Ten Questions for Mathematics Teachers

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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

2. PISA
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.
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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.

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?

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?

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.

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.

9. Teaching strategies
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.

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.

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

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

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

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

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.

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

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.

19. Cognitive activation
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

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.

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).

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

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

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

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.

27. Classroom climate
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.

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

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

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

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.

33. What do we
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,

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

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

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

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

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

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.

41. Can I help my
students learn
how to learn
mathematics?

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