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EQUATIONS AND
INEQUALITIES:
MAKING MATHEMATICS
ACCESSIBLE TO ALL
Andreas Schleicher
Chiara Monticone
Mario Piacentini
WHY MATHEMATICS MATTERS
FOR PEOPLE
2
0
10
20
30
40
50
60
70
80
Norway
Japan
Italy
Flanders
(Belgium)
Korea
SlovakRepublic
France
Denmark
Estonia
Netherlands
Austria
Spain
Sweden
Germany
OECDaverage
CzechRepublic
Ireland
Poland
England/Norther
nIreland(UK)
Finland
Australia
Canada
UnitedStates
Use or calculate fractions or percentages Use simple algebra or formula
Use advanced mathematics or statistics
%
Use of mathematics skills at work
Source: Figure 1.2,
OECD Survey of Adult Skills (PIAAC) (2012), Table 1.1a.
3
1.00 1.10 1.20 1.30 1.40 1.50 1.60
Is in good general health
Is in the top quarter of earnings
Has a job
4
Adults with good mathematics skills
earn higher salaries
Increase in the likelihood of the outcome related to an increase of one
standard deviation in numeracy, OECD average (22 countries)
Odds ratios
Source: Figure 1.3
OECD Survey of Adult Skills (PIAAC) (2012), Table 1.2
Adults with higher
numeracy (by 50 points)
are 53% more likely to
have high wages
0
50
100
150
200
250
300
350
Canada91
Portugal93
Tunisia26
Macao-China
HongKong-China
UnitedStates33
Mexico18
Iceland-10
NewZealand
Australia6
Japan18
Italy19
Denmark18
Latvia10
RussianFederation15
Belgium21
Brazil
Korea-33
OECDaverage13
Liechtenstein
Spain34
Indonesia-23
Greece22
France
Switzerland
Thailand-18
Luxembourg4
Norway33
Poland-7
Germany14
Ireland
CzechRepublic14
Sweden17
SlovakRepublic-18
Finland19
Turkey-28
Netherlands21
Austria-10
Uruguay-27
Hungary-13
2012 2003Minutes
Time spent in mathematics classes has
increased
Source: Figure 1.6
In 2012, the average 15
year-old student in an
OECD country spent 13
minutes more per
week in mathematics
classes than in 2003
5
Change in time spent in mathematics classes between 2012 and 2003
LearningQuality of
instruction
Opportunity
to learn
Ability
Perseverance
Aptitude
School learning has many facets
Students’
characteristics
Directly
shaped by
teachers/
schools /
systems
6
Opportunity to learn
refers to the content taught
in the classroom and the
time a student spends
learning this content
7
Many students have never heard of
basic mathematics concepts
OECD average
Source: Table 1.7
0 20 40 60 80 100
Vectors
Arithmetic mean
Linear equation
Never heard the concept
Heard the concept often/a few times
Know well/understand the concept
%
Conditioning
factors
• Characteristics
of the:
• Student
• Schools
• Systems
Opportunity
to learn
• Exposure to
tasks
• Familiarity
with
concepts
• Time in class
Outcomes
• Mathematics
performance
• Attitudes
towards
mathematics
Analytical framework of the report
Source: Figure 1.1
8
WHY ACCESS TO
MATHEMATICS MATTERS
AND HOW IT CAN BE
MEASURED
9
Applied mathematics
Working out from a <train timetable>
how long it would take to get from one
place to another.
Calculating how much more expensive a
computer would be after adding tax.
Calculating how many square metres of
tiles you need to cover a floor.
Understanding scientific tables
presented in an article.
Finding the actual distance between two
places on a map with a 1:10,000 scale.
Calculating the power consumption of
an electronic appliance per week.
Pure mathematics
Solving an equation like:
6x2 + 5 = 29
Solving an equation like
2(x+3) = (x + 3)(x - 3)
Solving an equation like:
3x+5=17
How PISA measures exposure to
applied and pure mathematics
10
R² = 0.05
-1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
-0.60 -0.40 -0.20 0.00 0.20 0.40 0.60
Indexofexposuretoappliedmathematics Weak relationship between exposure to
applied and pure mathematics
OECDaverage
Source: Figure 1.8
OECD average
Index of exposure to pure mathematics
Less
exposure
More
exposure
More
exposure
11
0
1
2
3
4
Sweden
Tunisia
Iceland
Argentina
Luxembourg
Brazil
CostaRica
Greece
Switzerland
Belgium
UnitedKingdom
Portugal
Uruguay
NewZealand
Ireland
Chile
Thailand
Australia
Mexico
France
SlovakRepublic
Liechtenstein
Lithuania
VietNam
Colombia
Poland
Italy
OECDaverage
Indonesia
Turkey
Finland
Malaysia
Denmark
Austria
Peru
CzechRepublic
Kazakhstan
Albania
Qatar
HongKong-China
Latvia
Israel
Hungary
Netherlands
Serbia
Spain
Romania
Bulgaria
Estonia
Montenegro
Germany
RussianFederation
Canada
Shanghai-China
ChineseTaipei
Croatia
Slovenia
UnitedStates
Korea
UnitedArabEmirates
Jordan
Japan
Singapore
Macao-China
Mean index
Large international differences in
familiarity with algebra….
Source: Figure 1.7
Sweden
Singapore
Macao-China
Never
heard the
concept
Heard the
concept
once
Heard the
concept few
times
Often heard
the concept
Knows
the
concept
well
0.83
2.85
3.04
12
0
1
2
3
4
Sweden
Iceland
Netherlands
Ireland
Germany
Austria
Switzerland
Liechtenstein
NewZealand
Argentina
Malaysia
Denmark
Brazil
Indonesia
Luxembourg
Tunisia
Lithuania
Finland
CostaRica
SlovakRepublic
Qatar
Australia
OECDaverage
UnitedKingdom
Thailand
Chile
Slovenia
Uruguay
Colombia
Spain
Israel
CzechRepublic
Peru
Mexico
HongKong-China
Japan
Montenegro
UnitedStates
Canada
Portugal
Poland
Italy
ChineseTaipei
Bulgaria
Korea
Estonia
Croatia
Hungary
France
UnitedArabEmirates
Jordan
Macao-China
Romania
Turkey
Kazakhstan
RussianFederation
Latvia
Belgium
VietNam
Serbia
Singapore
Greece
Albania
Shanghai-China
Mean index
… and in familiarity with geometry
Source: Figure 1.7
Sweden
Singapore
Shanghai-
China
Never
heard the
concept
Heard the
concept
once
Heard the
concept few
times
Often heard
the concept
Knows
the
concept
well
0.54
2.94
3.56
13
VARIATIONS IN STUDENTS’
EXPOSURE TO AND
FAMILIARITY WITH
MATHEMATICS
14
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
NewZealand
Portugal
Brazil
Qatar
Luxembourg
Tunisia
Jordan
Australia
Sweden
Belgium
Denmark
UnitedArabEmirates
Colombia
Argentina
ChineseTaipei
Chile
CzechRepublic
Turkey
Netherlands
Malaysia
Canada
SlovakRepublic
Austria
Indonesia
Romania
CostaRica
Thailand
Switzerland
Uruguay
Bulgaria
Latvia
Montenegro
OECDaverage
Serbia
Israel
France
Greece
Finland
Peru
Mexico
Germany
UnitedKingdom
Norway
Estonia
UnitedStates
Hungary
Ireland
Poland
VietNam
Japan
Shanghai-China
Iceland
Lithuania
Italy
Croatia
Kazakhstan
Slovenia
HongKong-China
RussianFederation
Spain
Liechtenstein
Singapore
Macao-China
Korea
Indexofexposuretopuremathematics
Bottom quarter (disadvantaged students) Second quarter Third quarter Top quarter (advantaged students)
Exposure to pure mathematics
increases with socio-economic status
Source: Figure 2.5b
15
0
5
10
15
20
25
Estonia
Malaysia
HongKong-China
Denmark
Finland
Jordan
Mexico
Tunisia
VietNam
Canada
Sweden
Israel
Macao-China
Kazakhstan
Latvia
Iceland
Greece
Qatar
UnitedArabEmirates
CostaRica
Indonesia
Argentina
Poland
RussianFederation
NewZealand
UnitedKingdom
Ireland
Lithuania
Australia
UnitedStates
Peru
Shanghai-China
Colombia
Montenegro
Romania
Spain
CzechRepublic
Luxembourg
OECDaverage
Singapore
Bulgaria
Uruguay
Serbia
Thailand
Turkey
Portugal
Italy
Japan
Croatia
Switzerland
Brazil
SlovakRepublic
Chile
Korea
Netherlands
ChineseTaipei
Belgium
Slovenia
Germany
Austria
Hungary
Liechtenstein
Variation explained by students' socio-economic status
Variation explained by students' socio-economic status and schools' socio-economic profile
Socio-economic profile explains 9% of the
variation in familiarity with mathematics
%
Source: Figure 2.2
Variation in familiarity with mathematics explained by socio-economic profile
16
Boy
Immigrant
Did not attend
Girl
Non-
immigrant
Attended pre-
primary
-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30
Gender
Immigrant background
Pre-primary education
Index of familiarity with mathematics
Girls, non-immigrants and students who
attended pre-primary education are more
familiar with mathematics
Source: Table 2.10
Note: OECD averages are computed only for countries with available data.
OECD average
17
0
2
4
6
8
10
12
14
16
18
20
0 20 40 60 80 100
Percentage of students in schools that engage in a given practice
Systems with more selective schools give
more unequal access to mathematicsAccesstomathematics
More
equal
More
unequal
Sources: Figures 2.10, 11, 21
Transferring low-achieving
students to another school
R2 = 0.42
Considering
academic
performance for
admission
R2 = 0.31Considering
residence for
admission
R2 =0.28
Variation in familiarity with mathematics explained by students' and
schools' socio-economic profile, OECD average
%
18
%
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
0.40
0.50
Macao-China
HongKong-China
VietNam
Greece
Korea
Tunisia
Austria
Brazil
Indonesia
CostaRica
UnitedStates
Estonia
Mexico
Bulgaria
Chile
Canada
Italy
Colombia
Luxembourg
Thailand
Argentina
Israel
France
CzechRepublic
Netherlands
Kazakhstan
Portugal
OECDaverage
Singapore
Qatar
Uruguay
Sweden
Latvia
UnitedArabEmirates
Shanghai-China
SlovakRepublic
Slovenia
Australia
Ireland
Germany
RussianFederation
Switzerland
NewZealand
ChineseTaipei
Hungary
Students in the last year of ISCED 2 Students in the first year of ISCED 3
Stronger relationship between familiarity and
socio-economic status as students progress
to upper secondary education
Source: Figure 2.13
Change in familiarity with mathematics associated with one-unit increase in
students’ socio-economic status
Index change
19
Earlier tracking associated with more
unequal access to mathematics
Accesstomathematics
More
equal
More
unequal
Source: Figure 2.15
Australia
New Zealand
Poland
United
Kingdom
Variation in familiarity with mathematics explained by students' and
schools' socio-economic profile, OECD countries
20
Austria
Belgium
Sweden
Chile
Czech Republic
DenmarkEstonia
Canada
Germany
Greece
Hungary
Ireland
Israel
Italy
Japan
Korea
Luxembourg
Mexico
Netherlands
Iceland
Portugal
Slovak Republic
Slovenia
Spain
Finland
Switzerland
Turkey
United States
OECD average
R² = 0.54
0
5
10
15
20
25
9 10 11 12 13 14 15 16 17
Students' age at first tracking, system level
% of the variation
Students in vocational schools are more
likely to be socio-economically and
academically disadvantaged
Source: Figure 2.16
Odds
ratios
Morelikelytobe
disadvantagedorlessfamiliar
Lesslikely
Change in likelihood of having less familiarity with mathematics or being socio-
economically disadvantaged associated with enrollment in vocational schools
21
0.00
1.00
2.00
3.00
4.00
5.00
6.00
Ireland
Croatia
Hungary
Spain
Korea
Slovenia
Serbia
Netherlands
Belgium
Montenegro
Italy
Macao-China
Portugal
Greece
OECDaverage
ChineseTaipei
Israel
Japan
Bulgaria
SlovakRepublic
France
Germany
Austria
RussianFederation
Chile
Shanghai-China
Uruguay
UnitedKingdom
Australia
Turkey
Argentina
CzechRepublic
Luxembourg
Thailand
Kazakhstan
Malaysia
Switzerland
Indonesia
CostaRica
Mexico
Colombia
UnitedArabEmirates
Being socio-economically disadvantaged Having less familiarity with mathematics
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
Austria
Switzerland
Germany
Turkey
HongKong-China
Croatia
Portugal
Macao-China
Greece
France
Chile
Argentina
Luxembourg
Iceland
Montenegro
Latvia
Australia
Jordan
RussianFederation
Qatar
Serbia
Kazakhstan
Lithuania
Italy
Singapore
OECDaverage
Netherlands
Tunisia
Belgium
Mexico
UnitedArabEmirates
Uruguay
CzechRepublic
Canada
Estonia
Brazil
Indonesia
Romania
Japan
SlovakRepublic
Finland
Spain
Denmark
Malaysia
Peru
Sweden
CostaRica
Slovenia
Colombia
Poland
Hungary
ChineseTaipei
Bulgaria
Thailand
VietNam
Korea
Shanghai-China
UnitedStates
Before accounting for gender, students' socio-economic status, and schools' socio-economic profile
After accounting for gender, students' socio-economic status, and schools' socio-economic profile
Weak relationship between ability grouping
and familiarity with mathematics
Source: Figure 2.18b
Ability grouping has a small
negative association with
familiarity once characteristics
of students and schools are
taken into account
Change in the index of familiarity with mathematics associated with ability grouping
Index change
22
-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
helps
students
learn from
mistakes
lets students
decide on
their own
procedures
makes
students
reflect on
the problem
gives
problems
that require
thinking for
an extended
time
gives
problems
that can be
solved in
different
ways
gives
problems
with no
immediate
solution
asks
students to
explain how
they solved a
problem
Disadvantaged schools Advantaged schoolsIndex change
The use of cognitive activation practices is associated with greater
performance and familiarity in socio-economically advantaged
schools than in disadvantaged ones
Source: Figure 2.23b
Change in the index of familiarity with mathematics associated with
use of cognitive activation strategies, OECD average
The teacher…
Higher
familiarity
Lowerfamiliarity
23
• Exposure to, and familiarity with, mathematics increase with
socio-economic status, and
• Vary by students gender, immigrant background, and pre-
primary education
…individual
characterist
ics
• Grade repetition, schools’ selection mechanisms, and
between-school tracking are associated with more unequal
access to mathematics
• Weak, negative relationship between ability grouping and
familiarity with mathematics for the average student
…how
systems
and schools
sort and
select
students
• Disadvantaged schools have a (slightly) lower student-to-
teacher ratio, but mathematics teachers in disadvantaged
schools tend to be less qualified
• The use of cognitive activation practices is associated with
greater performance and familiarity in socio-economically
advantaged schools than in disadvantaged ones
…teaching
resources
and
practices
Key messages:
How access to mathematics varies by…
24
EXPOSURE TO
MATHEMATICS IN SCHOOL
AND PERFORMANCE IN PISA
25
-30
-20
-10
0
10
20
30
40
Shanghai-China
ChineseTaipei
Albania
Japan
Macao-China
Korea
Kazakhstan
Malaysia
RussianFederation
Switzerland
SlovakRepublic
Indonesia
Poland
Latvia
Singapore
HongKong-China
Thailand
Liechtenstein
Portugal
Qatar
Uruguay
Bulgaria
Romania
Montenegro
Slovenia
Italy
Peru
CzechRepublic
Jordan
Mexico
Serbia
Denmark
Argentina
Luxembourg
Hungary
Chile
Iceland
VietNam
OECDaverage
Austria
Turkey
Tunisia
Belgium
France
Germany
Lithuania
Spain
Colombia
Australia
Canada
Estonia
NewZealand
UnitedArabEmirates
CostaRica
Sweden
Norway
Brazil
Croatia
Finland
Netherlands
Greece
Israel
UnitedStates
UnitedKingdom
Ireland
Score-point
difference
High-performing countries do relatively better on
problems requiring knowledge of geometry and algebra
Country's/economy's performance
on the subscale is higher than on
the overall mathematics scale
Country's/economy's performance
on the subscale is lower than on the
overall mathematics scale
Source: Figure 3.1
Relative performance on the “Space and Shape” sub-scale
26
Japan
Chinese Tapei
Shanghai-China
Ireland
-1.60
-1.40
-1.20
-1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
Uruguay
Spain
Hungary
Sweden
RussianFederation
SlovakRepublic
Portugal
France
OECDaverage
NewZealand
Denmark
Australia
CzechRepublic
Belgium
Finland
2012 2003Logit
Performance on tasks with a focus on geometry
deteriorated between 2003 and 2012
Source: Figure 3.3c
Performance
Change in performance on items in the space and shape sub-scale
between 2003 and 2012
(countries where the change is significant)
27
420
440
460
480
500
520
540
Less than 2 hours Between 2 and 4
hours
Between 4 and 6
hours
More than 6 hours
Mathematics Reading ScienceMean score
Longer class time up to four hours per week is
associated with a large improvement in
mathematics performance
Source: Figure 3.4
Hours per
week:
OECD average
28
-0.15
-0.10
-0.05
0.00
0.05
0.10
420
430
440
450
460
470
480
490
500
510
520
Less than 2 Between 2 and 4 Between 4 and 6 More than 6
Indexofdisciplinaryclimate
Mathematicsscore
Mathematics Disciplinary ClimateMean score
Hours per
week:
Instruction time above 6 hours a week is more
frequent in classes with poor disciplinary climate
Source: Figure 3.6
OECD average
29
-10
-5
0
5
10
15
20
Ireland
Iceland
Macao-China
Liechtenstein
Norway
Bulgaria
NewZealand
Greece
CostaRica
Sweden
Albania
Slovenia
Thailand
SlovakRepublic
Estonia
Brazil
UnitedStates
Spain
Chile
Hungary
Qatar
Finland
Canada
UnitedArabEmirates
Portugal
Peru
UnitedKingdom
Tunisia
Luxembourg
Mexico
Kazakhstan
RussianFederation
Denmark
OECDaverage
Uruguay
Argentina
Switzerland
Colombia
Lithuania
HongKong-China
Israel
Australia
Germany
Poland
Belgium
Jordan
France
Netherlands
Indonesia
Turkey
Montenegro
Italy
Latvia
Singapore
Japan
Malaysia
Serbia
VietNam
CzechRepublic
Austria
Romania
ChineseTaipei
Korea
Shanghai-China
Croatia
Score-point change
The relationship between time and performance is
much weaker after accounting for school characteristics
30
Croatia
Shanghai-China
Korea
Chinese Tapei
Romania
Austria
Italy
Japan
Indonesia
Netherlands
Turkey
Malaysia
Singapore
Relationship between time and performance among students in the same school and grade
Liechtenstein
Chech Republic
420
440
460
480
500
520
540
First quintile Second quintile Third quintile Fourth quintile Fifth quintile
Quintiles of exposure
Applied mathematics Pure mathematics
Exposure to pure mathematics is more strongly related
to performance than exposure to applied mathematics
Source: Figure 3.9
Mean score
31
-40
-20
0
20
40
60
80
100
120
140
160
180
SlovakRepublic
Shanghai-China
Albania
Serbia
Slovenia
Macao-China
Romania
Portugal
Bulgaria
Mexico
Brazil
Colombia
Austria
Tunisia
Indonesia
Estonia
CostaRica
Liechtenstein
Croatia
Australia
Hungary
Kazakhstan
Israel
Montenegro
Denmark
Ireland
Argentina
Canada
Peru
Thailand
Germany
Jordan
NewZealand
Uruguay
Chile
Latvia
OECDaverage
UnitedArabEmirates
HongKong-China
RussianFederation
ChineseTaipei
CzechRepublic
Netherlands
Turkey
Italy
UnitedKingdom
Poland
Lithuania
UnitedStates
Belgium
Sweden
Singapore
France
Finland
Norway
Switzerland
Qatar
Malaysia
Spain
Luxembourg
VietNam
Korea
Score-point
change
Exposure to pure mathematics is related to higher
performance, even after accounting for school
characteristics
Source: Figure 3.11
32
Relationship between exposure and performance among students in the same school
Charts Q1
Revolving Door Q2
R² = 0.39
1.00
1.20
1.40
1.60
1.80
2.00
2.20
2.40
2.60
2.80
300 400 500 600 700 800
Drip Rate Q1
Arches Q2
Stronger association between familiarity with
concepts and performance on more demanding tasks
Drip Rate Q1
Effectoffamiliarity
Higher
positive
effect
Lower
positive
effect
Source: Figure 3.12
Difficulty on the PISA scale
Revolving Door
Q2
33
Odds ratio
Familiarity with pure mathematics is enough to
solve procedural problems…
Scenario:
Nurses calculate the drip rate for infusions using
the formula:
𝐷𝑟𝑖𝑝 𝑟𝑎𝑡𝑒 =
𝑑𝑣
60𝑛
d is the drop factor in drops per mL
v is the volume in mL of the infusion
n is the number of hours the infusion is required
to run
Question:
Describe how the
drip rate changes if
n is doubled but the
other variables do
not change.
Drip Rate Question 1
34
Korea
OECD average
Indonesia
Malaysia
Qatar
Shanghai-China
Chinese-Taipei
R² = 0.57
-3.50
-3.00
-2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
-1.00 -0.50 0.00 0.50 1.00 1.50
LogitfortheitemDripRateQ1
Index of familiarity with mathematics
BEFORE accounting for countries’ performance on all the other tasks
Source: Figure 3.13
Familiarity with mathematics and performance on
Drip Rate Question 1: Country-level relationship
35
Familiarity with mathematics and performance on
Drip Rate Question 1 - Country-level relationship
Korea
OECD average
Netherlands
Luxembourg
Spain
R² = 0.22
-1.60
-1.40
-1.20
-1.00
-0.80
-0.60
-0.40
-0.20
0.00
-0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00
AFTER accounting for countries’ performance on all the other tasks
Index of familiarity with mathematics
LogitfortheitemDripRateQ1
Source: Figure 3.13
36
…but being familiar with mathematics content might
not be enough to solve problems that require to reason
mathematically
Scenario:
A revolving door
includes three wings
which rotate within a
circular-shaped space
and divide the space
into three equal
sectors. The two door
openings (the dotted
arcs in the diagram)
are the same size.
Possible air flow in
this position
200 cm
Question:
What is the
maximum arc length
in centimetres (cm)
that each door
opening can have, so
that air never flows
freely between the
entrance and the
exit?
Revolving Door Question 2
37
Familiarity with mathematics and performance on
Revolving Door Question 2 - Country-level relationship
Korea
OECD average
Indonesia
Malaysia
Qatar
Shanghai-China
Chinese-Taipei
R² = 0.18
-6.00
-5.00
-4.00
-3.00
-2.00
-1.00
0.00
-1.00 -0.50 0.00 0.50 1.00 1.50
BEFORE accounting for countries’ performance on all the other tasks
Logitfortheitem
RevolvingDoorQ2
Index of familiarity with mathematics
Source: Figure 3.14
38
Familiarity with mathematics and performance on
Revolving Door Question 2 - Country-level relationship
Korea
OECD average
Netherlands
Luxembourg
Spain
R² = 0.02
-5.00
-4.50
-4.00
-3.50
-3.00
-2.50
-2.00
-1.50
-1.00
-0.50
0.00
-0.80 -0.30 0.20 0.70
Logitfortheitem
RevolvingDoorQ2
Index of familiarity with mathematics
Source: Figure 3.14
AFTER accounting for countries’ performance on all the other tasks
39
-30
-20
-10
0
10
20
30
40
Macao-China
HongKong-China
Tunisia
Malaysia
Estonia
Mexico
Denmark
Israel
CostaRica
VietNam
Kazakhstan
Argentina
Latvia
Greece
Lithuania
Shanghai-China
Romania
Finland
Ireland
UnitedArabEmirates
Japan
SlovakRepublic
CzechRepublic
Bulgaria
RussianFederation
NewZealand
Poland
Indonesia
Sweden
UnitedKingdom
Uruguay
Jordan
Montenegro
Canada
Luxembourg
Iceland
Serbia
Singapore
OECDaverage
Slovenia
Qatar
Peru
Turkey
Colombia
Australia
Italy
ChineseTaipei
France
Netherlands
Chile
Spain
Croatia
Thailand
Portugal
Brazil
UnitedStates
Belgium
Hungary
Switzerland
Germany
Austria
Korea
% of score-point difference
Familiarity with mathematics explains 19% of the
socio-economic performance gap
Source: Figure 3.15
Percentage of the score-point difference between advantaged and
disadvantaged students explained by different familiarity with mathematics
40
0.30
0.40
0.50
0.60
0.70
0.80
0.90
Macao-China
Estonia
Thailand
Indonesia
Jordan
Iceland
Finland
Mexico
Serbia
HongKong-China
Argentina
Greece
Tunisia
Norway
Kazakhstan
Croatia
Korea
Malaysia
UnitedKingdom
Romania
Canada
Turkey
Germany
Latvia
Colombia
Denmark
CostaRica
Netherlands
UnitedStates
Qatar
Switzerland
Sweden
RussianFederation
Lithuania
CzechRepublic
Italy
Brazil
Montenegro
OECDaverage
VietNam
Ireland
Slovenia
Australia
Japan
Austria
Singapore
Poland
NewZealand
Portugal
UnitedArabEmirates
Spain
Belgium
Uruguay
Luxembourg
Hungary
France
Chile
Shanghai-China
Israel
Bulgaria
Peru
ChineseTaipei
SlovakRepublic
The task has a scientific context The task has a personal contextOdds ratio
41
Socio-economically disadvantaged students
perform better on “familiar” tasks
Socio-economicgapinperformance
Larger
gap
Smaller
gap
Source: Figure 4.18
• Countries where students have higher familiarity with geometry and
algebra perform better in all tasks and relatively better on tasks
requiring geometry and algebra
• Performance on tasks with a focus on geometry deteriorated between
2003 and 2012
Structure of
curriculum
• Increasing instruction time in mathematics beyond 6 hours a week
has no clear relationship with performance. The relationship differs
substantially across countries, and within countries according to the
quality of the disciplinary climate in the classroom
• Exposure to pure mathematics tasks (equations) is strongly related to
performance
• Exposure to and familiarity with mathematics concepts may not be
sufficient for solving problems that require the ability to think and
reason mathematically
Amount/type
of
mathematics
tasks and
performance
• Almost 20% of the performance gap of disadvantaged students is
explained by their lower familiarity with mathematics concepts.
• Disadvantaged students lag behind other students particularly in
those complex tasks requiring modelling skills and the use of
symbolic language.
Socio-
economic
disadvantage
and exposure
to
mathematics
Key messages
42
OPPORTUNITY TO LEARN
AND STUDENTS’ ATTITUDES
TOWARDS MATHEMATICS
43
Less than half of students enjoy
studying mathematics
0
10
20
30
40
50
60
70
80
90
Austria-4
Hungary
SlovakRepublic-5
Finland4
Belgium-5
CzechRepublic
Korea
Japan5
Norway
Netherlands
Luxembourg
Poland-4
Canada
UnitedStates
Sweden
Ireland4
Spain
OECDaverage
NewZealand
Latvia
Australia3
Germany-4
France-5
Macao-China
RussianFederation
Portugal
Italy
Iceland10
Switzerland-4
Uruguay
Greece8
Turkey-5
Mexico8
HongKong-China3
Liechtenstein
Brazil-4
Denmark
Tunisia-9
Thailand
Indonesia5
2012 2003%
Source: Figure 4.2
Percentage of students who agree with the statement I do mathematics
because I enjoy it"
44
The difference
between 2003 and
2012 is significant
45
Exposure to more complex mathematics is related to
lower self-concept, among students of similar ability
Mathematicsself-concept
Source: Figure 4.7
-0.40
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
0.40
Liechtenstein
Indonesia
Argentina
Thailand
Kazakhstan
Austria
Luxembourg
Netherlands
Tunisia
Qatar
Japan
Romania
Macao-China
Belgium
Germany
Bulgaria
SlovakRepublic
Malaysia
Brazil
Switzerland
HongKong-China
Shanghai-China
Latvia
Uruguay
VietNam
Estonia
CostaRica
Greece
Lithuania
Mexico
Israel
Denmark
RussianFederation
Peru
Sweden
Colombia
CzechRepublic
Chile
Montenegro
OECDaverage
NewZealand
Ireland
UnitedArabEmirates
Albania
UnitedKingdom
Turkey
Croatia
UnitedStates
Hungary
Jordan
Spain
Finland
France
Slovenia
Canada
Italy
Singapore
Portugal
Iceland
Poland
Australia
Serbia
ChineseTaipei
Korea
Before accounting for performance in mathematics After accounting for performance in mathematicsIndex change
Change in students’ self-concept associated with 1 unit change in familiarity
Exposure to more complex mathematics is also
related to greater anxiety among low-performing
students
Source : Figure 4.8
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
Malaysia OECD average Czech Republic
Bottom quarter by mathematics performance Top quarter by mathematics performanceIndex
Change in students’ anxiety associated with a change in familiarity, by
students' mathematics performance
46
MoreanxietyLessanxiety
Students with hard-working friends are more
motivated to learn, especially in schools where
students are least familiar with mathematics
0.00
0.50
1.00
1.50
2.00
2.50
I am interested in
the things I learn
in mathematics
I do mathematics
because I enjoy it
I look forward to
mathematics
lessons
Making an effort is
worthwhile for the
work I want to do
Mathematics is
important for
what I want to
study later on
Schools where students are more familiar with mathematics
Schools where students are less familiar with mathematics
Odds ratio
Source: Figure 4.11
Change in the probability that students agree with each statement, associated
with having friends who work hard on mathematics
47
High-performing students whose parents do not like
mathematics are more likely to feel helpless
1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40
OECD average
France
Top quarter of mathematics performance Bottom quarter of mathematics performance
Odds ratio
Source: Figure 4.14
Change in the probability that students feel helpless when doing mathematics
problems associated with having parents who do not like mathematics
Children whose parents dislike mathematics have higher anxiety
48
Students whose teachers provide feedback or specify
learning goals are more familiar with mathematics
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
The teacher gives
different work to
classmates who have
difficulties learning
and/or to those who
can advance faster
The teacher has us
work in small
groups to come up
with joint solutions
to a problem or task
The teacher gives
extra help when
students need it
The teacher
continues teaching
until the students
understand
The teacher asks
questions that make
us reflect on the
problem
The teacher gives
problems that
require us to think
for an extended time
Students more familiar with mathematics Students less familiar with mathematics
Index change
Source: Figure 4.16
Change in the index of mathematics self-concept associated with having mathematics
teachers who provide feedback or specify learning goals in every or most lessons
49
Teachers’ feedback practices have a different
relationship with anxiety depending on students’
familiarity with mathematics
Source: Figure 4.15
-0.10
0.00
0.10
The teacher gives me feedback on my strengths
and weaknesses in mathematics
The teacher tells us what is expected of us when
we get a test, quiz or assignment
Low familiarity students High familiarity studentsIndex change
Change in mathematics anxiety associated with having teachers who
engage in these practices, OECD average
MoreanxietyLessanxiety
50
Using a computer during mathematics lessons is
associated with higher motivation for learning
mathematics
0.00
0.10
0.20
0.30
0.40
0.50
Japan
Iceland
Denmark
Macao-China
Liechtenstein
Serbia
Mexico
Spain
Estonia
Germany
CzechRepublic
Austria
Belgium
Shanghai-China
RussianFederation
HongKong-China
Finland
Switzerland
Norway
Singapore
Croatia
Uruguay
Italy
Ireland
Netherlands
OECDaverage
CostaRica
Australia
Latvia
Slovenia
Chile
Portugal
Poland
Turkey
Sweden
Hungary
SlovakRepublic
Korea
Greece
NewZealand
Jordan
ChineseTaipei
Israel
After accounting for students' and schools' characteristics
Index change
Source: Figure 4.17
Change in intrinsic motivation for mathematics associated with using a
computer in mathematics class
51
• Exposure to more complex mathematics concepts is associated
with
• lower self-concept and higher anxiety among low-
performing students, and with
• higher self-concept/lower anxiety among high-performing
students
Opportunity
to learn and
attitudes
towards
mathematics
• Peers: Having hard-working friends can increase mathematics
self-concept, but students can develop lower beliefs in their own
ability when they compare themselves to higher-achieving peers
• Parents may transfer their feelings about mathematics to their
children, even high-performing ones
• Teachers’ practices can have a different relationship with
students self-concept and anxiety depending on students’
familiarity with mathematics
Mediating
factors
Key messages
52
WHAT DOES THIS MEAN
FOR POLICY?
53
Develop
coherent
standards
Develop
skills
beyond
knowledge
Reduce the
impact of
tracking
Support
teachers of
heterogenous
classes
Support
positive
attitudes
Monitor
Opportunity
to Learn
Develop coherent standards,
frameworks and instruction
material for all students
How:
• Cover core ideas more in
depth
• Increase connections between
topics
• Review textbooks and teaching
material accordingly
A policy framework to widen opportunities
to learn
A policy
programme in 6
points
In Singapore the mathematics
framework covers a relatively
small number of topics in depth,
following a spiral organisation
in which topics introduced in
one grade are covered in later
grades at a more advanced level
Develop
coherent
standards
Develop
skills
beyond
knowledge
Reduce the
impact of
tracking
Support
teachers of
heterogeno
us classes
Support
positive
attitudes
Monitor
Opportunit
y to Learn
Help students acquire
mathematical skills beyond
content knowledge
How:
• Replace routine tasks with
challenging, open problems
• Develop specific training for
teachers
• Integrate problem-solving
abilities into assessments
55
A policy framework to widen opportunities
to learn
A policy
programme in 6
points
Recent revisions of the
mathematics curricula in
England, Scotland,
Korea and Singapore
emphasise the
development of problem-
solving skills
Develop
coherent
standards
Develop
skills
beyond
knowledge
Reduce the
impact of
tracking
Support
teachers of
heterogeno
us classes
Support
positive
attitudes
Monitor
Opportunit
y to Learn
Reduce the impact of tracking on
equity in mathematics exposure
How:
• Consider possibilities to delay
tracking
• Improve quality and quantity of
mathematics instruction in non-
academic pathways
• Allow students to change tracks
56
A policy framework to widen opportunities to
learn
A policy
programme in 6
points
Sweden and Finland reformed
their education systems in the
1950-1970s: a later age at
tracking reduced inequalities in
outcomes later on.
Also Germany and Poland
reformed the tracking system to
reduce the influence of socio-
economic status on student
achievement
Develop
coherent
standards
Develop
skills
beyond
knowledge
Reduce the
impact of
tracking
Support
teachers of
heterogeno
us classes
Support
positive
attitudes
Respon-
sibility
Learn how to handle
heterogeneity in the classroom
How:
• Provide students with multiple
opportunities to learn key concepts
at different levels of difficulty
• Adopt student-oriented practices
such as flexible grouping or
cooperative learning
• Offer more individualized support
to struggling students
57
A policy framework to widen opportunities to
learn
A policy
programme in 6
points
In Finland, half of
children with
special education
needs are
mainstreamed and
assigned special
teachers, rather
than being in
special schools.
Develop
coherent
standards
Develop
skills
beyond
knowledge
Reduce the
impact of
tracking
Support
teachers of
heterogeno
us classes
Support
positive
attitudes
Monitor
Opportunit
y to Learn
Support positive attitudes
towards mathematics
through innovations in
curriculum and teaching
How:
• Develop, use and share
engaging tasks and learning
tools (including IT-based)
• Learn how to give effective
feedbacks to struggling
students
• Engage parents
58
A policy framework to widen opportunities to
learn
A policy
programme in 6
points
The 2011 revisions of the
mathematics curriculum in
Korea has reduced curriculum
content to give more time to
engaging activities that would
improve students’ motivation
Develop
coherent
standards
Develop
skills
beyond
knowledge
Reduce the
impact of
tracking
Support
teachers of
heterogeno
us classes
Support
positive
attitudes
Monitor
Opportunity
to Learn
Monitor and analyse
opportunity to learn
How:
• Collect and analyse data on
the implemented curriculum
both from teachers and
students
• Support multi-year research
and curriculum-development
programmes
• Analyse data on mathematics
teaching practices from video
studies
59
A policy framework to widen opportunities to
learn
A policy
programme in 6
pointsThe Teaching and Leaning
International Survey (TALIS)
study is piloting an
international video study of
teaching practices to provide
insights into effective teaching
practices
THANK YOU
60

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Equations and Inequalities - Making mathematics accessible to all

Editor's Notes

  1. Familiarity with algebra: exponential function, quadratic function and linear function.
  2. Familiarity with geometry: vector, polygon, congruent figure and cosine.
  3. Teachers can influence equity in access to mathematics content not only by grouping students of similar ability and by assigning different tasks to students, based on their ability, but also more directly: through the quantity and quality of the tasks, and by engaging in certain teaching practices. Cognitive-activation strategies tend to be used more often in socio-economically advantaged schools than in disadvantaged schools (Table 2.24a). On average across OECD countries, the effect of cognitive-activation strategies on opportunity to learn mathematics is mixed in advantaged schools. However, on average across OECD countries, no cognitive-activation strategy is associated with greater familiarity with mathematics among students in disadvantaged schools. Overall, these results suggest that teachers use cognitive-activation strategies to deepen the curriculum content and support the development of problem-solving abilities among students in socio-economically advantaged schools. By contrast, in disadvantaged schools, it appears that using strategies that emphasise thinking and reasoning for an extended time may take time away from covering important material in the curriculum. In less-favourable learning environments, there is a cost to having students engage more deeply in mathematics thinking: less material is covered.