Montag, 24. Juni 2019

“I am your calculator” - critical exploration and roleplay

Figure 1: Jan writes on the blackboard
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 (IayC), your calculating machine" mean? The reference schema illustrates important elements.
Figure 2: referencing scheme of the IayC
In reference to Piaget (1967), it is a conscious renunciation or relativization of a test method that merely makes fears and incompetence visible. JAN is positively provoked to act in roleplay.
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."

Figure 3: logo of the IayC
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. *Piaget: "However, a method is not good or bad in itself. It can only be judged according to the problems it is intended to solve, which in turn are oriented by more or less explicit epistemological perspectives").

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. The fantasy of number artists (arithmeticians) is sustained by the "poetry of possibilities" (cf. Sutton-Smith, 1986).

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; Eggenberger, 2021).
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" (cf. Moreno, 2007a, b; Smith & Roopnarine, 2018; Meyer, 2020). The same is true for research.
IayC with groups (October, 2021)
Let's recall the basic questions at IayC again: What is mental arithmetic exactly? What does "calculate like a flash" mean? What is the "calcolo ragionato"? How does mental arithmetic occur when a person solves arithmetic operations with his fingers? How do intuitions, comprehension, and competence in arithmetic operations occur when learners have little or no schemata for them?

The "I am your calculator" method positively provokes mind games from which answers to the posed questions can be generated. - The concern about transmission pedagogical methods is not discussed further (cf. Meyer, 2017). But a methodological note is presented, which differentiates the meaning of IayC.

The previous case studies and explanations described working relations in two-person situations or in the work of one teacher with two children. It turned out, despite all methodological precision, that it is difficult to recognize and overcome the linear-chronological teaching behavior (cf. Eggenberger, 2021). And this is one of the objectives of the IayC method. Walter Omar Kohan (Rio de Janeiro) shed light on this in the lecture "Paulo Freire more than (n)ever?" (cf. international conference 100 years of Paulo Freire from 15.10.2021 in Salzburg; Kohan, 2018) the following aspect. As a rule, transmission pedagogy works with the measurable concept of time of chronos. Liberating pedagogy reflects on a different concept of time, which focuses on lifetime or individual life (cf. GEMOLL: Aiòn: lifetime, life, generation, time, duration, period. Lifeless, long, unlimited time and eternity). The holistic view of life is already found in Heraclitus: "Aiòn (time) is a child, - a child at the board game; a child sitting on the throne" (B52, Heraclitus, 2007).

How can an educator free himself or herself from the dictates of Chronos? Through what does she learn to think, work, and feel with the liberating concept of time? The answer lies not in a romantic enchantment, but in the transformation of working and working relationships. Play pedagogy is a key to changing the way we work in pedagogy. This will be exemplified in the following scene looking at groups of learners. 
Figure 4: IayC as a puppet play (Image: S. Meyer).
Figure 4 opens the discussion.  Following the method of photovoice (cf. Wang, 2006), characters are engaged in a mind game (cf. Meyer, 2020), which is scripted as a propaedeutic libretto for real games. The stuffed animals might talk. Let's further imagine that they are in exchange with an adult who is enacting a stuffed animal mental arithmetic theater. She asks the group, "What does 1+1 give?" The fox replies, "2, that's obvious." The adult continues asking, "and 2+2 to that?" The sheep calls out, "wait: 2 plus 2 plus 2," and it counts with its fingers, "that gives six." In this way, they set tasks for each other until Skye (Paw Patrol) asks an intermediate question, "What does "count like a flash" actually mean?" The stuffed animals and the adults search for answers in a free conversation. At the end, the adult asks a metacognitive question, "How did you notice how many thoughts and examples could be portrayed? What do we need to pay attention to when we want to explain to someone what mental arithmetic is? Let's think about that some more."

This episode provides insight into the "I'm Your Calculator Mental Arithmetic Game." Game ideas, roles, tasks, playtime, and metacognition are developed, carried out, discussed, modified, and continued in free conversation. The IayC and the mental arithmetic game would be expressions of reciprocal maieutics and free operative conversation (cf. Dolci, 2011; Fasce & Vigilante, 2011; Piaget, abstraction réfléchissante; Meyer, 2006).

Referring to Mahoney et al. (2020, p. 3), it is clear that "I am your calculator" is a cultural tool. It aligns with systemic social and emotional learning (SEL), which is an approach to creating equitable learning conditions that actively engages all students from preschool through 12th grade in learning and practicing social, emotional, and academic skills.


Arrigo, G. (2014). Calcolo mentale-approssimato-strumentale. Bollettino dei docenti di matematica, 68, 53-62. 
Bodrova, E. & Leong, D. J. (2015). Vygotskian and Post-Vygotskian Views on Children’s Play. American Journal of Play, 7(3), 371–388. 

Cargnelutti, E., Tomasetto, C. & Passolunghi, M. C. (2017). The interplay between affective and cognitive factors in shaping early proficiency in mathematics. Trends in Neuroscience and Education8–9, 28–36.

Cuomo, N. (2007). Verso una scuola dell'emozione di conoscere. Il futuro insegnante, insegnante del futuro. Pisa: Edizioni ETS.  

Dolci, D. (2011). Dal trasmettere al comunicare (Terza edizione.). Casale Monferrato (Alessandria): Sonda. Verfügbar unter:

Duckworth, E. (2004). L’exploration critique dans la salle de classe. In J.-P. Bronckart & M. Gathier Thurler (Hrsg.), Transformer l’école (S. 79–98). Bruxelles : De Boeck & Larcier.  

Eggenberger, D. (2021). „Ich bin dein Taschenrechner“ (IbdT) - Kritische Exploration und Rollenspiel. Masterarbeit. Zürich: Interkantonale Hochschule für Heilpädagogik. Zugriff am 03.11.2021. Verfügbar unter:

Fasce, P. & Vigilante, A. (Hrsg.). (2011). Danilo Dolci e l’educarsi maieutico. Educazione Democratica2. Günter, M. (2003). Psychotherapeutische Erstinterviews mit Kindern. Winnicotts Squiggletechnik in der Praxis. Stuttgart: Klett-Cotta. 

Heraklit. (2007). Fragmente (14. Aufl.). Zürich: Artemis & Winkler.

Hodgen, J., Foster, C., Marks, R. & Brown, M. (2018). Evidence for Review of Mathematics Teaching: Improving Mathematics in Key Stages Two and Three: Evidence Review. London: Education Endowment Foundation. Verfügbar unter: improving-mathematics-in-key-stages-two-and-three/ 

Ingenkamp, K. & Lissmann, U. (2005). Lehrbuch der pädagogischen Diagnostik (5. völlig neu überarbeitete Aufl.). Weinheim: Beltz.

Inhelder, B., Sinclair, H., Bovet, M. (1974). Apprentissage et structures de la connaissanceParis: Presses Universitaires de France. 

Kohan, W. O. (2021, Oktober 15). “Paulo Freire more than (n)ever?”. Vortrag, hybrid gehalten auf der Internationalen Tagung 100 Jahre Paulo Freire, Salzburg (AT). Zugriff am 15.10.2021. Verfügbar unter:  

Mahoney, J. L., Weissberg, R. P., Greenberg, M. T., Dusenbury, L., Jagers, R. J., Niemi, K. et al. (2021). Systemic social and emotional learning: Promoting educational success for all preschool to high school students. The American Psychologist, 76(7), 1128–1142.

Mannoni, B. (1978). Mathe-Unterricht - Man ist schliesslich nicht zum Spass da! In M. Mannoni (Hrsg.), Ein Ort zum Leben. Die Kinder von Bonneuil (S. 80–99). Frankfurt a.M.: Syndikat. 

Meyer, S. (2006). Das flexible Interview. Zugriff am 20.1.2019. Verfügbar unter:

Meyer, S.L. (2020). Denken beim Apéro. Das dezimale Stellenwertsystem in unterhaltsamen Situationen meistern lernen. Didattica della Matematica, 8, 28-47.

Moreno, J. L. (2007a). Kanon der Kreativität und Analyse der Kreativitätscharta. In H.G. Petzold & I. Orth (Hrsg.), Die neuen Kreativitätstheorien. Handbuch der Kunsttherapie. Theorie und Praxis (4. Aufl., Bände 1-II, Band I, S. 187–188). Bielefeld und Locarno: Edition Sirius.

Moreno, J. L. (2007b). Theorie der Spontaneität-Kreativität. In H.G. Petzold & I. Orth (Hrsg.), Die neuen Kreativitätstheorien. Handbuch der Kunsttherapie. Theorie und Praxis (4. Aufl., Bände 1-II, Band I, S. 189–202). Bielefeld und Locarno: Edition Sirius.

Morgan, A. (2007). Using video‐stimulated recall to understand young children's perceptions of learning in classroom settings. European Early Childhood Education Research Journal, 15(2), 213-226. 

Nind, M., Kilburn, D., Wiles, R. (2015). Using video and dialogue to generate pedagogic knowledge: teachers, learners and researchers reflecting together on the pedagogy of social research methods. International Journal of Social Research Methodology, 1-16.

Piaget, J. (1967). Le jugement et le raisonnement chez l'enfant (6ème édition). Neuchâtel: Delachaux et Niestlé. 

van Oers, B., Wardekker, W., Elbers, E. & van der Veer, R. (2008). The Transformation of Learning. Advances in Cultural-Historical Activity Theory. New York: Cambridge University Press. 

Roth, W.-M., Walshaw, M. (2019). Affect and emotions in mathematics education: toward a holistic psychology of mathematics education. Educational Studies in Mathematics102(1), 111–125. 

Shakespeare, W. (2012). A Midsummer Night's Dream. Ein Sommernachtstraum. Stuttgart: Philipp Reclam jun.   

Smith, P. K. & Roopnarine, J. L. (Hrsg.). (2018). The Cambridge Handbook of Play: Developmental and Disciplinary Perspectives. Cambridge, United Kingdom ; New York, NY: Cambridge University Press.

Storch, M. (1996, 27. 09). Kreativität und Psychodrama. Vortrag gehalten auf der 53. Psychotherapie-Seminar vom 22. bis 27. September 1996, Freudenstadt.

Sutton-Smith, B. & S. (1986). Hoppe, hoppe Reiter...Die Bedeutung von Kinder-Eltern-Spielen. München: Piper. 

Vygotsky, L.S. (2019). Mind in Society. Cambridge MA: Harvard University Press. Kindle-Version. 

Wang, C. C. (2006). Youth Participation in Photovoice as a Strategy for Community Change. Journal of Community Practice14(1–2), 147–161. Routledge.  

Werfeli, B. & Meyer, S. (2019). Flexible Interviews besprechen. Fallstudie und Reflexion des flexiblen Interviews mit Hilfe von «sharing the video». Interkantonale Hochschule für Heilpädagogik. Zugriff am 21.5.2019. Verfügbar unter: 

Winnicott, D. W. (2002). Vom Spiel zur Kreativität (10. Aufl.). Stuttgart: Klett-Cotta. 

Zaslavsky, C. (1994). Fear of Math. How to get over it and get on with your life. New Brunswick: Rutgers University Press.

I thank Leslie Burri for the help with the translation.

Keine Kommentare:

Kommentar veröffentlichen