MATHEMATICS INSTRUCTION AND LEARNING IN SCHOOLS
AND UNIVERSITIES
RESEARCH PLAN of the two mathematics research and development groups
Didactical Mathematics in School and University
coordinated by Prof. Ilpo Laine
Constructing Mathematics in Computer Environment
coordinated by Dr. Sisko Repo
Project background, Local,
national and international cooperation
The programme above as well as the present project is based on the general
idea of improving the teaching of mathematics, resp. physics, on the basis
of didactically oriented research in the respective field with starting
point in the scientific structure and contents of mathematics, resp. physics.
This kind of research already has strong international traditions. On the
contrary, such a tradition in Finland is relatively short up to now. Due
to different cultures in instruction and learning, as well as different
curricula, national research is needed before transferring international
results to local school and university environments. Didactically oriented
mathematics, resp. physics research is interdisciplinary by definition,
needing connections to history and philosophy of science, pedagogy and
psychology, in addition to the field itself. In the present project, the
necessary infrastructure for such a multidisciplinary approach can be based
on the existing experience in the above Teacher Education Programme which
started in 1992.
The project is strongly bound with Mathematics and Physics Teacher Education.
In this decade, the subject teacher education curriculum has moved from
sequential (pedagogical learning and training after completed subject studies)
to simultaneous (pedagogical part starts in the second year, continuing
up to the end of studies). At the University of Joensuu, this change has
resulted in significant changes in both Faculties (Science and Education),
aiming to better coordination of subject studies and pedagogical ones.
These changes have improved the freedom of students to apply different
approaches to teaching, encouraging them to penetrate into the complex
relations of their base science and education to school practises. Therefore,
a substantial part of the present project aims to concentrate on university
level instruction and subject teacher education in particular. However,
to obtain more necessary depth, and making use of the universal nature
of mathematics and physics, some parts of the project are directed towards
primary school teacher education, polytechnics education and school teaching.
University Practice School at Joensuu: Jari Rokkila
The work of this team is based on the observation that beginning undergraduate
students frequently have an oversimplified or even false images of fundamental
mathematical notions, preventing them to learn understanding the true contents
of them as well as the cumulative nature of mathematics. Therefore, the
team plans to work with the following topics: (1) To improve the mathematical
literacy of students, by locating the sources of difficulties and misunderstandings,
and by developing instruction methods to overcome these problems. Through
this approach, the knowledge and understanding of the students on mathematics
as a part of the human culture will be improved. (2) To develop mathematically
oriented computer environments at different levels of instruction. More
precisely, the plans focus on an (abstract) algebra course for mathematics
students, a modelling based differential equations course for physics and
chemistry students, a discrete mathematics course for primary school teacher
students and a development project for geometry teaching at the training
school. (3) To cooperate with the Savonlinna team, see below, in the use
of a laptop computer based mathematics instruction at the university level.
In order to achieve best possible profits from the synchronization of
the subject and pedagogical studies, it is important to carefully observe
and encourage the interaction between them. The pedagogical studies provide
the becoming teachers with a diversity of general teaching and learning
methods, and a natural and authentic framework for reinforcing this can
be offered in their own subject studies. This has a vital influence on
the education of teachers and how they learn, and whence, how they teach
mathematics in the schools. Therefore, central parts of the subject con-tent
are taught in ways that support the mental maturation of students to socially
and pedagogically solid individuals. In a concrete level, we already have
a working system of guided excercise classes in calculus, and activating
courses directed both to students and inservice teachers.
The specific objects of the group are:
1. To improve the mathematical literacy of the students. We investigate
the reasons and forms of the difficulties to receive and understand exact
mathematical information, to communicate with formal and informal mathematical
language and to express one 2.
To develop mathematically oriented computer environments. There is a growing
educational literature indicating that difficulties in understanding abstract
mathematical objects can be overcome by making them concrete and lively
in a mathematically designed computer based learning environment. This
can mean, besides computer graphics animations, interactive study of mathematical
concepts and processes through programming activities. The environment
heavily makes use of the social interaction between students working in
small cooperative groups, in mutual responsibility of the learning. We
investigate the advantages and other con-sequences of this approach, especially
the quality of learning, the effects to mathematical literacy and attitudes
towards learning and teaching of mathematics.
3. To cooperate with the Savonlinna team on the laptop experiment. We
shall follow closely the work and results of the Repo research group described
below. We have plans to implement a similar portable lab to be used in
testing the com-puter implementations produces in this research project,
and in normal teaching as well.
Professor Ilpo Laine has the financial responsibility for the group.
He maintains general international relations, including the European intensive
course scheme.
Professor Tuomas Sorvali is in charge of the overall functioning of
the group.
Lic. Ph. Liisa Kinnunen does research among the first year university
mathematics students about the relationship between understanding the definitions
of mathematical concepts and the formal mathematical language. (1)
Ph.D. Martti E. Pesonen is developing (together with the physics group)
an undergraduate level computer based learning package on physically meaningful
differential equations, parts of which can be used in science education
in high schools. He also conducts the computer based abstract algebra course
development described below. (2)
M.Sc Jari Rokkila is doing postgraduate research based on his teaching
experiments of lower secondary school geometry implemented by the of Gabri
Geometry II. (2)
Tanja Terho is starting the postgraduate studies concerning the role
of computer based cooperative environments as a stimulator for the transformation
of explicit computerprogrammed processes to mental animations and schemes.
(1-3)
The members of the group have worked actively in planning and developing
the subject teacher education reform programme mentioned in Section 1.
In addition, Tuomas Sorvali has created a popular specializing curriculum
for pre-service - and later for in-service - elementary school teachers.
He has also been very active in the progression of geometry to schools.
Liisa Kinnunen is a Ph.D. student in mathematics. She has been carefully
improving the contents of the basic mathematics courses for pre-service
subject teachers and she has also attended to the implementing of the specializing
curriculum mentioned above.
Martti Pesonen has been active in preparing lecture notes to several
mathematics courses, and in introducing computer aided instruction into
math department. He used the learning by progamming approach in a course
on discrete mathematics. Tanja Terho will be graduated around the turn
of the year. Her masters thesis, supervised by Pesonen, is the planning
and partial organizing of an abstract algebra course supported by computer
activities in a mathematically oriented programming environment called
ISETL (Interactive Set Language).
to Local, national and international cooperation
The theoretical model: The experimental teaching will be planned to
conform to the teaching cycle developed by the research group headed by
Ed Dubinsky: theoretical analysis, the teaching process, evaluation. The
theoretical analysis involves an explicit investigation into the mathematical
content and the stages of concept formation. On the basis of an evaluation
of the learning results (information is gathered in a network environment),
an analysis of the concept formation of the following lesson is carried
out and new learning tasks are planned. Also the didactical structure of
the instruction is cyclical in nature: experimental work, classroom work,
and practice.
How the instruction is carried out: The students work in groups in costructing
the mathematical theory. The groups propose - on the basis of experimental
work - their suggestions for axiomatic systems and definitions of mathematical
concept, they suggest terms and notations - and finally an agreement is
reached which follows the current mathematical usage. The groups formulate
mathematical theorems and attempt to prove them by means of logical rules
of inference. In the groups, the students practise writing, reading and
interpreting mathematical notations, and the links between different modes
of representation are pointed out. After the stage of mathematical theory
construction, the groups present their results, the instructor comments
on them, stimulates discussion, and augments them into a mathematical theory.
M.Sc. Pekka Kekoni, Lecturer of mathematics and computer science: Pursues
post-graduate studies in the University of Tampere. Development constructivist
learning materials for primary grades.
M.Ed. Tuula Häsä, Teacher at Savonlinna Teacher Training:
Experimentation utilizing Pekka Kekoni s materials in the primary school.
Research fellow (Ph.D. student level) + research secretary (joint) in
Savonlinna for the years 1998-2000 (the job will be filled on the basis
of applications): Research associated with the experiment, co-ordinating
activities, planning, organizing and marketing and implementing of in-service
training for teachers, charting, gathering, reporting of WWW-materials
for school mathematics, planning computer assisted instruction.
Nationally, the teams participate in the activities of the Graduate
School for Teachers in Mathematics, Physics and Chemistry, coordinated
by the University of Helsinki. Presently, this frame covers in Joensuu
3 Ph.D. students in Mathematics and 2 in Physics as well as 3 Phil.Lic.
students in Mathematics and 8 in Physics, including postgraduate students
directly participating this research plan. Internationally, the existing
essential contacts to U.S.A., Russia (Carelia) and Germany will continue.
In particular, we plan to organize an intensive course in CAI in mathematics
instruction by Prof. Dubinsky (Georgia State University) at the early phase
of the project. Finally, the Department of Mathematics has been involved
with a plan to organize European intensive courses in mathematics teacher
education, with emphasis on mathematics applications and CAI in mathematics,
with participating institutions at least from Greece, Portugal, England
and Germany.
Members of the mathematics groups have attended several national and
international conferences on mathematics education, including the latest
meeting of the In-ternational Group for the Psychology of Mathematics Education
(PME 21 in Lahti 1997). There we met with many researchers, whose articles
had raised our interest in advance, and we became convinced that very fruitful
cooperation can be reached in that direction. So we feel that the PME group
is the central society for those who are seriously worried about the state
of learning mathematics.
To Mathematics Department Home Page
To the main page of Didactical Mathematics
Link to other related Projects
Project background
The Mathematics and Physics Teacher Education Programme at the University
of Joensuu has been one of the national Centres of Excellence in University
Teaching in 1995 and 1996. The present project intends to continue that
programme which was based in good cooperation between the Departments of
Mathematics and Physics, the subject teacher education at the Faculty of
Education and the Training Schools of the University.
Didactical Mathematics in School and University coordinated by Prof. Ilpo
Laine
Department of Mathematics: Ilpo Laine, Tuomas Sorvali, Liisa Kinnunen,
Martti Pesonen, Tanja Terho
The objectives and aims of the project
The general goal of this group is to find and implement successful ways
to improve the teaching of mathematics, in order to heighten the abilities
and achievements of the students in width and depth of knowledge, and to
provide the becom-ing mathematics teachers with a variety of successful
teaching methods.
The group members, their background and contributions
The members have the following roles in the project:
Constructing Mathematics in Computer Environment coordinated by Dr. Sisko
Repo
Savonlinna Department of Teacher Education: Sisko Repo, Pekka
Kekoni
Savonlinna Teacher Training School: Tuula Häsä
This team works at the Savonlinna Department of Teacher Education, focusing
at instruction of subject studies in mathematics and mathematics studies
in the primary school teacher education.The idea is to test the applicability
of the constructi-vistic learning method for university instruction in
mathematics. As a technical de-vice, a networked class of 16 laptop computers
will be applied, by organizing all mathematics instruction in question
in this class. In this framework, the following issues will be observed:
(1) Use of the laptop computers, except for the actual learning process,
also for evaluation of the process and for collecting the research data.
(2) Use of internet and hypermedia in mathematics instruction. (3) To introduce
a portfolio-based learning environment for each course in question. (4)
To cooperate with the Joensuu mathematics team, see above, in improving
the stu-dentshtml> understanding of mathematical culture.
Backround and significance
Starting in the autumn of 1997 mathematics teaching in Savonlinna department
of teacher education has followed a syllabus compatible with a constructivist
view of learning. On account of the small number of students, the Savonlinna
department has excellent possibilities for experimentation with and investigation
of innovative methods in mathematics instruction. For several years already,
mathematics courses have been taught on the basis of constructivist learning
theory. The ex-periments have been described in many mathematics education
seminars in Finland. Sisko Repo defended her dissertation about computerassisted
mathematics education in the constructivist environment last year. In the
Savonlinna Teacher Training School, constructivist teaching materials are
being tried out for the second year in a row.
Aims and methods
The aim of the project is to introduce new technology as an essential element
of mathematics teacher education and mathematics instruction. The new syllabus
will supply theoretical studies in education with a concrete content in
universitylevel mathematics teaching. In their practical training, the
teacher students teach mathe-matics based on the didactical model presented
in the university. In the experiment, all mathematics teaching is given
in the computer classroom. The computer is used in the experimental teaching
(symbolic manipulation programs), in evaluation, and in collecting of research
material. Hypermedia programmes dealing with school and university level
mathematics and mathematics teaching are utilized. In teaching, evaluation,
and research the WWW and the network environment is made use of. In the
autumn 1997 advanced mathematics classes for upper secondary high school
students will start in the computer classroom, and also ordinary lower
and upper secondary school mathematics lessons will be given there. Inservice
training in computer assisted mathematics teaching will be given for teachers
as well. The laptop computers are easily transferred from the classroom
to other places in which they are needed.
Constructing mathematics in the experiment.
The aims of the syllabus: 1) To give future teachers a picture of mathematics
as a science. 2) To acquaint students with the methods of constructing
the theory of mathematics. 3) To emphasize the significance of a social
group in the creation of knowledge. 4) To acquaint the students with mathematics
teaching compatible with aconstructivist view of learning. 5) To integrate
the use of the computer into a natural element of mathematics teaching.
Schedule of research associated with the experiment
The personnel involved in the experiment
Ph.D. Sisko Repo, Lecturer in mathematics: Development of teaching materials
for mathematics instruction in a constructivist learning environment, development
and evaluation of teaching methods, in service training of teachers, research
involving the opinion of mathematics teachers on mathematics and the development
of the concept of real number and the ability to perform mathematical proofs.
National and international co-operation
Sisko Repo has been active in an international working group dealing with
the de-velopment of symbolic calculation in mathematics teaching. The research
involved in the project is closely associated with the project of Erno
Lehtinen, and close co-operation with this group is envisioned. In the
project the hypermedia programme created by Simo Kivelä in the Helsinki
University of Technology will be tried out. The constructivist teaching
material for primary school is made on the basis of russian teaching materials.
Results
The project will furnish important research data for developing school
and university level mathematics instruction. In particular an attempt
will be made to increase the effectiveness of teaching methods, experimental
work and computer use in teacher education and consequently in school mathematics.
The project will feature planning and experimentation of constructivist
teaching materials for mathematics teaching and computerassisted mathematics
instruction at university and primary school levels. On the basis of the
research results and the experience gained from the experimentation, inservice
training for teachers will be provided as well as computer assisted mathematics
teaching for upper secondary school pupils (to be begun in 1998). The research
results will be utilizable in university level mathematics instruction
in general and particularity in teacher education. In this way innovative
methods will be used to a greater and greater extent also in school mathematics
instruction.
Local, national and international cooperation
Local cooperation between the three research teams is a continuation of
the existing tradition in the frame of the Mathematics and Physics Teacher
Education Programme. For practical implementations, a joint research secretary
is proposed. In addition, the physics team in particular proposes to keep
continuous contacts with the corresponding staff working with the chemistry
teacher education in Joensuu.