Forget Finland. Could Estonia help to reverse our dire results?
Here’s a 2 minute summary of this post, for those in a hurry:
- Don’t be fooled by the backslapping: the country’s GCSE results are pretty terrible (and no: warm-strict, knowledge-rich schools don’t seem to be doing much better).
- After 12 years of school, 75% of pupils got 60–100% of their answers wrong on this year’s maths exams.
- 42% of pupils — almost half of all entrants — could not score more than 17% on their maths paper, getting 83–100% of answers wrong (read full blog for more details).
- Even a score of 56% was a monumental struggle for most pupils, with only 7.4% reaching this level of confidence.
- Things are not much better in other core subjects.
- I do not blame schools (85% of which are judged ‘good’ or ‘outstanding’); I blame the curriculum, which is too big for large-scale mastery and runs counter to the main findings of cognitive science.
- The current behemoth of a curriculum is biased towards those with exceptionally high fluid intelligence and working memory, a bottle-neck which appears genetically determined.
- This in-built discrimination against those with poorer working memory means that the gap widens with each new objective taught. With a statutory duty to cover all the objectives, basic fluency soon becomes a distant dream for huge numbers of pupils.
- I agree with David Didau, that: “Due to their greater working memory capacity, someone with higher fluid intelligence will process more information in a given time and is more likely to retain more of it than someone with lower fluid intelligence.”
- The ‘given time’ in this sentence is important:
- At KS2, children have approximately 3 days to learn each maths objective to the point of solving reasoning questions with a high level of element interactivity (see blog for definition / examples).
- At GCSE, it is 1.9 days per objective to reach this level of mastery. And this is assuming that every school day is a maths teaching day. In reality, it appears to be more like 1.1 days per objective.
- Even though everyone is capable of learning all of the content, time is the mother of depth, and not all working memories can process the same information into long-term memory in the same amount of time.
- Even Michaela, where the conditions for knowledge accrual and retention seem almost perfect, not even 25% of pupils could reach 85% mastery on their maths exams
- NB: before you only read the summary and misread me, here is a direct quote from the blog: ‘I would send my own children to Michaela if I could, and I think their GCSE results constitute a triumph against the odds.’ For what it’s worth…even a quarter of Eton pupils couldn’t get above 75% on their tests, and half of pupils at (£60,000 per year) Dulwich College only managed 50% on theirs.
- One reason countries like China and South Korea do so much better than us in PISA tests is that they effectively increase the amount of teaching time per objective, with children often attending ‘cram schools’ until 10pm.
- Estonia, one of the poorest countries in Europe, occupies a fascinating position at the top of the PISA rankings just below the Asian powerhouses.
- School starts at age 7, the hours are similar to the UK and most teaching applicants come from the bottom half of university graduates…so what is their secret sauce?
- For one, their equivalent Year 10–11 maths curriculum has 83% fewer objectives.
- In addition, from Year 2 to Year 6, Estonia give each pupil 4 weeks of teaching time per maths objective (in contrast to our 3 days!)
- They also devote more time to maths lessons. To match the amount of teaching time per objective, we would have to increase our current lesson time by 488%.
- Their entire curriculum seems devoted to mastering threshold concepts, which seem to anchor more children in mathematical fluency.
- I believe we have reached the ‘Goldilocks Plateau’ (see blog for explanation)
- Until we re-calibrate the 3 levers of difficulty, number of objectives and teaching time per objective, our children’s knowledge will remain a mile wide and an inch deep.
- To see the two countries’ Year 10–11 curricula side by side, see the appendix at the end of this blog.
Something is Wrong
Imagine for a moment that you are at parent’s evening and you open up your child’s maths book. On page after page, 80 to 100% of their answers are wrong.
Confused (this is the end of the year and you thought your child was doing well), you decide to have a peek in the other maths books which are laid out on the table. Unbelievably, you notice that in almost half the other maths books, 80–100% of answers are wrong, too.
Now imagine that instead of one teacher’s maths books, these were the nation’s GCSE maths papers.
Come August, it is very easy to be fooled by the post-results backslapping. While the headlines fleet across a general upward trend in results, a more disturbing truth lies hidden in plain sight. Despite an increase in ‘passes’ and ‘top grades’, our children are doing badly. Very badly indeed. Dig beneath the surface language of improved outcomes and what you find is shocking.
Let’s start with maths. This year, 75% of GCSE entries were graded 5 or below. Depending on the exam board, the grade 5 passmark on the most popular paper was between 27% and 34% (or an equivalent score on the foundation paper), with a maximum score of 39–45%.
In other words, after 12 years of schooling, regular exam practice and weeks of cramming — at the very point when children should know the most, SEVENTY-FIVE PERCENT of students got around 60–100% of their maths questions wrong.
Dig a little deeper and the numbers get even more alarming. Taking the AQA results (which constitute about a third of maths entries and are virtually indistinguishable from any other exam board’s results), an eye-watering 42% — almost half of all students taking a maths GCSE — got 83–100% of questions wrong (or an equivalent score on the foundation paper).
The maths results are particularly dire, but it is a very similar picture in other core subjects too. While grade boundaries and distributions fluctuate slightly depending on the exam board and other factors, one finding remains almost constant: around half of all students get around half or more of all their questions wrong (with the number of wrong answers increasing sharply for the bottom 40% of pupils).
What’s the problem?
Let me make one thing very clear: I am categorically not blaming schools. It would certainly raise a few questions of Ofsted — who say that 85% of our schools are good or outstanding — if it was purely down to poor teaching. That seems one hell of a thing to miss!
My hypothesis is simple: the national curriculum, from Year 1 to Year 11, is too big for large-scale mastery.
Call me old fashioned, but I think the fundamental aim of a statutory curriculum — a body of knowledge that the nation has decided it wants to pass on to future generations — should be to get every student mastering 100% of it. To design it any other way would seem absurd — and a massive waste of time.
If I was setting a quiz on any other body of knowledge, which I do regularly, I’d want at least 85% success on the quiz from at least 85% of children. I don’t know a teacher who wouldn’t be looking for similar levels of understanding.
Yet almost no-one is reaching this level of mastery at any stage of English schooling. Not even close.
Nationally, the percentage of children in 2019 who reached 85% in their maths GCSE was 2.9%. Even a score of 56% was a monumental struggle for most pupils, with only 7.4% reaching this level of confidence.
And if you’re like me, you will have been greatly anticipating the GCSE results of knowledge-rich schools that use explicit instruction and enforce strict discipline. For me, these schools represent the best hope our children have of imbibing the knowledge that will open doors in later life. I want to get it on the record here, before I am accused of school-shaming, that I would send my own children to Michaela if I could, and I think their GCSE results constitute a triumph against the odds.
However, even Michaela, where conditions for knowledge accrual and retrieval are about as close to perfect as they could possibly be, not even 25% of children could reach 85% mastery on the test. And I have a very strong feeling that 85% would not be seen as unreachable on a Michaela quiz.
The ‘Goldilocks Plateau’ and the problem of working memory
Before I get to Goldilocks, I want to take a look at what I think is at the crux of the problem, namely that the size of the national curriculum is discriminatory towards those with poorer fluid intelligence and working memory, and will forever get in the way of mass fluency until it is reduced.
Another way of looking at it, perhaps, is that it is biased against those with strong fluid intelligence.
Fluid Intelligence and Working Memory: a 60 second reminder
Most people will have heard these terms before so I don’t want to get into detailed definitions here. Suffice it to say that fluid intelligence is the capacity to think logically and solve problems, completely independent of what we know (knowledge), which is known as ‘crystallised intelligence.’
Fluid intelligence correlates particularly well with working memory, which is that part of our memory concerned with processing the task at hand.
Here’s the thing: we know that in all populations (such as a maths class), there is a spectrum of fluid intelligence and working memory. Not all working memories are born equal.
In other words, even though we may all be capable of learning all of the content, some of us can process lots of stuff in an hour’s lesson, and some of us can’t. This is genetically determined and there isn’t anything we can do about it.
As David Didau puts it in his book, Making Kids Cleverer:
“Despite the claims of various brain training gurus, it doesn’t actually appear to be possible to increase working memory capacity or fluid intelligence — what you’ve got is what you get.”
And here’s the rub when it comes to the national curriculum:
“Due to their greater working memory capacity, someone with higher fluid intelligence will process more information in a given time and is more likely to retain more of it than someone with lower fluid intelligence.”
Now, it may sound like I am advocating a dumbing down of the curriculum just because some people can’t process things quickly. I’m not. This final quote of Didau’s is my mantra: “Given sufficient time, everyone can remember stuff.”
The ‘everyone’ here is important. It’s why most of us get up in the morning. But so is the ‘sufficient time’. We are losing track of the fact that there is, quite literally, a physical limit on how much information an entire group can process within a given time, and it appears that we are exceeding that limit for the vast majority of pupils. This means that teachers are forced to tear through content and make mastery almost impossible for the majority.
We are stuck between a rock and a hard place. While the content of the national curriculum is statutory and must be taught within the allotted time period, only the top 15% of pupils seem able to keep up and master it.
Teaching time per objective
At primary, in Key Stage 2, the aforementioned ‘alloted time’ is around 3 days per new maths objective. Three days to build up varied fluency and master a concept to the point of solving problems with a high level of element interactivity.
Examples of low vs. high element interactivity (see more on Greg Ashman’s blog here)
At GCSE, it is around 1.9 days per objective. And when I say ‘objective’ — here is an example of ‘one’ objective:
- order positive and negative integers, decimals and fractions
As you can see, this objective masks multiple different concepts. This is by no means atypical of the other 202 objectives.
Another thing: the 1.9 days per objective assumes that every single school day contains a maths lesson (with no polling booth days, trips, residentials, reward days etc).
In a speech in 2010, Michael Gove claimed that the average secondary school spent 116 hours a year teaching maths, which suggests that the reality is closer to 1.1 days per objective.
This figure also leaves out the reams of extra notes attached to certain objectives, as well as the appendices of formulae to be taught.
In a nutshell
The bottom half of students must be kept on the treadmill at a pace of learning that seems beyond their ability to process it into long-term memory, a crucial criterion for mastery. Therefore, teachers are racing through content and leaving children behind, unable to master the basics.
National results, from SATs to GCSEs, strongly suggest that our children’s knowledge is a mile wide but only inch deep.
This seems an extraordinary waste of time as well as a recipe for frustration, poor results and teacher burnout.
The Goldilocks Plateau
My starting point here is that every curriculum in the world can be placed on 3 key spectrums:
- DIFFICULTY (too easy = never going beyond number bonds to 10 → too difficult = only ever trying to solve the 7 Millenium Prize Problems from Year 1 upwards)
- NUMBER OF OBJECTIVES (too few = only 1 objective from Year 1 to Year 11→ too many = 5,000 objectives per year group)
- TEACHING TIME PER OBJECTIVE (too little = 30 seconds per objective → too much = 30 years per objective)
How would your year as a teacher look different if I only gave you one objective for the whole year? What would observations, Ofsted inspections, interventions, meetings, performance management meetings etc look and feel like with one objective for the whole year? What would your SATs or GCSE results look like if this was the only thing tested?
Don’t worry, that isn’t what I’m advocating. One objective is clearly ridiculous.
I’m going to give you 3 objectives for the whole year (yes, still ridiculous, bear with me). The pupils differ slightly in their ability, so it doesn’t feel quite as much of a doddle any more…but hey, 3 objectives — you’re still living the dream.
You find by the end of the year that every single child in your class has reached a level of mastery completely unheard of. What are the effects on the children? On your morale and theirs? How does this sense of conquest affect their behaviour and attitude in other lessons?
Ok, ok, still ridiculous.
Let’s start ramping it up…30 maths objectives…as hard as you try now, some children are struggling to keep up and differentiation is becoming a logistical burden…
60 objectives now. That gives you 3 days per objective. Every time you move on to something new, you know you’re leaving children behind and widening the gap in your classroom. But it’s statutory and you have to cover everything, so you plough on.
Right, let’s whack it up to 90 objectives, each one to be mastered to the point of answering reasoning questions with high element interactivity.
But don’t forget, each of these 90 objectives has to pass through the working-memory-bottleneck of 30 different children, some of whom had breakfast today and some of whom didn’t; some of whom stayed up last night playing Fortnite and some of whom didn’t.
And now, finally, let’s try 100 maths objectives. If you teach every single day, this will give pupils 1.9 days to master every single objective to the point of solving problems with a potentially high level of element interactivity.
You can see where I’m going: each layer that is placed on top of one objective adds a layer of complexity that begins to change EVERYTHING. Add enough layers and you’ll find that morale, meetings, monitoring and money go into complete free fall in pursuit of the impossible.
On the one hand, we have a system that demands no child is left behind, but on the other we have a curriculum which, by its very design, leaves huge numbers of children behind.
Starting with one objective and building up one layer at a time, we eventually hit what I think of as the ‘Goldilocks Plateau’: that point beyond which not everyone can master all of the content.
In other words, the point at which it feels ‘just right’.
Every statistic on English education that I can see suggests we reached this plateau long ago.
But Look at China!
I don’t want to get too bogged down in PISA rankings, given how obviously flawed the whole system is.
However, it is interesting to look at the top echelons of the maths league table. How do China, Singapore, Japan and Korea manage equally ginormous curricula and do so much better than us?
The answer is multifaceted, but one reason is glaringly obvious.
They have increased the amount of teaching time per objective. They have increased it by A LOT. Things became so serious in South Korea, for example, that they had to pass a law to prevent hagwons (after-hours ‘cram schools’) from closing after 10pm.
In China, it is not unusual to have classes 6.5 days a week from 7.30am to 9pm.
Is this what we want for our children? Mastery at the expense of their waking lives?
Estonia: the promised land?
There is one very interesting entry in the PISA top ten.
How has Estonia, one of the poorest countries in the EU, with some of the lowest paid teachers in the world, managed to land just beneath the Asian powerhouses in the global maths rankings? (they are, by the way, even higher for science and reading).
School starts at age 7, the hours are similar to the UK and most teaching applicants come from the bottom half of university graduates…so what is their secret sauce?
Well, I’d be willing to bet that it has a lot to do with their curriculum, which is, comparatively speaking, absolutely miniscule.
In maths, for example, their equivalent GCSE syllabus is almost 83% smaller than ours.
This leaves 10 days per maths objective at GCSE level (compared to our 1.9), with each one resembling a ‘threshold concept’ that will anchor pupils more firmly in mathematical fluency.
But that’s not all. To match their teaching time per objective, we’d have to increase ours by 488% (within comparable school hours). In their national curriculum, it is stipulated that there will be 13 maths lessons per week. The stats above assume that each lesson is 45 minutes long and would be even higher if they were one hour.
Oh, and there’s more.
From the age of 7 to 12, which spans Year 2 to Year 6 in England and Wales, there are only TWENTY-FOUR objectives. This leaves an absolutely whopping 23 days per objective.
That’s over 4 weeks of school devoted to each new maths concept.
I recognise a winning formula when I see one: boil the curriculum down to key threshold concepts, and spend more time mastering them.
Mark my words: until we radically reduce our curriculum, children’s fluency will forever be harmed.
That table again:
APPENDIX
To see the two countries’ curricula contrasted, see below (look for the flags to see where one ends and one begins)
WARNING: The English GCSE curriculum is almost 3,000 words long and would take you around 15 minutes just to read)
English GCSE Syllabus STARTS HERE (notes in bold):
order positive and negative integers, decimals and fractions
use the symbols =, ≠, <, >, ≤, ≥
*Notes:*including use of a number line.
apply the four operations, including formal written methods, to integers, decimals and simple fractions (proper and improper), and mixed numbers — all both positive and negative
understand and use place value (eg when working with very large or very small numbers, and when calculating with decimals)
*Notes:* including questions set in context.
Knowledge and understanding of terms used in household finance, for example profit, loss, cost price, selling price, debit, credit, balance, income tax, VAT and interest rate. See also
recognise and use relationships between operations, including inverse operations (eg cancellation to simplify calculations and expressions)
use conventional notation for priority of operations, including brackets, powers, roots and reciprocals
use the concepts and vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation theorem
apply systematic listing strategies
including use of the product rule for counting
*Notes:*including using lists, tables and diagrams.
use positive integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5
estimate powers and roots of any given positive number
*Notes:*including square numbers up to 15×15
Students should know that 1000=103 and 1 million = 106
calculate with roots, and with integer indices
calculate with fractional indices
calculate exactly with fractions
calculate exactly with multiples of /π/
calculate exactly with surds
simplify surd expressions involving squares and rationalise denominators
calculate with and interpret standard form/A/×10/n/, where 1≤/A/<10 and /n/ is an integer
*Notes:*with and without a calculator.
Interpret calculator displays.
work interchangeably with terminating decimals and their corresponding fractions
change recurring decimals into their corresponding fractions and vice versa
Notes:*including ordering.
identify and work with fractions in ratio problems
interpret fractions and percentages as operators
*Notes:*including interpreting percentage problems using a multiplier.
use standard units of mass, length, time, money and other measures (including standard compound measures) using decimal quantities where appropriate
*Notes*: know and use metric conversion factors for length, area, volume and capacity.
estimate answers
check calculations using approximation and estimation, including answers obtained using technology
*Notes*: including evaluation of results obtained.
round numbers and measures to an appropriate degree of accuracy (eg to a specified number of decimal places or significant figures)
use inequality notation to specify simple error intervals due to truncation or rounding
*Notes*: including appropriate rounding for questions set in context.
Students should know not to round values during intermediate steps of a calculation.
apply and interpret limits of accuracy
including upper and lower bounds
use and interpret algebraic notation, including:
* /ab/ in place of /a/×/b/
* 3/y/ in place of /y/+/y/+/y/ and 3×/y/
* /a/2 in place of /a/×/a/ , /a/3 in place of /a/×/a/×/a/ , /a/2/b/ in place of /a/×/a/×/b/
* /a/
*
* /b/
*
in place of /a/÷/b/
coefficients written as fractions rather than as decimals
brackets
*Notes*: it is expected that answers will be given in their simplest form without an explicit instruction to do so.
substitute numerical values into formulae and expressions, including scientific formulae
*Notes*: unfamiliar formulae will be given in the question.
See the Appendix for a full list of the prescribed formulae.
understand and use the concepts and vocabulary of expressions, equations, formulae, inequalities, terms and factors
to include identities
simplify and manipulate algebraic expressions by:
simplify and manipulate algebraic expressions (including those involving surds) by:
* collecting like terms
* multiplying a single term over a bracket
* taking out common factors
* simplifying expressions involving sums, products and powers, including the laws of indices
simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by:
* expanding products of two binomials
* factorising quadratic expressions of the form /x/2+/bx/+/c/, including the difference of two squares
* expanding products of two or more binomials
* factorising quadratic expressions of the form /ax/2+/bx/+/c/
understand and use standard mathematical formulae
rearrange formulae to change the subject
*Notes*: including use of formulae from other subjects in words and using symbols.
know the difference between an equation and an identity
argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments
to include proofs
where appropriate, interpret simple expressions as functions with inputs and outputs
interpret the reverse process as the ‘inverse function’
interpret the succession of two functions as a ‘composite function’
*Notes*: understanding and use of f(/x/) , fg(/x/) and f-1# (/x/# ) notation is expected at Higher tier.
work with coordinates in all four quadrants
plot graphs of equations that correspond to straight-line graphs in the coordinate plane
use the form /y/=/mx/+/c/ to identify parallel lines
find the equation of the line through two given points, or through one point with a given gradient
use the form /y/=/mx/+/c/ to identify perpendicular lines
identify and interpret gradients and intercepts of linear functions graphically and algebraically
identify and interpret roots, intercepts and turning points of quadratic functions graphically
deduce roots algebraically
deduce turning points by completing the square
recognise, sketch and interpret graphs of linear functions and quadratic functions
including simple cubic functions and the reciprocal function /y/=
1
/x/
with /x/≠0
including exponential functions /y/=/k//x/ for positive values of /k/, and the trigonometric functions (with arguments in degrees) /y/=sin/x/ , /y/=cos/x/ and /y/=tan/x/ for angles of any size
sketch translations and reflections of a given function
plot and interpret graphs, and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration
including reciprocal graphs
including exponential graphs
*Notes*: including problems requiring a graphical solution.
calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts
recognise and use the equation of a circle with centre at the origin
find the equation of a tangent to a circle at a given point
solve linear equations in one unknown algebraically
find approximate solutions using a graph
including those with the unknown on both sides of the equation
solve quadratic equations algebraically by factorising
find approximate solutions using a graph
including those that require rearrangement
including completing the square and by using the quadratic formula
solve two simultaneous equations in two variables (linear/linear) algebraically
find approximate solutions using a graph
including linear/quadratic
find approximate solutions to equations numerically using iteration
*Notes*: including the use of suffix notation in recursive formulae.
translate simple situations or procedures into algebraic expressions or formulae
derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution
solve linear inequalities in one variable
solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable
represent the solution set on a number line
represent the solution set on a number line, using set notation and on a graph
*Notes*: students should know the conventions of an open circle on a number line for a strict inequality and a closed circle for an included boundary.
In graphical work the convention of a dashed line for strict inequalities and a solid line for an included inequality will be required.
generate terms of a sequence from either a term-to-term or a position-to-term rule
*Notes*: including from patterns and diagrams.
recognise and use sequences of triangular, square and cube numbers and simple arithmetic progressions
including Fibonacci-type sequences, quadratic sequences, and simple geometric progressions # (/r//n/ where /n/ is an integer and /r/ is a rational number > 0)
including other sequences
including where /r/ is a surd
deduce expressions to calculate the /n/th term of linear sequences
including quadratic sequences
change freely between related standard units (eg time, length, area, volume/capacity, mass) and compound units (eg speed, rates of pay, prices) in numerical contexts
compound units (eg density, pressure)
in numerical and algebraic contexts
use scale factors, scale diagrams and maps
*Notes*: including geometrical problems.
express one quantity as a fraction of another, where the fraction is less than 1 or greater than 1
use ratio notation, including reduction to simplest form
divide a given quantity into two parts in a given part : part or part : whole ratio
express the division of a quantity into two parts as a ratio
apply ratio to real contexts and problems (such as those involving conversion, comparison, scaling, mixing, concentrations)
*Notes*: including better value or best-buy problems.
express a multiplicative relationship between two quantities as a ratio or a fraction
understand and use proportion as equality of ratios
relate ratios to fractions and to linear functions
define percentage as ‘number of parts per hundred’
interpret percentages and percentage changes as a fraction or a decimal, and interpret these multiplicatively
express one quantity as a percentage of another
compare two quantities using percentages
work with percentages greater than 100%
solve problems involving percentage change, including percentage increase/decrease and original value problems, and simple interest including in financial mathematics
solve problems involving direct and inverse proportion, including graphical and algebraic representations
use compound units such as speed, rates of pay, unit pricing
use compound units such as density and pressure
*Notes*: including making comparisons.
compare lengths, areas and volumes using ratio notation
scale factors
make links to similarity (including trigonometric ratios)
understand that /X/ is inversely proportional to /Y/ is equivalent to /X/ is proportional to
1/Y/
interpret equations that describe direct and inverse proportion
construct and interpret equations that describe direct and inverse proportion
interpret the gradient of a straight-line graph as a rate of change
recognise and interpret graphs that illustrate direct and inverse proportion
interpret the gradient at a point on a curve as the instantaneous rate of change
apply the concepts of average and instantaneous rate of change (gradients of chords and tangents) in numerical, algebraic and graphical contexts
set up, solve and interpret the answers in growth and decay problems, including compound interest
and work with general iterative processes
use conventional terms and notations: points, lines, vertices, edges, planes, parallel lines, perpendicular lines, right angles, polygons, regular polygons and polygons with reflection and/or rotation symmetries
use the standard conventions for labelling and referring to the sides and angles of triangles
draw diagrams from written description
use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle)
use these to construct given figures and solve loci problems
know that the perpendicular distance from a point to a line is the shortest distance to the line
*Notes*: including constructing an angle of 60°.
apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles
understand and use alternate and corresponding angles on parallel lines
derive and use the sum of angles in a triangle (eg to deduce and use the angle sum in any polygon, and to derive properties of regular polygons)
derive and apply the properties and definitions of: special types of quadrilaterals, including square, rectangle, parallelogram, trapezium, kite and rhombus
and triangles and other plane figures using appropriate language
*Notes*: including knowing names and properties of isosceles, equilateral, scalene, right-angled, acute-angled, obtuse-angled triangles. Including knowing names and using the polygons: pentagon, hexagon, octagon and decagon.
use the basic congruence criteria for triangles (SSS, SAS, ASA, RHS)
apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides, including Pythagoras’ theorem and the fact that the base angles of an isosceles triangle are equal, and use known results to obtain simple proofs
identify, describe and construct congruent and similar shapes, including on coordinate axes, by considering rotation, reflection, translation and enlargement
including fractional scale factors
including negative scale factors
describe the changes and invariance achieved by combinations of rotations, reflections and translations
*Notes*: including using column vector notation for translations.
identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference
including: tangent, arc, sector and segment
apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results
*Notes:*including angle subtended by an arc at the centre is equal to twice the angle subtended at any point on the circumference, angle subtended at the circumference by a semicircle is 90°, angles in the same segment are equal, opposite angles in a cyclic quadrilateral sum to 180°, tangent at any point on a circle is perpendicular to the radius at that point, tangents from an external point are equal in length, the perpendicular from the centre to a chord bisects the chord, alternate segment theorem.
solve geometrical problems on coordinate axes
identify properties of the faces, surfaces, edges and vertices of: cubes, cuboids, prisms, cylinders, pyramids, cones and spheres
interpret plans and elevations of 3D shapes
construct and interpret plans and elevations of 3D shapes
use standard units of measure and related concepts (length, area, volume/capacity, mass, time, money etc.)
measure line segments and angles in geometric figures, including interpreting maps and scale drawings and use of bearings
*Notes*: including the eight compass point bearings and three-figure bearings.
know and apply formulae to calculate: area of triangles, parallelograms, trapezia;
volume of cuboids and other right prisms (including cylinders)
know the formulae: circumference of a circle = 2/πr/=/πd/
area of a circle = /πr/2
calculate perimeters of 2D shapes, including circles
areas of circles and composite shapes
surface area and volume of spheres, pyramids, cones and composite solids
*Notes*: including frustums.
Solutions in terms of /π/ may be asked for.
calculate arc lengths, angles and areas of sectors of circles
apply the concepts of congruence and similarity, including the relationships between lengths in similar figures
including the relationships between lengths, areas and volumes in similar figures
know the formulae for: Pythagoras’ theorem, /a/2+/b/2=/c/2 and the trigonometric ratios,
apply them to find angles and lengths in right-angled triangles in two dimensional figures
apply them to find angles and lengths in right-angled triangles and, where possible, general triangles in two and three dimensional figures
know the exact values of sin/θ/ and cos/θ/ for /θ/ = 0°, 30°, 45° , 60° and 90°
know the exact value of tan/θ/ for /θ/ = 0°, 30°, 45° , 60°
know and apply the sine rule,
and cosine rule,
to find unknown lengths and angles
know and apply
/Area/=
1
2
/ab/sin/C/
to calculate the area, sides or angles of any triangle
describe translations as 2D vectors
apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors
use vectors to construct geometric arguments and proofs
record, describe and analyse the frequency of outcomes of probability experiments using tables and frequency trees
apply ideas of randomness, fairness and equally likely events to calculate expected outcomes of multiple future experiments
relate relative expected frequencies to theoretical probability, using appropriate language and the 0 to 1 probability scale
apply the property that the probabilities of an exhaustive set of outcomes sum to 1
apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to 1
understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size
enumerate sets and combinations of sets systematically, using tables, grids, Venn diagrams
including using tree diagrams
construct theoretical possibility spaces for single and combined experiments with equally likely outcomes and use these to calculate theoretical probabilities
calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions
*Notes*: including knowing when to add and when to multiply two or more probabilities.
calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams
infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling
interpret and construct tables, charts and diagrams, including frequency tables, bar charts, pie charts and pictograms for categorical data, vertical line charts for ungrouped discrete numerical data, and know their appropriate use
including tables and line graphs for time series data
*Notes*: including choosing suitable statistical diagrams.
construct and interpret diagrams for grouped discrete data and continuous data, ie histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use
interpret, analyse and compare the distributions of data sets from univariate empirical distributions through:
appropriate graphical representation involving discrete, continuous and grouped data
appropriate measures of central tendency (median, mean, mode and modal class) and spread (range, including consideration of outliers)
including box plots
including quartiles and inter-quartile range
*Notes*: students should know and understand the terms: primary data, secondary data, discrete data and continuous data.
apply statistics to describe a population
use and interpret scatter graphs of bivariate data
recognise correlation
know that it does not indicate causation
draw estimated lines of best fit
make predictions
interpolate and extrapolate apparent trends whilst knowing the dangers of so doing
Notes*: students should know and understand the terms: positive correlation, negative correlation, no correlation, weak correlation and strong correlation.
*GCSE Syllabus ENDS HERE*
Estonian GCSE Equivalent Syllabus STARTS HERE (notes in bold)
add, subtract, multiply, divide and rise to a power of rational numbers with exponent of natural number mentally, in writing and by means of a pocket calculator and use the order of operations;
write large and small numbers in standard format;
round off numbers to a given accuracy;
explain the meaning of involution with exponent of natural number and use involution rules;
explain the meaning of the square root of a number and find the square root mentally or by means
of a pocket calculator;
compile dataset on the basis of actual data, arrange them, compile tables of frequency and relative
frequency and characterise the dataset by arithmetic means; and
explain the meaning of probability and calculate classic probability in simplest cases.
Notes: Calculation with rational numbers. Degrees of 10 (including negative integer exponent). Standard format of numbers. Power with exponent of natural number. Square root of numbers.
Dataset and its properties (frequency, relative frequency and arithmetic mean). The term ‘probability’.
Use of computer programmes in order to practise required skills.
find the whole on the basis of a given partial rate in percentages;
express the quotient of two numbers in percentages;
find what percentage one number forms of another;
determine increase and decrease of quantity in percentages;
interpret quantities expressed in percentages in other subjects and in everyday life, including
expenses and dangers related to loans (simple interest only); and
discuss the importance of taxes in society.
arrange monomials and multinomials, add, subtract and multiply monomials and multinomials and divide monomials and multinomials by a monomial;
factorize multinomials (bring before brackets, use auxiliary formulas and factorize quadratic trinomials);
cancel and extend algebraic fractions and add, subtract, multiply and divide algebraic fractions; 4) simplify rational expressions with two operations;
solve linear and proportional equations using the basic properties of equations;
solve linear equation systems;
solve complete and incomplete quadratic equations; and
solve word problems by means of equations and equation systems.
Notes: Monomial and multinomial. Operations with monomials and multinomials. Formulas for difference of squares, sum squares and difference squares.
Basic properties of equations. Linear equation. Linear equation system. Complete and incomplete quadratic equation. Proportional equation. Proportional decomposition. Use of computer software for solving equations and linear equations systems.
Algebraic fraction. Operations with algebraic fractions.
Solving word problems by means of equations and equation systems.
explain the meaning of proportional dependence based on real-life examples;
draw graphs of functions by formula (both by hand and by means of a computer programme) and
read the values of functions and arguments from the graph;
explain the dependence of the position and form of the function’s graph (using dynamic drawings
made on a computer) on the coefficient in the function’s expression (in the case of quadratic
function, on the quadratic term’s coefficient and a constant member only);
explain the meaning of zeros of a function and find zeros on graphs and formulas;
read the text of a parabola from the drawing and calculate the coordinates of the vertex of the
parabola.
Notes: Variable quantity and function. Proportional and inversely proportional dependence. Practical work: determination of proportional and inversely proportional relation (i.e. distance, interval of time and speed). Linear function. Quadratic function.
draw and construct (both by hand and computer) plane figures on the basis of given elements;
calculate linear elements, perimeter and area and volume of figures;
know figures, the midline of a triangle and trapezium, the median of a triangle, the circumscribed
and inscribed circles of a triangle and the central angle and peripheral angles of triangle;
describe properties of figures and classify figures according to common properties;
dentify the ‘theorem’, ‘postulate’, ‘assertion’ and ‘proof’, explain train of thought of proving certain theorems;
solve open-end problems with geometrical content;
find the linear elements of a right-angled triangle;
use similarity between triangles and polygons when solving open-end problems; and
use technological tools in discovering regularities and formulating hypotheses.
Notes: Definition, theorem, assumption, assertion and proof. Polygons (triangle, parallelogram, trapezium and regular polygon), perimeter and area of polygons.
Circle and circumference. Central angle. Peripheral angle, Thales’ Theorem. Tangent of circumference. Inscribed and circumscribed circles of triangle and regular polygon. Criterion of parallel straight lines. Midline of triangle and trapezium. Median and centre of gravity of a triangle. Similarity properties of triangles. Similarity of polygons.
Planning of areas. Pythagoras’ Theorem. Trigonometric functions of acute angles. Solid figures (vertical parallelepiped, vertical prism, pyramid, cylinder, cone and sphere), their area and volume.
