Equations and Inequalities - Making mathematics accessible to all

EQUATIONS AND INEQUALITIES:

MAKING MATHEMATICS ACCESSIBLE TO ALL

Andreas SchleicherChiara MonticoneMario Piacentini

2

WHY MATHEMATICS MATTERSFOR PEOPLE

3

Nor

way

Japa

n

Italy

Flan

ders

(Bel

g...

Kor

ea

Slov

ak R

epub

lic

Fran

ce

Den

mar

k

Esto

nia

Net

herla

nds

Aus

tria

Spai

n

Swed

en

Ger

man

y

OEC

D a

vera

ge

Cze

ch R

epub

lic

Irela

nd

Pola

nd

Engl

and/

Nor

the.

..

Finl

and

Aus

tral

ia

Can

ada

Uni

ted

Stat

es

0

10

20

30

40

50

60

70

80

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.

4

Is in good general health

Is in the top quarter of earnings

Has a job

1.00 1.10 1.20 1.30 1.40 1.50 1.60

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 ratiosSource: 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

5

Canad

a 91

Tunisia

26

Hong Kong-China

Mexico

18

New Zea

land

Japan

18

Denmark

18

Russian

Federa

tion 1

5

Brazil

OECD avera

ge 13

Spain 3

4

Greece

22

Switzerl

and

Luxembourg

4

Poland -

7

Irelan

d

Sweden

17

Finland 1

9

Netherl

ands

21

Uruguay -

270

50

100

150

200

250

300

3502012 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

Change in time spent in mathematics classes between 2012 and 2003

Learning

Quality 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

Vectors

Arithmetic mean

Linear equation

0 20 40 60 80 100

Never heard the conceptHeard the concept often/a few timesKnow well/understand the concept

%

8

Conditioning factors

• Characteristics of the:

• Student • Schools • Systems


• Exposure to tasks

• Familiarity with concepts

• Time in class

Outcomes

• Mathematics performance

• Attitudes towards mathematics

Analytical framework of the report

Source: Figure 1.1

9

WHY ACCESS TO MATHEMATICS MATTERS

AND HOW IT CAN BE MEASURED

10

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

11

-0.60 -0.40 -0.20 0.00 0.20 0.40 0.60-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

R² = 0.0492741435508877

Inde

x of

exp

osur

e to

app

lied

mat

hem

atic

s

Weak relationship between exposure to applied and pure mathematics

OEC

D

aver

age

Source: Figure 1.8

OECD average

Index of exposure to pure mathematicsLess exposure

More exposure

More exposure

12

Swed

enTu

nisi

aIc

elan

dA

rgen

tina

Luxe

mbo

urg

Bra

zil

Cost

a R

ica

Gre

ece

Switz

erla

ndBe

lgiu

mU

nite

d K

ingd

omPo

rtug

alU

rugu

ayNe

w Z

eala

ndIre

land

Chile

Thai

land

Aus

tralia

Mex

ico

Fran

ceSl

ovak

Rep

ublic

Liec

hten

stei

nLi

thua

nia

Viet

Nam

Colo

mbi

aPo

land Italy

OEC

D a

vera

geIn

done

sia

Turk

eyFi

nlan

dM

alay

sia

Den

mar

kA

ustri

aPe

ruC

zech

Rep

ublic

Kaz

akhs

tan

Alb

ania

Qat

arHo

ng K

ong-

Chin

aLa

tvia

Isra

elH

unga

ryN

ethe

rland

sSe

rbia

Spai

nR

oman

iaBu

lgar

iaEs

toni

aM

onte

negr

oG

erm

any

Russ

ian

Fede

ratio

nC

anad

aSh

angh

ai-C

hina

Chi

nese

Tai

pei

Cro

atia

Slov

enia

Uni

ted

Stat

esKo

rea

Uni

ted

Ara

b Em

irate

sJo

rdan

Japa

nSi

ngap

ore

Mac

ao-C

hina

0

1

2

3

4Mean index

Large international differences in familiarity with algebra….

Source: Figure 1.7

Swed

en

Sing

apor

eM

acao

-C

hina

Never heard the concept

Heard the concept once

Heard the concept few times

Often heard the concept

Knows the concept well

0.83

2.853.04

13

Swed

enIc

elan

dNe

ther

land

sIre

land

Ger

man

yAu

stria

Switz

erla

ndLi

echt

enst

ein

New

Zea

land

Arg

entin

aM

alay

sia

Denm

ark

Bra

zil

Indo

nesi

aLu

xem

bour

gTu

nisi

aLi

thua

nia

Finl

and

Cost

a R

ica

Slov

ak R

epub

licQ

atar

Aust

ralia

OEC

D a

vera

geU

nite

d Ki

ngdo

mTh

aila

ndC

hile

Slov

enia

Urug

uay

Col

ombi

aSp

ain

Isra

elCz

ech

Rep

ublic

Peru

Mex

ico

Hon

g Ko

ng-C

hina

Japa

nM

onte

negr

oUn

ited

Stat

esCa

nada

Port

ugal

Pola

ndIta

lyC

hine

se T

aipe

iBu

lgar

iaKo

rea

Esto

nia

Croa

tiaHu

ngar

yFr

ance

Unite

d A

rab

Emira

tes

Jord

anM

acao

-Chi

naRo

man

iaTu

rkey

Kaza

khst

anRu

ssia

n Fe

dera

tion

Latv

iaBe

lgiu

mVi

et N

amSe

rbia

Sing

apor

eG

reec

eAl

bani

aSh

angh

ai-C

hina

0

1

2

3

4Mean index

… and in familiarity with geometry

Source: Figure 1.7

Swed

en

Sing

apor

eSh

angh

ai-

Chi

na

Never heard the concept

Heard the concept once

Heard the concept few times

Often heard the concept

Knows the concept well

0.54

2.943.56

14

VARIATIONS IN STUDENTS’ EXPOSURE TO AND FAMILIARITY WITH

MATHEMATICS

15

New Zea

land

Brazil

Luxembourg

Jord

an

Sweden

Denmark

Colombia

Chinese T

aipei

Czech

Rep

ublic

Netherl

ands

Canad

a

Austria

Romania

Thailan

d

Uruguay

Latvia

OECD avera

geIsr

ael

Greece

Peru

German

y

Norway

United Stat

es

Irelan

d

Viet N

am

Shanghai-

China

Lithuan

ia

Croati

a

Slovenia

Russian

Federa

tion

Liechten

stein

Macao

-China

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

Bottom quarter (disadvantaged students) Second quarterThird quarter Top quarter (advantaged students)

Inde

x of

exp

osur

e to

pur

e m

ath-

emat

ics

Exposure to pure mathematics increases with socio-economic status

Source: Figure 2.5b

16

Estonia

Hong K

ong-C

hina

Finlan

d

Mexico

Viet Nam

Sweden

Macao

-China

Latvi

a

Greece

United A

rab Emira

tes

Indone

sia

Poland

New Zeala

nd

Irelan

d

Australi

aPeru

Colombia

Romania

Czech

Rep

ublic

OECD avera

ge

Bulgaria

Serbia

Turkey Ita

ly

Croatia

Brazil

Chile

Netherla

nds

Belgium

German

y

Hungar

y0

5

10

15

20

25

Variation explained by students' socio-economic statusVariation 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

17

Gender

Immigrant background

Pre-primary education

-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30

Boy

Immigrant

Did not attend

Girl

Non-im-migrant

Attended pre-primary

Index of familiarity with mathematics

Girls, non-immigrants and students who attended pre-primary education are more familiar with mathematics

Source: Table 2.10Note: OECD averages are computed only for countries with available data.

OECD average

18

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14

16

18

20

Percentage of students in schools that engage in a given practice

Systems with more selective schools give more unequal access to mathematics

Acc

ess

to m

athe

mat

ics

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 admissionR2 =0.28

Variation in familiarity with mathematics explained by students' and schools' socio-economic profile, OECD average

%

%

19

Macao

-China

Viet Nam

Korea

Austria

Indone

sia

United S

tates

Mexico Chile Ita

ly

Luxe

mbourg

Argenti

na

France

Netherla

nds

Portug

al

Singapo

re

Urugu

ayLa

tvia

Shang

hai-C

hina

Slovenia

Irelan

d

Russian Fe

dera

tion

New Zeala

nd

Hungar

y-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

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

20

Earlier tracking associated with more unequal access to mathematics

Acc

ess

to m

athe

mat

ics

More equal

More unequal

Source: Figure 2.15

AustraliaNew ZealandPoland United Kingdom

Variation in familiarity with mathematics explained by students' and schools' socio-economic profile, OECD

countries

9 10 11 12 13 14 15 16 170

5

10

15

20

25

OECD average

Austria

Belgium

Sweden

Chile

Czech Republic

DenmarkEstonia

Canada

Germany

Greece

Hungary

Ireland

Israel

Italy JapanKorea

Luxembourg

Mexico

Netherlands

Iceland

PortugalSlovak Republic

Slovenia

Spain

Finland

SwitzerlandTurkey

United States

R² = 0.541186620995712

Students' age at first tracking, system level

% of the variation

21

Students in vocational schools are more likely to be socio-economically and academically disadvantaged

Source: Figure 2.16

Odds ratios

Mor

e lik

ely

to b

e di

sadv

anta

ged

or le

ss

fam

iliar

Le

ss

likel

y

Change in likelihood of having less familiarity with mathematics or being socio-economically disadvantaged associated with enrollment in vocational schools

1 2 3 4 5 6 7 8 9 101112131415 16171819202122 232425262728290.00

1.00

2.00

3.00

4.00

5.00

6.00

Being socio-economically disadvantaged Having less familiarity with mathematics

22

Austria

German

y

Hong K

ong-C

hina

Portug

al

Greece

Chile

Luxe

mbourg

Monten

egro

Australi

a

Russian F

edera

tion

Serbia

Lithu

ania

Singapo

re

Netherla

nds

Belgium

United A

rab Emira

tes

Czech

Rep

ublic

Estonia

Indone

siaJa

pan

Finlan

d

Denmar

kPeru

Costa R

ica

Colombia

Hungar

y

Bulgari

a

Viet Nam

Shang

hai-C

hina

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

Before 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

23

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08Disadvantaged 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…

Hig

her

fam

iliar

ity

Low

er f

amili

arit

y

24

• Exposure to, and familiarity with, mathematics increase with socio-economic status, and

• Vary by students gender, immigrant background, and pre-primary education

…individual characteri

stics

• 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…

25

EXPOSURE TO MATHEMATICS IN SCHOOL

AND PERFORMANCE IN PISA

-30

-20

-10

0

10

20

30

40

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

27

Urug

uay

Spai

n

Hung

ary

Swed

en

Russ

ian

Fede

ratio

n

Slov

ak R

epub

lic

Portu

gal

Fran

ce

OEC

D av

erag

e

New

Zea

land

Denm

ark

Aust

ralia

Czec

h Re

publ

ic

Belg

ium

Finl

and

-1.60

-1.40

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20 2012 2003Logit

Performance on tasks with a focus on geometry deteriorated between 2003 and 2012

Source: Figure 3.3c

Perf

orm

ance

Change in performance on items in the space and shape sub-scale between 2003 and 2012

(countries where the change is significant)

28

Less than 2 hours Between 2 and 4 hours

Between 4 and 6 hours

More than 6 hours

420

440

460

480

500

520

540

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

29

Less than 2 Between 2 and 4

Between 4 and 6

More than 6 420

430

440

450

460

470

480

490

500

510

520

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15Mathematics Disciplinary Climate

Mat

hem

atic

s sc

ore

Inde

x of

dis

cipl

inar

y cl

imat

e

Mean 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

Irelan

d

Macao

-China

Norway

New Zeala

nd

Costa R

ica

Albania

Thail

and

Estonia

United S

tates

ChileQatar

Canada

Portug

al

United K

ingdo

m

Luxe

mbourg

Kazak

hstan

Denmar

k

Uruguay

Switzer

land

Lithu

ania

Israel

German

y

BelgiumFra

nce

Indone

sia

Monten

egro

Latvi

aJa

pan

Serbia

Czech

Rep

ublic

Romania

Korea

Croatia

-10

-5

0

5

10

15

20

Score-point change

The relationship between time and performance is much weaker after accounting for school characteristics

Source: Figure 3.5 Equations

30

CroatiaShanghai-

ChinaKoreaChinese

TapeiRomaniaAustria

Italy

Japan

IndonesiaNetherlandsTurkey

MalaysiaSingapore

Relationship between time and performance among students in the same school and grade

Liechtenstein

Chech Republic

31

First quintile Second quintile

Third quintile Fourth quintile Fifth quintile420

440

460

480

500

520

540

Applied mathematics Pure mathematics

Quintiles of exposure

Exposure to pure mathematics is more strongly related to performance than exposure to applied mathematics

Source: Figure 3.9

Mean score

32

-50

0

50

100

150

200

Score-point change

Exposure to pure mathematics is related to higher performance, even after accounting for school characteristics

Source: Figure 3.11

Relationship between exposure and performance among students in the same school

33

300 400 500 600 700 8001.00

1.20

1.40

1.60

1.80

2.00

2.20

2.40

2.60

2.80

Charts Q1

Revolving Door Q2

R² = 0.39372476738353

Drip Rate Q1

Arches Q2

Stronger association between familiarity with concepts and performance on more demanding tasks

Drip Rate Q1

Effec

t of

fam

ilia

rity

Higher positiveeffect

Lower positiveeffect

Source: Figure 3.12

Difficulty on the PISA scale

Revolving Door Q2

Odds ratio

34

Familiarity with pure mathematics is enough to solve procedural problems…

Scenario:Nurses calculate the drip rate for infusions using the formula:

d is the drop factor in drops per mLv is the volume in mL of the infusionn 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

35

-1.00 -0.50 0.00 0.50 1.00 1.50-4.00

-3.00

-2.00

-1.00

0.00

1.00

2.00

Korea

OECD average

Indonesia

MalaysiaQatar

Shanghai-China

Chinese-TaipeiR² = 0.571370808831112

Logi

t fo

r th

e it

em D

rip

Rat

e Q

1


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

36

Familiarity with mathematics and performance on Drip Rate Question 1 - Country-level relationship

-0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00-1.60

-1.40

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

KoreaOECD average

Netherlands

Luxembourg

Spain

R² = 0.223220613403386

AFTER accounting for countries’ performance on all the other tasks


Logi

t fo

r th

e it

em D

rip

Rat

e Q

1

Source: Figure 3.13

37

…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

38

Familiarity with mathematics and performance on Revolving Door Question 2 - Country-level relationship

-1.00 -0.50 0.00 0.50 1.00 1.50-6.00

-5.00

-4.00

-3.00

-2.00

-1.00

0.00

Korea

OECD averageIndonesia

MalaysiaQatar

Shanghai-China

Chinese-Taipei

R² = 0.18013358168691

BEFORE accounting for countries’ performance on all the other tasks

Logi

t fo

r th

e it

em

Rev

olvi

ng D

oor

Q2

Index of familiarity with mathematicsSource: Figure 3.14

39

Familiarity with mathematics and performance on Revolving Door Question 2 - Country-level relationship

-0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80-5.00

-4.50

-4.00

-3.50

-3.00

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

Korea

OECD average

Netherlands

Luxembourg

Spain

R² = 0.0163224692151583Logi

t fo

r th

e it

em

Rev

olvi

ng D

oor

Q2


Source: Figure 3.14

AFTER accounting for countries’ performance on all the other tasks

40

Macao

-China

Tunisia

Estonia

Denmark

Costa Rica

Kazak

hstanLatv

ia

Lithuan

ia

Romania

Irelan

dJa

pan

Czech

Republic

Russian

Federa

tion

Poland

Sweden

Uruguay

Monteneg

ro

Luxembourg

Serbia

OECD avera

geQata

r

Turkey

Australi

a

Chinese T

aipei

Netherl

ands

Spain

Thailan

dBraz

il

Belgium

Switzerl

and

Austria

-30

-20

-10

0

10

20

30

40% 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

41

Mac

ao-C

hina

Esto

nia

Thai

land

Indo

nesi

aJo

rdan

Icel

and

Finl

and

Mex

ico

Serb

iaH

ong

Kong

-Chi

naAr

gent

ina

Gre

ece

Tuni

sia

Nor

way

Kaza

khst

anC

roat

iaKo

rea

Mal

aysi

aU

nite

d Ki

ngdo

mR

oman

iaC

anad

aTu

rkey

Ger

man

yLa

tvia

Col

ombi

aD

enm

ark

Cos

ta R

ica

Net

herla

nds

Uni

ted

Stat

esQ

atar

Switz

erla

ndSw

eden

Rus

sian

Fed

erat

ion

Lith

uani

aC

zech

Rep

ublic

Italy

Braz

ilM

onte

negr

oO

ECD

ave

rage

Viet

Nam

Irela

ndSl

oven

iaAu

stra

liaJa

pan

Aust

riaSi

ngap

ore

Pola

ndN

ew Z

eala

ndPo

rtuga

lU

nite

d Ar

ab E

mira

tes

Spai

nBe

lgiu

mU

rugu

ayLu

xem

bour

gH

unga

ryFr

ance

Chi

leSh

angh

ai-C

hina

Isra

elBu

lgar

iaPe

ruC

hine

se T

aipe

iSl

ovak

Rep

ublic

0.30

0.40

0.50

0.60

0.70

0.80

0.90

The task has a scientific context The task has a personal contextOdds ra-tio

Socio-economically disadvantaged students perform better on “familiar” tasks

Soc

io-e

cono

mic

gap

in p

erfo

rman

ce

Larger gap

Smaller gap

Source: Figure 4.18

42

• 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 performanc

e

• 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

43

OPPORTUNITY TO LEARN AND STUDENTS’ ATTITUDES TOWARDS MATHEMATICS

44

Less than half of students enjoy studying mathematics

Aust

ria

-4Hu

ngar

y

Sl

ovak

Rep

ublic

-5

Finl

and

4

Belg

ium

-5

Czec

h Re

publ

ic

Kore

a

Ja

pan

5

Norw

ay

Neth

erla

nds

Luxe

mbo

urg

Pola

nd

-4Ca

nada

Un

ited

Stat

es

Swed

en

Irela

nd

4Sp

ain

OEC

D av

erag

e

Ne

w Z

eala

nd

Latv

ia

Aust

ralia

3

Ger

man

y

-4Fr

ance

-5

Mac

ao-C

hina

Ru

ssia

n Fe

dera

tion

Portu

gal

Italy

Ic

elan

d 1

0Sw

itzer

land

-4

Urug

uay

Gre

ece

8

Turk

ey

-5M

exic

o

8Ho

ng K

ong-

Chin

a

3Li

echt

enst

ein

Braz

il -

4De

nmar

k

Tu

nisi

a -

9Th

aila

nd

Indo

nesi

a 5

0102030405060708090

2012 2003%

Source: Figure 4.2

Percentage of students who agree with the statement I do mathematics because I enjoy it"

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

Mat

hem

atic

s se

lf-c

once

pt

Source: Figure 4.7

Liec

hten

stei

n

Arg

entin

a

Kaz

akhs

tan

Luxe

mbo

urg

Tuni

sia

Japa

n

Mac

ao-C

hina

Ger

man

y

Slov

ak R

epub

lic

Bra

zil

Hon

g K

ong-

Chi

na

Latv

ia

Viet

Nam

Cos

ta R

ica

Lith

uani

a

Isra

el

Rus

sian

Fed

erat

ion

Swed

en

Cze

ch R

epub

lic

Mon

tene

gro

New

Zea

land

Uni

ted

Ara

b Em

irate

s

Uni

ted

Kin

gdom

Cro

atia

Hun

gary

Spai

n

Fran

ce

Can

ada

Sing

apor

e

Icel

and

Aus

tral

ia

Chi

nese

Tai

pei-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40Before accounting for performance in mathematics Index

change

Change in students’ self-concept associated with 1 unit change in familiarity

46

Exposure to more complex mathematics is also related to greater anxiety among low-performing students

Source : Figure 4.8

Malaysia OECD average Czech Republic-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

Bottom quarter by mathematics performance Index change

Change in students’ anxiety associated with a change in familiarity, by students' mathematics performance

Mor

e an

xiet

y Le

ss

anxi

ety

47

Students with hard-working friends are more motivated to learn, especially in schools where students are least familiar with mathematics

I am interested in the things I learn in math-

ematics

I do mathemat-ics because I

enjoy it

I look forward to mathemat-

ics lessons

Making an ef -fort is worth-while for the

work I want to do

Mathematics is important for what I want to study later on

0.00

0.50

1.00

1.50

2.00

2.50

Schools where students are more familiar with mathematicsSchools 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

48

High-performing students whose parents do not like mathematics are more likely to feel helpless

OECD av-erage

France

1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40

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

49

Students whose teachers provide feedback or specify learning goals are more familiar with mathematics

The teacher gives different work to class-

mates 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 so-

lutions to a problem or task

The teacher gives extra help when students

need it

The teacher continues teach-

ing until the students under-

stand

The teacher asks questions that make us reflect on the

problem

The teacher gives problems that require us to think for an extended time

0.00

0.05

0.10

0.15

0.200.25

0.30

0.35

0.40

0.45

Students more familiar with mathematics Students less familiar with mathematicsIndex 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

50

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

Low familiarity students High familiarity studentsIndex change

Change in mathematics anxiety associated with having teachers who engage in these practices, OECD average

Mor

e an

xiet

yLe

ss a

nxie

ty

51

Using a computer during mathematics lessons is associated with higher motivation for learning mathematics

Japan

Denmark

Liechten

stein

Mexico

Estonia

Czech

Republic

Belgium

Russian

Federa

tion

Finland

Norway

Croatia

Italy

Netherl

ands

Costa Rica

Latvia

Chile

Poland

Sweden

Slovak R

epublic

Greece

Jordan

Israe

l0.00

0.10

0.20

0.30

0.40

0.50After 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

52

• 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


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

53

WHAT DOES THIS MEAN FOR POLICY?

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 skills

beyond knowledg

e

Reduce the

impact of tracking

Support teachers

of heterogen

ous classes

Support positive

attitudes

Monitor Opportuni

ty 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



pointsRecent revisions of the mathematics curricula in England, Scotland, Korea and Singapore emphasise the development of problem-solving skills


Develop skills

beyond knowledg

e

Reduce the

impact of tracking

Support teachers

of heterogen

ous classes

Support positive

attitudes

Monitor Opportuni

ty 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



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 skills

beyond knowledg

e

Reduce the

impact of tracking

Support teachers

of heterogen

ous 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



points

In Finland, half of children with

special education needs are

mainstreamed and assigned

special teachers, rather than being

in special schools.


Develop skills

beyond knowledg

e

Reduce the

impact of tracking

Support teachers

of heterogen

ous classes

Support positive

attitudes

Monitor Opportuni

ty 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



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 skills

beyond knowledg

e

Reduce the

impact of tracking

Support teachers

of heterogen

ous 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



pointsThe Teaching and Leaning International Survey

(TALIS) study is piloting an international video

study of teaching practices to provide

insights into effective teaching practices

60

THANK YOU

Equations and Inequalities - Making mathematics accessible to all

Education

Transcript of Equations and Inequalities - Making mathematics accessible to all