Педагогика высшей школы

63

Introduction to University Calculus for “Bunkei’’ Students
in Waseda
Takuya Sobukawa – PhD, Prof. of Global Education Center, e-mail: sobu@waseda.jp
Satoru Takagi – PhD, Assoc. Prof. of Global Education Center, e-mail: satoru@waseda.jp
Waseda University, Tokyo, Japan
Address: Nishi-Waseda 1-6-1, Shinjuku, Tokyo, 169-8050, Japan
Hiroaki Uesu – PhD, Assist. Prof. of Math. and Science Education Research Center, e-mail: uesu@
neptune.kanazawa-it.ac.jp
Kanazawa Institute of Technology, Kanazawa, Japan
Address: 7-1 Ohgigaoka Nonoichi, Ishikawa, 921-8501, Japan
Abstract. In these days, mathematics is one of the most important tools for any person. However,in Japan, many students hate and avoid mathematics in their high school age. In Waseda University, one of the biggest and oldest private universities in Japan, special curriculum is constructed
for such “Bunkei’’ students who major in social or human science.
Keywords: math education, economy, commerce, “Bunkei” students, high school math, higher
school math, curriculum for «Bunkei», social science, Humanities, “Rikei”, natural science
Cite as: Sobukawa, T., Takagi, S., Uesu, H. (2019). Introduction to University Calculus for“Bunkei’’ Students in Waseda. Vysshee obrazovanie v Rossii = Higher Education in Russia. Vol. 28.
No. 2, pp. 63-68.
DOI: https://doi.org/10.31992/0869-3617-2019-28-2-63-68
Introduction
First, we shall discuss about background andour aim in this paper.
“Bunkei” and “Rikei”. In Japan, in manycases, each senior high school is divided into 2
courses, “Bunkei” and “Rikei” for the preparation of the entrance examination of universities. In “Bunkei” course, they study social andhuman sciences, mainly. In “Rikei”, they study
natural science, mainly. At the end of first gradeof senior high school (=tenth grade), they haveto choose one of them. Teachers recommend“Rikei” to excellent students, not after their interest but the score of mathematics. So, almost
all “Bunkei” students hate or avoid mathematics. Yes, they do not need math for their entrance examination at university. Hence, at theentrance of university, many “Bunkei” students
have very few (or sometimes no) knowledge ofmathematics.
New plan in math education as liberal arts
in 21st century. Waseda university, one of theoldest, biggest and strongest private universities

in Japan, has constructed a new plan on liberalarts1. It has five divisions – academic writing,English, information (and communication)technology, statistics and mathematics. In ourmath division, we teach in four parts: 1) calculus,2) linear algebra, 3) mathematical structures, 4)mathematics, logic and communication.
Such “Bunkei” students, 80 percent of Waseda’s students, need to learn high school mathagain, but we decided that we DO NOT give“high school math” itself. It should be some kindof “university math”.
In 2007, the second author S. Takagi started
to plan the course, especially, parts 1 and 2.
He named them as “Introduction to University Math” and planned that we teach not only
high school math but also some application toeconomy or commerce. So, in part 1, we teachcalculating interest and its preparation in highschool/university math, and linear algebra and1
In Waseda university, that plan is named “Waseda
Next 125” after the 125th anniversary year in 2007 [1].

64Высшее образование в России • № 2, 2019linear programming, in part 2. Later, the third
author H. Uesu reinforced such application inpart 1 after his knowledge of math education.
The first author, T. Sobukawa has constructedpart 4 in 2014 and other version of part 1 in
2017. Part 3 is just in preparation.
In this paper, we shall introduce the historyof part 1, as a new suggestion on math educationin university.
Main part: On calculus – Basic policy
They say math is an important tool for“Bunkei” studies. So, classes of basic mathematics, calculus and linear algebra in many cases,
have been organized at each university. However, in general mathematics classes, emphasis isoften placed on solving mathematical problems
without the point of contact with the real world.
Hence, the motivation for learning mathematics
is reduced for “Bunkei” students who wish to use
mathematics as a tool.
Therefore, we decide the goals of this course
are the following: 1) simple and compound interest, 2) install deposit, 3) repayment of loan, 4)
mini/max problems of 2 variable functions, usedin mathematical economy.
In high school in Japan, they learn the following topics in calculus and related fields:
1) linear and quadratic function, 2) exponential functions, 3) logarithmic functions, 4)
trigonometric functions, 5) other functions, 6)
sequence and series (arithmetic, geometric etc.),7) recursive relation in sequence, 8) limit, 9) derivative/differentiation, 10) integration.
We pick up useful topics of them to study ourgoal. So, we have constructed 3 courses.
Version 1.0, Takagi’s plan (2008–2014)
A basic plan of the course is “prepare first,
then apply”.
Course A. Takagi created the elementarylevel “Course A” for “Bunkei” students who especially hate or avoid mathematics until now. In
2008, as the first version 1.0, he put the following
units of this course:
1) concept of numbers and sets,2) exponential and logarithmic functions,

3) sequence and series,4) limit,5) binomial theorem,
6) Napier’s number and proof of existence
the number,7) concepts of interest, simple and compound interest, continuous compound interest,
calculation of simple and compound interest.
The flow of learning that deals math and economics separately is as usual.
He also wrote a textbook dedicated to theselearning contents [2], and made video lectures
based on it. The main feature of this textbook
is that the notations of University Math are
explained as the first section for elementary“Bunkei” students. After the first quarter, most
students felt hard to study especially the proofof existence of Napier’s number. Therefore, herevised the course units each year. Finally, hearranged this course as version 1.4 with the following units:
1) exponential functions,2) logarithmic functions,
3) sequence and series,4) binomial theorem,5) limit and Napier’s number,6) concepts of interest, simple and compound interest, continuous compound interest,7) calculation of simple and compound interest.
This revised textbook [3] contains the appendix that deals with not only economics but
other applications, pyramid scheme, magnitudeof earthquake, hydrogen-ion exponent (pH),equilibrium state of heat etc. Recurring decimal, the ε-N method of limit, the continuity ofreal line are also discussed in appendix. Anotherfeature of this revised textbook is used squareroot, exponential and logarithm tables. These
would support that students understand mathdeeply.
Course B. In 2009, Takagi set up a new intermediate level “Course B” for those who took
the elementary level “Course A”. This “Course
B” has the following units:
1) concepts of interest, revisited,
2) useful functions (revisited),Педагогика высшей школы
3) limit and continuity of functions4) recursive definition of arithmetic and geometric sequences,5) recursive definition by affine transform,6) equal principal/total payment,7) annual rate and real annual rate.
One of the features of this course is that theconcepts of interest are explained in detail that
depends upon the Japanese language. He wrote
a textbook dedicated to this course [4]. In thistextbook, recursive definition between 3 terms
is not treated in the main course. Fibonacci sequence is considered in appendix, only. As to thiscourse, there are few revisions of contents [5].
For the “Bunkei” students with some mathematics skills, Takagi also put a pretest which canjudge which course should be taken. Even if the“Bunkei” students do not take the elementarylevel “Course A”, they can take the intermediate level “Course B” without taking “Course A”
if they pass the pretest. The pretest consists ofthe following five questions of five alternatives2:
1. Which one is correctly calculated( 2) ?
1236

2. When expressing 1,1x = 2 using logarithm,
which one is correctly represented?
1 n
3. What is the limit value lim 1 +
?
n
n →∞

( )

4. When a person who always walks at
4 km/h is walking y km for x hours without rest,
which one does correctly represent the relationship between x and y?
5. Which is the correct limit as x → 0 of thefunction f(x) = –1(x < 0), 0(x = 0), 1(x >0)?
Course C. As the advanced class, for introduction to calculus, the followings were given3:
1) functions,2) definition of derivative,3) derivative of polynomial functions,4) (local) maximum and minimum, convexity and concavity,
2

Here, we omit the choice of five alternatives.
This course was given by Dr. R. Shinjo, Associate
Professor of Kokushikan University, former AssistantProfessor of Waseda University.
3655) partial differential of 2 variable functions,6) extreme problems of 2 variable functions,7) utility function method.
There three courses have a big feature that
trigonometric functions and integrations are
NOT treated.
Minor version up has been given every year.
Finally, we got version 1.4.
Version 2.0, Uesu’s plan (2014–)
The plan of this version is “studying mathamong economy”.
H. Uesu arranged Takagi’s version (ver.1.4)
and rewrote the textbook of calculus [6–8]. The
big difference between ver.1.4 and ver. 2.0 is theintroduction of each learning content. Takagi
wrote textbooks (like ordinary mathematicaltextbooks) in the order of “definition, theorem,
proving”. Uesu had considered that such textbook style would be very tough for mathematics learning for “Bunkei” students who are notused to studying mathematics. Therefore, hewrote mathematical contents after introducing
economic examples so that the mathematicalmeaning can be understood smoothly. In otherwords, he wrote examples of economics and themathematics used there, rather than suddenly
introducing mathematical concepts.
In addition, he composed Course A and
Course B as a series of courses. As with ver.1.4,
Course A is an elementary level and Course B
is an intermediate level, but Uesu wrote a textbook with the aim of learning “Napier’s number” through Courses A and B. For that reason,
he advised “Bunkei” students to take Courses A
and B in order.
Course C is for students who wish to learn
mathematics more. The difference from ver.1.4
is that the first concrete examples are placed in
the middle of the course. As with Courses A andB, he wrote students to make understand basic
mathematical concepts through concrete examples.
The contents of each course are as follows.
Course A. Course A’s subject goal is “to acquire important basic knowledge of calculus”.
We explained the fundamental of the sequence,

66Высшее образование в России • № 2, 2019

exponential function, and logarithmic function,
by taking an example of the compound interest
calculation. It consists of the following 7 lectures:
1) simple interest and sequence;2) simple interest and arithmetic series.
Through these 2 lectures, students learn about
the basis of a sequence (general term of sequence, sum of sequence, etc.) and arithmetic
progression;3) compound interest, then geometric progression and series;4) annual/half year compound interest, thenexponential function;5) quarterly or shorter compound interest
and comparison of simple/compound interest. As an example of compound interest in thelength of the period, students learn about themathematical characteristics of exponentialfunctions and linear functions;6) compound interest and logarithmic function. Students learn about logarithmic function
after an example of number of years reachingtarget amount in compound interest;7) loan repayment, then recursive definitionof sequence.
Course A includes important items of basic mathematics. Students can acquire usefulknowledge in real life such as simple/compoundinterest calculation and loan calculation.
Course B. Course B’s subject goal is “to understand Napier’s number”. We explained therecurrence relations of the sequence, the limit,
and Napier’s number, by taking an example ofthe loan calculation. The following lectures areincluded:1) simple and compound interest (revisited);2) the system of repayment of loan, equal
principal/equal total repayment, add on repayment;3) equal total repayment and recursive definition of sequence, real annual rate;4) monthly equal total repayment. In these
two lectures, student also learns how to solve
the recurrence relations of a sequence;5) limit of sequence to consider continuouscompound interest;6) binomial theorem, as an example of total
amount in compound interest calculation;7) Napier’s number and continuous compound interest.
We explain about the knowledge for advanced students such as loan calculation andloan repayment.
Course C. Course C’s subject goal is “methodof Lagrange multiplier”. We explained the differentiation of one variable and the differentiation of two variables, by taking an example ofthe utility maximization problem. The course
includes the following lectures:
1) definition of derivative, exponential/logarithmic functions;
2) formulae of derivative and mean value
theorem;3) higher derivative and graph of functions;4) application to economy I, utility function
and budget constraint and maximization problem;5) function of 2 variables, definition, continuity and partial derivative;6) implicit function theorem and conditional
extremal problems of 2 variable functions, then,the method of Lagrange multiplier;7) utility limit, indifference curve and utility maximization, an example of multivariablecalculus.
We composed Course C as ver.1.4 redesign.
The goal of the subject is to learn the representative solution of the conditional maximization
problem “method of Lagrange multiplier”. As
an example, we explain the utility maximization
problem, etc.
We consider that there are two significancesof learning these courses as follows: 1) mathematical economy is studied deeply; 2) every
mathematical tool is explained among economy. So, this course is, so to say, good for the“Rikei” students, because they have no chance
to study such topics.
About the former, mathematics is necessary
to model the problem and to clarify its mechanism and to suggest a solution. From thesecourses, students are able to learn it with reasonable pace.

67Педагогика высшей школы
About the latter, in general “Rikei” mathematics classes, emphasis is often placed on
solving mathematical problems without
much emphasis on the point of contact withthe real world. Therefore, it is very meaningful for students to learn the application ofmathematics.
Version 1.5, Sobukawa’s planin English (2017–)
After the policy of Japanese government,
Waseda is preparing many lectures in English.
Some students, educated in foreign countries,
need such math courses. They have each mathhistory. Therefore, it is difficult to constructin the method of version 2.0, depending on the
education in Japan or Japanese language.
Therefore, T. Sobukawa has constructed thecourse NOT after Uesu’s idea but Takagi’s.
Course A. This course consists of:1) exponential functions,2) logarithmic functions,
3) sequence and series,4) binomial theorem,5) limit and Napier’s number,6) simple and compound interest,7) install deposit.
This course has several features. Before theversion 1.4, notation of university math was explained [3], because the student who has studiedin Japan does not use it. However, it is closely
related to the expression in English. This version is given for the students studied abroad.
So, we omit it. Moreover, many mathematicalconcepts are treated slightly, e.g. the definitionof limit. We need it for pure math but not in
economy. On the other hand, we add the theoryof install deposit in lecture 7 to use the theoryof arithmetic/geometric series. In the version
before 1.4, we cannot find the reason why they
study them. We DO NOT treat the definitionof limit of sequence/function rigidly than otherversions.
Course B. This course consists of the following lectures:
1) useful tools, exponential/logarithmic
functions, sequence and series, revisited;

2) install deposit, revisited;3) linear and quadratic approximation ofdata;4) sequence, recursive definition by affine
transform;5) recursive definition among 3 terms;6) equal principal payment;7) total equal payment.
This course has several features: in the lecture 3, we treat Lagrange (linear and quadratic) interpolation (approximation). It is justthe preparation to correlation coefficient on
statistics. Recursive definition among 3 termsof sequence has treated in the lecture 5. It is apreparation to numerical analysis.
Sobukawa claims these courses have to givesuch preparation for other studies.
Conclusion
We have offered three courses on math education for “Bunkei” students at university. They
are also useful for “Rikei” students for theirstudy.
References
1. Waseda Next 125, Statement of the executive
board of Waseda University. 2007. Available at:
http://www.waseda.jp/keiei/next125/common/
pdf/vision.pdf (In Japanese)
2. Takagi, S. (2008). Suugaku kiso plus (calculatinginterest), Waseda University Press, 101 p., ISBN
978-4-657-08902-1. (In Japanese)
3. Takagi, S. (2013). Suugaku kiso plus α (calculating interest), Waseda University Press, 173 p.,
ISBN 978-4-657-13008-2. (In Japanese)
4. Takagi, S. (2009). Suugaku kiso plus β (calculating interest), Waseda University Press, 244 p.,
ISBN 978-4-657-09909-9. (In Japanese)
5. Takagi, S. (2013). Suugaku kiso plus β (calculating interest), (Japanese), Waseda University
Press, 242 p., ISBN 978-4-657-13010-5. (In Japanese)
6. Uesu, H. (2014). Suugaku kiso plus γ (calculus)
2014, Waseda University Press, 93 p., ISBN 9784-657-14003-6. (In Japanese)
7. Uesu, H. and Takagi, S. (2015). Suugaku kiso plus
α (calculating interest) 2015, Waseda University
Press, 185 p., ISBN 978-4-657-15004-2. (In Japanese)

68Высшее образование в России • № 2, 20198. Uesu, H. and Takagi, S. (2015). Suugaku kiso plus
β (calculating interest) 2015, Waseda University
Press, 109 p., ISBN 978-4-657-15006-6. (In Japanese)

Acknowledgement. The paper is prepared on
the basis of the report at the 5th International
Scientific Conference “Functional spaces. Dif-

ferential operators. Problems of mathematical education” (Moscow, People’s Friendship
University of Russia, 26-29 November 2018).
The authors express their gratitude to the colleagues for the discussion of the manuscript.
The paper was submitted 09.01.19
Accepted for publication 13.01.19

Введение в курс математики для гуманитариев в Университете Васэда
Такуя Собукава – PhD, проф., Глобальный образовательный центр. E-mail: sobu@waseda.jpУниверситет Васэда, г. Токио, Япония
Адрес: Nishi-Waseda 1-6-1, Shinjuku, Tokyo, 169-8050, JapanСатори Такаги – PhD, доцент, Глобальный образовательный центр. E-mail: satoru@waseda.jpИнститут технологий Канагавы, Канагава, Япония
Адрес: 7-1 Ohgigaoka Nonoichi, Ishikawa, 921-8501, JapanХироаки Уесу – PhD, доцент, Исследовательский центр математических и естественных
наук. E-mail: uesu@neptune.kanazawa-it.ac.jpИнститут технологий Канагавы, Канагава, Япония
Адрес: 7-1 Ohgigaoka Nonoichi, Ishikawa, 921-8501, JapanАннотация. В наши дни математика является важным инструментом для любого человека. Между тем в Японии многие студенты ненавидят математику и избегают ею заниматься со времен обучения в средней школе. В Университете Васэда, одном из старейшихи крупнейших частных высших учебных заведений Японии, для студентов-гуманитариев
(«Bunkei»), специализирующихся в социальных и экономических науках, разработано триспециальных практико-ориентированных курса по математике. В статье представлено содержание каждого курса.
Ключевые слова: преподавание математики, курс математики для студентов-гуманитариев, Bunkei, Университет ВасэдаДля цитирования: Sobukawa, T., Takagi, S., Uesu, H. Introduction to University Calculus for“Bunkei’’ Students in Waseda // Высшее образование в России. 2019. Т. 28. № 2. С. 63-68.
DOI: https://doi.org/10.31992/0869-3617-2019-28-2-63-68
Благодарности. Статья подготовлена на основе доклада, сделанного на 5-й Международной конференции «Функциональные пространства. Дифференциальные операторы. Проблемы математического образования» (Российский университет дружбы народов, Москва, 26-29
ноября 2018 г.). Авторы выражают признательность коллегам за конструктивное обсуждение идей доклада.
Статья поступила в редакцию 09.01.19
Принята к публикации 13.01.19