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Age group:!MN@A!
Published:!I&3$&'6&/!;<<L!
Reference no:!<?<<OP!
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www.ofsted.gov.uk
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X&2&/&-)&!-1C!<?<<OP!
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Y!W/19-!W138/(0%$!;<<L!
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Contents
Executive summary 4
Key findings 6
Recommendations 7
Part A: Mathematics in primary and secondary schools 9
:)%(&4&'&-$!#-.!*$#-.#/.*! A
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Part B: Every child’s mind should matter in mathematics 35
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D,3(7*H!&-[18'&-$!#-.!4(&9*!12!'#$%&'#$()*! =;
Conclusion 55
Notes 56
Further information 57
Annexes 61
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Mathematics: understanding the score
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Key findings
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Recommendations
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!
Mathematics: understanding the score
A
!
Part A: Mathematics in primary and secondary schools
Achievement and standards
Pupils’ performance in tests and examinations
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!
Mathematics: understanding the score
@<!
!
Table 1: Pupils reaching the expected attainment thresholds in mathematics for
each key stage in 2001, 2004 and 2007
! Percentage of pupils
achieving selected
threshold indicators
2001 2004 2007
Government
target (and
target date)
Foundation
Stage
Within the Early
Learning Goals
-h#! -h#! OO! !!
Key Stage 1 Level 2+ A@! A<! A<! !
Level 4+ ?@! ?M! ??! L=!e;<<Of!Key Stage 2
Level 5+ ;=! P@! PP! !!
Level 5+ OO! ?P! ?O! L=!e;<<?f!Key Stage 3
Level 6+ MP! =;! =O! !
Grade C+ =@! =P! =?! !Key Stage 4
(GCSE)
Grade B+ P<! P;! PM! !
!
MC 5%&!('3/14&'&-$*!'#.&!(-!U&8!I$#0&!PR!%19&4&/R!#/&!-1$!6,(7$!1-!*,22()(&-$78!.,/(-0!
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2&9!8&#/*C!G-!;<<?R!?Ab!12!3,3(7*!9%1!%#.!/&#)%&.!V&4&7!O!#$!U&8!I$#0&!P!9&-$!1-!
$1!3#**!cWIB!#$!0/#.&!W!1/!%(0%&/R!#-.!;Ob!.(.!*1!2/1'!V&4&7!=C!5%&*&!3/131/$(1-*!
#/&!',)%!719&/!$%#-!$%&!)1//&*31-.(-0!2(0,/&*!21/!B-07(*%!#-.!*)(&-)&C!^1/!
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$&*$*!#$!$%&!&-.!12!U&8!I$#0&!P!(*!*,22()(&-$!$1!3/&3#/&!3,3(7*!21/!$%&(/!2,$,/&!*$,.8!12!
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%#4(-0!2#77&-!*%#/378!$1!6&719!M=R<<<!(-!;<<;!#-.!;<<PC!5%&!c14&/-'&-$H*!$#/0&$!12!
=OR<<<!&-$/(&*!68!;<@M!-19!#33&#/*!$1!6&!9($%(-!/&#)%C!`&4&/$%&7&**R!&-$/(&*!#/&!*$(77!
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12!618*!#-.!0(/7*!(*!6/1#.78!&S,#7C!T19&4&/R!*$,.&-$*!2/1'!*1'&!'(-1/($8!&$%-()!
0/1,3*!#-.!$%1*&!&7(0(67&!21/!2/&&!*)%117!'*!#/&!,-.&/!/&3/&*&-$&.!#$!:!7&4&7C!D#**!
/#$&*!#$!:I!7&4&7!%#4&!('3/14&.!*(0-(2()#-$78!2/1'!OAb!(-!;<<@!$1!L@b!(-!;<<?R!6,$!
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!
Mathematics: understanding the score
@@
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8&#/*R!2,/$%&/!'#$%&'#$()*!&-$/(&*!%#4&!(-)/&#*&.!68!MMb!#$!:I!#-.!M<b!#$!:!7&4&7C!
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!
Mathematics: understanding the score
@;!
Teaching and learning
The quality of teaching and learning in mathematics
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$%#-!(-!*&)1-.#/8!*)%117*!e^(0,/&!;fC!
Figure 2: Quality of teaching and learning in mathematics in the schools surveyed
(percentages of schools)!
7
10
47
56
41
32
5
1
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(-#.&S,#$&!(-!-&#/78!@<b!12!*&)1-.#/8!'#$%&'#$()*!7&**1-*!e^(0,/&!PfC!
Figure 3: Progress in mathematics lessons in the schools surveyed (percentages of
lessons)!
5
10
44
51
42
37
9
1
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What does good or outstanding teaching look like?
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!
Mathematics: understanding the score
@P
!
Prime practice:
teaching for
understanding
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;PC 5%&!2/&S,&-)8!#-.!S,#7($8!12!%1'&91/\!4#/(&.!9(.&78C!5%&!$#*\*!*&$!21/!%1'&91/\!
/#/&78!)#3$,/&.!3,3(7*H!('#0(-#$(1-!1/!&E$&-.&.!$%&(/!7&#/-(-0R!)1-)&-$/#$(-0!(-*$&#.!
1-!3,3(7*!3/#)$(*(-0!$#,0%$!*\(77*C!J%(7&!$%(*!(*!('31/$#-$R!*(-)&!3,3(7*!-&&.!$1!6&!
27,&-$!(-!*\(77*!(2!$%&8!#/&!$1!%#4&!$%&!(-$&77&)$,#7!*3#)&!21/!$%(-\(-0!9%&-!$%&8!$#)\7&!
'1/&!)1'37&E!1/!,-,*,#7!3/167&'*R!($!*%1,7.!-1$!6&!3,3(7*H!1-78!&E3&/(&-)&!12!
(-.&3&-.&-$!91/\C!:-!&E#'37&!12!%1'&91/\!6&(-0!,*&.!)1-*$/,)$(4&78!9#*!$%&!
!
Mathematics: understanding the score
@L!
*&$$(-0!12!#!*'#77!#'1,-$!#2$&/!&4&/8!7&**1-C!5%(*!%&73&.!3,3(7*!$1!/&27&)$!#-.!6,(7.!
1-!9%#$!$%&8!%#.!7&#/-&.!#-.R!(-!$%&!217719(-0!7&**1-R!$1!#*\!21/!%&73!9($%!#-8!
.(22(),7$(&*R!&-*,/(-0!$%#$!$%&8!.(.!-1$!2#77!6&%(-.C!I1'&!$&#)%&/*!,*&.!$%&!3/167&'*!
31*&.!(-!$%&!)71*(-0!'(-,$&*!12!#!7&**1-!)/&#$(4&78R!/&S,(/(-0!3,3(7*!$1!91/\!1-!$%&'!
6&21/&!$%&!-&E$!7&**1-R!9%&-!$%&8!6&)#'&!$%&!*$#/$(-0!31(-$!12!$%#$!.#8H*!7&#/-(-0C!
a-&!3,3(7!*#(.R!dJ&!.1-H$!0&$!#-8!%1'&91/\!(-!1,/!*&$C!J&!9&/&!*,331*&.!$1!,*&!
o#-!1-7(-&!/&4(*(1-!*&/4()&p!6,$!9&!.(.-H$R!*1!$%&!$&#)%&/!0#4&!,3CH!
Prime practice:
primaries
involving
parents
591!&E#'37&*!12!3/('#/8!*)%117*!*&&\(-0!$1!(-4174&!3#/&-$*!(-!
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9&/&!-1$!6&(-0!/&(-21/)&.!*,22()(&-$78R!&*3&)(#778!#$!%1'&C!^1/'#7!%1'&91/\!9#*!-1$!
*&&-!$1!6&!$%&!#-*9&/!6&)#,*&!$%&!)%(7./&-!9&/&!81,-0C!5%&!$&#)%&/*!%#.!$%&!(.&#!12!#!
d0#'&*!7(6/#/8HR!#))&**(67&!$1!#77!3#/&-$*!&#)%!9&&\C!c#'&*!9&/&!&E)%#-0&.!(-!$%&!
*#'&!9#8!#*!7(6/#/8!611\*C!5%&!0#'&*!9&/&!)171,/Q)1.&.!21/!#0&!#-.!#6(7($8!*1!3#/&-$*!
\-&9!9%()%!9&/&!*,($#67&!21/!$%&(/!)%(7./&-C!5%&!(-$/1.,)$(1-!12!$%&!7(6/#/8!9#*!
)1-*(.&/&.!$1!6&!#!*,))&**!*$1/8C!D#/&-$*!#)$&.!#*!7(6/#/(#-*!#-.!$%&!7(6/#/8!9#*!13&-!
1-!2(E&.!.#8*!#$!$%&!&-.!12!$%&!#2$&/-11-R!9%()%!9#*!)1-4&-(&-$!21/!3#/&-$*C!5%&!
%&#.$&#)%&/!*#(.!$%#$!)%(7./&-!714&.!%#4(-0!*,)%!#!9(.&!4#/(&$8!12!0#'&*!#-.!$%#$!$%&/&!
%#.!6&&-!#!-1$()C&!('3/14&'&-$!(-!$%&!3/10/&**!#-.!)1-2(.&-)&!12!$%1*&!9%1!37#8&.!
9($%!$%&'!9($%!$%&(/!3#/&-$*C!G-!#..($(1-R!$%&!)%(7./&-!9&/&!6&)1'(-0!2#'(7(#/!9($%!$%&!
(.&#!12!7(6/#/(&*!#*!#!4#7,#67&!/&*1,/)&!21/!$%&'*&74&*!#-.!#.,7$*C!
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(-21/'#$(1-!&4&-(-0*C!5%&!3,3(7*!.&4(*&.!0,(.#-)&!1-!)#7),7#$(1-!*$/#$&0(&*R!*,)%!#*!
.1,67(-0!#-.!%#74(-0C!5%(*!9#*!,371#.&.!1-$1!$%&!*)%117H*!9&6*($&!9%&/&!($!9#*!
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#-.!(.&-$(28(-0!9%#$!$%&8!-&&.&.!$1!.1!$1!('3/14&R!#7$%1,0%!3,3(7*!.(.!-1$!#79#8*!
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9%#$!)1,7.!6&!'#/\&.!68!$%&!3,3(7*!'(0%$!%#4&!'#.&!$%&!'#/\(-0!'1/&!)1-*$/,)$(4&!
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!
Mathematics: understanding the score
@A
What is not good enough about ‘satisfactory’ teaching?
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)%#/#)$&/(*$()*!12!*#$(*2#)$1/8!$&#)%(-0!(-!'#$%&'#$()*!9($%!$%1*&!12!011.!$&#)%(-0C!
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*&)1-.#/8!1-&*C!!
Figure 4: Quality of teaching in mathematics lessons in the schools surveyed
(percentages of lessons)
6
12
45
54
41
33
8
1
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Weaker
factors: rote
methods
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5%&!$&#)%&/!.(*)1,/#0&.!%&/R!6,$!*%&!(-*(*$&.!1-!$&77(-0!%('R!&4&-!$%1,0%!%&!.(.-H$!9#-$!
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!
Mathematics: understanding the score
;<!
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How might
it be
improved?
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schools surveyed (percentages of schools)
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Mathematics: understanding the score
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Prime practice:
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diagnostic
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Teachers’ subject expertise
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!
!
Mathematics: understanding the score
MP
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Weaker
factors:
knowledge
of
geometry
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Mathematics: understanding the score
MM!
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Weaker factors:
gaps in subject
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How might it be
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Mathematics: understanding the score
M=
Weaker factors:
poor use of
subject
expertise
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How might it
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Assessment for understanding: the teacher as detective
Why is assessment in lessons important?
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!
Mathematics: understanding the score
MO!
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Prime practice:
building
understanding
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371$$(-0!*$/#(0%$!7(-&!0/#3%*!)1//&)$78R!8&$!$%&8!)1-$(-,&!$1!91/\!1-!$%&'!,--1$()&.!
21/!$11!71-0C!G-!*1'&!)7#**/11'*R!6&)#,*&!12!$%&!9#8!$%&!.&*\*!9&/&!#//#-0&.!1/!
6&)#,*&!3/('#/8!3,3(7*!9&/&!*&#$&.!)71*&78!$10&$%&/!1-!$%&!)#/3&$!(-!2/1-$!12!$%&!
!
Mathematics: understanding the score
M?
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(-.(4(.,#778!1/!(-!0/1,3*!6,$!$%(*!9#*!12$&-!(-!/&*31-*&!$1!3,3(7*!9%1!%&7.!,3!$%&(/!
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(-21/'&.!#61,$!$%&(/!7&#/-(-0C!!
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:)$,#778R!$%&!3,3(7*H!611\*!#-.!.(*),**(1-*!(-.()#$&.!$%#$!'#-8!9&/&!,-)7&#/C!I,)%!
7&**1-*!'(0%$!%#4&!6&&-!'1/&!&22&)$(4&!(2!$%&!$&#)%&/!%#.!)(/),7#$&.!$1!)%&)\!1-!
3,3(7*H!611\*!#-.!.(*),**(1-*R!31*(-0!S,&*$(1-*!$1!4&/(28!9%&$%&/!3#/$(),7#/!3,3(7*!
9%1!9&/&!7(\&78!$1!%#4&!.(22(),7$8!.(.R!(-!2#)$R!,-.&/*$#-.C!^1),*&.!S,&*$(1-*!$1!
3#/$(),7#/!3,3(7*R!$%&!,*&!12!'(-(!9%($&61#/.*!#-.!.(*),**(1-!9($%!(-.(4(.,#7*!9&/&!
9#8*!$%(*!9#*!#)%(&4&.!(-!$%&!'1/&!*,))&**2,7!7&**1-*C!
Weaker
factors:
teacher not
circulating
:!7&**1-!*$#/$&/!(-!9%()%!$%&!$&#)%&/!9#*!,-#9#/&!$%#$!3,3(7*H!3/10/&**!
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%#.!9/($$&-!@RP<<!#-.!#$$&'3$&.!$%&!/&'#(-(-0!S,&*$(1-*!(-)1//&)$78!#*!#..($(1-*C!J%(7&!
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12!/&*31-*&*!*%19&.!$%#$!$%&!3,3(7*!9%1!#7/&#.8!\-&9!%19!$1!.1!$%(*!91/\!9&/&!-1$!
&E$&-.&.!#-.!$%1*&!9%1!.(.!-1$!\-19!0#(-&.!7($$7&!6&-&2($C!!
How might
it be
improved?
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$%&!$%/&&!-,'6&/*C!W1-$(-,(-0!$1!)(/),7#$&!#*!3,3(7*!91/\&.!91,7.!
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!
Mathematics: understanding the score
ML!
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S,&*$(1-R!81,!.1-H$!$&77!'&!$%&!#-*9&/C!n1,
!
Mathematics: understanding the score
MA
12$&-!#/&!-1$!.(/&)$78!/&7#$&.!$1!.&*)/(3$(1-*!#$!$%&!719&/!7&4&7*C!a-&!#33/1#)%!$1!
d)1''&-$!1-78H!317()(&*!9#*!$1!'#/\!.(#0-1*$()#778R!21),*(-0!1-!$13()*!9%&/&!
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S,#7($8!12!$%&!$#*\*C!G-!*1'&!)#*&*R!$%&!91/\!$&#)%&/*!*&$!#-.!$%&!S,&*$(1-*!$%&8!
#*\&.!9&/&!-1$!)%#77&-0(-0!1/!3/16(-0!&-1,0%!#-.!$%&/&21/&!.(.!-1$!*$(',7#$&!,*&2,7!
#**&**'&-$C!!
Pupils’ self-assessment
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7&#/-(-0R!(*!('3/14(-0!*71978C!G$!(*!'1/&!#.4#-)&.!(-!3/('#/8!*)%117*!6,$!12$&-!*$(77!(-!
($*!&#/78!*$#0&*C!c11.!3/#)$()&!.,/(-0!$%&!*,/4&8!(-)7,.&.!'&#-(-02,7!/&2&/&-)&!$1!
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/&0,7#/!#**&**'&-$*!$1!#(.!3,3(7*H!,-.&/*$#-.(-0!12!$%&(/!3/10/&**!#-.!#$$#(-'&-$C!
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!
Mathematics: understanding the score
=<!
3,3(7*!9&/&!)1-2(.&-$!(-!d$%(-\(-0!#71,.H!#-.!9&/&!-1$!#2/#(.!$1!%#4&!$%&(/!'(*$#\&*!
,*&.!$1!%&73!1$%&/*C!
Prime
practice:
discussion
:-!(-$&/&*$(-0!#33/1#)%!$1!/#$(1!#-.!3/131/$(1-!9($%!n&#/!O!3,3(7*!9($%!
71$*!12!.(*),**(1-C!
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&-)1,/#0(-0!.(*),**(1-!#-.!#/0,'&-$!(-!3#(/*!,-$(7!#-!#-*9&/!9#*!#0/&&.C!:!/&4&/*&!
#33/1#)%!$1!*174(-0!3/167&'*!9#*!&22&)$(4&!(-!0&$$(-0!3,3(7*!$1!$%(-\!#61,$!)7#/($8!12!
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!
Mathematics: understanding the score
=@
Weaker factors:
pseudo
investigation
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$%/&&!/,7&*!21/!#..($(1-!ealakBR!alBkaR!BlBkBf!6,$!/&#*1-*!9%8!$%&!/,7&*!91/\&.!
#-.!7(-\*!6&$9&&-!$%&!/,7&*!9&/&!-1$!'#.&!)7&#/C!
How might
it be
improved?
5%&!$&#)%&/H*!S,&*$(1-*!)1,7.!%#4&!6&&-!3%/#*&.!(-!#-!13&-!9#8+!
dJ%#$!%#33&-*!9%&-!81,!#..!1/!*,6$/#)$!$91!1..!-,'6&/*ZH!217719&.!
7#$&/!68+!d>1&*!$%(*!#79#8*!%#33&-ZH!
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0/&#$&/!(-.&3&-.&-)&!68!-1$!#**,'(-0!$%#$!#!/,7&!%#.!$1!&E(*$!#-.!68!
3/14(.(-0!3/#)$()#7!&S,(3'&-$!*,)%!#*!(-$&/71)\(-0!),6&*!*1!$%#$!$%&8!
)1,7.!/&3/&*&-$!1..!#-.!&4&-!-,'6&/*!4(*,#778C!D,3(7*!)1,7.!$%&-!
(77,*$/#$&!$%&(/!&E37#-#$(1-*!#-.
!
Mathematics: understanding the score
=;!
$&)%-(S,&*!#-.!/&)#77(-0!2#)$*C!I1'&!3/('#/8!*)%117*!#-.!*&)1-.#/8!.&3#/$'&-$*R!
12$&-!9&77!7&.!#-.!/&27&)$(4&!1-!$%&(/!3/#)$()&R!9&/&!(-$/1.,)(-0!#33/1#)%&*!$%#$!
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'#$%&'#$()*H!#-.R!$%&/&68R!(-!$&#)%(-0!21/!,-.&/*$#-.(-0C!
Prime practice:
teaching
mathematical
thinking
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-,'6&/!12!31**(6(7($(&*!#-.!$%#$!%#4(-0!$%/&&!7&$$&/*!$%&!*#'&!/&.,)&.!$%&!-,'6&/!
2,/$%&/R!6,$!/(*&.!$%#$!$%(*!-&&.&.!'1/&!$%1,0%$C!
!
Pupils’ enjoyment and views of mathematics
@PLC 5%&!'#[1/($8!12!3/('#/8!3,3(7*!&-[18&.!7&#/-(-0!'#$%&'#$()*R!)1''&-$(-0R!dG!7(\&!$%&!
$&#)%&/*C!5%&8!'#\&!7&**1-*!2,-HR!#-.!*&&(-0!($*!('31/$#-)&+!dn1,!)#-!,*&!($!(-!/!
7(2&CH!:*!9($%!*&)1-.#/8!3,3(7*R!$%&8!7(\&.!7&**1-*!$%#$!9&/&!(-$&/#)$(4&R!4#/(&.!9($%!
3/#)$()#7!#-.!1$%&/!(-$&/&*$(-0!#)$(4($(&*!#-.!9%()%!,*&.!GW5C!n&#/!O!3,3(7*R!%19&4&/R!
12$&-!*#(.!$%&8!%#.!7($$7&!%#-.*Q1-!&E3&/(&-)&!12!GW5!(-!)1'3#/(*1-!9($%!3/&4(1,*!
!
Mathematics: understanding the score
=P
8&#/*!#$!$%&(/!*)%117R!)($(-0!$%&!-&&.!$1!3/&3#/&!21/!$&*$*!#*!$%&!/&#*1-C!D,3(7*!12!#77!
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*&)1-.#/8!*)%117*R!%19&4&/R!$%&*&!&-[18#67&!2&#$,/&*!.(.!-1$!1)),/!/&0,7#/78!(-!
'#$%&'#$()*!7&**1-*C!D,3(7*H!&E3&/(&-)&!12!$%&'!4#/(&.!9(.&78!9($%(-!$%&!*#'&!
*)%117R!6&$9&&-!$&#)%&/*R!#-.!2/1'!1-&!\&8!*$#0&!$1!#-1$%&/C!!
@PAC "1*$!*&)1-.#/8!3,3(7*!9&/&!/&7#$(4&78!#'6(4#7&-$!#61,$!'#$%&'#$()*C!583()#7!
)1''&-$*!(-)7,.&.R!dG$H*!011.!9%&-!81,!,-.&/*$#-.!($H!#-.!dG$H*!61/(-0C!G!3/&2&/!
#)$(4&!1/!)/&#$(4&!*,6[&)$*R!7(\&!.#-)&!#-.!#/$CH!"1*$!*#(.!$%&8!dS,($&!&-[18!($H!6,$!2&9!
*&)1-.#/8!3,3(7*!)($&.!'#$%&'#$()*!#*!$%&(/!2#41,/($&!*,6[&)$R!&4&-!$%1*&!9%1!9&/&!
.1(-0!4&/8!9&77C!"1*$!&E3/&**&.!2&&7(-0*!12!*#$(*2#)$(1-!9%&-!$%&8!9&/&!#67&!$1!/&#)%!
)1//&)$!#-*9&/*!1-!#!3#/$(),7#/!$13()!#-.!7#)\!12!&-[18'&-$!9%&-!$%&8!)1,7.!-1$C!
5%&(/!&-[18'&-$!$%&/&21/&!4#/(&.R!12$&-!%(-0(-0!1-!%19!9&77!$%&8!)1,7.!d.1H!#!
3#/$(),7#/!$13()C!
@M<C :!/&'#/\#67&!.&0/&&!12!)1-*(*$&-)8!&E(*$&.!(-!',)%!12!9%#$!3,3(7*!*#(.!#61,$!$%&(/!
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2/1'!4(77#0&!3/('#/8!$1!7#/0&!(--&/Q)($8!)1'3/&%&-*(4&C!5%&/&!9#*!*$/1-0!#0/&&'&-$!
1-!$%&!2&#$,/&*!12!'#$%&'#$()*!7&**1-*!$%#$!$%&8!&-[18&.+!dG$H*!2,-!91/\(-0!(-!
0/1,3*H!#-.!dJ1/\(-0!9($%!*1'&1-&!&7*&!%&73*!81,!,-.&/*$#-.R!&*3&)(#778!(2!$%&8!#*\!
81,!S,&*$(1-*CH!5%&8!.(*7(\&.!$11!',)%!$#7\!68!$%&!$&#)%&/!#-.!1$%&/!3,3(7*H!
'(*6&%#4(1,/C!"#-8!3,3(7*R!&*3&)(#778!(-!*&)1-.#/8!*)%117*R!.&*)/(6&.!#!7#)\!12!
4#/(&$8R!9%()%!$%&8!21,-.!.,77C!583()#778R!$%&(/!7&**1-*!)1-)&-$/#$&.!1-!$%&!
#)S,(*($(1-!12!*\(77*R!*17,$(1-!12!/1,$(-&!&E&/)(*&*!#-.!3/&3#/#$(1-!21/!$&*$*!#-.!
&E#'(-#$(1-*C!dB4&/8!7&**1-R!81,!%#4&!$1!#-*9&/!S,&*$(1-*!2/1'!$%&!$&E$611\C!G$!0&$*!
61/(-0CH!5%&8!)1-$/#*$&.!$%(*!9($%!1))#*(1-#7!7&**1-*!$%&8!&-[18&.!9%&/&!$%&8!.(.!
(-4&*$(0#$(1-*R!$#)\7&.!3,jj7&*R!*1'&$('&*!91/\(-0!(-!0/1,3*R!#-.!,*&.!GW5!
(-.&3&-.&-$78C!a2$&-!*,)%!7&**1-*!%#33&-&.!#$!$%&!&-.!12!$&/'!#-.!9&/&!/&0#/.&.!
#*!&-.Q12Q$&/'!#)$(4($(&*!/#$%&/!$%#-!6&(-0!d/!'#$%*HC!!
@M@C G$!*%1,7.!6&!12!*&/(1,*!)1-)&/-!-#$(1-#778!$%#$!*1!'#-8!*&)1-.#/8!3,3(7*!*&&'&.!$1!
#))&3$!$%#$!$%(*!9#*!9%#$!7&#/-(-0!'#$%&'#$()*!*%1,7.!6&!7(\&R!.&*3($&!$%&(/!
/&)10-($(1-!$%#$!$&#)%(-0!#-.!7&#/-(-0!(-!1$%&/!*,6[&)$*!9&/&!-1$!$%&!*#'&C!"#-8!
3,3(7*R!(-)7,.(-0!*1'&!(-!n&#/!OR!*#(.!$%&8!&E3&)$&.!$1!2(-.!'#$%&'#$()*!.(22(),7$!#-.!
$%#$!$%&8!91,7.!%#4&!$1!3/#)$(*&!#-.!'&'1/(*&!$%&!'&$%1.*!$%&8!9&/&!*%19-!(2!$%&8!
9&/&!$1!*,))&&.!(-!$&*$*!#-.!&E#'(-#$(1-*C!5%&8!9&/&!3/&3#/&.!$1!.1!$%(*!6&)#,*&!
$%&8!\-&9!*,))&**!(-!'#$%&'#$()*!9#*!('31/$#-$C!a-&!n&#/!;!3,3(7!*#(.R!dGH'!-1$!
711\(-0!21/9#/.!$1!6&(-0!(-!n&#/!OC!5%&8!%#4&!$1!91/\!/̊!%#/.!21/!$%&!$&*$*CH!
@M;C J%&-!#*\&.R!'1*$!3,3(7*!/&)10-(*&.!$%&!.(22&/&-)&!6&$9&&-
!
Mathematics: understanding the score
=M!
Weaker
factors: doing
well but
without
understanding
:!3,3(7!)1//&)$78!)#7),7#$&.!$%&!#/&#*!12!)(/)7&*!12!/#.(,*!=)'!#-.!
?)'R!68!#3378(-0!$%&!*$#-.#/.!21/',7#
2
rA
π
= C!
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-&#/78!$9()&!#*!',)%!#*!$%&!2(/*$R!*%&!(''&.(#$&78!#**,'&.!%&/!#-*9&/!
',*$!6&!9/1-0R!#*!*%&!9#*!-1$!,*&.!$1!6&(-0!#*\&.!$1!(-$&/3/&$!%&/!
#-*9&/*C!:2$&/!2,/$%&/!.(*),**(1-R!($!6&)#'&!)7&#/!$%#$!*%&!%#.!7&#/-$!
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)1,7.!-1$!*#8!9%#$!9#*!'&#-$!68!$%&!#/&#!12!#!)(/)7&C!5%&!2&9!)(/)7&*!
./#9-!(-!%&/!611\!9&/&!#77!$%&!*#'&!*(j&C!I%&!%#.!7&#/-$!#!'&$%1.!$1!
16$#(-!#-*9&/*!$1!#!3/167&'!*%&!.(.!-1$!,-.&/*$#-.C!!
!
How might it
be improved?
5%&!3,3(7H*!,-.&/*$#-.(-0!91,7.!%#4&!6&&-!6&$$&/!(2!$%&!$&#)%&/!%#.+!
&*$#67(*%&.!#$!$%&!6&0(--(-0!%19!9&77!&#)%!3,3(7!,-.&/*$11.!$%&!
)1-)&3$!12!#/&#!!
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#)$,#7!*(j&!!
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#33/1E('#$&!$%&!#/&#!12!#!)(/)7&R!21/!&E#'37&!68!*#-.9()%(-0!($!
6&$9&&-!*S,#/&*!#-.h1/!317801-*C!
!
]-.&/*$#-.(-0!91,7.!#7*1!%#4&!6&&-!*$/&-0$%&-&.!(2!$%&!3,3(7!%#.!
6&&-!#*\&.!$%&!*1/$*!12!S,&*$(1-*!$%#$!91,7.!%#4&!'#.&!%&/!$%(-\!
#61,$!9%#$!*%&!9#*!7&#/-(-0!#-.!%19!$1!(-$&/3/&$!/&*,7$*C!!
!
@MPC "1*$!3,3(7*!)1-*(.&/&.!$%#$!$%&(/!91/\!9#*!0&-&/#778!#$!#61,$!$%&!/(0%$!7&4&7!12!
.(22(),7$8R!#-.!*#(.!$%&8!&-[18&.!($!9%&-!($!9#*!)%#77&-0(-0C!T19&4&/R!$%&(/!4(&9*!1-!
#-.!&E3&/(&-)&*!12!3/167&'!*174(-0!9&/&!'1/&!'(E&.C!I1'&!3/&2&//&.!$%&!)1'21/$!12!
/1,$(-&!&E&/)(*&*!9%&/&!$%&8!\-&9!9%&-!$%&8!%#.!01$!$%&!/(0%$!#-*9&/*R!6,$!17.&/!
3,3(7*R!#-.!'1/&!#67&!3,3(7*!12!#77!#0&*R!$&-.&.!$1!/&7(*%!$%&!)%#77&-0&!12!*174(-0!
3/167&'*_!1-&!*(E$%Q21/'!*$,.&-$!*#(.R!dG!7(\&!$%&!*#$(*2#)$(1-!12!0&$$(-0!*1'&$%(-0!
/(0%$!&4&-$,#778!9%()%!G!'#8!-1$!%#4&!,-.&/*$11.!#$!2(/*$CH!D,3(7*!91,7.!7(\&!'1/&!
$%#$!#..&.!(-$&/&*$!#-.!/&7&4#-)&!$1!$%&(/!7&#/-(-0!68!/&7#$(-0!$1!&4&/8.#8!*($,#$(1-*C!
I1'&!*&)1-.#/8!3,3(7*R!3#/$(),7#/78!$%&!'1/&!#67&R!*#(.!$%&8!&-[18&.!$%&!)%#77&-0&!
12!#70&6/#C!
@MMC "#$%&'#$()*!#$!#.4#-)&.!7&4&7!(*!)1''1-78!3&/)&(4&.!$1!6&!'1/&!.(22(),7$!$%#-!'1*$!
1$%&/!*,6[&)$*C!"#-8!%(0%&/Q#$$#(-(-0!n&#/!@@!3,3(7*!(-.()#$&.!*1'&!#-E(&$8!#61,$!
$%&(/!#6(7($8!$1!)13&!9($%!:Q7&4&7!'#$%&'#$()*C!W%11*(-0!$1!*$,.8!($!/&7#$&.!'1*$78!$1!
($*!,*&2,7-&**!(-!*,331/$(-0!*,6[&)$*!$%&8!9#-$&.!$1!/&#.!#$!,-(4&/*($8C!W1-*(.&/#67&!
4#/(#$(1-!/&'#(-*!(-!$%&!#.4()&!*)%117*!3/14(.&!21/!31$&-$(#7!:I!#-.!:Q7&4&7!*$,.&-$*C!
I1'&!*)%117*!.(*)1,/#0&!#77!6,$!$%&!%(0%&*$Q#$$#(-(-0!3,3(7*_!1$%&/*!#7719!9(.&/!
#))&**!6,$!$%&-!12$&-!.1!-1$!#.#3$!$&#)%(-0!#33/1#)%&*!#-.!*,331/$!$%&!*$,.&-$*!
9&77!&-1,0%R!*1!2#(7,/&!#-.!./13Q1,$!/#$&*!.,/(-0!$%&!)1,/*&!#/&!%(0%C!5%&!3#$%9#8*!
$19#/.*!S,#7(2()#$(1-*!$%#$!3,3(7*!217719!#2$&/!$#\(-0!cWIB!&#/78!#/&!-1$!#79#8*!
$%1,0%$!$%/1,0%!)#/&2,778!&-1,0%C!!
!
Mathematics: understanding the score
==
@M=C "#-8!3,3(7*R!3#/$(),7#/78!*&)1-.#/8!3,3(7*R!*31\&!12!2(-.(-0!91/\!.(22(),7$!(-!7&**1-*!
#-.!-&&.(-0!$1!#*\!21/!&E$/#!%&73!#2$&/9#/.*C!dG!.1-H$!7(\&!'#$%*!6&)#,*&!GH'!-1!011.!
#$!($C!F,$!$%&!%&73!2/1'!$&#)%&/*!(*!011.CH!5%(*!-&&.!/&4*!$%&!(-*&),/($8!12!$%&(/!
7&#/-(-0!(-!7&**1-*C!5%&!3,3(7*!3/#(*&.!$%&(/!$&#)%&/*H!.&.()#$(1-!(-!3/14(.(-0!$%(*!
*,331/$R!#*!9&77!#*!/&4(*(1-!21/!$&*$*!#-.!&E#'(-#$(1-*C!"#-8!3,3(7*!(-.()#$&.!$%#$!
$%&8!9&/&!'1/&!13&-!9($%!&#)%!1$%&/!e#-.!(-*3&)$1/*f!#61,$!.(22(),7$(&*!(-!
,-.&/*$#-.(-0!91/\!(-!7&**1-*!$%#-!$%&8!9&/&!9($%!$%&(/!$&#)%&/*C!5%(*!/&27&)$*!
3,3(7*H!3&/)&3$(1-*!12!$%&!('31/$#-)&!12!/(0%$!#-*9&/*!(-!'#$%&'#$()*+!6&(-0!*$,)\!(*!
4(&9&.!-&0#$(4&78!/#$%&/!$%#-!#*!#!)%#77&-0&!$1!6&!14&/)1'&C!
@MOC 5%&/&!(*!-1!/&#*1-!9%8!3,3(7*!*%1,7.!-1$!61$%!&-[18!#-.!,-.&/*$#-.!'#$%&'#$()*+!
$%&8!#/&!-1$!)1-$/#.()$1/8C!"#-8!3,3(7*!7#)\!)1-2(.&-)&!(-!$%&(/!19-!#6(7($(&*!#-.!
91//8!#61,$!0&$$(-0!#-*9&/*!9/1-0R!&4&-!9%&-!$%&(/!'&$%1.*!#/&!/(0%$C!5%&8!)#-!
2(-.!$%&!'1/&!13&-Q&-.&.!#-.!,-,*,#7!3/167&'*!.#,-$(-0C!`&4&/$%&7&**R!*%1/$Q$&/'!
*#$(*2#)$(1-!0#(-&.!2/1'!#!3#0&!12!$()\*!21/!)1//&)$!#-*9&/*!$1!/&3&$($(4&!&E&/)(*&*!(*!
2/#0(7&!(2!$%&!'#$%&'#$()*!($*&72!(*!-1$!,-.&/*$11.!9&77!&-1,0%!$1!6&!#337(&.!
(-.&3&-.&-$78!$1!,-,*,#7!3/167&'*C!I,6*&S,&-$!7&#/-(-0!(*!$%&-!6,(7$!1-!)1-)&3$,#7!
*#-.C!!
@M?C I1'&!*)%117*!%#4&!6&0,-!$1!)177&)$!3,3(7*H!4(&9*!12!$%&(/!&E3&/(&-)&!12!7&#/-(-0!
'#$%&'#$()*C!5%(*!(*!#!31*($(4&!'14&R!#-.!#)$(1-!$1!/&*31-.!$1!$%&'!%#*!$%&!
31$&-$(#7!$1!'#\&!7&#/-(-0!'#$%&'#$()*!'1/&!2,-!21/!3,3(7*!#-.!$%&(/!$&#)%&/*C!:!
3,3(7!*#(.+!d"#$%*!'#\&*!81,!$%(-\!N!81,/!'(-.!0/19*!(-$&77&)$,#778C!F,$!*1'&$('&*!
81,!7&#/-!'1/&!2/1'!81,/!2/(&-.*!$%#-!81,/!$&#)%&/C!BE37#(-(-0!6,(7.*!,3!1,/!
)1-2(.&-)&CH!
Conclusion
@MLC 5%(*!/&31/$!(*!3,67(*%&.!#$!#!$('&!12!)1-*(.&/#67&!)%#-0&!(-!'#$%&'#$()*R!21,/!8&#/*!
1-!2/1'!$%&!I'($%!/&31/$!#-.!*%1/$78!#2$&/!$%&!3,67()#$(1-!12!$%&!2(-.(-0*!12!$%&!
J(77(#'*!/&4(&9C!`&9!),//(),7#!#-.!-&9!S,#7(2()#$(1-*!#/&!&'3%#*(*(-0!$%&!-&&.!21/!
3,3(7*!$1!6&!'#$%&'#$()#778!2,-)$(1-#7C!5%(*!(*!&E#)$78!$%&!$('&!$1!0&$!$&#)%(-0!#-.!
7&#/-(-0!/(0%$!(-!'#$%&'#$()*+!($!(*!9%#$!),//&-$!#-.!2,$,/&!0&-&/#$(1-*!12!)%(7./&-!
#-.!81,-0!3&137&!-&&.!(2!$%&8!#/&!$1!6&!3/13&/78!'#$%&'#$()#778!&S,(33&.C!
@MAC 5&#)%(-0!#-.!7&#/-(-0R!$%&!),//(),7,'R!#-.!7&#.&/*%(3!#-.!'#-#0&'&-$!12!
'#$%&'#$()*!#/&!#77!*$/1-0&/!(-!3/('#/8!*)%117*!$%#-!(-!*&)1-.#/8!*)%117*C!"#-8!
*&)1-.#/8!*)%117*!2#)&!*(0-(2()#-$!)%#77&-0&*!(-!2(-.(-0!011.!$&#)%&/*!12!
'#$%&'#$()*C!T19&4&/R!)1-$(-,1,*!('3/14&'&-$!(*!%#/.!$1!#)%(&4&!#-.!$%&!*%#/3!
/(*&!(-!$%&!-#$(1-#7!$&*$!/&*,7$*!21/!3/('#/8!*)%117*R!9%()%!6&0#-!#!.&)#.&!#01R!%#*!
-19!'1/&!1/!7&**!/&#)%&.!#!37#$&#,C!"#-8!3/('#/8!$&#)%&/*!*$(77!-&&.!6&$$&/!
\-197&.0&!12!'#$%&'#$()*!(2!$%&8!#/&!$1!6&!&-#67&.!61$%!$1!%&73!$%&!719&*$!#$$#(-(-0!
3,3(7*!/&#)%!$%&!&E3&)$&.!*$#-.#/.*!#-.!$1!)%#77&-0&!$%&!%(0%&*$!#$$#(-&/*C!G-!
*&)1-.#/8!*)%117*R!.&*3($&!$%&!#$$&-$(1-!0(4&-!$1!cWIB!'#$%&'#$()*R!'#-8!3,3(7*!
#/&!-1$!.1(-0!#*!9&77!#*!$%&8!*%1,7.C!D,3(7*H!7&#/-(-0!(*!6#*&.!$11!',)%!1-!$%&(/!
#)S,(*($(1-!12!'&$%1.*R!/,7&*!#-.!2#)$*R!#*!3#/$!12!$%&!*$/1-0!21),*!1-!$&*$*!#-.!
&E#'(-#$(1-*R!#-.!$11!7($$7&!1-!$%&(/!,-.&/*$#-.(-0!12!$%&!,-.&/3(--(-0!)1-)&3$*R!1-!
)1--&)$(1-*!9($%!$%&(/!&#/7(&/!7&#/-(-0!#-.!1$%&/!$13()*!(-!'#$%&'#$()*R!#-.!1-!
!
Mathematics: understanding the score
=O!
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Mathematics: understanding the score
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Further information
Advisory Committee on Mathematics Education (ACME)
999C#)'&Q,\C1/0h!
:W"B!(*!#-!(-.&3&-.&-$!)1''($$&&!9%()%!#)$*!#*!#!*(-07&!41()&!21/!$%&!'#$%&'#$()#7!
)1'',-($8R!*&&\(-0!$1!('3/14&!$%&!S,#7($8!12!&.,)#$(1-!(-!*)%117*!#-.!)177&0&*C!G$!#.4(*&*!
c14&/-'&-$!1-!(**,&*!*,)%!#*!$%&!),//(),7,'R!#**&**'&-$R!#-.!$%&!*,3378!#-.!$/#(-(-0!12!
'#$%&'#$()*!$&#)%&/*C!G$*!'1*$!/&)&-$!)1-2&/&-)&!9#*!1-!d"#$%&'#$()*!(-!I5B"+!#!317()8!
3&/*3&)$(4&HC!G$!%#*!3,67(*%&.!4#/(1,*!31*($(1-!3#3&/*R!(-)7,.(-0!
"#$%&'#$()*!#-.!7&4&7!P!
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Department for Children, Schools and Families (DCSF)
999C.)*2C014C,\!
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(-)7,.(-0!$%&!91/\!12!$%&!`#$(1-#7!I$/#$&0(&*!e7(-\*!6&719fC!!
5%&!/&31/$!2/1'!$%&!c14&/-'&-$H*!(-S,(/8!(-$1!31*$Q@M!'#$%&'#$()*!&.,)#$(1-!
"#\(-0!
'#$%&'#$()*!)1,-$+!5%&!/&31/$!12!D/12&**1/!:./(#-!I'($%H*!(-S,(/8!(-$1!31*$Q@M!
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R!;<<MR!)#-!6&!21,-.!#$!
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Department for Innovation, Universities and Skills (DIUS)
999C.(,*C014C,\h!
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-,'&/#)8!#$!%$$3+hh0&$1-C.(/&)$C014C,\h!1/!(-21/'#$(1-!21/!3#/&-$*!#$!
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Further Mathematics Network
999C2'-&$91/\C1/0C,\!
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Mathematical Careers
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Mathematics: understanding the score
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National Association of Mathematics Advisers (NAMA)
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National Centre for Excellence in the Teaching of Mathematics (NCETM)
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7&#/-(-0H!#/&!#$!
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National Strategies
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c14&/-'&-$H*!3/10/#''&!21/!/#(*(-0!&.,)#$(1-#7!*$#-.#/.*!(-!7(-&!9($%!$%&!B4&/8!W%(7.!
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Qualifications and Curriculum Authority (QCA)
999CS)#C1/0C,\!
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Mathematics: understanding the score
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Office of the Qualifications and Examinations Regulator (Ofqual)
999C12S,#7C014C,\!
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Learning and Skills Improvement Service (LSIS)
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g,#7($8!G'3/14&'&-$!:0&-)8!21/!V(2&71-0!V&#/-(-0!egG:fC!G$!9(77!21),*!1-!7&#/-&/*!#-.!
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Royal Society
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Subject associations
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Mathematics: understanding the score
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Annexes
Annex A. Schools visited for this survey
Primary schools Local authority
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Mathematics: understanding the score
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Mathematics: understanding the score
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Mathematics: understanding the score
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Annex B. The age profile of the pupils in the schools surveyed
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