. They can transform graphs and know the gradient conditions for two straight lines to be parallel or perpendicular. Pupils use the equation of a circle in solving problems. When working with data, pupils select sampling techniques based on their knowledge of the population. Pupils interpret regressions lines for bivariate data.

Level 18

A pupil operating at DCL18 uses well-reasoned mathematical argument that is accurate, appropriate and concise. They use proof by deduction, proof by exhaustion, disproof by counter example and proof by contradiction. Pupils understand and can use exponential and logarithmic functions and their graphs. They know the laws of logarithms and can use them to solve equations. Pupils can perform vector addition and multiplication by scalars. At DCL18, pupils are beginning to specialise in areas of mathematics that will enable them to continue to study the discipline at a high level. If choosing to pursue a mechanics / dynamics path, pupils understand and use Newton's first law and second law for motion in a straight line under gravity. If choosing to pursue a statistics path, pupils understand informal interpretations of correlation and can describe why correlation does not imply causation. Their analysis of data is further refined by sophisticated use of standard deviation.

Level 19

A pupil operating at DCL19 is beginning to understand the implications of The Calculus. They understand and can use differentiation from first principles for small positive integer powers of x. They can find the second derivative and know that this represents a rate of change of gradient. They apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points. This allows them to identify a function as increasing or decreasing. Pupils know the Fundamental Theorem of Calculus. If pursuing a mechanics / dynamics path, pupils understand and can use Newton's third law. They can calculate the distance between two points represented by position vectors. If pursuing a statistics pathway, pupils can express solutions through correct use of 'and' and 'or', or through set notation. They know the binomial expansion of (a + bx)n and reliably use the notations n! and nCr.

Level 20

A pupil operating at DCL20 is a young mathematician who is continuing to understand implications of The Calculus. They understand integration as being anti-differentiation and can integrate expressions. They can find a definite integral and know this to represent the area under a curve. If pursuing a mechanics / dynamics path, pupils will expertly use vectors to solve problems in in context, including forces. If pursuing at statistics path, pupils can clean data and undertake statistical hypothesis testing. They select an appropriate probability distribution for a context, including when not to use the binomial or Normal model.

Level 21

A pupil operating at DCL21 is expanding their mathematical knowledge base to enable later study of more complex situations to be addressed with calculus. They work with increasing decreasing and periodic sequences, understand and use sigma notation, understand finite geometric series and the sum to infinity of a convergent geometric series. Pupils reliably use modulus notation. Pupils accurately perform algebraic division in simple cases and understand the modulus of a linear function. They can use composite functions; inverse functions and their graphs and are able to determine combinations of transformations. Their geometrical communication is enhanced by their use of radian measure and they know the exact values of sine, cosine and tangent of special radian measures. Pupils understand secant, cosecant and cotangent, arcsin, arccos and arctan. If pursuing a mechanics / dynamics path, pupils use vectors in three dimensions to solve problems in kinematics. They use trigonometric functions context, including problems involving vectors, kinematics and forces. Pupils use formulae for constant acceleration for motion in 2 dimensions using vectors and use calculus in kinematics for motion in 2 dimensions using vectors. If pursuing a statistics path, pupils extend the binomial theorem to any rational n and know its uses in approximation.

Level 22

A pupil operating at DCL22 is a mathematician ready to advance to higher education study in mathematics or a closely related field. Their competence in working with a significant range of mathematical problems is secure. They are articulate and able to communicate mathematically with precision and elegance. Their study of The Calculus means they can reliably differentiate and integrate expressions containing a wide variety of functions including trigonometric, exponential and logarithmic terms. They know the connection between the second derivative and convex and concave sections of curves and points of inflection. When differentiating, they use the product rule, the quotient rule and the chain rule. When integrating, they use integration by substitution, integration by parts and integration using partial fractions. They know integration as the limit of a sum. They understand limitations of methods and are able to deploy appropriate numerical methods such as Newton-Raphson. Pupils understand parametric equations of curves and can find the first derivative of simple functions defined implicitly or parametrically. Pupils produce clear and elegant geometrical proofs using, for example, double angle formulae and other trigonometric identities. Pupils can construct simple differential equations in context. If pursuing a mechanics / dynamics path, pupils can model motion under gravity in a vertical plane, resolve forces in 2 dimensions and understand equilibrium of a particle under coplanar forces. Pupils use addition of forces. They understand the impact of and can model motion of a body on a rough surface. They understand and use moments in simple static contexts. If pursuing a statistics path, pupils understand the conditional probability formula and use this when modelling with probability. They use Normal distribution as a model and can interpret a given correlation coefficient using a given p-value or critical value. A pupil operating at DCL22 has studied mathematics with determination and flair throughout their time at school and should be encouraged and supported by the mathematics teacher to follow a career as a mathematician.

So now you know more about the DCL framework and the broad phases that pupils pass through as they become more and more mathematically mature.

How then do these cognitive Demand Level Criteria inform the attainment metrics in Complete Mathematics?

Our MathsAge has been carefully mapped against the DCL and against other useful metrics, including GCSE, National 5, Core Maths, A Level, Higher and more.

Let us take a look at how the DCL line up with some of our most commonly used metrics.

DCL CM Stage Maths Age GCSE (Eng) GCSE (Wal) Nat 5 Core Maths A Level Higher NC Level
1 1 6 1 Low G Fail N/A N/A N/A 1
2 2 7 1 High G Fail N/A N/A N/A 2
3 3 8 1 Low F Fail N/A N/A N/A 3
4 4 9 2 High F Fail N/A N/A N/A 3
5 5 10 2 Low E Fail N/A N/A N/A 4
6 6 11 3 High E Fail N/A N/A N/A 5
7 6 11 3 Low D D N/A N/A N/A 5
8 7 12 4 High D D N/A N/A N/A 5
9 8 13 4 Low C C N/A N/A N/A 6
10 9 14 5 High C C E N/A N/A 7
11 9 14 5 Low B B E N/A N/A 8
12 10 15 6 High B B D N/A N/A 8
13 10 15 7 Low A A D N/A N/A 8
14 11 16 8 High A A C N/A N/A Above NC Levels
15 11 16 8 Low A* Upper A C N/A N/A Above NC Levels
16 11 16 9 High A* Upper A C N/A N/A Above NC Levels
17 12 16 Above GCSE Above GCSE Above Nat 5 B E N/A Above NC Levels
18 12 16 Above GCSE Above GCSE Above Nat 5 B D D Above NC Levels
19 12 17 Above GCSE Above GCSE Above Nat 5 A C C Above NC Levels
20 12 17 Above GCSE Above GCSE Above Nat 5 N/A B B Above NC Levels
21 13 18 Above GCSE Above GCSE Above Nat 5 N/A A A Above NC Levels
22 13 18 Above GCSE Above GCSE Above Nat 5 N/A A* Upper A Above NC Levels

As you will be able to see, the DCL often span multiple grades in a metric system. It is not the intention to convey the sense that a DCL or a grade can be pinned down accurately to a certain question of task – many tasks span multiple DCL and grades are a reflection of the performance of the population. Rather, what we are interested in is the pupil’s own development as a mathematician, the knowledge and skillset they acquire along the way and how these are articulated through the way in which a pupil behaves mathematically.

So, perhaps the next time you are looking at the MarkBook inside the Complete Mathematics platform, you can notice the attributes and dispositions those individual pupils exhibit in the classroom and see them as maturing gradually and know that, no matter what their current stage, they can continue to grow to become a successful young mathematician.

Using the Complete Mathematics Platform

Written by Abbie Brownlie Thursday, 26 March 2020

During this ongoing period of disruption in education, pupils and teachers have been working incredibly hard to keep on track.

Teachers and schools have done an incredible job with the support and provision they put in place in an exceptionally short time under remarkably challenging circumstances.

Our school support team have published a series of short training videos to show exactly how the Complete Mathematics platform can be used by both teachers and students to maximise the impact of blended learning.

This includes episodes on:

  1. Adding Pupils
  2. Lesson Planning
  3. Assigning Work
  4. Times Table Practice
  5. Remote Quizzes
  6. The Pupil App
  7. Accessing Support

Find out more about Complete Mathematics subscription with a demo from one of our team!

You can watch the whole ‘Using Complete Mathematics for Remote Learning’ playlist here!

Adding Pupils

A video covering how to easily add pupils to the platform, the generation of their login details for school and home access, as well as how your pupils log in.
Remember, you get unlimited, free pupil accounts with a Complete Mathematics subscription!

Lesson Planning

This video will show you how to create daily maths lessons from school or home, with appropriate support materials, resources and tutorial videos (including how to add your own). Plus, how your pupils can access these planned lessons on the platform, be it in the lesson or remotely.

Assigning Work

This talkthrough video will cover assigning content for your classes as both classwork and homework and your ability to do that from school or at home. While also taking a look at how pupils can access and interact with this assigned work, including: submission, adding their reflections, and asking you questions!

Times Table Practice

In this video you will see how pupils can take daily times tables quizzes with multiple representation of times table facts. Then find out how teachers can track progress and see their pupil's progression.

Remote Quizzes

This video will show you how to create informed, automatically marked, low-stakes quizzes based on your planning. See how pupils find and complete their assigned quizzes online and the opportunities for independent work via their results analysis view, either at home or in class.

The Pupil App

This video provides your pupils with a walkthrough of how exactly they can use the Complete Maths platform. This video is made to be shared directly with your pupils, which will show them where to find upcoming lessons, their assignments and quizzes that are due. Using the timetable to explore past and future lessons, and the quiz results page.

Accessing Support

We will discuss where to find our knowledge base, using the platform support hub, and other available sources of support.

Models & Didactics on the Complete Mathematics Platform

Written by Mark McCourt Friday, 29 May 2020

Pupils make sense of new mathematical ideas from a range of models, metaphors, examples and instruction. This means teachers need to be equipped with a set of approaches that they can call upon in the classroom.

Well established models, tried and tested over many years, such as the use of Cuisenaire rods, double number lines, algebra tiles, Dienes blocks, geoboards and many more are central to the subject specific pedagogical knowledge for everyone teaching mathematics.

We are adding these models to Complete Mathematics, tagged against the appropriate mathematical ideas so that teachers have immediate reminders of powerful pedagogies for the lessons they are planning.

Examples of models being added to Complete Mathematics

The specific technical details behind a mathematical idea – the building blocks of the idea itself – are critically important to understand and communicate precisely if pupils are able to make connections across the universe of mathematics as they grow and learn more. These technical details are known as the didactics of the mathematics.

Didactics act as a bridge between the teaching process and the learning process. An understanding of didactics enables teachers to translate the mathematical competencies they themselves have – the mathematics the teacher understands and can work with easily and without having to think – in to a communicated form of the mathematics such that pupils who have never before encountered it can make meaning and grip the idea at hand.

As Jim Fey writes in the seminal book ‘Didactics of Mathematics as a Scientific Discipline’ (1994) preparing mathematics for teaching can be conceived of as elementarization, that is, “the translation of mathematical concepts, principles, techniques, and reasoning methods from the forms in which they are discovered and then verified by formal reasoning to forms that can be learned readily by a broad audience of students.”

We are adding the didactics of mathematics to Complete Mathematics so that all teachers can readily access the technical detail of the ideas they are communicating and see how these mature over time.

Examples of didactics being added to Complete Mathematics

These and further updates to the curriculum will be live on the Complete Maths platform in the next few weeks. Stayed tuned for further investigations into this release soon.

La Salle Education raises £1M to make high quality maths tuition accessible to all students

Tuesday, 20 April 2021

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.

For more information email This email address is being protected from spambots. You need JavaScript enabled to view it.

The Most Important Academic Year in Generations

Written by Mark McCourt Friday, 27 March 2020

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.

For more details about our online CPD for teachers, visit the CPD page.

For more details about our private tuition for all pupils, visit our virtual school page.

For more details about the Complete Mathematics platform and how to join, visit our platform page.

Mastering Mastery: Making the cycle work!

Written by Gary Lamb Monday, 24 February 2020

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.

What Next?

We regularly run our hugely popular Mastery in Mathematics CPD course across the UK. Discover upcoming dates and book your tickets here.

What's New in Autograph 5

Thursday, 17 October 2019



2D Page

'Label objects' setting in the Axes Menu

  • Clicking 'Label objects' — objects will be labelled from now on.
  • Clicking 'Label Points only' – this will not label other objects (lines, etc)
  • Any new 2D page opens with 'Label Objects' off
  • New right-click options on selected points, lines etc:
    • 'Show Labels'
    • 'Hide Labels'
    • 'Edit draw options' (e.g. in the case of points, change their shape)
  • New right-click options on single point:
    • 'Edit Label'
  • Note: Labels are important when using the CALCULATOR or XY ATTRIBUTE POINT as both refer to host objects by label
Labelling objects

Line Tests

  • Select two lines/rays/segments:
    • Parallel or perpendicular. If the lines or line segments are parallel, single arrows are placed at the mid-point (+ a bit). If lines are perpendicular a right-angle sign is placed at the intersection
  • Select two or more segments:
    • As above together with test for equality which is displayed as marks at mid-point
  • Select any line with ticks & ’Remove Line Tests’:
    • Deletes all test marks in this and associated lines
Line tests

Fixed length line from point

  • Right click > line > Fixed Length Line
    • Enter length (‘d’) of line. Autograph then draws and labels a line horizontally to the right, which can then swing round a circle of radius ‘d’
Fixed length line

Create rectangle from two points

  • With 2 points selected, right click and create > Rectangle. Enter height; plotted clockwise from the 2nd point
  • Attributes: side, height, Perimeter = 2 * (side + height), Area = side x height
  • Status bar: Rectangle a x b, Area =
Rectangle creation

Angle from two points

  • Right click > Create > Angle
    • Enter angle (‘θ’). Autograph then draws and labels an angle ('B'), option for 'clockwise'
Angle creation

Ray from two points

  • Right click > Line > Ray
Ray creation

Quadratic from three points

  • x=f(y) option
Quadratic from 3 points

Unit gradient triangle from point on graph

  • With point on graph selected Right click > Line > Unit Gradient
Unit gradient triangle

New transformations

  • Rotation, Enlargement, translation, etc
  • Clockwise option for rotation

Regression Lines

  • y-on-x & x-on-y
  • Show dotted < xmin and > xmax
Regression line

New intersection icon

  • Equivalent to pressing CTRL to find intersections, max, min
New intersection icon

New click and hold options

  • With daya set drawn. Click-hold-drag as alternative to CTRL-drag to select/move points in a data set. For use on white-boards and touch interfaces
  • In select mode. Click (marquee select) and Hold (drag scene)
  • In zoom mode. Click-release: Zoom in/out, Click-drag: select zoom area
Move with click-and-hold

Points recorded as data

  • Create an “XY” attribute point from two objects (Here the two attributes are x-coordinate of the point on curve and the value of the slope of the gradient). Tick “Record to Results Box”
  • With ‘A’ selected click Animation and set the step to 0.25 Step along the curve, and note the values recorded in the Results Box.
Points recorded as data

Major revision of plotting procedures

  • Floor(x) and ceil(x) and int(x) now plot with no verticals
  • Plotting a discontinuity e.g. y = (x² − 4)/(x − 2)
  • 2D plotting revised and improved, e.g. y = sinx cos(1/x)
Floor(x) and ceil(x) and int(x) now plot with no verticals

Gradient function and polynomial best fit now behave like entered functions

  • So you are now able to enter point on curve, draw tangent, create area (2 pts); draw gradient function
  • Also Animation Controller available to animate order of polynomial
Gradient function now behaves like an entered function

New Area options

  • ‘General’ and ‘Mid-interval Rule’ added
  • ‘General’ is Simpson's Rule with 50 divisions and standard object fill
  • Rectangles (left) was (-) and Rectangles (right) was (+). “left” means the height used is the left of the interval, right for right of the interval.
  • Can calculate 'area between': Select 2 points (on or off graph) + TWO functions (in order)
New area options

Arc Length from two points on graph

  • With 2 points selected on a graph Right click > Arc Length, option to show centroid
Calculate arc length


  • Enter using eg a_2 for a₂ in the range 0 … 9. Manage constants redesigned: subscripts for lower-case and uppercase constants (single upper-case constants not permitted)

Miscellaneous Small Changes

  • Can change 'UI theme' in preferences
  • New option “Accessibility” to adjust colours in preferences. Used for selection by those hard of sight, or selection not showing well on whiteboards.
  • Attributes of polynomials >3 renumbered: e.g. quartic is a1x⁴+ a2x³+ a3x²+ a4x + a5c => should be a4x⁴, a3x³, a2x², a1x, a0c
  • Text Box and Calculator: MathML converts to single-line notation when copying
  • Origin circle no longer gets too small if window size reduced
  • “Create data set from graph” does not have “Join Points” ticked
  • Image attached to a point: takes its gradient at the point, not the middle
  • With 2 points selected Right click > Point > Ratio is now the calculatd the other way round
  • Pallet of font colours has been extended
  • In start-up options: Manual limits can take constants
  • Polygons – allow (integer) constants for number of sides
  • 2-point Vector: text box says AB () instead of “Join Vector”
  • Can 'nudge' images with arrow keys (and Ctrl)
  • XY Data`: Column Headers ticked by default
  • Import and export data (2D and Stats): CSV or TAB options + saves heading(s)
  • 2D page: CTRL-MOUSE-WHEEL zoom speeded up
  • Default Animation parameters changed from 0 … 10 and step of 1 to –3 … 3 and step of 0.1
  • In Edit Axes, if the variables have been changed to 't' and 'v' in ranhes the options now say't' and 'v' not 'x' and 'y'
  • Accept two ‘ (prime symbol) as equivalent to “ (double prime symbol) when entering 2nd order differential equations

3D Page

Arc Length and Surface of Revolution

  • With 2 points selected on a 2D graph Right click > Arc Length
  • Option to draw axis of revolution in the form y = k (plot as 2D)
  • With Arc length selected: Right click > Surface of revolution: enter axis of revolution (If not drawn and selected)
  • Arc length can be a line segment (must be in x-y plane, so z = 0))
Surface of revolution

The 3D “Create” Sub-menu

  • Parallelepiped
    • With 1 point and 3 vectors selected, right click > 'Create' > 'Parallelepiped'
    • Attributes: Volume = (a x b ) • c
  • Sphere
    • With 1 point selected, right click > 'Create' > 'Sphere' > enter radius
    • With 2 points selected, right click > 'Create' > 'Sphere' (center at first point)
    • Attributes: radius, surface area, volume
  • Cylinder
    • With 2 points selected, right click > 'Create' > 'Cylinder' > enter radius
    • Attributes: radius, height, surface area, top/bottom area
  • Cube
    • With 2 points selected on a plane, right click > 'Create' > 'Cube' (choice of positive or negative orientation and clockwise/anti rotation)
    • Attributes: Side length, Surface area, Volume, Inside/Outside radius
  • Tetahedron
    • With 2 points selected on a plane, right click > 'Create' > 'Tetrahedron' (choice of positive or negative orientation and clockwise/anti rotation)
    • Attributes: Side length, Height, Surface A, Volume, Inside/Outside radius
  • Polygon
    • With 2 points selected on a plane, right click > 'Create' > 'Regular polygon' > enter number of sides
    • With 3 points selected, right click > 'Create' > 'Polygon'
    • With ≥ 3 points selected on a plane, right click > 'Create' > 'Polygon'
    • Attributes: Sides, perimeter, area, centre, (Additionally if regular: side length, inside/outside radius, int angle)
  • Right pyramid
    • With 2 points selected on a plane, right click > 'Create' > 'Regular Pyramid' > enter number of sides and height
    • With 1 polygon selected on a plane or 1 plane side of a cube, tetrahedron, pyramid or prism selected, right click > 'Create' > 'Right Pyramid' > enter height
    • Attributes: Sides, side length, height, slant height, Surface area, volume, inside/outside radius, slant height, base area
  • Right prism
    • With 1 polygon selected on a plane or 1 plane side of a cube, tetrahedron, pyramid or prism selected, right click > 'Create' > 'Right Prism' > enter height
    • Attributes (regular pol): sides, height, Surface area, volume. base area, centre, side length, inside/outside radius
  • Cuboid
    • Enter and select 4 points on a plane in a rectangle, right click, 'Create' > 'Polygon'
    • With the polygon selected, right click, 'Create' > 'Right Prism' > (enter height)
    3D Shapes

3D parametric plotting improved

3D parametric plotting

Statistics Page

Dynamic Raw Data

  • 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.
Move raw data points

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.
Box plots

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”:
    1. Data is first sorted in ascending order and numbered 1 to n
    2. The mean and SD of the dataset are calculated
    3. 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.
Q-Q normal plot


  • 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

Complex Number Page

New object: Circle

  • Right click on complex number page > 'Circle' > input radius

New creations

  • With one complex number selected you can now create a vertical line, horizontal line and circle from single complex number
  • With two complex numbers selected you can now create a straight line, line segment, perp. bisector and ellipse (enter sum of radii)
  • With three complex numbers selected you can now create a circle and an ellipse (1st 2 CNs are the foci, then point on the ellipse)

New Complex transformations

  • for lines, circles and ellipse, and a single point


  • With a single point and its complex transformation selected, right click > 'Locus'
Complex transformation loci

Show/hide vector to origin

  • With single point selected, right click > 'Edit complex number' > Uncheck 'Show Vector to Origin'

Calculations on complex numbers

  • Including: 'Multiply by Scalar', 'Raise to Power', "Multiply by i', 'Complex Conjugate', 'Real Part', 'Imaginary Part', 'nth Roots'
  • Labels are automatically added

Axes Auto scaling

  • Always includes origin


Revised level selector on start-up

New level selection on load

Using Complete Mathematics to Support Implementing a Mastery Curriculum

Written by Chris McGrane Thursday, 19 September 2019

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.

The Mastery Learning Cycle by Mark McCourt, visuals by Oliver Caviglioli

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.

All of school mathematics — primary, secondary, FE — carefully mapped in the Complete Mathematics Curriculum

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.

Expert teachers

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.

Support materials within each granular objective of the Complete Mathematics Curriculum

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.

Assessments completed by pupils on the CM platform are accessible, responsive, and automatically marked.

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.

As part of results anayltics available on the platform, find a list of objectives ordered by priority for improvement based on recent assessments.

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.

Teaching progression through the Units of your scheme of work.

What Next?

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.

Additionally, you might be interested in organising a Bespoke CPD series for your institution to learn more about implementing a mastery approach with one of our expert team. Click here for further detail on this, or wanted to register your interest, please get in touch with the school support team at This email address is being protected from spambots. You need JavaScript enabled to view it..

Dynamic Example Questions from Complete Mathematics

Written by Tom Valsler Monday, 20 May 2019

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

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

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

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

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

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

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

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

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.

Dynamic Worksheet Generation from Complete Mathematics

Written by Chris McGrane Tuesday, 02 April 2019

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

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

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

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

x2 – 11x + 24 = (x – 8)(x – 3)

x2 – 14x + 24 = (x – 2)(x – 12)

x2 + 10x – 24 = (x – 2)(x + 12)

x2 – 10x – 24 = (x + 2)(x – 12)

x2 + 5x – 24 = (x – 3)(x + 8)

x2 – 5x – 24 = (x + 3)(x – 8)

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

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


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.

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

Written by Chris McGrane Thursday, 31 January 2019

I’ve used Cuisenaire occasionally in my career. Much of the time it was as an aide in the teaching of fractions to younger secondary pupils. However, this fabulous resource has so much more potential. It can be used to introduce the very basics of arithmetic such as additive relationships, or extended into harder topics such as simultaneous equations, Pythagoras and equation of a straight line.

Allow me to share a reflection of “a learning episode”.

This evening my five-year-old son, who is as inquisitive as children of that age tend to be, spotted a small bag of Cuisenaire rods on my desk. He was immediately drawn to them. “What are those daddy? Can I see them?” The verb “to see”, for a five-year-old is not just an interaction of the eyes and brain. It is a tactile action, it involves touching the object and interacting with it in some way.

He poured all of the blocks over the table, gazing in my direction to ensure that this was OK. Immediately, he began to play with them. He built little patterns and began to group the rods. There is something about these little rods that is inherently enticing.

An initial arrangement

Mark McCourt had told me that young children will begin to behave mathematically with these blocks, given enough opportunity to play with them. I was stunned, when, after just a few minutes, my son said “Maybe after this I could do it by sizes”. The level of categorising went beyond the first level I’d expected him to consider; colour. Instead it was a mathematical idea. I let him play with them for a while. I was minding my own business, leaving him to it and not prompting him in any way.

All of a sudden, a loud announcement, coloured with the excitement and joy of a profound revelation: “Its colour is its size!” In that moment, these little rods had gone from being toy blocks to being something else. It’s impossible to make inferences about the connections he was making. However, what was to follow demonstrates, to me, that he was thinking hard.

Some organisation

“Orange is the biggest one!”

I’d resisted the urge to prompt or direct him until now, but I couldn’t help myself, I wanted to play too. Displaying a little bit of shock for his benefit I asked him “Is it really bigger than the blue?”

He was, correctly, adamant that it was. Having his conjecture challenged, he did what any mathematician would do – he sought out a proof! Carefully lining up the blue and orange he showed me that there was a gap. “Look – you can put a white one there”.

He’d just modelled a number bond to ten. While he can already “do” addition he hadn’t yet recognised that the calculations he does at school were synonymous with his demonstration with these little rods. I think that will come in time – after all, the pace of progress in his use of the rods is startlingly fast.

He continued to play freely with the rods. He made some domino trails. This is the beauty of this manipulative – there is fun to be had with it! A short while later I saw him looking at the purple and dark green. “This is four more taller than purple”. I was perplexed with this idea of four, as the green is only two blocks more than the purple. I chose not to judge, but instead try to understand his interpretation of the situation. I asked him to show me why.

He motioned with his finger four equal steps from the end of the green to the end of the purple. I suggest that there were two possible thought patterns here: the first is that there was some unit of measurement, known only to him, which was his point of reference. Alternatively, he hadn’t quite grasped the relative size of the white block to the others.

Maybe in asking why, I challenged him in a way that made him reconsider things. He presented me, absolutely delighted with himself, the following set up:

Comparative thinking

The mathematics is simply pouring out of this free play. These are exactly the sort of comparative models I watched Mark McCourt share with teachers yesterday!

The free play continued with “now I want to count them all”. This was going really well. He had counted past 50 when, all of a sudden, his twin sister appeared. He continued to count but her presence (she was asking me about the rods) put him off a little. He said he thought he’d counted properly, but wanted me to double check. His sister volunteered – she was keen to get involved too. Midway through counting I heard her brother say to her “you’ve missed out all of the fifties and sixties”. He had been listening intently. They decided to count them again together, this timing getting the correct total. I didn’t check the total for them. They have the knowledge between them to be sure of succeeding.

They began to discuss the orange rod. He told her how it was the biggest one. She replied, clearly insulted that he thought she hadn’t realised this “I know! Look – it’s two yellows”. She lined up the rods to show him her thinking. I hoped they’d follow this line of inquiry further, so offered a suggestion “how many white ones to get the orange?”. The guesses were wildly inaccurate. One thousand is the phrase they like to use for “lots of something”, so this was the figure they last mentioned. They each made their own models, slowly and deliberately placed the whites against the orange. This was a real test for their fine motor skills.

“The big one is the same as ten.” I noticed that neither of them said “ten whites”. Could it be that they had stumbled upon the standard numerical values of the rods? I was about to offer another prompt when my son asked me for a pencil, so he could measure it. They have done a little bit of measuring in school recently. Did the number ten resonate with him in some way as to remind him of this?

The first attempt at measuring
A further attempt

Before long the pencil was cast aside and a box was to be measured. This looks like a potentially intuitive introduction to the idea of perimeter. Yet more rich mathematical activity.

Measuring round the box

All of the above happened in less than 30 minutes. With no direct instruction from me a whole wealth of possible starting points for further exploration have been encountered. Cuisenaire is an incredibly powerful and versatile manipulative. The extent of how it can be used to support learning and teaching is vast. You can learn more about this by coming along to one of our Concrete, Pictorial, Abstract and Language CPD days.

Meaningful Times Tables Practice

Thursday, 16 November 2017

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.

Suggested uses

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.

Visit the Complete Mathematics Times Tables app page now.

EEF - Improving Mathematics in Key Stages Two and Three

Friday, 03 November 2017

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.

The full report can be found on the EEF website.

Complete Mathematics Research Schools

Tuesday, 06 September 2016

Dear Colleague

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. 



  • 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!



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.



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.

To apply, please fill in the Research Schools Application Form.  We will be in touch as soon as possible.

I do hope you can join the programme and help us to keep Complete Mathematics growing and relevant to classroom practitioners.

Kind Regards 

Mark McCourt

Maths teacher shortage?  A La Salle Mentor can help.

Thursday, 20 August 2015

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.