MAking MAThEmatics TEAchers MObile
Summary Report of the Teacher
and Student Visitors’ Experiences.
by Leo Rogers, Project Evaluator
PART I - Preparation Before the Teacher/Student Exchanges
Part II - The Visit, the Teaching, and the Experiences
- Language preparation course
- Transition and contacts with host teachers
Part III - Reflections and Suggestions from Returning Students and Teachers
- In Host Preparation Observation and Teaching
- Models of teaching observed in host country
- Teaching of Mathematics
- Use of Text Books and Exercise Books
- Curriculum and Observation
- Organising Practical Work
- Use of Language in Class
- Lesson plans and changes: adapting to the host’s practices
- Use of Equipment
- Participants’ Concluding Observations and Suggestions
- Final Observations and Suggestions from the Students
IntroductionThe following report is a collation and summary of the separate reports submitted by the ten teachers and student teachers (trainees) who participated in the exchanges and taught mathematics in a foreign language in schools in the five project partners.
The report also contains a summary of the interviews held by members of the partner teaching teams with the teachers and students on their return from the host country.
The administrative and organisational details pertaining to the overall management and arrangements of the individual partners can be found in the main body of the account of the project.
This report is arranged in three parts: Preparation in the home institutions; Reception, observation and teaching in the partners’ schools; Reflections and Suggestions on return home.
PART I - Preparation Before the Teacher/Student Exchanges
General pre-visit preparationThe partners made great efforts to provide practical preparation and suitable language support in advance of the exchange visits. There was, however, a wide variation in the pre-visit training that could be provided due to timetabling difficulties, institutional inertia, and the normal commitments of students and staff. There was universal praise from the students and teachers and general satisfaction with the preparation programmes provided.
Concerns before the visitA natural expectation of the students was that, being in a new situation, relations between themselves, the class teacher, and the pupils, might be less relaxed than they were used to. There were anxieties also about the planning and implementation of teaching. In fact, in general, this turned out not to be the case. All participants reported that pre-visit relations with their host staff and pupils were most pleasant and productive.
2. Language preparation courseIn the Czech Republic for example, a course in teaching mathematics in a foreign language is a standard part of the curriculum, whereas special arrangements had to be made for time-tabling a similar provision in other countries. Organising a language training course specifically for teaching mathematics at short notice ran into many organisational and institutional difficulties. While on the one hand it was possible to organise some such teaching in Italy, the other partners were not so fortunate.
Probably the most fortunate were the French students who were able visit the “Liceo Leonardo da Vinci” in Paris where they observed some II Media classes and had a chance to speak to the Liceo teachers about their teaching and classroom organisation. One of the teachers provided a useful French-Italian supplement to the ‘tool-box’ list already provided by the training team. The French visitors were also introduced to typical Italian textbooks and had a chance to appreciate some of the differences in technical language and mathematical notation involved in the preparation of the their chosen teaching topic. Through this experience they were prepared for their visit to the Santa Caterina Institute in Pisa.
In Italy, the first visitor to Austria was able to attend a course for teaching mathematics in a foreign language at the University of Pisa and later was able to work with a mother tongue teacher of German.
On the other hand, due to sudden last-minute problems, a substitute visitor had to be found and her work was prepared on the basis of her former experience, and her discussions with the Italian project coordinator. Luckily, she had nearly twenty years experience of teaching Italian in a school in Germany that gave her a knowledge of the language both at oral and written level, good enough to provide a secure base for classroom communication. While she may have been modest about her mathematical knowledge, her language competence enabled her to manage her class teaching admirably, and after a fruitful exchange of e-mails with her host she was able to organize her own teaching activity.
There were many instances among the partners where students found time to cooperate in learning and practising the language of the host country and presenting their teaching units before the visits.
3. Transition and contacts with host teachersIn all cases, the contact with the host teachers was most friendly and informative. The hosts found accommodation, helped with transport, sent timetables and curriculum documents, and information about their pupils. Timetables and lesson plans for the visitors were prepared jointly with the help of both home and host teaching staff.
The willingness and availability of the host teachers who collaborated throughout each visit was a vital contribution to the serenity of the visiting students.
Part II - The Visit, the Teaching, and the Experiences
1. In Host Preparation Observation and Teaching All the visitors reported that their hosts gave them a great deal of support, organised material that was needed for the lessons and did not impose their views about teaching methods, but helped the visitors to understand their way of teaching, and the customs and expectations about preparation and organisation.
On a few occasions there were communication problems since the host teacher was not always available for consultation, due to clashes on the timetables of the visitor and host, and this had to be resolved outside teaching time.
School and curriculum organisation:As pointed out above, differences in expectations, school and class organisation and curriculum demands were only to be expected. Visitors were clearly more comfortable when they met conditions and conventions similar to those experienced at home; on the other hand, it made them examine their own preconceptions about education.
When pupils have become used to the ways in which particular teachers address them or the conventions of the school (like standing when a teacher come into class, or moving about the classroom without asking permission) it is not surprising that visitors have to take time to adjust to these differences.
Thinking that the same class organisation prevailed throughout a particular school was soon challenged, and it was realised that whole class and group teaching were adopted for different age groups or different reasons. Furthermore, the organisation of mixed ability classes offered a very different kind of challenge to the visitors.
A common problem (no matter whether it be in their own country or another) was that the visitors did not complete all the aspects of their lesson plans with the host pupils. In one case the visitors decided to change their plans completely and adopt group work instead of whole-class teaching, which they supported intensively by giving pupils different tasks and helping them individually. In this case it was only possible because there were two people working together in the class.
The greatest contrast and challenge remarked on by the visitors was between the Danish and Austrian schools systems. They said the schools appeared so different that they claimed it was not possible to compare the school in Skårup to any school in Austria. It was also claimed that an Austrian teacher, is not trained to handle the kind of pupils who need special support as found in the Danish class. This culture shock was mostly expressed by the Austrian visitors who were quite unused to managing mixed ability classes that called for a radical change in teaching styles. This problem was not anticipated before the visit and the trainee teachers had some difficulties adjusting to the situation.
Generally, the atmosphere in the classrooms was positive and productive; most pupils worked and tried to participate in the activities.
2. Models of teaching observed in host country
a) General organisationHost teachers were very good at organising their lessons, using mathematical terminology and explaining things to the pupils. Many of the lessons observed seemed to be conducted on more or less the same pattern.
b) Teacher driven lessonsFrom the majority of observations it appeared that the primary way of organizing teaching is in the form of whole-class teacher-directed work where the focus is on the teacher and on what happens at the blackboard.
The general format was:
In this model the teacher stands at the blackboard, possibly with a ‘good’ pupil who can be the same one during the entire lesson. No autonomous work by the pupils was observed.
- checking homework
- working at exercises with little demand for independent thinking
- giving homework
c) Teacher led lessonsHere, the format was similar to the above, but pupils were included as the teacher invited them to give answers to questions, and to come forward and make calculations on the blackboard.
d) Teacher organised lessons: group and individual workA mixture of whole-class teaching and individual work was observed, with occasional group work; but not to the extent that individual and group work are normally used in Denmark.
e) Use of materials, equipment and teaching aids by host teachersApart from frequent use of the blackboard, usually in a systematic way, by setting out the problems, calculations and formulae in a clear and legible manner, there was virtually no mention of any other equipment being used.
There seems to have been one occasion when a teacher used an overhead projector.
f) Practical activitiesVery few observations of practical activities were recorded, though more may well have happened, and the since host teachers supported the practical activities of the visitors, we assume that teachers did work in this way on occasion.
g) Teachers teaching other subjectsA significant characteristic of the Italian teachers is that they usually teach not only mathematics, but other science subjects like physics, chemistry or biology. This could be considered useful for many reasons such as being able to show examples of the practical application of mathematics in other subject areas.
3. Teaching of MathematicsVisiting students discovered that in spite of the popular view that mathematics is the ‘same’ everywhere, there are subtle differences in the terms used, and significantly, the language of teaching mathematics is not the same in all countries. This leads to unsuspected differences in the understanding of some mathematical concepts.
There were many examples of unexpected differences in classroom conventions that affected the presentation of problems and of practical exercises. Differences in notation between the work prepared and what was customary in the host classroom or found in the text-book, and orientation and labelling of diagrams were typical of the day-to-day problems the visiting students had to face.
The visitors had to learn to think ‘on their feet’ and adapt to different situations, and in many cases the pupils were most helpful and wanted to cooperate. This applied particularly in the presentation of practical exercises
4. Use of Text Books and Exercise BooksIn some countries there is a custom that pupils write notes and examples in their exercise books during the lesson; in other situations the textbooks are used for reference during the lesson discussion and are then used for work at home or outside the class. A considerable variation in these possibilities was observed.
One visitor commented that the teachers often gave about ten exercises to be completed for the next class. In most cases, the textbook gives the answer to the exercise, so it’s up to the students to find a method for the solutions.
Textbook format and presentationInterestingly, the textbooks displayed certain national characteristics. The layout and general format was quite different, and the text on the pages was more dense with some in comparison with others.
In a French textbook, the topic is presented by guided discovery and the stress is put on pupils’ independence. There are more tasks including the use of Cabri software and there are fewer exercises.
In the textbook used at home by one of the Czech students, the chapter on Pythagoras starts with the theorem and then, there are many exercises. This student remarked on how they preferred the discovery part of the French textbook and fewer exercises.
In an Austrian textbook the theory section of each topic is very short, with the main concept contained in a box; the student can obtain other important correlated concepts from the exercises. This is different from the typical Italian books.
Part III - Reflections and Suggestions from Returning Students and Teachers
1. Curriculum and ObservationThe visitors stressed the necessity of becoming well acquainted with the country and local school curricula and were pleased that they had the opportunity to make essential observations before their own teaching sessions. The time available for classroom observation was extremely useful for all visitors, although limited by the structure of the host school timetable and the number of times available to observe classes. A number of the visitors anticipated that there would be no problems with the mathematics, but they soon discovered the many difficulties that have already been noted. In some classes observed the teacher did not use any formal mathematical writing, most of the ideas were described in informal language.
Visitors’ teaching in the host countryVisitors had to learn to come to terms with the conflict between a carefully prepared lesson plan and the actual lesson situation.
“I had to construct everything in order to help the pupils understand what I wanted them to do. I think that the problem was not the language, but that they were used to the procedures where the teacher told them really everything”.
“Pupils did not pay sufficient attention to my instructions which may not have been as clear and simple as the instructions of their teacher. In my teaching, I am used to some independence of pupils.”
Customs, habits and expectationsThere are broad differences in approach to classroom teaching in different countries, adopted either through accepted cultural traditions, through individual institutional organisation, or government policies. The act of teaching is a personal idiosyncratic style that may or may not be informed by influences of tradition or policy. Furthermore, student teachers are well-known to be influenced by the ‘role-models’ they experienced when they themselves were school pupils, so it was no surprise to find many remarks from the visitors about how different the ‘teaching culture’ was in other countries, and yet “the structure was not so strict and old-fashioned as we had assumed” and once the visitors had settled in the situation became more relaxed.
The delivery of a lesson ‘from the blackboard’, as against organising groups of pupils in self-study units were experienced, and in a few cases the visitors had found it difficult to adjust. Other differences like the number of exercises given to pupils at any one time might well have depended upon situations unrealised by the visitors, like an impending examination or the necessity to practise a particular technique.
Expectations of pupils’ behaviour provoked some interesting comments.
Some visitors were surprised by the ‘freedom’ of the host pupils where they could, within reasonable limits, stand up, move about the room, and talk to each other, but they would participate in an atmosphere that was calm, where the pupils felt free to take part and ask questions. In a different country, pupils stood up when the teacher entered the class and did not sit down until they were told to.
The visitors expressed surprise that pupils in the host classrooms were much the same as in their own classes, and that the students were very willing to cooperate and carry out the activities and exercises given. In some cases given this kind of experience it was even reported that “teaching mathematics in a foreign language is not such a hard task to fulfil”.
Expectations of pupils’ work habits often assumed that pupils would conform.
Comments here often included the word ‘discipline’ and this refers both to the general behaviour of the pupils as a group, and to the individual work habits of the pupils.
It was reported that pupils generally worked more seriously with the tasks given, both individually and in class when the class was used to a more formal organisational structure. More open and relaxed classrooms gave the impression that pupils were not used to concentrating on the lesson, and in one case, by observing the pupils it was noticed that about half the class worked on the tasks given, while others appeared to do anything else without getting disciplined while those who did not work, were sometimes ignored.
Other, more individual customs were commented on, for example whether pupils were in the habit of using notebooks or exercise books to take notes during the lessons, and to what extent they relied on their textbooks for the answers to exercises or whether they were likely to work problems out for themselves.
Expectations of pupils’ achievement were often quite ad odds with the reality.
In some cases the pupils’ academic level and knowledge turned out to be much higher (or apparently lower) in comparison to a peer group in the visitors’ own country. In one situation it was remarked that pupils were much better at mental arithmetic because they knew ‘by heart’ their tables and also knew squares and square roots. It was suggested that this was because they did not use calculators.
In contrast, in some contexts, the subject matter is studied in more detail and more theoretically which in the visitors’ opinion, gives the pupils a more secure mathematical background.
Some regrets were expressed by visitors about the situation in their own country where teachers often wish the pupils would stop focusing on the solution and rather concentrate on the implied mathematical principles
2. Organising Practical WorkPupils unused to practical work were often confused about how to begin. If they were not used to working independently they needed hints or even a demonstration, and manual dexterity was sometimes an inhibiting factor. Even simple constructions like the division of an angle were a problem for some pupils. Some exercises seemed to be too difficult for pupils and visitors needed much more time for explanation. The language was not the problem, but their lack of mathematical background
Teachers wanting to use practical activities as part of the presentation of a lesson encounter extra problems when using a foreign language. What is customary in one country may not be so in another. Typically, geometry, and in this case Pythagoras’ Theorem caused particular problems. Some of the difficulties encountered were:
- it always takes more time that you anticipate;
- instructions have to be clearly written down;
- it is important to demonstrate the activity before pupils start work;
- take the process slowly step by step;
- all equipment and accessories are easily available;
- be prepared to offer alternative explanations or instructions.
3. Use of Language in ClassVisitors were often more worried by their language competence than the mathematics or the classroom situation and sometimes had problems in understanding pupils’ reactions in the colloquial language. In the majority of cases they did understand what was going on in class.
Even common formulae and expressions that had been repeated over and over again in the learning sessions at home were suddenly difficult to remember when the visitors were standing in front of the class.
There were some interesting comments from a French teacher who was also a Spanish speaker who went to work in an Italian school. During his teaching he tried to keep speaking Italian but when a word did not come to mind, he spoke in Spanish almost by reflex. Sometimes he did not notice he was speaking Spanish. He says that in these moments, “I was a bit surprised, but now I suppose it’s probably due to the proximity of the Italian, Spanish and French maths vocabularies.”
Common use of a foreign language in teachingAnother interesting situation arose when the common language of instruction was foreign to both visiting teachers and pupils. With Czech and Danish visitors, the language of instruction was English, and in both countries, pupils were being taught in English. The problems here were reciprocal.
Visitors experienced difficulties with the pupils’ comprehension of the English language and also with their own comprehension of the pupils’ use of English. In some lessons, a few pupils tended to drop out, because their comprehension and fluency was not sufficient and they found it difficult to express themselves in English.
In spite of these difficulties, visitors observed that pupils were able to profit from the teaching because they claimed pupils understood the mathematical terms and the written language, mathematics, is common to everyone. This led to a conclusion by some that both the pupils and the teacher have to adjust to the situation.
Strategies for managing language problemsSome strategies tried out were: adapting the lesson presentation to one’s capacity for speaking the language, including group work in the lesson organisation to enable closer contact with pupils and avoiding repetition, producing clear written directions for pupils, and organising PowerPoint presentations where both diagrams and language appear on the screen.
Translation and interpretationSome visitors remarked that they thought it was not necessary to have a perfect command of a foreign language, but on the other hand it is vital to be well acquainted with the way of giving instructions and using mathematical terms and expressions, especially the words for unusual mathematical notation, and classroom language. Visitors agreed that if they had had more time together with the pupils, then it would have been possible to solve the many of the linguistic problems.
4. Lesson plans and changes: adapting to the host’s practicesLesson observation enabled visitors to change their point of view about planning the sessions to be presented in the host country, and teaching units were modified by adopting a balance between their professional practice in the home country and the teaching practices observed in the host classes.
This showed a need to adapt the teaching sequence to the actual needs of the respective classes, showing professional skills that enabled them to overcome unpredicted obstacles.
Some visitors had the opinion that it was not necessary to plan the lessons in detail before the visit. They stressed the necessity to be flexible and sensitive to the environment. This depended on the variety of their own experience and how secure they were with the language of instruction.
Practical activitiesAs described above, in organising practical activities, explaining the activity was a problem because it was necessary to add additional explanation. The discussion itself with groups was fruitful and during the individual approach, the students became more attentive, and worked willingly.
5. Use of Equipment
Blackboard workOne visitor, who was used to writing a lot on the board, remarked that in her situation the rooms had small blackboards and could not contain everything written in a lesson. So she had to write until the board was full and then to wipe it off and carry on writing. There was no comment about the pupils’ use of their notebooks here.
Overhead ProjectorA presentation using an overhead projector was attempted, but unfortunately, the poor manual skills of a significant majority of pupils made the visitor decide to give up the construction of a dynamical model with transparencies.
PowerPointA PowerPoint presentation was used as a device for managing the language problem; it enabled the student to have some control over the situation and to have some appropriate words planned for each slide.
Software and use of computersA presentation of the Pythagorean Theorem was prepared as a computer activity using GeoGebra to overcome some of the language problems. However, this is open to difficulties if the are problems with organisation in the computer room or with the software, and different technical terms are needed.
6. Participants’ Concluding Observations and SuggestionsReturning to their home country, participants were involved in a discussion and review of their experiences of teaching in another language with members of the project team, and their remarks have been included in this summary.
Home preparation course contents and organisationThe course was created for the purpose of preparing the participants to go to a particular school and teach a particular topic in a particular language. This made it relatively easy to have focused course content and some preparation of organising foreign language teaching specifically for mathematics exchanges. This faced a number of difficulties as outlined in Part I of this report.
The idea of a ‘tool box’ that included a list of words and phrases was vital and good contact with the hosting teacher before the visit was also very important
Preparation and observation in the host countryAll visiting students and teachers agreed that they needed more time to observe and adjust. It was important for the visiting teachers to have watched the regular teacher in the classroom before they had to engage in the actual teaching episodes for the reasons already explained.
Watching the host teachersThe role of the host teacher as advisor and mentor is very important. In their lessons they know how to combine the management of space and their circulation in the classroom, with specific help for the pupils either individually or in the class situation. They also maintain the dialogue with their pupils. The host teachers are also able to compare the visitors’ lesson plans with the process of implementation in the real classroom and suggest alternative organisation and teaching strategies.
Summarising what has been learned:
a) Mobility and experienceThese are some of the remarks volunteered by the students:
- “The mobility of teachers is feasible and an excellent experience.”
- “The visit has been a proof of the importance of learning a foreign language.”
- “Teaching mathematics in a foreign language is a joyful and achievable task, given a well organised and focused preparation course.”
- “The experience allowed me to grow professionally in a different school context in both language and teaching style.”
b) LanguageThe importance of working in a different language showed that:
- knowing the language is fundamental: the more it is mastered, the more it supports the work of the visiting teacher.
- language knowledge allows the teacher to make contact easily with pupils and cope with any kind of requests situation.
- students tolerate the mistakes made in their mother tongue during the early lessons with a non-native teacher and are able to focus on the lesson itself.
c) Lesson planningMeeting new contexts and cultures showed that it was:
- necessary to decide the most important parts of the lesson and give them priority over other activities;
- essential to prepare lessons in a much more detailed way than normally because they felt unable to improvise in a new situation;
- possible to adapt their own methods to the new situation.
d) Subject matterBy teaching mathematics in another country and in a foreign language it is possible to gain deeper insight into the subject matter and the experience of discovering new teaching strategies was enriching and helpful in personal career development.
The experiences the students and teachers had undergone enabled them to:
- understand that school mathematics is taught differently, has different terminology and notation, and is presented differently in other countries;
- compare different curricula, textbooks, and approaches to teaching mathematics;
- discuss with their host teachers different pedagogical approaches and ways of organising teaching;
- realise that there are different academic levels across classes in the same school, and see the difficulties of comparing abilities of pupils from one country to another;
- confront other teaching models, adapt to new teaching practices, and use ideas from other countries for use at home;
- discover the problems of organising group or practical work in contexts where pupils are not used to working autonomously.
7. Final Observations and Suggestions from the Students
The planning of the activity: a more direct and longer co-operation between the hosting and visiting teachers. Video recording with a commentary of the host teacher working would be an advantage.
More information about the subject matter: style of presentation in the host textbooks, notations and conventions of setting out and modes of working.
The Toolbox: As well as the essential vocabulary already established, a selection of relevant mathematical notations should be included. Also an extract of the school curriculum (or sample from a textbook) concerning the chosen teaching subject would be useful.
Formal organised language provision: apart from the partners who either had a course established or were able to provide a reasonable programme, institutions need to be able to provide a specialist course in the teaching of mathematics in at least one of the main European languages.
The observation of the class-group in the host country: more time is needed to obtain meaningful information about the pupils’ academic level and practical abilities.
The implementation: under the given circumstances three lessons of fifty minutes were not sufficient for the discussion of the chosen topic. More lessons are needed to have time to explain, discuss, do exercises and assess the pupils.
Video Recordings: making recordings of sample lessons for post-experience analysis would be a useful aid to evaluation.
The terms ‘student’ or ‘visitor’ refer to the visiting student teachers
or full-time teachers working in the host countries. The term ‘teacher’ or
‘host’ refer to the full-time teachers who are in the country or classroom being
visited. Unless otherwise designated, the term ‘home staff’ refers to the
teaching teams of the project partners.