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Maths
Mathematics is not about numbers, equations, computations, or algorithms:
it is about understanding. (William Paul Thurston)
Definition of Subject
Mathematics is the language of pattern, measurement and logical rules, it is the art of interpreting, quantifying and working with error and uncertainty and it is concerned with using imagination, intuition and reasoning to find new ideas and solve puzzling problems. Engaging in mathematical activity is part of what it means to be human. The study, discovery and use of the subject goes back to ancient times.
Nature of Subject
The core aspects of mathematics can be grouped under the headings of Number, Ratio and proportion, Algebra, Geometry and Measure, and Statistics. Although on the surface they seem unconnected, there are many links that connect these different topics. Throughout schemes of learning we explicitly link these varying areas of the curriculum.
Mathematical development starts with students learning how to manipulate different types of mathematical object and building fluency with a variety of essential procedures and relationships. Over time, they learn how to combine and use this knowledge to in more complex ways. A widely accepted aim of mathematical development is for students to be able to solve unseen problems by identifying and combining appropriate mathematical skills.
Purpose of Subject
Mathematics can be regarded as the queen and servant of the curriculum. It is a language used in most areas of human endeavour to model and predict. At the same time, many find an innate beauty in the abstract logic and relationships described by the subject.
There are many reasons for studying mathematics. It is a tool of thought that is essential for evaluating information and making good decisions in many areas of life from the financial to the practical. In addition to this, study of mathematics exposes individual to the beauty of logical reasoning that make it part of our intellectual heritage that has survived through the ages as ‘something worth knowing’. Finally, skills and qualifications in mathematics give students access to a wide variety of industries and areas of further study.
Mathematics has been fundamental to human development. It is underneath all modern technical advancement that has improved the living conditions of millions. Without it we would not have defeated diseases or put people on the moon.
Design of Mathematics Curriculum
Mathematics contains a small core of declarative knowledge; technical vocabulary, key formulas and definitions. On top of this is built a well defined and extensive area of procedural knowledge. This knowledge is hierarchical in nature in that most of it can only be accessed if earlier knowledge has been learnt sufficiently well. As well as this, the order of content is based on the following principles:
- A strategic KS2 topic overlap ensuring that students receive further teaching in areas that they find challenging. For example, proportion, fractions, ratio and percentage.
- Early introduction of ‘challenging’ topics to ensure that students have time to become familiar with new or intimidating notation.
- Priority and time given to high impact topics that develop wider mathematical skills and have multiple application.
- A balance of difficulty and strands (number, ratio & proportion, algebra, geometry, probability and statistics) of mathematics to ensure variety for students and support spaced practice.
Through this we aim for all students to become expert mathematicians who have acquired a strong understanding of mathematics that they can apply across the curriculum and beyond.
Kingsbridge Community College follows the ESW maths curriculum which also has the following features:
- Explicit support to ensure that all students can make links between different concepts. For example, students are explicitly taught how to apply their knowledge of forming and solving equations in the different contexts of area, perimeter and angles.
- Learning outcomes that are broken down into small steps to support direct instruction techniques (and teaching resources that support this approach).
- Priority given to key concepts that are essential for good progress.
- Support for regular low stakes quizzes and responsive feedback repair.
- Strategic assessment which is used by teachers to inform planning and intervention, and leaders to inform monitoring, CPD and curriculum development
- Common teaching objectives for all abilities at KS3 to ensure students are given a broad diet of mathematics and are not held back.
- Three streams of difficulty at KS4 to support students as they work towards the foundation or higher tier level at GCSE.
- Features to build students capacity and ability for independent study.
Home learning is an important part of the maths curriculum. At Key Stage 3 & 4 students get a personalised homework every week that has a proportion of new topics and prior learning topics, Years 7-10 through Sparx. Detailed data enables teachers and school leaders to monitor engagement and intervene where needed. In Year 11 & Key Stage 5 we have a mixture of topic based home learning to develop fluency and deepen understanding of learning outcomes along with longer mixed learning outcomes tasks to provide spaced practice across the two year course.
Extension of Subject
The extension of our maths curriculum begins in our classrooms with students being provided with a rich diet of questions which extend and deepen their understanding of a topic beyond the core learning outcomes. We also participate in the annual UKMT Maths Challenge and AMSP programs. Some students have also been selected to participate in the Mathematical Enrichment Community run by Exeter Maths School.
Mathematics opens the doors for many opportunities at university. Many courses will require a passing grade (or higher) at GCSE before viewing an application. For students taking Mathematics at A-Level their options at university include: Mathematics, Engineering, Computer Science, Medicine, Economics, Physics, Statistics, Dentistry, and Sports Science amongst many others.
When it comes to careers, it is hard to be specific as mathematics qualifications are valued in a huge number of different sectors (mathscareers.org.uk). In addition to this, numerous studies have shown that people with higher Mathematics qualifications benefit from a significant wage premium.
ESW Mathematics Curriculum Principles
Introduction
We believe that with the right teaching, the majority of students can make exceptional progress and are capable of achieving a strong pass or better at GCSE and can achieve strong passes on Level 3 courses. To achieve this goal requires high quality teaching supported by a well-structured curriculum in the context of strong whole school systems. This document explains how the ESW Mathematics curriculum supports the teaching of Mathematics. It does this by describing 5 desirable characteristics of good teaching and how they are linked to specific features of the curriculum.
Document outline
- Logical topic progression
- Responsive teaching
- Teaching with stretch and challenge
- Building fluency and recall
- Reducing workload and improving lesson planning
- Logical topic progression
Maths is a hierarchical subject where the order of teaching is an important factor in helping students to understand complex ideas. However, the order in which topics are taught can also help students to transfer their learning to long term memory. This is because subsequent topics can offer opportunities to practice and make links within and between previously learnt concepts.
The ESW Mathematics curriculum supports a logical topic progression in the following ways:
- A reasoned topic progression, which is communicated to teachers through notes on the ESW scheme of learning and the scheme overview. This is reviewed regularly in the light of practice.
- Explicit teaching objectives (and resources) to combine current topics with previously learnt ones (interleaving/spaced practice)
Topic progression posters communicate the sequencing of learning to teachers, students and parents (These will be introduced in early 2020).
It is important to note that a scheme topic progression is not sufficient for all students as some may have missed prior teaching for a variety of reasons or not been successful in acquiring knowledge when it was taught for the first time. This leads to the second element of the curriculum.
- Responsive teaching
Successful teaching is always based on an accurate knowledge of what students know and understand. Responsive teaching refers to the responsibility of all teachers to find out what their students already know before teaching a topic and a responsibility to evaluate the effectiveness of their instruction as they go along.
The ESW Mathematics curriculum supports responsive teaching in the following ways:
- A generous time allocation for teaching topics. This means that teachers can teach prior knowledge objectives outside of the current teaching order as and when they are needed. For some groups this time is used for extension focused on depth rather than breadth.
- Diagnostic materials including short tests are provided for key topic areas to allow teachers to assess prior knowledge and plan repair lessons or episodes before moving on.
- Mastery tests are provided for teachers to edit and use to discover how effective their teaching has been for each learning objective.
- Support objectives to assist teachers of very weak groups or individuals to identify and find materials that support the teaching of required prior knowledge.
- A clear direction in all guidance that teachers are expected to respond to the needs of the class.
- Teaching with stretch and challenge
It is easy to underestimate what children are capable of; setting is not exact and students mature at different rates. Teaching with stretch and challenge requires teachers to be ready to present students with complex concepts and open to developing their abilities to explain and teach them. We expect all students to be provided with a curriculum which inspires them to make outstanding progress.
The ESW Mathematics curriculum supports teaching with stretch and challenge in the following ways:
- A strategic choice of which KS2 topics to re-teach in year 7 and which not to teach. For example, basic fractions and percentages are important concepts that most students struggle with and is therefore given curriculum teaching time. In contrast, place value is a key topic but many students gain a strong understanding of this in KS2 and do not require re-teaching in KS3 so this is left as an element for teachers to respond to (see section 2).
- Common (and challenging) teaching objectives for all abilities in years 7 to 9 are programmed. Time is allowed for teachers to work on pre-requisite learning that is not secure. Material is provided to deepen the understanding of more able students.
- Granular assessment by teachers in all lessons. This might be using any of the following techniques; cold call, mini whiteboards, hinge question, exit pass, Sparx Classroom data, LSQ..
- Common assessments with high levels of challenge and a systematic approach towards preparing students for them.
- Teaching objectives that interleave previously learnt concepts with the current area of study. This gives students practice as well as the opportunity to engage with complex multi-step problems.
- Teaching for fluency and recall
Learning is a combination of understanding and remembering. While understanding can assist remembering, teachers should use specific strategies for helping students to retain their knowledge and understanding in the long term.
The ESW Mathematics curriculum supports teaching for fluency and recall in the following ways:
- Provision of do now booklets for use every lesson containing practice of previously learnt concepts. These are editable by teachers to allow them to meet the exact needs of their classes.
- Four annual cumulative assessments that contain a high proportion of prior learning.
- Pre-assessment preparation homework materials that assist students and teachers with revision of prior learning before each assessment.
- Inclusion (or interleaving) of topics. This provides spaced practice for students for previously learnt concepts as well as developing links between areas of knowledge. For example, in the year 7 unit on percentages, there are teaching objectives and materials that focus on using percentage with area. This helps students to recall their understanding of area while learning about percentages.
- Reducing workload and improving lesson planning
We recognise that workload is a challenge that can impact on our ability to recruit and retain high quality teachers. It is important to ensure that teachers’ time is used carefully and in ways that have maximum impact on the progress of students. One of the key areas to reduce workload and at the same time improve the quality of lesson planning is in the provision of resources. This allows teachers to focus their efforts on customising their lesson for their class rather than spending their time looking for or creating resources.
The ESW Mathematics curriculum aims to reduce workload by:
- Providing a clear selection of trusted resources to reduce or remove the need for teachers to create or find resources.
- Ensuring that where possible, resources support a systematic and logical approach to introducing and teaching a topic.
- Providing resources that support effective teaching techniques that have a strong research base (low stakes quizzing, direct instruction, variation theory, silent teacher, atomisation, interleaving and spaced practice)
Sparx and the ESW Maths Curriculum
Purpose of this document
To provide a clear reasoning behind our investment in and use of Sparx homework and classroom to support maths teaching in Kingsbridge Community College and ESW schools.
What are Sparx homework and classroom?
Sparx classroom is best thought of as an electronic maths textbook that monitors students. Teachers select objectives for their lesson and Sparx provides a carefully designed sequence of practice questions for these objectives. The questions are grouped by difficulty into introduce, strengthen and deepen. Teachers use these questions in the same way that they would use a worksheet or textbook with the following differences:
- All students get the same form of question with different numbers to eliminate copying
- Students get immediate feedback when they input an answer (as does the teacher)
- All questions have custom help videos tailored to the specific question
- The interface is uncluttered and designed to reduce cognitive load
- Students get small rewards at the end of task sections
- Teachers get instant feedback on which questions students have done correctly, how many attempts and if they have accessed the help video.
- Teachers can lock all devices to help students focus on their teaching.
Sparx homework
- Automatically set, every week.
- Personalised for each student based on their progress on previous homework.
- Instantly marked
- Students have to get every question correct (they are encouraged to seek help if they get a question wrong 3 times in a row)
- Help videos for all question to support
- Well-designed questions
- Completion data enables teachers and leaders to focus on students not engaging.
How Sparx supports our classroom teaching strategies
Teaching strategy |
Sparx support |
Impact notes |
Responsive teaching informed by effective in lesson AFL mechanisms. |
Sparx classroom keeps the teacher informed of exactly which questions each student has completed at all times. This enables the teacher to respond appropriately. |
This works well. Teachers using Sparx are able to keep track of where students are in their work. Wrong answers are flagged up and the teacher can respond. It has been particularly effective for picking up on students who engage in the appearance of work without the substance. Students can’t hide in a Sparx lesson. |
Variation in practice questions to build understanding and avoid students creating false generalisations by following a pattern. |
The order of Sparx classroom questions is carefully planned to scaffold understanding. Questions sets are sequenced logically. |
Questions are high quality. For weaker groups the difficulty can sometimes ramp up too quickly and the teacher has to provide fluency building material to support. However, students using Sparx are less likely to face worksheets containing a lot of very similar questions. |
Able students are supported to deepen their knowledge and understanding. |
Sparx lessons contain an ambitious amount of content that allows the hard-working student to deepen their understanding of a topic. Support videos means that they can succeed with on this with minimal teacher support. Consequently, the teacher is able to work with those who are struggling. |
This is a very successful element of Sparx classroom. It has eliminated the situation of children waiting for their teacher when they finish a task. |
Independent study skills are developed |
Support videos for everything mean that students have an option for help when the teacher is unavailable. |
This is a useful feature and students have increased their ability to learn from examples. However, we have found a need to encourage students not to use this support in the first instance. This is undesired behaviour but still better than doing nothing when stuck. |
Teacher time is used efficiently |
The availability of Sparx lessons significantly reduces time spent searching for or making resources. |
The effectiveness of this depends on the amount of use an individual teacher makes of Sparx. This is not consistent across teams yet. |
How Sparx supports our homework strategies
Homework strategy |
Sparx support |
Impact notes |
Centrally set homework ensures a consistent level of quality for all students. |
Sparx homework is set for each year group every week on the same day. |
This has been successful – there is now no difference in quality, quantity or regularity between classes. |
Homework should be achievable for all students |
All students get the same quality of homework although the amount and difficulty is matched to the child. |
This has generally worked well but students and parents have required educating about the adaptive nature of the homework. The personalisation algorithm has been misled by parental assistance. Some teachers have found the reduced control over the content of homework frustrating. |
Automatically marked homework creates time for teachers to chase non-completion |
Students get instant feedback on their answers and have to get all questions correct to complete their homework. |
Instead of marking, teachers’ time is now used to chase homework and to plan teaching based on the homework insight summary available from Sparx. |
Clear data on homework completion allows students and teachers to be supported. |
Completion data for individuals, classes and year groups allows engagement with homework to be monitored and improved. |
Data is supporting school and department leadership to intervene with individuals and groups to increase engagement. |
Students should have access to homework support when they are stuck. |
All homework questions have support videos that are matched to the question. |
This has been popular with students, parents and anyone who has tried to help students with their homework. |
Teacher time is used efficiently |
Teachers do not have to spend much time setting homework and no time marking it. Homework insights support teachers in planning further teaching. |
This has been a big contribution to the workload agenda and has focussed our attention on the students who are not doing their homework rather than the quality of teacher marking. |
ESW Maths curriculum reasoning
Principles:
- Common teaching objectives for all abilities up to the end of year 9.
Setting is not exact and students mature at different rates. Pre-requisite learning that is not secure can be taught in context or if the teacher considers it necessary can be given lesson time.
- Common assessment for all abilities up to the end of year 9.
Setting is not exact. Younger students have the opportunities to show how they directly compare with their peers. Teachers will be able to more directly compare their impact.
- Do now and homework designed to assist recall.
This is continuing the work done on the do now activities and extending it to homework. Both of these things should be tied to the scheme to reduce teacher planning. There will need to be different difficulty levels of homework.
- Time allowed for teaching each topic will allow teachers to use the mastery cycle and do their repair as part of their first teaching.
Most people need a bit of time to get things the first time – we should recognise and allow for this.
- Opportunities for interleaving will be clearly identified and resourced.
We should make it easy for teachers to combine topics and use the opportunity for extending the ability of students to use their learning in different contexts (aka problem solving)
- Sparx friendly
The Sparx classroom product is an extremely powerful teaching tool and we should ensure that our scheme has as few barriers to its use as possible.
Weak students, the same teaching objectives for all and the overlap with key stage 2
Students come to secondary having had years of teaching on things like written methods, place value etc. This sort of KS2 pre-requisite knowledge will be identified on the scheme (but not as teaching objectives) so the teacher is aware of it. If the teacher considers that the class has an area of weakness they will do the following:
- In the first instance, the teacher will attempt to repair/develop the KS2 knowledge in the context of the topic being taught (this can be done on an individual/small group or whole class basis depending on the need).
- If there is a serious lack of understanding then the teacher will devote teaching time for the whole class before resuming the main topic objectives. This is where professional judgement will be required.
Cognitive load, notation and calculators
The teaching guidance should make it clear where teachers should be aware of inadvertently making a challenging topic unnecessarily more difficult. An example of this might be teaching expanding brackets with a group that struggles with their times tables. The scheme should encourage teachers to use calculators where appropriate to reduce cognitive load and then deliberately introduce non-calculator work as an additional complication when students understand the main concept.
Similarly, where possible students should be introduced to ideas before notation is added. Examples of this would include:
Pythagoras - where a2+b2=c2 should be introduced after students understand that solving Pythagoras consists of 1)squaring the sides, 2)deciding whether to add or subtract and 3)square rooting the answer.
The parallel line angle facts – where students should understand that the two sets of angles on each parallel line are identical (and that there is no proof or reason for this) before the idea is complicated by the naming conventions for alternate, corresponding, allied, co-interior, z and f angles etc.
Mastery
From the national curriculum document (italics mine):
Decisions about progression should be based on the security of pupils’ understanding and their readiness to progress to the next stage[objective]. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent should consolidate their understanding, including through additional practice, before moving on.
In practice, the teacher will be constrained by the time allowed on the scheme.
Fitting everything in
Leaving aside students who have a lot of sickness or join the school after year 7, we should be able to be certain which topics students have been taught. It is unfair to leave the decision of whether or not to teach something to a given student down to the class teacher and where possible this decision should be made by the scheme of work.
Time allocations should allow enough time for the average teacher to teach the average class the given objectives. The objective lists should be given in difficulty order so that teachers of the weakest groups know which objectives (if any) they can leave off. More able groups should be stretched by being given more complex teaching material around the given objectives (possibly material that involves other areas of maths that have already been taught).
Y11 should be the time when the teacher is required to use assessment evidence to prioritise which topics to repair and develop for the final GCSE exams.
KS2 knowledge areas that will not be identified for specific teaching. The scheme should identify opportunities for teachers to look for understanding of these things so they can do repair work if necessary. Do Now tasks should also contain questions covering the overlap between KS2 and KS3.
- Written methods for multiplication, division, addition and subtraction of integers and decimals. Students should continue to practice and improve this area of KS2 learning but we should only apply non calculator methods to a topic after it has been learnt, as a strategy to reduce cognitive load.
- Equivalent fractions (introduced in year 3) – this should be checked as part of the fractions of an amount and adding/subtracting fractions section.
- FDP (Year 4/5) – should be taught as part of the percentage multiplier work.
- Money and time calculations (all through KS2).
- Rounding to whole numbers and powers of 10 (years 4&5) – we should be teaching rounding as a generalised skill and getting students to see the general process for rounding to any degree of accuracy.
- Knowledge of unit conversions (metric) and the ability to convert between units (a year 5 objective). This should be covered as part of area and perimeter.
- Bar charts and pictograms – students will have first met these in year 3
- Names of triangles and quadrilaterals (and their properties) – this comes from the year 4 programme of study. Students should be reminded of these alongside the teaching of area.
- Basic angle facts (from year 5) – around a point, on a line and in a triangle. These are naturally covered in the study of harder angle facts.
- Factors, multiples and primes (year 5) – these should be taught alongside prime decomposition, fractions and referred to where appropriate (eg/ when factorising).
Pre-A level topics that don’t really get developed further at GCSE. These are split into two groups, difficult topics that should be fully taught to high ability students who have mastered the rest of the curriculum and topics that are an easy way for mid-ability students to get extra marks
Hard
- Fractional indices – very little benefit to GCSE algebra. Extremely important at A level. Some opportunity to enjoy interesting numbers and practice roots and powers. Do at end.
- Transformations of graphs and common graph shapes. – Transformations mainly useful as prep for A level. Common graph shapes is a memory issue.
- Geometric sequences – a pre-A level topic with very limited application at GCSE.
- Functions and function notation – pre-A level
Easy
- Recurring decimals to fractions – a curiosity at GCSE. Useful for students going on to A level as it provides a first view of a strategy for dealing with infinite sequences.
- Combinations – another pre-a level idea. Not complex at GCSE.
Random topics with low long-term impact. Things that do not develop general mathematical skills such as algebraic or proportional reasoning. These should generally be taught closer to the exam as there are likely to be fewer opportunities to develop them in the context of other topics.
- Stem and leaf – rarely used outside of GCSE. Useful only for the exam. Easy to teach and understand.
- Iteration – unusual notation that is only used in this topic.
- Vector proof – useful as it reinforces ratio and fractions in its harder form but not a big idea.
- Compass constructions – develops students’ dexterity and supports understanding of angles but not a basis for much else at GCSE.
- Loci – confusing for many and a topic that doesn’t really support the development of other mathematical ideas.
Topics to do after the higher/foundation split
These are topics that only appear on the higher tier and represent the end of the GCSE progression and the start of pre-A level work.
- Bounds
- Surds
- Quadratics/linear simultaneous equations
- Equation of a tangent to a circle
- Geometric sequences
- Gradient of curves
- Proof (the harder stuff)
- Cosine and sine rule
- 2D inequalities
Topics that provide level 6/7 students with practice of key skills while extending level 8/9 students
- Quadratics/linear simultaneous equations. With scaffolding these provide a lot of practice simplifying and solving quadratics.
- Equation of a tangent to a circle. If scaffolded, this provides great practice of most of the straight line coordinate geometry at GCSE.
- 2D inequalities – lots of straight line drawing practice for weaker students.
- Some algebraic proof – lots of practice expanding double brackets and simplifying.