# 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.

### About the Author

#### Mark McCourt

Mark is the UK's leading authority on teaching for mastery. He has trained over 2000 schools in mastery models for schooling in the UK and overseas.

A leading figure in mathematics education, Mark has led many large-scale government education initiatives, both in the UK and overseas. Mark was a Director at the National Centre for Excellence in the Teaching of Mathematics (NCETM) and has also been a school leader, an Advanced Skills Teacher, a school inspector and a teacher trainer. He founded and was Chairman of the Teacher Development Trust.

Mark has extensive experience of mathematics teaching and learning across all age and ability groups, having taught students from age 3 to PhD!