Mathematics Curriculum Overview Statement
The art of doing mathematics consists in finding that special case which contains all the germs of generality.”
― David Hilbert
The Mathematics curriculum at Poole Grammar School has three elements:
1) Intent: What do we learn?
2) Implementation: How do we learn it?
3) Impact: How do we know we have been successful?
The mathematics programme of study is designed around the four main strands of the National Curriculum; namely, “number”, “algebra”, “geometry and measures” and “statistics and probability.” Recently, “Ratio and Proportion” has been identified as a separate strand, due to the importance and power of proportional reasoning in problem solving. In our curriculum, “Ratio and Proportion” questions are connected to number, algebra, probability and shape.
We aim to build fluency, confidence and appreciation of mathematics as well as mastery of its core techniques and key concepts. We do this through an emphasis on “mathematical fundamentals” in Years 7 and 8 across the strands described above. We seek to make students fluent and confident in the language of mathematics so that, as they progress, they can tackle problems that require “mathematical decision making”.
Our aim is to establish an unthreatening climate for learning in which pupils are prepared to take risks and see their mistakes as part of their learning process. They should also appreciate that being able to explain how they approached a problem and being able to produce a mathematically coherent argument is more important than simply obtaining an answer with little working to reinforce its validity. In brief, successful learners will select appropriate knowledge and skills, break a problem into simpler steps, arrange these steps into a coherent overall method and solve each successive step accurately to find a final solution. They will check the reliability of this solution against reasonable expectations and ideally, appreciate the satisfaction of solving the problem successfully.
Ultimately, our intent is to equip students with the skill to apply their learning in mathematics to solve problems, sometimes from real life, sometimes more abstract and sometimes unfamiliar. This aim applies within mathematics itself and across the curriculum when students need to apply their learning in mathematics in other subjects.
As students take their mathematics further they will be able to appreciate the beauty of mathematical patterns, the power of mathematical models and the overall fascination of the subject.
Students’ practice of key techniques in years 7 and 8 gives them confidence at working with fractions, decimals and percentages and allows them to apply these skills in various contexts: for example, find the area and perimeter of a rectangle with fractional dimensions. We might also use a similar example with dimensions given in standard index form. In addition to this “numerical fluency”, students develop basic algebraic proficiency involving skills such as index laws, collecting similar terms and expanding brackets. Our focus as teachers is on the use of correct mathematical language with a gradual increase in rigour and we expect students to master the use of this language in their own explanations. Words such as “coefficient”, “integer”, “term” and phrases such as “Express in terms of”, “Expand and simplify” ,“Show that” or “Find the general term” are an integral part of our mathematical discussions in lessons.
When we tackle higher level problem-solving we encourage students to ask their own questions: “What topics are involved?” , “Where should we start?”, “What do I know already that can be applied in this context?”, “Is there a simpler problem that is relevant?”, “How can I prove this conjecture?”
As the confidence and mathematical proficiency of students grow, richer problems, connecting a variety of topics, are introduced. This practice fulfils a variety of key aims: it helps to make learning coherent, connecting prior and current learning; it is a useful revision technique; it develops students’ discussions skills as they work on problems together and share their strategies.
As teachers, we make use of important educational research that supports successful curriculum implementation (in other words, “good teaching”). For example, Barak Rosenshine’s “Principles of instruction” which help to connect learning and build powerful learning techniques. Regular review, the breaking down of new content, intelligent practice, assessment of progress and independent practice are all part of the lesson planning behind successful implementation. These teaching techniques have much in common with the “Mastery Curriculum”, in which our interest has led to teams of teachers in the department planning collaboratively and observing each other’s lessons.
Our students’ enthusiasm for mathematics, their enjoyment of lessons, their performance in class, their success in exams and in life beyond school are all a product of a high quality mathematical curriculum that is well- delivered. The students’ enthusiasm and success is itself a product of our department’s passion for mathematics, our teachers’ drive to engage learners and their persistent pursuit of excellence in the classroom.
The impact of the mathematical curriculum is that students can apply their knowledge and skill in mathematics to problems posed in diverse settings.
By the end of Y8 the skills of algebraic simplification and arithmetic processing enable the development of higher-level mathematical modelling, where students need to derive their own algebraic expressions or form and solve their own equations.
A firm understanding of ratio and proportional reasoning at KS3 leads students onto harder challenges at KS4 involving forming and solving equations from proportionality statements, both direct and inverse. Students should also be able to draw graphs that illustrate how two variables are connected when they are either directly or inversely proportional to each other
Similarly, an understanding of the fundamental properties of angles allows students to progress to more complex geometry; perhaps involving an appreciation of geometric proof, the connections between side lengths and angles in triangles, vectors or the circle theorems.
The work in Statistics at KS3 provides the base for accessing problems involving comparison between statistical distributions. Detecting statistical outliers, presenting data using an appropriate statistical graph, comparing sets of data using both measures of dispersion and location are all involved in higher-level statistical problem solving at KS4.
The gradual increase in GCSE exam standard questions (and Further Maths Level 2 questions for high ability students) throughout years 9, 10 and 11 means that students are continuously recalling prior skills and concepts, applying them to new problems in new contexts and analysing and checking their results against their estimations and expectations. To develop exam technique, students sometimes need to draw on all of these abilities under time pressure.
All of the skills described above are not only essential for high performing students in mathematics but also invaluable tools that will serve students well as they progress to higher level studies or professional work.
In conclusion, our intent is to achieve an implementation whose impact is both academic and non-academic, measurable and immeasurable, in school and beyond school: an enjoyment of learning, an ambition to succeed, a good set of examination results and a set of values, skills and all round intelligence for success in life.
Finally, two more quotes from, David Hilbert (23 January 1862 – 14 February 1943), widely recognised as one of the most influential and universal mathematicians of the 19th and early 20th centuries.
Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.
Wir müssen wissen — wir werden wissen! “We must know — we will know!”