L earning from J apan:
A study of pedagogy that has led to high levels of
mathematical thinking aptitude among J apan’s
lower secondary school students
John McLean
Yasuda Women’s University, Japan
Abstract: Tests of secondary school students’ mathematical aptitude have long played a
central role in large-scale international comparative studies. While some researchers
argue that this trend is due to the fact that mathematics possesses an agreed body of
knowledge which can be easily compared between nations, a more plausible
explanation comes from the undeniable significance that mathematical aptitude plays
in commerce, industry, technology and science on which the wealth of all industrialised
nations are built. Bearing that significance in mind, it is therefore no surprise that
Japan’s economic boom in the latter half of the twentieth century was preceded by a
movement in its educational system typified by measurably high standards of
mathematical aptitude. Extensive research into this phenomenon commonly concludes
that it has been the result of constructivist pedagogy or unique socio-cultural
circumstances; however, in this paper I present empirical evidence that challenges that
assumption. I argue that much of what researchers in the West know about
mathematics education in Japan has been coloured by linguistic barriers and the
official line presented by Japan’s Ministry of Education. More to the point, based on my
research findings, I explain how factors fundamental to Japan’s educational success can
be assimilated into almost any educational system.

The West has long shown an interest in Japan’s educational system: as far
back as 1900, just thirty years after Japan emerged from feudal rule, the
Office of Special Reports of the Board of Education in London published a

paper on ‘Commercial Education in Japan’. However, prior to Japan’s
economic boom in the 1980s, the West’s interest in Japan’s educational
system revolved around a belief that it was something that needed shaping
and improving, and not something to learn from (Cummings et al., 1986).
Examination of Japan’s educational achievements reveals that while
students rarely excel on international tests of science or literacy—first or
foreign language, they consistently excel on tests of mathematics. In fact,
evidence suggests that Japan’s lower secondary school students have
outperformed their G8 counterparts in every major test of mathematical
aptitude conducted since 1964. Unfortunately, however, according to Horio
(1990), the involvement of Japan’s Ministry of Education in Western
research aimed at understanding this phenomenon has led to a body of
research that is either misinformed or prejudiced. One such example of this
is the unsubstantiated claims made by Her Majesty’s Inspectorate (1991: 26)
from the UK who reported that ‘the West has little to learn from the
Japanese and their education system because of, among other reasons,
cultural differences and their more homogeneous society.’
In order to understand the significance of factors that have led to
Japan’s lower secondary school students achieving a high level of
mathematical aptitude, I first briefly detail the cognitive factors
fundamental to the development mathematical aptitude. I explain how
mathematical aptitude can be measured and consider how the wealth of a
nation can be impacted by the standard of mathematical aptitude achieved
by its lower secondary school students. I then turn my attention to
mathematics education in Japan and show how various factors such as the
intervention of Japan’s Ministry of Education have led to a substantial
amount of unclear and conflicting research findings. In the second part of
this paper, I present independently collected survey data relating to the
formal and non-formal mathematics education of lower secondary school
students in 109 state, national and private schools in Hiroshima prefecture. I
explain how my findings show that Japan’s high instances of mathematical
aptitude are not the product of sophisticated constructivist pedagogy as
many believe but rather the outcome of a cognitive process that has occurred
due to a combination of formal and non-formal mathematics instruction.
Finally, I conclude this paper with a tentative proposal for how to create a
similar learning environment within a formal educational system as a means

of raising mathematics standards.

W hat is mathematical aptitude?
In spite of the fact that various schools of thought currently exist with regard
to what pedagogical techniques best encourage the development of
mathematical aptitude, common ground can still be found in the cognitive
process—hereinafter referred to as mathematical thinking—that is believed
to constitute that aptitude. Barnard (2002: 115), for example, defines
mathematical thinking as:
…proceeding via operations involving a small number of ‘items’ at any one time,
an important feature is the phenomenon in which a section of mathematical
structure may be mentally held as a single unit, processing an interiority that
can be subsequently expanded without loss of detail and trigger connections
with other parts of cognitive structure.

Tall (1994) similarly explains that mathematical thinking involves the
construction of a mental object that can be manipulated in the mind by
analogy with actions on objects experienced in the external world. More
comprehensively, I would define mathematical thinking as the cognitive
process that occurs when consolidated mathematical knowledge is applied
indirectly to a problem by mapping the conceptual structure of one set of
ideas into another (See Figure 1).
F igure 1: Mathematical Thinking (MT)
Exposure to mathematical knowledge
Consolidation of mathematical knowledge

MT

Application of mathematical knowledge

How is mathematical aptitude measured?

The International Association for the Evaluation of Educational
Achievement (IEA) and the Organization for Economic Cooperation and
Development (OECD) currently dominate the world of large-scale
international studies of secondary school students’ mathematical ability.
However, while each organization’s studies bare a resemblance in that they
centre around a test of mathematical ability, the source of the questions that
appear in the respective tests differ: the IEA, on the one hand, draw
questions from the curriculum of participating nations, whereas the OECD,
in contrast, draw questions from real-world situations that do not appear in
the curriculum of any of the participating nations.
Based on an understanding that a student who exhibits a high level
of mathematical aptitude should be able to apply their mathematical
knowledge to real-world situations using mathematical thinking, it can be
argued that the OECD test is the more appropriate measure of that ability.
However, that being said, this alone does not prove that the IEA test is not
also equally as good at measuring mathematical aptitude. Owing to the fact
that mathematical thinking is an essential element of mathematical
modeling (See Figure 2) which is fundamental to most high-tech jobs on
which the wealth of industrialised nations are built (Gainsburg, 2006), and
considering that today’s secondary school students will be part of tomorrow’s
workforce, it can be argued that the most accurate test of mathematical
aptitude is one that closely correlates to the per capita wealth of a nation
(See Table 1).
F igure 2: The Role of Mathematical Thinking (MT) in Mathematical Modeling
Real-world Problem
MT
Mathematical Problem
MT
Mathematical Solution
MT
Real-world Solution

Table 1: Correlation of I E A and OE CD mathematics studies against Gross National

I ncome (GNI ) per capita

Year

Organisation

Mathematical

P earson correlation

Study

measured against
GNI per capita

1995

IEA

TIMSS

0.350

1999

IEA

TIMSS-R

0.575(**)

2000

OECD

PISA

0.453(*)

2003

IEA

FIMSS

0.504(**)

OECD

PISA

0.621(**)

** Correlation is significant at the 0.01 level (2-tailed).
* Correlation is significant at the 0.05 level (2-tailed).
n = 37

TIMSS: Third International Mathematics and Science Study
TIMSS-R: Third International Mathematics and Science Study-Repeat
PISA: Program for International Student Assessment
FIMSS: Fourth International Mathematics and Science Study
Source: Based on information from: the US Department of Education, National Center
for Education Statistics. Available at: http://www.nctm.org; and W orld Development
I ndicators database, W orld B ank

As can be seen from Table 1 above, not only do the results of the
OECD PISA 2003 test strongly correlate with participating nations’ GNI per
capita but so too do the IEA TIMSS-R 1999 and FIMSS 2003 tests; hence
indicating the reliability of test data from both OECD and IEA studies as an
accurate measure of lower secondary schools students’ mathematical
aptitude.
Although the Pearson coefficient does not allow for any direct
conclusion about the causality of a correlation to be made, as Field (2005)
explains it is possible to measure the variability of one variable that is
explained by the other by squaring the coefficient and converting the
resulting figure into a percentage (Figure 3). Consequently, as can be seen
from Figure 3, the level of mathematical aptitude achieved by lower
secondary school students is so closely tied to the wealth of a
nation—accounting for anything up to 39 percent of GNI per capita—that

improving mathematics standards should be high on the list of priorities of
any policymaker.
F igure 3: A measure of the impact lower secondary school students’ mathematical
aptitude has on GNI per capita
GNI per capita in 1999

25.4%
Lower secondary schools students’
mathematical aptitude as measured
by the IEA TIMSS-R 1999 test
GNI per capita in 2003

GNI per capita in 2003

33.1%

38.6%

Lower secondary schools students’

Lower secondary schools students’

mathematical aptitude as measured

mathematical aptitude as measured

by the IEA FIMS 2003 test

by the OECD PISA 2003 test

M athematical Aptitude in J apan
Bearing in mind that Japan, a country with few natural resources, emerged
as the second largest economy in the latter half of the twentieth century on
the back of its workforce, it comes as no surprise that the mathematical
aptitude of its school students is measurably high. As previously noted, the
mean average performance of Japan’s lower secondary school students has
been in excess of their G8 counterparts in every IEA and OECD test of
mathematical aptitude conducted since 1964 (See Table 2 and Graph 1).

Table 2: Mean average performance of J apan’s lower secondary school students on I E A

and OE CD’s tests of mathematical aptitude
1. IEA FIMS 1964

2nd out of 12 countries

2. IEA SIMS 1981

1st out of 20 countries

3. IEA TIMSS 1995

3rd out of 41 countries

4. IEA TIMSS- R 1999

5th out of 38 countries

5. IEA FIMSS 2003

5th out of 46 countries

------------------------------------------------------------------------------------------------------------------------1. OECD PISA 2000

1st out of 32 countries

2. OECD PISA 2003

6th out of 41 countries

FIMS: First International Mathematics Study
SIMS: Second International Mathematics Study

Source: Based on information from: the IEA available at http://www.iea.nl and MEXT
(2005: 14)
Graph 1: G8 countries mean average performance of lower secondary school students in
I E A and OE CD’s comparative studies of mathematical aptitude conducted since 1995
60 0

50 0

40 0

C anada
T IM S S

1995

UK
T IM S S

F ran ce
19 99

G erm a ny
F IM S S

It a l y
200 3

J apan

R u s s ia
PIS A

1999

US
P IS A

2003

Source: Based on information from: the US Department of Education, National Center
for Education Statistics. Available at: http://www.nctm.org (See Appendix A for related
data)

While there is no doubt about the high standard of mathematical
aptitude achieved by Japan’s lower secondary school students nor about the

impact those high standards have had on the Japanese economy, a number of
issues of contention exist with regard to what factors have enabled that
aptitude to develop; issues concerning the nature of pedagogy in secondary
schools, the role of textbooks, the quality of educators, the motivational
levels of students and the impact of Japan’s non-formal educational system.
As previously mentioned, a number of inaccuracies in the findings of
Western scholars about mathematics education in Japan have emerged due
to the involvement of the Ministry of Education (Horio, 1990). What is more,
these inaccuracies are often compounded by one or more of the following
factors:
l Scholars limited command of the Japanese language
l Preconceived ideas about Japanese culture
l Limited understanding of mathematical thinking
l Preconceived ideas about what factors must be present in Japan
in order for mathematical thinking to develop
For ease of future reference, I have collated summarised existing literature
about mathematics education in Japan’s lower secondary schools under the
following headings: General/factual information (see Table 3);
Non-conflicting positive findings (see Table 4); Non-conflicting negative
findings (see Table 5); and Conflicting findings (see Table 6).
Table 3: General/factual information about mathematics education in J apan
l

Japan’s lower secondary school students have outperformed their G8 counterparts
in every IEA and OECD test of mathematical aptitude since 1964

l

Japan’s lower secondary school mathematics lessons follow a national curriculum

l

Japan’s lower secondary schools offer 105 (50 minute) mathematics lessons a year,
which equates to 263 hours of mathematics instruction over the three years of
lower secondary school education (As a point of comparison, approximately 330
hours of mathematics instruction is offered over three years of lower secondary
school education in the UK)

l

Mathematics is taught in mixed-ability classes

l

The Japanese government invests less in education—as a percentage of
GDP—than the OECD average (Japan 3.6 percent; OECD average 5.4 percent)

Table 4: Non-conflicting positive findings about education in J apan

l

Japan’s elementary school mathematics education is of the highest quality (Jacobs
and Morita, 2002; Whitburn, 2000; Stevenson and Lee, 1997; Rohlen, 1997)

Table 5: Non-conflicting negative findings about education in J apan
l

Approximately 50 percent of Japan’s lower secondary school students attend
private cram schools (MEXT, 2005; Shimizu, 2001)

l

Japan’s ministry of education is keen to reform its current educational system
(MEXT, 1998)

Table 6: Conflicting findings about education in J apan (Compare positive with negative)
P OSI TI VE F I NDI NGS
l

Mathematics

lessons

at

NE GATI VE F I NDI NGS
lower

l

Teachers at secondary schools in

secondary schools are taught using

Japan use pedagogical techniques

constructivist teaching methodology

that encourage memorization and rote

(Stigler and Hiebert, 1999; Jones,

learning (Nemoto, 1999; White, 1987)

1997; Whitman, 1991)

l

Teachers at secondary schools teach to
the test rather than trying to inspire
students to think creatively (Schoppa,
1991)

l

Japan’s

non-formal

education

compensates for the inflexibility of its
formal educational system (Guo, 2005;
Russell,

1997;

Cummings,

1986;

Kitamura, 1986)
l

Textbooks

are

only

used

in l

Teachers at secondary schools rely

approximately 2 percent of lower

heavily on textbooks (Schmidt, 1993;

secondary school mathematics lessons

Koyama, 1985)

(Jones, 1997)
l

Japan has a wealth of highly qualified
educators

working

at

its

lower

secondary schools (Barrett, 2001)

l

Japan’s

state

secondary

schools

employ an excessively large number of
part-time teachers—in some cases up
to

one

third

of

the

academic

staff—many of whom are not qualified
to work fulltime (Sakamoto, 2005)
P OSI TI VE F I NDI NGS

NE GATI VE F I NDI NGS

l

Entrance exams for upper secondary l

Japan’s

lower

schools provide students with the

students have extremely low intrinsic

extrinsic motivation required to attain

and

a high level of mathematical aptitude

mathematics (OECD, 2004; Shimizu,

(Guo, 2005; Rohlen, 1997; Shimahara,

2001; Nemoto, 1999; Robitaille, 1993)

extrinsic

secondary
motivation

school
towards

1997)

M E THOD
This study is the first part of a larger study that aims to provide valid and
reliable informative about formal and non-formal mathematics education in
Japan. In this paper, I investigate the following research questions:
(1) Are comparatively high instances of mathematical thinking
among Japan’s lower secondary school students the result of
constructivist teaching methodology and limited use of textbooks
in mathematics lessons?
(2) In what way has non-formal education in Japan aided the
development of lower secondary school students’ mathematical
thinking?

P articipants and procedure
Data for this study was collected using a questionnaire survey comprising 16
closed answer questions (see Appendix B) about participants’ experiences
and opinions of lower secondary school formal and non-formal mathematics
instruction. The participants came from 9 different academic departments at
two universities in Hiroshima (see Appendix C) and between them graduated
from a total of 143 different lower secondary schools across Japan: 111 of
those schools—104 state, 5 private and 2 national—are located in Hiroshima
prefecture; 12 schools—11 state and 1 national—are located in Yamaguchi
prefecture; and 20 schools are located in 13 other prefectures.
The sample was chosen from university students for the following
reasons:
l Data about pedagogy in a large number of lower secondary
schools can be collected with a relatively small sample

l A survey can be carried out without the cooperation of local
boards of education
l It is easy to guarantee university students the anonymity
necessary for them to feel at ease to comment freely about
mathematics teaching methodology in their lower secondary
schools
l Due to the passing of time university students are more likely to
objectively reflect on their lower secondary school experiences as
a whole rather than subjectively comment on their current or
most recent experience

Measures
The questionnaire was divided into the following sections and subsections:
Section 1:
Section 2:

About Yourself
About Your Secondary Education

Subsection 1:
Subsection 2:
Section 3:

Lower Secondary School: General Information
Lower Secondary School: Mathematics

About Non-Formal Education

Each question on the questionnaire can be categorised as classification,
behavioural or attitudinal . Classification questions—Q1-Q4, Q7-Q9 and
Q13—split the sample into smaller subgroups for comparison. B ehavioural
questions—Q11-Q13—provide information about the mathematics teaching
methodology and use of textbooks in participants’ lower secondary schools;
and about the participants’ attendance of non-formal mathematics education.
Attitudinal questions— Q5-Q6, Q10a/b Q14-Q16—provide information about
participants’ opinions concerning formal and non-formal mathematics
education.
Due to the arbitrary nature of the values pertaining from the
behavioural questions, I summarised the numeric data using frequencies. I
then separated the sample into subgroups using the classification questions
and tested for significance between two or more variables using Pearson’s
chi-square test. Finally, where a relationship was evident in a 95 percent
confidence interval (p < 0.05), I used the odds ratio to measure the size of
that effect. In the behavioural question relating to textbook use participants

were asked to categorise the frequency that textbooks were used in their
lower secondary school mathematics lessons as Never, Rarely, S ometimes,
Most L essons or E very L esson. In the question relating to teaching
methodology participants were asked to tick the lesson plan—type A or
B—that best resembled their typical lower secondary school mathematics
lesson: type A lesson is based on a classical deductive mathematics lesson
commonly practiced in the West, and type B lesson is based a constructivist
mathematics lesson described by Jacobs and Morita (2002: 167) as Japanese
teachers’ ideal script for a mathematics lesson; a third option, C was also
presented for any participants who did not think that his or her mathematics
lesson could be characterised by type A or type B.
After analysing participants’ responses to the behavioural questions,
I turned my attention to the attitudinal questions as a means of adding
insight into the role that formal and non-formal mathematics education has
played in raising standards of mathematical thinking. As with the
behavioural questions, I tested for significance between two or more
variables using Pearson’s chi-square test and measured the size of the effect
using the odds ratio.

RE SU LTS
As can be seen from Graph 2, the key findings with regard to the use of
textbooks in participants’ lower secondary school mathematics lessons were
as follows: only 3 percent of the entire sample reported that textbooks were
rarely used in their mathematics lessons; a minimum of 80 percent or the
participants in the four main subgroups—Hiroshima state schools,
Hiroshima private schools, Yamaguchi private schools and schools in other
prefecture—claimed that textbooks were used in most or every mathematics
lesson; the most common (mode) response of participants in every
statistically significant subcategory was that textbooks were used in every
lesson; and the most typical (median: 50% line) response in significant
subcategories was that textbooks were used in most mathematics lessons.
Finally, in a 95 percent confidence interval no statistically significant
difference was found between the responses of participants in any of the
subcategories.

Graph 2: A horizontal bar chart showing the frequency of textbook use in

participants’ lower secondary school mathematics
TOTAL
47%

36%

14%

3%

15%

2%

Hiroshima: State
45%

37%
Hiroshima: Private

52%

33%

14%

Yamaguchi: State
58%

33%

8%

Other Prefectures
60%

0%

10%

Every Lesson

20%

30%

20%

40%

Most Lessons

50%

60%

70%

10%

80%

Sometimes

90%

10%

100%

Rarely

The key findings with regard to mathematics teaching methodology
were as follows (see Table 7): every participant that responded indicated that
their typical lower secondary school mathematics lesson was either
deductive or constructivist; only 6 percent of the entire sample indicated that
their lower secondary school mathematics lessons were constructivist
whereas 93 percent indicated that their lessons were deductive in nature; a
maximum of 9 percent of participants in any statistically significant
subcategories indicated that their mathematics lessons were constructivist;
and no statistically significance difference was found between the responses
of the participants in any of the subcategories.
Based on the above findings, it follows that as there is a 3 in 50
chance of a lesson picked at random from the sample being constructivist
and a 3 in 100 chance of a lesson being one in which a textbook is rarely used,
there is only 9 in 5000 chance that a lesson picked at random is
constructivist in which a textbook is rarely used. This figure is supported by
the fact that only one of the 264 participants in the sample indicated that his
or her mathematics lessons satisfied both of those criteria.

Table 7: A contingency table showing how the participants categorised the teaching
methodology in their lower secondary school mathematics lessons

Hiroshima:State

Hiroshima:Private

Hiroshima:National

Yamaguchi
State

Yamaguchi:
National

OtherPrefectures

Other(Unspecified)

mathematics
lesson

Typeof

A: L ower Secondary School by region/type

Total

187

20

2

11

1

19

5

245

% of A Total

93%

95%

67%

92%

100%

95%

100%

93%

Constructivist

13

1

1

1

1

17

% of A Total

6%

5%

33%

8%

5%

6%

Deductive

No E ntry
% of A Total

2

2

1%

1%

Total Count

202

21

3

12

1

20

5

264

% of Total

77%

8%

1%

5%

<1%

8%

2%

100%

Other notable findings include:
l

Participants who were taught mathematics using constructivist
methodology were no more statistically likely to like or dislike their
lesson than participants who were taught using deductive

l

methodology
67 percent of the sample studied mathematics at cram school when

l

they were lower secondary school students
Participants who studied mathematics at cram school were 2.35
times more likely to think that their school mathematics lessons
were easier than those who did not

l

l

62 percent of the participants who attended cram school think that
their cram school mathematics lessons were more interesting than
their lower secondary school mathematics lessons
84 percent of the participants that attended cram school think that
their cram school mathematics lessons were more useful in helping
them to prepare for the upper secondary school entrance exams than

l

their lower secondary school mathematics lessons
81 percent of the participant who attended cram school think that
their cram school lessons were easier to understand than their lower
secondary school mathematics lessons

l

l

32 percent of the students who went on to study a major at university
in which mathematical aptitude is important were able to do so
without studying mathematics at cram school
Participants who studied mathematics at cram school were no more
likely to go on to study a major at university in which mathematical
aptitude is important than those who did not

DI SCU S SI ON
Based on the data collected in this study, the simple answer to the question,
‘Are comparatively high instances of mathematical thinking among Japan’s
lower secondary school students the result of constructivist teaching
methodology and limited use of textbooks in mathematics lessons?’ is no; at
least not in Hiroshima prefecture for which this study includes information
about teaching practices in 70 percent of the state schools. What is more,
with 93 percent of the sample indicating that their mathematics lessons
were deductive in nature (see Table 7) and 83 percent (see Graph 2) that
textbooks were used in most or every mathematics lesson, it can be argued
that the lower secondary school mathematics lessons experienced by the
participants in this study were, for all intents and purposes, the same as the
mathematics lessons experienced by their counterparts in the West.
After dismissing the idea that constructivist pedagogy has been
fundamental to Japan’s high standards of mathematical thinking, attention
must turn to understanding the importance of non-formal mathematics
instruction at private cram schools which approximately 50 percent of lower
secondary school students attend (MEXT, 2005). If the findings of this
questionnaire survey are taken at face value, it would appear that
mathematics instruction in Japan’s private cram schools is more interesting,
easier to understand, and prepares students for high school entrance exams
better than the mathematics instruction at lower secondary schools.
However, this kind of statement overlooks the fact that mathematics
instruction at private cram schools is not a primary source of educational
input. As a secondary source of educational input cram schools benefit from

the fact that most students arrive with prior mathematical knowledge that
aids in making the learning experience more interesting and easier to
understand. Furthermore, while there is no doubt that some cram schools
employ dynamic teachers who teach using sophisticated pedagogical
practices, the vast majority of teachers in cram schools are not so called ‘star’
teachers; in fact, as Roesgaard (2006) points out more than 80 percent of
teachers in cram schools have never had any kind of formal teaching
training.
Analysis of the importance of mathematics instruction at cram
schools may look like a daunting task owing to its extensive variety in terms
of class size, use of streaming, use of technology and the experience or
qualifications of the teachers; however, it is that variety that makes it so
much easier to identify the function it serves in raising students
mathematical thinking. After eliminating all inconsistencies, only three
significant factors that influence the learning experience of students
attending private crams schools remain:
1. Change in environment
2. Change in classroom dynamic
3. Change in perspective
Consequently, it can be argued that the act of transposing mathematical
knowledge from the primary learning environment to the secondary learning
environment where that knowledge must then be used to interact with a
similar mathematical problem from a potentially different perspective in a
classroom with a different dynamic requires the mapping of conceptual
structure of one set of ideas into another and vice versa (see Figure 4); hence
aiding in the consolidation of mathematical knowledge through a basic form
of mathematical thinking that would not be possible if students were to
spend more time study the same mathematical problems in the primary
learning environment.
Finally, it should be noted that while some students are naturally apt
at mapping the conceptual structure of one set of ideas into another, the vast
majority are not. Therefore, if any nation hopes to benefit from the obvious
advantages of a workforce with high levels of mathematical thinking without
developing a large non-formal educational sector, it needs to consider how to

incorporate a change of environment, perspective and classroom dynamic
into its current educational system.

F igure 4: A map of mathematical thinking (MT) in J apan
Formal school:

Base MT

mathematics knowledge

Formal school:

Non-formal school:
mathematics problem

Base MT

Non-formal school:

mathematics problem

mathematics solution

Formal school:

Mathematical

mathematics solution

knowledge/aptitude
MT
Real World Problem

I M P L I CATI ONS, L I MI TATI ONS, AND
SU GGE S TI ONS F OR F U TU RE RE SE AR CH
The results of this study suggest that comparatively high instances of
mathematical thinking among Japan’s lower secondary school students are
not the result of constructivist teaching methodology and limited use of
textbooks in mathematics lessons as some researchers believe, but rather the
result of a combination of formal and non-formal mathematics instruction. I
use the results to explain how a combination of studying mathematics in two
different environments, with two different classroom dynamics and from two
potentially different perspectives requires students to map the conceptual
structure of one set of ideas into another, which is fundamental to the
development of mathematical thinking. The most significant implication of
this paper results from the identification of underlying factors that have
enabled mathematical thinking to develop in Japan is that it provides other

nations a means of creating a comparable learning environment within their
own formal educational system.
Due to the limitations of time and space, I have only presented one
aspect of a much larger study about Japan’s mathematics education in this
paper. In a follow up to this study, I confirm the findings presented in this
paper with questionnaires, interviews and classroom observations involving
lower secondary school students, teachers and members of the local boards of
education; I compare and contrast the contents of the mathematics
curriculum taught in Japan’s lower secondary schools with other G8 nations;
I discuss the significance of the high number of Japan’s lower secondary
school students staying on in fulltime education until they are 18 years old;
and I look in detail at how Japan has benefited from having a system of
examinations at the entrance to almost every stage of education.

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Appendix A
IEA

IEA

IEA

OECD

OECD

TIMSS 1995

TIMSS-R 1999

FIMSS 2003

PISA 2000

PISA 2003

(41)

(38)

(46)

(32)

(41)

Canada

521(11)

531(10)

533(5)

532(7)

England

498(14)

496(20)

529(7)

France

517(10)

511(17)

Germany

490(20)

503(20)

Italy

491(16)

479(22)

484(22)

457(26)

466(29)

Japan

581(3)

579(5)

570(5)

557(1)

534(6)

Russian Federation

524(10)

526(12)

508(10)

478(22)

468(28)

United States

492(15)

502(19)

504(15)

493(19)

483(27)

Source: Based on information from: the US Department of Education, National Center
for Education Statistics. Available at: http://www.nctm.org

Appendix B

About Yourself
Q1. Sex:

Male

Female

Q2. Academic Year:

First

Second

Q3. University:

Yasuda Women’s University

Hiroshima City University

Q4. Department:

English

Business

Third

Psychology

Design

Nutrition

Primary Education

Japanese Literature

Intelligent Systems

Information Machines and Interfaces
Q5. Do you think that mathematical aptitude is important for your major?
Yes

No

Q6. Do you think that mathematical aptitude is important for your future career?
Yes

No

About Your L ower Secondary School E ducation
Q7. What lower secondary school did you go to?
Q8. What prefecture is it in?
Hiroshima

Other (please specify)

Q9. What type of school is it?
State

National

Other (please specify)

Private

Q10.a. Did you like your mathematics lessons in lower secondary school?
Yes

No

Q10.b. Why? (Tick as many boxes as is appropriate)
The lesson was interesting

The lesson was boring

The lesson was easy

The lesson was difficult

You liked the teacher

You disliked the teacher

Other (please specify)
Q11. How often was a textbook used in the lesson?
Never

Rarely

Sometimes

Most Lessons

Every Lesson

Q12. Which lesson plan best describes your lower secondary school mathematics lesson?
Type A

Type B

Other (Please Specify)

Type A Mathematics L esson
The

teacher

tells

the

class

Type B Mathematics L esson
what The teacher poses a new mathematical

mathematics they are going to study.

problem on the board.

The teacher explains how to solve that type In

groups:

Students

discuss

the

of mathematics problem using an example problem until they agree on a solution.
from the textbook.
The teacher writes a similar mathematics Students present their solution to the
question on the board which students work problem in front of the class.
on individually.
A small number of students are called to the The teacher guides the students to the
front to write their answer on the board best solution.
which is then checked by the teacher and
used to explain how to answer the question.
Students

work

individually

similar questions in the textbook.

answering Students use what they have learned
to solve similar problems individually.

About Non-F ormal E ducation
Q13. Did you study mathematics at a private cram school when you were in lower
secondary school?
Yes

No

If you answered yes to the above question, in which year of study at lower secondary
school did you also attend private cram school? (Tick as many boxes as is appropriate)
First

Second

Third

I f you answered yes to Q13 please answer Q14~Q16
Q14. Which mathematics lesson did you enjoy more? (Tick one only)
Cram School

Lower Secondary School

Q15. Which mathematics lesson was more helpful in preparing for the upper secondary
school entrance exam? (Tick one only)
Cram School

Lower Secondary School

Q16. Which mathematics lesson was easier to understand? (Tick one box)
Cram School

Lower Secondary School

Appendix C
F irst Year
F emale

Second Year

Male

Third Year

F emale

Male

F emale

Male

HCU I ntelligent Systems

11

13

24

HCU I nfo Machines and I nterfaces

1

23

24
8

Total

YW U E nglish

38

13

59

YW U J apanese L iterature

16

1

17

YW U P sychology

22

8

30

YW U P rimary E ducation

25

25

YW U B usiness

13

6

19

YW U Design

21

6

27

YW U Nutrition

28

11

39

Total

150
150

HCU = Hiroshima City University
YWU = Yasuda Women’s University

0

64
100

36

14
14

0

264