“I am your calculator” - critical exploration and roleplay

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.

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."

Illustration by S.L.Meyer

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).

 

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. Meyer, 2020).

References

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 Education, 8–9, 28–36. https://doi.org/10.1016/j.tine.2017.10.002

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: https://danilodolci.org/maieuticareciproca/

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: https://zenodo.org/record/5638606#.YYJD7xwxmUl

Fasce, P. & Vigilante, A. (Hrsg.). (2011). Danilo Dolci e l’educarsi maieutico. Educazione Democratica, 2. 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: https://educationendowmentfoundation.org.uk/evidence-summaries/evidencereviews/ 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 connaissance. Paris: 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: http://paulofreire100jahre.sbg.ac.at/  

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: https://www.interview.hfh.ch/

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 Mathematics, 102(1), 111–125. https://doi.org/10.1007/s10649-019-09899-2 

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

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 Practice, 14(1–2), 147–161. Routledge. https://doi.org/10.1300/J125v14n01_09  

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: http://www.interview.hfh.ch/page005.htm 

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.