Developing The competences of primary school teacherS VIA cOLLECTIVE REFLECTION
Alena Hospesova,University of South Bohemia, Ceske Budejovice
Marie Ticha,Mathematical Institute, Academy of Sciences of the Czech Republic
Jana Machackova, Primary school, Prague
Abstract: Teachers’ ability in creating a challenging climate in mathematics lessons positively influences the quality of pupils´ mathematical knowledge in all phases of its acquisition. Considering the complexity of the teacher’s task, it is necessary to see teachers’ competence as a structure that consists of, among other things, subject-didactic and pedagogical competences. Teachers are supposed to cultivate their own competences systematically during their whole professional life. There are definite advantages for the improvement of teachers’ competences in using the collective reflection of their own or other teachers’ teaching. This not only enables the sharing of effective teaching practices, but could also produce deeper views on different questions about teaching (the mathematical content and different ways of its didactic elaboration, classroom discussion, analysis of unexpected pupils´ input, etc.). In this submitted contribution we are introducing our experience from in-service courses for primary school teachers based on collective reflection of videotaped teaching/learning episodes. Collective reflection, in our opinion, helps the participants to understand better the processes going on during mathematics teaching in the social context of the classroom and to perceive the possibilities of using collectively-acquired knowledge for the preparation of person-centered approaches to education.
The competences of the mathematics teacher
The mathematics teacher’s practice is characterized by great complexity and unpredictability, so the teacher must havethe ability to be able to react to situations that arise in the classroom and reflect on them. There are various approaches on how to specify the competences. J. Slavik and S. Sinor (1993, p. 156) suggest understanding teachers’ competences as “the teachers’ readiness to cope with the demands of their professional role and, at the same time, to preserve a necessary measure of the authenticity of their personality”. V. Spilkova (2001) emphasizes a dynamic conception of competences, which is expressed by the orientation towards, and the coming nearer to, the desirable parameters of the teaching profession.
With respect to mathematics education the importance of (a) the subject-didactic competence[1] (comprising the knowledge of the mathematical contents with an orientation towards their educational meaning and possible didactic interpretation), and (b) the pedagogicalcompetence (consisting of creating conditions for the pupils’ cognition) is generally stressed (e.g. Bromme, 1994, Harel and Kien, 2004, Helus, 2001). An often-cited reference is L. S. Shulman (1986, 1987), who emphasized that a teacher’s knowledge of the content includes “subject matter content knowledge”, “pedagogical content knowledge” and “curriculum knowledge”. He emphasised that “pedagogical content knowledge” is characteristic for the teaching profession because it covers subject-specific content and the didactical approaches to it.
In this context, we would like to mention something written by Comenius (1946, XVI., p. 26) more than three hundred years ago, but still very relevant (our translation): “The teacher is required ... 1. to know himself what he is to teach to others; 2. to be able to teach others what he himself knows (in other words, to be a gifted teacher and to be able to survive ignorant pupils patiently and also to be able to banish the ignorance efficiently, etc.); finally 3. to want to teach what he knows and be able to teach it (in other words, to be active and diligent and to long for bringing others to the source of light in which he himself delights).”
It is evident that teachers’ competences create in the course of their formation and improvement the framework of synergic elements. E. Fennema and M. L. Franke (1992, p. 162) propose: “Within a given context, teachers’ knowledge of content interacts with knowledge of pedagogy and students cognition and combines with beliefs to create a unique set of knowledge that drives classroom behavior.”
Among other ways of improving teachers’ competences many authors consider self-reflection and collective reflection as being very important (Climent and Carrillo, 2001, Jaworski, 2003, Krainer, 1996).
The Comenius Project
We have concerned ourselves with self-reflection and collective reflection in the framework of the Socrates-Comenius project “Understanding of mathematics classroom culture in different countries”. (For more information about the project, see In the studies, which originated within the project, we have concentrated so far on the mutual influence between pedagogical self-reflection and qualified reflection on the one side and the level of teachers’ competences on the other (Hospesova and Ticha, 2003).
The basis of our reasoning was the characteristic of mathematics teaching formulated by P. Scherer and H. Steinbring (published in 2003, p. 1):
“The problems, which are connected to the profession of teaching mathematics, can be described as a contrast between two poles:
-The instruction activity is subject to the tension between the teacher’s immediate involvedness and critical distance. During the classroom events, the teacher is bound directly into the interaction with his or her students and cannot play the role of a distanced observer at the same time.
-As a science, mathematics, and thus eventually also school mathematics is a consistent and ready worked out knowledge stock. In the instructional interaction, however, the mathematical knowledge only develops together with the students, so to speak new and independently.”
Our work started from the belief that it is not possible to influence teaching by the mere sharing and reproduction of effective teaching practices. “Good teaching” consists in sensitive guidance of cognizing pupils. It is necessary for the teachers (a) to form for themselves sensitive approaches towards pupils´ ways of thinking, (b) to create the ability to recognise the moments that are valuable from the point of view of students’ cognition, and have the capability to use them, i.e., to cultivate a better way of teaching.
This idea was also the starting point for the preparation of courses for teachers, which were part of our work on the Comenius project. There we acquired valuable further experience of the selection of content and ways of realization of courses for in-service primary teachers. The work in the meetings with teachers was based on multiple introduction and presentation of videotaped short episodes from lessons (the introductions consisted of explanations of the chosen topics from various subject-didactic points of view). Reflection came about not only from our stimuli but also from participants’ experiences.
We can illustrate our work with the following example.
The base for collective reflection
Teaching episode
The session was based on the reflection and analysis of a short (videotaped) teaching episode from the 2nd grade, in which the teacher wanted to promote understanding of subtraction algorithms up to 100. The teacher asked her pupils to explain whether the procedure recorded on the blackboard “63 – 8 = 60 – 5” is correct or not. A part of the discussion between the teacher and a girl (Mary), persuading the teachers and other pupils that the computation is correct, was used here.
Mary
/ Blackboard: 63 – 8 = 60 – 5I have found it out now. 63 is broken down into parts ... a bit. (She points in front of the number 63.) And from the eight…
Teacher / You can break it down, too. Write on the blackboard, or do anything you want. Tell us, show us, write it down…
Mary / And from the 8 ... (She points at the number 8.) 3 is taken away.
Teacher / Show it, which 3?
Mary / These ones. (She points at the digit 3.)
Teacher / I see…From those 8.
Mary / From this eight (She points at the number 8.) 3 is taken away. (She points at the digit 3). It equals 5. (She points at the number 5.)
Teacher / So if I take it away, does it equal 5?
Mary / No. (Laughter) From 8. 8 – 3 = 5.
And then from here. (She points at the number 60.) No. 8 (She points at the number 5.) And here was the number 3. (She points at the digit 0 in the number 60.) And it was taken away. And then 5 was left. 60 – 5. And it equals 55. As well as 60 – 3. Ehm, ehm. (She shakes her head.) 63 – 8 = 55
Teacher / So in your opinion, it is counted correctly.
Mary / Hm.
Questions, challenges, discussion
According to our opinion it is necessary to start with the analysis from the point view of the content:
-What are the goals of the topic “Subtraction up to 100”?
-What strategies can pupils use in subtraction up to 100? (This could be summarized along the lines of (e.g.) Heirdsfield and Cooper, 1995.)
-Can the strategies be visualised? If so, how?
-Is it better to introduce only the “most effective” strategy or explain several different strategies? ... Etc.
After the video recording is presented the problem assigned by the teacher and the teacher/pupil interaction can be analysed from different points of view:
-Could the problem itself promote an understanding of the concept of subtraction?
-Is the formulation of the problem appropriate for 7-year-olds?
-What are the pivotal moments in the communication between the teacher and Mary?
-Did Mary understand the holistic “levelling” strategy?
-Did the teacher (in your opinion) anticipate Mary’s answer? ... Etc.
Discussion and Concluding Remarks
We found out that collective reflection of chosen videotaped teaching episodes supports teachers’ professional development and improves their competences. Thorough analysis of short video-taped parts of lessons is one of the ways of influencing teachers’ beliefs and their subject-didactic competence and it leads to better teaching. Although some teachers found it difficult to discuss and present their opinions, their comments on our courses were very positive: “I appreciate the opportunity to discuss problems of mathematics teaching with my colleagues.” “There are many things I hadn’t realised – they had never occurred to me.” “Thanks to this course I have the opportunity to explore the teaching of mathematics more deeply…“
Collective reflection gives researchers and teacher-trainers a good opportunity to influence practising teachers, directly and indirectly, and to inform them about the latest research results in mathematics education. It opens ways for recognising where the causes of failure of some teachers are, why problems appear in teaching and how to remove them. Teacher-trainers can be clearer about which aspects (subject and didactic) should be emphasised in the education of future teachers and practising teachers.
Collective reflection leads to the understanding of processes going on during mathematics teaching in the social context of the classroom. It helps to show the possibilities of using collectively-acquired knowledge for the preparation of person-centred approaches to education and strengthens the constructivist philosophy and, therefore, the constructivist approach.
References
Bromme, R.: 1994, Beyond subject matter: a psychological topology of teachers’ professional knowledge, in R. Biehler at al. (eds.), Didactics of Mathematics as a Scientific Discipline, Kluwer Academic Publishers, Dordrecht, Boston, London, pp. 73-88.
Climent, N. and Carrillo, J. (2001). Developing and Researching Professional Knowledge with Primary Teachers. In: J. Novotna (ed.), CERME 2. European Research in Mathematics Education II, Part 1.Prague: Charles University, Faculty of Education, pp. 269 – 280.
Fennema, E. and Franke, M.L.: 1992, Teachers´ knowledge and its impact, in D.A. Grows (ed.) Handbook of Research on Mathematics Teaching and Learning, MacMillan, New York, pp. 147-165.
Harel, G. and Kien, L.H.: 2004, Mathematics teachers´ knowledge base: preliminary results, in: Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, Vol. 3, pp. 25-32
Heirdsfield, A.M. and Cooper, T.J.: 1995, Teaching Implications from Research on Children's Mental Computation for Addition and Subtraction, in Proceedings of the 15th National Conference of the Australian Association of Mathematics Teachers. Darwin, NT.
Helus, Z.: 2001, Ctyri teze ktematu „zmena skoly“ (Four Theses on School Reform, in Czech), Pedagogika, vol. 51(1), pp. 25-41.
Hospesova, A. and Ticha, M.: 2003, Self reflection and improvement of mathematics classroom culture, Proceedings of Third Conference of theEuropean Society for Research in Mathematics Education, February, 28 – March, 3, 2003, in Bellaria, Italy, CERME3/draft/proceedings_draft/ TG11_draft/index.htm
Jaworski, B.: 2003, Research Practice into/influencing Mathematics Teaching and Learning Development: Towards a Theoretical Framework Based on Co-learning Partnerships, Educational Studies in Mathematics, v. 54(2-3), pp. 249-282.
Komensky, J.A. (Comenius): 1946, Didaktika analyticka (In Czech, translation from Latin H. Businska), Samcovo knihkupectví, Praha.
Krainer, K.: 1996, Some considerations on problems and perspectives of in service mathematics teacher education, in: C. Alsina et al. (Eds.), 8th International congress on Mathematics Education: Selected Lectures, Sevilla: SAEM Thales, pp. 303-321.
Scherer, P. and Steinbring, H.: 2003, The professionalisation of mathematics teachers’ knowledge – teachers commonly reflect feedbacks to their own instruction activity, Proceedings of Third Conference of theEuropean Society for Research in Mathematics Education, February, 28 – March, 3, 2003, Bellaria, Italy, Scherer.pdf
Shulman, L.S.: 1986, Those who understand: knowledge growth in teaching, Educational Researcher, Vol. 15, pp. 4-14.
Shulman, L.S.: 1987, Knowledge and teaching: foundations of the new reform, Harvard Educational Review, 57(1), pp.1-22.
Slavik, J. and Sinor, S.: 1993, Kompetence ucitele v reflektovani vyuky (In Czech: The teacher’s competence in reflection of teaching), Pedagogika, Vol. 43 (2), pp. 155-164.
Spilkova, V.: 2001, Professional development of teachers and student teachers through reflection of practice, The New Hampshire Journal of Education, Vol. 4, pp. 9-14.
Acknowledgements
This research was partially supported by the Socrates/Comenius project 87636-CP-1-2000-1-CZ-Comenius-C31 and GACR 406/02/0829.
Help on translation was given by Mr. Brian Espiner.
[1]We use the term competence in a similar sense to other authors’ speaking about knowledge.