Nicolina A. MALARA, Giancarlo NAVARRA, ArAl
Project. Arithmetic pathways
towards favouring pre-algebraic thinking, 2003,
pp. 257, € 25.00, ISBN 88-371-1383-8
Introduction:
This
publication is the result of six years (1997-2002) research and
experimental work, involving some one hundred teachers and more than 5000
pupils aged from 6 to 14 years.
1.1
Presentation
International
research in the field of algebraic learning and the difficulties connected
to its understanding demonstrates the existing crisis of traditional
teaching methods in relation to this field. Most recent studies tend to
place, within the pre-algebraic field, main cognitive obstacles, underlining that
these obstacles often appear unexpectedly from arithmetical contexts, which then tend to create conceptual blocks
that hinder the development of algebraic thinking.
In
other words: algebra must be constructed slowly as a tool for thinking,
instead of emphasising its manipulative mechanisms and calculation aspects.
However, without the awareness of arithmetic procedures and of the way
they were created, one cannot grasp the conceptual basis which then leads
to algebraic knowledge. Most students lack the appropriate arithmetical
structures from which they can then generalise. The
ArAl Projects objective is to approach algebra initially as a language.
We believe that the natural language learning process is analogous to that
of the algebraic language. We use the babbling
metaphor
to explain this point of view. When
a child is learning his/her natural language, he grasps meanings and rules
because they accompany and support him step by step. He grasps them
ingenuously, through trial, error and imitation, until he reaches school
age, when he learns to read and reflect upon grammatical aspects and
language syntax. On
the other hand, within traditional algebraic language teaching, a child
begins by studying rules; in other words, formal manipulation. Therefore,
grammatical aspects, procedures and syntax come before the understanding
of meanings. There is a tendency to teach algebraic syntax
and
to neglect algebraic semantics.
The
hypothesis is that our mental structures and algebraic thinking processes
begin from primary school, in parallel
with our arithmetical structures and thoughts. This means we can teach and
think of arithmetic
in an algebraic way,
through the creation of experience fields that encourage an autonomous
processing of what we call algebraic
babbling. By
means of inspiring methods, this new language and the gradual acquisition
of its rules take place. The applied didactics are tolerant of initial
trials that favour above all a sensitivity to the meanings
of
the algebraic language.
1.2.
Project Structure
The
ArAl Project is put forward as an integrated
training system within which teachers:
-
Participate in
the Projects initial fine-tuning;
-
Work
periodically in primary school joint classes, together with researchers;
-
Compile meeting
diaries of these joint classes;
-
Assess on a monthly basis, together with co-ordinators, the progress
status of the Project;
-
Attend, together with GREM researchers, meetings aimed at deepening the
theoretical references;
-
Contribute to
the ongoing organisation of teaching materials, as well as the final
versions (i.e. the Teaching Units);
-
Present the Project at various conventions.
After
the experimental stage, all ArAl Project activities are organised into Teaching
Units. The entire process can be summarised as follows:
a.
Selection
of Contents
During
seminars taking place at the beginning of each school year, teachers are
presented with themes and work outlines, around which the experimental
activities of joint classes will be developed.
b.
Joint
Classes and Meeting Diaries
Each
year some 120-140 joint classes take place, in which both teachers and
researchers participate. These joint classes are recorded by teachers (often
with audio or video equipment), who are sometimes helped by students from
teachers training colleges. Class diaries are a key tool for analysing the
teaching/learning process within the Project.
c.
From
the Diaries to the Units
After
being transferred to computers by teachers and reorganised by researchers,
the class diaries are periodically assessed by a group of
teachers-experimenters. At the end of each school year, the diaries are
reorganised by the GREM researchers into Teaching Units, which will
subsequently be tested with participating classes.
d.
The
Units in their final version
After
these checks, the Units - consisting of some 25-30 pages - are published,
and the relevant teaching materials are made available on the Net. Since
2002 and still within the framework of the theoretical perspective of the
ArAl Project, a new project is being experimented for an introduction to
arithmetical thinking. Kindergarten teachers and pupils also participate
in this project.
1.3.
Features and Structure of the Units
The
Units are structured in such a way as to make the teaching process
transparent in relation to the problem situation being examined (methodology
choices, activated class dynamics, key elements of the process, extensions,
potential behaviour of pupils and difficulties they may encounter). The
aim is that of offering teachers outside the Project innovative teaching
models, in a pre-algebraic environment, which may encourage them to
reflect on their own knowledge and modus
operandi in the classroom, even before providing them with pathways
that they should follow precisely with their pupils.
