From: steve@hubcap.clemson.edu ("Steve" Stevenson)
Newsgroups: comp.research.japan
Subject: An interesting review....
Date: 1 Oct 92 17:29:21 GMT
Distribution: comp
Organization: Clemson University

I got this in the mail today....I thought it might be interesting reading:

>From: Allan Pinkus <MAPRX99%TECHNION.TECHNION.AC.IL@TAUNIVM.TAU.AC.IL>
>Subject:      AT-NET Bulletin 8
>Comments: To: "Approximation Theory Network" <at-net@technion.technion.ac.il>
...
---------------------------------
From: askey@math.wisc.edu (Richard Askey)

3) Review for The College Mathematics Journal:

Japanese Grade 7 Mathematics, Kunihiko Kodaira, editor, translated by
Hiromi Nagata from the 1984 edition, University of Chicago School
Mathematics Project, 5835 S. Kimbark Avenue, Chicago, IL 60637, 1992,
$19.00, 185 pp. Also Grade 8 and Grade 9 books, same information except
205 pages and 199 pages, respectively.

Harold Stevenson and Karen Bartsch [10] have written an interesting paper
comparing Japanese and American textbooks.  They were trying to find some
of the reasons why our students understand mathematics less well than
students in other industrialized countries.  For example, in long term
comparisons between elementary school students in Minneapolis; Sendai,
Japan; and Taipei, Taiwan, Stevenson has found that the average
knowledge of mathematics in fifth grade in Minneapolis in the best of the
20 schools he studied is lower than the average knowledge
in all of the corresponding 20 schools in Sendai and the 20 in
Taipei.  In his hour address to the MAA in Baltimore, Stevenson
said that if you take the amount of mathematics known by the average 5th
grade student in Minneapolis, you have to go to the bottom 1 1/2% before
finding students in Taipei who know less.  Once you realize this, it is
clearly important to find out what goes on in these classrooms.

Stevenson and Stigler have a general trade book [12] giving some of their
conclusions based on more than 10 years of work.  There is also a very
interesting video, "The Polished Stones" [8], which I recommend be seen
by all current and prospective elementary school teachers and those
who teach them.  Two technical books on the tests which led to the
above conclusions have been published [11], [13].  However, except
for [10], none of this deals with text books.  Benjamin Duke's book
"The Japanese School: Lessons for Industrial America" [4] contains sample
problems from some Japanese texts, and an entrance exam for a Japanese
high school is included in [1].  Up until now, it was necessary to be
able to read Japanese to really see what the school texts are like.  Now,
thanks to the long time collecting efforts of Izaak Wirszup of the
University of Chicago and financial support from the Amoco Foundation,
we have translations of one series of Japanese books from
seventh to ninth grades.

Before looking at these in some detail, a little knowledge of Japanese
education is useful.  There is an outlined national curriculum which is
followed in the texts.  There is no tracking, so all students follow this
curriculum through ninth grade, which is the end of compulsory education.
There is a splitting of students in high school, with some going to elite
high schools, most going to academic high schools with lower but still
very high standards, and some going to technical high schools.  A good
description of these different types of schools can be found in
T. Rohlen, "The Japanese High School" [9].  Over 90% of all Japanese
children graduate from high school.  A very useful survey of
Japanese education is in [6].

The texts under review are similar to those used by all students in
Japan. They are "lean" but by American standards most would not say
"lively".  A better second word would be "serious".

Each of these three books has a number of short chapters, 7 in the 7th
and 9th grade books, 8 in the 8th grade book.  Each chapter starts with a
description of a problem or a setting in life where the mathematics of
this chapter is used.  For example, Chapter 6 of the 7th grade book
starts with this problem:

The family of Mr. A raises cows.  Every morning, Mr. A goes to the cow
shed carrying buckets of feed, and on his way back he goes down to the
stream, washes the buckets, and returns to the house.  Looking for
different places to wash the buckets, he noticed that the distance he
walked to get to those places varied. Then he decided he would like to
keep the distance he had to walk as short as possible, so he used the
diagram below to figure out where on the river he should wash the save
buckets.

How was he able to decide which was the shortest distance to walk?

The text in this chapter introduces straight lines, angles, circles and
sectors of them, arc length of an arc of a circle, area of a circle and of a
sector of a circle in 8 pages of facts, examples and problems.  Some of the
examples are done completely, both for the ideas used and so students can
learn how to write mathematics.  Other examples are only outlined, with the
student expected to fill in the missing steps.  There are about a dozen
problems in these eight pages, and the students are expected to do, and be
able to do next year, problems similar to those in the text.  There are eight
more pages of text, on ruler and compass constructions of such things as
angle bisectors and perpendicular lines, including how to find the shortest
line segment from a point to a line not through this point.  Since the
students go to school more than 10 months a year, the students and teachers
can spend more than a month on the 16 pages, and a couple of pages of
exercises that make up Chapter 6.  Neither the problems nor the end of the
chapter exercises have answers at the back of the book.  There are additional
"Computation Exercises" and "Review Exercises" at the end of each book, and
brief answers are given for these problems.

Other topics covered in the 7th grade book are integers, including common
divisors and common multiples, the sieve of Eratosthenes to generate all the
primes less than a given number (it is not given a name here), and positive
and negative numbers including a very nice explanation of the fact that the
product of two negative numbers is positive.  These make up the first two
chapters.  Some of this is review at a more sophisticated level than the
students saw in primary school, but there is the supposition that if material
was covered in previous years, this work is known.  For example, on page 12
the following paragraph is included.


"Multiples that are shared among integers are the common multiples of those
numbers.  This you learned in elementary school."

When was the last time you could say to a class that they know something in
algebra or geometry because they learned this in high school, and have them
know it because they learned it rather than memorized it?

There are three chapters on expressions with letters, the first an
introduction to these objects, then one on linear equations in one variable,
and finally one where students study some more general functions and their
graphs.  In the chapter on linear equations, students learn to set up
equations and learn how to solve them.  This is done in 17 pages and then
three further pages of exercises are given.  The last problem is the
following:

"It takes 12 minutes to fill a tank with water using pipe A and 18 minutes
using pipe B.  First pipe A was opened, and with the tank still filling, pipe
A was closed and pipe B was opened.  When pipe B had been open for 3 minutes
longer than pipe A, the tank was full.  How many minutes was pipe A opened?"

There is no problem exactly like this which the students can copy, unlike
most of the problems in our text books.

The last chapter deals with figures in space.  These include the usual
objects: spheres, parallelopipeds, lines and planes, cones, cylinders, and
pyramids.  The regular polyhedrons are also introduced, both by two
dimensional pictures of the three dimensional objects, and two dimensional
pictures of their faces unfolded so that the student can cut them out and
make them.  A picture of the cube formed by joining midpoints of adjacent
faces of the regular octahedron is given, and the students are asked to tell
what figure is formed by connecting the midpoints of adjacent faces of a
cube.  There is a very useful section on how to look at and draw geometric
solids.  Finally, surface areas and volumes of some solids are treated.  A
useful picture of how to cut a cube into three rectangular pyramids of the
same size and shape is given, which helps explain the area formula for a
pyramid and cone.  These are not derived yet, nor is the Pythagonean theorem
included in either of these chapters.  There is only a brief start at locally
deductive geometry, but there is a start.

The next two books are similar.  They start with algebra; linear inequalities
and simultaneous equations in the 8th grade, square roots and quadratic
equations in 9th grade.  Then there is a chapter on functions and their
graphs; linear in 8th and quadratic in 9th.  Then there are chapters on
geometry about which I will write a bit more below.  Each book ends with a
chapter on data.  In the 8th grade this is on how to organize data, and
includes frequency diagrams, cumulative frequency and means.  In 9th grade
this is extended and various discrete probability problems are introduced.
Here's and example,

250 white and red balls are mixed up in a sack.  When we take 20 balls at
random from this sack, we get 8 red balls.  Can we deduce approximately how
many red balls were in this sack?

There is a superficial resemblance to the NCTM Standards [3].  Topics are
integrated in each year, probability and statistics are introduced well
before they are treated in our current texts, and many topics are introduced
by a real world problem.  However, there are many differences as well.
Calculators and computers are conspicuous by their absence.  Partly this is
because the texts were published in 1984, but that is not the only reason.
In [5] there is a comparison between science education in Japan and the
United States.  One surprising fact is that the difference between how well
boys do than girls do is larger in the U.S. than in Japan.  As we have come
to expect, the Japanese students do better than ours do, sometimes strikingly
so.  There is also a set of correlations between how well students do on an
international test and facts from their background.  For ninth grade
students, having many books in the home and finding science easy to learn
have the highest positive correlations.  Of 25 items compared, all except one
had positive correlations in both countries or negative correlations in both
countries.  The exception was "Use of hand calculators".  This was positively
correlated with achievement in the U.S., but had the highest negative
correlation of any of the items in Japan.  This was -.174, and the star which
should have been there to denote significance at the .05 level was omitted by
mistake according to one of the authors.  Another of the authors gave a reson
for this high negative correlation.  In more idiomatic English (checked with
one of the authors) he said that calculators are not used in elementary or
junior high schools because the primary emphasis is on helping students
develop their mental abilities.

The sections on geometry are worth considering in some detail, for the
difference between the Japanese texts and NCTM Standards is very evident
here. Consider the most important theorem from elementary geometry, the
Pythagorean theorem.  Jacob Bronowski, who was an algebraic geometer before
the second world war, when his interests broadened, called this the most
important theorem in all of mathematics in his television series, The Ascent
of Man.  See [2] for a printed version.  I have tried to figure out which if
any students who study a program based on the NCTM Standards will be expected
to be able to prove this important result, but I have been unable to.  I
think very few, as is the case now.  Only one student out of 22 could prove
this theorem at the start of the semester in an honor section of second
semester calculus in the fall of 1991, and a reasonable fraction of these
students said they had never seen a proof.  In the eighth grade Japanese book
there are two proofs given with details, two different proofs are outlined,
and five further proofs are given by pictures at the end of the book.  I can
not say for sure how many of these students will be able to give a proof when
they start college, but I have seen Japanese college entrance exams and the
students can do geometry problems which are significantly harder, so most
probably still understand and can give a proof of this result.

Why am I so concerned about this?  Many of the early theorems in Euclid's
treatment of geometry are intuitively obvious, so it is hard to convince
students that these "obvious" facts need proofs.  It is often hard for the
students to see why these facts are useful.  The Pythagorean theorem is
clearly useful, and it is not obvious to most students.  One aspect of good
teaching is trying to get students interested in understanding important
facts which are not obvious.  Proofs do not always lead to understanding, but
in this case there are a number of different proofs which will help our
students understand this gem.  This illustrates one of my concerns with the
NCTM Standards.  They recommend a lot of experimenting, which is fine, but
too frequently things stop there before there is any real understanding.
Mathematics is partly informed guesses, but it is much more.  We seem to have
gone from one extreme to another.  In the past all that most of our students
learned was the statements of some important facts.  The new proposals are to
emphasize the discovery of simple facts along with letting a calculator do
the numerical and graphical work, and later have a computer algebra system
take over all techniques.  What is missing is a deeper understanding of
mathematics, and especially the ideas behind the techniques which the best of
our students used to learn along with the facts which most others thought was
all there was to mathematics.  I don't believe it is possible to teach deep
ideas without teaching the techniques which put these into action.  The
Japanese have both intuitive understanding and a deeper understanding with
the emphasis on intuitive understanding in the early grades and a deeper
understanding when the students are older.  We postpone the deeper
understanding until graduate school for most students, which means almost
none realize that mathematics should be understood at a deeper level.

Proofs in geometry in these books are local.  They start with a fact or two
which the students feel is clear, and then a derivation is given.  No
explicit axioms are introduced, and the proofs are written in sentences, not
the tabular "statement-reason" pattern which leads so many students to think
this is the only way something could be proved, and so that proofs only exist
in geometry.

There are some minor errors in the translation.  In the seventh grade book,
on page 158, in line 2, "at the top of the next page" should be "below".  In
the 8th grade book, on page 68, in problem 4, one of the two occurrences of
"decreased" should be replaced by "increased".  A number of the geometric
illustrations are not drawn accurately.  Sides of triangles which are marked
as equal are not drawn this way.  Two examples appear on page 126 of the 8th
grade book.  The problem came from computer aided drawing in the translation,
not in the original.  On page 127 in the 9th grade book, the first
illustration is missing the label of 100m on the bottom side.  The
publication information says these books were originally published by the
Japanese Ministry of Education.  This is incorrect.  They were published by
Tokyo Shoseki Company, Ltd.

Current plans are to translate the 10th grade book, which is used by all
students, and the easier of the 11th grade books.  This is used by 40% of the
students, and contains probability and statistics, vectors, exponential,
logarithmic and trigonometric functions and an informal treatment of
differentiation and integration.  The other 60% of the students use two
texts.  One is a more comprehensive treatment of coordinate geometry in two
and three dimensions, vectors and matrices.  The other is a more through
treatment of trigonometry and differential and integral calculus.  Only about
25% of the students study mathematics in 12th grade.  They take a further
course on probability and statistics and a rigorous course on calculus.

I suspect the reason for not translating the texts of the more serious 11th
grade course and the 12th grade course is that the material is not taught in
American high schools, at least at the level it is taught in Japan.  This is
an error, for we have traditionally expected less of our students than we now
see is not only possible, but is probably necessary if they are to compete
with students elsewhere.  Having translations of these books will help us set
better goals.  To be honest, if the 12th grade texts are as good as the three
I have seen, I would like to use them in a couple of my university courses.

In addition to the Japanese texts mentioned above, a complete set of Russian
texts from first grade through high school is being translated.  Texts for
the first three grades have been published by the same organization who
published the Japanese texts.  I hope the rest of these translations will
appear rapidly, and be seriously studied by people who are responsible for
education in the United States.  Some of the things I have learned from these
Japanese texts follow.

A national curriculum is usually thought to have all students in the same
grade studying the same topics by the same methods at the same time.  That is
clearly not the case in Japan.  The curriculum emphasizes important topics,
and does not spend time on peripheral topics.  The texts are short and the
teacher must supplement them with extra explanations and problems.  For this
to work, the teachers must know the subject well, have time to do this, and
will probably use supplementary books written to help them with this added
work.  It would be useful to have at least one of these books translated, so
we can see what help is provided for the teachers.

In contrast to the Japanese texts, our texts are too long.  The extra costs
involved in multicolor printing make it hard for publishers to publish books
that do not have a large projected market, so it is hard to try out new ideas
for books.  Our current texts have far too many worked examples which our
students use as crutches, solving problems by changing numbers in an example
which is essentially the same as the assigned problem.  This can not be done
in the Japanese books, since there are many fewer worked examples.  Students
want to have worked examples and answers in the back of the book, but then
children want more candy than is good for them and responsible parents see
that these wishes are not granted too often.  We, as responsible teachers,
should have the good sense not to give in to such requests too often.  As I
regularly tell my students, in real life there are no answers in the back of
the book so they will have to learn to think of ways to check if their
answers are correct, or if the problems are very complicated, whether the
answers are reasonable.

Some of our industrial firms are learning a lesson from Japan, which the
Japanese seemed to have learned after the Second World War from Edward
Demming - get it right the first time.  This is also their philosophy in
education.  Learn something well the first time so that you do not have to
redo it time after time in later grades.  Our philosophy has been different.
We cycle through topics, expecting a small percent of the students to learn
it the first time, and others to be exposed to it so that when we teach it
again, supposedly at a higher level, the best students will learn it at the
higher level, the others will finally learn it at the lower level and be
exposed to it at a higher level, so that ... .  The message I am learning
from our current students is that our method is not working as well now as it
used to work. It is time to try something else for now we know different
methods can work when there is a national will.  Without a national
curriculum we will not be able to suppose that students know any specific
idea or fact, and so will continue to have to teach with too much review.  As
long as students know material will be reviewed time and again, too many of
them will say that it does not matter if they learn it, for the teacher never
seems to assume they know it.  This is a depressing note to end a review on,
but I am not depressed by the international comparisons which show that other
students learn much more than our students do.  We have many very serious
problems in the world, and the fact that many more students can be educated
to a high level to help find solutions to these problems is reassuring.  I
hope for the sake of our students that we start to provide them a better
education, for the world is much more interesting when you understand more
about it.  That should be the real goal of education.

I have discussed educational problems with a colleague, Philip Miles, for
many years.  Among other responsibilities, he supervises our placement
program, and so has tracked the changes in our new students for a number of
years.  His comments in general, and specifically on an earlier draft of this
review, are greatly appreciated.

Richard Askey University of Wisconsin - Madison

[1] J.P. Becker, Mathematics Teaching in Japanese Elementary and Secondary
Schools - A Report of the ICTM Japan Mathematics Delegation (1988), 1989.
Available from the author at C&I Dept., Southern Illinois Univ., Carbondale,
IL $4

[2] J. Bronowski, The Ascent of Man, Little Brown, Boston, 1973.

[3] Curriculum and Evaluation Standards for Mathematics, Working Groups of
the Comm.  on Standards for School Math. of the NCTM, NCTM, Reston, VA, 1989.

[4] B. Duke, The Japanese School: Lessons for Industrial America.  Prager,
New York, 1986.

[5] W.J. Jacobson, S. Takemura, and others, Science education in Japan, in
[7], 135-172.

[6] R. Leestma, et al, Japanese Education Today, U.S. Government Printing
Office, Washington, D.C., 1987.

[7] R. Leestma and H.J. Walberg, Japanese Educational Productivity, Michigan
Papers in Japanese Studies, No.22, Center for Japanese Studies, The Univ. of
Michigan, Ann Arbor, 1992.

[8] The Polished Stones, 35 minute video available from National Council of
Teachers of Mathematics, Reston, VA.

[9] T.P. Rohlen, Japan's High Schools, University of California Press,
Berkeley, 1983.

[10] H.W. Stevenson and K. Bartsch, An analysis of Japanese and American
textbooks in mathematics, in [7], 103-133.

[11] H.W. Stevenson et al, Making the Grade in Mathematics, National Council
of Teachers of Mathematics, Reston, VA, 1990.

[12] H.W. Stevenson and J.W. Stigler, The Learning Gap, Summit Books, New
York, 1992.

[13] J.W. Stigler, Shin-Ying Lee and H.W. Stevenson, Mathematical Knowledge,
National Council of Teachers of Mathematics, Reston, VA, 1990.
-- 
===============================================================================
Steve (really "D. E.") Stevenson           steve@hubcap.clemson.edu
Department of Computer Science,            (803)656-5880.mabell
Clemson University, Clemson, SC 29634-1906