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PRINCIPLES AND TOOLS FOR TEACHERS’ EDUCATION AND THE ASSESSMENT OF THEIR

PROFESSIONAL GROWTH

Nicolina A. Malara, Giancarlo Navarra University of Modena and Reggio Emilia (Italy)

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In the field of arithmetic and algebra - where we conduct our studies - most of the teachers believe that the teaching of arithmetic precedes the teaching of algebra

CERME 9 - TWG 18 Prague February 3-8 - Nicolina A.Malara, Giancarlo Navarra - Italy

Teachers change and early algebra

The change needed to a teacher to become an inquiry-oriented teacher is not simple to achieve

the procedural point of view prevails over the relational one

Our Studies

Since the nineties we have addressed questions of the renewal of this teaching area

Our aims To individuate the conditions of real applicability in our schools of didactical innovations centered on:

• algebraic problem solving

• generalization

• modeling

• justifying and proving in the frame of a socio-constructive teaching

Our Studies

• In this frame the role of natural language is central

It is the main didactical mediator for the slow construction of syntactic and semantic aspects of algebraic language

Verbalization, argumentation, discussion, exchange, favour both the understanding and the critical review of ideas

At the same time, through the enactment of processes of translation, natural language sets up the bases for both producing and interpreting writings in symbolic language

Our Studies

In this kind of approach

the teacher has a key role

She needs to set up a teaching where space is given to:

• the linguistic aspects • the representation of information • the meta-cognitive aspects

All this requires a deep restructuring of the teachers’ conceptions about both the contents to be taught and the teaching methodology in the classroom

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(5) Analysis of one’s own practice

(7) Development of theoretical framework, methodologies, materials

leads to

influences

(6) Critical shared reflection among teachers, tutors, maths educators on the

classroom processes

leads to

(1) Theoretical study

(2) Analysis of practice of others

(3) Planning action

(4) Classroom activity

influences

Methodology of work with/for teachers

A glossary shared with the teachers

supports the educational process

Forwards Early Algebra

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arising from the shared reflections with the teachers on the enacted experimental paths about:

CERME 9 - TWG 18 Prague February 3-8 - Nicolina A.Malara, Giancarlo Navarra - Italy

• the construction of the mathematical content

Key tools are

the Multi-Commented transcipts (MTs)

• the quality of the teacher’ behavior

• attitudes, knowledge, ways of thinking emerged in the students

The Multicommented Transcripts


Transcript

Other commentators

Teacher

E-tutor

A teacher • records a lesson • sends its Commented Transcript (MT) to an E-tutor

The E-tutor • comments the MT • sends the new version to other members of the team

The other members • write their comments too.

The MT so reached become a powerful tool for teschers’ reflection and learning

Our Studies

Our main aim in this educational process is to lead the teachers:

• to become increasingly able to interpret the complexity of class processes through the analysis of the inner micro-situations

• to reflect upon the effectiveness of their own role and become aware of the effects of their own micro-decisions

• to be in a better and finer control of both behaviours and communication styles they use

• to notice, during classroom activity, the impact of the critical-reflective study undertaken on pupils’ behaviour and learning

The evaluation of the teachers is complex also because what we want to evaluate is

a process continuously becoming


Evaluation of the teachers change

How to evaluate the teacher change?

Our studies concentrate on a formative, dinamic evaluation of what arises in the MTs which pays attention to


How to assess

in which she often shows the co-existence of pre-existing beliefs and possibly hesitating opening attitudes, induced by the training

micro-decisions that the teacher makes in just as many micro-situations


How to assess

Attitudinal Indicators

• The teacher’s shift forward relational and metacognitive aspects in teaching

• The teachers’ ability to keep silent, allowing the

students reflection and interaction

• The teacher’s appropriate reference to the theory (use of theoretical linguistic constructs, focus on the translations and interpretations processes, …)


How to assess

• Indicators about the practice

In the class • the pupils’ interventions • the modalities in which the teacher interacts with the

pupils (her ways to formulate the questions and the lexicon she uses)

• the teacher’s ability to involve and guide all the students

In the community of inquiry • the comments expressed by the teacher in the MTs

and his/her reflections in the sharing of his/her MTs within the community of inquiry.

2 examples are given in the report

1 example is given in the report

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How teachers interact with pupils In the 2nd example there are many elements that let us assess positively, at different levels, the teacher’s action: (a) mathematically: she has introduced the pupils to the

use of letters, to the priority of operations within expressions, to the use of parentheses;

(b) linguistically: she has fostered the organization of meaningful, complete sentences;

(c) metalinguistically: she has promoted the reflection on the mathematical writings and their comparison;

(d) socially: by inviting pupils into discussion, she has let them interact without her influence, listening to each other and having spontaneous dialogues;

(e) methodologically: she has shared the theoretical framework with the class, by spreading words such as ‘represent’ and ‘translate’.

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Conclusion

Conclusive considerations

Teachers change can be reached in a medium/long-term (two/three years or more) under these conditions:

• enacting an educational project with several supporting tools

• avoiding separation between theory and practice

• building an environment in which effective circular relationship occurs between what happens in the classroom and the joint reflection of teachers-tutors-maths educators on classroom events

• Involving teachers in the construction of ‘theoretical tools’ from their practices


Questions for the discussion

• Which mathematics education should future teachers receive in order to improve their sensitiveness towards those micro-situations that allow students to ‘see algebra within arithmetic’ and to conceive algebra as a tool for thinking?

• What kind of initiatives can we realize with the aim of changing teachers’ dominant beliefs about algebra

teaching and, more in general, of spreading a vision of the teacher professionalism which merges theory and practice?

• What are the obstacles (cultural, social political constraints) against the spreading among teachers of this vision of their professionalism?

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