Contributed by:

The findings and recommendations in this report are based on the academic research literature on mathematics education, on data from 2012 assessment and from the questionnaires distributed to participating students and school principals, and on teacher data from 2013. Keep in mind that the teaching and learning strategies discussed in this report were not actually observed; students were asked about the teaching practices they observed from their current teachers only, and teachers were asked to report on the strategies they use. PISA and TALIS are cross-sectional studies – data are collected at one specific point in time – and they do not – and cannot – describe cause and effect. For these reasons, the findings should be interpreted with caution.

1.
Teaching

strategies

Cognitive

activation

Lessons

drawn

Classroom

climate

Memorisation

Pure & applied

maths

Control

Socio-economic

status

Elaboration

strategies

Ten Questions

for Mathematics

... and how PISA can

help answer them

strategies

Cognitive

activation

Lessons

drawn

Classroom

climate

Memorisation

Pure & applied

maths

Control

Socio-economic

status

Elaboration

strategies

Ten Questions

for Mathematics

... and how PISA can

help answer them

2.
PISA

Ten Questions for

Mathematics Teachers

... and how PISA can help answer them

Ten Questions for

Mathematics Teachers

... and how PISA can help answer them

3.
This work is published under the responsibility of the Secretary-General of the OECD. The opinions

expressed and the arguments employed herein do not necessarily reflect the official views of the

OECD member countries.

This document and any map included herein are without prejudice to the status of or sovereignty

over any territory, to the delimitation of international frontiers and boundaries and to the name of

any territory, city or area.

Please cite this publication as:

OECD (2016), Ten Questions for Mathematics Teachers ... and how PISA can help answer them,

PISA, OECD Publishing, Paris,

http://dx.doi.or /10.1787/9789264265387-en.

ISBN 978-9264-26537-0 (print)

ISBN 978-9264-26538-7 (online)

Series: PISA

ISSN 1990-85 39 (print)

ISSN 1996-3777 (online)

The statistical data for Israel are supplied by and under the responsibility of the relevant Israeli

authorities. The use of such data by the OECD is without prejudice to the status of the Golan Heights,

East Jerusalem and Israeli settlements in the West Bank under the terms of international law.

Latvia was not an OECD member at the time of preparation of this publication. Accordingly, Latvia is

not included in the OECD average.

Corrigenda to OECD publications may be found on line at: www.oecd.org/publishing/corrigenda.

© OECD 2016

You can copy, download or print OECD content for your own use, and you can include excerpts from

OECD publications, databases and multimedia products in your own documents, presentations,

blogs, websites and teaching materials, provided that suitable acknowledgement of OECD as source

and copyright owner is given. All requests for public or commercial use and translation rights should

be submitted to rights@oecd.org. Requests for permission to photocopy portions of this material for

public or commercial use shall be addressed directly to the Copyright Clearance Center (CCC) at info@

copyright.com or the Centre français d’exploitation du droit de copie (CFC) at contact@cfcopies.com.

expressed and the arguments employed herein do not necessarily reflect the official views of the

OECD member countries.

This document and any map included herein are without prejudice to the status of or sovereignty

over any territory, to the delimitation of international frontiers and boundaries and to the name of

any territory, city or area.

Please cite this publication as:

OECD (2016), Ten Questions for Mathematics Teachers ... and how PISA can help answer them,

PISA, OECD Publishing, Paris,

http://dx.doi.or /10.1787/9789264265387-en.

ISBN 978-9264-26537-0 (print)

ISBN 978-9264-26538-7 (online)

Series: PISA

ISSN 1990-85 39 (print)

ISSN 1996-3777 (online)

The statistical data for Israel are supplied by and under the responsibility of the relevant Israeli

authorities. The use of such data by the OECD is without prejudice to the status of the Golan Heights,

East Jerusalem and Israeli settlements in the West Bank under the terms of international law.

Latvia was not an OECD member at the time of preparation of this publication. Accordingly, Latvia is

not included in the OECD average.

Corrigenda to OECD publications may be found on line at: www.oecd.org/publishing/corrigenda.

© OECD 2016

You can copy, download or print OECD content for your own use, and you can include excerpts from

OECD publications, databases and multimedia products in your own documents, presentations,

blogs, websites and teaching materials, provided that suitable acknowledgement of OECD as source

and copyright owner is given. All requests for public or commercial use and translation rights should

be submitted to rights@oecd.org. Requests for permission to photocopy portions of this material for

public or commercial use shall be addressed directly to the Copyright Clearance Center (CCC) at info@

copyright.com or the Centre français d’exploitation du droit de copie (CFC) at contact@cfcopies.com.

4.
A TEACHER’S GUIDE TO MATHEMATICS TEACHING AND LEARNING . 3

A teacher’s guide to mathematics

teaching and learning

Every three years, a sample of 15-year-old students around the world

sits an assessment, known as PISA, that aims to measure how well

their education system has prepared them for life after compulsory

schooling. PISA stands for the Programme for International Student

Assessment. The assessment, which is managed by the OECD, in

partnership with national centres and leading experts from around

the world, is conducted in over 70 countries and economies. It

covers mathematics, science and reading.

PISA develops tests that are not directly linked to the school curriculum; they

assess the extent to which students can apply their knowledge and skills to

real-life problems. In 2012, the assessment focused on mathematics. The results

provide a comparison of what 15-year-old students in each participating country

can or cannot do when asked to apply their understanding of mathematical

concepts related to such areas as quantity, uncertainty, space or change. As part

of PISA 2012, students also completed a background questionnaire, in which

they provided information about themselves, their homes and schools, and their

experiences at school and in mathematics classes in particular. It is from these

data that PISA analysts are able to understand what factors might influence

student achievement in mathematics.

While many national centres and governments try to ensure that the schools and

teachers participating in the assessments get constructive feedback based on PISA

results, most of the key messages published in the PISA reports don’t make it back

to the classroom, to the teachers who are preparing their country’s students every

day. Until now.

A teacher’s guide to mathematics

teaching and learning

Every three years, a sample of 15-year-old students around the world

sits an assessment, known as PISA, that aims to measure how well

their education system has prepared them for life after compulsory

schooling. PISA stands for the Programme for International Student

Assessment. The assessment, which is managed by the OECD, in

partnership with national centres and leading experts from around

the world, is conducted in over 70 countries and economies. It

covers mathematics, science and reading.

PISA develops tests that are not directly linked to the school curriculum; they

assess the extent to which students can apply their knowledge and skills to

real-life problems. In 2012, the assessment focused on mathematics. The results

provide a comparison of what 15-year-old students in each participating country

can or cannot do when asked to apply their understanding of mathematical

concepts related to such areas as quantity, uncertainty, space or change. As part

of PISA 2012, students also completed a background questionnaire, in which

they provided information about themselves, their homes and schools, and their

experiences at school and in mathematics classes in particular. It is from these

data that PISA analysts are able to understand what factors might influence

student achievement in mathematics.

While many national centres and governments try to ensure that the schools and

teachers participating in the assessments get constructive feedback based on PISA

results, most of the key messages published in the PISA reports don’t make it back

to the classroom, to the teachers who are preparing their country’s students every

day. Until now.

5.
4 . TEN QUESTIONS FOR MATHEMATICS TEACHERS

USING PISA TO SUPPORT MATHEMATICS TEACHERS

The PISA student background questionnaire sought information about students’

experiences in their mathematics classes, including their learning strategies and

the teaching practices they said their teachers used. This information, coupled

with students’ results on the mathematics assessment, allow us to examine how

certain teaching and learning strategies are related to student performance in

mathematics. We can then delve deeper into the student background data to

look at the relationships between other student characteristics, such as students’

gender, socio-economic status, their attitudes toward mathematics and their

career aspirations, to ascertain whether these characteristics might be related to

teaching and learning strategies or performance. PISA data also make it possible

to see how the curriculum is implemented in mathematics classes around the

world, and to examine whether the way mathematics classes are structured varies

depending on the kinds of students being taught or the abilities of those students.

This report takes the findings from these analyses and organises them into ten

questions, listed below, that discuss what we know about mathematics teaching and

learning around the world – and how these data might help you in your mathematics

Questions included in this report:

How much should I direct As a mathematics Can I help my

student learning in my teacher, how important students learn

mathematics classes? is the relationship I have how to learn

with my students? mathematics?

1

3 5

Teaching Cognitive Classroom

strategies activation climate Memorisation Control

2 4

Are some mathematics What do we know about

teaching methods more memorisation and

effective than others? learning mathematics?

USING PISA TO SUPPORT MATHEMATICS TEACHERS

The PISA student background questionnaire sought information about students’

experiences in their mathematics classes, including their learning strategies and

the teaching practices they said their teachers used. This information, coupled

with students’ results on the mathematics assessment, allow us to examine how

certain teaching and learning strategies are related to student performance in

mathematics. We can then delve deeper into the student background data to

look at the relationships between other student characteristics, such as students’

gender, socio-economic status, their attitudes toward mathematics and their

career aspirations, to ascertain whether these characteristics might be related to

teaching and learning strategies or performance. PISA data also make it possible

to see how the curriculum is implemented in mathematics classes around the

world, and to examine whether the way mathematics classes are structured varies

depending on the kinds of students being taught or the abilities of those students.

This report takes the findings from these analyses and organises them into ten

questions, listed below, that discuss what we know about mathematics teaching and

learning around the world – and how these data might help you in your mathematics

Questions included in this report:

How much should I direct As a mathematics Can I help my

student learning in my teacher, how important students learn

mathematics classes? is the relationship I have how to learn

with my students? mathematics?

1

3 5

Teaching Cognitive Classroom

strategies activation climate Memorisation Control

2 4

Are some mathematics What do we know about

teaching methods more memorisation and

effective than others? learning mathematics?

6.
A TEACHER’S GUIDE TO MATHEMATICS TEACHING AND LEARNING . 5

classes right now. The questions encompass teaching strategies, student learning

strategies, curriculum coverage and various student characteristics, and how they

are related to student achievement in mathematics and to each other. Each question

is answered by the data and related analysis, and concludes with a section entitled

“What can teachers do?” that provides concrete, evidence-based suggestions to help

you develop your mathematics teaching practice.

WHAT DO WE MEAN BY TEACHING AND LEARNING STRATEGIES?

In simple terms, teaching strategies refer to “everything teachers do or should do in

order to help their learners learn”.1 Also called teaching practices in this book, they

can include everything from planning and organising lessons, classes, resources and

assessments, to the individual actions and activities that teachers engage in during

their classroom teaching.

Learning strategies are the behaviours and thoughts students use as they attempt

to complete various tasks associated with the process of learning a new concept or

acquiring, storing, retrieving and using information.2

Should I encourage Should my teaching emphasise What can teachers learn

students to use mathematical concepts or how from PISA?

their creativity in those concepts are applied in

10

mathematics? the real world?

6 8

Elaboration Socio-economic Pure & applied Students’ Lessons

strategies status maths attitudes drawn

7 9

Do students’ backgrounds Should I be concerned about

influence how they learn my students’ attitudes towards

mathematics? mathematics?

classes right now. The questions encompass teaching strategies, student learning

strategies, curriculum coverage and various student characteristics, and how they

are related to student achievement in mathematics and to each other. Each question

is answered by the data and related analysis, and concludes with a section entitled

“What can teachers do?” that provides concrete, evidence-based suggestions to help

you develop your mathematics teaching practice.

WHAT DO WE MEAN BY TEACHING AND LEARNING STRATEGIES?

In simple terms, teaching strategies refer to “everything teachers do or should do in

order to help their learners learn”.1 Also called teaching practices in this book, they

can include everything from planning and organising lessons, classes, resources and

assessments, to the individual actions and activities that teachers engage in during

their classroom teaching.

Learning strategies are the behaviours and thoughts students use as they attempt

to complete various tasks associated with the process of learning a new concept or

acquiring, storing, retrieving and using information.2

Should I encourage Should my teaching emphasise What can teachers learn

students to use mathematical concepts or how from PISA?

their creativity in those concepts are applied in

10

mathematics? the real world?

6 8

Elaboration Socio-economic Pure & applied Students’ Lessons

strategies status maths attitudes drawn

7 9

Do students’ backgrounds Should I be concerned about

influence how they learn my students’ attitudes towards

mathematics? mathematics?

7.
6 . TEN QUESTIONS FOR MATHEMATICS TEACHERS

You’ll also find some data in this report from the Teaching and Learning

International Survey, or TALIS, an OECD-led survey in which 34 countries and

economies – and over 104,000 lower secondary teachers – took part in 2013.

(Lower secondary teachers teach students of approximately the same age as the

students who participate in PISA.) TALIS asked teachers about themselves, their

teaching practices and the learning environment. These data provide information

about how certain teaching strategies or behaviours might influence you as a

teacher. In other words, could certain actions that you take actually improve your

own feelings of self-confidence or your satisfaction with your work?

THE BOTTOM LINE

Teaching is considered by many to be one of the most challenging, rewarding and

important professions in the world today. As such, teachers are under constant

pressure to improve learning and learning outcomes for their students. This report

tries to give you timely and relevant data and analyses that can help you reflect

on how you teach mathematics and on how your students learn. We hope that you

find it useful in your own development as a mathematics teacher.

ABOUT THE DATA

The findings and recommendations in this report are based on the academic research

literature on mathematics education, on data from the PISA 2012 assessment and

from the questionnaires distributed to participating students and school principals,

and on teacher data from TALIS 2013. Keep in mind that the teaching and learning

strategies discussed in this report were not actually observed; students were asked

about the teaching practices they observed from their current teachers only, and

teachers were asked to report on the strategies they use. PISA and TALIS are cross-

sectional studies – data are collected at one specific point in time – and they do not

– and cannot – describe cause and effect. For these reasons, the findings should be

interpreted with caution.

The OECD average is the arithmetic mean of 34 OECD countries: Australia, Austria,

Belgium, Canada, Chile, the Czech Republic, Denmark, Estonia, Finland, France, Germany,

Greece, Hungary, Iceland, Ireland, Israel, Italy, Japan, Korea, Luxembourg, Mexico, the

Netherlands, New Zealand, Norway, Poland, Portugal, the Slovak Republic, Slovenia,

Spain, Sweden, Switzerland, Turkey, the United Kingdom and the United States. Latvia

acceded to the OECD on 1 July 2016. It is not included in the OECD average.

You’ll also find some data in this report from the Teaching and Learning

International Survey, or TALIS, an OECD-led survey in which 34 countries and

economies – and over 104,000 lower secondary teachers – took part in 2013.

(Lower secondary teachers teach students of approximately the same age as the

students who participate in PISA.) TALIS asked teachers about themselves, their

teaching practices and the learning environment. These data provide information

about how certain teaching strategies or behaviours might influence you as a

teacher. In other words, could certain actions that you take actually improve your

own feelings of self-confidence or your satisfaction with your work?

THE BOTTOM LINE

Teaching is considered by many to be one of the most challenging, rewarding and

important professions in the world today. As such, teachers are under constant

pressure to improve learning and learning outcomes for their students. This report

tries to give you timely and relevant data and analyses that can help you reflect

on how you teach mathematics and on how your students learn. We hope that you

find it useful in your own development as a mathematics teacher.

ABOUT THE DATA

The findings and recommendations in this report are based on the academic research

literature on mathematics education, on data from the PISA 2012 assessment and

from the questionnaires distributed to participating students and school principals,

and on teacher data from TALIS 2013. Keep in mind that the teaching and learning

strategies discussed in this report were not actually observed; students were asked

about the teaching practices they observed from their current teachers only, and

teachers were asked to report on the strategies they use. PISA and TALIS are cross-

sectional studies – data are collected at one specific point in time – and they do not

– and cannot – describe cause and effect. For these reasons, the findings should be

interpreted with caution.

The OECD average is the arithmetic mean of 34 OECD countries: Australia, Austria,

Belgium, Canada, Chile, the Czech Republic, Denmark, Estonia, Finland, France, Germany,

Greece, Hungary, Iceland, Ireland, Israel, Italy, Japan, Korea, Luxembourg, Mexico, the

Netherlands, New Zealand, Norway, Poland, Portugal, the Slovak Republic, Slovenia,

Spain, Sweden, Switzerland, Turkey, the United Kingdom and the United States. Latvia

acceded to the OECD on 1 July 2016. It is not included in the OECD average.

8.
A TEACHER’S GUIDE TO MATHEMATICS TEACHING AND LEARNING . 7

This publication was written by Kristen Weatherby, based on research and

analysis by Alfonso Echazarra, Mario Piacentini, Daniel Salinas, Chiara Monticone,

Pablo Fraser and Noémie Le Donné. Giannina Rech provided analytical and

editorial input for the report. Judit Pál, Hélène Guillou, Jeffrey Mo and Vanessa

Denis provided statistical support. The publication was edited by Marilyn Achiron,

and production was overseen by Rose Bolognini. Andreas Schleicher, Montserrat

Gomendio, Yuri Belfali, Miyako Ikeda and Cassandra Davis provided invaluable

guidance and assistance.

1. Hatch, E., and C. Brown (2000), Vocabulary, Semantics and Language Education, Cambridge University

Press, Cambridge.

2. Dansereau, D. (1985), “Learning Strategy Research”, in J. Segal, S. Chipman and R. Glaser (eds.), Thinking

and Learning Skills, Lawrence Erlbaum Associates, Mahwah, New Jersey.

This publication has

Look for the StatLinks at the bottom of the tables or graphs in this book. To download the

matching Excel® spreadsheet, just type the link into your Internet browser, starting with the http://dx.doi.org

prex, or click on the link from the e-book edition.

This publication was written by Kristen Weatherby, based on research and

analysis by Alfonso Echazarra, Mario Piacentini, Daniel Salinas, Chiara Monticone,

Pablo Fraser and Noémie Le Donné. Giannina Rech provided analytical and

editorial input for the report. Judit Pál, Hélène Guillou, Jeffrey Mo and Vanessa

Denis provided statistical support. The publication was edited by Marilyn Achiron,

and production was overseen by Rose Bolognini. Andreas Schleicher, Montserrat

Gomendio, Yuri Belfali, Miyako Ikeda and Cassandra Davis provided invaluable

guidance and assistance.

1. Hatch, E., and C. Brown (2000), Vocabulary, Semantics and Language Education, Cambridge University

Press, Cambridge.

2. Dansereau, D. (1985), “Learning Strategy Research”, in J. Segal, S. Chipman and R. Glaser (eds.), Thinking

and Learning Skills, Lawrence Erlbaum Associates, Mahwah, New Jersey.

This publication has

Look for the StatLinks at the bottom of the tables or graphs in this book. To download the

matching Excel® spreadsheet, just type the link into your Internet browser, starting with the http://dx.doi.org

prex, or click on the link from the e-book edition.

9.
Teaching strategies

How much should

I direct student

learning in my

mathematics

classes?

How much should

I direct student

learning in my

mathematics

classes?

10.
HOW MUCH SHOULD I DIRECT STUDENT LEARNING IN MY MATHEMATICS CLASSES? . 9

The traditional view of a classroom that has existed for generations

TEACHING STRATEGIES

in schools around the world consists of students sitting at desks,

passively listening as the teacher stands in the front of the class

and lectures or demonstrates something on a board or screen. The

teacher has planned the lesson, knows the content she needs to

cover and delivers it to the students, who are expected to absorb

that content and apply it to their homework or a test. This kind

of “teacher-directed” instruction might also include things like

lectures, lesson summaries or question-and-answer periods that

are driven by the teacher. This form of teaching isn’t limited to

mathematics, necessarily, and it’s a teaching strategy that everyone

has experienced as a student at one time or another.

For decades now, educationalists have encouraged giving students more control

over their own learning; thus student-oriented teaching strategies are increasingly

finding their way into classrooms of all subjects. As the name indicates, student-

oriented teaching strategies place the student at the centre of the activity, giving

learners a more active role in the lesson than in traditional, teacher-directed

strategies. These student-oriented teaching strategies can include activities such

as assigning student projects that might take a week or longer to complete or

working in small groups through which learners must work together to solve a

problem or accomplish a task.

Which type of teaching strategy is being used to teach mathematics in schools

around the world? And which one should teachers be using? Data indicate a

prevalence of teacher-directed methods, but deciding how to teach mathematics

isn’t as simple as choosing between one strategy and another. Teachers need

to consider both the content and students to be taught when choosing the best

teaching strategy for their mathematics lessons.

The traditional view of a classroom that has existed for generations

TEACHING STRATEGIES

in schools around the world consists of students sitting at desks,

passively listening as the teacher stands in the front of the class

and lectures or demonstrates something on a board or screen. The

teacher has planned the lesson, knows the content she needs to

cover and delivers it to the students, who are expected to absorb

that content and apply it to their homework or a test. This kind

of “teacher-directed” instruction might also include things like

lectures, lesson summaries or question-and-answer periods that

are driven by the teacher. This form of teaching isn’t limited to

mathematics, necessarily, and it’s a teaching strategy that everyone

has experienced as a student at one time or another.

For decades now, educationalists have encouraged giving students more control

over their own learning; thus student-oriented teaching strategies are increasingly

finding their way into classrooms of all subjects. As the name indicates, student-

oriented teaching strategies place the student at the centre of the activity, giving

learners a more active role in the lesson than in traditional, teacher-directed

strategies. These student-oriented teaching strategies can include activities such

as assigning student projects that might take a week or longer to complete or

working in small groups through which learners must work together to solve a

problem or accomplish a task.

Which type of teaching strategy is being used to teach mathematics in schools

around the world? And which one should teachers be using? Data indicate a

prevalence of teacher-directed methods, but deciding how to teach mathematics

isn’t as simple as choosing between one strategy and another. Teachers need

to consider both the content and students to be taught when choosing the best

teaching strategy for their mathematics lessons.

11.
10 . TEN QUESTIONS FOR MATHEMATICS TEACHERS

WHERE DOES MATHEMATICS TEACHING FALL IN THE TEACHER- VS. STUDENT-

DIRECTED LEARNING DEBATE?

In PISA, students were asked about the frequency with which their teachers use

student-oriented or teacher-directed strategies in their lessons. Findings indicate

that today, teacher-directed practices are used widely. For instance, across OECD

countries, eight out of ten students reported that their teachers tell them what

they have to learn in every lesson, and seven out of ten students have teachers

who ask questions in every lesson to check that students understand what they’re

On the other hand, the student-oriented practice that teachers most commonly

use is assigning students different work based on their ability, commonly called

differentiated instruction. However, according to students, this practice is

used only occasionally, as fewer than one in three students in OECD countries

reported that their teachers use this practice frequently in their lessons. Figure

1.1 shows the reported frequency of both teacher-directed and student-oriented

instructional strategies for mathematics.

WHERE DOES MATHEMATICS TEACHING FALL IN THE TEACHER- VS. STUDENT-

DIRECTED LEARNING DEBATE?

In PISA, students were asked about the frequency with which their teachers use

student-oriented or teacher-directed strategies in their lessons. Findings indicate

that today, teacher-directed practices are used widely. For instance, across OECD

countries, eight out of ten students reported that their teachers tell them what

they have to learn in every lesson, and seven out of ten students have teachers

who ask questions in every lesson to check that students understand what they’re

On the other hand, the student-oriented practice that teachers most commonly

use is assigning students different work based on their ability, commonly called

differentiated instruction. However, according to students, this practice is

used only occasionally, as fewer than one in three students in OECD countries

reported that their teachers use this practice frequently in their lessons. Figure

1.1 shows the reported frequency of both teacher-directed and student-oriented

instructional strategies for mathematics.

12.
HOW MUCH SHOULD I DIRECT STUDENT LEARNING IN MY MATHEMATICS CLASSES? . 11

Figure 1.1 Teacher-directed and student-oriented instruction

TEACHING STRATEGIES

Percentage of students who responded “in every lesson” or “in most lessons”, OECD average

a. Teacher-directed strategies

At the beginning of a lesson, the teacher

presents a short summary of the previous lesson

The teacher asks me or my classmates to present

our thinking or reasoning at some length

The teacher sets clear goals for our learning

The teacher asks questions to check whether

we have understood what was taught

The teacher tells us what we have to learn

0 10 20 30 40 50 60 70 80 90 %

b. Student-oriented strategies

The teacher assigns projects that require at

least one week to complete

The teacher asks us to help plan classroom

activities or topics

The teacher has us work in small groups to come

up with joint solutions to a problem or task

The teacher gives different work to

classmates who have difficulties and/or who

can advance faster

0 10 20 30 40 %

Note: The OECD average includes all member countries of the OECD except Latvia.

Source: OECD, PISA 2012 Database, adapted from Echazarra, A. et al. (2016), “How teachers teach and students learn: Successful

strategies for school”, OECD Education Working Paper, no. 130.

Statlink: http://dx.doi.org/10.1787/888933414750

Figure 1.1 Teacher-directed and student-oriented instruction

TEACHING STRATEGIES

Percentage of students who responded “in every lesson” or “in most lessons”, OECD average

a. Teacher-directed strategies

At the beginning of a lesson, the teacher

presents a short summary of the previous lesson

The teacher asks me or my classmates to present

our thinking or reasoning at some length

The teacher sets clear goals for our learning

The teacher asks questions to check whether

we have understood what was taught

The teacher tells us what we have to learn

0 10 20 30 40 50 60 70 80 90 %

b. Student-oriented strategies

The teacher assigns projects that require at

least one week to complete

The teacher asks us to help plan classroom

activities or topics

The teacher has us work in small groups to come

up with joint solutions to a problem or task

The teacher gives different work to

classmates who have difficulties and/or who

can advance faster

0 10 20 30 40 %

Note: The OECD average includes all member countries of the OECD except Latvia.

Source: OECD, PISA 2012 Database, adapted from Echazarra, A. et al. (2016), “How teachers teach and students learn: Successful

strategies for school”, OECD Education Working Paper, no. 130.

Statlink: http://dx.doi.org/10.1787/888933414750

13.
12 . TEN QUESTIONS FOR MATHEMATICS TEACHERS

The PISA survey also indicates that students may be exposed to different

teaching strategies based on their socio-economic status or gender. For example,

girls reported being less frequently exposed to student-oriented instruction in

mathematics class than boys did. Conversely, disadvantaged students, who are

from the bottom quarter of the socio-economic distribution in their countries,

reported more frequent exposure to these strategies than advantaged students

did. Teachers might have reasons for teaching specific classes in the ways they

do; and other factors, such as student motivation or disruptive behaviour, might

be at play too. Ideally, however, all students should have the opportunity to be

exposed to some student-oriented strategies, regardless of their gender or social

status. Also, when considering an entire country, the more frequently teacher-

directed instruction is used compared with student-oriented instruction, the more

frequently students learn using memorisation strategies (Figure 1.2).

Figure 1.2 How teachers teach and students learn

Results based on students’ reports

More

Students in Ireland reported the most frequent

use of teacher-directed instruction compared to

United Kingdom student-oriented instruction Ireland

New Zealand

Uruguay Australia

Norway Israel

Netherlands Austria

France

Learning

Iceland Canada Belgium

Indonesia Singapore

Chile Spain Japan Luxembourg R² = 0.10

Costa Rica United States Finland

United Arab Bulgaria Germany Hong Kong-China Hungary

Portugal OECD average

Emirates Greece

Brazil

Sweden Denmark Czech Estonia Shanghai-China

Turkey

Argentina Republic Korea

Switzerland

Thailand Romania Croatia

Jordan Macao-China Slovenia

Montenegro Italy Poland

Qatar Colombia Mexico

Peru Latvia Viet Nam

Malaysia

Chinese Taipei

Tunisia Serbia Slovak

Albania Republic Lithuania

Kazakhstan Russian Federation

More More

student-oriented Teaching teacher-directed

instruction instruction

Source: OECD, PISA 2012 Database.

Statlink: http://dx.doi.org/10.1787/888933414765

The PISA survey also indicates that students may be exposed to different

teaching strategies based on their socio-economic status or gender. For example,

girls reported being less frequently exposed to student-oriented instruction in

mathematics class than boys did. Conversely, disadvantaged students, who are

from the bottom quarter of the socio-economic distribution in their countries,

reported more frequent exposure to these strategies than advantaged students

did. Teachers might have reasons for teaching specific classes in the ways they

do; and other factors, such as student motivation or disruptive behaviour, might

be at play too. Ideally, however, all students should have the opportunity to be

exposed to some student-oriented strategies, regardless of their gender or social

status. Also, when considering an entire country, the more frequently teacher-

directed instruction is used compared with student-oriented instruction, the more

frequently students learn using memorisation strategies (Figure 1.2).

Figure 1.2 How teachers teach and students learn

Results based on students’ reports

More

Students in Ireland reported the most frequent

use of teacher-directed instruction compared to

United Kingdom student-oriented instruction Ireland

New Zealand

Uruguay Australia

Norway Israel

Netherlands Austria

France

Learning

Iceland Canada Belgium

Indonesia Singapore

Chile Spain Japan Luxembourg R² = 0.10

Costa Rica United States Finland

United Arab Bulgaria Germany Hong Kong-China Hungary

Portugal OECD average

Emirates Greece

Brazil

Sweden Denmark Czech Estonia Shanghai-China

Turkey

Argentina Republic Korea

Switzerland

Thailand Romania Croatia

Jordan Macao-China Slovenia

Montenegro Italy Poland

Qatar Colombia Mexico

Peru Latvia Viet Nam

Malaysia

Chinese Taipei

Tunisia Serbia Slovak

Albania Republic Lithuania

Kazakhstan Russian Federation

More More

student-oriented Teaching teacher-directed

instruction instruction

Source: OECD, PISA 2012 Database.

Statlink: http://dx.doi.org/10.1787/888933414765

14.
HOW MUCH SHOULD I DIRECT STUDENT LEARNING IN MY MATHEMATICS CLASSES? . 13

WHICH TEACHERS USE ACTIVE-LEARNING TEACHING PRACTICES IN

TEACHING STRATEGIES

The TALIS study asked mathematics teachers in eight countries about their

regular teaching practices. The study included four active-learning teaching

practices that overlap in large part with student-oriented practices: placing

students in small groups, encouraging students to evaluate their own progress,

assigning students long projects, and using ICT for class work. These practices

have been shown by many research studies to have positive effects on student

learning and motivation. TALIS data show that teachers who are confident in

their own abilities are more likely to engage in active-teaching practices. This is a

somewhat logical finding, as active practices could be thought of as more “risky”

than direct-teaching methods. It can be challenging to use ICT in your teaching

or have students work in groups if you are not confident that you have the skills

needed in pedagogy, content or classroom management.

Figure 1.3 How teachers’ self-efficacy is related to the use of active-learning instruction

Teachers with lower self-efficacy Teachers with higher self-efficacy

More

Active learning instruction

Less

Mexico Australia Latvia Romania Portugal Singapore Spain Finland

Notes: All differences are statistically significant, except in Portugal and Singapore.

Teachers with higher/lower self-efficacy are those with values above/below the country median.

The index of active-learning instruction measures the extent to which teachers use “information and communication technologies in the

classroom”, let “students evaluate their own progress”, work with “students in small groups to come up with a joint solution to a problem”

or encourage students to work on long projects.

The index of self-efficacy measures the extent to which teachers believe in their own ability to control disruptive behaviour, provide

instruction and foster student engagement.

Countries are ranked in descending order of the frequency with which teachers with higher self-efficacy use active-learning instruction.

Source: OECD, TALIS 2013 Database.

Statlink: http://dx.doi.org/10.1787/888933414779

WHICH TEACHERS USE ACTIVE-LEARNING TEACHING PRACTICES IN

TEACHING STRATEGIES

The TALIS study asked mathematics teachers in eight countries about their

regular teaching practices. The study included four active-learning teaching

practices that overlap in large part with student-oriented practices: placing

students in small groups, encouraging students to evaluate their own progress,

assigning students long projects, and using ICT for class work. These practices

have been shown by many research studies to have positive effects on student

learning and motivation. TALIS data show that teachers who are confident in

their own abilities are more likely to engage in active-teaching practices. This is a

somewhat logical finding, as active practices could be thought of as more “risky”

than direct-teaching methods. It can be challenging to use ICT in your teaching

or have students work in groups if you are not confident that you have the skills

needed in pedagogy, content or classroom management.

Figure 1.3 How teachers’ self-efficacy is related to the use of active-learning instruction

Teachers with lower self-efficacy Teachers with higher self-efficacy

More

Active learning instruction

Less

Mexico Australia Latvia Romania Portugal Singapore Spain Finland

Notes: All differences are statistically significant, except in Portugal and Singapore.

Teachers with higher/lower self-efficacy are those with values above/below the country median.

The index of active-learning instruction measures the extent to which teachers use “information and communication technologies in the

classroom”, let “students evaluate their own progress”, work with “students in small groups to come up with a joint solution to a problem”

or encourage students to work on long projects.

The index of self-efficacy measures the extent to which teachers believe in their own ability to control disruptive behaviour, provide

instruction and foster student engagement.

Countries are ranked in descending order of the frequency with which teachers with higher self-efficacy use active-learning instruction.

Source: OECD, TALIS 2013 Database.

Statlink: http://dx.doi.org/10.1787/888933414779

15.
14 . TEN QUESTIONS FOR MATHEMATICS TEACHERS

HOW CAN A VARIETY OF TEACHING STRATEGIES BENEFIT STUDENT

ACHIEVEMENT?

When looking at students’ mean mathematics scores on the PISA assessment

alongside their exposure to the teaching strategies discussed in this chapter,

another reason for using a variety of teaching strategies emerges. Let’s look first

at the most commonly used teaching practices in mathematics, teacher-directed

strategies. The data indicate that when teachers direct student learning, students

are slightly more likely to be successful in solving the easiest mathematics

problems in PISA. Yet as the problems become more difficult, students with more

exposure to direct instruction no longer have a better chance of success. Figure

1.4 shows the relationship between the use of teacher-directed strategies and

students’ success on mathematics problems of varying difficulty.

Figure 1.4 Teacher-directed instruction and item difficulty

Odds ratio, after accounting for other teaching strategies, OECD average

Receiving teacher-directed instruction is associated with an increase

in the probability of success in solving a mathematics problem

Easy problem

Difficult

problem

Odds ratio

R2 = 0.24

Receiving teacher-directed instruction is associated with a decrease

in the probability of success in solving a mathematics problem

300 400 500 600 700 800

Difficulty of mathematics items on the PISA scale

Notes: Statistically significant odds ratios are marked in a darker tone.

Chile and Mexico are not included in the OECD average.

Source: OECD, PISA 2012 Database, adapted from Echazarra, A. et al. (2016), “How teachers teach and students learn: successful

strategies for school”, OECD Education Working Paper, no. 130.

Statlink: http://dx.doi.org/10.1787/888933414786

HOW CAN A VARIETY OF TEACHING STRATEGIES BENEFIT STUDENT

ACHIEVEMENT?

When looking at students’ mean mathematics scores on the PISA assessment

alongside their exposure to the teaching strategies discussed in this chapter,

another reason for using a variety of teaching strategies emerges. Let’s look first

at the most commonly used teaching practices in mathematics, teacher-directed

strategies. The data indicate that when teachers direct student learning, students

are slightly more likely to be successful in solving the easiest mathematics

problems in PISA. Yet as the problems become more difficult, students with more

exposure to direct instruction no longer have a better chance of success. Figure

1.4 shows the relationship between the use of teacher-directed strategies and

students’ success on mathematics problems of varying difficulty.

Figure 1.4 Teacher-directed instruction and item difficulty

Odds ratio, after accounting for other teaching strategies, OECD average

Receiving teacher-directed instruction is associated with an increase

in the probability of success in solving a mathematics problem

Easy problem

Difficult

problem

Odds ratio

R2 = 0.24

Receiving teacher-directed instruction is associated with a decrease

in the probability of success in solving a mathematics problem

300 400 500 600 700 800

Difficulty of mathematics items on the PISA scale

Notes: Statistically significant odds ratios are marked in a darker tone.

Chile and Mexico are not included in the OECD average.

Source: OECD, PISA 2012 Database, adapted from Echazarra, A. et al. (2016), “How teachers teach and students learn: successful

strategies for school”, OECD Education Working Paper, no. 130.

Statlink: http://dx.doi.org/10.1787/888933414786

16.
HOW MUCH SHOULD I DIRECT STUDENT LEARNING IN MY MATHEMATICS CLASSES? . 15

Therefore, just as one teaching method is not sufficient for teaching a class of

TEACHING STRATEGIES

students with varying levels of ability, a single teaching strategy will not work for

all mathematics problems, either. Past research into the teaching of mathematics

supports this claim too, suggesting that teaching complex mathematics skills

might require different instructional strategies than those used to teach basic

mathematics skills.1 More recent research furthers this argument, saying that

more modern teaching methods, such as student-oriented teaching strategies,

encourage different cognitive skills in students.2

Some countries, such as Singapore, are taking this research to heart and are

designing mathematics curricula that require teachers to use a variety of

teaching strategies (Box 1.1). Yet rather than doing away with more traditional,

teacher-directed teaching methods altogether, these methods should be used in

tandem. In other words, teachers need a diverse set of tools to teach the breadth

of their mathematics curriculum and to help students advance from the most

rudimentary to the most complex mathematics problems.

Therefore, just as one teaching method is not sufficient for teaching a class of

TEACHING STRATEGIES

students with varying levels of ability, a single teaching strategy will not work for

all mathematics problems, either. Past research into the teaching of mathematics

supports this claim too, suggesting that teaching complex mathematics skills

might require different instructional strategies than those used to teach basic

mathematics skills.1 More recent research furthers this argument, saying that

more modern teaching methods, such as student-oriented teaching strategies,

encourage different cognitive skills in students.2

Some countries, such as Singapore, are taking this research to heart and are

designing mathematics curricula that require teachers to use a variety of

teaching strategies (Box 1.1). Yet rather than doing away with more traditional,

teacher-directed teaching methods altogether, these methods should be used in

tandem. In other words, teachers need a diverse set of tools to teach the breadth

of their mathematics curriculum and to help students advance from the most

rudimentary to the most complex mathematics problems.

17.
16 . TEN QUESTIONS FOR MATHEMATICS TEACHERS

Box 1.1 TEACHING AND LEARNING STRATEGIES FOR MATHEMATICS IN SINGAPORE

The objective of the mathematics curriculum in Singapore is to develop students’ ability to

apply mathematics to solve problems by developing their mathematical skills, helping them

acquire key mathematics concepts, fostering positive attitudes towards mathematics and

encouraging them to think about the way they learn. To accomplish this objective, teachers

use a variety of teaching strategies in their approach to mathematics. Teachers typically

provide a real-world context that demonstrates the importance of mathematical concepts

to students (thereby answering the all-too-common question: “Why do I have to learn

this?”). Teachers then explain the concepts, demonstrate problem-solving approaches, and

facilitate activities in class. They use various assessment practices to provide students with

individualised feedback on their learning.

Students are also exposed to a wide range of problems to solve during their study

of mathematics. In this way, students learn to apply mathematics to solve problems,

appreciate the value of mathematics, and develop important skills that will support their

future learning and their ability to deal with new problems.

Singapore Mathematics Curriculum Framework

Beliefs Monitoring of

Interest M one’s own thinking

et

Appreciation

des ac

og

Self-regulation of

Confidence titu ni learning

Perseverance At tio

n

MATHEMATICAL

Numerical calculation Reasoning,

PROBLEM SOLVING

Algebraic manipulation communication and

s

esse

Skil

Spatial visualisation connections

Data analysis Applications and

Proc

ls

Measurement modelling

Use of mathematical tools Thinking skills and

Estimation heuristics

Concepts

Numerical

Algebraic

Geometric

Statistical

Probabilistic

Analytical

Source: Ministry of Education, Singapore

Box 1.1 TEACHING AND LEARNING STRATEGIES FOR MATHEMATICS IN SINGAPORE

The objective of the mathematics curriculum in Singapore is to develop students’ ability to

apply mathematics to solve problems by developing their mathematical skills, helping them

acquire key mathematics concepts, fostering positive attitudes towards mathematics and

encouraging them to think about the way they learn. To accomplish this objective, teachers

use a variety of teaching strategies in their approach to mathematics. Teachers typically

provide a real-world context that demonstrates the importance of mathematical concepts

to students (thereby answering the all-too-common question: “Why do I have to learn

this?”). Teachers then explain the concepts, demonstrate problem-solving approaches, and

facilitate activities in class. They use various assessment practices to provide students with

individualised feedback on their learning.

Students are also exposed to a wide range of problems to solve during their study

of mathematics. In this way, students learn to apply mathematics to solve problems,

appreciate the value of mathematics, and develop important skills that will support their

future learning and their ability to deal with new problems.

Singapore Mathematics Curriculum Framework

Beliefs Monitoring of

Interest M one’s own thinking

et

Appreciation

des ac

og

Self-regulation of

Confidence titu ni learning

Perseverance At tio

n

MATHEMATICAL

Numerical calculation Reasoning,

PROBLEM SOLVING

Algebraic manipulation communication and

s

esse

Skil

Spatial visualisation connections

Data analysis Applications and

Proc

ls

Measurement modelling

Use of mathematical tools Thinking skills and

Estimation heuristics

Concepts

Numerical

Algebraic

Geometric

Statistical

Probabilistic

Analytical

Source: Ministry of Education, Singapore

18.
HOW MUCH SHOULD I DIRECT STUDENT LEARNING IN MY MATHEMATICS CLASSES? . 17

TEACHING STRATEGIES

WHAT CAN TEACHERS DO?

Plan mathematics lessons that strive to reach all levels of learners in a class.

The benefits of differentiating instructions for students of different abilities are widely

acclaimed across the research literature of all subject areas. Teachers should take this into

consideration when planning mathematics lessons. Make sure each lesson or unit contains

extension activities that are available for those students who finish their work quickly or

are ready to move on to more challenging subjects. Think about planning time during each

week for you – or your more advanced students – to offer support to those learners who

might be struggling. Propose research or project-based problems that provide a variety of

activities and roles for students with different abilities and interests.

Provide a mix of teacher-directed and student-oriented teaching strategies.

In mathematics especially, it is easy for teachers to rely on a textbook in their lessons, using

it as a guide to explaining concepts to students and then assigning the exercises supplied

by the publisher as students homework. This kind of lesson only provides teacher-directed

instruction to students, and doesn’t allow for much student input into their own learning.

(It also doesn’t account for differences in students’ abilities and motivation.) Try to move

beyond the textbook-provided lectures and homework and add new activities to lessons

that allow students to work together or use new tools, such as technology and games, to

cement their understanding of mathematical concepts.

Let the difficulty of the mathematics problem guide the teaching strategy.

When you are thinking about which strategies to use to reach different students in your

class, spend a moment thinking about the strategies that work best for problems of different

levels of difficulty. You may want to reserve your teacher-directed lessons for simpler

mathematical concepts, and research other strategies for teaching more difficult concepts.

1. Schoenfeld, A.H. (1992), “Learning to think mathematically: Problem solving, metacognition, and

sense-making in mathematics”, in D. Grouws, (ed.) Handbook for Research on Mathematics Teaching

and Learning, MacMillan, New York, pp. 334-370.

Schoenfeld, A.H. (ed.) (1987), Cognitive Science and Mathematics Education, Erlbaum, Hillsdale,

New Jersey.

2. Bietenbeck, J. (2014), “Teaching practices and cognitive skills”, Labour Economics, Vol. 30, pp. 143-153.

TEACHING STRATEGIES

WHAT CAN TEACHERS DO?

Plan mathematics lessons that strive to reach all levels of learners in a class.

The benefits of differentiating instructions for students of different abilities are widely

acclaimed across the research literature of all subject areas. Teachers should take this into

consideration when planning mathematics lessons. Make sure each lesson or unit contains

extension activities that are available for those students who finish their work quickly or

are ready to move on to more challenging subjects. Think about planning time during each

week for you – or your more advanced students – to offer support to those learners who

might be struggling. Propose research or project-based problems that provide a variety of

activities and roles for students with different abilities and interests.

Provide a mix of teacher-directed and student-oriented teaching strategies.

In mathematics especially, it is easy for teachers to rely on a textbook in their lessons, using

it as a guide to explaining concepts to students and then assigning the exercises supplied

by the publisher as students homework. This kind of lesson only provides teacher-directed

instruction to students, and doesn’t allow for much student input into their own learning.

(It also doesn’t account for differences in students’ abilities and motivation.) Try to move

beyond the textbook-provided lectures and homework and add new activities to lessons

that allow students to work together or use new tools, such as technology and games, to

cement their understanding of mathematical concepts.

Let the difficulty of the mathematics problem guide the teaching strategy.

When you are thinking about which strategies to use to reach different students in your

class, spend a moment thinking about the strategies that work best for problems of different

levels of difficulty. You may want to reserve your teacher-directed lessons for simpler

mathematical concepts, and research other strategies for teaching more difficult concepts.

1. Schoenfeld, A.H. (1992), “Learning to think mathematically: Problem solving, metacognition, and

sense-making in mathematics”, in D. Grouws, (ed.) Handbook for Research on Mathematics Teaching

and Learning, MacMillan, New York, pp. 334-370.

Schoenfeld, A.H. (ed.) (1987), Cognitive Science and Mathematics Education, Erlbaum, Hillsdale,

New Jersey.

2. Bietenbeck, J. (2014), “Teaching practices and cognitive skills”, Labour Economics, Vol. 30, pp. 143-153.

19.
Cognitive activation

Are some

mathematics

teaching methods

more effective than

others?

Are some

mathematics

teaching methods

more effective than

others?

20.
ARE SOME MATHEMATICS TEACHING METHODS MORE EFFECTIVE THAT OTHERS? . 19

It’s so easy, as a teacher, to forget how important it is to give students

COGNITIVE ACTIVATION

– and ourselves – the time to think and reflect. With the pressures

of exams, student progress, curriculum coverage and teacher

evaluations constantly looming, it is often easier to just keep moving

through the curriculum, day by day and problem set by problem set.

Teachers may have become accustomed to teaching a certain way

throughout their careers without taking a step back and reflecting

on whether the teaching methods they are using are really the best

for student learning. It’s time for all of us to stop and think.

As the previous chapter discusses, using a variety of teaching strategies is

particularly important when teaching mathematics to students with different

abilities, motivation and interests. But student data indicate that, on average

across PISA-participating countries, the use of cognitive-activation strategies

has the greatest positive association with students’ mean mathematics scores.1

These types of teaching strategies give students a chance to think deeply about

problems, discuss methods and mistakes with others, and reflect on their own

learning. Teachers should understand the importance of this kind of teaching and

should have a strong grasp of how to use these strategies in order to give learners

the best chance of success in mathematics.

WHAT IS COGNITIVE ACTIVATION IN MATHEMATICS TEACHING?

Cognitive activation is, in essence, about teaching pupils strategies, such as

summarising, questioning and predicting, which they can call upon when solving

mathematics problems. Such strategies encourage pupils to think more deeply in

order to find solutions and to focus on the method they use to reach the answer

rather than simply focusing on the answer itself. Some of these strategies will

require pupils to link new information to information they have already learned,

apply their skills to a new context, solve challenging mathematics problems

that require extended thought and that could have either multiple solutions

or an answer that is not immediately obvious. Making connections between

mathematical facts, procedures and ideas will result in enhanced learning and a

deeper understanding of the concepts.2

It’s so easy, as a teacher, to forget how important it is to give students

COGNITIVE ACTIVATION

– and ourselves – the time to think and reflect. With the pressures

of exams, student progress, curriculum coverage and teacher

evaluations constantly looming, it is often easier to just keep moving

through the curriculum, day by day and problem set by problem set.

Teachers may have become accustomed to teaching a certain way

throughout their careers without taking a step back and reflecting

on whether the teaching methods they are using are really the best

for student learning. It’s time for all of us to stop and think.

As the previous chapter discusses, using a variety of teaching strategies is

particularly important when teaching mathematics to students with different

abilities, motivation and interests. But student data indicate that, on average

across PISA-participating countries, the use of cognitive-activation strategies

has the greatest positive association with students’ mean mathematics scores.1

These types of teaching strategies give students a chance to think deeply about

problems, discuss methods and mistakes with others, and reflect on their own

learning. Teachers should understand the importance of this kind of teaching and

should have a strong grasp of how to use these strategies in order to give learners

the best chance of success in mathematics.

WHAT IS COGNITIVE ACTIVATION IN MATHEMATICS TEACHING?

Cognitive activation is, in essence, about teaching pupils strategies, such as

summarising, questioning and predicting, which they can call upon when solving

mathematics problems. Such strategies encourage pupils to think more deeply in

order to find solutions and to focus on the method they use to reach the answer

rather than simply focusing on the answer itself. Some of these strategies will

require pupils to link new information to information they have already learned,

apply their skills to a new context, solve challenging mathematics problems

that require extended thought and that could have either multiple solutions

or an answer that is not immediately obvious. Making connections between

mathematical facts, procedures and ideas will result in enhanced learning and a

deeper understanding of the concepts.2

21.
20 . TEN QUESTIONS FOR MATHEMATICS TEACHERS

HOW WIDELY USED ARE COGNITIVE-ACTIVATION STRATEGIES?

The good news is that, across countries, cognitive-activation strategies are

frequently used in mathematics teaching (Figure 2.1). Data indicate that the most

frequently used practice in this category is asking students to explain how they

solved a problem. Over 70% of students around the world reported that their

teachers ask them to do this in most lessons or in every lesson.

Figure 2.1 Cognitive-activation instruction

Percentage of students who reported their teachers use cognitive-activation

strategies “in every lesson” or “most lessons”, OECD average

The teacher asks us to decide on our own procedures for

solving complex problems

The teacher presents problems for which there is no

immediately obvious method of solution

The teacher gives problems that require us to think for an

extended time

The teacher presents problems in different contexts so that

we know whether we have understood the concepts

The teacher asks questions that make us reflect on

the problem

The teacher gives problems that can be solved in several

different ways

The teacher helps us to learn from mistakes we have made

The teacher presents problems that require us to apply what

we have learned to new contexts

The teacher asks us to explain how we have solved a problem

0 10 20 30 40 50 60 70 80 %

Source: OECD, PISA 2012 Database, adapted from Echazarra, A. et al. (2016), “How teachers teach and students learn: Successful

strategies for school”, OECD Education Working Paper, no. 130.

Statlink: http://dx.doi.org/10.1787/888933414798

In addition, more than 50% of students across the surveyed countries also

reported that their teachers use other cognitive-activation strategies, such as

those that require students to apply or recognise concepts they have learned in

different contexts, reflect on how to solve a problem – possibly for an extended

time – or learn from their own mistakes.

HOW WIDELY USED ARE COGNITIVE-ACTIVATION STRATEGIES?

The good news is that, across countries, cognitive-activation strategies are

frequently used in mathematics teaching (Figure 2.1). Data indicate that the most

frequently used practice in this category is asking students to explain how they

solved a problem. Over 70% of students around the world reported that their

teachers ask them to do this in most lessons or in every lesson.

Figure 2.1 Cognitive-activation instruction

Percentage of students who reported their teachers use cognitive-activation

strategies “in every lesson” or “most lessons”, OECD average

The teacher asks us to decide on our own procedures for

solving complex problems

The teacher presents problems for which there is no

immediately obvious method of solution

The teacher gives problems that require us to think for an

extended time

The teacher presents problems in different contexts so that

we know whether we have understood the concepts

The teacher asks questions that make us reflect on

the problem

The teacher gives problems that can be solved in several

different ways

The teacher helps us to learn from mistakes we have made

The teacher presents problems that require us to apply what

we have learned to new contexts

The teacher asks us to explain how we have solved a problem

0 10 20 30 40 50 60 70 80 %

Source: OECD, PISA 2012 Database, adapted from Echazarra, A. et al. (2016), “How teachers teach and students learn: Successful

strategies for school”, OECD Education Working Paper, no. 130.

Statlink: http://dx.doi.org/10.1787/888933414798

In addition, more than 50% of students across the surveyed countries also

reported that their teachers use other cognitive-activation strategies, such as

those that require students to apply or recognise concepts they have learned in

different contexts, reflect on how to solve a problem – possibly for an extended

time – or learn from their own mistakes.

22.
ARE SOME MATHEMATICS TEACHING METHODS MORE EFFECTIVE THAT OTHERS? . 21

HOW CAN THE USE OF COGNITIVE-ACTIVATION STRATEGIES BENEFIT

COGNITIVE ACTIVATION

STUDENT ACHIEVEMENT?

PISA data indicate that across OECD countries, students who reported that their

teachers use cognitive-activation strategies in their mathematics classes also

have higher mean mathematics scores. The strength of the relationship between

this type of teaching and student achievement even increases after the analyses

also take into account teachers’ use of other teaching strategies in the students’

mathematics classes. As Figure 2.2 shows, when students’ exposure to cognitive-

activation instruction increases, their performance improves.

The use of cognitive-activation teaching strategies makes a difference no matter

how difficult the mathematics problem. In fact, the odds of student success are

even greater for more challenging problems. Students who are more frequently

exposed to cognitive-activation teaching methods are about 10% more likely to

answer easier items correctly and about 50% more likely to answer more difficult

items correctly.

IN WHAT ENVIRONMENT DOES COGNITIVE ACTIVATION FLOURISH?

Studies in education as well as data collected from PISA give us a picture of the

kinds of schools and classrooms in which cognitive activation thrives. Students

in academically-oriented schools (as opposed to vocational schools) reported

more exposure to cognitive-activation strategies. Socio-economically advantaged

students reported more exposure to these strategies than disadvantaged students;

and when cognitive-activation strategies are used, the association with student

performance is stronger in advantaged schools than in disadvantaged schools

(Figure 2.3).

If these strategies are so beneficial, why isn’t every teacher using them more

frequently? PISA data suggest that certain school and student characteristics

might be more conducive to using cognitive-activation strategies. These types of

teaching strategies emphasise thinking and reasoning for extended periods of

time, which may take time away from covering the fundamentals of mathematics.

Thus, using cognitive-activation strategies might be easier in schools or classes

in which students don’t spend as much time focusing on basic concepts. It might

also be difficult for a teacher to use cognitive-activation strategies in a class

that is frequently disrupted by disorderly student behaviour (see here for more

information on how classroom climate can affect the teaching of mathematics).

HOW CAN THE USE OF COGNITIVE-ACTIVATION STRATEGIES BENEFIT

COGNITIVE ACTIVATION

STUDENT ACHIEVEMENT?

PISA data indicate that across OECD countries, students who reported that their

teachers use cognitive-activation strategies in their mathematics classes also

have higher mean mathematics scores. The strength of the relationship between

this type of teaching and student achievement even increases after the analyses

also take into account teachers’ use of other teaching strategies in the students’

mathematics classes. As Figure 2.2 shows, when students’ exposure to cognitive-

activation instruction increases, their performance improves.

The use of cognitive-activation teaching strategies makes a difference no matter

how difficult the mathematics problem. In fact, the odds of student success are

even greater for more challenging problems. Students who are more frequently

exposed to cognitive-activation teaching methods are about 10% more likely to

answer easier items correctly and about 50% more likely to answer more difficult

items correctly.

IN WHAT ENVIRONMENT DOES COGNITIVE ACTIVATION FLOURISH?

Studies in education as well as data collected from PISA give us a picture of the

kinds of schools and classrooms in which cognitive activation thrives. Students

in academically-oriented schools (as opposed to vocational schools) reported

more exposure to cognitive-activation strategies. Socio-economically advantaged

students reported more exposure to these strategies than disadvantaged students;

and when cognitive-activation strategies are used, the association with student

performance is stronger in advantaged schools than in disadvantaged schools

(Figure 2.3).

If these strategies are so beneficial, why isn’t every teacher using them more

frequently? PISA data suggest that certain school and student characteristics

might be more conducive to using cognitive-activation strategies. These types of

teaching strategies emphasise thinking and reasoning for extended periods of

time, which may take time away from covering the fundamentals of mathematics.

Thus, using cognitive-activation strategies might be easier in schools or classes

in which students don’t spend as much time focusing on basic concepts. It might

also be difficult for a teacher to use cognitive-activation strategies in a class

that is frequently disrupted by disorderly student behaviour (see here for more

information on how classroom climate can affect the teaching of mathematics).

23.
22 . TEN QUESTIONS FOR MATHEMATICS TEACHERS

Figure 2.2 Mathematics performance and cognitive-activation instruction

Score-point difference in mathematics associated with more frequent use of cognitive-

activation instruction

Albania

Romania Higher score

Iceland when teacher

Kazakhstan uses cognitive- Before accounting

Argentina activation for other teaching

Jordan instruction more strategies

Thailand frequently

United States After accounting

Mexico

Peru for other teaching

Czech Republic strategies

Macao-China

United Arab Emirates

Qatar

Finland

Canada

Brazil

Bulgaria

Turkey

Tunisia

Portugal

Uruguay

Montenegro

Serbia

Indonesia

Netherlands

Spain

Greece

Colombia Notes: Statistically significant values

Singapore before accounting for other teaching

Australia strategies are marked in a darker tone. All

Costa Rica

Estonia values after accounting for other teaching

Slovak Republic strategies are statistically significant.

Ireland Other teaching strategies refer to the PISA

Norway indices of teacher-directed, student-

Russian Federation

OECD average oriented and formative-assessment

New Zealand instruction.

Lithuania The index of cognitive-activation instruction

Croatia measures the extent to which students

Luxembourg

Hong Kong-China reported that teachers encourage

France them to acquire deep knowledge

Sweden through instructional practices such as

Hungary giving students problems that require

Chile them to think for an extended time,

United Kingdom presenting problems for which there is

Korea no immediately obvious way of arriving

Austria

Malaysia at a solution, and helping students

Japan to learn from the mistakes they have

Germany made.

Latvia Countries and economies are ranked

Denmark

in ascending order of the score-point

Switzerland

Chinese Taipei difference in mathematics performance,

Poland after accounting for other teaching

Belgium Lower score strategies.

Slovenia when teacher

Source: OECD, PISA 2012 Database,

Israel uses cognitive-

Viet Nam adapted from Echazarra, A. et al. (2016),

activation

Italy instruction more “How teachers teach and students learn:

Shanghai-China frequently Successful strategies for school”, OECD

Liechtenstein Education Working Paper, no. 130.

-20 -10 0 10 20 30 40 Statlink: http://dx.doi.org/

Score-point difference 10.1787/888933414800

Figure 2.2 Mathematics performance and cognitive-activation instruction

Score-point difference in mathematics associated with more frequent use of cognitive-

activation instruction

Albania

Romania Higher score

Iceland when teacher

Kazakhstan uses cognitive- Before accounting

Argentina activation for other teaching

Jordan instruction more strategies

Thailand frequently

United States After accounting

Mexico

Peru for other teaching

Czech Republic strategies

Macao-China

United Arab Emirates

Qatar

Finland

Canada

Brazil

Bulgaria

Turkey

Tunisia

Portugal

Uruguay

Montenegro

Serbia

Indonesia

Netherlands

Spain

Greece

Colombia Notes: Statistically significant values

Singapore before accounting for other teaching

Australia strategies are marked in a darker tone. All

Costa Rica

Estonia values after accounting for other teaching

Slovak Republic strategies are statistically significant.

Ireland Other teaching strategies refer to the PISA

Norway indices of teacher-directed, student-

Russian Federation

OECD average oriented and formative-assessment

New Zealand instruction.

Lithuania The index of cognitive-activation instruction

Croatia measures the extent to which students

Luxembourg

Hong Kong-China reported that teachers encourage

France them to acquire deep knowledge

Sweden through instructional practices such as

Hungary giving students problems that require

Chile them to think for an extended time,

United Kingdom presenting problems for which there is

Korea no immediately obvious way of arriving

Austria

Malaysia at a solution, and helping students

Japan to learn from the mistakes they have

Germany made.

Latvia Countries and economies are ranked

Denmark

in ascending order of the score-point

Switzerland

Chinese Taipei difference in mathematics performance,

Poland after accounting for other teaching

Belgium Lower score strategies.

Slovenia when teacher

Source: OECD, PISA 2012 Database,

Israel uses cognitive-

Viet Nam adapted from Echazarra, A. et al. (2016),

activation

Italy instruction more “How teachers teach and students learn:

Shanghai-China frequently Successful strategies for school”, OECD

Liechtenstein Education Working Paper, no. 130.

-20 -10 0 10 20 30 40 Statlink: http://dx.doi.org/

Score-point difference 10.1787/888933414800

24.
ARE SOME MATHEMATICS TEACHING METHODS MORE EFFECTIVE THAT OTHERS? . 23

The OECD teacher survey, TALIS, also suggests that teachers’ own collaboration

COGNITIVE ACTIVATION

with colleagues makes a difference in the teaching strategies they use and can

even influence student performance (Box 2.1).

Figure 2.3 Cognitive-activation strategies and students’ performance in mathematics,

by schools’ socio-economic profile

Score-point difference in mathematics associated with the use of each cognitive-activation

strategy, OECD average

Disadvantaged schools Advantaged schools

20

Higher score when teacher uses cognitive-

activation instruction more frequently

15

10

5

0

Lower score when teacher uses cognitive-activation

instruction more frequently

-5

ak rn

tim ire

ur e

le ts

w be

ex s

bl w

lu ith

ex t

nt ha

nt m

ed cid

ob n

ro ho

ist a

ed qu

so s w

es

e

es

m

s

ts

em

n

ts

co ble

pr de

nt n

co w

m s le

ay

oc e

tio

re ca

a p ain

nd re

pr s d

w ply

e tu

ia lem

nt ro

m nt

ffe at

te at

th s s

l

re p

n nt

ed p

ne ap

fro ude

di th

ex th

ed b

lv ex

s

on ke

w de

ffe nt

m pro

te

to o

in ms

n s

ct ma

t

so to

ro u

d st

di e

ra m

ss

ei st

s

ed le

im s

fo ble

ne nt

in pre

ey ts

lp

e

lv b

th lets

th en

no giv

ar e

he

so pro

ng pro

le tud

ud

fle

s

ve s s

st

in es

ve

re

ha ask

th giv

ks

on

gi

as

ki

ey

th

Cognitive-activation strategies used in mathematics lessons

Notes: Statistically significant values for disadvantaged schools are marked in a darker tone. All values for advantaged schools are

statistically significant.

Disadvantaged (advantaged) schools are those schools whose mean PISA index of economic, social and cultural status is statistically lower

(higher) than the mean index across all schools in the country/economy.

Source: OECD, PISA 2012 Database, adapted from OECD (2016), Equations and Inequalities: Making Mathematics Accessible to All, OECD

Publishing, Paris.

Statlink: http://dx.doi.org/10.1787/888933377210

The OECD teacher survey, TALIS, also suggests that teachers’ own collaboration

COGNITIVE ACTIVATION

with colleagues makes a difference in the teaching strategies they use and can

even influence student performance (Box 2.1).

Figure 2.3 Cognitive-activation strategies and students’ performance in mathematics,

by schools’ socio-economic profile

Score-point difference in mathematics associated with the use of each cognitive-activation

strategy, OECD average

Disadvantaged schools Advantaged schools

20

Higher score when teacher uses cognitive-

activation instruction more frequently

15

10

5

0

Lower score when teacher uses cognitive-activation

instruction more frequently

-5

ak rn

tim ire

ur e

le ts

w be

ex s

bl w

lu ith

ex t

nt ha

nt m

ed cid

ob n

ro ho

ist a

ed qu

so s w

es

e

es

m

s

ts

em

n

ts

co ble

pr de

nt n

co w

m s le

ay

oc e

tio

re ca

a p ain

nd re

pr s d

w ply

e tu

ia lem

nt ro

m nt

ffe at

te at

th s s

l

re p

n nt

ed p

ne ap

fro ude

di th

ex th

ed b

lv ex

s

on ke

w de

ffe nt

m pro

te

to o

in ms

n s

ct ma

t

so to

ro u

d st

di e

ra m

ss

ei st

s

ed le

im s

fo ble

ne nt

in pre

ey ts

lp

e

lv b

th lets

th en

no giv

ar e

he

so pro

ng pro

le tud

ud

fle

s

ve s s

st

in es

ve

re

ha ask

th giv

ks

on

gi

as

ki

ey

th

Cognitive-activation strategies used in mathematics lessons

Notes: Statistically significant values for disadvantaged schools are marked in a darker tone. All values for advantaged schools are

statistically significant.

Disadvantaged (advantaged) schools are those schools whose mean PISA index of economic, social and cultural status is statistically lower

(higher) than the mean index across all schools in the country/economy.

Source: OECD, PISA 2012 Database, adapted from OECD (2016), Equations and Inequalities: Making Mathematics Accessible to All, OECD

Publishing, Paris.

Statlink: http://dx.doi.org/10.1787/888933377210

25.
24 . TEN QUESTIONS FOR MATHEMATICS TEACHERS

Box 2.1 THE RELATIONSHIP BETWEEN TEACHER CO-OPERATION AND USE OF

COGNITIVE-ACTIVATION STRATEGIES IN MATHEMATICS

Data from the TALIS 2013 teacher survey demonstrated that teachers who collaborate

with colleagues reap many benefits themselves, such as higher levels of job satisfaction

and confidence in their own abilities as teachers. The impact of teacher collaboration on

mathematics teaching practices was examined when TALIS 2013 data was combined with

data from the PISA 2012 assessment. The analyses indicated that the more a mathematics

teacher co-operates with colleagues from the same school, the more likely he or she is to

regularly use cognitive-activation practices in teaching mathematics. The figure below

shows the relationship between teachers’ reported collaboration with fellow teachers and

their use of cognitive-activation practices in their mathematics classes.

Figure 2.4: How teacher co-operation is related to the cognitive-activation

instruction

Teachers who co-operate less frequently Teachers who co-operate more frequently

More

Cognitive-activation instruction

Less

Portugal Mexico Romania Spain Latvia Australia Finland Singapore

Notes: All differences are statistically significant, except in Mexico and Romania.

Teachers who co-operate more/less are those with values above/below the country median.

The index of cognitive-activation instruction measures the extent to which teachers challenge their students, such as by

expecting them to “think about complex problems” or encouraging them “to solve problems in more than one way”.

The index of teacher co-operation measures the frequency with which teachers “observe other teachers’ classes and provide

feedback” or “teach jointly as a team in the same class”.

Countries are ranked in descending order of the extent to which teachers who co-operate more frequently use cognitive-activation

instruction.

Source: OECD, TALIS 2013 Database.

Statlink: http://dx.doi.org/10.1787/888933414810

Box 2.1 THE RELATIONSHIP BETWEEN TEACHER CO-OPERATION AND USE OF

COGNITIVE-ACTIVATION STRATEGIES IN MATHEMATICS

Data from the TALIS 2013 teacher survey demonstrated that teachers who collaborate

with colleagues reap many benefits themselves, such as higher levels of job satisfaction

and confidence in their own abilities as teachers. The impact of teacher collaboration on

mathematics teaching practices was examined when TALIS 2013 data was combined with

data from the PISA 2012 assessment. The analyses indicated that the more a mathematics

teacher co-operates with colleagues from the same school, the more likely he or she is to

regularly use cognitive-activation practices in teaching mathematics. The figure below

shows the relationship between teachers’ reported collaboration with fellow teachers and

their use of cognitive-activation practices in their mathematics classes.

Figure 2.4: How teacher co-operation is related to the cognitive-activation

instruction

Teachers who co-operate less frequently Teachers who co-operate more frequently

More

Cognitive-activation instruction

Less

Portugal Mexico Romania Spain Latvia Australia Finland Singapore

Notes: All differences are statistically significant, except in Mexico and Romania.

Teachers who co-operate more/less are those with values above/below the country median.

The index of cognitive-activation instruction measures the extent to which teachers challenge their students, such as by

expecting them to “think about complex problems” or encouraging them “to solve problems in more than one way”.

The index of teacher co-operation measures the frequency with which teachers “observe other teachers’ classes and provide

feedback” or “teach jointly as a team in the same class”.

Countries are ranked in descending order of the extent to which teachers who co-operate more frequently use cognitive-activation

instruction.

Source: OECD, TALIS 2013 Database.

Statlink: http://dx.doi.org/10.1787/888933414810

26.
ARE SOME MATHEMATICS TEACHING METHODS MORE EFFECTIVE THAT OTHERS? . 25

COGNITIVE ACTIVATION

WHAT CAN TEACHERS DO?

Use cognitive-activation strategies. Data indicate that the use of these strategies is

related to improved student achievement for problems of all levels of difficulty, and that

these strategies are especially effective as problems become more challenging. This makes

sense: students should be able to learn from their mistakes, work together, and reflect on

problems that are both simple and more advanced.

Find ways to use cognitive-activation strategies in all of your classes. Challenging

students might be easier in quiet classrooms with more advanced students, but you can

also see it the other way round: challenging and “activating” your students may be the

most effective way of creating a positive learning environment in your classroom. There

are also ways to encourage students to be creative and critical in seemingly disorganised

environments. Genuine creative and critical thinking often blooms in less-structured

settings, for instance when students are asked to work in small groups, debate with their

peers or design their own experiments.

Look at what the research says about how students best learn mathematics. Many

teachers will have studied how students learn mathematics during their initial teacher

education, but that may have been years ago. Teachers may have developed other teaching

habits tailored to the curriculum or to the culture of the school, some of which could

be enriched by incorporating the findings of new research. It is worth refreshing your

knowledge of the research in teaching and learning of mathematics to make sure your

beliefs are aligned with your teaching practices.

Collaborate with other teachers. Collaborating with your colleagues, both inside and

outside of school, can help you acquire new learning tools and gain confidence in using

them. Your students will benefit as a result.

1. Echazarra, A., et al. (2016), “How teachers teach and students learn: Successful strategies for school”,

OECD Education Working Papers, No. 130, OECD Publishing, Paris.

2. Burge, B., J. Lenkeit and J. Sizmur (2015), PISA in practice - Cognitive activation in maths: How to use it

in the classroom, National Foundation for Educational Research in England and Wales (NFER), Slough.

COGNITIVE ACTIVATION

WHAT CAN TEACHERS DO?

Use cognitive-activation strategies. Data indicate that the use of these strategies is

related to improved student achievement for problems of all levels of difficulty, and that

these strategies are especially effective as problems become more challenging. This makes

sense: students should be able to learn from their mistakes, work together, and reflect on

problems that are both simple and more advanced.

Find ways to use cognitive-activation strategies in all of your classes. Challenging

students might be easier in quiet classrooms with more advanced students, but you can

also see it the other way round: challenging and “activating” your students may be the

most effective way of creating a positive learning environment in your classroom. There

are also ways to encourage students to be creative and critical in seemingly disorganised

environments. Genuine creative and critical thinking often blooms in less-structured

settings, for instance when students are asked to work in small groups, debate with their

peers or design their own experiments.

Look at what the research says about how students best learn mathematics. Many

teachers will have studied how students learn mathematics during their initial teacher

education, but that may have been years ago. Teachers may have developed other teaching

habits tailored to the curriculum or to the culture of the school, some of which could

be enriched by incorporating the findings of new research. It is worth refreshing your

knowledge of the research in teaching and learning of mathematics to make sure your

beliefs are aligned with your teaching practices.

Collaborate with other teachers. Collaborating with your colleagues, both inside and

outside of school, can help you acquire new learning tools and gain confidence in using

them. Your students will benefit as a result.

1. Echazarra, A., et al. (2016), “How teachers teach and students learn: Successful strategies for school”,

OECD Education Working Papers, No. 130, OECD Publishing, Paris.

2. Burge, B., J. Lenkeit and J. Sizmur (2015), PISA in practice - Cognitive activation in maths: How to use it

in the classroom, National Foundation for Educational Research in England and Wales (NFER), Slough.

27.
Classroom climate

As a mathematics

teacher, how

important is the

relationship I have

with my students?

As a mathematics

teacher, how

important is the

relationship I have

with my students?

28.
AS A MATHEMATICS TEACHER, HOW IMPORTANT IS THE RELATIONSHIP I HAVE WITH MY STUDENTS? . 27

Every teacher has great teaching days. These are the days when

CLASSROOM CLIMATE

your lesson works, and the students are motivated to learn and are

engaged in class activities. Think back to your last great teaching

day: how was the learning environment in your classroom? Did you

continually have to discipline students because of their behaviour?

Were students late for class or causing other disruptions? Or were

learners staying on task, actively participating and treating you and

their peers with respect? This kind of positive classroom climate,

with minimal interference, gives teachers more time to spend on

teaching, and makes those great teaching days possible. Teachers

don’t have to spend time addressing disruptions, and the classroom

becomes an environment in which learning can take place. What’s

more, the quality of the learning environment is not only related to

how teachers are able to teach, but also how they feel about their

jobs and their own abilities as teachers.

WHAT IS A GOOD CLASSROOM ENVIRONMENT FOR MATHEMATICS TEACHING

AND LEARNING?

A positive classroom climate, good classroom management and strong

relationships between teachers and learners should be considered prerequisites

for high-quality teaching. In general, more teaching, and presumably learning,

occurs when there is a positive school environment, including support from

teachers and good classroom management. In addition, the disciplinary climate

of the classroom is related to what and how teachers are able to teach. For

example, it might be easier for teachers to use cognitive-activation strategies, such

as encouraging students to be reflective in their thinking, in classrooms where

students stay on task and disruptions are kept to a minimum.

PISA data suggest a link between the behaviour of students in a class and their

overall familiarity with mathematics in general. As Figure 3.1 indicates, in most

countries, a better disciplinary climate is related to greater familiarity with

mathematics, even after comparing students and schools with similar socio-

economic profiles.

Every teacher has great teaching days. These are the days when

CLASSROOM CLIMATE

your lesson works, and the students are motivated to learn and are

engaged in class activities. Think back to your last great teaching

day: how was the learning environment in your classroom? Did you

continually have to discipline students because of their behaviour?

Were students late for class or causing other disruptions? Or were

learners staying on task, actively participating and treating you and

their peers with respect? This kind of positive classroom climate,

with minimal interference, gives teachers more time to spend on

teaching, and makes those great teaching days possible. Teachers

don’t have to spend time addressing disruptions, and the classroom

becomes an environment in which learning can take place. What’s

more, the quality of the learning environment is not only related to

how teachers are able to teach, but also how they feel about their

jobs and their own abilities as teachers.

WHAT IS A GOOD CLASSROOM ENVIRONMENT FOR MATHEMATICS TEACHING

AND LEARNING?

A positive classroom climate, good classroom management and strong

relationships between teachers and learners should be considered prerequisites

for high-quality teaching. In general, more teaching, and presumably learning,

occurs when there is a positive school environment, including support from

teachers and good classroom management. In addition, the disciplinary climate

of the classroom is related to what and how teachers are able to teach. For

example, it might be easier for teachers to use cognitive-activation strategies, such

as encouraging students to be reflective in their thinking, in classrooms where

students stay on task and disruptions are kept to a minimum.

PISA data suggest a link between the behaviour of students in a class and their

overall familiarity with mathematics in general. As Figure 3.1 indicates, in most

countries, a better disciplinary climate is related to greater familiarity with

mathematics, even after comparing students and schools with similar socio-

economic profiles.

29.
28 . TEN QUESTIONS FOR MATHEMATICS TEACHERS

Figure 3.1 Disciplinary climate and familiarity with mathematics

Change in students’ familiarity with mathematics associated with a better

disciplinary climate in class

Less familiarity with Greater familiarity

mathematics with mathematics

Liechtenstein

Finland Greater familiarity with

Tunisia mathematics when

Indonesia students reported a better

Kazakhstan disciplinary climate

Chile

Poland

Iceland

Estonia

Mexico

Sweden

Hong Kong-China

Montenegro

United Kingdom

Denmark

Colombia

Macao-China

Latvia

Switzerland

Argentina

Russian Federation

New Zealand

Brazil

Thailand

Slovak Republic

Uruguay

Malaysia

Portugal

Luxembourg

Canada

Ireland

Peru

Austria

OECD average

Serbia

Australia Note: Statistically significant values are

Germany

marked in a darker tone.

Italy

Costa Rica The index of disciplinary climate is based

Viet Nam on students’ reports of the frequency with

Lithuania which interruptions occur in mathematics

Netherlands

class. Higher values on the index indicate

Czech Republic

a better disciplinary climate.

Albania

Greece The index of familiarity with mathematics

Japan is based on students’ responses to 13

Hungary items measuring students’ self-reported

Israel

familiarity with mathematics concepts,

France

Croatia such as exponential function, divisor and

Jordan quadratic function.

United Arab Emirates Countries and economies are ranked in

United States ascending order of the change in the index

Bulgaria

of familiarity with mathematics associated

Shanghai-China

Chinese Taipei with a one-unit increase in the index of

Romania disciplinary climate.

Turkey Source: OECD, PISA 2012 Database,

Slovenia

adapted from OECD (2016), Equations

Singapore

and Inequalities: Making Mathematics

Belgium

Qatar Accessible to All, OECD Publishing, Paris.

Spain Statlink: http://dx.doi.org/

Korea 10.1787/888933377232

Figure 3.1 Disciplinary climate and familiarity with mathematics

Change in students’ familiarity with mathematics associated with a better

disciplinary climate in class

Less familiarity with Greater familiarity

mathematics with mathematics

Liechtenstein

Finland Greater familiarity with

Tunisia mathematics when

Indonesia students reported a better

Kazakhstan disciplinary climate

Chile

Poland

Iceland

Estonia

Mexico

Sweden

Hong Kong-China

Montenegro

United Kingdom

Denmark

Colombia

Macao-China

Latvia

Switzerland

Argentina

Russian Federation

New Zealand

Brazil

Thailand

Slovak Republic

Uruguay

Malaysia

Portugal

Luxembourg

Canada

Ireland

Peru

Austria

OECD average

Serbia

Australia Note: Statistically significant values are

Germany

marked in a darker tone.

Italy

Costa Rica The index of disciplinary climate is based

Viet Nam on students’ reports of the frequency with

Lithuania which interruptions occur in mathematics

Netherlands

class. Higher values on the index indicate

Czech Republic

a better disciplinary climate.

Albania

Greece The index of familiarity with mathematics

Japan is based on students’ responses to 13

Hungary items measuring students’ self-reported

Israel

familiarity with mathematics concepts,

France

Croatia such as exponential function, divisor and

Jordan quadratic function.

United Arab Emirates Countries and economies are ranked in

United States ascending order of the change in the index

Bulgaria

of familiarity with mathematics associated

Shanghai-China

Chinese Taipei with a one-unit increase in the index of

Romania disciplinary climate.

Turkey Source: OECD, PISA 2012 Database,

Slovenia

adapted from OECD (2016), Equations

Singapore

and Inequalities: Making Mathematics

Belgium

Qatar Accessible to All, OECD Publishing, Paris.

Spain Statlink: http://dx.doi.org/

Korea 10.1787/888933377232

30.
AS A MATHEMATICS TEACHER, HOW IMPORTANT IS THE RELATIONSHIP I HAVE WITH MY STUDENTS? . 29

This finding is especially important as students’ familiarity with mathematics

CLASSROOM CLIMATE

and their access to mathematics content at school can affect not only their

performance in school but also their social and economic situation later in life.

PISA data show large variations within countries in students’ awareness of and

access to mathematical content in schools; some of these variations could stem

from the quality of the classroom learning environment.

HOW DOES THE LEARNING ENVIRONMENT IN MY CLASSROOM INFLUENCE MY

TEACHING AND MY STUDENTS’ LEARNING?

Whether students feel supported and listened to by their teachers is important

to their schooling experience for many reasons, both social and academic. In

mathematics, there appears to be a link between how a teacher teaches and the

relationships he or she has with students. According to PISA data, students say

that their teachers are more likely to use all teaching practices if there is a better

disciplinary climate (except for student-oriented strategies), a system of classroom

management in place, and students feel supported by their teachers and have

good relations with them.1 Other PISA findings also show that the disciplinary

climate in mathematics lessons and student performance go hand-in-hand.2

It’s not just students who benefit from improvements in classroom management

and more positive relationships between teachers and learners; teachers

themselves profit in many ways. TALIS 2013 asked teachers about both the climate

of their classroom and their relationships with their students. Their responses

revealed important connections between the quality of the learning environment

and teachers’ job satisfaction, as well as their confidence in their own abilities as

teachers. For example, as Figure 3.2 shows, on average across countries, teachers’

job satisfaction is lower when there are higher percentages of students in their

classes with behavioural problems. In many countries, having more students with

behavioural problems is also associated with teachers feeling less confident about

their own teaching abilities.

These results are perhaps understandable. Dealing with challenging classrooms

of students all day can be difficult and might make teachers feel more negative

towards their job, school or chosen career. Such demanding classes might also

cause a teacher to question his or her own abilities, especially in the area of

classroom discipline. But having strong, positive relationships with students

can help. TALIS data also indicate that the detrimental effects that challenging

This finding is especially important as students’ familiarity with mathematics

CLASSROOM CLIMATE

and their access to mathematics content at school can affect not only their

performance in school but also their social and economic situation later in life.

PISA data show large variations within countries in students’ awareness of and

access to mathematical content in schools; some of these variations could stem

from the quality of the classroom learning environment.

HOW DOES THE LEARNING ENVIRONMENT IN MY CLASSROOM INFLUENCE MY

TEACHING AND MY STUDENTS’ LEARNING?

Whether students feel supported and listened to by their teachers is important

to their schooling experience for many reasons, both social and academic. In

mathematics, there appears to be a link between how a teacher teaches and the

relationships he or she has with students. According to PISA data, students say

that their teachers are more likely to use all teaching practices if there is a better

disciplinary climate (except for student-oriented strategies), a system of classroom

management in place, and students feel supported by their teachers and have

good relations with them.1 Other PISA findings also show that the disciplinary

climate in mathematics lessons and student performance go hand-in-hand.2

It’s not just students who benefit from improvements in classroom management

and more positive relationships between teachers and learners; teachers

themselves profit in many ways. TALIS 2013 asked teachers about both the climate

of their classroom and their relationships with their students. Their responses

revealed important connections between the quality of the learning environment

and teachers’ job satisfaction, as well as their confidence in their own abilities as

teachers. For example, as Figure 3.2 shows, on average across countries, teachers’

job satisfaction is lower when there are higher percentages of students in their

classes with behavioural problems. In many countries, having more students with

behavioural problems is also associated with teachers feeling less confident about

their own teaching abilities.

These results are perhaps understandable. Dealing with challenging classrooms

of students all day can be difficult and might make teachers feel more negative

towards their job, school or chosen career. Such demanding classes might also

cause a teacher to question his or her own abilities, especially in the area of

classroom discipline. But having strong, positive relationships with students

can help. TALIS data also indicate that the detrimental effects that challenging

31.
30 . TEN QUESTIONS FOR MATHEMATICS TEACHERS

classrooms have on teachers’ job satisfaction are mitigated when teachers also

report having strong interpersonal relationships with their students.

Figure 3.2 Teachers’ job satisfaction and students with behavioural problems

Lower secondary teachers’ job satisfaction by the percentage of students with behavioural problems

More satisfied

Havin

g few

er stu

de

greate nts with

r job be

satisfa havioural

Teachers' job satisfaction

ction p

amonroblems is

g tea a

chers ssociated

with

None 1% to 10% 11% to 30% 31% or more

Less satisfied

Percentage of students in class with behavioural problems

Notes: Data on students with behavioural problems are reported by teachers and refer to a randomly chosen class they currently teach

in their weekly timetable.

To assess teachers’ job satisfaction, TALIS asked teachers to indicate how satisfied they feel about their job (on a four-point scale ranging

from “strongly disagree” to “strongly agree”) by responding to a number of statements about their work environment (“I would like to

change to another school if that were possible”; “I enjoy working at this school”; “I would recommend my school as a good place to

work”; and “All in all, I am satisfied with my job”) and the teaching profession (“The advantages of being a teacher clearly outweigh the

disadvantages”; “If I could decide again, I would still choose to work as a teacher”; “I regret that I decided to become a teacher”; and “I

wonder whether it would have been better to choose another profession”).

The analysis is based on the average of the countries participating in the TALIS survey.

Source: OECD, TALIS 2013 Database.

Statlink: http://dx.doi.org/10.1787/888933414826

classrooms have on teachers’ job satisfaction are mitigated when teachers also

report having strong interpersonal relationships with their students.

Figure 3.2 Teachers’ job satisfaction and students with behavioural problems

Lower secondary teachers’ job satisfaction by the percentage of students with behavioural problems

More satisfied

Havin

g few

er stu

de

greate nts with

r job be

satisfa havioural

Teachers' job satisfaction

ction p

amonroblems is

g tea a

chers ssociated

with

None 1% to 10% 11% to 30% 31% or more

Less satisfied

Percentage of students in class with behavioural problems

Notes: Data on students with behavioural problems are reported by teachers and refer to a randomly chosen class they currently teach

in their weekly timetable.

To assess teachers’ job satisfaction, TALIS asked teachers to indicate how satisfied they feel about their job (on a four-point scale ranging

from “strongly disagree” to “strongly agree”) by responding to a number of statements about their work environment (“I would like to

change to another school if that were possible”; “I enjoy working at this school”; “I would recommend my school as a good place to

work”; and “All in all, I am satisfied with my job”) and the teaching profession (“The advantages of being a teacher clearly outweigh the

disadvantages”; “If I could decide again, I would still choose to work as a teacher”; “I regret that I decided to become a teacher”; and “I

wonder whether it would have been better to choose another profession”).

The analysis is based on the average of the countries participating in the TALIS survey.

Source: OECD, TALIS 2013 Database.

Statlink: http://dx.doi.org/10.1787/888933414826

32.
AS A MATHEMATICS TEACHER, HOW IMPORTANT IS THE RELATIONSHIP I HAVE WITH MY STUDENTS? . 31

CLASSROOM CLIMATE

WHAT CAN TEACHERS DO?

Focus time and energy on creating a positive classroom climate. If classroom

management and discipline are of particular concern to you, find a way to get additional

support. Speak to or observe other teachers in your school to learn successful classroom-

management strategies. Ask your school leadership if you can look for ongoing professional

development on this issue.

Invest time in building strong relationships with your students. This is particularly

demanding for those teachers who see upwards of 150 students each day, but it could

make a difference to both your students’ learning and your teaching – not to mention your

own well-being as a teacher. Students want to feel that their teachers treat them fairly,

listen to them and will continue teaching them until they understand the material. In

addition, learning about students’ lives outside of school might help you to connect topics

in mathematics with real-world situations that are meaningful to your students.

1. Echazarra, A., et al. (2016), “How teachers teach and students learn: Successful strategies for school”,

OECD Education Working Papers, No. 130, OECD Publishing, Paris.

2. OECD (2016), Equations and Inequalities: Making Mathematics Accessible to All, PISA, OECD Publishing,

Paris, http://dx.doi.org/10.1787/9789264258495-en.

CLASSROOM CLIMATE

WHAT CAN TEACHERS DO?

Focus time and energy on creating a positive classroom climate. If classroom

management and discipline are of particular concern to you, find a way to get additional

support. Speak to or observe other teachers in your school to learn successful classroom-

management strategies. Ask your school leadership if you can look for ongoing professional

development on this issue.

Invest time in building strong relationships with your students. This is particularly

demanding for those teachers who see upwards of 150 students each day, but it could

make a difference to both your students’ learning and your teaching – not to mention your

own well-being as a teacher. Students want to feel that their teachers treat them fairly,

listen to them and will continue teaching them until they understand the material. In

addition, learning about students’ lives outside of school might help you to connect topics

in mathematics with real-world situations that are meaningful to your students.

1. Echazarra, A., et al. (2016), “How teachers teach and students learn: Successful strategies for school”,

OECD Education Working Papers, No. 130, OECD Publishing, Paris.

2. OECD (2016), Equations and Inequalities: Making Mathematics Accessible to All, PISA, OECD Publishing,

Paris, http://dx.doi.org/10.1787/9789264258495-en.

33.
What do we

know about

memorisation

and learning

mathematics?

know about

memorisation

and learning

mathematics?

34.
WHAT DO WE KNOW ABOUT MEMORISATION AND LEARNING MATHEMATICS? . 33

Every mathematics course involves some level of memorisation. The area

MEMORISATION

of a circle is pi times radius squared. The square of the hypotenuse is equal

to the sum of the square of the other two sides. As teachers, we encourage

our students to commit some elements, such as formulas, to memory

so that they might be effortlessly recalled to solve future mathematics

problems. PISA data suggest that the way teachers require students

to use their memory makes a difference. Are we asking students to

commit information to memory and repeatedly apply it to many similar

problems? Or do we expect our students to memorise, understand and

apply the concepts they have learned to problems in different contexts?

Data indicate that students who rely on memorisation alone may be

successful with the easiest mathematics problems, but may find that a

deeper understanding of mathematics concepts is necessary to tackle

more difficult or non-routine problems.

HOW PREVALENT IS MEMORISATION AS A LEARNING STRATEGY IN

Teachers and students alike are familiar with the technique of memorisation:

to learn something completely so that it can later be recalled or repeated. In

mathematics classes, teachers often encourage students to use their memories

through activities such as rehearsal, routine exercises and drills. To find out how

students around the world learn mathematics, PISA asked them which learning

strategy best described their own approach to the subject. Students were asked

whether they agreed with statements that corresponded to memorisation strategies.

PISA findings indicate that students around the world often use memorisation

to learn mathematics. On average in almost every country, when students were

asked about the learning strategies they use, they agreed with one of the four

possible memorisation-related statements (Figure 4.1). These statements are listed

in Box 4.1.

That most students use memorisation to a greater or lesser degree is not surprising,

given that memorisation does have some advantages as a learning strategy,

Every mathematics course involves some level of memorisation. The area

MEMORISATION

of a circle is pi times radius squared. The square of the hypotenuse is equal

to the sum of the square of the other two sides. As teachers, we encourage

our students to commit some elements, such as formulas, to memory

so that they might be effortlessly recalled to solve future mathematics

problems. PISA data suggest that the way teachers require students

to use their memory makes a difference. Are we asking students to

commit information to memory and repeatedly apply it to many similar

problems? Or do we expect our students to memorise, understand and

apply the concepts they have learned to problems in different contexts?

Data indicate that students who rely on memorisation alone may be

successful with the easiest mathematics problems, but may find that a

deeper understanding of mathematics concepts is necessary to tackle

more difficult or non-routine problems.

HOW PREVALENT IS MEMORISATION AS A LEARNING STRATEGY IN

Teachers and students alike are familiar with the technique of memorisation:

to learn something completely so that it can later be recalled or repeated. In

mathematics classes, teachers often encourage students to use their memories

through activities such as rehearsal, routine exercises and drills. To find out how

students around the world learn mathematics, PISA asked them which learning

strategy best described their own approach to the subject. Students were asked

whether they agreed with statements that corresponded to memorisation strategies.

PISA findings indicate that students around the world often use memorisation

to learn mathematics. On average in almost every country, when students were

asked about the learning strategies they use, they agreed with one of the four

possible memorisation-related statements (Figure 4.1). These statements are listed

in Box 4.1.

That most students use memorisation to a greater or lesser degree is not surprising,

given that memorisation does have some advantages as a learning strategy,

35.
34 . TEN QUESTIONS FOR MATHEMATICS TEACHERS

Box 4.1 MEASURING THE USE OF MEMORISATION STRATEGIES IN MATHEMATICS

LEARNING

To calculate how often students use memorisation strategies, they were asked which

statement best describes their approach to mathematics using four questions with three

mutually exclusive responses to each: one corresponding to a memorisation strategy,

one to an elaboration strategy (such as using analogies and examples, or looking for

alternative ways of finding solutions) and one to a control strategy (such as creating a

study plan or monitoring progress towards understanding). The index of memorisation,

with values ranging from 0 to 4, reflects the number of times a student chose the following

memorisation-related statements about how they learn mathematics:

a) When I study for a mathematics test, I learn as much as I can by heart.

b) When I study mathematics, I make myself check to see if I remember the work I have

already done.

c) When I study mathematics, I go over some problems so often that I feel as if I could solve

them in my sleep.

d) In order to remember the method for solving a mathematics problem, I go through

examples again and again.

Statement a) assesses how much students use rote learning, or learning without paying

attention to meaning. The remaining three statements come close to the ideas of drill,

practice and repetitive learning.

particularly when it is not just mechanical memorisation. Memorising can lay the

foundation for conceptual understanding by giving students concrete facts on

which to reflect. It can also lead to mathematics “automaticity”, speeding up basic

arithmetic computations and leaving more time for deeper mathematical reasoning.

WHO USES MEMORISATION THE MOST?

There are many reasons why students use particular learning strategies, or a

combination of them, when learning mathematics. Among students who mainly

use memorisation, drilling or repetitive learning, some may do so to avoid intense

mental effort, particularly if they are not naturally drawn to mathematics, are not

familiar with more advanced problems, or do not feel especially confident in their

own abilities in the subject. To some extent, PISA results support this hypothesis.

They indicate that, across OECD countries, persevering students, students with

positive attitudes, motivation or interest in problem solving and mathematics,

students who are more confident in their mathematics abilities, and students

who have little or no anxiety towards mathematics are somewhat less likely

to use memorisation strategies. Boys, too, are less likely than girls to use these

Box 4.1 MEASURING THE USE OF MEMORISATION STRATEGIES IN MATHEMATICS

LEARNING

To calculate how often students use memorisation strategies, they were asked which

statement best describes their approach to mathematics using four questions with three

mutually exclusive responses to each: one corresponding to a memorisation strategy,

one to an elaboration strategy (such as using analogies and examples, or looking for

alternative ways of finding solutions) and one to a control strategy (such as creating a

study plan or monitoring progress towards understanding). The index of memorisation,

with values ranging from 0 to 4, reflects the number of times a student chose the following

memorisation-related statements about how they learn mathematics:

a) When I study for a mathematics test, I learn as much as I can by heart.

b) When I study mathematics, I make myself check to see if I remember the work I have

already done.

c) When I study mathematics, I go over some problems so often that I feel as if I could solve

them in my sleep.

d) In order to remember the method for solving a mathematics problem, I go through

examples again and again.

Statement a) assesses how much students use rote learning, or learning without paying

attention to meaning. The remaining three statements come close to the ideas of drill,

practice and repetitive learning.

particularly when it is not just mechanical memorisation. Memorising can lay the

foundation for conceptual understanding by giving students concrete facts on

which to reflect. It can also lead to mathematics “automaticity”, speeding up basic

arithmetic computations and leaving more time for deeper mathematical reasoning.

WHO USES MEMORISATION THE MOST?

There are many reasons why students use particular learning strategies, or a

combination of them, when learning mathematics. Among students who mainly

use memorisation, drilling or repetitive learning, some may do so to avoid intense

mental effort, particularly if they are not naturally drawn to mathematics, are not

familiar with more advanced problems, or do not feel especially confident in their

own abilities in the subject. To some extent, PISA results support this hypothesis.

They indicate that, across OECD countries, persevering students, students with

positive attitudes, motivation or interest in problem solving and mathematics,

students who are more confident in their mathematics abilities, and students

who have little or no anxiety towards mathematics are somewhat less likely

to use memorisation strategies. Boys, too, are less likely than girls to use these

36.
WHAT DO WE KNOW ABOUT MEMORISATION AND LEARNING MATHEMATICS? . 35

Figure 4.1 Students’ use of memorisation strategies

MEMORISATION

Based on students’ self-reports

Less Memorisation More

Uruguay 23

Ireland 28 Percentage of students who

United Kingdom 37 reported that they learn by heart

Netherlands 22

Spain 19

Indonesia 23 Above the OECD average

New Zealand 35

At the same level as the OECD average

Chile 22

Australia 35 Below the OECD average

United Arab Emirates 13

Thailand 46

Israel 14

Jordan 14

Belgium 24

Norway 28

Luxembourg 13

Hungary 17

United States 29

Finland 32

Portugal 27

Austria 13

Greece 20

Singapore 22

Canada 26

Brazil 30

Turkey 13

OECD average 21

Bulgaria 11

Estonia 14

Shanghai-China 25

Czech Republic 25

Sweden 31

Argentina 21

Costa Rica 19

Montenegro 13

France 19

Croatia 9

Peru 22

Romania 16

Tunisia 10

Slovenia 11

Korea 17

Qatar 13

Japan 12 Note: The index of memorisation

Germany 17 strategies is based on the four questions

Iceland 23 about learning strategies in the student

Colombia 26 questionnaire. In each question,

Latvia 22 students were asked to choose among

Italy 10

three mutually exclusive statements

Denmark 28

Hong Kong-China 10 corresponding to the following

Chinese Taipei 16 approaches to learning mathematics:

Kazakhstan 22 memorisation, elaboration and control.

Lithuania 14 Countries and economies are ranked

Viet Nam 5

in descending order of the index of

Liechtenstein 17

Malaysia 12 memorisation strategies.

Poland 9 Source: OECD, PISA 2012 Database,

Mexico 19 adapted from Echazarra, A. et al. (2016),

Switzerland 13

“How teachers teach and students learn:

Albania 12

Slovak Republic 11 Successful strategies for school”, OECD

Serbia 11 Education Working Paper, no. 130.

Russian Federation 16 Statlink: http://dx.doi.org/

Macao-China 15 10.1787/888933414832

Figure 4.1 Students’ use of memorisation strategies

MEMORISATION

Based on students’ self-reports

Less Memorisation More

Uruguay 23

Ireland 28 Percentage of students who

United Kingdom 37 reported that they learn by heart

Netherlands 22

Spain 19

Indonesia 23 Above the OECD average

New Zealand 35

At the same level as the OECD average

Chile 22

Australia 35 Below the OECD average

United Arab Emirates 13

Thailand 46

Israel 14

Jordan 14

Belgium 24

Norway 28

Luxembourg 13

Hungary 17

United States 29

Finland 32

Portugal 27

Austria 13

Greece 20

Singapore 22

Canada 26

Brazil 30

Turkey 13

OECD average 21

Bulgaria 11

Estonia 14

Shanghai-China 25

Czech Republic 25

Sweden 31

Argentina 21

Costa Rica 19

Montenegro 13

France 19

Croatia 9

Peru 22

Romania 16

Tunisia 10

Slovenia 11

Korea 17

Qatar 13

Japan 12 Note: The index of memorisation

Germany 17 strategies is based on the four questions

Iceland 23 about learning strategies in the student

Colombia 26 questionnaire. In each question,

Latvia 22 students were asked to choose among

Italy 10

three mutually exclusive statements

Denmark 28

Hong Kong-China 10 corresponding to the following

Chinese Taipei 16 approaches to learning mathematics:

Kazakhstan 22 memorisation, elaboration and control.

Lithuania 14 Countries and economies are ranked

Viet Nam 5

in descending order of the index of

Liechtenstein 17

Malaysia 12 memorisation strategies.

Poland 9 Source: OECD, PISA 2012 Database,

Mexico 19 adapted from Echazarra, A. et al. (2016),

Switzerland 13

“How teachers teach and students learn:

Albania 12

Slovak Republic 11 Successful strategies for school”, OECD

Serbia 11 Education Working Paper, no. 130.

Russian Federation 16 Statlink: http://dx.doi.org/

Macao-China 15 10.1787/888933414832

37.
36 . TEN QUESTIONS FOR MATHEMATICS TEACHERS

strategies; in fact, in no education system did boys report more intensive use of

memorisation when learning mathematics than girls (Figure 4.2).

When looking at students’ self-reported use of memorisation strategies across

countries, the data also show that many countries that are amongst the highest

performers in the PISA mathematics exam are not those where memorisation

strategies are the most dominant. For example, fewer students in East Asian countries

reported that they use memorisation as a learning strategy than did 15-year olds in

some of the English-speaking countries to whom they are often compared. These

findings may run against conventional wisdom, but mathematics instruction has

changed considerably in many East Asian countries, such as Japan. (Box 4.2).

Figure 4.2 Who’s using memorisation?

Correlation with the index of memorisation, OECD average

Less Memorisation More

Higher self-efficacy

in mathematics

More openness to

problem solving

Score higher in

mathematics

More interested in

mathematics

Student is a boy

Better self-concept Students with

in mathematics greater anxiety

towards

More instrumental motivation mathematics use

for learning mathematics memorisation

more frequently

More perseverance

Greater mathematics

anxiety

Note: All coefficient correlations are statistically significant.

Source: OECD, PISA 2012 Database.

Statlink: http://dx.doi.org/10.1787/888933414846

strategies; in fact, in no education system did boys report more intensive use of

memorisation when learning mathematics than girls (Figure 4.2).

When looking at students’ self-reported use of memorisation strategies across

countries, the data also show that many countries that are amongst the highest

performers in the PISA mathematics exam are not those where memorisation

strategies are the most dominant. For example, fewer students in East Asian countries

reported that they use memorisation as a learning strategy than did 15-year olds in

some of the English-speaking countries to whom they are often compared. These

findings may run against conventional wisdom, but mathematics instruction has

changed considerably in many East Asian countries, such as Japan. (Box 4.2).

Figure 4.2 Who’s using memorisation?

Correlation with the index of memorisation, OECD average

Less Memorisation More

Higher self-efficacy

in mathematics

More openness to

problem solving

Score higher in

mathematics

More interested in

mathematics

Student is a boy

Better self-concept Students with

in mathematics greater anxiety

towards

More instrumental motivation mathematics use

for learning mathematics memorisation

more frequently

More perseverance

Greater mathematics

anxiety

Note: All coefficient correlations are statistically significant.

Source: OECD, PISA 2012 Database.

Statlink: http://dx.doi.org/10.1787/888933414846

38.
WHAT DO WE KNOW ABOUT MEMORISATION AND LEARNING MATHEMATICS? . 37

MEMORISATION

Box 4.2 RECENT REFORMS IN MATHEMATICS TEACHING IN JAPAN

Mathematics teaching in Asian countries has historically been regarded as highly

traditional, particularly by many western observers. Whether accurate or not, the typical

image of Japanese education often includes highly competitive entrance exams, cram

schools and rote memorisation.

However, Japanese education has gradually evolved into a system that promotes the

acquisition of foundational knowledge and skills and encourages students to learn and

think independently, which is one of the ideas behind the “Zest for Living” reform. In

Japanese education today, academic and social skills refer to the acquisition of basic

and foundational knowledge and skills; the ability to think, make decisions and express

oneself to solve problems; and being motivated to learn.1 For example, the policy “Period

for Integrated Studies”, which asks teachers and schools to develop their own cross-

curricular study programmes, encourages students to participate in a range of activities,

including volunteer activities, study tours, experiments, investigations, and presentations or

discussions, with the aim of developing students’ ability to recognise problems, learn and

think independently and improve their problem-solving skills.

WILL MEMORISATION HELP OR HURT MY STUDENTS’ PERFORMANCE IN

Some experts in mathematics education consider memorisation to be an

elementary strategy that is better suited to solving routine problems that require

only a shallow understanding of mathematics concepts.2 PISA results reinforce

this view. They show that students who reported using memorisation strategies

are indeed successful on easier mathematics tasks. For example, one of the

easiest mathematics problems in the PISA 2012 assessment was a multiple-

choice question involving a simple bar chart. Some 87% of students across PISA-

participating education systems answered this question correctly. Students who

reported that they use memorisation strategies to learn mathematics had about

the same success rate on this easy item as students who reported using other

learning strategies.

Although memorisation seems to work for the easiest mathematics problems, its

success as a learning strategy does not extend much beyond that. According to the

data, as problems become more challenging, students who use memorisation are

less likely to be able to solve them correctly. Results are even worse for the most

challenging mathematics problems. Only 3% of students answered the most difficult

question on the 2012 PISA exam correctly. Solving this problem required multiple

MEMORISATION

Box 4.2 RECENT REFORMS IN MATHEMATICS TEACHING IN JAPAN

Mathematics teaching in Asian countries has historically been regarded as highly

traditional, particularly by many western observers. Whether accurate or not, the typical

image of Japanese education often includes highly competitive entrance exams, cram

schools and rote memorisation.

However, Japanese education has gradually evolved into a system that promotes the

acquisition of foundational knowledge and skills and encourages students to learn and

think independently, which is one of the ideas behind the “Zest for Living” reform. In

Japanese education today, academic and social skills refer to the acquisition of basic

and foundational knowledge and skills; the ability to think, make decisions and express

oneself to solve problems; and being motivated to learn.1 For example, the policy “Period

for Integrated Studies”, which asks teachers and schools to develop their own cross-

curricular study programmes, encourages students to participate in a range of activities,

including volunteer activities, study tours, experiments, investigations, and presentations or

discussions, with the aim of developing students’ ability to recognise problems, learn and

think independently and improve their problem-solving skills.

WILL MEMORISATION HELP OR HURT MY STUDENTS’ PERFORMANCE IN

Some experts in mathematics education consider memorisation to be an

elementary strategy that is better suited to solving routine problems that require

only a shallow understanding of mathematics concepts.2 PISA results reinforce

this view. They show that students who reported using memorisation strategies

are indeed successful on easier mathematics tasks. For example, one of the

easiest mathematics problems in the PISA 2012 assessment was a multiple-

choice question involving a simple bar chart. Some 87% of students across PISA-

participating education systems answered this question correctly. Students who

reported that they use memorisation strategies to learn mathematics had about

the same success rate on this easy item as students who reported using other

learning strategies.

Although memorisation seems to work for the easiest mathematics problems, its

success as a learning strategy does not extend much beyond that. According to the

data, as problems become more challenging, students who use memorisation are

less likely to be able to solve them correctly. Results are even worse for the most

challenging mathematics problems. Only 3% of students answered the most difficult

question on the 2012 PISA exam correctly. Solving this problem required multiple

39.
38 . TEN QUESTIONS FOR MATHEMATICS TEACHERS

steps and involved substantial geometric reasoning and creativity. An analysis of

PISA results shows that students who reported using memorisation the most when

they study mathematics – those who chose the memorisation-related statement

for all four questions – were four times less likely to solve this difficult problem

correctly than students who reported using memorisation the least (Figure 4.3).

Indeed, PISA results indicate that no matter the level of difficulty of a mathematics

problem, students who rely on memorisation alone are never more successful in

solving mathematics problems. This would suggest that, in general, teachers should

encourage students to go beyond rote memorisation and to think more deeply

about what they have learned and make connections with real-world problems.

But PISA results also show a difference in students’ performance based on the types

of memorisation activities used. Students who practice repetitive learning (drilling)

are more successful in solving difficult problems than those who simply learn

something by heart (rote memorisation). Repetitive learning can ease students’

Figure 4.3 Memorisation strategies and item difficulty

Odds ratio across 48 education systems

Greater success

Easy problem Using memorisation strategies is associated with an increase in the

probability of successfully solving a mathematics problem

Me

m

suc orisat

ces io

s as n is a

pro ssoc

ble i

ms ated w

bec

om ith les

em s

ore chan

diffi ce o

cul f

t

R ² = 0.81

Using memorisation strategies is associated with a decrease

in the probability of successfully solving a mathematics problem

Difficult problem

300 400 500 600 700 800

Less success Difficulty of mathematics items on the PISA scale

Notes: Statistically significant odds ratios are marked in a darker tone.

Chile and Mexico are not included in the OECD average.

Source: OECD, PISA 2012 Database, adapted from Echazarra, A. et al. (2016), “How teachers teach and students learn: Successful

strategies for school”, OECD Education Working Paper, no. 130.

Statlink: http://dx.doi.org/10.1787/888933414854

steps and involved substantial geometric reasoning and creativity. An analysis of

PISA results shows that students who reported using memorisation the most when

they study mathematics – those who chose the memorisation-related statement

for all four questions – were four times less likely to solve this difficult problem

correctly than students who reported using memorisation the least (Figure 4.3).

Indeed, PISA results indicate that no matter the level of difficulty of a mathematics

problem, students who rely on memorisation alone are never more successful in

solving mathematics problems. This would suggest that, in general, teachers should

encourage students to go beyond rote memorisation and to think more deeply

about what they have learned and make connections with real-world problems.

But PISA results also show a difference in students’ performance based on the types

of memorisation activities used. Students who practice repetitive learning (drilling)

are more successful in solving difficult problems than those who simply learn

something by heart (rote memorisation). Repetitive learning can ease students’

Figure 4.3 Memorisation strategies and item difficulty

Odds ratio across 48 education systems

Greater success

Easy problem Using memorisation strategies is associated with an increase in the

probability of successfully solving a mathematics problem

Me

m

suc orisat

ces io

s as n is a

pro ssoc

ble i

ms ated w

bec

om ith les

em s

ore chan

diffi ce o

cul f

t

R ² = 0.81

Using memorisation strategies is associated with a decrease

in the probability of successfully solving a mathematics problem

Difficult problem

300 400 500 600 700 800

Less success Difficulty of mathematics items on the PISA scale

Notes: Statistically significant odds ratios are marked in a darker tone.

Chile and Mexico are not included in the OECD average.

Source: OECD, PISA 2012 Database, adapted from Echazarra, A. et al. (2016), “How teachers teach and students learn: Successful

strategies for school”, OECD Education Working Paper, no. 130.

Statlink: http://dx.doi.org/10.1787/888933414854

40.
WHAT DO WE KNOW ABOUT MEMORISATION AND LEARNING MATHEMATICS? . 39

anxiety towards mathematics by reducing the subject to a set of simple facts, rules

MEMORISATION

and procedures that might seem less challenging for the least-confident students to

master. Drilling can also free up time for more advanced mathematics by gradually

reducing the mental effort needed to complete simple tasks.

WHAT CAN TEACHERS DO?

Encourage students to complement memorisation with other learning strategies.

Memorisation can be used for some tasks in mathematics, such as recalling formulas or

automating simple calculations to speed up problem solving. This will help students free

up time for deeper thinking as they encounter more difficult problems later on. However,

you should encourage your students to go beyond memorisation if you want them to

understand mathematics, and solve real complex problems later in life.

Use memorisation strategies to build familiarity and confidence. Students may practice

or repeat certain procedures as this helps consolidate their understanding of concepts

and builds familiarity with problem-solving approaches. These activities don’t have to be

boring; teachers can find free interactive software or games on line to make such practice

activities more interesting to students.

Notice how your students learn. Learners who are less confident in their own

mathematical abilities or more prone to anxiety may rely too much on memorisation. Urge

those students to use other learning strategies as well by helping them make connections

between concepts and real-world problems and encouraging them to set their own goals

for learning mathematics. Also, remember that the way you teach concepts and assess

students’ understanding can influence how students approach mathematics.

1. National Center for Education Statistics (2003), Third International Mathematics and Science Study

1999: Video Study Technical Report, Volume 1: Mathematics, Washington, DC.

OECD (2013), Lessons from PISA 2012 for the United States, Strong Performers and Successful Reformers

in Education, OECD Publishing, Paris, http://dx.doi.org/10.1787/9789264207585-en.

Souma, K. (2000), “Mathematics Classroom Teaching”, Journal of Japan Mathematics Education

Institution, Vol. 82/7/8.

Takahashi, A. (2006), “Characteristics of Japanese Mathematics Lessons”, paper presented at the APEC

International Conference on Innovative Teaching Mathematics through Lesson Study, January 14-20, Tokyo.

2. Boaler, J. (1998), “Open and Closed Mathematics: Student Experiences and Understandings”, Journal

for Research in Mathematics Education, Vol. 29/1, pp. 41-62.

Hiebert, J. and D. Wearne (1996), “Instruction, understanding, and skill in multidigit addition and

subtraction”, Cognition and Instruction, Vol. 14/3, pp. 251-283.

Rathmell, E. (1978), “Using thinking strategies to teach the basic facts”, NCTM Yearbook, Vol. 13/38.

anxiety towards mathematics by reducing the subject to a set of simple facts, rules

MEMORISATION

and procedures that might seem less challenging for the least-confident students to

master. Drilling can also free up time for more advanced mathematics by gradually

reducing the mental effort needed to complete simple tasks.

WHAT CAN TEACHERS DO?

Encourage students to complement memorisation with other learning strategies.

Memorisation can be used for some tasks in mathematics, such as recalling formulas or

automating simple calculations to speed up problem solving. This will help students free

up time for deeper thinking as they encounter more difficult problems later on. However,

you should encourage your students to go beyond memorisation if you want them to

understand mathematics, and solve real complex problems later in life.

Use memorisation strategies to build familiarity and confidence. Students may practice

or repeat certain procedures as this helps consolidate their understanding of concepts

and builds familiarity with problem-solving approaches. These activities don’t have to be

boring; teachers can find free interactive software or games on line to make such practice

activities more interesting to students.

Notice how your students learn. Learners who are less confident in their own

mathematical abilities or more prone to anxiety may rely too much on memorisation. Urge

those students to use other learning strategies as well by helping them make connections

between concepts and real-world problems and encouraging them to set their own goals

for learning mathematics. Also, remember that the way you teach concepts and assess

students’ understanding can influence how students approach mathematics.

1. National Center for Education Statistics (2003), Third International Mathematics and Science Study

1999: Video Study Technical Report, Volume 1: Mathematics, Washington, DC.

OECD (2013), Lessons from PISA 2012 for the United States, Strong Performers and Successful Reformers

in Education, OECD Publishing, Paris, http://dx.doi.org/10.1787/9789264207585-en.

Souma, K. (2000), “Mathematics Classroom Teaching”, Journal of Japan Mathematics Education

Institution, Vol. 82/7/8.

Takahashi, A. (2006), “Characteristics of Japanese Mathematics Lessons”, paper presented at the APEC

International Conference on Innovative Teaching Mathematics through Lesson Study, January 14-20, Tokyo.

2. Boaler, J. (1998), “Open and Closed Mathematics: Student Experiences and Understandings”, Journal

for Research in Mathematics Education, Vol. 29/1, pp. 41-62.

Hiebert, J. and D. Wearne (1996), “Instruction, understanding, and skill in multidigit addition and

subtraction”, Cognition and Instruction, Vol. 14/3, pp. 251-283.

Rathmell, E. (1978), “Using thinking strategies to teach the basic facts”, NCTM Yearbook, Vol. 13/38.

41.
Can I help my

students learn

how to learn

mathematics?

students learn

how to learn

mathematics?