‘FUNDAMENTALS OF
MATHEMATICS’
a challenge in Japanese CURRICULUM OF
MATHEMATICS
Shigeru Iitaka
Gakushuin University, Tokyo
contents
1. Overview of academic ability of
2.
3.
4.
5.
6.
Japanese high-school students
Math test conducted by the
governments in 2002
Math curriculum from 2003
‘Fundamentals of math’
Text books of ‘fundamentals of math’
conclusions
1
Overview of academic ability
of Japanese high-school
students
through mathematics test
(comprehensive survey of academic ability)
conducted by the government
Mathematics test
1. Took place on November in 2002
2. Nationwide
3. 2% random sampling from students
aged 17(about 30000)
4. Math I(learned when they were 15)
5. By the curriculum research center.
Mathematics I
(old､untill 2002),
1.Quadratic functions
2.number of cases
(combination &permutation)
3.Probability
4.Trigonometry and measuring
Samples of problems:
Basic level
Choose graphs of quadratic functions
from 4 given graphs.
‘ y=(x-3)2
‘ y=x2-3
How much percentage do
ordinary high-school students get
in your country?
Just guess please.
Scores obtained :
‘ y=(x-3)2
(Average score 58 % ; 70 % expected )
‘ y=x2-3
(Average score 75.5 % ; 75 % expected )
count the number of rectangules containing A
A
How much percentage do
ordinary high-school students get
in your country?
Just guess please.
count the number of rectangules containing A
A
Average score=39.5 % (55 expected)
Required to give a proof of
sine law.
1. Sine law and cosine law are taught in
math I.
2. To prove sine law is not easy for
students.
1. Please guess how much percent do
students prove sine law in simple form.
percent of students who prove sine law
in simple form.
The average score is 24 %(expected
score is 40%) .
8 big problems
all together
15 problems
score
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
total
numbers
%
420
730
983
1118
1122
1045
950
993
948
889
937
907
873
943
999
1551
15408
2
4
6
7
7
6
6
6
6
5
6
5
5
6
6
10
Math I
12
10
scores
8
6
4
2
0
scores
Results
15408 students(almost 1 % of students
aged 17)
2. Average score=8(full score 15)
3. S.D. = 4.5
4. The barchart suggests that there at least
two groups. Students belonging to the
former group have difficulty in learning
math.
1.
2
Attitudes of students toward
math
1.
2.
3.
4.
5.
Q:Do you think
learning math is important ?
Affirmative 20%
Almost
34%
Half
20%
almost
20%
negative
6%
Learning math is important
40
35
30
25
20
15
10
5
0
affirmative
a little
half
a little
negative
in favor of math
well
almost
half
poor
very poor
percent
14
23
18
40
4
45
40
35
30
25
20
15
10
5
0
系列1
in favor
of math
well
almost
half
poor
very
poor
non
understanding of math courses
well
almost
half
poor
very poor
non
7
28
30
22
11
1
non
poor
almost
understanding
of math
courses
35
30
25
20
15
10
5
0
Œn—
Some (many) students hate
mathematics. They are not
interested in math.
Their academic skills in math
are very poor and they have
difficulty in learning equations
and computing algebraic
expressions.
In many countries, most boys
and girls over 15 attend
senior high-schools
(enrollment ratio is 97% in
Japan). But some part of
them have difficulty in
learning traditional math
courses. They start to dislike
and hate math courses.
To ease agonies of students in
the courses of math who have
boring time in the math
courses, new subject should be
introduced,
, in which mathematics is
taught to help students
understand
1)the relationships between
mathematics and human
beings and 2)the roles which
mathematics plays in society.
To enhance students' interests
and concerns in mathematics
is much more important than
to maintain academic skills.
New subject was named
‘FUNDAMENTALS OF
MATHEMATICS’ , which started
from April 2003.
3
CURRICULUM OF
MATHEMATICS FOR THE
UPPER SECONDARY SCHOOL
(ages from 15 to 17.)
Revised by the Ministry of Education in 1999,
and introduced in 2002 (2003 in highschools).
It consists of 7 Subjects:
FUNDAMENTALS OF
MATHEMATICS
(2 units)
‘FUNDAMENTALS OF
MATHEMATICS’ is a new
subject.
Others are rather traditional.
MATHEMATICS I (3 units)
MATHEMATICS II (4 units),
MATHEMATICS III (3 units)
(core)
MATHEMATICS A , B,and C
(each 2 units)
(optional)
1. Either MATHEMATICS I or
FUNDAMENTALS OF
MATHEMATICS is compulsory.
2. MATHEMATICS II,and III,
MATHEMATICS A , B,and C are
elective.
Objectives of
FUNDAMENTALS OF
MATHEMATICS
(according to guideline)
To help students
1) understand
the relationships between
mathematics and human beings,
2) and the roles which mathematics
plays in society,
a) to enhance students' interests
and concerns in mathematics, and
b)
to appreciate mathematical ways of
observing and thinking, and to
foster attitudes which seek to apply
mathematics.
NB. Here is nothing to encourage
students to acquire basic
knowledge and to master skills.
It is more important to enhance
students' interests and concerns
in mathematics than to master
academic skills.
In the objectives of
MATHEMATICS I, it is
important to acquire basic
knowledge and to master
academic skills.
4
Look at a cover sheet of
actual text book of
‘Fundamentals of mathematics’
Published by Tokyo book co.ltd.
From Guideline of
‘Fundamentals of mathematics’
2. Content
(1) Mathematics and human
activities
To enable students to understand that
concepts and principles of numbers,
quantities, geometrical figures, etc.
have been developed in response to
human activities, and to enhance
students' interests and concerns in
mathematics.
a). Numbers and human
beings
b). Geometrical figures
and human beings
Look at
The first chapter of a text book.
Princes Komachi’s arithmetic
Onono Komachi is a famous poet, who was most
beautiful in ancient Japan(11 th century)
PROBLEM: Using 1,2,3,4,5,6,7,8,9 in the order and
addition and subtraction, make 100.
For example ,
123-45-67+89=100
Problem : prove that using only addition, cannot make
100.
From Guideline of
‘Fundamentals of mathematics’
(2) Mathematical
considerations in society
To enable students to know the usefulness
of mathematics, and to enrich
mathematical ways of observing and
thinking through mathematical
considerations of familiar phenomena and
situations where mathematics is used
in society,
a).
Society and mathematics
b).
Mathematical considerations
of familiar phenomena
Text book translated:
Society and mathematics,
sales tax of books
(ＶＡＴ：3% , 5% now),
compound interests.
Now, interests are very low according to
longtime deflation:
1 year deposit 0.04 %
(When the text book was written, it was 0.12 %)
3 year deposit 0.07 %(each year)
Difficult to understand the effect of
compound interests.
Long time ago, interests were pretty high.
Mathematics of loans
In 16th century, interests were very high;
1/8 for every 3 months.
A money lender outside house seemed
obedient and respectful to noble
women inside the house. But, he was
merciless.
Mathematics in the game using
cards
When playing cards, one can find lots
of arithmetic mod 53.
By using arithmetic mod N, you can
tell the place of your favorite card.
Theory of substitutions and games
called amida, which means a name
of god.
It’s a beginning of group theory.
amida
1
2
3
4
5
1
2
3
4
5
6
6
From Guideline of
‘Fundamentals of mathematics’
(3) Elementary statistics
To enable students to collect
appropriate data according to their
purposes, to arrange data by using
tables and graphs, and thereby to
grasp the trends in data by means of
the representative values, etc., and to
understand the concepts of statistics
and to apply them.
a). Arrangement of data
b). Grasp trends in data
From Guideline of
‘Fundamentals of mathematics’
3. Treatment of Contents
(I) As for the content (1), the teacher
should deal with topics like
mathematical history, closely
associating the process of the
mathematical conceptualization
and of realization of mathematical
principles and laws with human
beings and their cultures.
(II)
As for the content (2), the
teacher should consider
dealing with familiar
examples showing the
relationship between
society and mathematics.
(III)
As for the content (3), the teacher
should deal with the basic concepts
of statistics, and should consider
making use of computers, etc.
in the students' learning.
(IV)
In teaching this subject, the
teacher should instruct in such a
manner that every student can
learn to deal with the familiar
examples independently, and
should be careful not to proceed
too far in theoretical
considerations.
Attention please
Here are five copies of ‘fundamentals of
mathematics’ which were brought from
Tokyo,Japan.
If you want to keep these, please tell me after the
lecture.
First five persons will get them.
5
From Guideline of
Mathematics I,II,III
II. MATHEMATICS I
from 2003
(1)Equations and inequalities
(2) Quadratic functions
(3) Geometrical figures and measuring.
MATHEMATICS II(4 units)
(1) Algebraic expressions and proofs,
and higher degree equations
a. Algebraic expressions and proofs
b. Higher degree equations
(2) Geometrical figures and
equations
a. Points and lines
b. Circles
(3) Various functions
a. Trigonometric functions
b. Exponential functions and logarithmic
functions
(4) Ideas of differentiation
and integration
a. Ideas of differentiation
b. Ideas of integration
MATHEMATICS III
(1) Limits
a. Limits of sequences
b. Functions and their limits
(2) Differential calculus
a. Derivations
b. Applications of derivations
(3) Integral Calculus
From Guideline of
Mathematics A,B,C
MATHEMATICS A
(1) Plane geometry
(2) Sets and logic
(3) Number of cases and probability
a. Permutations and combinations
b. Probability and its basic laws
c. Independent trials and probability
MATHEMATICS B
(choose 2 from 4 items below)
(1) Sequences
(2) Vectors
(3) Statistics and computers
(4) Numerical computations and
computers
MATHEMATICS C
(choose 2 from 4 items below)
(1)
(2)
(3)
(4)
Matrices and their applications
Algebraic expressions and curves
Probability distributions
Statistics
6
Statistics on numbers
of sold copies of math
textbooks
in 2003.
numbers of copies
fundamentals
math I
math II
Math A
Math B
math III
Math C
thousand
38
1382
1055
975
538
264
234
nu
fu
I
m
at
h
II
M
at
h
A
M
at
h
B
m
at
h
III
M
at
h
C
h
m
at
ls
es
ta
co
pi
am
en
of
nd
rs
m
be
1600
1400
1200
1000
800
600
400
200
0
Œn
‘Fundamentals of M.’ in 2004
1. Few copies sold (1 % of Math 1)
2. Very few students (0.2 % ) took the course.
3. Teachers are reluctant to teach
fundamentals of math’.(main reason useless
for entrance examination and hard to teach)
4. 5 different text books were published by 5
publishers, each of which seems excellent.
5. But revised versions are not published.
‘Fundamentals of M.’ in 2004
(continued)
1. An idea of ‘fundamentals’ seems
good,but not successful for
unhappy students in the course of
math.
2. Needs more new ideas and efforts.
Thank you
851051@gakushuin.ac.jp
The lecture is over.
Here are five copies of ‘fundamentals of
mathematics’.
Please come here and take five copies
immediately.