The Course
Aim of the course
To sensitise, motivate and encourage the teachers on the course and to enhance their competencies in order to improve the teaching of mathematics in their classrooms.
Objectives of the project
To sensitise the teacher to:
· a student’s non-standard response – not imitative nor reproductive
· individuality – individual solving strategies, algorithms, conceptual understandings, ...
· a student’s misunderstandings, misconceptions, ...
To motivate and encourage the teacher to:
· undertake experimental work
· seek and implement new approaches to specific topics – content
· seek and implement new teaching strategies – methods
· become a “reflective” teacher – constantly reviewing and analysing his/her teaching performance
· change pupil/students’ attitudes towards mathematics
· investigate, assess and understand a student’s thinking processes in mathematics.
To enhance a teacher’s competency in:
· understanding a student’s cognitive structures, processes, relationships
· diagnosing the developmental level of a student’s knowledge
· re-educating a student who has misconceptions/misunderstandings
· enhancing the students’ confidence in their mathematical competence
· creating a mathematically positive climate in the classroom.
Target group and languages
The course is aimed at Primary and Secondary school teachers, teacher-educators and Mathematics advisers. It will be dependent on them being able to trial an agreed teaching experiment with their students and to reflect on the outcomes of this experiment with respect to themselves and their students. In turn new methods of working in the classroom and new knowledge gained will enhance the teachers competence and greatly benefit their students not only in the understanding of mathematics but in all those subjects which use mathematics. The course language is English with translators into the Czech language available.
Research reports and certificates
To complete the course successfully, participants are asked to write up a report which gives details of the experimental work/research/topic, participants will undertake together with the analyses of their own and pupils’/students’ reactions and learning. Participants may wish to include examples of photographs of their students’ work as an appendix. Guidelines for the report in each module will be given to participants during the course. It will be expected that participants follow the guidelines to be able to attain a satisfactory level.
Two certificates will be given to participants.
(1) a certificate of satisfactory completion of attendance, contribution to the discussions and practical work in both modules,
(2) a certificate of competence of achievement a satisfactory level of their experimental
work and resulting reports.
Course structure
The course comprises of a seven-day residential period followed by a further period of time undertaking an agreed teaching experiment in the classroom. Following this a written report could be presented. |Contact with tutors is available at all times through e-mail.
During the residential period, participants will undertake work in all the offered modules. Towards the end of this period, an opportunity is given for participants to discuss with tutors possible scenarios for the teaching experiment they will do in their schools. An outline of the work to be done will be agreed with a tutor.
Below we will present the three modules which will be offered to you on this course. First, the common aims will be given.
Aims
The aims of each module are to enable the participants to:
· experience similar kinds of learning situations which we hope they will employ in their own classrooms to improve the mathematics lessons;
· enjoy the emotional satisfaction of extending their mathematical knowledge and understanding through this methodology;
· analyse both the teaching strategies used and the content of their work and be able to criticise constructively;
· undertake action research in their own classroom by choosing a ‘topic’ which they will teach to their pupils and write up their experiences, including a critical analysis of their work and that resulting from the pupils.
The three modules to be offered are:
A. Using pupils’ experiences in and out of school
B. Grid-paper – the bridge between geometry and arithmetic
C. Projects for pupils in school mathematics
Outlines of the modules
Module A: Using Children’s Experiences in and out of School
1. Introduction
Mathematics is all around us and yet so many children’s understanding of the subject is that it is only that part of the curriculum which is labelled ‘mathematics’ on the school timetable. However children come to school having gained a wealth of mathematical experiences whilst in the home, playing with their friends, watching television and in other curriculum subjects at school.
We believe that the reason for this is that the majority of teachers do not make much, if any, use of the mathematics the children meet, either in other subjects in the school curriculum or that which they meet outside the school environment. There could be many reasons for this, including the teacher’s own insecurity with mathematics, the limitations imposed by some syllabi and the very formal teaching strategies used by some teachers.
We hope that this module encourages teachers to be more adventurous in their teaching, to stimulate their pupils’ enthusiasm for the subject and above all, to show that mathematics is a universally applicable subject rather than a sterile abstract one
2. Rationale
2.2.1 Pupils/students need to see the usefulness of mathematics in situations that they are likely to encounter in everyday life. This can be done by taking experiences, which the pupils have had outside the mathematics classroom and showing how their knowledge level of mathematics at different stages can be used to enhance their understanding of the subject and its use. Other curriculum subjects can be used as vehicles for the introduction of new mathematical ideas.
2.2 Pupils/students need to understand the mathematics which they do in the classroom. Practical projects and group work using out-of-school experiences or the work within another curriculum subject can help with that understanding and give an underpinning for the theoretical basis for mathematics.
2.3 Pupils/students need to enjoy their mathematics. If the work they do in the classroom is firmly based on their experience and given in contexts which are relevant to them, this will develop the understanding and usefulness of the subject for the pupils. This will give them confidence and enjoyment in their mathematics.
2.4 Teachers need to develop teaching strategies for mathematics which allow the pupils to experiment with ideas, to develop their own strategies for the solution of problems and to work together. This means a constructivist approach to teaching which will enable the teacher to be come a facilitator and enable the development of the communication of mathematics both to and by the pupil.
3. Content Outline
The module will begin with a discussion of the mathematical topics which students at various levels of mathematical development might meet in the world outside the classroom and in other subjects. A list of these mathematical ideas will be kept throughout the module. The mathematical knowledge needed for these topics will be discussed and where necessary new approaches will be shared with the participants. During this part of the module, emphasis will be put on differentiation of the work, including examples of problems which can be solved by different mathematical means for different ages and levels of ability.
The participants will be given a topic, such as ‘A visit to the Supermarket’, and asked to brainstorm the mathematics which might be derived from such a topic. They will then explore some of these ideas in the role of pupils, working in groups of two or three. This may necessitate a visit to the local supermarket for reality. The groups will have to define their objectives for their explorations and determine the equipment needed before they start the work.
The mathematics that the tutors expect will emerge from such a topic would be considerable work on measures – length, volume, capacity, mass/weight, money – percentages and algebra and analysis for the older students. Relevant applications could be explored, such as packaging, which would involve the geometry of nets, tessellations, economics and aesthetics.
At the end of this period the groups will present their findings so that the different approaches can be shared. This will be followed by an in-depth analysis of the work they have done and the place of such work in a mathematical structure. As a part of this analysis they will decide whether their objectives have been met and discuss their own reactions to working in this way.
The participants will decide on an agreed topic to be carried out by them with their own pupils. Each member of the group will have a tutorial to discuss the topic, its relevance to their pupils and the methodology they might use. This research will be presented in the form of a report. Contact with the tutors may be maintained by e-mail or by other means.
Module B: Grid paper – the bridge between geometry and arithmetic
1. Introduction
Mathematics is not only a tool for understanding the world, but also a part of the cultural development of mankind. As a school subject, it can enhance the development of thinking and communication of students to a great extent. The contemporary state of mathematical education (as far as we know it in our countries and from reports of our colleagues from other countries) does not satisfy the goals set out above. Within this module, we would like to discuss possibilities of improving the situation. We hope that ideas coming from different parts of Europe, different schools and teachers, might create a good motivating climate and help all of us to think about new teaching experiments and new ways of teaching. The mathematical environment which will be used as a ”playground” for our work is grid paper.
2. 2. Rationale
2.1. The starting point for any human activity is motivation. If we want to motivate a child/student towards the intellectual activity, we have to
- give him/her appropriate challenging problems to solve,
- create a stimulating climate which helps him/her to use his/her own ways of solving problems, argumentation and communication, etc.
2.2. We teachers, have to keep in mind the differences among students in
- their motivation,
- their mathematical knowledge, mathematical abilities and intellectual experiences,
- their thinking and learning styles.
2.3. Geometry and arithmetic on the primary and lower secondary level are usually treated as two different parts of mathematics. However, the linkage between these two fields can be introduced very early. A suitable tool for creating this linkage is square grid paper with which even 7-year-old students have their own experience. This environment allows pupils to look at the given situation either from the geometrical or arithmetic point of view, thus it enables them to chose an approach to a given problem which is most appropriate for them.
3. Content outline
We will introduce several topics which will concern for example
different descriptions of grid figures,
new understanding of various properties of grid figures,
new ways of argumentation using the structure of the grid,
constructions on the grid, without the use of a compass,
comparing the lengths of segments without the use of any instruments,
measurements of length of segments, areas of figures, and angles without the use of any instruments,
discovering of geometrical patterns, relations and possibly even formulas.
Each of these topics can be used by a teacher as effective preparatory work for a deeper understanding of some mathematical concepts (e.g. straight line, co-ordinates, measure, symmetry) or for ‘discovering’ some deep mathematical ideas (e.g. Theorem of Pythagoras, trisection of a given segment, construction of a similar triangle, Pick’s formula).
Our experience shows that the grid environment can be successfully used for all students. We can start with very easy tasks in grade one. The difficulty of problems can be up graded step-by-step to the secondary level. This educational strategy, based on the solving sequences of graded problems rather than on traditional lecturing, is good motivation for the mathematical development of university primary teacher-students.
Each topic will be discussed on several levels according to the participants’ choice. The module work will start with individual or group work and the solution of some problems at a chosen level of difficulty. This will be followed by a discussion aimed at the possibility of using these ideas in the classroom. Stress will be put on looking for as many ways as possible of mathematical and didactical solutions for the problems concerned.
Module C: Pupil/student projects in school mathematics
1. Introduction
Our approach to the teaching of mathematics is based on our beliefs that the teacher’s role is not merely to transmit ready-made knowledge to students but rather to guide them in the way which will enable them to discover knowledge for themselves. The thesis “to learn something” is replaced by the thesis “to learn how to learn”.
The student should become active in his/her learning and in order to achieve this, the teacher should refrain from interfering with the pupil’s efforts (he\she should not ‘work’ for the student). The teacher should not put across the questions from the curriculum at the expense of the student’s questions and substitute his/her words for the students’ words. One of the means of reaching such a change is through the project teaching.
There are a lot of different interpretations of the project method. Traditionally, projects comprise several subjects and mathematics is considered to be a tool for solving problems in them. In our interpretation, mathematics itself becomes the core of student projects which then comprise various activities in which students discover mathematical concepts and laws and/or in which they learn about possibilities of the use of mathematical concepts outside mathematics. We can distinguish two basic types of student projects: mathematical projects (hereinafter MAPR) and interdisciplinary projects (hereinafter IDPR). Moreover, besides the projects prepared by teachers, we have experience with projects which are prepared (to a certain extent) by students themselves. Students often even suggest the topic for their project.
2. Rationale
2.1 We offer an interpretation of the students’ projects which enables us to create ‘purely’ mathematical projects, i.e. projects which can be solved within mathematics. Some mathematical topics can be elaborated into a series of mutually connected problems whose solutions bring joy to students from “doing real mathematics” and enable them to come up with their own problems, look for their solutions and interpret them.
2.2 In other types of projects, mathematics is seen in connection with other subjects or with real life. Pupils learn how to organise work towards a given goal, how to plan it and look up necessary information.
2.3 The project method can be productive both for able and less-able students. Students can work at their own pace, each of them can reach different stages of the student project.
2.4 Communicative skills are developed both during the elaboration of the project by a student and by the presentation of his/her work to others.
3. Content outline
We expect that the work during the course will be on three levels:
1. Participants will play the role of students and work on student projects prepared by the tutors.
2. Through the process of preparing an interdisciplinary project, participants will play both the role of constructors of the student projects and students solving the project.
3. Participants will start elaborating their own student projects which will be the basis of their experimental work.
After presenting some basic information on the history of project teaching and its role in education, participants will work on one mathematical project. They will play the role of students while doing this. They will solve given problems, pose and solve their own problems and write down a brief report of their work (i.e. different kinds of solutions, blind alleys, …). These reports will be then presented to other groups. One member of each group will talk about interesting results, strategies, new problems etc. they got while working on MAPR. During the discussion, participants themselves will try to elicit basic characteristics of mathematical projects, how to prepare and record them.
For the second part, participants will choose one topic and elaborate it in groups. Their task will be to ask and answer questions concerning matters necessary for the construction of the interdisciplinary project on the given topic. They will also prepare the mathematical background for the topic and try to formulate problems for students whose solutions would require the use of knowledge from other subjects and/or real life. They will have to anticipate students’ possible reactions to the problems they pose. The groups will either write a short report and/or prepare a poster of their results.
We expect that at the end of the residential week, participants will have personal experience of a student working on a project, self-reflection on their work on MAPR and IDPR and information of constructing projects.
