In a diagnostic report at the beginning of her master-studies of special
needs education, S.G. (teacher) wrote: “Testing
with the BesMath 2 was not possible because JAN, a girl in the 2nd grade of
primary school, shut herself off. She could no longer solve the simplest
calculations from the 2nd grade (e.g. counting backwards from 20). After 30
minutes I stopped the exercise. I prepared the BesMath for first graders and
wanted to do this test with her. I almost despaired, because JAN couldn't and
didn't want to solve a single task, which she usually would have solved very
well with manipulatives. I called off the whole exercise and asked for advice."
In the case discussion I drew S.G.'s attention to the flexible interview
and encouraged her to experiment with this method. Moreover, if JAN is so
closed and anxious, emotions, thinking and mathematical competences will have
to be freed through role-playing (cf. Bodrova & Leong, 2015; Zaslavsky, 1994). The high
learning speed and the pressure to perform got the girl out of step.
A week later, S.G. reported, "I played JAN's calculator. She had to program me. She found this
role-playing very amusing. I wrote down her solutions and computational paths
that she had dictated. It was very
remarkable how JAN opened up and did well. As soon as the calculations became
complex, she said, "Let's figure out together."
What does the combination of the flexible
interview with the role-play "I am your calculator, your calculating
machine" mean? The reference schema illustrates important
elements.
Piaget
described the relativization and simultaneous transgression of the clinical
method of psychiatry as "méthode critique". This was the point I made
to S.G.: she should learn to critically overcome her own testing method.
The introduction to the work of Inhelder,
Sinclair & Bovet (1974, p. 35) continued to define the clinical method: « La méthode
clinique - plus significativement appelée (méthode d'exploration critique) -
provoque dans les milieux des expérimentalistes stricts des réactions allant de
l'étonnement amusé jusqu' au scepticisme le plus profond."
In this respect, the roleplay "I am your
calculating machine" means that pedagogical diagnostics will be equipped
with methods of critical exploration.
This is not only about JAN's mathematical
competencies, but primarily about the pedagogical and methodical competencies
of the special needs teacher.
The transformation of JAN’s and S.G.’s
behavior was spurred by role-playing and differentiated through critical
exploration. The roleplay of the special needs teacher
works like a transitional object (cf. Winnicott, 2002; Günter, 2003; Storch, 1996). Promoting
development by role-play, JAN can overcome her dependence and helplessness,
becoming more independent and competent. Conversely, this also applies to the special needs
teacher's dependence on methods. The role play and the critical exploration
provoke the "incessant back and forth between theory and practice" in
a positive way, it "prevents one from falling into methods that would
ultimately block any work" (cf. Mannoni, 1978, p. 99). It is also an interaction of spontaneity and creativity (see Moreno, in Storch, 1996). By blocking methods we also understand "lessons" based on the concept of cultural conserves (see Moreno, in Storch, 1996).
Can we conclude that the critical method is better than testing? Piaget was open-minded: «Or, une méthode n'est pas bonne ou mauvaise en soi. Elle ne peut être jugée qu'en fonction des problèmes qu'elle est appelée à résoudre et qui, à leur tour, sont orientés par des perspectives épistémologiques plus ou moins explicites » (cf. Piaget in Inhelder, Sinclair & Bovet, 1974, p. 35; see Duckworth, 2004).
Vygotsky (2019, p. 88-89) specifies the meaning of test methodologies as follows: "Formerly, it was believed that by using tests, we determine the mental development level with which education should reckon and whose limits it should not exceed. This procedure oriented learning toward yesterday’s development, toward developmental stages already completed. The error of this view was discovered earlier in practice than in theory. It is demonstrated most clearly in the teaching of mentally retarded children. Studies have established that mentally retarded children are not very capable of abstract thinking. From this the pedagogy of the special school drew the seemingly correct conclusion that all teaching of such children should be based on the use of concrete, look-and-do methods." From the meta-analysis of Hodgen, Foster, Marks and Brown (2018) it can be concluded that the evidence of this method is high because an ensemble of pedagogic approaches, learning forms and tools is used.
In the case of JAN, a pedagogical and an epistemological problem had to be solved: What means insight in mathematical education and how can it be developed in the field? Flexible interviews and role-playing games like "I am your calculator" are rich systemic tools. The triangulation of methods (roleplay, critical exploration, thinking arithmetic – arithmetic thinking (see Arrigo “calcolo ragionato”), semi-written arithmetic, critical exploration, zone of proximal development, see Bodrova & Leong, 2015; Duckworth, 2004; Winnicott, 2002; Cuomo, 2007; Lillard, Lerner, Hopkins, Dore, Smith & Palmquist, 2012; Vygotsky, 2019) dynamizes pedagogy and research. This case analysis does not only describe developments in the here and now of JAN, but also current developments of the pedagogue.
Future case studies will examine experiences and development work at different levels. "Sharing the video" is a promising method to explore the meaning of the experiences with the talking and acting calculator in a "video-stimulated-dialogue" (see Morgan, 2007; Nind, Kilburn, Wiles, 2015; Werfeli & Meyer, 2019).
Can we conclude that the critical method is better than testing? Piaget was open-minded: «Or, une méthode n'est pas bonne ou mauvaise en soi. Elle ne peut être jugée qu'en fonction des problèmes qu'elle est appelée à résoudre et qui, à leur tour, sont orientés par des perspectives épistémologiques plus ou moins explicites » (cf. Piaget in Inhelder, Sinclair & Bovet, 1974, p. 35; see Duckworth, 2004).
Vygotsky (2019, p. 88-89) specifies the meaning of test methodologies as follows: "Formerly, it was believed that by using tests, we determine the mental development level with which education should reckon and whose limits it should not exceed. This procedure oriented learning toward yesterday’s development, toward developmental stages already completed. The error of this view was discovered earlier in practice than in theory. It is demonstrated most clearly in the teaching of mentally retarded children. Studies have established that mentally retarded children are not very capable of abstract thinking. From this the pedagogy of the special school drew the seemingly correct conclusion that all teaching of such children should be based on the use of concrete, look-and-do methods." From the meta-analysis of Hodgen, Foster, Marks and Brown (2018) it can be concluded that the evidence of this method is high because an ensemble of pedagogic approaches, learning forms and tools is used.
In the case of JAN, a pedagogical and an epistemological problem had to be solved: What means insight in mathematical education and how can it be developed in the field? Flexible interviews and role-playing games like "I am your calculator" are rich systemic tools. The triangulation of methods (roleplay, critical exploration, thinking arithmetic – arithmetic thinking (see Arrigo “calcolo ragionato”), semi-written arithmetic, critical exploration, zone of proximal development, see Bodrova & Leong, 2015; Duckworth, 2004; Winnicott, 2002; Cuomo, 2007; Lillard, Lerner, Hopkins, Dore, Smith & Palmquist, 2012; Vygotsky, 2019) dynamizes pedagogy and research. This case analysis does not only describe developments in the here and now of JAN, but also current developments of the pedagogue.
Future case studies will examine experiences and development work at different levels. "Sharing the video" is a promising method to explore the meaning of the experiences with the talking and acting calculator in a "video-stimulated-dialogue" (see Morgan, 2007; Nind, Kilburn, Wiles, 2015; Werfeli & Meyer, 2019).
William Shakespeare wrote in "A Midsummer Night's Dream (Act V, Scene 1; 2012):
Lovers and madmen have such seething brains,
Such shaping fantasies, that apprehend
More than cool reason ever comprehends.
The lunatic, the lover, and the poet, are
Of imagination all compact (from childhood on).
Look at the children, at their shaping fantasies, and how they "are of imagination all compact." Therefore I wrote "from childhood on." Teaching mathematics needs the fantasy, the imagination and the apprehension "more than cool reason."
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I thank Leslie Burri for the help with the
translation.
