Strand 3:
Technology in teacher education
Jaime Carvalho e Silva
Coimbra, Portugal
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Plenary lecture: |
Teacher training: The role of technology |
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Practical aspects of CAS using sinusoidal functions |
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Investigating teachers’ perceptions on their preparation to use IT in classroom instruction |
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Using computers in mathematics teacher training programs: A reflection upon an experiment |
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A modern approach to limit processes |
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Internet as a tool in the preparation of future mathematics teachers |
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Changing the classroom practices — The use of technology in mathematics teaching |
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Integrated teaching mathematics with elements of computer science |
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Innovations in mathematics, science and technology teaching—IMST² — Initial outcome of a nation-wide initiative for upper secondary schools in Austria |
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Current educational theories and New Technologies: Development of a training program for mathematics teachers in the Philippines |
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Integrating ICT into the teaching and learning of discrete mathematics |
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What teachers can request from CAS-designers |
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Computers in school mathematics — A pilot course for school teachers of mathematics in Estonia |
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Computers in mathematics education — An ongoing experience |
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Using Derive for beginner courses of recursion theory |
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Internet materials in mathematics teaching |
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CASCADE-IMEI: Web site support for student teachers learning — Realistic mathematics education in Indonesia |
The Strand ”Technology in teacher education” included 17 presentations by specialists from 12 countries. There were two kinds of presentations: ones dealing directly with the preparation of teachers (pre-service or in-service) and others dealing with new aspects that showed up in mathematics teaching with the emergence of new technological tools.
Among the presentations dealing with the preparation of teachers, most deal with national projects in their own countries (Austria, Estonia, Indonesia, Philippines, Portugal and Poland). These presentations were very interesting, showing different methodological approaches to rather similar problems. All these projects show the difficulties of introducing technology in an effective way in day-by-day teaching.
It is not a trivial task to change the practice of mathematics teachers that do not use already technology in their teaching; technology is not just a new chapter that may be added some day or some year. Not only the topics to be taught may change (with more discrete mathematics and study of algorithms, for example), but also the use of technology forces changes in the methodologies of the classroom (much more experimentation and conjecturing, for example). The experiences of these six countries give good ideas of what can be done and how.
The papers dealing with new topics (or new aspects of old topics) discussed different subjects: what Computer Algebra Systems (CAS) may give to teachers beyond the present developments, how the teaching of Discrete Mathematics (Combinatorics, Recursion, etc.) may improve, how the mastering of CAS is important to physics students, how the study of limits may profit from new approaches, how the Internet may open new windows to teaching and learning.
All the presentations of this Strand were followed by discussions that were not more complete because there was a schedule to be satisfied. But these lively discussions showed that there is a wide interest for these topics. Certainly the preparation of new teachers and the in-service preparation of teachers to face the new challenges of technology and the needed improvement of mathematics teaching is an indispensable part of any educational system, and so we should all be concerned with it.
The first presentation was made by Jaime Carvalho e Silva describing an experience with future mathematics teachers using an Internet mailing list; other participants shared their experience with similar activities, and all agreed that they had an enormous potential.
Konrad Krainer presented the project with the same name as an Austrian town, Imst, launched when Austria did not obtain good results in the TIMSS achievement test. This project builds around some assumptions that include supporting teams of teachers, strengthening reflection and networking, making existent innovations visible, recognizing that innovations are continuous processes and that schools should maintain ownership of their innovations. The participants of this strand found the idea genuine and promising. Very much debated was the role of technology in this project, strategies of spreading the idea from some schools to the whole school system, strategies of linking theory and practice within the project, the importance of a good balance between autonomy and networking and the funding of the project.
Bogdan Zoltowski presented the point of view of two physicists that need a lot of mathematics in their teaching and that have difficulties with that. They consider that students do not know some basic mathematical ideas (like angle, amplitude and addition of sines) and cannot use CAS for their basic or tedious calculations. They gave very interesting examples of how CAS is useful in physics classes. There was a very heated discussion on this topic. Mathematicians argued that they need more time to teach CAS, a time they don’t have. Zoltowski argued that students are not using trigonometric identities anyway, so why not use CAS instead? Some people argued that much more cooperation will be needed between mathematicians and physicists and that mathematicians who teach engineers should be more aware of this issue. It was felt that much more debate would be needed on this topic (maybe a strand to discuss the pedagogical implications of CAS in a next ICTCM meeting).
Rein Prank and Eno Tonisson presented a very complete national program in Estonia with courses to prepare secondary school teachers to integrate computers in their teaching. The presenters felt that now teachers in Estonia are ready to organize other courses themselves. Some practical difficulties were analyzed like the big number of students (30) in each class that make it difficult to use the computer, and the existence of informatics taught as separate subject (and only sometimes informatics teaching connects in some way with mathematics teaching). They said that a needed next step would be the work with textbook authors.
Walther Neuper, a teacher in a technological high school and researcher in CAS, presented some views on what teachers might expect from future developments, stressing that the needs in education (like a tutorial, a debugger) conflict somehow with other interests (like in industry that needs speed above all) so educators should get more involved in the software development of CAS packages.
Zulkardi presented a project, supervised by Jan de Lange, to help student teachers in Indonesia to use Realistic Mathematics Education (RME) in their classes. The project uses a web site as a support. The present difficulties were discussed (like having 40-50 students per class and teachers being used to the rule/example/problem routine). It was discussed whether technology was really the best way to introduce RME (Indonesia is a very big country) and whether RME was a good idea (Indonesian students hate math, they feel math is not really useful).
Maria Zajac presented a methodology to organize technological resources on a web page so they could be more easily found and presented some plans to assess in the future the quality of the materials indexed on that web page.
Primo Brandi and Anna Salvadori presented a new approach to the teaching of limits using technology, which is a part of a more general project aimed at improving the teaching of calculus to engineering students. There was an interesting debate on this topic, namely on the student reaction. The presenters told that the best students in high schools do not react very well, but medium level students do. The time needed for such an approach was also debated; it was argued that more time is needed but it is not a waste of time because students acquire the concepts more easily.
Ana Rosendo and Jaime Carvalho e Silva presented how the teaching of student teachers is organized at the University of Coimbra in a course whose goal is to prepare students to integrate technology in their future teaching. The debate on this presentation focused on the previous experience of students (they only used computers to learn computer science that they don’t like very much), the difficulties of assessing a course like this one (oral examinations based on the projects they made) and the kind of support materials needed for such a course (no textbook is used).
Auxencia Limjap described the Summer Institute of Graduate Studies, a first step of a national teacher training program, and how technology was included in it. It was debated whether it would be better to use computers (more efficient machines) or graphing calculators (rather efficient but much less expensive and more easily replaced). Other issues debated included the integration of technology in mathematics courses, the low motivation of teachers and the small emphasis on mathematical modeling.
Luiz Carlos Guimarães presented the enormous task of preparing teachers in a country where there are 47 million students (from the 5th to the 12th grade) and an estimate of 150 thousand mathematics teachers. A course for the preparation of teachers was described and how good the reaction was, teachers wanting to learn more and asking for more advanced courses. Other projects were described, namely a teacher oriented master course and a dynamical geometry piece of software with an on-line tool.
Adnan Baki described an experience of integration of computer based mathematical activities in university teaching of student teachers, concluding that first hand experience is essential but not sufficient to ensure that the implementation of an innovation will be successful.
Eva Milková presented a university course in Discrete Mathematics for student teachers where students make software projects that are used to teach other students.
Nelson Urrego showed how the teaching of Recursion Theory can improve using a CAS tool like Derive, very simple to program. Somebody suggested that the same might be achieved with ISETL that is simple, free and handles formulas very well.
Isabel Fevereiro and Maria do Carmo Belchior explained how the Ministry of Education in Portugal organized a NET of 80 teachers that received special preparation to prepare all other teachers for a curriculum change that took place in 1997. Meetings were organized with teachers from all secondary schools to discuss ways of implementing these changes (and to convince teachers to use a more active methodology that includes, for example, mathematical modeling, use of graphing calculators and a gradual approach to formal proof). The presenters were asked lots of questions about national examinations (that include questions of a mathematical modeling type and requiring the use of graphing calculators), textbooks (that are more or less in agreement with the new orientations), how teachers are supported (with publications and a web site http://www.mat-no-sec.org ), and why such a change could be carried out (there is a strong mathematics teachers association, APM).
Henryk Kakol presented a new program for polish secondary schools, created by a team he coordinated, that aims at an integration of the study of Mathematics and Computer Science. A whole set of books and teaching aids have been created (including computer tests and little computer programs). There were lots of questions about this new approach that began in 14 schools and is now spreading. The fusion of Mathematics and Computer Science is a choice the schools can make; only mathematics teachers teach this course. The preparation of teachers is provided through the activities of the SNM, the polish association of teachers of mathematics, that organizes meetings and workshops for teachers.
Teacher training: The role of technology
Branca Silveira
Porto, Portugal
1. Teacher training: The role of technology
We can't have a change in our schools without teachers and teachers don't change if they are not convinced that change is going to improve something. The world is changing, society is changing, pupils are changing, and the schools? How are schools coping with this? Technology is everywhere. No discussion about that. Everyday we see new software, new computers, new calculators, etc. Are the schools ready for this? Does technology play a significant role in the change of the curriculum?
How do teachers face this? Are they prepared to use technology effectively? Which kind of difficulties do teachers face? Some teachers have been using technology, did they really changed their methodologies or are they using them in an inadequate environment? What about teacher training? Which kind of training is more effective? Initial training? In-service training? But, what should we do so that teachers include technology in their practice? More computers? More training? A different schedule for the classroom? Making the use of technology compulsory? In Portugal the use of graphic calculators is compulsory in secondary schools, so everybody has to use them. Should we do the same with computers and other technology? What about Internet? How should we train teachers for the use of Internet in the classroom? How can teachers develop the ability to analyse and integrate in an intelligent way, in their teaching, the future technological developments (software, hardware, communication...)?
Those are some of the questions we are going to discuss in this talk, based on the Portuguese experience, focusing my point of view as a teacher, as a teacher trainer and as a member of the board of directors of APM (the Portuguese "Association of Teachers of Mathematics").
1 Teacher training: The role of technology
We all know that technology is not the solution for all the problems we face in our classroom and all of us are aware, that teaching with technology is not going to make our job easier, on the contrary. But we know, as well, that we can't live without technology and should take the best out of it. It has been said that schools can't compete with society and they shouldn't even try to. But what should we do to make teachers use technology in their practice? More computers? More training? A different schedule for the classroom? Making the use of technology compulsory? And what about initial training?
Our high schools and universities that do initial training, most of them don't include computers in the curriculum. The students spend a few hours learning how to use a word processor, or a database, in some cases a spreadsheet, and so on, but teachers don't use technology in their classes, as a tool. Lots of researchers in education said that the new teacher tends to reproduce the teaching he got. So the future teachers leave high school without seeing how to include technology in their practice, without having experienced technology as students. In some cases they have lots of theory about the subject, lots of papers to read, but ... just talk.
Luckily in some schools this is changing
What about in-service training? In Portugal teachers do in-service training in training Centres. In order to be recognised their professional development they have to get a certain amount of credits taking courses in these training centres. When we analyse the programs of these Centres we saw that all of them perform courses in technologies. These sessions are usually those who have the maximum number of participants. Teachers enjoy being there, but they go back to their schools and don't make any use of what they have been learning. We saw, as well, that these courses are not set up for a particular subject area, but they are mainly set up to teach how to use a particular technological tool.
APM (Association of Teachers of Mathematics) has a training Centre as well, but what we try to do, is not to train teachers in technology just in technical aspects, but mainly focusing on the use of technology as a pedagogical tool and how to integrate it in the classroom. All our courses are centred in problems that teachers can use with their pupils and we try to put teachers in the same situation, as children will be.
In Portugal every basic or secondary school has computers and an ISDN Internet connection. The Ministry of Science and Technology is trying to connect all primary schools to the Internet till the end of 2001. Some of them, mainly secondary schools, have already a good ratio pupil/computer but teachers are scarcely using them. They use computers for their personal work, but not in the classroom.
It has been very difficult to change that. Good equipment and teacher training are essential if we want to change the way things are. Teachers keep complaining about the number of computers (in some schools this is a fact, in many schools there is only one computer connected to the Internet) and the need of training in ICT etc… From my readings and from my contacts with teachers from other countries I think that is a bit the same almost everywhere (even if in some countries there is no lack of computers). The main problem is that teachers are not confident in the use of ICT and they must have another attitude regarding teaching. Using computers makes them to cut with old habits and change their methodologies. For many teachers this is very hard to cope with. But we can't have a change in our schools without teachers and teachers don't change if they are not convinced that change is going to improve something.
The theme of this strand is "Technology in teacher education" and what I'm going to talk about is mainly what is going on in Portugal, focusing on the following items:
National projects
Initial training
In-service training
Curriculum reorganisation
And now?
2 National projects
The Minerva project
In Portugal some national projects have been set up in order to introduce ICT in schools. We had in the 80´s a national program aiming at the introduction of technologies in schools. It was called Minerva project and the Ministry of Education sponsored it. It was a big project and it had a huge impact in our schools and in our society. Some of our Universities and high schools, in different regions of Portugal, had a group of teachers just to train other teachers from the schools chosen to be in the project. These trainers were recruited among teachers, in schools, that had some experience with computers. I made part of that group working in an Institution in Porto.
It was an "open" project. We had some goals to achieve but each Institution had its own plan and different ways of working. In this one I was working with, after identifying the needs of our schools we ran small courses basically about word processor, databases, spreadsheets. We were in 198 ... .
Almost every school wanted to publish its own newspaper and we told them how to do it, not only the technical part but also running workshops and promoting seminars with professionals from some of our local newspapers. Every time we came across some software that could be used in the classroom we explored it and showed it to the teachers, always trying to make lesson plans that might give suggestions for the classroom. At that time it was almost the only way teachers could do their own training. They knew that if they wanted to try something new we were there to give the support they needed. The Minerva team was always available to help schools to carry on their projects keeping in mind that our main goal was the introduction of ICT in the classroom and in the school daily life.
The project ended in 1993 when the Minerva project ended teachers were left alone in schools with technology out of date, coping with technical problems and without any kind of support. (I must say that even now, schools don’t have a technician to take care of computers). The number of pupils who had computers at home increased and they began to complain about the old hardware and software the school had. Some teachers continue to use computers with their pupils but most of them simply refused to go on.
After the initial boom and all the enthusiasm they experienced, they realised that it was very hard to use computers in classroom without any kind of support.
The Nónio project
Three years after, the Ministry of Education set up a new national project, called Nónio Sec.XXI that is still going on. The aim of this program is "to find diversified answers which are appropriate to the new scientific and technological phase of evolution we are crossing, with the objective of creating an "informed school" which is open to the world".
The philosophy of this project is completely different from the Minerva Project. First of all some institutions (universities, high schools and training Centres) were recognised as Competence Centres. After that, the schools who were interested had to submit a project, monitored by a Competence Centre. Some of these projects got financial support to buy equipment and to carry on their projects. In Portugal schools are not very used to be involved in projects (now, this is changing a bit. They have to, because it is the only way to get more money!) So, schools that had some experience in setting up a project were the schools that managed to get financial support.
There are big differences between Minerva and Nónio projects
In Minerva, schools were
chosen by the Centres, to integrate the project; in Nónio, schools must set up a project and apply for funding.
In Minerva, schools were
"divided" by the Centres and were "connected" with a
Centre, based on the region; in Nónio,
schools can choose the Competence Centre they want, even if that particular
Centre is far away (not very common but it happened). The objective was to be
accompanied by a Competence Centre whose project was closer to their project
and their needs.
In Minerva, schools didn't
choose the training that their teachers needed. In those days we were starting
to use computers and most of them didn't even know what they wanted; in Nónio the Competence Centres do the
training schools need, and ask for, to carry on their projects.
In Minerva, schools didn't
choose the equipment they got: in Nónio they have to choose and buy all the
equipment and make their own contracts with the suppliers.
In Minerva, schools didn't
pay for training and for support; in Nónio
schools have to establish some financial contracts with the Competence Centres
(maybe this particular aspect restricted the development of the project).
Those were the two big national projects we had.
3 Initial training
In 1998 APM published a report (Matemática 2001) based on a research made by a working group. It was a national research focusing on the teaching of Mathematics and the teacher's attitudes, performances, professional development, needs,… . In this report it was referred that the use of technology was the main area in which teachers declared to need more training. Also, in a work carried on by Ponte and Serrazina (1998), about the initial training courses, it is said that the abilities and knowledge in the use of ICT of the future teachers were not in a satisfactory level.
Abilities required of a teacher in ICT
Ponte and Serrazina (1998) identified the main abilities a teacher must have in this particular area:
Being aware of ethical and social implications
of ICT
Being able to use general software
Being able to use and evaluate educational
software
Being able to integrate ICT in the
teaching/learning process
More than being a curricular area, ICT plays an important role in the teaching/learning process and all teachers must be able to work with them. They also refer that the success of integrating ICT in schools depends, first of all, on teacher training. So, Institutions who provide initial training have a huge responsibility to train future teachers in order to develop in them an open mind to changes, the pleasure of long-life learning and the desire to be aware of pedagogical innovations. The initial training must promote in future teachers a good relationship with ICT, making them able to use them in the classroom.
Recommendations to promote ICT for teachers
Some recommendations of Ponte and Serrazina (1998):
1- Institutional training projects
The initial training Institutions must analyse the way ICT can be used in initial training (specific subjects, educational subjects, general aspects…), reviewing the ICT role in their curricula, promoting interdisciplinary work, keeping permanently up to date, and creating links between them.
2- Knowledge and abilities
Some subjects like word processor, e-mail, Internet, databases, electronic presentations, must be seen as indispensable tools for the new teacher. So, the Institutions must find the most suitable moments and subjects so the students can learn how to use them and how to make educational use of them.
3- Resources (material and human)
Institutions must have updated equipment, software and human resources. They need:
To analyse the existent specific software for
each area
To improve the quality of the equipment and
software
To increase the number of human resources
qualified on ICT use
4- Accreditation and evaluation
The accreditation of initial training courses must pay careful attention to the ICT role in the curricula. In Portugal, an Institution for the accreditation of initial training courses has been created recently. It is called INAFOP (Instituto Nacional de Acreditação da Formação de Professores) but its action is still in the initial set-up steps. They already produced documents concerning
“The Standards in Initial Teacher Education”;
“Regulation of the accreditation process for
initial pre-school, basic and secondary school teacher education programs”,
“General Performance Profile of pre-schools and
primary schools teachers”,
“Specific
teaching profiles: of pre-school teachers and of primary school teachers”.
5- The role of the Government
The Ministry of Education and the Ministry of Science and Technology should take an important role, supporting institutions development projects, funding the acquisition of hardware and software and promoting teacher training in general. In the other end initial training Institutions should be committed in updating their institutional projects, curricula and inter-institutional links.
Everybody agrees that initial training is one of the keys to be a successful teacher and as I said before teachers tend to reproduce the model they experienced when students. If we don’t have teachers able enough to use new materials, to use and discuss technologies in the classroom, we can say that our high schools are also responsible for that.
And what about other teachers?
4 In-service training
A Portuguese survey on teacher training in ICT
This year the Ministry of Education presented the report of a survey made in training Centres all over the country (Santos, 2001). The results show that in 1998 and 1999 the number of courses and other sessions of teacher training carried out by the training Centres involving ICT was 23% of the total, and the number of teachers involved in this training was 27%, in 98 and 30%, in 99. This means that more and more teachers are interested in that subject.
In Portugal there are different kinds of training sessions. The most popular are courses, but now workshops that have an experimental component, that is, teachers have to make some work in the classroom with their pupils, became more popular. In 98 and 99 we had mainly courses.
The participation in ICT training courses
The chart in Figure 1 shows the % of trainers in ICT in training Centres.
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Fig. 1 |
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The subjects of the ICT training sessions
The report studied the subjects of the training sessions in ICT and divided them in three groups; for the results see Figure 2:
1º- Thematic (that is, sessions focusing on a particular subject, ex. Mathematics, Biology, Foreign languages, etc)
2º- Educational (focusing on educational themes, but not leading to a particular area of the curriculum)
3º- General (focusing only on the technology)
Analysing this chart we realise that teachers are using more technology in their schoolwork, but not in the classroom.
Links between in-service training and teaching in class
Another area of this study aimed to identify links between the training offered by the training Centres and the projects carried out in the schools. They divided the projects into four groups; for the results see Figure 3:
A- Projects of different Ministries
namely Education, Science, etc and European projects
B- Nónio projects
although Nónio is a ministerial project it is sufficiently important to be treated independently
C- School projects
D- Other actions without projects
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Fig. 3 |
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Analysing this chart we think that our training centres are now going in the right direction, making their plans according the schools needs. This report shows that 30 000 teachers were involved in ICT training sessions. This means that, regarding the lines of the EC documents for the education in the Society of Information and knowledge, eEurope and eLearning and the national guidelines, that point that all Portuguese teachers (around 150 000) have to be engaged in ICT training sessions till 2002, we see that we have a long way to go just to keep up with this target.
A European resolution on eLearning
If we take a look at the Draft Council resolution on eLearning we notice that The Council of European Union invites the member states:
i) To continue their efforts concerning the effective integration of ICT in education and training systems, ….
ii) To capitalise on the potential of Internet, multimedia and virtual learning environments for a better and faster realisation of lifelong learning as a basic educational principle.
iii) To promote the necessary provision of ICT learning opportunities within education and training systems by accelerating integration of ICT and revision of school and higher education curricula ….
iv) To continue their efforts in the initial and in-service training of teachers and trainers in the pedagogical use of ICT….
v) To encourage those responsible for educational and training establishments … to acquire the necessary understanding of the potential offered by ICT for enhancing new ways of learning and pedagogical development ….
vi) To accelerate the provision of equipment and of a quality infrastructure for education and training, taking into account technical progress ….
vii) To encourage the development of high-quality digital teaching and learning ….
viii) To support the development and adaptation of innovative pedagogy that integrates the use of technology within broader cross-curriculum approaches ….
ix) To exploit the communication potential of ICT to foster European awareness, exchanges, and collaboration at all education and training levels, and especially in schools….
x) To support and stimulate virtual meeting places for co-operation and exchange of information, experience and good practice ….
xi) To capitalise and build on the experiences gained in the framework of initiatives such as the European School-net and European Network of Teacher Education Policies.
xii) To foster the European dimension of joint development of ICT-mediated and ICT-complemented curricula in higher education ….
xiii) To enhance research in eLearning, in particular on how to improve learning performance through ICT, ….
xiv) To promote partnership between the public and private sectors as a contribution to the development of eLearning…. .
xv) To monitor and analyse the process of integration and use of ICT in teaching, training and learning.
What to do to keep up to these goals?
We don’t know what our Ministry of Education intends to do. The previous report recommends the Ministry to carry on a national training plan where the training must be included in the regular activity of the teachers. To generalise the pedagogical use of ICT it will be necessary to define a basic profile regarding new abilities related with distance learning and long-life learning.
Last year the Ministry of Education launched a workshop for trainers. This workshop involved 105 trainers, in groups of 15 in seven regions of Portugal. The main objective was to discuss a basic curriculum in ICT for training teachers and to make materials for the training sessions. Each group had face-to-face sessions and part of the work was made at distance.
I was in this workshop, from November 2000 till February 2001. As usual, the platform we used had features that allowed us to put our works on the Internet, to have Chat sessions with all the groups and to set up discussions in the forum. It was not clear to us if the materials we were building will be used in a future “massive” training program, but we believe this was the original idea of the Ministry of Education. Till now we don’t know what they are going to do with the materials we built in that workshop.
5 Changes in our curriculum
Past changes
In 1997/98 Mathematics had new programs. Some subjects have been changed but the main change was in the proposed methodologies. In them, the technology plays an important role. The use of graphic calculators was made compulsory, and the use of computers has been strongly recommended. The creation of Mathematics' laboratories was seen as a goal that every school should achieve. Of course, graphic calculators, computers and educational software were included in the list of indispensable material.
In that particular year was created by the Ministry of Education the figure of accompanying teacher. These teachers were divided in pairs and each pair was responsible for a group of schools. Their main work was to help teachers in schools to carry on the new program. After some training, they performed meetings in schools to give teachers some guidelines and to discuss with them these guidelines, to discuss the use of materials, to create new ones and working sheets, but mainly to discuss and to talk with them about new methodologies.
The Future
Traditionally in basic and secondary schools we have four mathematics' classes of 50 minutes each, per week. Depending on the school, mathematics' classes could join two periods in the same day, and if the number of pupils is greater than 22, the teacher could divide them in two groups, during one of those periods. That is, in the class the pupil has only mathematics four times a week but the teacher has five. This timetable is not compatible with the methodologies we want to implement in the classroom. We want to give pupils the opportunity to do small research activities, to use materials and technology and all of us know that this takes time, and with pupils running every 50 minutes from one class to another the time to perform these kind of activities is not enough.
Next year 2001/2002 we are going to have a big change in terms of curriculum in basic schools and in 2002/2003 in secondary schools. We are going to have 90 minutes classes instead of 50 minutes. Although this change is worrying lots of Portuguese teachers, this is an old wish of many mathematics teachers who want to change their methodologies and include technology in the classroom.
The new curricula, for basic and secondary schools, points technology as a subject that crosses the entire curriculum. Till now the pupils who want to, could choose the subject “technology” as an independent area. It was called ITI (Introduction to Information Technologies). In the curriculum to come this area disappears and technology must be part of all areas. This means that every teacher must know how to use it and as we saw lots of teachers are still far away from them.
6 And now?
The quality of teaching
We are always trying to improve the quality of teaching, but how to do it? In the words of Figueiredo (1998), the quality of teaching has to do with:
The quality of the institution
The quality of the teacher
The quality of the subject
The quality of the context
The quality of the pupil
I will just refer to the quality of the teacher that Figueiredo based in Ernest Boyer proposed:
Discovering:
focusing on the research activity, criticising, searching for knowledge.
Teaching: teacher must be able: to
promote knowledge in his pupils; to stimulate an active learning; to develop in
them a critical attitude and make them autonomous learners.
Integration:
the capacity to make links between his knowledge and other’s knowledge, and
between discovering activities of teaching and their applications.
Application:
the capacity to apply his knowledge trying to solve real world problems.
He said that the big challenge for XXI century is not to prepare teachers in how to use technologies in the classroom, but to keep an interdisciplinary reflection always update about how to face the opportunities and threats of the information society (Figueiredo, 1998) The shape of the teacher training will have to be significantly influenced by a new approach. It must not be the delivery of speeches to relatively passive audiences. It must take the form of project work that fully engages the activity and creativity of the teachers. It should be offered within the motivational impulse of large mobilising projects. And after he says that a possible way of attracting teachers to deep engagement in curriculum development projects might be:
To show them new problems and solutions
To prepare and make materials that creates
further awareness of those problems and solutions
To engage teachers in action/research projects
concerning those problems and solutions
To reward their efforts and achievements,
professionally
Which abilities must a teacher have to be able to teach using technology?
As I told before, there are some attempts to define a basic curriculum in ICT for teachers (ex: European Computer Driving License) a bit everywhere, and some projects are still going on to define a teacher profile in this area.
Very recently I have been asked to comment a document that is going to be the final product of a project, funded by the Socrates program concerning the attitudes and competencies in ICT a teacher must have. PICTTE (Profiles in ICT for teacher education) is a project, which involves eight partners from Portugal, Germany and Spain. In this project the partners are trying to agree on a common teacher profile, in order to design an ODL course. In the introduction it is written: “The speed of technological changes turns difficult a precise definition of the new competencies teachers require. If we are not aware of this the competencies profile won’t last long.” They distinguished attitudes from competencies and skills for the teachers, and for the competence profile they adapt the document “Expected Outcomes for teachers- England, Wales &: Ireland" and divided it in:
General teaching competencies: When and how to
integrate ICT in different teaching phases, from planning to assessment; how to
use ICT to improve the school dynamics.
Subject teaching competence: How to integrate ICT
in subject teaching, knowing and evaluating educational software.
ICT skills: How to explore the existing
resources at the school level; be familiar with the equipment and software
We still have a long way to go. I have been training teachers in “ICT” and in “ICT in Mathematics teaching” for quite a long time. This is my regular work for the moment, and I enjoy it. As a teacher in a secondary school I began to use technology in the classroom fourteen or fifteen years ago and if I go back to the school again, I will continue to do it. I never had a training course to do that. I’m what we could call a “self made” ICT trainer. I just find out how technology can be useful, not only in our daily life but also in our work. That’s the idea I try to pass to my trainees, but it is not an easy thing to do.
Hargreaves wrote, “The rules of the world are changing. It is time for the rules of teaching and teachers’ work to change with them”, but changes come very quickly and it is not easy to cope with them. In the final report of a project “Teacher’s ICT skills and knowledge needs” (Robert Gordon University - Aberdeen) I enjoyed particularly a comment made by a secondary teacher: “I need to keep up with developments. It’s a fast growing aspect of education. You are kind of running just to stand still”
References
APM (1998) Matemática 2001 – Relatório. APM.
http://www.apm.pt/apm/2001/2001_a.htm
Figueiredo, A.D. (1995) What are the Big Challenges of Education for the XXI Century: Proposals for Action. White Book of Education and Training for the XXI Century. Eurydice program.
Figueiredo, A.D. (1998) Como renovar a qualidade do ensino. Invited conference. Universidade de Coimbra.
Hargreaves, A. (1994) Changing Teachers, Changing Times. Cassell, London.
INAFOP - www.inafop.pt
PICTTE - European project.
Ponte, J.P., Serrazina, L. (1998) As Novas Tecnologias na Formação dos Professores. Ministério da Educação.
Santos, H. (2001) As Tecnologias de Informação e Comunicação na Formação Contínua de Professores. Ministério da Educação.
Practical aspects of CAS
using sinusoidal functions
George Adie and Bogdan Zoltowski
Kalmar, Sweden; Lodz, Poland
2. Use of sinusoidal functions in physics
CAS is revolutionising physics, science and engineering teaching. In these subjects we expect our students to be able to use CAS. This is highlighted using the example of alternating current theory where manipulation of sine and cosine functions is important. New ways of teaching using CAS are shown.
We conclude that we are putting new demands on basic mathematics courses to include the use of CAS and drop repetitive manual calculations.
1 Introduction
The use of CAS in physics teaching allows us to avoid areas of conventional mathematics that students often find difficult. It allows more efficient use of contact time. The teacher can concentrate on the concepts of physics instead of long and time consuming mathematical based proofs.
This is a very good theory. Practice is different. What we find is that students are not learning to use CAS properly in mathematics. The time we save by not going through mathematical proofs in physics is spent teaching students to use CAS instead.
We want to show how CAS is changing the teaching of physics and we will do this by highlighting the use of sinusoidal functions in the theory of alternating currents. There is a symbiosis between our subjects and the parameters upon which this symbiosis is based are changing because of CAS. That is why we want to make mathematics teachers aware of these changes.
From our point of view, we would like to see conventional mathematics courses upgraded to include CAS and concentrate on a deeper understanding of mathematics instead of being based on repetitive mechanical calculations.
2 Use of sinusoidal functions in physics
There are a number of areas where we use sinusoidal functions in physics.
Simple harmonic motion
Rotation
Waves and interference
Alternating current (AC) theory
In order to study these, students need to understand and manipulate sinusoidal functions. We use the following mathematical concepts.
The ideas of amplitude, frequency, period, phase
and phase shift
The relationship between sine, cosine and
tangent
Angles in degrees and radians
Angles less than zero and greater than 2p radians
Addition and subtraction of sine and cosine
functions
Multiplication of sine and cosine functions
Differentiation and integration of sine and
cosine functions
These are all standard topics in basic mathematics courses, but we often find that students’ knowledge of these areas in mathematics is shallow and in particular not tied to the use of CAS. We will show how the ideas are used in the introduction of AC theory using a calculator with CAS, TI89/TI92. This is because all of our students (Kalmar) have these calculators.
3 AC – theory. Using CAS
We shall study a circuit comprising a resistor, capacitor and inductor coupled in series to an alternating current power supply. The circuit diagram is shown and we shall use the symbols introduced here. Small letters used for the voltages (v) and current i imply that they are time dependant.
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The current is i = i0·sin(2·p·f·t) where f is the frequency. The definitions for R, C and L give the following.
A common alternative is to express these as functions of v. |
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Fig. 1 |
Task 1
Our first task is to represent the current and voltages mathematically so that the students can get a handle on them. The conventional pre-CAS method is to evaluate the above equations and discuss the phase differences.
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Using CAS, the equations are fed into the calculator as functions from y1 to y4 as shown on the right. The variable x is used for time, i0 is the amplitude of the current and f is the frequency. From a physics point of view, it is very satisfactory to use the integral and differential in functions y3 and y4 as these stem directly from the laws of physics. |
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Fig. 2 |
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The analytical functions can also be obtained by the students. See Fig. 3 to the right. |
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Fig. 3 |
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Using appropriate values for i0, f, L, R and C the students can plot pairs of graphs and study the phase differences. An example of this is shown on the right where the dotted line shows the voltage across the resistor and the full line shows the voltage across the inductor. |
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Fig. 4 |
Commentary 1
The important point is that the students use physics definitions directly without the need to agonise over (what are for some) difficult integrals and differentials. They understand the theory and ideas from the graphs instead of from the mathematical expressions. The mathematical functions are available for reference as well. We want students to be able to express differentials and integrals (with and without limits) using CAS.
Task 2
The next step is to look at the power developed over each of the components. The instantaneous power is p = v·i for each of the components. These products are conventionally worked out using algebraic methods for solving products of sines and cosines. These are transformed to highlight frequency doubling. Discussion is based on the graphs of the functions.
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Using CAS the functions are easily fed into the calculator. The screen on the right shows how this is fed into the calculator as y5, y6 and y7 for the three components R, C and L. If necessary, the resulting functions can be studied algebraically. (See next screen.) What is even better here is that the functions can be studied graphically so that the students get a real picture of what is happening. The graphs below show how power is developed over the resistor and capacitor. The dotted line shows the current and the full line shows the power. |
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Fig. 5 |
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Fig. 6 |
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Power developed over the resistor is always positive and changes as a displaced negative cosine with twice the frequency of the current; see Fig. 7a. Power developed over the capacitor averages over a long period to zero; see Fig. 7b. The graph for the inductor is similar to that for the capacitor and shows that there is no power developed over a perfect inductance. Commentary 2We want the students to be able to analyse and interpret the graphs. The algebraic derivation of the functions is far less important than the understanding of the parameters. In particular 4·p·f gives double the frequency. |
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Fig. 7a: Power developed over the resistor |
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Fig. 7b: Power developed over the capacitor |
Task 3
We define the r m s value of the current as the constant value irms that gives the same power output as the alternating current over a long time interval (with a whole number of oscillations). We have seen previously that all the power is generated over the resistor.
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The r m s value is conventionally calculated using standard integrals and algebraic methods. The r m s value can easily be calculated using CAS. This leaves time to discuss the underlying ideas. The solution is shown on the right.
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Fig. 8 |
Commentary 3
There are three important aspects of using CAS in the above.
Defining an upper limit that is a whole number of oscillations.
Setting up the integral from the definition.
Solving the integral using CAS and the – solve - function.
All of these are fundamental skills that are often lacking in our students.
The problem with integrals is especially serious. Our students have studied mathematics and can perform exceedingly complicated integrations. Unfortunately they have not got a clue about how to set up an integral. They do not understand what an integral is or how it can be used. The use of CAS allows the mathematics teacher to concentrate on these fundamental ideas instead of repetitive calculations. This gives the student a better base for modern scientific/engineering studies.
Task 4
The next stage is to look at amplitudes and the phase of different voltages compared to the current and the voltage over the resistor. (As shown earlier the voltage over the resistor is in phase with the current.)
In physics we are interested in;
v = vR + vC + vL compared with vR
vR + vC compared with vR
vR + vL compared with vR
This is conventionally done by working out the sums algebraically and then using phasors.
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Using CAS, the equations are fed into the calculator as shown on the right with y8, y9 and y10. Using appropriate values, the graphs can easily be studied, analysed and interpreted. Here is an example. |
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Fig. 9 |
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The graphs showing vR dotted and vC as a thin line together with (vR + vC ) as a thick line are shown on the right. Students see immediately that the resulting voltage has the same frequency as the other two and that there is a new phase and amplitude. The relationships between the amplitudes and phases can be studied directly from the graph. Students can change input parameters to examine other cases. |
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Fig. 10 |
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Once students have established this understanding, they can verify their results analytically on the HOME screen. The screen on the right shows how the amplitude and phase for y8. = (vR + vC ) can be studied. |
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Fig. 11 |
Commentary 4
Manual addition of sine functions with different amplitudes and phases is difficult for our students. It takes a lot of time and they lose track of the underlying physics. These methods using CAS allow them the opportunity to understand what they are doing.
4 Conclusion
CAS is revolutionising physics teaching and this means that we are putting new demands on our colleagues who teach mathematics. We have chosen a small area of physics to highlight these changes, but they are prevalent throughout the rest of physics, science and engineering. Most of our students will use mathematics as a tool. Very few of them will ever perform manual calculations in their working lives, they will use computers. We would like to propose that the ability to perform repetitive manual calculations is becoming redundant. Students need to learn to use modern technology as a part of mathematics in order to be able to deal with science/engineering studies.
We would like to see more basic mathematics courses where the emphasis is on using modern handheld technology or computer programs with CAS. This should help students gain a deeper understanding of the underlying mathematics so that they can formulate problems mathematically and interpret results.
References
Adie, G. (1998) The impact of the graphics calculator on Physics Teaching. Phys. Educ. 33(1).
Adie, G. (1999) Graphical Calculators and Mathematics in Physics Teaching. Shaping the Future. Physics in a mathematical mood. IoP.
Adie, G. (2000) Using the TI-89 in Physics. bk-teachware 2000.
Adie, G. and Zoltowski, B. (1998) Graphing calculator based activities in the student physics laboratory. XII Conference on Teaching Physics at Technical Universities, Poznań.
Adie, G. and Zoltowski, B. (1999a) Differential equations in practical physics teaching. ICTMT 4. Plymouth.
Adie, G. and Zoltowski, B. (1999b) Mathematical aspects of using the calculator as a demonstration tool in physics. ICTMT 4. Plymouth.
Adie, G. and Zoltowski, B. (2000a) The Impact of Handheld Technology on Physics Teaching for Engineers. PTEE 2000. Budapest Hungary.
Adie, G. and Zoltowski, B. (2000b) Handheld Technology in the Undergraduate Physics Laboratory. PTEE 2000. Budapest Hungary.
Investigating teachers’ perceptions on their preparation to use IT in classroom instruction
Adnan Baki
Karadeniz, Turkey
The researcher taught a two-term required course within mathematics teacher education program to train student teachers and to investigate perceptions on their preparation to use computers in their own teaching. This paper describes issues emerging from the analysis of the course. Data were gathered through questionnaires. Students who felt prepared made the link between computer-based mathematical activities and school mathematics, and had more experience on the instructional software during the course than the others. The implications of these results to the design and implementation of computer-based undergraduate courses and for further research in this field are discussed.
Will new teachers entering their classrooms be prepared to teach mathematics with computers? Unfortunately, the answer may be “no” far more often then it will be “yes”. Mathematics student teachers often find very little modelling of the use of technology during their faculty years. The need for an improved use of computer-based educational technology in mathematics education was indicated by the researcher (Baki, 1994). By indicating the importance of training teachers to teach with computers the researcher proposed to the Faculty of Education at the Karadeniz Technical University a two-term required pre-service course called “computer-based mathematics teaching,” to help prospective mathematics teachers improve their use of educational computing in the classroom. So, the course was to allow mathematics student teachers to have first hand experience on the applications of some instructional software and begin to think their current preparation for using computers in mathematics teaching.
The traditional way of teaching is telling the students the answer and therefore keeping them happy. On the other hand, a new approach is leading them to discover the solutions or structures themselves and possibly causing frustration. The traditional approaches to mathematics teaching generally have also failed to foster the skills in thinking mathematically. In order to accomplish developing skills in thinking mathematically, the structure of the conventional classroom should be changed.
However, significant changes in mathematics education will only be achieved if there are marked changes in teachers’ conceptions about the effectiveness of innovative curriculum and approaches. As a result, teacher’s crucial role in educational change is becoming increasingly accepted. Therefore, teachers should have practical experiences with new innovative curriculum, approaches, materials and activities that they are expected to employ when they teach. This means that teacher education is the key to improving the use of computers in classroom instruction.
1 Methodology
Instrumentation: The research was undertaken with student teachers pursuing a 4-year undergraduate program at the Faculty of Education in Karadeniz Technical University. The data source consisted of fifty-six student mathematics teachers enrolled in the course in the academic years 1998-1999. The data was gathered through questionnaires. Pre-post questionnaires and participants’ writings were administered to the participants to identify if any significant changes occurred in the perceptions on their preparation for using computers in mathematics teaching.
Questionnaires: The pre-post questionnaires both included the identical items. The questionnaire consists of two parts. The first part included four items to elicit initial information about teachers’ previous computer experiences. Looking at the responses of the participants to the questions in the first questionnaire, the participants were divided into two groups as computer-literate and computer-illiterate. The rest of the items in both questionnaires were designed to gather the perceptions of the student teachers on their preparation to use computers in the classroom. These items were a revised form of computer attitude scale developed by Gressard and Loyd. The post questionnaire was administered during the final class meetings. Comparing the responses of the participants to these three questions the researcher tried to determine to what extent student teachers feel prepared to teach mathematics with computers before and after the course.
Procedures: In response to the need for teacher development in educational technology, the course has been given since 1995 at the Faculty of Education in Karadeniz Technical University. The course was offered as a two term required course at the last year of a four year undergraduate program providing Bachelor’ degree with specialisation in mathematics education. The central focus was not just the computer but on the learning of mathematics through computers. In order to provide mathematics teachers with such experience, Logo and Excel have been chosen as means of operationalising an alternative in practice. Why Logo and Excel? They are user friendly and have special features for mathematics. The rationale for using Logo emerged from an extensive literature (Baki,1994). There are convincing arguments made by Hoyles and Noss that Logo offers possibilities for mathematical exploration and provides an environment in which the learner can construct mathematical models and ideas. Excel is very widely used throughout the world of mathematics education especially for numerical and iterative methods in mathematics. Excel allows the display of numerical data and text in tabular form on a grid of cells as an example of mathematical matrices. Excel is designed as a database package to record, classify and sort large amounts of data. It also includes graphical facilities, which enable aspects of the tables to be displayed rapidly in graphical form. The curriculum of the course has been designed with the aim of providing a model for classroom practice. Therefore, the course involved and dealt with numbers, calculus, trigonometry, geometry and algebra. In the final weeks of the second term the students were motivated to develop their own projects (40% of the course grade in the second term was allocated to the projects).
Analysis: Questionnaires and participants’ writings were the sources of data in this study. Quantitative data emerged from the questionnaires. The large amount of qualitative data collected from participants’ writings was used to elaborate the findings emerged from quantitative data about the perceptions of the participants on preparation for teaching mathematics with computers. During the interpretation of qualitative data, the researcher tried to identify some salient aspects and recurring themes in the reflections.
2 Findings
The four questions about computer-literacy in the first part of the pre questionnaire were designed to test the participants’ experiences with computer.
Question A1: What currently do you know about computers?
Although the modal response is “much”, 18 students chose “very little” or “nothing”. This indicates that computers were new to some of them (33%). On the other hand, of the 56 student teachers, 38 chose “very much” or “much”. This indicates that majority of them (67%) felt that they knew enough about computers.
Question A2: Have you used a computer before?
Although the modal response is “several times”, the results of the question A2 were almost similar to the first question. Of the 56 student teachers, 18(32%) had never used a computer, 6(11%) had used it at least once while and 32 (57%) used a computer at least several times.
Question A3: What do you know about hardware?
The participants might suppose that hardware could be some parts of computers like monitor, case and keyboard without considering their interrelations and functions. The modal response to this question is “I know something about it”. The results of this questions indicated that the participants’ previous knowledge about hardware seemed to be based on some presumptions rather than real experiences.
Question A4: Do you know what software is?
As in the question A3, responses to this question seemed to be based on their presumptions. While 26 (46 %) chose “computer programs”, 5 (9%) chose “diskettes”, 15 (28%) marked “monitor & keyboard” as a correct answer, and 10 (17%) reported that they did not have any idea about software. From the responses to the four questions as a whole, it seemed that the student teachers appeared to have different experiences on computers. According to the criterion used by Gressard and Loyd, 41% student teachers began the course as novices (computer-illiterate), and only 59% appeared to be computer-literate. After using the computer, writing small programs in Logo, using functions and their graphs and constructing mathematical models with Excel during the course, majority of the student teachers appeared to have gained ideas about hardware and software. While there was no significant change in the computer-literate group, there was a significant change in the computer-illiterate group in terms of computer experiences as a result of the course. According to the criterion used by Gressard and Loyd, 93% of student teachers completed the course to be computer-literate. This does not mean that the course prepared almost all student teachers to teach mathematics with computer. In the following, the researcher elaborates this issue in light of responses to the second part of the questionnaire.
Question B1: Do you feel that the computer will affect the quality of your life?
In the pre questionnaire, the modal response of the computer literate group is “little” and the modal response of the computer-illiterate group is “very much”. The results of the pre questionnaire related to the question B1 indicated that the computer-illiterate group felt that computers would improve the quality of their life to a great extent (84%) than did the computer-literate group (41.4%). Most (68%) of the computer-illiterate group felt that computer technology would dramatically change their life and improve it. The results from the pre questionnaire related to the question B1 indicated that at the beginning of the course the computer-illiterate group appeared to glorify the computer and see it as a panacea solving every problem in education and life. Their writings at the beginning of the course also illustrated similar preconceptions about the role of computer technology. Most of them in their first writings suggested that the computer is a contemporary need, it should be used in all fields in order to catch up with developed countries. Positive statements made by the computer-illiterate group clearly indicated that the computer was typically construed as a way of being “modern”, and was seen as important for the future of the teaching profession and “improving the quality of life”. In the post questionnaire, the modal response of the computer literate group is again “little” and the modal response of the computer-illiterate group is also “little”.
In the computer-literate group, the modal response did not change as a result of the course. The percentage of the modal response only shifted from 54.6% to 56%. There was no significant change in both questionnaires. The computer-literate group seemed to think that computer would not affect their life too much, and believe that computers would not improve the quality of their lives as much as the computer-illiterate group believed in the pre questionnaire. The modal response of the computer-illiterate group shifted from “very much” (68%) to “little” (60%). Towards the completion of the course the computer-illiterate group significantly changed their views so that they matched the computer-literate group. This case is not indicating a pessimistic change in the computer-illiterate group. This may be interpreted that as a result of their experiences with Logo and Excel they reached at a reasonable level to evaluate appropriately the potential of the computer in their professional life.
Question B2: How do you feel to learn to use computer?
This question was asked to see how the participants felt they were capable of learning computer before and after the course. The modal response of the computer-literate group is “very quickly” (74.7%). The modal response of the computer-illiterate group is “slowly” (66.8%). Additionally, 11% of the computer-illiterate group chose “not at all”. This highlights that the computer-illiterate group as in the question B1 glorified the computer and felt that they were not capable of learning to use the computer. The results from the two questionnaires about the question B2 indicated that the computer-literate group felt prepared to learn computer very quickly or quickly. The range of their responses to this question did not change significantly in both questionnaires. On the other hand, the computer-illiterate group went from a slowly response to a quickly response after completing the course. This indicates a positive attitude change towards learning to use computer. It is hoped that this change may increase interest in using the computers as learning and teaching tools.
Question B3: How do you feel prepared to use Logo and Excel in mathematics teaching?
Although at the beginning of the course the modal response is “little”, only 26% of the computer-literate group felt prepared (very much + much) to use Logo and Excel in mathematics teaching. On the other hand, the modal response of the computer-illiterate group is “not at all”. Thus, 94.7% of the computer-illiterate group did not feel prepared (little + not at all) to use the software in their teaching before the course. The modal response of the computer-literate group is “much”. The modal response of the computer-illiterate group is “little”. In the post-questionnaire, 84% of the computer-literate group felt prepared (very much + much) to use the software (Logo and Excel) in their own teaching. Only 25% of the computer-illiterate group felt prepared (very much + much) to use the software (Logo and Excel) in their own teaching. In the computer-literate group, the modal response to this question before the course is “little” whereas after the course the modal response is “much”. On the other hand, the modal response of the computer-illiterate group to the question B3 is “not at all” before the course and the modal response is “little” after the course. Although this clearly indicates a positive view about their computer experiences, this is not implying that the computer-illiterate group gained adequate enough experience about using computers in mathematics teaching.
3 Conclusions
The course provided student teachers with an understanding of the philosophy of mathematics education, as well as training in the computer-based environment. The relationship between computer-based mathematical activities and the needs of students in school mathematics was thoroughly discussed. From the findings, we can say that taking this kind of pre-service course did have an impact on the feeling prepared to use the computer for learning and teaching. If the statement of “we teach as we are taught” is true, then education faculties must consider this implication. Asking education faculties to consider this implication requires professional development at the university level. Designing and providing appropriate technology experiences means that the faculty itself must develop comfort with and an awareness of the technology, which is currently being used in, schools (Novak and Knowles, 1991). More importantly it means that education faculties must model the use of the educational software in their own teaching programs. The current literature and the perceptions of student teachers described in this study suggest several possibilities and directions for further research in this field:
We do not have convincing information about what education faculties
in Turkey believe about their student preparation for educational technology.
We need an investigation of the relationship between what student
teachers say during the course and do during their actual teaching. This
indicates a cross-sectional analysis.
The course remains an active and ongoing catalyst for teacher reflections and professional considerations. The research process continues in that the model created for research and course structure are being used to consider other software (such as Mathematica, Cabri and Derive) across the mathematics curriculum.
References
Baki, A. (1994). Breaking with Tradition: A study of Turkish student teachers’ experiences within a Logo-based mathematical environment. Ph.D. Thesis, Institute of Education, University of London.
Gooler, D. (1989) Preparing teachers to use technologies: Can universities meet the challenge? Educational Technology, 29 (3), 18-21.
Gressard, C. and Loyd, B. A. (1986) Validation studies of a new computer attitude scale. Association of Educational Data System Journal, 19, 295-301.
Hoyles, C. (1992). Mathematics Teaching and Mathematics Teachers: A Meta-case study. For the Learning of Mathematics, 12(3), 32-44.
Hoyles, C. and Noss, R. (1992). Learning Mathematics and Logo. MIT Press, Cambridge.
Novak, D. and Knowles, J. K. (1991). Beginning elementary teachers’use of computers in classroom instruction. Action in Teacher Educ. 13 (2), 43-48.
Rabson, C. (1993) Real world research: a resource for social scientists and practitioner-researchers. Oxford: Blackwell.
Using computers in mathematics teacher training programs:
A reflection upon an experiment
Elizabeth Belfort, Luiz Carlos Guimarães, and Rafael Barbastefano
Rio de Janeiro; Getúlio Vargas, Brazil
2. Analysis of the instructional materials developed by the teachers
4. Medium term consequences on teacher's work
As part of a graduate course, secondary teachers attend a discipline exploring the use of computers for teaching mathematics. They are asked to produce their own instructional materials, using an educational computer package. The authors have designed this discipline and have also ministered it for a number of years. In this article, we analyse qualitatively the materials produced during the course by the teachers taking it, as well as some of the consequences of these activities in their own subsequent classroom work.
During the first semester of a graduate course, secondary Mathematics teachers revisit basic contents in geometry and functions, in disciplines supported by computer laboratory lessons. They continue to use computers as tools for teaching and learning Mathematics in a subsequent discipline, called 'Computers for Mathematics Teaching' (CMT), designed to provide a counterpoint to the previous subject matter disciplines of the course, where computers are used to teach them. With their experience as learners to rely upon, they are asked to discuss, as well as exercise, the possibilities of the computer as a teaching tool. A series of laboratory lessons were planned, where secondary Mathematics classroom situations are simulated. In them the teachers are encouraged to suggest ways in which softwares could be used to facilitate the teaching of sample materials taken from the secondary school Mathematics curriculum. Participants are asked to produce an essay for assessment in CMT, in which they must present their own computational instructional materials to other mathematics teachers, and justify its use for classroom work. In this article, we discuss the outcomes of a research project designed to evaluate these materials, as well as the influence of CMT activities in teachers' future work. We analyse the materials produced by the teachers and report on interviews conducted with teachers a year after they had completed the course.
1 Background and methodology
The CMT course design was based on previous in-service course experiences. Some of the reasons why we have decided to include laboratory lessons in these courses are: (1) we have been using laboratory lessons as part of the methodology in our undergraduate disciplines for many years, and we are convinced they can be efficacious; and (2) "to convince teachers that computers can be used as tools for teaching mathematics, it is necessary to furnish them with some interesting computer assisted learning experiences" (Belfort et al, 1998, 376).
While working in the laboratory environment during the in-service courses, secondary mathematics teachers were less troubled about asking questions related to their subject matter. The fact that they were obviously not expected to know how to use the educational packages made them less self-conscious, and they could freely show their doubts on content.
We believe that there are clear benefits stemming from providing teachers with opportunities to revisit subject matters related to their classroom practice. In support of this opinion, we can point out several research reports that discuss the need for a solid teacher's subject matter knowledge (see, for instance, Ball et al, 1988, Ball, 1991; Leinhardt et al., 1991). When designing CMT, we were also influenced by research projects investigating those special qualities exhibited by a teacher’s understanding of subject knowledge which seem to relate to good teaching practices. Ball (1988) discusses the issue and, more recently, Ma (1999) defined profound understanding of fundamental mathematics (PUFM) as "an understanding ... that is deep, broad and thorough" (120). According to her, elementary teachers with PUFM displays, in their practice, "connectedness, multiple perspectives, awareness of the basic ideas and longitudinal coherence" (122).
Based on this framework, CMT intends to provide opportunities for teachers to deepen their subject knowledge and to establish connections among related concepts. To compensate for a perhaps overly strong reliance on textbooks, teachers are asked to produce their own instructional materials, and we strongly encouraged them to use additional reference sources. Recommended readings include classical geometry texts (Heath, 1956; Legendre, 1823), materials specially elaborated for teacher training (Guimarães & Belfort, 1999, Santos et al, 1998), and articles on use of mathematical educational softwares.
2 Analysis of the instructional materials developed by the teachers
We analysed the materials developed by the teachers from three perspectives:
UCS: usage of the computational software as a
didactical resource;
SMK: subject matter knowledge and consistent
mathematical reasoning; and
ADP: appropriateness of the didactical proposal,
considering the targeted year group.
The balance among these three factors was considered the main criterion for success in the evaluation of the materials, but it was perceived only in roughly one third of them. The majority was oriented towards one of the three perspectives, mostly ignoring the others. We briefly describe here the main features of these categories and provide some examples of the developed materials.
UCS oriented instructional materials
The materials classified in this group (about 10% of them) reflect their authors' focus of interest as being the process of mastering the use of the software. They presented a sophisticated use of the package resources but give little room for students' exploration. Concepts are neither explained nor justified. The material developed by Marcos to draw geometric loci is an extreme example in this category. Marcos acquired advanced skills as a D.G. software user, but his use of it did not provide opportunities for the students to construct the geometrical proprieties shared by the points in any of the presented loci. Students are, at best, treated as spectators: all they have to do is to 'click the mouse' over the animation button.
SMK oriented instructional materials
The materials in this group (about 10% of them) reflect their authors' focus of interest: to revisit subject contents. They show their interest in mathematical results and, in general, in the correct justifications for them. On the other hand, only the simplest resources of the software were used for the development of the materials. As for the didactical proposal, again very little is open to students' experiences, as these material are usually ready visualisations of mathematical results. José used a DG Software to study the points of intersection of the cevians of a triangle. The written essay is correct, presenting definitions, theorems and well-organised proofs in a consistent logical order. On the other hand, the use of the software is naive and the students’ job is reduced to move the vertices of the triangles to verify that the intersection points remain unique. The written material developed by José to support the laboratory lesson did not ask the students to discuss the geometrical reasons for this observed result. On the other hand, not all teachers presenting SMK oriented materials were worried about definitions and theorems. Pedro decided to develop material on conic sections. His essay can only be described as a huge list of results, which were presented without previous definitions, justifications or connections among them.
ADP oriented instructional materials
Almost half of the instructional materials presented by the teachers can be classed in this category. Their authors focused on designing interesting computer assisted learning experiences. Even if the use of the resources is at an intermediate level, they usually fail to provide a correct and complete mathematical development. These materials are usually strongly influenced by the textbook approaches. The proposed experiences display a fragmented vision of the topic and a clear hurry in getting to a “formula”. They fail to provide connections and multiple perspectives. Mariana’s materials, on area measurement, can be considered a typical example. While using her first sketch, pupils are asked how many squares area units are needed to fill one rectangle. The following sketches repeat the experience, using larger rectangles. Mariana expected the students to inductively conclude the formula for the area of the rectangle, as a means to avoid the repetitive job. On the other hand, all the rectangles used by her have integer side measures. It is to be noted that the year group targeted by her is the very one studying operations with fractions in our schools. That doesn’t seem to be relevant for many textbook authors either, as a quick check on their examples of area indicates.
Well balanced instructional materials
An instructional material was classified as well balanced whenever all three perspectives were integrated by the teacher/author. The examples in this category typically provide the students with a sequence of computer activities aiming at developing a mathematical concept. There is always a concern with proper definitions and with clear justifications for the results.
3 Comments and reflections
There were substantial differences in the outcomes of teachers’ work in CMT, reflecting different views on mathematics teaching. Teachers who produced well balanced materials, in which concepts were treated as connected parts of a body of knowledge, usually were made comprehensive use of the bibliography, and seemed to have established clear objectives for the developed materials. On the other hand, some materials seemed to reinforce the role of the teacher as the knowledge keeper (and teller), while others provided experiences with no clear objectives and no connections with other experiences. Also, the lack of deep, broad and thorough subject matter knowledge on the part of the teachers resulted in fragmented materials, where formulas were overestimated, and on strong reliance on simplified approaches found in some textbooks available in the market. As expected, the discussions promoted in classroom on the characteristics of their produced materials made these teachers more conscious of some critical educational issues related to Mathematics teaching and learning and, in most cases, made them also willing to make the effort to overcome their difficulties.
4 Medium term consequences on teacher's work
We conducted interviews with seven subjects of the first group enrolled in the CMT discipline, a year after they concluded their courses. Three of them live and work in the city of Rio de Janeiro, while the others work in small towns within the state. They constitute a representative sample of all tendencies during the CMT discipline, and all of them reported positive consequences to their subsequent careers. One of them is regularly using computers, but four others report they usually request one computer in their classroom to illustrate their teaching. Two teachers interviewed (one in a small town and another in Rio) recently started work as assistant lecturers in local small colleges. Two of the teachers from small towns, who used to be in charge only of the initial secondary year groups, are now working preferentially with senior secondary students.
All teachers declare they have changed the way they used to teach. One of them says that the interactivity and dynamics of the ideas she experienced in laboratory lessons reflects even in her use of the blackboard in the classroom. Perhaps surprisingly, this is so even though she works at a school in a deprived neighbourhood in Rio, with no computers available. Three of these teachers are part of a group of former students who started a career of research and teacher training. They have participated in local in-service course efforts and regularly present articles and workshops in local and national mathematics education meetings.
5 Conclusion
Our results suggest that to provide their students with learning experiences using computers is an appealing idea for mathematics teachers. They also indicate that, in general, they are not prepared to do so in a consistent and connected way as a result of insufficient initial preparation. Nevertheless, it is possible to help them to overcome content knowledge difficulties, and to develop a critical awareness of materials for classroom and/or laboratory work. In this sense, computer packages can be useful to support content subject disciplines in laboratory lessons during graduate or undergraduate teacher training courses, and there is also the need to give them room to reflect on their use in elementary and secondary education.
Mathematical computer educational software can be a powerful tool for teachers. Nevertheless, as it happens with any other tool, it is the way it is used that determines the final outcomes. If we expect teachers to fully understand the potential of these packages we'd better start to use them ourselves as a tool in teacher training courses.
References
Ball, D. L. (1988) The Subject Matter Preparation of Prospective Teachers: Challenging the Myths. (Research Report). National Center for Research on Teacher Education, East Lansing, MI.
Ball, D. L. (1991) Research on teaching mathematics: Making subject matter knowledge part of the equation. Brophy, J. (ed.) Advances in research on teaching vol. 2. JAI Press, Greenwich, CT, 1-48).
Ball, D. L. and Feiman-Nemser, S. (1988) Using Textbooks and Teachers' Guides: A Dilemma for Beginning Teachers and Teacher Educators. Curriculum Inquiry 18 (4), 401-423.
Belfort, E., Guimarães, L.C. (1998) Uma Experiência com Software Educativo na Formação Continuada de Professores. Ann. VI ENEM (Braz. Nat. Meet. of Math. Educ.), Unisinos, São Leopoldo, RS, 376-378.
Guimarães, L. C. and Belfort, E. (1999) Roteiros de Laboratório de Geometria. Ed. IM-UFRJ, Rio de Janeiro.
Heath, T. L. (1956) Euclid: The Thirteen Books of the Elements. Dover, New York.
Legendre, A. M. (1823) Éléments de Géométrie - 12ª edition. Librarie de Firmin Didot Frères, Paris.
Leinhardt, G., Putnam, R. Stein, M. and Baxter, J. (1991) Where subject matter knowledge matters. Brophy, J. (ed.) Advances in Research on teaching vol. 2. JAI Press, Greenwich, CT, 87-113.
Ma, L. (1999) Knowing and Teaching Elementary Mathematics: teacher's understanding of fundamental mathematics in China and the United States.LEA, Mahwah.
Santos, A. R. e.a. (1998) Introdução às Funções Reais. Ed. IM-UFRJ, Rio de Janeiro.
A modern approach to limit processes
Primo Brandi and Anna Salvadori
Perugia, Italy
2. Introduction to limit process
1 Background
Limit
is the key concept of Calculus. Since it isn’t an easy subject, very often
teachers choose to pass over the definition and present only the idea and the
applications (e.g. derivatives, continuity, series).
The
main difficulty is that students get the intuition straight away, but then they
don’t accept the definition as a natural translation of the concept.
To
overcome this difficulty, we proposed a new approach with the aim of driving
students from perception to epsilon-delta definition gradually.
This process involves the three fundamental aspects: geometric,
numeric and algebraic. Moreover, it proves very useful in dealing with critical
examples, as we will show.
To
supply the graphic support a software ad hoc is implemented.
Before
entering into details we wish to mention that our proposal is a part of a more
general reform project on teaching and learning Calculus[1].
The project Progetto Innovamatica promoted by P.Brandi in 1994, is devoted to students of 16-19 years hold and has the main aim of supporting the admission at University and reducing the large failure rate. This experimentation in teaching methodology has involved about 4.000 students and 300 teachers of the Secondary Schools in Umbria (Italy) and about 3.000 students of first-year Calculus in the Faculties of Sciences, Engineering and Agriculture of the University of Perugia. The main target is to conciliate the necessary renew of the traditional Calculus (emphasizing intuitive arguments, mathematical modeling and active learning) with a wise preservation of the mathematical exactitude.
2 Introduction to limit process
The traditional presentation of limit process (from the intuition directly to the formal definition) is here substituted by a well-constructed approach which translates the idea into various mathematical formulations (all equivalent) which gradually increase in abstraction till the usual epsilon-delta formalization.
Students have an active role in this path and appreciate this presentation which helps them to enter into the concept deeply.
Our proposal is summarized by the following scheme.
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Fig. 1 |
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Step 1 – The intuition of the concept
We start by proposing concrete problems taken from everyday real-life situations which stimulate the necessity of this new subject.
The concept is focalized first intuitively, through graphical and numerical technology-assisted explorations or laboratory experiments.
Easy examples show that different situations may occur.
Step 2 – The natural translation of the idea
We stimulate student’s suggestions for a first mathematical description of the phenomenon which is a natural translation of the intuition.
Given a function f: [a,b] → R and a point x0 Î [a,b], the limit process describes the behaviour of the function f “near” the point x0. More precisely, it discusses how the values f(x) vary as x approaches the point x0. To describe the region “near” x0 It is natural to adopt the balls[2]
Note that (Bn) is a decreasing sequence such that the smaller the radius is the closer the points of Bn are to x0 i.e.
Since we are interested on the values attained by the function, we consider the images of the balls. It is easy to see that the point x0 itself is not involved in the process since we search for the behavior “near” the point. Moreover, the function is not necessarily defined on the point x0 as the examples will show. Thus we take the images of the balls deprived of the point x0:
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The sequence (En) is still decreasing moreover, as n increases En contains the values f(x) attained on points, which are closer and closer to x0. In this way we get a “dynamic” sequences of images which contains the secret of the limit-behavior. |
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Fig. 2 |
How can we capture it? The natural answer is: intersect the images!
Indeed, to be sure of getting a non empty set we have to intersect the closure of the images
The set E contains exactly the information we are looking for.
Definition: The function f admits a limit for x approaching x0 if the set E contains only one element, i.e. E = {l}. In this case we write
.
If E has more that one element, the function f has no limit for x approaching x0..
Let us see some critical examples.
Example 1: Let f: R → [0,1] be the Heaviside function
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For n sufficiently big we have
and hence
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Fig. 3 |
Example 2: Let f: R -{0} → R -{0} be the hyperbola
.
Let us
consider the points x0=0 and x0= ±
.
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Since
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we have
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Fig. 4 |
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and we conclude that
Example 3: Let f: R -{0} → [-1, 1] be the function f[x] = sin 1/x.
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Let us consider first the point x0=0. It is easy to see that En = [-1,1] "n Î N and hence E = [-1, 1] and the limits does not exist. |
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Fig. 5 |
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Now
let us consider the points x0=± Since
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for every x > 0 it is easy to see that
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thus E = {0}
x0=± |
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Fig. 6 |
In conclusion we have
Example 4: Let f: R+ → {1, 2} be the function
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For n sufficiently big we have
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Fig. 7 |
More complicate limit behavior can be discussed by means of a suitable computer simulation (see Corsi di innovazione & sperimantazione didattica in the web site www.innovamatica.it).
Step 3 – An existence result
As the examples have focalized, the limit of a function in a given point may not exists. Thus, before trying to compute it is necessary to know if it exists.
Note that it is not necessary to determine the shape of the set E exactly (which can be an hard problem), but it is sufficient to detect if it is a singleton or not.
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Moreover, every set En admits a lower bound and an upper bound, i.e.
which control the “dimension” of the set. The two monotone sequences
control as sliders the dynamic process of the images
The sets
are separate and the following results holds. |
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Fig. 8 |
Theorem (Existence):
The limit exists Û the two sets Lf and Uf are contiguous, i.e. sup Lf = inf Uf.
The different behavior of these sliders in the various examples is very well illustrated by our animation (see Corsi di innovazione & sperimantazione didattica in the web site http://www.innovamatica.it/).
Step 4 – A first equivalent definitions. Numerical approach
After the discussion on the existence, we have to face the problem of the determination of the limit value. The couple of sets Lf and Uf are still fundamental in this direction.
In fact if they are contiguous, the separating element itself gives the value of the limit.
Proposition 1: If Lf and Uf are contiguous then
Note that the above characterization is fundamental also for a numerical approximation of the limit. Indeed the elements of Lf approaches the limit from below, while those of Uf approaches the limit from above. Moreover, since the two sequences
are monotone, the (n+1) - approximation improves the n - one.
Step 5 – An other equivalent definitions. Formal approach
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The following characterization of the separating element of a couple of contiguous sets is useful. Proposition 2: An
element l Î R
È {± "Vl neighbourhood of l there exist
Recalling that the sequence (En) is decreasing, we deduce that an integer nV exists such that
Theorem (Usual epsilon-delta definition): The function f admits l as limit for x approaching x0 , i.e.
This last formulation of limit process, is less intuitive, but is still the most efficient for the algebra of the limits. In other words is the key formulation for developing the calculus of limits. |
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Fig. 9 |
References
Brandi, P., Salvadori, A. (1993 )Appunti ed esercizi di Analisi Matematica. Aracne, I. (ed.). Roma.
Brandi, P., Salvadori, A. (1995) Sulla definizione di limite. Giornate di didattica, storia ed epistemologia della matematica. Trieste1995.
Brandi, P., Salvadori, A. (1996) Un nuovo approccio al concetto di limite. Atti Convegno Nazionale Mathesis. Verona 1996.
Brandi, P., Salvadori, A. (1997) Un moderno approccio al concetto di limite. Dall’idea intuitiva alla definizione con l’ausilio di computer-graphic, Proc. “Matematica ed arte, un sorprendente binomio”, Vasto.1997.
Brandi, P., Salvadori, A. (1997) Percorsi di Analisi Matematica. Libreria Athena Ed. Perugia.
[1] See the web site: http:www.innovamatica.it
[2] If = ± , we
consider Bn
= [n, +] or Bn = [-, n] respectively.
Internet as a tool in the preparation
of future mathematics teachers
Jaime Carvalho e Silva, José Carlos Balsa, and Maria José Ramos
Coimbra, Mira, Portugal
In this paper we describe a project that was developed with two groups of future mathematics teachers (7th-12th grades) that worked in two different schools (30km apart). They sent messages with weekly reports of their activities, comments, and files to a mailing list organized using the eGroups/Yahoo!® Groups system. The participation was considered to be very fruitful, and these future mathematics teachers became more aware of activities outside their daily routine, developing at the same time their communication skills; they exchanged more than 100 messages and more than 50 files (mainly with activities and exams). All considered this project to be a very important part of their preparation to be teachers of mathematics, showing how they can get new ideas and fight their isolation using the Internet. This project showed that the Internet is a very powerful tool for the preparation of teachers and should be used more frequently.
1 Background
Pre-service and in-service preparation of teachers is of key importance to any educational system. In Portugal, future mathematics teachers from grades 7 to 12 are prepared in the University. They must follow mathematics courses (usually 75% of the total), education courses and didactics for four years; the fifth year is a stage at a school where they teach one or two classes during one complete school year. The stage in the school is accompanied and assessed by two advisors: one teacher from the school and a professor from the University (in some universities in Portugal the structure varies a little).
In this paper we describe a project that was developed with two groups of seven future mathematics teachers (7th-12th grades) that worked in two different schools (30km apart). The first author was the scientific advisor of the two groups, and the other authors were the local advisors in each school. These seven student teachers and the three advisors subscribed an electronic mailing list of the eGroups/Yahoo!® Groups system: http://groups.yahoo.com/ . The seven future teachers sent messages with weekly reports of their activities, with comments about their successes and difficulties, and with files to this mailing list. One of the interesting characteristics of this mailing list is that the messages were simultaneously sent to all subscribers and archived in the Internet page of the list. This Internet page contains some other features, like zones for files, links, chat sessions, etc (see Fig. 1). This mailing list is easily customisable and exists in several languages (the Portuguese language was obviously used). The electronic list “FloresMira” was thus created using this very simple to use system. The messages were archived in
http://groups.yahoo.com/group/FloresMira
in the English and
http://br.egroups.com/group/FloresMira
in the Portuguese language.
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Fig. 1: The main page of the list |
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The plan of activities was the following: each week one of the student teachers from each group would send a message to the list describing (with some detail) the activities of the group in that week. Other messages could also be exchanged whenever there was some interesting topic. In the files zone they would put the files they would find potentially interesting for the others. Being student teachers they might not be enough at ease to carry out public discussions, so this list was made private. Only the description of the list, the links and some statistical data were available to the public. The main activities of this discussion list used the zones for the messages, the files and the links. Other zones were not used.
Messages: Every week, between November and July, each group send a message detailing which classes were taught that week and which other activities were carried out (personal work, seminars, visits with students, Pi day, etc). A total of 120 messages have been sent. Half of the messages discussed other topics not included in the weekly reports.
Files: A total of 61 files were transferred to the files zone of the list, occupying more than 14 MB (of a maximum of 20MB allowed by the system). Most of the files can be classified into four types:
worksheets (20)
tests (13)
school activities (12)
Geometer’s
Sketchpad files (10)
Links: This zone contains only 9 links to pages related with the official mathematics syllabus, other student teacher groups and some mathematics pages in Portuguese. This zone was rarely used, probably because there are other starting pages for mathematics that are well known.
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Fig. 2: The first messages exchanged on the list |
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2 A first assessment
For this first assessment each student teacher was asked their written opinion. All considered this project very favourably. They think the exchange of experiences and documents was very positive especially because they did rarely meet personally. Some students complained that although there are more than 100 mathematics student teachers at the university of Coimbra (divided into more that 30 groups in 30 different schools, some distant more than 100km from the city of Coimbra), they rarely meet and so cannot exchange ideas and points of view about this new work for them, at their schools. One of the students noticed that "projects like this one can fight the isolation of most groups of student teachers").
They also noticed that they were not all teaching the same grades, so they noticed details and difficulties of grades they were not teaching; one of the student teachers wrote he liked to become aware of "the difficulties faced by my colleagues and also the students". Also, the ones that were teaching the same grades noticed that "it was interesting to see how the same topic can be taught successfully in different ways".
They all wrote that the exchange of files was a very positive aspect. This exchange of files (worksheets, tests and other files related directly to the teacher activity) had such a big impact that they proposed, for a future project, live discussions, in person or through videoconference, at least once a month. That may not be easily carried out; nevertheless a chat session is easy to set up.
Another obvious consequence of this project was the opportunity it gave (or it forced) the student teachers to think about what they were doing. Some of them showed at first some difficulty in writing what they did and what they felt. But they would gradually write more complete and better reports. One of the student teachers remarked that "the fact that we had to write and read our reports was in a way a means of making a weekly summary of the activities and to have a better conscience of what was done or could not be done." Personal comments were always encouraged and when reading the whole set of messages we clearly see that this aspect was a success because all student teachers made interesting comments, quite appropriate in a lot of situations.
Finally, it should also be noted that these students became more proficient in the use of email and the Internet. They all considered that mastering these new tools was very important for their future professional activity.
3 Conclusion
This project would be impossible without the Internet. In fact, the regularity of the exchanges, the fact that everybody was receiving each message exactly when it was sent (and not some days or weeks later) would seriously undermine the impact of a project like this. The archiving characteristic made it impossible to loose a message or document. If an accident happened, the Internet archive would always have the missing message or file. Also, this set of files can be shown as an example to other groups of student teachers. We think this project proves that the Internet can easily furnish us the tools to solve some of the problems we face (at least in Portugal) in the pre-service preparation of teachers:
the
isolation of the different groups of student teachers;
the
reduced number of activities and projects that the student teachers contact
with, in one school year;
the
lack of contact with activities related to grades that are not taught in the
school where the student teacher works;
the
lack of analysis about the professional activities of the future mathematics
teacher;
the
lack of follow-up of the future mathematics teacher.
One of the student teachers noticed: "with this discussion list I could become aware that the Internet is not only a vehicle of a lot of information that allows a lot of searches, but is also, and mainly, a way to promote dialogue, discussion and confrontation of ideas, the exchange of experiences, the exchange of files, the information about activities,…"
If we think these benefits should be widespread, the university and the relevant departments should get more involved in the preparation of future teachers; that depends only on the fact that the preparation of future teachers is (or is not) considered an important priority of the educational system. Is it important enough?
References
Carvalho e Silva, Jaime (1997) A formação de professores em novas tecnologias da informação e comunicação no contexto dos novos programas de Matemática do Ensino Secundário. 2º Simpósio "Investigação e Desenvolvimento de Software Educativo", D.E.I., Coimbra.
http://www.mat.uc.pt/~jaimecs/pessoal/matnti.html
da Ponte, João Pedro (1998) Da formação ao desenvolvimento profissional. Actas do ProfMat 98, APM, Lisboa.
http://www.educ.fc.ul.pt/docentes/jponte/docs-pt/98-Profmat.rtf
da Ponte, João Pedro (1994) O Desenvolvimento Profissional do Professor de Matemática. Educação e Matemática (APM) Nº 31, 9-12 e 20.
http://www.educ.fc.ul.pt/docentes/jponte/docs-pt/94-Educ&Mat.rtf
Changing the classroom practices -
The use of technology in mathematics teaching
Isabel Fevereiro and Maria do Carmo Belchior
Lisboa, Portugal
1 Mathematics teachers’ NET
In 1997/98, 98/99 and 99/2000 an adjustment to the Mathematics curriculum guidelines for secondary grades (10th, 11th, 12th) has been implemented in order to improve and reinforce:
An
experimental teaching/learning process;
Learner
– centred classroom practices;
Solving
problems;
The
use of technology in Mathematics teaching.
From the beginning, in the Ministry of Education we realized that it was very important:
To
train the teachers in order to improve new practices in the classroom and use
the appropriate technology;
To
make the use of graphic calculators compulsory in the teaching/learning process
and in the final exams in 12th grade.
In order to achieve that, a Net of 80 teachers, each of them teaching two classes of secondary grades, was created. Those teachers are distributed throughout the country in order to attend all schools. They are grouped by district and they work in small groups of two or three. Each group of teachers (the Net Teachers) is responsible for a small group of schools (10 to 30). They have received a special training for this particular work (66h each year). All over the year Net Teachers organised meetings, debates and workshops with their peers from neighbouring schools. In those meetings they read and discussed the official syllabus, emphasizing the methodological suggestions for each content and they catalyse discussions about lesson plans, technological resources (especially graphic calculators), assessment and classroom practices. They share practices, experiences, concerns and exam questions.
In June 2000, with the national final exams in 12th grade, the first cycle of the Net came to an end. Nevertheless, in the Ministry of Education, we felt that the conditions for carrying on with this project were very good and there was a lot of work that could still be done with the teachers in schools. In addition, in September 2002 , a new curriculum for Math will be implemented. That means another adjustment to the actual syllabus and a new syllabus for Social Science Studies. So, in September 2000, a new cycle started for the Net teachers, and the incentives for workshops and training courses were reinforced. In the workshops, teachers worked essentially with mathematical models, graphic calculators, computers, sensors, and modes of assessment. In the training courses, teachers have to produce classroom materials and experiment innovative activities using technology.
2 The use of technology
Graphic calculators
Net teachers and teachers in schools are using the G.C. when they are working with functions, statistics and probabilities. We will take a look at two exam questions that appeared in the final exams in 2000 and 2001, as well as at one of the different activities that were experimented in some schools.
Final exam - 12th grade - June 2000
In Lisbon, the time between sunrise and sunset, in day n, is given by:
f (n) = 12.2 + 2.64 sin (p (n-81) / 183) n Î {1, 2, ......366}
1. On 24 March, sunrise occurred at six thirty in the morning. At what time did sunset happen? ( give the result rounded to the nearest unit)
2. In some days of the year, the time between sunrise and sunset is greater than 14.7 hours. Using the calculator, find out how many days that happen. Explain your reasoning.
Scoring criteria: In question 2 the students are expected to write the problem through an inequality f(n) > 14.7 and solve it by using the G.C.
Final exam - 12th grade - June 2001
f is a function, domain R+ so that: f (x) = 3x – 2 ln x
1. Using only analytical methods :
study the existence of asymptotes in the graph of f.
show that f has only one minimum.
2. The graph of the function f has one unique point so that its y-coordinate is equal to the square of its x-coordinate. Using the calculator, find an approximate value for the x-coordinate (round to one decimal place). Show what you have done.
Scoring criteria: In question 2 students are expected to interpret and write the problem through an equation (3x – 2 ln x = x2 or equivalent) and they have to justify the method they have used to solve graphically the equation. A sketch of the function f and the parabola is expected, clearly showing the intersection point, or a sketch of the function 3x – 2 ln x – x2 showing its zero.
These exam questions are two good examples of the kind of activities developed in schools using the G.C. Teachers that have corrected the exams say that it is easy to distinguish students that worked with the G.C. in classroom, from students that are not familiarised with its use. The first ones show clearly a different vision of the problem by using more easily connections between the graphical and analytical representations to justify their reasoning.
A typical activity - Which triangle has the biggest area?
Fold a sheet of paper so that the upper corner touches the lower side. Which is the largest triangle in the lower corner of the sheet? Study this problem and do a report on your research.
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Fig. 1a |
Fig. 1b |
This is a typical research activity, where the teacher can discuss with the students an analytical resolution and/or an experimental one; it can be applied to different contexts, especially when they are studying cubic functions and curves that best fit a set of given points. When students work on an experimental resolution they have to collect data; represent data in a table; get a graph and find an expression that best fits the graph.
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0 |
2 |
4 |
6 |
8 |
10 |
12 |
14 |
16 |
18 |
20 |
21 |
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Area |
0 |
10.4 |
20.2 |
28.8 |
36 |
40.5 |
42.6 |
40.6 |
35.2 |
25.2 |
10 |
0 |
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When analysing the graph and because the problem is about the triangle area, at first students may think this is a quadratic function representation. So, when looking for the curve that best fits the data, they will probably experiment the quadratic regression. |
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Fig. 2a |
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Fig. 2 b |
Fig. 2c |
Afterwards they can experiment the cubic regression and see it is the most appropriate. We cannot forget this is an intuitive study and an investigative activity.
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Fig. 3a |
Fig. 3b |
They can also make an analytical study. They will find a cubic expression and get the graph in the calculator.
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h2 + x2 = (21 - h)2 h = (441 - x2) / 42 A = ( x (21 - x) (21 + x) ) / 84 |
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Fig. 4 |
The discussion in the classroom will be an excellent opportunity to debate the different strategies used.
The computers
At present, just a small number of teachers are using computers in the classroom. Net teachers are organising some training courses in that area and are working essentially with Dynamic Geometry (Cabri and GSP) for geometry and functions, modelling problems (Modellus) and Statistics (Excel).
We will take a look at some typical activities worked in the classroom:
Filling an octahedron – Dynamic geometry (GSP)
Represent an octahedron so that one of the spatial diagonals will be
vertical;
Simulate its filling with a liquid;
Make a conjecture and construct a sketch (using paper and pencil) for
each one of the functions:
- The perimeter of the liquid section, as a function of the liquid height;
- The area of the liquid section, as a function of the liquid height
- The volume of the liquid, as a function of the liquid height
Using the computer, build those function graphs, and compare them with
the sketches you made using the pencil.
In order to solve the problem, students are expected to improve their competence in using technology (in particular the GSP), in using the right properties of spatial and plane geometry and they can develop their competence in graphic and analytical representation by establishing the right relationships between both representations.
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Fig. 5 |
The model – Modellus
The photo shows the movement of a ball on the rebound. Use it to determine the model of the curve.
i) Do the right measurements to get the equation representing the parabola.
ii) Find the model and implement it.
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To do this activity students should know the parabola properties and do the right measurements, in order to find the adequate parameters to get a good model of the curve.
The Sensors
Net teachers are working with sensors and are promoting its use. Here are some examples of activities with sensors and the context where they are being used.
The pendulum - Mathematical models describing oscillating objects: Functions with radicals.
Cooling a liquid - Mathematical models describing the temperature variation with the time variable: Exponential functions
Light intensity - Mathematical models describing the light intensity variation with the time variable and with the distance variable. Periodical functions / Trigonometric functions
Walking- Mathematical models development that best fit graphs relating the variation of the distance, the velocity and the acceleration with the time variable.
3 After four years of Net
What is changing? A large number of teachers are implementing new practices in the classroom and they are working with technology namely the Graphic Calculator. So, in the teaching/learning process, teachers and students are using technology for:
doing numerical problems approaching;
modelling; simulating; and solving problems;
using visual scenarios to illustrate mathematics concepts;
using visual methods to solve equations and inequalities;
doing mathematics experiences; making and analysing conjectures:
studying and classifying the behaviour of different function classes;
anticipating differential calculus concepts;
investigating and exploring different relations between different
representations for a problem situation.
Mathematics teachers are implementing and reinforcing a collaborative work with their peers in schools, and between neighbouring schools. They are sharing materials and experiences, and they are reflecting upon doubts and difficulties they face.
We know that there are some difficulties in the technology implementation in the classroom because some teachers do not feel comfortable when using it and/or they do not understand its pedagogical use. Therefore they do not believe its use can help improve the mathematics teaching/learning process. The other problem with the technology implementation is the equipment of schools, especially computers and sensors. Since 1997, some schools have been applying, to educational projects, in order to obtain finance resources to get the adequate equipment for a math laboratory. But that is not enough. In the Ministry of Education we are trying to supply schools with the necessary equipment. The new syllabus that will be implemented in 2002 is very clear about the importance of the use of the technology in Mathematics teaching. We must make all efforts to make that possible in all schools.
Integrated teaching mathematics with elements of computer science
Henryk Kakol
Cracow, Poland
When in the 80s first PCs appeared in Polish schools teachers were very enthusiastic about that fact. This modern technical instrument was believed to change school teaching. Especially maths teachers had high hopes for the computer. They wrote computer programs, gave demonstration lessons. Training courses, scientific conferences for maths teachers were organized and special magazines were published. It was generally believed that this new teaching instrument would change the image of mathematics at school. Instead of static, dull mathematics, disliked by most pupils, a completely new mathematics would appear, and school results would improve automatically.
A dozen or so years passed. Modern computers appeared and were found at schools. At present nearly all Polish schools have computer rooms well equipped while teaching mathematics generally is traditional. During school lessons chalk and blackboard are still teaching instruments frequently applied. A handful of enthusiasts of using computers in maths teaching are not able to change this picture of the school; the enthusiasts who have completed postgraduate studies in computer science and try to “smuggle” mathematical content to computer classes; the enthusiasts who having completed various training courses try to bring the computer to maths lessons and apply new maths programs.
What are the reasons for such a situation? There are many of them. Let us list some of them.
Maths
teachers (as well as teachers of other subjects) have no access to a computer
room.
Majority
of them do not feel like using computers during maths lessons.
In
general, they cannot operate this appliance, do not know good professional
maths programs.
There
is not enough relevant teaching literature, particularly in Polish.
Teacher
training in this field is not satisfactory.
Under such circumstances the integrated Program of Teaching Mathematics with Elements of Computer Science in Gymnasium has been created under my direction. It eliminates many of the above-mentioned problems that hinder introduction of new computer techniques to teaching mathematics. It offers teaching mathematics and elements of computer science in the form of one thematic block. Let us explain its main assumptions.
In teaching mathematics we introduce in turn, of course in a spiral way, individual topics covered by the compulsory curriculum in a given country (see Figure 1).
Fig. 1
We teach the above using remarkable achievements of maths didactics and traditional teaching instruments while, as a rule, we do not apply new computer technologies during the course of organizing this process.
In teaching computer science, like in teaching mathematics, we also teach relevant sections (see Figure 2) while using mathematics, which is a natural resource for teaching particular issues, but mathematics plays here only an auxiliary role.
Fig. 2
While introducing, for instance, the spreadsheet, we give number based examples, use algebraic expressions, the concept of function and its graph like while teaching other issues included in the program of teaching computer science.
The Program of Teaching Mathematics with Elements of Computer Science in Gymnasium offers the following way of implementation of mathematical and computing content. The picture below shows, for instance, the use of spreadsheet in individual topics.
Fig. 3
Applying the spreadsheet for the first time with topics related to number based calculations we teach its construction and applications but only within the frame in which it is necessary at the time being. We come back to it with algebraic topics, teach its next features and apply it when we think it should be applied. We do similarly with next topics: Functions, Statistics and Probability. In this way, on one hand, the pupil learns gradually, in a spiral way, the features and applications of the spreadsheet and, on the other hand, he or she learns how to apply it in the process of learning mathematics.
The general assumption of this program is to use the computer in the course of teaching mathematics when the teacher comes across various difficulties, often impossible to overcome with the use of traditional teaching methods and instruments previously applied. In this conception the computer in the process of teaching mathematics is treated mainly as an aid to accomplish the planned goals. It does not mean that during the implementation of this teaching block computer content is not concerned. On the contrary, in order to use the computer effectively in teaching mathematics a block of computer classes has been introduced. During such classes pupils get familiar with computers and their operation, learn how to use computer program and network resources. All these actions are conducive to create a comprehensive computing culture among pupils and, as a consequence, an appropriate attitude towards mathematics.
Teaching mathematics and computer science in one subject block is integrated. Mathematical education, where a great emphasis is put on proper forming mathematical concepts, mathematical reasoning, solving problems and forming mathematical language, takes advantage of enormous calculating, graphic and simulating possibilities of the computer. On the other hand, computing education on a stage which should prepare young people for life in a computing society is carried out on the basis of mathematical material which is an excellent material to develop various forms of activity, not only mathematical but also those which are essential in everyday life. Hence the program suggests teaching mathematics and computer science both during lessons in mathematical and computer rooms.
The program and materials for its implementation already prepared (student’s and teacher’s books, computer programs) prefer problem teaching. In such a system of work the teacher is not any more a transmitter of ready-to-learn mathematical and computing knowledge but he or she becomes an organizer of pupils’ self-learning, creates appropriate conditions, manages the process skilfully and controls its effects.
For the implementation of the content covered by the program 6 hours weekly are provided:
4 hrs in the maths room;
2 hrs in the computer room with division in
groups.
The maths room should be equipped, except for traditional teaching aids, with one computer with a projector or a big TV monitor. The computer room should be furnished with an appropriate hardware to ensure the installation and application of basic computer programs such as: text editors, spreadsheets, and auxiliary programs for teaching geometry, graphic editors or other educational programs.
The integrated teaching mathematics with elements of computer science raises many questions and research problems.
Does
such teaching stimulate pupils’ interest in mathematics?
To
what extent are pupils’ difficulties in learning mathematics soothed and
liquidated during the implementation of the above content? Or, on the contrary,
do additional difficulties in learning mathematics appear?
Does
learning elements of computer science while learning mathematics enable pupils
to use the computer successfully in other educational fields, as well as in
their everyday life?
Will
the use of computer in such a wide range result in a change in style of
teaching, i.e. departure from methods of passing knowledge to searching ones,
in the direction of problem teaching?
Integrated teaching mathematics with elements of computer science, as many specialists think, will improve the results of teaching mathematics which neither parents nor educational authorities, nor teachers or, finally, pupils themselves, are satisfied with. The knowledge acquired will be, undoubtedly, an important element of comprehensive education of the young people who can not only understand but also change the surrounding reality.
Innovations in mathematics, science and technology teaching — IMST²
Initial outcome of a nation-wide initiative for upper
secondary schools in Austria
Konrad Krainer
Klagenfurt,
Austria
2. Results of the IMST research project on the status quo
3. Innovations in Mathematics, Science and Technology Teaching – a nation-wide initiative
4. Initial outcome of the initiative IMST² (pilot-year 2000-01)
Following the poor results of Austrian high school students in the TIMSS achievement test, a research project was set up in which the results were analysed and additional investigations into the situation of mathematics and science teaching were started. As a consequence, the initiative IMST² was launched to support teachers’ efforts in raising the quality of learning and teaching in mathematics and science. In the school year 2000-01, 126 Austrian schools participated in total with about one quarter collaborating more intensively with the IMST²-team and documenting one or more innovations at their school. The concept, initial experiences and findings of IMST² are presented and discussed here.
1 Preliminary remarks
Whereas in the last few decades many countries launched reform initiatives in mathematics and science instruction, similar systematic steps in Austria did not really happen. There is a big gap between intended and implemented instruction, in particular with regard to upper secondary schools. Although the promotion of understanding, problem solving, independent learning, etc. and the use of manifold forms of instruction and didactic approaches are regarded as important, in reality, teacher-centered instruction and application of routines dominate. At the same time, Austria does have a variety of dedicated teachers and innovative teaching approaches across the country, supported by some promising initiatives. Nevertheless, there is no adequate nation-wide support system for mathematics and science teaching that would promote systematic professional communication among teachers, and among teacher educators, as well as between these two groups. As is now clear, the educational system needed an external and publicity-driven impulse which appeared in the form of the Third International Mathematics and Science Study (TIMSS).
Whereas the results concerning the primary and the middle school were rather promising, the poor results of the Austrian high school students (grades 9 to 12 or 13), in particular with regard to the TIMSS advanced mathematics and physics achievement test shocked the public. The ranking lists showed Austria as the last (advanced mathematics) and the last but one (advanced physics) of 16 nations (see e.g. Mullis et al. 1998, 129 and 189). As a reaction, the responsible ministry launched the research project “Innovations in Mathematics and Science Teaching" (IMST)[1] in the year 1998. The task was to analyse the situation and to work out suggestions for the further development of mathematics and science teaching in Austria. The universitary project team started from the assumption that international comparative studies like TIMSS are good starting points for analyses of the national situation. However, due to differences in the general cultural, societal and economic conditions that lead to different curricula or to typical regional patterns of instructional practices (see e.g. Cogan & Schmidt 1999) etc., the comparability is limited. Each test constructs its own “test reality” (with specific attainment goals) which differs from the various “teaching realities” in different countries. Achievement tests alone might give some insights into differences in students’ achievement, however, they do not contribute to our knowledge on the process of learning and teaching and, very importantly, to its further development. Therefore, the project team decided not only to analyse the results of TIMSS but also to carry out additional analyses. The research project IMST (1998-1999) aimed at contributing to the following tasks (see e.g. Krainer 2002): analysis of TIMSS-items and achievement results, suggestions for using the available TIMSS-items, contributions to an analysis of the state of Austrian upper secondary mathematics and science teaching, brief description of exemplary reform initiatives in other countries, and suggestions for consequences on the basis of the national and international analyses.
2 Results of the IMST research project on the status quo
Below the main findings of the IMST project on the status quo of mathematics and science teaching at upper secondary schools in Austria are summarised (for further details see e.g. Krainer et al. 2000):
The absolute ranking lists based on the TIMSS-achievement results
are questionable because they do not take into account the specific sample of
students that were involved in the test. For example, Austria – unlike most
countries – sent many students to the advanced mathematics and physics
achievement test who had not been taught these subjects in that school year.
Other (more fair) comparisons (such as the TOP 5% or 10% students’ achievements)
show slightly better results, nevertheless, the picture remains disappointing.
In all the TIMSS tests, literacy and advanced as well as mathematics
and science, Austria is among those nations with the biggest achievement
differences between boys and girls.
Again from the TIMSS tests, Austrian (and German) students show poor
results in particular with regard to items which refer to higher levels of
thinking (see e.g. the analysis concerning mathematics literacy in Baumert et
al. 1998).
In their response to the item in the TIMSS-questionnaire concerning
reasoning tasks in lessons[2], less than a third of Austrian students felt that they
are involved in reasoning tasks in most or every mathematics lesson(s),
resulting in the last but one place in the international ranking of 16 nations.
In physics the figures were half and last place (see e.g. Mullis et al. 1998,
165 and 221).
Interestingly, Austrian (and German) students who feel that they are
asked to do reasoning tasks every mathematics lesson – on average – do not have
significantly better test results than those who feel they do reasoning tasks
in some lessons. The involvement of students seems to be done less effectively
than in other countries. This seems to be an outcome of a special kind of teacher-centered
instruction in German speaking countries. This „fragend-entwickelnder
Unterricht“ aims at leading a whole class to the intended goal of the lesson
through posing (mostly „small step“) questions to the students. However,
interaction studies have shown for a long time (see e.g. Voigt 1984) that
during the process more and more students cannot or will not follow, but even a
few students might give the teacher the feeling of successful teaching. In the
long run, such a method leads students to question whether their active
reasoning really has an impact on the process and the outcome of instruction.
The answers to a written questionnaire by Austrian teachers, teacher
educators and representatives of the education authorities supported the
results from the TIMSS-data. For example, teachers were predominantly seen as
dedicated and as having a lot of pedagogical and didactic autonomy. However,
this autonomy is sometimes restricted by themselves or by general conditions
and therefore often not passed on to the students. The analysis further showed
that students’ active involvement in the teaching and learning process is seen
as a major weakness of mathematics instruction in upper secondary schools.
An analysis of web sites at schools in Carinthia (southern part of
Austria) showed that schools aim at convincing the public with regard to the
quality of their work with a variety of initiatives. However, mathematics and
science initiatives are extremely rare, whereas information technology and
(predominantly English) language initiatives seem to attract much energy from
students, teachers and principals.
This concern has been underlined in a workshop with principals at
upper secondary schools who pointed out that mathematics and science teachers
in general don’t belong to the „powerful“ groups of teachers. This has a
magnifying impact on many questions, for example, whether a school decides to
set a focal point on mathematics and science teaching.
Mathematics education and in particular science education are poorly
anchored at Austrian universities. In chemistry education, for example, no
university has a university professor for that scientific domain. Teacher
education is dominated by subject experts, the collaboration with educational
sciences and schools is – with exception of a few cases – underdeveloped. A
competence center like the Freudenthal Institute at the University of Utrecht
in The Netherlands or the Institute for
Science Education at the University of Kiel in Germany does not exist.
The picture with regard to documented innovations in mathematics and
science teaching was ambivalent. On the one hand, it was astonishing how many
creative initiatives were carried out by individuals, groups or institutions.
On the other hand, it was irritating to see how unlinked these activities were,
and that a networking structure was missing. This impression is repeated when
looking at the whole educational system (two different pre-service teacher
education systems that are nearly unconnected, a variety of different kinds of
schools with corresponding administrative bodies in the ministry and the
institutions for in-service education, etc.). This shows a picture of a
„fragmentary educational system“ with people from schools, teacher education
institutions, administration, etc. which form a loosely-coupled,
self-reproducing system of lone fighters. The consequence is a high level of
(individual) autonomy and action, however, less reflection and networking (see
e.g. Krainer 2001).
Thus in the IMST analyses a complex picture of diverse problematic influences on status and quality of mathematics and science teaching has emerged. It was the background for the IMST-team to suggest the launch of a long-term nation-wide initiative IMST² – Innovations in Mathematics, Science and Technology Teaching involving the subjects biology, chemistry, mathematics and physics. The addition of "Technology" in the project title is to express the fundamental importance of technologies for mathematics and science teaching. The four-year initiative, starting with a pilot-project IMST² in the school year 2000-01, is being financed by the Federal Ministry of Education, Science and Culture.
In the following, the initiative’s goals, tasks and intervention assumptions are briefly described.
3 Innovations in Mathematics, Science and Technology Teaching (IMST²) – a nation-wide initiative
The long-term goals of the IMST² initiative are:
Better basic education – higher quality of
understanding, problem solving, reasoning and reflection
Bigger variety of teaching and learning styles –
creativity, independence, gender sensible teaching and learning, supported by
new media and technology
More, better designed forms of professional exchange
of experiences among teachers, contributing also to the further development of
the whole school
Setting up and further developing a network that
supports carrying out and evaluating innovations, and for communicating these
in various forms to a wider public
Improved „image“ – more favourable perceptions
and expectations with regard to mathematics and science in schools and society
In the pilot-project 2000-01 the main tasks were to work out a detailed master-plan for the continuation of the initiative and to start supporting innovations at schools. In the following three years the support of innovations at schools will be continued and the establishment of a support system will be started.
Four priority programmes (S1 – S4) have been established with the following reasoning:
Basic education
(S1): The unclear expectations concerning qualifications, knowledge and
contents that students need when leaving secondary school. The four S1-teams
(biology, chemistry, mathematics and physics) support initiatives at schools
that reflect such expectations and they aim at working out (interdisciplinarily
interconnected) concepts for basic education at the upper secondary level for
the four subjects. These concepts – intended to be generated by theoretical
considerations and by practical experiences from the collobaration with schools
and thus negotiated by a wider form – are expected to be a key element for a
support system for mathematics and science teaching. It is assumed that
teachers’ clearer view on the importance of goals and content might raise the
quality of learning and teaching.
School development (S2): The relatively low status of the subjects biology, chemistry,
mathematics and physics at schools, in comparison to their importance in
society and the economy, might lead, in times of greater autonomy of schools,
to a situation where, in general, these subjects are left behind when schools
change their profile. The S2-team supports schools that set a focal point on
mathematics and science teaching and tries to establish a network of such
schools. In parallel, and using the practical experiences, it aims at working
out a concept that reflects the initiation, support and evaluation of school
development processes that (partially) focus on the enhancement of mathematics
and science teaching. This concept is also to be supposed as an important
element of a future support system. It is assumed that organisational
development (often underestimated in subject didactics) – when fairly linked
with classroom development – makes a crucial contribution to the quality of
learning and teaching.
Teaching and learning processes (S3): The dominance of relatively passive forms
of learning, not suffiently taking into account the individual needs of
students in general, and the low interest and the poor results of Austrian
girls in the TIMSS-achievement test in particular. The S3-team both supports
innovations at schools focusing on situation-appropriate teaching and learning
processes and aims at working out a concept for generating, analysing and
evaluating such processes. Such a concept, supplemented by material like a CD
with video-clips of real teaching that is intended to be used in pre- and
in-service teacher education, should support teachers’ growth in planning and
reflecting on their own teaching. It is assumed that such an increased
competence has a deep impact on teaching and learning processes.
Practice-oriented research (S4): The lack of well-developed practice-relevant research and development
in mathematics education and in science education in particular. The S4-team
initiates, finances and supports teams of school teachers or universitary
teacher educators (or mixed teams) who carry out investigations into their own
teaching (action research) or classical research projects. Following the IMST
analyses, the promotion of students’ independent learning is seen as a major
goal, hence the projects focus on that issue. The team also aims at working out
a concept for the promotion of subject-didactic research and culture. Through
raising teachers’ and teacher educators’ interest and competence in
practice-relevant research, the network of researchers in mathematics and
science education is expected to grow, both in quality and quantity. A stronger
mathematics and science education, where theoreticians and practitioners
collaborate more intensively, is expected to be a fundamental part of a support
system for school practice.
This shows that each of the four priority programme teams has two important – closely interconnected – tasks: firstly, to support innovations at schools (and in S4 also in teacher education) and secondly, to work out concepts that help better to plan, describe and understand such innovations.
Innovations are the key feature of the way of IMST² towards establishing a nation-wide support system. The corresponding basic assumptions behind this intervention into the educational system are:
Starting
from strengths: Innovations are initiated, supported and made visible, thus motivating others to
join in, stimulating “attraction” instead of generating “pressure”.
Innovations
are not regarded as singular events that replace an ineffective practice but as
continuous processes that lead to a
natural further development of practice.
Participation
is voluntary, teachers and schools
have the ownership of their
innovations.
There is no
“best practice” which might be defined by an external authority. For each
learning and teaching different
approaches to „good practice“ exist. Innovations are planned steps towards
a “good practice”.
Through
innovations and reflections teachers
construct their own professional growth (likewise the students are seen as
active learners).
Writing down the experiences in a systematic way means a second cycle of reflection
and opens the opportunity for more people to learn from those experiences.
The
dissemination of innovations passes along personal
relationships and experiences.
One powerful
strategy for widespreading innovations to a whole system is to initiate regional networks and to promote their
communication with other networks.
Another important feature of IMST² is the emphasis on supporting teams of teachers from one school. The background for that approach is the experience that working with single teachers from different schools may often cause considerable progress for individual teachers but does not necessarily have any impact on other teachers in their school (see e.g. Loucks-Horsley 1998; Borasi, Fonzi, Smith, & Rose 1999; Krainer 2001). If professional communication among teachers is not an important feature of the culture of a school, innovations by individual teachers remain limited to their own heads and classrooms. Even a pair of colleagues co-operating successfully might not be enough as a critical mass. Of great importance is the support of the principal. In IMST², therefore, the teams of priority programmes sign contracts with teams of teachers, and these documents which define the goals and content of the collaboration are also signed by the principal.
It is taken for granted that schools have different starting points concerning interests, resources, time, etc. IMST² schools can therefore choose their intensity of participation (within one school year), getting the status of an information school, contact school, colloboration school or focus school. For example, to become a contact school, one subject or interdisciplinary team of teachers has to collaborate in IMST². To become a focus school, two teams in one school collaborate in one of the priority programmes and a steering group is involved in the further development of mathematics and science teaching at that school.
The role of team members working with schools is to support the teachers’ struggle for professional growth, to generate new knowledge about this supporting process and about teacher’s growth, and to apply this new knowledge in forthcoming support processes.
Evaluation is an integral part of the IMST² initiative whereby three different functions have been defined:
The process-oriented
evaluation should generate in a continuous feedback process steering
knowledge for the project management and the project teams in order to further
develop the internal structures and processes. Sample instruments are
interviews with team members on their view on the strengths, weaknesses,
opportunities and threats of the project or feedback by an advisory board
(consisting of representatives from theory and practice).
The outcome-oriented
evaluation should work out the impact of the project at different levels of
the educational system (students, teachers, schools, teacher education
institutes, etc.). Sample instruments are case studies about teachers’
professional growth or questionnaires for schools (e.g. assessing the clarity
of the project goals).
The knowledge-oriented
evaluation should generate new theoretical and practical knowledge which
will form a basis for improving support to innovations at schools.
4 Initial outcome of the initiative IMST² (pilot-year 2000-01)
The project started at a time where public discussions about teachers’ work and other topics of education policy led to a rather passive behaviour by several Austrian schools. Furthermore, when the 582 upper secondary schools got the first information about the project (beginning November 2000), the school year was nearly two months old. This meant that many schools had started a lot of other activities. Nevertheless, 22% of all target schools expressed an interest in participating in IMST². During the school year 2000-01, 32 focus and collaboration schools and two universitary teacher education teams were supported, carrying out 38 innovations and research projects (36 at schools, 2 at universities). Given the fact that it needed some time for the participants to become familiar with the IMST² approach, to develop first plans for activities, and to coordinate their plans with other teachers and the principal, not much time remained neither for carrying out and reflecting on the innovations nor for the opportunities of the project team to support the teachers’ activities.
Results of a questionnaire
In February 2001, a questionnaire was sent to 86 contact, collaboration and focus schools in order to get a preliminary feedback (see e.g. Specht in IFF 2001). 63 questionnaires (73%) were sent back and showed a representative distribution concerning the four priority programmes. It was not surprising to see that the decisions for collaborating in IMST² were mostly taken by single persons (52% teachers, 12% principals). In no case was the decision made during an official school meeting, neither in a teacher conference nor in a school partnership forum. This reflects our experience that only few schools have established subject-related teams and fora where teachers regularly meet and share experiences. The responses to another question (previous forms of collaboration among teachers) show a similar picture: only 30% of schools reported that they had already systematic collaboration among their subject group or in an interdisciplinary context; 59% regarded the collaboration as informal, and 11% even felt that there was no subject-related collaboration among colleagues at their school at all. Both results support our view that Austrian teachers to a large extent stay as lone fighters in their schools. Considering the four dimensions of professional practice – action, reflection, autonomy and networking (see e.g. Krainer 2001), there is much individual autonomy and much action, but less reflection and networking among teachers. This underlines both the necessity and the big challenge of IMST² to work with teams of teachers (and not individuals) in order to contribute to the establishment of a culture of professional communication and collaboration among teachers. 31% of the schools reported that they did not carry out previous or recent initiatives for the further development of mathematics or science teaching, thus taking IMST² as the first opportunity to jointly share experiences and get external support. This means that the project reaches a considerable amount of teachers that had not been involved in joint activities concerning mathematics and science teaching at their school so far. The schools’ reasons for participating in IMST² are predominatly pedagogical and intrinsic in origin: “raising students’ interest and understanding”, “further developing the culture of teaching and assessing”, and “improving students’ achievements” were the most commonly named motives, whereas for example “proposal by the principal” was ranked last.
Examples of innovations at schools
Four innovations at IMST²-schools that teachers planned and carried out during the school year 2000-01 are briefly sketched. They all relate to the use of technology in mathematics teaching and each stems from one of the four priority programmes:
S1 Basic education: The project „Promoting talented students in mathematics teaching“ at a Higher Vocational School (HAK) supports grade 10-students’ work on the topics „interpolation“ and „regression“. The students work almost independently in pairs using Mathematica and MathSchoolHelp.
S2 School development: The project „Mathematics with Derive and Excel in a notebook class“ at a Grammar School (Gymnasium) takes advantage of the fact that all students of a grade 9 class have received a notebook. The students work on topics like „linear quadratic functions“ and „circumcentre of a triangle“.
S3 Teaching and learning processes: Following a variety of preparatory initiatives, the project „How do CAS and intelligent calculators change mathematics teaching?“ at a Higher Technical School (HTL) will investigate school-leavers’ beliefs concerning the impact of CAS, with a particular focus on the gender aspect.
S4 Practice Research: Within the project „Trigonometry“ at a Grammar School (Gymnasium) grade 10-students independently work in a sequence of „stations“ with different learning activities. At one station the students (using a TI 85 or TI 92) were supported by a student of grade 12.
Some sample teachers’ comments from their written reports on the use of technology are:
The
use of technology promotes a variety of learning styles („makes math less dry“)
It promotes independent and active
learning (Students like this kind of work and it gives
them a lot of freedom and space for initiative.).
It
means new roles for students and teachers (The teacher
becomes a facilitator for his/her students on several levels: e.g. in the case
of problems with hard- and software as well as with the students’ independent
study of mathematical problems. This situation significantly enhanced the
relationship between students and teachers.
It
is labour-intensive and generates high expectations on the teacher (“The students expect that we know an answer to every question and a
solution to every problem, and besides we should help each student quite
individually ...”).
The
challenge of the use of technology is to find a good balance between the
learning of high/low achievers and girls/boys because technology tends to widen
the gap (“Low-achievers have a smaller chance than in traditional
lessons to regain lost ground through hard work and practice.”).
It might be argued that such results are not new at all and can be read in several publications. However, whether a specific teacher really finds in such research reports the viable support with regard to his or her context and situation is questionable. It is the basic assumption of IMST² that teachers through starting from their own questions, investigating relevant aspects of their practice, collaborating with other teachers at their own school, getting support from teacher educators, and writing down their findings, have a better chance to construct the own local knowledge they need to meet the challenges of their practice.
References
Baumert, J. and Watermann, R. (2000). Institutionelle und regionale Variabilität und die Sicherung gemeinsamer Standards in der gymnasialen Oberstufe. Baumert, J., Bos, W. and Lehmann, R. (eds.): TIMSS/III. Dritte Internationale Mathematik- und Naturwissenschaftsstudie – Mathematische und naturwissenschaftliche Bildung am Ende der Schullaufbahn, Band 2. Opladen, Leske & Budrich, 317-372.
Borasi, R., Fonzi, J., Smith, C.F. and Rose, B. (1999) Beginning the Process of Rethinking Mathematics Instruction: A Professional Development Program. Journal of Mathematics Teacher Education 2 (1), 49-78.
Cogan, L. & Schmidt, W. (1999) An Examination of Instructional Practices in Six Countries. Kaiser, G., Luna, E. and Huntley, I. (eds.): International Comparisons in Mathematics Education, 68-85. Falmer, London & Philadelphia.
IFF (ed.) (1999). Zwischenberichte und Endbericht zum Projekt IMST – Innovations in Mathematics and Science Teaching. Im Auftrag des BMUK. IFF, Klagenfurt.
IFF (ed.) (2001). Zweiter Zwischenbericht zum Projekt IMST² – Innovations in Mathematics, Science and Technology Teaching. Im Auftrag des BMBWK. IFF, Klagenfurt.
Krainer, K. (2000). Das Projekt IMST² als Ausgangspunkt für eine Reforminitiative zur Weiterentwicklung des österreichischen Mathematik- und Naturwissenschaftsunterrichts. Mitteilungen der Gesellschaft für Didaktik der Mathematik, vol. 70, Juni 2000, 77-85.
Krainer, K. (2001) Teachers’ Growth is More Than the Growth of Individual Teachers: The Case of Gisela. Lin, F.-L. and Cooney, T. (eds.): Making Sense of Mathematics Teacher Education, 271-293. Kluwer, Dordrecht, Boston & London.
Krainer, K. (2002) Ausgangspunkt und Grundidee von IMST². Reflexion und Vernetzung als Impulse zur Förderung von Innovationen. Krainer, K., Dörfler, W., Jungwirth, H., Kühnelt, H., Rauch F. and Stern, T. (eds.): Lernen im Aufbruch: Mathematik und Naturwissenschaften. Pilotprojekt IMST². StudienVerlag, Innsbruck-Wien-München-Bozen.
Krainer, K., Dörfler, W., Jungwirth, H., Kühnelt, H., Rauch F. and Stern, T. (eds.) (2002) Lernen im Aufbruch: Mathematik und Naturwissenschaften. Pilotprojekt IMST²: StudienVerlag, Innsbruck-Wien-München-Bozen.
Loucks-Horsley, S., Hewson, P.W., Love, N. & Stiles, K.E. (1998). Designing Professional Development for Teachers of Science and Mathematics. Thousand Oaks, California: Corwin Press.
Voigt, J. (1984) Interaktionsmuster und Routinen im Mathematikunterricht. Beltz, Weinheim & Basel.
[1] The full project title is in English in order to indicate the importance of thinking in international dimensions. At the same time, the abbreviation IMST is the name of a town in Austria in order to highlight the importance of seriously taking into account the specific regional context.
[2] Students taking the advanced mathematics and physics test were asked how often they have to do reasoning tasks in their lessons. The possible answers were „never or almost never“, „in some lessons“, „most lessons“ or „every lesson“.
Current educational theories and New Technologies:
Development of a training program for mathematics teachers in the Philippines
Auxencia A. Limjap
Manila, Philippines
2. A current teacher training program
3. A close look at the students of the SIGS program
4. A proposed model teacher training program
5. A technology based SIGS program of the De La Salle University
1 Introduction
Large-scale in-service training programs have been provided by the Philippine government in recognition of the need to improve the quality of teaching school mathematics and the sciences in the Philippines. Various private sectors in the Philippine society including educational institutions also initiated intervention programs to upgrade teacher competencies. Concerns were raised though, on the forms of delivering in-service training, and their capability to effect changes both in the whole educational programs of our country, and in the teacher’s classroom practice. Bernardo points out that “in-service programs are conceived in formal models, wherein teachers upgrade their teaching knowledge by enrolling in formal graduate degree programs” in teacher training institutions. This works under the presumption that the training providers truly explore and develop new and improved ways of teaching.
Reform movements on mathematics education in different parts of the world point out to the need to adopt a cognitivist view of instruction that focuses on the nature and process of mathematics learning. Proponents advocate constructive learning and gear teaching towards the development of meaningful quantitative and qualitative thinking. They adhere to the social origins of cognition and situate learning in realistic settings. They harness technology as a learning resource that provides both context and support for meaningful problem solving activities. Consequently, learner centred educational theories proliferated with the advances in educational technologies.
In the Philippines, research shows that while “the teachers [a]re acquainted and appreciative of more contemporary learner-oriented and constructivist perspectives to teaching and learning, they also cl[i]ng strongly to beliefs closely intertwined with the traditional transmission models of teaching and learning”. This further puts into question the effectivity of in-service training programs designed to help the teacher implement instructional innovations. This also brings to light other factors that may affect the teacher’s pedagogical practices.
Developments in pedagogy and didactics pose a big challenge to school mathematics teachers especially those who have neither experienced the constructive process of learning mathematics in the classroom, nor employed the current educational technologies. Besides, even if some teachers utilize new technologies such as computer programs and applications and hand held calculators in the classrooms in the Philippines, there is no guarantee that these technologies are used to generate mathematical ideas or facilitate understanding. According to Rapatan “as far as the Philippines situation is concerned, existing studies found in several graduate schools reflect a predominance of the behaviourist paradigm” in the use of technology. In this paradigm, technology delivers and transmits information to students, and provides drills for practice and mastery in a behaviourist sense.
This paper develops a teacher-training program that aims to infuse new technologies with mathematics content and learner centred pedagogy. This program initially seeks to understand the thought processes of teachers and adopt appropriate educational technologies that sharpen such thinking skills as inquiry and problem solving. The end product of the program is the development of models of classroom mathematics instruction that employ technology to facilitate student thinking and teacher’s assessment of learning.
Specifically, this paper looks at a current teacher training program to make a quick assessment of the teachers’ (1) level of diligence as they comply with the program’s requirements (2) perception about the appropriate learning environment that can generate thinking and meaningful learning, (3) perception about the role that technologies play in the student’s learning environment, (4) perception about the role that technologies play in facilitating student’s understanding of concepts, (5) perception about the role that technologies play in teacher assessment, (6) level of utilization of educational technologies in his/her classroom and (7) appropriate context for the teacher training program on the use of new educational technologies (8) appropriate curriculum goals and content for such teacher training programs. This paper shows how the proposed model teacher-training program can be concretely applied to an existing program.
2 A current teacher training program
Certain groups in the Philippine society initiated intervention programs to address the need to improve the quality of teaching school mathematics and the sciences. Aware of the need to strengthen the mathematics proficiency of schoolteachers, numerous short-term in-service seminars and workshops were organized by government agencies, academic, and professional organizations. While the effectiveness of such programs has been doubted, some government and private institutions offered scholarship programs to allow teachers to undergo more intensive training. Teachers have been awarded scholarships to do graduate studies in pure mathematics or in mathematics education. One such program is the Summer Institute of Graduate Studies (SIGS) at the De La Salle University. The institute designed non-thesis master’s degree programs for teachers primarily at the tertiary level. It contributes in its own way to the professional empowerment of mathematics teachers at different levels, across the nation.
Modularised lecture classes run for 7 days each for 6 hours per day. Modularised classes with laboratory run for 7 days for 10 hours per day. Each program offers 3 courses every summer for 4 consecutive summers except for those programs, which run for 5 summers. This design was adopted in recognition of the teachers’ heavy workload and tight schedule during the school year, and to allow teachers from various parts of the country to join the program.
The Colleges of Education and Science offer the degree Master of Education major in Mathematics. This degree program requires 6 units of education, 3 units of research, and 27 units of mathematics courses. The thesis requirement is waived in favour of additional major courses of specialization, integrating seminars, a project paper and comprehensive examination. The main goal is to upgrade the teachers’ mastery of mathematics.
The education courses include the History and Philosophy of Science and Mathematics, the Use of Technology in Teaching Mathematics, Research and Statistics and a Graduate Seminar. The major courses handled by the mathematics department of the College of Science include Algebra and Trigonometry, two Calculus courses, Set theory and Logic, Probability and Statistics, Linear Algebra, Abstract Algebra and Graph Theory.
3 A close look at the students of the SIGS program
As the SIGS director in school years 1997-1998, and 1998-1999 I got the chance to take a close look at the academic programs and the needs of the students. In the interview that I conducted among the mathematics majors, the students identified the problem on information overload as the most pressing one. Since classes run for only 7 days for 6 hours per day, the students have difficulty accommodating, and assimilating a relatively large amount of information during the summer term. They find it very difficult to meet all the requirements of each course. Modules for a one-term course on Set Theory and Logic were developed and used in the summer of school year 1999-2000 for the second batch of SIGS students.
Research showed that the modules helped the SIGS students learn the course effectively. They read the modules and performed the activities diligently. Consequently, they gained confidence in the subject matter even on the first day of the seven-day sessions for the course. This confidence motivated them to perform well in class. The modules on Set Theory and Logic encouraged academic achievement among the learners. The students’ diligence in reading the modules and in doing the exercises correlated significantly with their performance in class. The students found the modules on Set Theory and Logic very helpful in learning the course. This showed that when properly used, modules could provide the desired scaffolding for learning such abstract concepts as logic and set theory.
While the main weakness of the program is the fast scheme that it adopts in scheduling the course, this research shows that the strength is the seriousness of purpose of the students. In spite of all the constraints like proximity of the university, lack of library resources in their place of work, and heavy teaching load, the students persevere to make the most out of the opportunity that they are given for professional growth. They study hard and adapt to the requirements of their course. This supports the possibility of using open and distant learning using technology in the program.
4 A proposed model teacher training program
The model teacher-training program aims to harness the potential of technology to enhance mathematics learning. This advocates the use of the constructivist inspired teaching theories that promote collaborative learning in a problem based, inquiry based environment.
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Learning theory |
Features of the curriculum |
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Building on the learner’s prior knowledge |
Probing questions to map learner’s concept Single goal problem types Real world problems Paradigm shift on beliefs about teaching and learning and the role of modern technology in education. |
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Active construction of learning Multiple representation Situated learning Strategic thinking Information and communication technologies as allies in the teaching and learning processes. |
Strong discipline content knowledge. Full appreciation of the constructivist philosophy Mathematical explorations and investigation Generalization of patterns and constraints Study of structures Mathematical modelling Project based learning Acquisition of skills to use new pedagogical strategies like the internet activities. Strategies that cater to individual differences and learning styles. Ability to design inquiry based, problem based interactive classroom activities. Search professional development resources. Creation of web pages Design and conduct of participatory action research project Practice of alternative assessment strategies Reflect upon the benefits and limitations of teaching with technology |
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Mathematical community of learners Electronic Collaboration Social context Negotiated meaning Distributed expertise |
Constructivist pedagogy using modern technology. Cooperative learning Environment of mutual support Established norms Shared data for collaborative research Group based projects Sustained focus Student accountability for learning Engage in participatory action research Leadership training on advocacy of the constructivist instruction in their own regions. |
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Cognitive and metacognitive tools including human and technological support |
Understanding modern technology’s process, tools, and product Technological tools that include hand held technology, multimedia laboratory, software, hardware, World Wide Web technologies and virtual learning environment. Problem schema Analogical transfer Distance learning Open examinations Portfolio assessment Feedback |
With the goal to integrate curriculum content and pedagogy with technology, this model can be easily applied to the SIGS program of the De La Salle University. An additional feature of the new SIGS program is the development of the teachers’ capability to plan, design, and produce realistic and effective approach to teaching mathematics with technology. The immersion takes place during the entire 4 years of the SIGS program. Distance learning is employed during each school year while direct interaction among the teacher and students occur in summer. The university becomes the learning centre for distant and open learning.
5 A technology based SIGS program of the De La Salle University
The first summer of the SIGS mathematics program will be devoted to building on the prior knowledge of the teachers. Web based assessment of their knowledge of Algebra and Trigonometry through problem-based activities will be designed. There will be activities in the course that use the graphics calculator and/or Mathematica. Consequently, they will learn all the basic principles of the use of technology while enhancing their skills on problem solving and mathematical representations. It includes on line learning on the web or on virtual learning environment such as the Integrated Virtual Learning Environment (IVLE). They learn simulation, self-assessment, inquiry, and multiple assessments in the process. They are going to design teaching materials using technology for during the following school year.
The courses in the succeeding summer are posted in the IVLE from the start of the school year. Students will be instructed to complete the requirements of the calculus courses in December. The professors will evaluate their responses during the span of the two terms. Basic principles of the participatory action research are also posted in the IVLE by the research advisers. During the succeeding school year, the courses on Introduction to Modern Geometry, Linear Algebra, and Abstract Algebra will be posted in the IVLE in the first, second and third terms respectively. The last school year will be devoted to a review of all the courses included in the written comprehensive examination. Review questions on different areas will be posted at different times of the school year in the IVLE. The professors who will give the written comprehensive examinations are expected to respond to the inquiries of the students via the Internet. They will also take the course History and Philosophy of Science and Mathematics through the IVLE. The students are also expected to come up with the final draft of their research papers for the oral defence.
6 Conclusion
This paper shows a model teacher training program that integrates the current educational theories with appropriate new technologies in teaching and learning mathematics content. The goal is to empower teachers with strong mathematics content and a capability to use constructivist based teaching strategies that employ modern technology. This also provides an equitable learning environment for teachers all over the Philippines who are constrained by distance and workload to go to the university for graduate studies. To prove its viability, the model was applied to an existing teacher-training program that offers full scholarships and other benefits to qualified teachers in the Philippines.
There are other universities in the Philippines that have the capability to provide a technology based training program for teachers. The manpower and human resources available in the country can be harnessed to expand the project at a nationwide level and at a wider scope. While the participants may not be able to use the acquired expertise immediately in their classes due to lack of facilities, the immersion will help them appreciate both the mathematics content and the pedagogical principles that are modelled.
Integrating ICT into the teaching and learning
of discrete mathematics
Eva Milková and Milan Turčáni
Hradec Králové; Nitra, Czech Republic
3. Graph theory and combinatorial algorithms
The integration of modern information and communication technology into teaching allows an entirely new approach to education. There are a lot of possibilities of exploring suitable teaching packages. Also, the rich intellectual potential of some of our students allows the creation of many useful programs. It is in the hands of the teacher to take advantage of the student’s abilities, to motivate them and to lead them into creating a program, which may be useful for them, the teacher and other students.
The subject Discrete Mathematics as taught at our university is divided into three parts: Combinatorics, graph theory and combinatorial algorithms. Teaching with the help of visualization becomes interesting and more understandable. Because our faculty can use good and modern equipment and there are several students who are able to prepare nice programs, we decided to improve the lectures of Discrete Mathematics with student made teaching packages. In this respect, programs created by the students within their thesis were welcomed. We will describe some of them briefly in this paper.
1 Introduction
The best way to teach has always been the teaching with concrete objects. To make the classes more interesting and effective, and the explained subject matter more visual, it is important to have suitable presentations and teaching material. Information and communication technology can help us a lot in this way. In this paper we will mention our teaching approach and effort to provide the students with teaching materials corresponding to the third millennium. We will show how we guided them to use modern information and communication technology effectively. We will also show how we are trying to prepare interesting and object lessons by these means. In the subject Discrete Mathematics we focus on three fields, closely connected with each other. These fields are combinatorics, graph theory and combinatorial algorithms.
2 Combinatorics
In the part devoted to combinatorics first of all we deal with the basic configurations such as permutations, variations and combinations, and all these with and without repetitions. Most students know these concepts since they have attended secondary school, but they are only slightly familiar with them. There is no much time to devote to them at this level in schools. Therefore and because combinatorics is a strong tool for the increase of logical thinking we work hardly with all concepts and solve a lot of practical examples. With each configuration we deal first with tasks classified from the easiest to more difficult. The next step is to practice all these configurations in various examples. We always try very fairly to analyse each example and use different approaches to it, whenever possible. What is very important is the fact that each wrong answer students give must be explained either by the teacher or by the other students. Namely, each student needs be aware what mistake he did and the teacher should know why his student thought in such a wrong way. After getting through the basic combinatorics configurations, we solve examples using the principle of inclusion and exclusion and the Dirichlet principle.
Later, in the part of graph theory, students use the gained knowledge to try themselves or learn how to determine the number of graphs with n vertices, to find out the number of various graph configurations, of which the most interesting is probably the determination of number of spanning trees in complete graphs and in complete bipartite graphs.
Among students, especially among those who will be future teachers at secondary schools and will teach informatics and mathematics, there were and still are those who enjoy combinatorics so much that they choose it as the major field of their thesis. These students make themselves familiar with multimedia and presentation software in the subject “Multimedia” and “Presentation graphics” and they would like to use these gained facilities to create their own programs. So various teaching materials are made as part of students’ thesis. Also the programs “Combinatorics” and “Combinatorics like game” created in the environment of the Macromedia Director began as part of the thesis „Multimedia support of teaching combinatorics at secondary schools” last year.
The above-mentioned program “Combinatorics” is a presentation and it offers the students the possibility to review the subject matter, which has been explained, and to practice it in examples. The program consists of an explanatory part and solutions of some examples of the given topic. These are ordered according to their level from the easiest to the more difficult. The program also contains a collection of unsolved examples with references to their results. The program is first introduced to the students at the seminars where students try to solve the examples given in the program “Combinatorics” and subsequently these examples are modified to get other ones and students solve them as well.
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Fig. 1 |
Fig. 2 |
The program “Combinatorics like game” is also a presentation. With very nice animations the principle of inclusion and exclusion and the Dirichlet principle are explained there.
3 Graph theory and combinatorial algorithms
The theory of graphs is a wonderful, practical discipline, often offering little puzzles. Informatics takes a big part in its development. Both areas influence and penetrate each other. This can perhaps best be seen in the part concerning the design of computer algorithms. The big interest in efficiency and accuracy of solutions motivates mathematicians to research more deeply not only new problems but also problems that had already been solved. This research enables a new look at problems. The creation of algorithms forms an inseparable part of the basic skills of our students whose specialisation is informatics. For them it is important to be able to think algorithmically, to develop logical thinking and to gain wider and deeper insight into solving the given problem. And it is graph theory together with combinatorial optimisation, which enables them to formulate and illustrate a lot of interesting practical tasks.
Mutual relations between individual algorithms
When explaining algorithms we put emphasis on mutual relations between individual algorithms. On one hand there are more algorithms which all can use to solve the same task while on the other hand using effective modifications of some algorithms we can obtain different methods to solve other tasks. For example we give our students four methods used to solve the well-known optimisation problem, the minimum spanning tree problem. We show them three classical algorithms (Borůvka, Jarník (Prim) and Kruskal algorithms) and also for comparison one dual algorithm (dual Kruskal algorithm). All methods are described as an edge colouring process.
The minimum spanning tree problem
Given a connected graph G = (V, E) having n vertices and m edges. For each edge let us have a real number w (e). Our task is to find a minimum spanning tree of the graph G. Borůvkas algorithm to solve it:
1 Initially all edges of the graph G are uncoloured and let each vertex of the graph G be a blue tree (we suppose a blue forest which consists of n blue trees).
2 We will repeat the following colouring step until there is only one blue tree, the minimum spanning tree:
3 Colouring step: For each blue tree T we will select the minimum-weight uncoloured edge incident to T (i.e. edge having one vertex in T and the other not). Then we will colour all selected edges blue.
Notes
Otakar Borůvka (10.5.1899 – 22.7.1995) is
an outstanding personality in the history of Czech and Slovak mathematics.
After finishing his studies in Brno, he spent two years (1926 and 1929) in
Paris and one year in Hamburg. Borůvka’s scientific work reflects the main
streams of the developments of the 20th century mathematics, in particular new
methods in differential geometry, algebra, and differential equations.
Vojtĕch Jarník (22.12.1897 – 22.9.1970) was
outstanding Czech mathematician. He studied mathematics and physics at the
Charles University in Prague, then spent the years 1923 – 1925 and 1927 – 1929
in Göttingen. His main fields of research were number theory, real analysis and
its foundations. In the thirties, Jarnik became an international mathematician
(Jarnik’s Minkowski problem is being quoted till today).
Then our students can see the relation between Jarník's algorithm and the other algorithms. We show them how it is possible to obtain the searching algorithms, the Breadth-First-Search and Depth-First-Search, from it and we also display its modification into Dijkstra’s algorithm looking for the shortest way. Then we present the modification of the Breadth-First-Search and Depth-First-Search algorithm into other methods, for example modification of Depth-First-Search algorithm into finding an Eulerian circuit or an Eulerian trail.
Visual presentations help understanding
Visual presentations prepared with the aid of modern information and communication technology help to understand better all explained algorithms. Implementation of these presentations into the lectures makes them interesting, illustrative and understandable, and their location within the visual studying environment enables the students to get, to complete, to test and to further deepen their knowledge. That’s also why it’s nice when the students want to take part in the creation of such programs.
One of the programs explains and illustrates all above-mentioned algorithms for finding the minimum spanning tree and also the Breadth-First-Search and Depth-First-Search algorithms. The program was written in the environment Delphi and is divided into three parts: comment, visualisation and testing. In the first one the solution of a problem is described. In the second one the solution is illustrated with a graph created by the user himself or with graphs chosen from the library of graphs. In the third one the student can test himself if he understands the algorithms well. Another useful program is the visualisation of algorithms to find either an Eulerian circuit or an Eulerian trail in Eulerian graphs. The program was created in the Delphi environment as well. Users can also create themselves a graph or choose a graph from the library of graphs
Both programs provide the user with numerous possibilities of setting up the environment in which the presentation of the given algorithms runs. The students-authors had good opportunity to test their ability to make nice useful and quite big programs.
4 Conclusion
Our students find the Internet very handy when looking up things of their own interest. The teachers should take advantage of this fact and should try to create conditions and a virtual study environment to enable students to find all necessary study material there. The teacher should try to change gradually the traditional way of teaching into a modern educational process. The use of information and communication technologies is inevitable as a support for projects, as well as for the development of independent and creative work of students.
References
Milková, E. (2001) Problém minimální kostry grafu. Gaudeamus, Hradec Králové.
Nešetřil, J., Milková, E., Nešetřilová, H. (2001) Otakar Borůvka on minimum spanning tree problem (Translation of both the 1926 papers, comments, history). Discrete Mathematics 233, 3.
Pražák, P. (2000) Základy matematiky I. Gaudeamus, Hradec Králové.
What teachers can request from CAS-designers
Walther A. Neuper
Graz, Austria
3. How it works, and tasks for teachers
The basic functionality of computer algebra systems (CAS), increasingly introduced to math classes, is considered not yet optimal for education: CAS show up with the final result in one go, and do not show their built in knowledge. A concept for re-engineering the interactive features of CAS is presented from the users point of view: An example session illustrates what a teacher (and a student!) can request w.r.t. the assistance in modelling and specifying a problem, and w.r.t. the user-guidance in stepwise solving a problem. Brief explanations point out, how the concept presented makes the example session work; and tasks for teachers are mentioned.
1 Introduction
Within a few years high-school students will be the great majority of the users of computer algebra systems (CAS), apparently. CAS producers already provide for special hardware (e.g. TI92) and special user interfaces for math education --- but the basic design of the CAS is still for experts, who know about the methods employed by CAS-functions, and about their limitations.
A student, however, who wants to learn about the steps of a method (as a 'white box') towards a solution, doesn't get any help, but just gets the solution in one go, and is seduced to be satisfied with the method as a 'black box' (Buchberger 1992). And if the student wants to use a CAS-function as a 'black box' solving a sub problem of a more complex problem, the CAS does not help in specification (except complaining on wrong arguments, or simply returning the input unchanged).
If a CAS really would be designed for educational use, given the present state of the art: which features would be the most important? The answer given here is: Assistance in modelling and specifying problems, and a stepwise execution of the few basic functions ('solve', 'simplify', 'differentiate', etc.), where each step is justified by a theorem, and where each step can be supplied either by the system or by the user.
2 An example session
The following example session illustrates the interaction between a user (called 'student') who may input a tactic towards a solution, a formula, or a command pushing a < button > , and a re-engineered CAS (called 'system') with output tactic, formula, request ? or message ! --- i.e. the interaction is designed as for partners on an equal base.
Model a formalized representation
from a textual description, and eventually by a figure, comprises the first phase in solving a problem:
A coil with a circle-shaped section and radius R should get a cross-shaped kernel (two equal bars with length b and width a) of iron, see Fig. 1. Determine a and b such that the area A of the kernels section is maximal for a given R .
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Fig. 1: Coil with a cross-shaped kernel to be maximized |
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The student first will try to transfer that information into some formulae. The system assists by presenting the following frame for input, as soon as the example is selected:
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given: |
constant_values : [ ] |
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find: |
maximum : |
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other_values : [ ] |
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relate: |
relations : [ ] |
where [ ] indicate that lists of formulae are to be input. Let us assume the student isn't sure how to start the calculation and pushes < yourTurn>
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student: |
< yourTurn > |
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system: |
constant_values ;[R= arbitrary_but_fixed ] |
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student: |
< accept> |
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system: |
maximum A |
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... |
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Now let us assume the student has become self confident enough to proceed on her or his own, and wants to input the values to be 'found', and relations capturing the shape of the kernel:
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... |
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student: |
other_values ;[a,b] |
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system: |
accept ! |
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student: |
relations [A=a b + a b] |
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( forgets to subtract the square in the middle of the kernel) |
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system: |
A=a b + a b does not belong to this example ! |
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student: |
A=a b + a b - a2 |
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system: |
accept ! |
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student: |
Apply_Method 'maximum_by_calculus' |
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(tries to finish the specification too early ) |
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system: |
relations [A=a b + a b – a2] is incomplete ! |
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student: |
< yourTurn > |
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system: |
[A=a b + a b - a2, a/2 = R sin a] |
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student: |
< howComes > |
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system: |
definition of sin ! |
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... |
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The model-phase may result in the following:
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given: |
constant_values : [R = arbitrary_but_fixed] |
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find: |
maximum : A |
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other_values : [a, b] |
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relate: |
relations : [A = a b + a b - a2, a/2 = R sin a, b/2 = R cos a] |
This is the input to the next phase in problem solving.
Specify domain, problem and method
The specify-phase adds structures capturing the goal-directedness of problem solving. The goal is approached in two steps common to mathematics and software technology: The first Step describes what the goal is, and the second step describes how to achieve the goal. Specification comprises three items, the domain, the problem and the method.
With respect to the example given, the student may be led by the observation of the function constants sin and cos , or the numerical constant 0.0 to the input
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... |
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student: |
Specify_Domain R |
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system: |
accept ! |
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student: |
... |
At this point the student can search the hierarchy of problem-types available in the system:
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… |
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optimization_problems: |
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maximum_with_additional_conditions |
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minimum_with_additional_conditions |
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linear_optimization |
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… |
Let the user mistakenly select the third problem-type, and the check of its precondition causes a message:
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... |
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student: |
Specify_Problem 'linear_optimization' |
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system: |
is_linear_in (A = 2ab - a2) [a, b] is not true |
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student: |
Specify_Problem 'maximum_with_additional_conditions' |
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system: |
Accept ! |
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... |
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Now the specified problem-type is instantiated by the formulae already input, and the precondition ('where') is checked[1].
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problem |
‘maximum_with_additional_conditions’ |
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given: |
constant_values: [R = arbitrary_but_fixed] |
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where: |
0.0 £ R |
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find: |
maximum: A |
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other_values: [a, b] |
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with: |
A = 2ab - a 2 Ù a/2 = R sin a Ù b/2 = R cos a Ù |
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" A', a', b', a'. A = 2 a' b' - (a')2 Ù a’/2 = R sin a' Ù b’/2 = R cos a' Þ A'£A |
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relate: |
relations: [A = a b + a b - a 2 Ù a/2 = R sin a ; Ù b/2 = R cos a ] |
Finally the method is specified (eventually by another search in the knowledge base) and applied:
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... |
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student: |
Specify_Method 'maximum_by_calculus' |
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system: |
accept ! |
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student: |
Apply_Method 'maximum_by_calculus' |
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... |
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Solve the problem
is the phase which is generally most emphasized in maths courses at highschools. Let us proceed with the system being the active partner:
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… |
A = 2ab – a2 |
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system: |
Substitute a → 2 R sin a |
|
student: |
< accept > |
|
system: |
A = 2(2R sin a ) b - ( 2R sin a ) 2 |
|
student: |
< accept > |
|
… |
|
And now the DG starts to involve the student more and more.
|
... |
|
||
|
student: |
we substitute → 2 R sin a |
||
|
... |
A = 2(2 R sin a )b - (2 R sin a )2 |
||
|
system: |
Substitute b → 2Rcos a gives which formula ? |
||
|
student: |
A = 8R2 sin a cos a - 4R2 (sin a )2 |
||
|
system: |
accept ! |
||
|
|
Subproblem differentiate l a . A = 8 R 2 sin a cos a - 4 R 2(sin a)2 = : |
||
|
|
A' = d/da (8 R 2 sin a cos a - 4 R 2(sin a)2 ) |
||
|
|
|
Which rule would you apply ? |
|
|
|
|
#1: (diff_bdv, d/dx x = 1) |
|
|
|
|
#2: (diff_const, d/dx x = 0) |
|
|
|
|
#3: (diff_sum, d/dx (u+v) = … ) |
|
|
|
|
… |
|
Notice that the system skipped the whole model phase and specification phase (due to a reasonable decision of the course designer).
|
student: |
<#3> |
|
system: |
A' = d/dx ( 8R 2 sin a cos a ) - d/dx ( 4R 2(sin a)2) |
|
|
gives which formula? |
|
student: |
A = 8R 2 cos a sin a - 8R 2 cos a |
|
system: |
cannot derive the formula ! |
Now, as the system has more and more involved the student, he has failed. And the general 'dialog-strategy' supplied by the course-designer will decide how to react to the students failure, for instance:
|
|
which rule did you apply ? |
|
student: |
< dontKnow > |
|
system: |
Rewrite diff_prod d/dx (u ´ v) = ( d/dx u) ´ v + u ´ ( d/dx v) |
|
student: |
< accept > |
|
system: |
A' = ( d/dx 8R 2 sin a) cos a + ….. - d/dx 4R 2(sin a)2 fill in … |
|
student: |
…+ 8R 2 sin a d/dx cos a - … |
|
… |
|
3 How it works, and tasks for teachers
The above sample session is based on research and development done at the Institute for Software Technology, TU Graz, in close contact to the Research Insitute for Symbolic Computation, University Linz. The key-concept developed is the separation of the mathematics engine and the dialog module from the math knowledge. The knowledge is described in a human readable format and structured along three axes for (1) domains, (2) problem-types and (3) methods. The current state of development of a prototype can be viewed at http://www.ist.tu-graz.ac.at/research/edu/isac.
A conclusive presentation of the concepts exceeds the space limits of this article; it can be found in (Neuper 2001b) together with related work and connexions to didactics. Here we briefly explain how the concepts make the sample session work, relate the concepts to existing CAS features, and point out some tasks for teachers.
Modeling a problem
in a traditional CAS is done by providing the appropriate arguments to its functions. Modeling can be supported by a CAS-like system only if a formalized representation of the problem has been prepared. For the above example this can be at least three variants:
|
(1) |
[[R = arbitrary_but_fixed], |
0.0 £ a £ R/2, |
[A = 2ab-a2, (a/2)2 + (b/2)2 = R2]] |
|
(2) |
[[R = arbitrary_but_fixed], |
0.0 £ b £ R/2, |
[A = 2ab-a2, (a/2)2+( b/2)2 = R2]] |
|
(3) |
[[R = arbitrary_but_fixed], |
0 £ a £ p/2, |
[A = 2ab-a2, a/2=R sin a, b/2=R cos a]] |
The preparation of this 'formalization' usually would be done by an author or a course designer, i.e a teacher is free to add his or her own examples for students independent learning (or for written exams~?), it is not much effort. For exercises the formalization may be hidden from the student.
Modeling is the phase in problem solving least suited for mechanized assistance. The systems reply (1) on p.pageref{def-sin, for instance, really needs adaption to the student -- either by specific addons by the teacher, or the teacher personally; for such tasks a human teacher knowing her students is indispensible.
If someone wants to solve a new problem, i.e. without a prepared formalization, the process starts immediately with the following.
Specification of domain, problem-type and method
is done automatically and tacitly by a traditional CAS. To be more precise: The specification of the domain (type of numbers cal R,C etc.) is not necessary there, because commercially available CAS are type-less (and some functions, for instance solve, have switches for cal R,C etc.) The specification of the problem-type and the method is tacitly coupled (where a student should explicitly decide on the method~!) in traditional CAS, and done by hughe case distinctions and pattern matching.
A re-engineered CAS would exmploy the same thechniques in the math engine, but applies them to a separate 'hierarchy of problem-types' (Neuper 2001b). This hierarchy is extensible, can be viewed by the user, and searched for the problem-type most appropriate for solving the problem at hand -- and this search can be done by the system as well. For the example above the problem-type could be
|
problem-type 'maximum_with_additional_conditions' |
|
|
given: |
constant_values : cs |
|
where: |
foldl op $Ù$ true (map $0.0 £ cs) |
|
find: |
maximum : m |
|
|
other_values : os |
|
with: |
let p = foldl op$Ù$ true rs; vs = (vars p) cs; vs' = map prime vs |
|
|
in " vs' $. $l vs.rs (vs') $ Þ prime m $ £ m |
|
relate: |
relations : rs |
where $ constructs the predicates during instantiation of the problem-type Such a problem-type as in section 2 is quite general, for instance this one should capture (almost) all 'Extremwertaufgaben' found in Austrian textbooks for seventh-grade high-school math. Thus a teacher rarely would want to extend the hierarchy.
Actually, the design of the hierarchy of problem-types is considered the most comprehensive task in re-engineering CAS (Neuper 2001a) to be done by experts of computer math.
Solve the problem
is a task done by some C or Lisp-code in a traditional CAS. Again, a re-engineered CAS describes the steps of a method towards the solution separated in so-called 'scripts'. The script for the example is
|
Script Maximum_value (fix_::bool list) (m_::real) (rs_::bool list) |
|
|
|
(v_::real) (itv_::real set) (err_::bool) = |
|
|
(let e_ = (hd o (filter (Testvar m_))) rs_ |
|
|
t_ = (if #1 < Length rs > |
|
|
then (Subproblem (Reals,[make,function],no_met) m_ v_ rs_) |
|
|
else (hd rs_)) |
|
|
mx_ = Subproblem (Reals,[on_interval,max_of,function], |
|
|
maximum_on_interval) t_ v_ itv_ |
|
|
in (Subproblem (Reals,[find_values,tool],find_values) |
|
|
mx_ (Rhs t_) v_ m_ (dropWhile (ident e_) rs_))) |
providing the subproblems [make,function], [on_interval,max_of,function] and [find_values,tool] with the actual arguments. The script is general enough to solve (almost) all problems to the type 'maximum_with_additional_conditions'; subproblems (e.g. for different equations) may be solved by different methods, i.e. by their respective scripts.
A script is interpreted in a way that passes control to the user at each 'tactic'[2]. The most common tactics relate to 'rewriting', a basic technique in symbolic computation. And rewriting by theorems like
d/dx (u+v) = d/dx u + d/dx v
is just what a teacher usually explains to a student. Thus it is likely to show the rewrites step by step to the user. There is already an educational system at http://www.mathpert.com/ doing like that, however without supporting explicit specification.
4 Conclusion
In the near future the great majority of CAS-users will be students in math and science classes at high schools. Thus it is the right and the duty of teachers, to request a re-engineering of CAS for education: stepwise, interactive execution of functions, and assistance in specification. Didactics of mathematics will hopefully support the teacher’s requests and strengthen research on educational design of math knowledge, and on dialog-design for re-engineered CAS. And institutions already concerned with CAS in education will find an important role in coordinating and representing the requests from the teaching practitioners. Their involvement in early phases of re-engineering and testing is highly desirable. A question is, who will begin with re-engineering CAS: established CAS-producers, or the academic world --- the relevant knowledge is available in institutes of Austrian universities, and cooperation with other European universities seems desirable.
References
Buchberger, Bruno (1992) The white box / black box principle. Technical report, RISC-Linz, Johannes Kepler University Linz.
Neuper, Walther A. (2001a) Re-engineering algebra systems for education. Technical report, IICM - Inst. f. Software technology. Technical University Graz.
Neuper, Walther A. (2001b) Reactive User-Guidance by an Autonomous Engine Doing High School. Math. PhD thesis, Technical University Graz.
[1] The postcondition ('with') is the most characteristic discriminant for problem-types; whether it should be presented to the student or not, is questionable.:
[2] Note, that the scripts are not involved in details of the dialog - this is the task of the dialog-module.
Computers in school mathematics –
A pilot course
for
school teachers of mathematics in Estonia
Rein Prank and Eno Tonisson
Tartu, Estonia
1. Some facts about the computerization of estonian schools
2. The situation of software for teaching mathematics
January to December 2000, a pilot course on the use of computers in school mathematics teaching was held at the Faculty of Mathematics of the University of Tartu, Estonia, for mathematics teachers. The course (total 480 hours, including 144 hours of auditory training) comprised 9 modules primarily dealing with the use of different programs.
The need for and the possibility of holding such a relatively extensive course were occasioned by the fairly high-level and homogenous situation of both the existing educational software at schools and its application conditions. At the beginning of the article, we give a rather detailed description of this situation. Chapter 1 describes the computerization of Estonian schools in more general terms. Chapter 2, an overview of the mathematics software used in Estonian schools, is directly linked with the above-mentioned course; in fact, the course was prepared bearing in mind the software used in mathematics teaching and learning. The rest of this article is a description of the different aspects of the course: its objectives, financing, the structure of its modules, its participants, results, and feedback. Finally, some concluding remarks are made.
1 Some facts about the computerization of Estonian schools
At present, there are approximately 215,000 students in Estonian schools, Grades 1-12. Due to low population density, many of the schools are small. In 2000, there were 240 high schools (mainly Grades 1-12), 294 junior high schools (Grades 1-9) and 172 elementary schools (Grades 1-6 or 1-4) in Estonia. The country is divided into fifteen counties; approximately one third of the population lives in the capital Tallinn.
In 1997, a school computerization project, the “Tiger Leap” (www.tiigrihype.ee), was launched. Since then, state and local funding has resulted in the schools having one computer per 25 students in average. Almost all high schools and many junior high schools have a computer classroom equipped with 10-15 computers; at the same time, the Tiger Leap policy has been to supply only new computers. About 75 per cent of Estonian schools have on-line access to the Internet, many others have dial-up access. Seventy per cent of the teachers have completed a 40-hour basic computer course. Fairly popular among them are the ECDL module exams.
Almost all high schools and many junior high schools teach informatics as a separate subject. While the primary focus of the subject is on the use of general-purpose software, integration with other subjects is also emphasized (preparation of essays and reports on computer, searching for the required information on the Internet, using learning software). Estonian schools employ more than 200 teachers qualified for teaching informatics and being trained, to a certain extent, for using computers in teaching other subjects.
As concerns teaching software, the Tiger Leap project has financed both new software development projects and the purchase and translation into Estonian, in cooperation with the authors, of software produced elsewhere. In the field of new software development, the subjects directly linked with Estonia, such as the Estonian language, Estonian culture, history, geography, and natural science, have been given priority, since no software dealing with these is produced elsewhere. However, other software has been developed apart from that, such as an original test system APSTEST. Of purchased items, mention may be made of school or class licenses for several larger programs, such as Europlus for teaching English, Chemistry Set, and StudyWorks, a mathematics program. An English-Estonian electronic dictionary and some encyclopedias and atlases have been put to use. The software developed under the Tiger Leap project is free for all schools and other non-commercial educational institutions while the availability of the purchased and translated software is dependent on the contracts concluded with the respective companies and authors.
In the period under study, the situation in Estonia has also been influenced by the Phare project ISE (Information Systems in Education, www.ise.ee). (The Phare Program is the main channel for the European Union's financial and technical cooperation with the countries of central and eastern Europe.) Under the ISE project, one high school in each county was selected as a pilot school. From Phare funds, a computer classroom with 15 computers and a data projector has been established in each pilot school. A set of software for teaching various subjects (mainly in the English and Russian languages) has also been purchased for these schools. Study trips to Scandinavian schools have been organized for pilot school teachers. The pilot schools constitute a good local basis for training teachers of different subjects.
The Phare ISE project has done good cooperation with the Tiger Leap project. For instance, the Phare project has given rise to the issuing of 5 CD’s, which have been sent free of charge to all Estonian schools. These disks have also been used for the dissemination of the entire Tiger Leap educational software. Another important software channel for school teachers is the Internet portal “Õpetaja võrguvärav” [Teacher Net Gate] (www.opetaja.ee), which offers systematized material being of interest for teachers.
2 The situation of software for teaching mathematics
By the end of the third year of the Tiger Leap project (1999), Estonian schools possessed the following software for mathematics teaching:
StudyWorks (Modules 3 and 7).
StudyWorks, created by the MathSoft company, may be called a school version of
MathCAD. It enables the performance of many mathematical operations in both
numerical and symbolic form (simplification, factorization, equation solution,
graphics, simulation, etc.). It is accompanied by a large amount of materials
in English dealing with mathematics as well as physics, chemistry and biology.
StudyWorks is currently available at 370 Estonian schools.
Funktion, GeomeTricks,
TableTalk, Pattern (Module
1). These relatively small programs, created by the Danish educational software
developer Viggo Sadolin, have found a favorable reception in Estonia.
GeomeTricks is a program dealing with dynamic geometry. Funktion lends itself
to making graphics. Tabletalk is a special table processing program suited for
the study of compound interest increments and decrements. Pattern is a program
for handling the filling of a plane with certain figures; its applications in
school mathematics are more limited than those of the other three. The use of
these programs is free of charge thanks to the benevolence of V. Sadolin and
the support of the Tiger Leap project.
Geolog (Module 4). Geolog, an intelligent
tutoring system created by Gerhard Holland, professor of Giessen University, is
designed for solving proof problems, arithmetical problems and construction
problems in plane (congruence) geometry. The student solves problems step by
step. The program checks each component of the step but is also capable of
giving advice in each phase. The use of Geolog is free of charge thanks to the
benevolence of professor Holland.
Tõenäosusteooria [Probability Theory] (Module 6). This program, which describes the basic principles of
the theory of probabilities, has been written by some students of the University
of Tartu under the supervision of Professor Kalev Pärna. It allows to simulate
many experiments with many different parameter values. The program is very easy
to use and therefore suitable for relatively young students.
Some smaller drill programs designed
primarily for elementary school students. In addition to the above mentioned
programs, the test system APSTEST
(Module 2) constitutes a substantial educational resource. Many schools also
utilize Microsoft Office, which allows to apply Excel (or its StarOffice analogue) (Module 5) in mathematics
teaching.
V. Sadolin’s programs and the Geolog interface have been translated into Estonian in cooperation with the authors. While the above mentioned packages provide a fairly good coverage of a number of mathematical topics, Estonian schools have practically no graphic calculators and computer algebra systems (unless we regard StudyWorks as one).
In school work, virtually all of the above mentioned programs need to be supplemented with worksheets, problem sets or other materials created by teachers. The programs enable the development of new materials or the amendment or correction of old ones, if necessary. On the one hand, this allows to translate existing materials and adjust them to the curricula of Estonian schools or to the goals set by a particular teacher. On the other hand, the teachers first need to jointly create part of the required materials. Then, however, the teachers need to be able to critically assess, in terms of both practical skills and the subject matter, the quality of the materials being used at school and alter them if necessary.
Different projects have resulted in the preparation and distribution to teachers of teaching materials in Estonian suitable for use in, for instance, the Sadolin programs, StudyWorks, and other programs. Fairly extensive question banks for use in APSTEST have been compiled by teachers of informatics and mathematics.
3 Courses
Goals, financing and organizers
In late 1999, an idea presented itself to the Computer Science Institute of the Faculty of Mathematics of the University of Tartu to hold a course on the use of computers in mathematics teaching for more capable teachers in order for them to be able to train other teachers in their respective counties, even if on a somewhat smaller scale. Another objective was to enable the participants to undertake small projects for preparing worksheets and problems to supplement math teaching programs. After assessing our computer classrooms and istructors, we considered it possible to work with up to 40 teachers. (The cost of the course per participant turned out to be 10,780 Estonian kroons = 690 euros). We were able to raise funds for the course from two sources. The Tiger Leap project promised to pay 50 percent of the cost of a course for 40 teachers. The Phare ISE project agreed to pay the other half of the cost of a course for 30 teachers (on the condition that geographical representation be as extensive as possible). As stipulated by the sponsors, we were obligated to make the study materials created during the course available to schools, free of charge.
The principal organizers were the authors of this article. We also involved other teachers from the University of Tartu and from schools.
The scope and structure of the modules
The course comprised nine modules with the following scopes (1 cu equaling 40-hour work done by one participant).
|
|
Module |
Size [cu] |
|
1. |
GeomeTricks, Funktion, Tabletalk, Pattern |
1,5 |
|
2. |
APSTest |
1,5 |
|
3. |
StudyWorks |
2 |
|
4. |
Geolog |
1 |
|
5. |
Excel |
1,5 |
|
6. |
Probability Theory and Math. Statistics Software |
1 |
|
7. |
Computer Algebra Systems |
1,5 |
|
8. |
Ressources in Web and Distance Education possibilities |
1 |
|
9. |
School Mathematics and Software |
1 |
Table 1
Modules 1-6 were designed for instruction in the individual existing programs. The following two modules focused on future needs and provided a description of the resources so far unused in Estonian schools. The last module was intended for an overview of the topics included in the mathematics syllabus, in view of the applicability or inapplicability of particular software to the teaching of one or the other topic.
Auditory work was conducted in eight weekend sessions and one three-day session in August. There were 3-4-week breaks between sessions held in consecutive months to give time for homework. Table 2 presents the hours of auditory work done in each session of a module. Each of the nine sessions also included a lecture or a seminar on more general topics: computer use methods in a teacher’s work, the situation of educational software in Estonia and worldwide, graphic calculators, an overview of the ICME congress, a meeting with the chairman of the Mathematics Council of the Ministry of Education, etc.).
Each session dealt with 2-4 modules; in addition, the final seminar of another module was sometimes held, with the demonstration of the newly completed materials.
|
Modules |
Sessions Month |
||||||||||
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
Sum |
||
|
1. |
GeomeTricks, Funktion, ... |
5 |
7 |
4 |
1 |
|
|
|
|
|
17 |
|
2. |
APSTest |
6 |
6 |
3 |
3 |
|
|
|
|
|
18 |
|
3. |
StudyWorks |
|
|
|
4 |
4 |
3 |
|
|
|
11 |
|
4. |
GeoLog |
|
|
|
|
5 |
4 |
2 |
|
|
11 |
|
5. |
Excel |
|
|
6 |
5 |
4 |
|
2 |
|
|
17 |
|
6. |
Probability & Statistics Software |
|
|
|
|
|
6 |
4 |
1 |
|
11 |
|
7. |
Computer Algebra Systems |
|
|
|
|
|
6 |
7 |
4 |
|
17 |
|
8. |
Web and Distance Education |
1 |
|
|
|
|
|
|
5 |
7 |
13 |
|
9. |
School Mathematics and Software |
|
|
|
|
|
2 |
1 |
3 |
6 |
12 |
|
|
Additional Themes |
3 |
2 |
2 |
2 |
2 |
2 |
|
2 |
2 |
|
|
|
Sum |
15 |
15 |
15 |
15 |
15 |
23 |
16 |
15 |
15 |
144 |
Table 2
A typical module consisted of the following:
Opening
lecture
Practical
training (work in the student’s role, technical skills)
Homework
(technical skills + preparation of the graduation paper)
A
seminar or lecture on application possibilities
Practical
training (more complex examples)
A
lecture on deeper working principles of the program
Homework
(initial version of the graduation paper)
If
a module lasted for 4 sessions, then on the third session: typical errors,
collection of student results
Closing
seminar – demonstration of homework results.
The structure of the StudyWorks course, at which distance training played a very important role, was somewhat different. This module was conducted in the WebCT environment, which is specifically designed for web-based training.
Independent work and communication with instructors
Of the total hours of our course, a mere 30 per cent was designated for sessions (144 out of 480). In planning the scope of work, we doubted that we would be able to maintain such a proportion without losing participants through dropout. The questionnaire conducted at the end of the year showed that we had managed to distribute the workload fairly accurately. The average estimate given by the participants to the time spent on learning was consistent with our timetable. Few participants had been able to work faster. A greater expenditure of time by some participants was due to their lesser skills in working with a computer. Apart from the intent of the problems, we describe here the organization of our work and the means of communication intended by us to support independent work.
During the course, general information was posted on the course’s website, which had been opened already before the beginning of the course. The information contained brief overviews of the modules, session curricula, homework submission procedure, module organizers’ contact data, etc. Communication was performed by e-mail and the so-called WebCT pseudo course, which enabled the WebCT means of communication but not the posting of materials on the Internet.
The structure of a typical module has been presented under the previous item. Homework may conditionally be divided into two groups: technical exercises concerning the use of a particular program, and the preparation of a module’s graduation paper. In the technical exercises, a typical requirement was to compile or solve, in fairly great detail, problems as prescribed by the instructor of practical training for the purpose of testing the different possibilities of the program under study in practice. The exercises were required to be performed individually; however, they were usually the same for all the participants. The participants requested help by e-mail, from both the instructors of the practical training and their more experienced course-mates. One hour before the beginning of the first lesson of each session, the instructors were available for consultation, and this opportunity was seized by those in difficulty.
Each module ended with a graduation paper, in which the participants were to compile a certain type of teaching material to be used by the particular program at school. We considered it important that credit points for a module be given in this form. Our experience has shown that a compilation assignment is much more acceptable to teachers than a test. At the same time, it enables to produce materials that are of somewhat better quality than individual work and that can be used by other teachers as well. In our case, making the teaching material created at the course available to all schools was also a condition stipulated by the sponsors of the course. It seems that the possibility and obligation to publicize their work had a stimulating effect on the participants.
In a technical sense, the graduation papers concerning most of the programs were essentially worksheets, the exceptions being question sets and tests composed on their basis concerning APSTEST and a set of proof and construction problems concerning Geolog. In most of the modules, the participants were also required to submit lesson notes comprising instructions for the use of the computerized material created. In the summary module, the participants were required to present an analysis of a chapter from the mathematics syllabus, in which they were to identify the items that could be taught/learnt by using the known programs and their appropriate forms, the domains requiring new software and the items that would be more practical to be taught without using computers. In most of the modules, the graduation paper was prepared in teams of two or three.
In many modules, graduation paper topics were disclosed at the very first session, with the participants being required to have reviewed the respective topics in the textbooks by the second session. For learning purposes, even those topics in which the use of computers is difficult or less natural were included.
Homework and auditory work were closely linked. In a number of modules, a seminar was held in the second session, in which the participants were to present their ideas, with the other participants and the supervisor commenting on them and making suggestions. Thereafter, a rough copy was prepared and submitted to the supervisor for review. Occasionally, a graduation paper was sent back and forth between the supervisor and a participant for 3-4 times. This was time-consuming for the supervisor (sometimes up to 2 hours per paper). Cooperation between the participants themselves was also intensive. In some cases, the papers were mutually reviewed and commented, with typical mistakes being discussed in the lecture room. At the concluding seminar of a module, the works done were demonstrated. Among other benefits, the demonstrations enabled the participants to practice the use of a data projector while making presentations. Data projectors are still lacking in many schools. As a rule, corrections were made to the papers after the closing seminar in response to suggestions from other participants.
Participants
When planning the course, our position was to take the same people through all the modules. The sponsors also suggested that different modules be taught for different people. In the end, many participants went through all the modules, and, in our opinion, gained more from that than from just going through specific modules – they gained an integrated overview. The course was attended by a total of 40 people, of whom 31 opted for all the modules. Lesson attendance was very good (which was even a little surprising, considering the fact that the participants had full-time workload as teachers, often even greater). The dropout was fairly small and apparently caused by objective reasons (for instance, bad health). As required by the sponsors, practically all the counties were represented. It is noteworthy that there were only 2 participants from Tallinn, which is inhabited by approximately one third of Estonia’s population.
A topic worth separate treatment might be the motivation of the teachers to participate in the course. To a certain extent, the motivation might have been occasioned by the fact that the teachers having completed the course would gain a better ranking at the evaluation of their professional abilities. On the other hand, this group of participants seemed to have been a group of expectant people waiting for their call, so to say. They had an interest in things like that. The following groups may not necessarily be as interested.
Feedback
A questionnaire was held at the end of each module as well as the entire course. In this article, we only present the average grades the respondents gave to the modules in the final questionnaire (possible grades 1-5).
|
Module |
How well did the module |
||
|
provide particular skills for using particular programs? |
provide general knowledge about a particular field? |
enable you to apply the material learned in it to your current work? |
|
|
GeomeTricks, Funktion, ... |
4,64 |
3,88 |
4,33 |
|
APSTest |
4,20 |
3,90 |
3,67 |
|
StudyWorks |
4,38 |
4,06 |
4,16 |
|
GeoLog |
3,91 |
3,39 |
2,52 |
|
Excel |
4,33 |
4,09 |
4,30 |
|
Probability & Statistics Software |
4,38 |
4,00 |
3,91 |
|
Computer Algebra Systems |
3,69 |
3,87 |
3,26 |
|
Web and Distance Education |
3,94 |
4,22 |
3,66 |
|
School Math. and Software |
3,71 |
3,87 |
3,57 |
Table 3
The grades given are relatively high, which shows, in our opinion, that the course as a whole and each of the modules were properly prepared and provided the participants, in their judgment, with good knowledge and skills. Some modules were perceived by the participants as having a theoretical bias while others were perceived as more practical. As a matter of fact, the three last modules were designed to provide a more perspective overview, and this was how the participants viewed them. The slightly lower grades given to the module on GeoLog are probably due to a small proportion of proof problems in the current mathematics curriculum.
4 Results and summary
It may be maintained that the course as a whole yielded the following results:
40
teachers received training enabling them to use computer software in
mathematics teaching, worksheet preparation and software assessment,
These
teachers are ready to organize similar courses themselves, and several of them
have already done so,
These
teachers are able to make a better contribution to various projects concerning
computers and software,
Module
graduation papers provided a large amount of Estonian-language teaching
material, which was made available to all the Estonian schools on CD’s and over
the Internet,
The
course helped to determine more clearly than before the kind of software needed
at schools (for instance, there is a need for software on algebraic
variations).
The impediments to the use of teaching software in a mathematics class at Estonian schools have developed along typical lines. Shortage of hardware and lack of teaching software are no longer the primary problems. Further development is clearly dependent on the training of mathematics teachers, which, if inadequate, will pose the third impediment. Feedback from teachers having received better training (participants of the course presented in this article, and a number of active mathematics teachers having recently obtained the qualifications of an informatics specialist in addition to their professional qualifications) suggests, however, that there will also be a fourth impediment – lack of secondary software. Mathematics teaching programs are available, but they need to be supplemented with the necessary worksheets, problems sets and other resources adjusted to our textbooks and being in the Estonian language, and even able teachers lack time to produce these supplements. The course we organized was a good step towards overcoming both the third and the fourth impediments.
Computers in mathematics education -
An ongoing experience
Ana Isabel Rosendo and Jaime Carvalho e Silva
Coimbra, Portugal
In this paper we describe how future mathematics teachers are being prepared to use technology in mathematics teaching (at the Mathematics Department of the University of Coimbra, Portugal).
The first formal course on Computers in Education in a Portuguese university was created in 1987 at the mathematics department of the university of Coimbra and is being offered since then with the name "Computing in Mathematics Education". This was the only course in our Department where computers were discussed having mathematics education in mind. At first we worked only with computers but since 1992/93 graphics calculators were also discussed. Since 1999/2000 the discussion of the educational uses of graphics calculators began being done in another course ("Foundations and Teaching of Analysis") and so only computers are discussed again in the course "Computing in Mathematics Education".
In our department we prepare schoolteachers (for grades 5-12) since 1972. Before that date students had to graduate first in mathematics and only afterwards they would begin their preparation to be teachers. Since 1972 this changed but they still have the first two years in common with pure and applied mathematics students. In the two following years they have more courses on several Mathematics topics, on Mathematics Education and on Psychology; in the fifth and last year they have a stage in a secondary school where they are supervised by a schoolteacher and a university professor.
The Ministry of Education created a national project in Portugal (the Minerva project) in order to introduce computers in primary, basic and secondary schools (grades 1-12). From the beginning we saw that the emphasis was too much in hardware and programming languages (including Basic, Pascal and also Logo) and not much on the particular aspects and uses of each scientific discipline. And in fact this national project ended in 1993 and we can say very few teachers used computers in their teaching at that time (we could easily estimate they were less than 5%). A new national project was launched in 1996 Portugal (the Nónio Séc. XXI project) and things have improved a little since then, but the use of computers in the classroom is still under a desirable level.
We think that the right way to prepare future schoolteachers is to put them at ease with what they will teach later. If they are at ease doing mathematics with the computer then they will be better prepared to integrate it in their teaching. This is a general principle we think is true for all the preparation of teachers to use technology. Lajean Thomas, a professor of education at Louisiana Tech University and the International Society for Technology in Education's (ISTE) representative to the National Council for Accreditation of Teacher Education (NCATE-USA), speaking about teacher preparation for the use of technology said: "You have to look for courses that (…) provide the kind of technology you would use in the classroom. We in education would like to think the ideal way for teachers to learn how to use technology is by using it in their content area." No doubt this is a very difficult task. So we think the preparation of future teachers must begin while they still are students at the university.
This was the main reason why we proposed the creation of the course named "Computers in Mathematics Education" in 1987. Fortunately the Mathematics Department accepted it and it has been for 14 years a compulsory course for future schoolteachers in the second semester of their fourth year of studies. The course "Foundations and Teaching of Analysis" was created 7 years ago and since 1999/2000 it includes work with graphics calculators. We have been the two responsible for these courses over most of the years and we would like to discuss briefly how they were conducted. A first report was presented at ICTCM 7 in 1994.
In "Computers in Mathematics Education" we have 2 hours a week of theory and demonstrations and 3 hours a week of practical work in a computer lab. In this lab we have now 16 personal computers (this changed a lot over the years). We have 90-110 students each year that are divided into four groups of 15-25 students working in groups of two for the sessions in the computer lab.
In the theoretical course we discuss what computers are, what software is, how the computer can be used in the mathematics classroom, how mathematics is taught in a different way with the computer with a bigger involvement from the part of the student, which types of software are available (we make an abstract classification of educational software), and we give examples of how things work (using a portable computer and a data display).
We put a lot of emphasis on the connections with the Portuguese curriculum being used each year in grades 7 to 12. There were major changes in 1991 and 1997 in the official mathematics syllabus. In the mathematical syllabus for secondary school (from 1997) we can read:
"The computer, with its potential, namely in the areas of graphics representation of functions and simulation, allows activities not only of exploration and research but also of remedial teaching and development, and so is a valuable support to teachers and students, and its use must be regarded as compulsory in this syllabus."
The new Portuguese curriculum also encourages the use of problem solving, mathematical modelling, writing of reports, investigations and projects; the official documents are discussed in class and are available to the students at the Department's Library. Several papers from Portuguese journals are also available to the students and we give them a list of references where they can find more information about the topics discussed in the classroom.
In the practical sessions, students work with different pieces of software. They experiment with the software, they write small reports about the software, they execute small projects chosen in advance or they have to plan small projects. At first students do not feel at ease with the computers. Before that they only worked theoretically in abstract computer science with terminals of some big computer. In our course it is the first time they are in front of the "beast". After the first two weeks of classes students feel enormously at ease with the computer and most of them discuss with interest a lot of aspects that are related with the use of computers in the mathematics classroom.
In the practical sessions there is space for pure exploration where the student can experiment the software at ease, and space for the execution of worksheets written for secondary school students (some of them have already been experimented with students). These worksheets include for example geometry activities with the Dynamic Geometry software Cabri-géomètre and activities with sequences with Modellus, a simulation and modelling tool.
The assessment is made of two parts each worth 50%. The students must write 4 small projects (some individually and some in groups of two) during the semester and they have a final examination (with a written part and an oral part). The oral examination consists of the discussion of the projects made by each student. The projects must be made outside the sessions of practical work. The students can work freely in the computer lab outside these hours.
They face situations where explorations with computing tools give some positive insight and allow them to discuss meaningful mathematics or imagine how their students might react to a particular problem.
Over the years the course changed a little. In the beginning it was more theoretical and philosophical. Now it stresses very much the concrete experiences with the students and the connections with the curriculum the students will have to teach in the future. We discuss less the computer in itself and we practice more with the computer. In the last years the main change has been the use of more powerful tools like the Dynamic Geometry tools and simulation and modelling.
The students react very well to this type of course. They see that it is very close to the reality they will have to face afterwards, and they see that the discussions include significant mathematics they will teach afterwards (and sometimes they were not very happy with some of that mathematics, when they were secondary school students). The students come regularly to the classes and participate actively. In their final year of graduation they try to use the computer in their classroom (during the stage in a secondary school), although schools are not very well equipped with computers (the computers are used in technical disciplines and not all colleagues view the computer as a very valuable tool for more classic disciplines).
We make a very positive balance of our experience teaching this course. In the future we plan to do some kind of follow-up to see how our students react later as teachers.
In the course "Foundations and Teaching of Analysis" we have 3 hours a week of theory and demonstrations and 1.5 hours a week of practical work in a normal classroom. All the practical work is done with graphics calculators. We have 90 students each year that are divided into three groups. Students practice the use of graphics calculators, the new possibilities, and the pitfalls and do some experiments with sensors and mathematical modelling. Each student has to write a report on an individual project taken from a book edited by the Ministry of Education (Teixeira, P. et al.) so that all the work has a real connection to what students will find afterwards as teachers.
Scientific calculators are compulsory since 1992 in Portugal in secondary school and graphic calculators from 1997, so students feel at the outset that this course is useful. The reaction has been very good, and the students work hard to master the use of the graphics calculators. Of course we cannot think that students can learn everything they need about calculators in a single course; topics like programming, CAS and regression are barely discussed; also they do not work directly with secondary school students. So some kind of follow-up, to have some measure of the real impact of this course in schools, is needed.
References
Rosendo, Ana Isabel and Silva, Jaime C. (1994) Computers in Mathematics Education - An Experience. Electronic Proceedings of the 7th International Conference on Technology in Collegiate Mathematics – ICTCM.
http://archives.math.utk.edu/ICTCM/abs/7-FC14.html
Silva, Jaime C. (1996) Are graphing calculators the catalyzers for a real change in mathematics education? Gómez, P. and Waits, B. (eds.) Roles of calculators in the classroom.
http://ued.uniandes.edu.co/servidor/em/recinf/tg18/Carvalho/Carvalho-1.html
Teixeira, P. et al. (1997-99) Funções, 10o ano, 11o ano, 12o ano. ME-DES, Lisboa.
Teodoro, V. D. - Modellus™ experiments with mathematical models.
http://phoenix.sce.fct.unl.pt/modellus/
Weiner, Rebecca S. (2001) Online Courses to Improve Teacher Technology Skills. New York Times, April 25.
http://www.nytimes.com/2001/04/25/technology/25EDUCATION.html
Using Derive for beginner courses of recursion theory
Nelson Urrego P.
Bogotá, Colombia
3. Primitive recursive functions
4. General recursive functions
In this Paper, the author gives a short introduction to the main concepts of Recursive Functions and some examples of this sort of functions are programmed using DERIVE. These kind of exercises can motivate students at the collegiate level in the design of algorithms for solving rigorous arithmetic problems such as the implementation of a procedure to generate a 1-1 Primitive Recursive correspondence between N2 and N.
1 Introduction
Recursion Theory is one of the most important branches of Mathematics because, when considering Recursive Functions as those applications which can be carried out by a computer machine, we are dealing with a powerful discipline that supplies the limits of mechanical procedures (algorithms).
Towards the beginning of the 20th century, mathematicians began being concerned with solving problems, in different branches of Mathematics, where a more precise concept of the algorithm notion was required.
Intuitively speaking, we can define an algorithm saying that it is a process, where no creative invention is involved, that searches for the solution of a given problem by a finite number of steps. However, this ambiguous idea was not sufficient if the challenge was to solve theoretical problems such as Hilbert’s tenth problem (to give an algorithm which will tell of a given polynomial Diophantine equation with integer coefficients whether or not it has a solution in integers); as well as the famous Decision Problem to give a mechanical procedure to determine the truly of falsity of a sentence of the Predicate Calculus.
In 1931 Kurt Gödel published his famous Incompleteness theorems which resulted in the end of some important basic goals of Hilbert’s Program. Gödel proved that there is no way of constructing a consistent first-order formal system so that every true statement representable within the system is provable and that the consistency of such system cannot be proved with methods involved on it. As an important element in his proof, Gödel introduces the formal and rigorous notion of Recursive Function, which permits to define those functions whose values can be calculated by a computer machine.
This basic concept (Recursive Function) is very important for students at the collegiate level of Computer Sciences or Logic; that is why it is so important to encourage them to use computing tools to program different algorithms in order to represent several kinds of theoretical problems that involve not only Primitive Arithmetic functions but more complex situations in Formal Number Theory, Algebra or Computability.
We shall give a short introduction to the main concepts of recursion theory.
2 Arithmetic functions
A function f from Nn to N, where Nn is the n-fold cartesian product of N with itself is called Arithmetic. We can define new Arithmetic Functions from other by means of the following rules:
Composition
If f is an arithmetic function with m arguments and each of the g1, g2,..., gm are functions of n arguments, the function h(x1,..., xn) = f(g1(x1,..., xn),..., gm(x1,..., xn)) is obtained from f and g by composition.
Primitive Recursion
If g is a function with n arguments and h is a function with n + 1 arguments, then the function f defined as follows
f(x1,..., xn, 0) = g(x1,..., xn)
f(x1,..., xn, y+1) = h(x1,..., xn, y, f(x1,..., xn, y))
is obtained from g and h by Primitive Recursion.
Minimalization
If g is a function with n+1 arguments, the function h defined as follows h(x1,..., xn) = my[g(x1,..., xn, y)=0], the smallest y such that g(x1,..., xn, y)=0 is obtained from g by minimalization.
Here minimalization is restricted to those cases in which there is
at least one y for any áx1,..., xnñ such that g(x1,..., xn, y) = 0.
In general, for the predicate P(x1,..., xn, y), the notation myP(x1,..., xn, y) indicates the least y such that P(x1,..., xn, y) is true if there is any y.
We shall call the following arithmetic functions the Initial functions:
The
Successor Function: s(x) = x+1
The
Zero Function: Z(x) = 0
The
Identical Function: Iin (x1,..., xn)
= xi (1 £
i £ n).
3 Primitive recursive functions
An Arithmetical Function is said to be Primitive Recursive if it is one of the Initial Functions or can be obtained from them by finite applications of rules 1 - 2.
Examples 1-14 are Primitive Recursive Functions (Table 2)
DERIVE supplies a very simple and useful symbolic language in order to determine the algorithm that computes the respective Primitive Recursive function. Initial functions can be programmed easily with DERIVE as follows:
Examples 1-14 also can be defined in DERIVE and no advanced knowledge is required for this task:
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Ex. 1: Predecessor Function:
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Ex. 2: Addition Function:
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Ex. 3: Product Function:
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Ex. 4: Exponent Function:
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Ex. 5: Factorial Function:
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Ex. 6: Difference Modified Function: |
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x |
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Ex. 7: |
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sg(x) = |
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Ex. 8: sg*(x) = 1 |
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Ex. 9: Absolute Difference Function: ½x - y½=add(x |
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Ex. 10: Minimum Function (two arguments) minim(x, y) = x
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Ex. 11: Maximum Function (two arguments) maxim(x, y) =
add(y, x |
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Ex. 12: Remainder Function (remainder upon division of y by x)
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Ex. 13: Quotient Function (quotient upon division of y by x)
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Ex. 14: Number of divisors of x:
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y)
Table 1: Some examples of Primitive Recursive Functions
DERIVE package has internal functions for each of examples 3-13, except maybe for 8 and 11. The objective is to be able to design every algorithm without the support of those internal functions as well as it is possible. Examples 12-14 require some of the functions defined previously:
4 General recursive functions
An arithmetic function f is said to be General Recursive (or simply Recursive) if f can be obtained by finite applications of rules 1-3. We should note that every Primitive Recursive Function is General Recursive. Examples 15 and 16 are recursive functions and can be defined with DERIVE as indicate expressions 44 and 46:
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Ex. 15:
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Ex.16: |
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5 Recursive predicates
A predicate P(x1,..., xn) is an expression where variables áx1,..., xnñ can be replaced by natural numbers áa1,..., anñ, so it is possible to determine the truth or falsity of P(a1,..., an) e.g. the predicate P(x, y) := x+y = 5 is true for the pair á2,3ñ, but is false for á3,4ñ.
The Characteristic Function of P is an arithmetical function defined by:
A predicate P(x1,..., xn) is recursive if its characteristic function is recursive. In Table 2 we can find some examples of Recursive Predicates.
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Predicate |
Description |
Characteristic Function |
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Ex. 17: |
x | y |
x divides to y |
divi(x,y) = sg(rem(x,y)) |
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Ex. 18 : |
x < y |
x is less than y |
min(x,y)
= sg* (y |
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Ex. 19: |
x = y |
x is equal to y |
eq(x,y) = sg*|x-y| |
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Ex. 20: |
Pr(x) |
x is a prime number |
P(x) = sg[(D(x)
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Table 2: Examples of Recursive Predicates
Using some Recursive Functions defined above with DERIVE we have programmed the following Characteristic Functions:
Now we can define a Recursive Function to determine the nth prime number starting from 0:
This function can be programmed with DERIVE as follows:
Finally, we shall construct a function that generates a Primitive Recursive 1-1 correspondence between N2 and N.
Pairs of natural numbers can be enumerated on a list as follows:
Here pairs in kth group (starting from 1) are ordered in this form:
(0, k+1), (k+1, 0), (k+1, 1), (1, k+1),..., (k, k+1), (k+1, k), (k+1, k+1).
If x = y, then (x, y) is the (x+1)2 pair of the (x + 1)th group. On the other hand, if x < y, then (x, y) appears before (y, x) and it is the (2x+1)th pair of the (y+1)th group. The function s(x, y) determines the place of (x, y) on the list. The pair (0, 0) occupies the 0th place on the list:
s(x,
y)
= (sg(x 
y)).(x2+2y+1) + (sg*(x 
y)).(y2+2x)
The functions a(n) and b(n) such that a(s(x,y)) = x and b(a(s(x,y)) = y are defined bellow and determine the inverse function y(n) = (a(n), b(n)) which denotes the nth pair on the list:
This correspondence can be programmed in DERIVE as follows:
We can write VECTOR(y(n), n, 0, k) for obtaining the first k pairs on the list :
6 Conclusion
There are a large number of applications for the functions defined in this paper. Many of them are very useful in trying to perceive the intuitive connection between Computable Functions and Recursive Functions especially for the student that faces for the first time a rigorous course about Formal Number Theory. At the end of a course supported by computing tools everybody could be more than illustrated with using a simple but powerful tool in mathematical work such as DERIVE in order to complement theoretical and practical concepts in one important branch of mathematics such as Recursion Theory.
Acknowledgement
Special thanks to Dr. Iván Castro for encourage the using of DERIVE as fundamental part in mathematics courses in Javeriana University with his Seminary on Mathematics Supported by Computer and also to Dr. Guillermo Arias for introducing me in studying Recursion Theory.
References
Boolos, G. S. and Jefferey, R.C. (1993) Computability and Logic. 3rd ed. Cambridge University Press.
Castro, I. (1992) Cómo hacer Matemáticas con DERIVE. Reverté. Bogotá.
Mendelson, E. (1979) Introduction to Mathematical Logic. 3rd ed. Van Nostrand.
Urrego, N. (2000) Some applications of Post and Turing Machines in Mathematics Teaching. Proceedings of the 4th Int. Conf. DERIVE TI 89/92. John Moores University, Liverpool.
Internet materials in mathematics teaching
Maria Zajac
Krakow, Poland
There are different forms of implementing IT in the teaching process. One of the recent ways is the Internet. It becomes more and more popular and more common. The schools and the students get the access to the Internet and that is relatively cheap. The teachers, especially mathematics teachers, are the group of educators who understand this value very well. The number of teachers, who use IT in their everyday work, is growing systematically. But one of the frequently asked questions is: “Where can I find useful materials for my work?”
The amount and variety of the resources offered by the web pages are huge. They range from big commercial portals to single individual pages created by the teachers or, more often, by their students. The choice of HTML-processors offered by different software companies makes the task easy even for non-experienced users. Such a variety has its advantages and disadvantages. To find the staff one really needs is often a big trouble.
One of the important problems is the quality of the web sites. It happens that some pages contain pieces of information, which are misleading or incomplete or even false. Moreover, commercial portals usually are not free of advertisements, offered by the sponsors and not always linked to the main subject of a page. That causes a so called “information noise” and makes the way of finding the things we are looking for much more difficult, than it could really be. We can obviously use a computer program called a search engine in order to find pieces of information linked to the keyword we have indicated. But, I suppose everyone who has tried to find anything in the Internet using such a program can give at least a few examples of strange results of their search. You put the word “Pythagoras” for instance and you get 500 matches. How many of them have anything to do with this famous mathematician?
Another obstacle is the language. There are extremely interesting web pages, which offer lots of ideas, useful tasks and hints both for the teachers and for the students, but they are in English. It goes without saying that the Polish teacher cannot use them directly during the lesson even if he or she is familiar with that language.
On the other hand, quite often individual pages are created by people, who are very keen on the subject they present on their web sites, they know it very well and they have collected a great amount of interesting ideas and solutions to them.
At the Pedagogical University in Krakow, where I work, we teach teachers. During the postgraduate courses the teachers broaden their knowledge and share their experience. They are always eager to learn new web addresses and to check them. That gave us the idea of creating an educational web site, located on our university home page, free of any sort of commercials and aimed at the teachers, who could find there the information they are looking for. The resources of our page mostly come from the collaboration with the teachers we teach, and who stay in touch with us, even when they leave the university. By the end of the course they are obliged to prepare some teaching materials using the techniques they have learned during their studies. It can be a Power Point presentation, a web site, a computer program or simply a scenario of a lesson based on the use of a computer. The collection of most valuable “products” is stored on the web site. We have called this page “An Internet Educational Bulletin.” The materials we put in there imply the structure of a web site. Figure 1 shows its home page.
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Fig. 1: Internet Educational Bulletin – home page |
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The main menu offers the choice either by the form of the resources (presentation, web page, scenario, computer program) or by the school subject. Both ways lead to the same result: one can find an appropriate piece of information linked to the particular subject. Figure 2 shows a mathematics page. All the math resources stored in a database are listed in an alphabetical order. They are aimed at different school levels (primary, secondary 1 and 2) and they have various forms.
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Fig. 2: A view of a page “School subjects – mathematics” |
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The resources can be viewed in the Internet browser, can be run on-line on the remote server or can be copied to the teacher’s computer for later offline use. Anything you want to do, you simply press the appropriate button. The decision depends on the user, although it is strongly recommended to download the files one has chosen instead of running them directly from the server.
Not all the resources are placed on the university server. Sometimes we only give the description and the address, where one can find something useful. There are pages we recommend and the contacts with people who have created them.
As our intention was to collect possibly complete information for teaching purposes we added a fistful of links to educational institutions like the Ministry of National Education or The Centre of Teachers Education. People can find there the syllabuses for school subjects, the announcements about currently organised courses and studies, or all the legal regulations newly imposed. The addresses to organisations like, for instance, the Association of Mathematics Teachers as well as the addresses of the schoolbook editors are also accessible through this page. Many of them include on their web pages supplementary materials for the books they sell, in the form of lesson scenarios or sets of tasks and hints for the teachers. For those, who are not afraid of foreign languages we give some addresses to similar organisations or groups in other countries, like Nrich Club in Great Britain or an international initiative called European Schoolnet.
Everyone can include his or her own materials on our web page. An option in the menu leads to the help on how to add new resources: what information is needed, what format of files is required and what are the rules one has to obey.
The content of our web site will be updated regularly. Everybody interested in getting information about the current changes can subscribe to a mailing list by clicking a button on the right hand side of a home page and filling out an appropriate form.
We cannot forget about the quality of our resources. It is planned that all the materials will be revised by specialists prior to their inclusion. At the Pedagogical University Krakow we teach all the school subjects. At present we are at the stage of creating a group of professionals, who would collaborate with us on revising the stuff we have already collected and we will get in the coming future. It has to be stressed that the teachers are not only interested in getting access to electronic materials they can use at school, but also many of them are eager to share their own “products” and to publish them in the Internet. It would obviously be very convenient to have a group of tutors that would co-ordinate the process of creating the materials but the solution is not easy because the whole process is dispersed in time and space. If our initiative gets a broad acceptance among the teachers we consider launching an interdisciplinary project to supervise it.
As the use of computer based teaching materials became very popular in recent years I hope it will be useful to exchange our ideas with educators from other countries. Although the resources cannot be applied directly because of different languages we use, some aspects of teaching are common for teachers and their students all over the world.
The web site presented can be viewed at the following location: http://www.ap.krakow.pl/ibe/ - any comments and remarks can be addressed to the author via e-mail.
CASCADE-IMEI:
Web site support for student teachers learning - Realistic mathematics education in Indonesia
Zulkardi and Nienke Nieveen
Sriwijaya, Indonesia;
Twente, The Netherlands
3. Description of the learning environment
CASCADE-IMEI is a learning environment in the form of a face-to-face course and a web site (www.cascadeimei.com) which aims to support student teachers in Indonesia to learn Realistic Mathematics Education (RME). RME is an instructional theory in mathematics education that was originally developed in the Netherlands. So far, two prototypes of the learning environment have been developed and evaluated both by student teachers in Indonesia and several experts in the Netherlands. This paper presents the origins of the learning environment (with an emphasis on its web site) as well as the results of the first two cycles of its prototyping process.
1 Introduction
CASCADE-IMEI (Computer Assisted Curriculum Analysis, Design and Evaluation for an Innovation in Mathematics Education in Indonesia) is a learning environment that aims to support mathematics student teachers in Indonesia who are planning to implement lessons that are based on the realistic mathematics education (RME) approach. RME is a well-known teaching-learning theory in mathematics education that has been proved to be successful in reforming the mathematics curriculum especially in the Netherlands (Lange, 1993; Gravemeijer, 1994). Contrary to the teaching approach that is common in Indonesia, RME uses contextual problems or applications as a source as well as a starting point for mathematics teaching. In this study RME is assumed to be a promising approach to improve mathematics education in Indonesia. The learning environment consists of two main parts that is a face-to-face course and a web site. The course stresses on the activities such as learning about the main ideas behind RME, its curriculum materials and how to use the materials in the classroom. On the other hand, the web site focuses on supporting course participants with RME materials, training opportunities focuses on how to teach RME materials using RME approach in the school and communication tools that can be used by participants when implementing RME lessons (Nieveen, 1997; Khan, 1997).
This paper first presents theoretical framework of the study that is theory of Realistic Mathematics Education that is used as a guidelines in designing the learning environment and the strategies how to introduce RME as well as web site technology as innovation to the prospective teacher of school mathematics in Indonesia. The description of the learning environment is briefly presented. Then, the research methodology and the results of the study are discussed. A discussion and a conclusion complete the article.
2 Theoretical framework
RME: philosophy, principles and characteristics
The learning environment, including both the course and the web site, is designed based on the RME philosophy and principles. The philosophy of RME is mostly determined by Freudenthal's view on mathematics (Freudenthal, 1991). Two of his important points of view are: (1) mathematics must be connected to reality; and (2) mathematics should be seen as a human activity. First, in order to start from reality that deals with phenomena that are familiar to the students, Freudenthal’s didactical phenomenology that learning should start from a contextual problem is used. Second, by the guided reinvention principle through progressive mathematizations, students are guided didactically and efficiently from one level to another level of thinking through mathematization. These two principles and the concept of self developed models (Gravemeijer, 1994) can be used as design principles both in developing the course materials and the web site. Furthermore, these principles are operationalized into five basic characteristics of realistic mathematics education or five tenets of RME (de Lange, 1987; Gravemeijer, 1994). In short those are:
Use of contextual problems (contextual problems figure as applications and as starting points
from which the intended mathematics can come out).
Use of models or bridging
by vertical instruments (broad attention is paid to
development models, schemas and symbolization rather than just offering the
rule or formal mathematics right away).
Use of students' contribution
(large contributions to the course are coming from
student's own constructions, which lead them from their own informal to the
more standard formal methods).
Interactivity (explicit negotiation, intervention, discussion, cooperation and
evaluation among pupils and teachers are essential elements in a constructive
learning process in which the student's informal strategies are used as a lever
to attain the formal ones).
Intertwining of learning
strands (the holistic approach implies that learning
strands can not be dealt with as separate entities; instead, an intertwining of
learning strands is exploited in problem solving).
Strategies for introducing RME in teacher education in Indonesia
De Lange (1993) pointed out several problems that were faced when Dutch experts introduced the RME approach in Dutch secondary education. The general problem was: How to educate teachers to learn using new materials and to teach with materials using the new approach? This appeared to be a complex endeavor due to several changes: (1) the mathematics materials differ from former ones; (2) the role of teachers change from teaching to 'unteaching' ; and (3) the level of thinking needed changes from low level thinking towards middle-level and high-level order thinking. These changes need to be taken into account when considering implementation strategies for RME in Indonesia.
As a consequence of reform, alternative curricula, alternative teaching materials, and alternative forms of assessment are essential (Van den Heuvel-Panhuizen, 1996; Fullan, 1991). Selter (1997) pointed out that these are all mediated through the teacher, specifically through teacher's beliefs about how to organize and facilitate pupil's learning of mathematics. From this perspective, teacher education (pre-service as well as in-service) plays a vital role. One key strategy in this situation is to engage teachers or student teachers in their professional development using the following strategies (Loucks-Horsley, Hewson, Love & Stiles, 1998): (1) a short course (for building knowledge by teachers or student teachers); (2) curriculum development (by adapting the innovative materials into the school practice); and (3) using technology (in order to provide teachers or student teachers with a sustainable tool which provides rich information about the new approach).
In this study, RME will be introduced to Indonesian student teachers by developing a learning environment in which a face-to-face RME course and a web site both contain background information and RME curriculum materials.
3 Description of the learning environment
This section provides a brief description on both components of the learning environment: the course and the web site (see Figure 1).
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Fig. 1: The components of the learning environment |
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The course and its materials
The RME course is a part of learning environment that is developed in order to make student teachers understand what RME is and how to implement RME in the classroom. The main contents of this course include: (1) overview of the RME theory; (2) learning what are RME materials and how to redesign them; (3) learning how to teach using RME approach in the classroom; and (4) learning how to assess the students in the RME classroom.
The course materials are adapted from RME books that were developed by Freudenthal Institute experts as well as materials from the "Mathematics in Context" books (mathematics books for student grade 5-8 in USA) that were developed during the collaboration project between Freudenthal Institute and University of Wisconsin-Madison (Romberg, & de Lange, 1998). Especially the examples, curriculum level, context of mathematics topics and number of mathematics problems on each topic needed to be adapted to the Indonesian context. At this moment, five topics were adapted and used in the course that is side-seeing, symmetry, linear equation system, four cubes and matrices.
The web site
The web site, http://www.cascadeimei.com/, is developed in order to support the course participants in a sustainable way. In order to do so, the following options are available:
Online Info-base or task. The online info-base is the main component of the web site and
consists of exemplary RME materials such as student materials and teacher
guides (Fig. 2); student productions from RME classes; applet programs and
mathematical games; links to web sites that have relationship with mathematics
education in general and RME.
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Fig. 2: Online RME lesson materials |
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Online Tutor. In order to inspire student teachers before they conduct teaching
practice in the school, an online tutor was designed. This moment, the online tutor
consists of theory on how to use RME materials in the classroom. In the next
coming months, a number of video clips that illustrate critical moments of
teaching using RME materials in the classroom will be made available For
example, how to start the lesson, how to organize and to manage groups or class
discussions and how to close the lesson.
Online Talk. In order to provide student teachers a support in which they can
discuss their problems and their experiences, the online talk elements of the
web site was provided. Two kinds of communication tools can be used are e-mail
and a mailing list.
Online Test. In order to facilitate the users or student teachers with a number
of RME problems, an online test called problem
of the month (see figure 3) was developed. It contains not only example of
RME problems but also a guide how and when to use them in the classroom
practice.
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Fig. 3: RME problem of the month |
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4 Methodology
Research phase
This study uses a development research approach (van den Akker, 1999). With this method, the learning environment is developed and evaluated in three main phases: preliminary study, prototyping stage and assessment stage. This paper only discusses the first two phases.
The preliminary study phase
In the first phase, a review of the relevant literature on RME and curriculum development has been conducted and resulted in preliminary prototype of RME curriculum materials. Also a preliminary prototype of the learning environment has been designed and evaluated with several experts in the Netherlands. After revising this preliminary prototype and adapting it to the Indonesian context it was implemented to the target group in teacher education in Bandung, Indonesia.
The prototyping phase
In this phase, two prototypes of the learning environment have been developed and evaluated both in the Netherlands and in Indonesia. Based on results and suggestions, the third prototype was developed and evaluated from February 2001 to May 2001.
The results of the formative evaluation of third prototype are discussed in the remainder of this paper. The formative evaluation was geared towards the following questions:
What
is the perception of participants with respect to the learning environment?
What
is the practicality of the third prototype of the learning environment?
What
knowledge and skills did participants learn after using the learning
environment?
Participants
The participants of the formative evaluation of the learning environment were 8 pre-service student teachers (1 man and 7 women) at the Department of Mathematics Education, Indonesian Educational University in Bandung, Indonesia. All of them were fourth-year students and have no teaching experience before. They can use window-based computer but without access to the internet. The average of their age is 22 years old.
Instruments and activities
The instruments that were used in order to evaluate both the course and the web site are summarized in Table 1.
Table 1: Activities, research questions and data collection on effects
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Activities |
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Research question |
Whole program |
Cooperative evaluation of the web site |
Document analysis |
Teaching practice in the schools |
Seminar |
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Participants’ perceptions |
Questionnaire |
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Practicality of the learning environment |
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Interview schema |
Analysis of e-mails and reflective journals |
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Participants’ learning gains |
Questionnaire and test |
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Analysis of teaching preparation |
Lesson observation |
Group discussion/ seminar check list |
Procedure
RME Course
The course was implemented in the teacher education college in the period Feb 2001 - May 2001. The activities were conducted within a frame time of eight blocks of two-hours for training and two blocks of two hours for seminar a reflection. After the participants filled out the entry questionnaire the course started by giving the participants information about the basic principles and characteristics of RME. Then some examples of realistic mathematics problems were given and discussed in groups in order to get an idea of each characteristic. Next, the student teachers were given a number of RME problems in five topics (side-seeing, symmetry, linear equation system, four cubes and matrices). In this case they were treated as learners while the researcher acted as a teacher. After they solved the problems, they were guided in discussing the various strategies and in several cases they were invited to present their answers in front of the class. At the end of the course they were tested to see their performance in solving the problems and redesigning a number of contextual problems into a short lesson. In addition, they filled out the final questionnaire.
All participants demonstrated their RME teaching skills in front of their peers. Here, they gained input and feedback from their peers as well as the researcher. Then, they used the materials in the school classroom. Due to time limitation only three participants were followed when they implemented the RME lessons in their classroom. The researcher observed their lessons.
The web site
The following activities were held regarding the web site. First, participants got some technical training. They were trained how to access the internet and were guided in making a personal e-mail. In order to get e-mail experience they were invited to send some personal background to the researcher as well as to other participants. Here, they learned how to compose a new e-mail, to send it and to receive e-mail. In addition they were taught how to use the attachment facility. Then, they were asked to access the web site of CASCADE-IMEI (www.cascadeimei.com), to give comments and to send those to the developer by e-mail. Further, they were asked to communicate their experiences and problems in the course with the developer and other participants using e-mail. Finally, the web site was evaluated using a cooperative evaluation, during which four student teachers were invited to work aloud while using the web site. All comments were recorded on tape for analysis.
5 Results and discussion
Due to limited space, only the main results are presented in this section.
Result 1:
What is the perception of participants with respect to the learning environment?
The data resulting from the questionnaire are summarized in the table 2. In general, their overall perception is that the learning environment is an helpful and interesting innovation for them. Both the RME course and the web site materials that perceived to be relevant to their needs as a student teacher.
Table 2: Participant's perceptions to the learning environment
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Overall perception to the learning environment |
N |
Mean |
S.d. |
Max |
Min |
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Helpful for me as a math student teacher |
8 |
4.9 |
0.35 |
5 |
4 |
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Rich with new math information |
8 |
4.6 |
0.52 |
5 |
4 |
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Interesting |
8 |
4.6 |
0.52 |
5 |
4 |
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Educative |
8 |
4.1 |
0.35 |
5 |
4 |
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Consistent with the needs |
8 |
4.0 |
0.53 |
5 |
3 |
Note: 5 = highly positive, 1 = highly negative
Result 2:
What is the practicality of the third prototype of the learning environment?
Based on the data from the questionnaire, interviews and reflective journals, the third prototype of learning environment seems to be practical for student teachers (easy to use, helpful, easy to understand and the content is consistent with the school curriculum). However, there are several comments from users which suggest some parts of the web site need to be improved such to add more topics on RME materials and more example of RME problems.
Result 3:
What knowledge and skills did participants learn after using the learning environment?
In order to answer this question four kind of results are used. First of all, the assessment of the understanding of student teachers of the RME theory. All participants were able to describe the philosophy of RME, its principles and its characteristics. It was important for them to gain this understanding before they dealt with the materials and how to use them in the classroom. Second, based on the analysis of the preparation plans it may be concluded that the student teachers were able to design the materials based on the RME tenets, of course, with support from the developer (such as a number of ready made lessons and guidelines how to make them). Then, the teaching skills of student teachers were observed by the researcher both in the simulated situation and in the real classroom. An overall impression was that they were able to teach using realistic materials. They used their knowledge from the course such as how to starting the lesson, making groups of student, guiding group and class discussions and closing the lesson. However, they also met several problems such as how to motivate the students to get involved in the discussion and how to conclude the lesson.
Finally, the reflections of the student teachers during the seminar were observed and discussed. Here they invited to present their experiences in front of the teacher educators and the researcher. In this activity the researcher was able to observe and discuss the performance of the student teachers with regard to their knowledge of RME theory, their attitudes and their skills related to RME.
6 Conclusion
Based on the results of the formative evaluation of the RME learning environment section it can be concluded that:
the
learning environment was perceived to be an interesting program by student
teachers in teacher education in Bandung.
the
web site as a part of learning environment was judged to be practical by
student teachers
student
teachers gained the following knowledge and skills:
- how to use e-mail facilities for communicating with peers or other users
- how to access or retrieve data from web site or internet
- background theory of RME and its materials
- how to redesign RME materials
- how to use RME materials in the real classroom and
- how to assess pupils during and after each lesson.
Acknowledgment: This study is funded by World Bank Indonesian Secondary Teacher Development (PGSM) project IBRD Loan No. 3979-IND. We wish to thank the Indonesian student teachers, the Indonesian co-supervisor and the supervisors from both the University of Twente and the Freudenthal Institute for their support.
References
De Lange, J. (1987) Mathematics, Insight and Meaning. OW &OC, Utrecht.
De Lange, J. (1993) Innovation in Mathematics Education. de Lange, J., Keitel, C., Huntley, I. and Niss, M. (eds) Innovation in Maths Education by Modelling and Applications Ellis Horwood, Chichester, 3-17.
Freudenthal, H. (1991) Revisiting Mathematics Education. China Lectures. Kluwer, Dordrecht.
Fullan, M. (1991) The new meaning of education change. Teacher College Press, New York.
Gravemeijer, K.P.E. (1994) Developing Realistic Mathematics Education. CD-ß Press, Freudenthal Institute, Utrecht.
Khan, B. H. (1997) Web-based instruction (WBI): What is it and why is it? Khan, B. H. (ed.), Web-based instruction. Educational Technology Publications, Englewood Cliffs, NJ 5-18.
[WWW document http://www.gwu.edu/~etlalex/khan/wbitc.html ]
Loucks-Horlsey, S., Hewson, P.W., Love. N. and Stiles, K.E.(1998) Designing professional development for teachers of science and mathematics. Corwin Press, Inc., Thousand Oaks.
Nieveen, N.M.(1997) Computer support for curriculum developers: A study on the potential of computer support in the domain of formative curriculum evaluation. Dissertation. The University of Twente, Enschede.
Romberg, A. and de Lange, J. (1998) Mathematics in Context: Get the Most Out of it. Grade 8 Algebra Strand. Britannica Mathematics system, Chicago.
Selter, C. (1997) Instructional design for teacher education. Beishuizen, M. Gravemeijer, K.P.E. Lieshour, E.C.D.M.van (eds.) The Role of Contexts and Models in the Development of Mathematics Strategies and Procedure. CD-B Press Freudenthal Institute, Utrecht, 55-77.
van den Akker, J.J.H. (1999) Principle and methods of development research. Akker, J.J.H van den , Branch, R., Gustafson, K. Nieveen, N. and Plomp, Tj. (eds), Design approaches and tools in education and training. Kluwer, Dordrecht, 1-14.
