flexible Interview and mental arithmetic

A game like Halfplay is thought to build relations to rational numbers in a human way. It is obvious that natural numbers (N) take an important part in many games. Why rational numbers (Q) should not be integrated too? Skill, co-construction, fortune, frankness, probability, ease, social relations and sometimes conflicts are characteristics of playing. Halfplay generates such experiences. These experiences offer fundaments and materials for effective processes of mathematizing and realistic problem solving with rational numbers. You come to like symbolic, mental and social relations between rules and the meanings of actions.

Das Halbespiel Einleitung Diese Spielanlage ergänzt den Umgang mit Bruchzahlen auf spielerische Weise. Die natürlichen Zahlen sind in Spielen häufiger vertreten. Warum sollen die rationalen Zahlen nicht auch Platz haben? Geschicklichkeit, das Ko-Konstruieren, das Regeln, Schicksal, Offenheit, Leichtigkeit und sozialer Bezug sind Eigenschaften, welche wir beim Spielen schätzen. Das Halbespiel kann solche Erfahrungen erzeugen. Diese liefern Grundlagen für nachhaltige Prozesse des Mathematisierens, des Übens und der Denkschulung im Bereich der rationalen Zahlen.

Esperienze ludiche concernenti rappresentazioni e operazioni di frazioni sulla retta numerica. " Halfplay" is a game to build relations to rational numbers in a natural and human way. It is obvious that natural numbers take an important part in many games. Why rational numbers (Q) should not be integrated in games too? Skill, fortune, frankness, probability, ease, socia! relations and sometimes conflicts are characteristics of playing. "Halfplay" creates experiences that offer fundaments and materials for effective processes of mathematisizing and realistic problem solving with rational numbers.

Das Halbespiel ist ein Gesellschaftsspiel, mit dem Schülerinnen und Schüler Erfahrungen zum Zählen und Operieren mit Bruchzahlen sammeln können. Es vereinigt Spielfreude, Erarbeitung, Konstruktion, Regelung, Übung und den Willen zum Mathematisieren. Das Halbespiel ist für 2-4 Spieler geeignet. " Halfplay" is a game to build relations to rational numbers in a natural and human way. It is obvious that natural numbers take an important part in many games. Why rational numbers (Q) should not be integrated in games too? Skill, fortune, frankness, probability, ease, socia! relations and sometimes conflicts are characteristics of playing. "Halfplay" creates experiences that offer fundaments and materials for effective processes of mathematisizing and realistic problem solving with rational numbers.

(Kamii, 1986) reported the gradual structure of the insight of the decimal system in the period from the second to the fifth grade. She interviewed one hundred first to fifth grade children from Geneva. In our study (cf. Brugger & Sidler, 2005) we wanted to check how the development tendencies behaved in German speaking Switzerland.

Im Aufsatz wird ein flexibles Interview von Sharon Ross (1986; zit. nach Kamii, 1989) über das dezimale Stellenwertsystem vorgestellt. Seit 1991 habe ich es in Workshops und in der schulpsychologischen Diagnostik im Kanton Solothurn eingesetzt. Seit 2001 wurden an der HfH verschiedene Studien und Tausende von Fallstudien durchgeführt, in denen diese Aufgabe repliziert und weiterentwickelt wurde (Brugger et al., 2007). Die Aufgabe wurde 2013 im MKT-2 einer repräsentativen Stichprobe von 267 Kindern aus der deutschsprachigen Schweiz und dem Fürstentum Liechtenstein vorgelegt. // The essay presents a flexible interview from Sharon Ross (1986) about the decimal place value, which was published by Constance Kamii (1989). Since 1991 I used it in workshops and diagnostics in the School Psychology of the Canton of Solothurn. Since 2001, various studies and thousands of case studies have been carried out at the HfH, in which this task has been replicated and further developed (Brugger et al., 2007). The task was presented in the MKT-2 to a representative sample of 267 children from German-speaking Switzerland and the Principality of Liechtenstein. 44% of second graders were able to complete it at the end of the school year (see Meyer & Wyder, 2017). The essay explains how to conduct a flexible interview in a pedagogical situation. Subsequently, metacognitive questions are suggested, which are carried out in free conversation with the children or with groups. Thus, the child's maximally possible thinking operations are explored and fostered in a school of thinking cf. Adey, 2008). This procedure expresses the dynamic character of the flexible interview or, as it is also called, the method of critical exploration.

In diesem Aufsatz über ein mathematikdidaktisches Unterrichtsprojekt wird der Frage nachgegangen, ob Kinder den Zehnerübergang auf eigenen Wegen entdecken und festigen können. Verschiedene Fallbeschreibungen aus zwei Einführungsklassen geben Einblick, wie die Kinder ausgehend von ihren eigenen Lösungswegen (singuläre Phase) in die Auseinandersetzung mit Klassenkameraden und –Kameradinnen begleitet worden sind (divergierende Phase).

Die Fallstudie untersuchte die Wirkungen der Analyse eines flexiblen Interviews mit der Methode «sharing the video». Im Mai 2017 wurde mit Ada (Name geändert; 3. Klasse Primarschule) ein flexibles Interview zum Thema Geld im Zahlenraum 1000) durchgeführt. Am 22.6.2017 schauten Ada und die Schulische Heilpädagogin die Videoaufnahme an. Sie besprachen, wie die Situation empfunden worden war und was dabei gedacht worden war. The case study examines the effects of analyzing a flexible interview using the "sharing the video"-method. In May 2017 a flexible interview was conducted with Ada (name changed; 3rd grade primary school) about money in the number space 1000). On 22.6.2017 Ada and her special needs teacher watched the video recording. They discussed how they felt about the situation and what had been thought.

Il gioco «Dado-differenza» è stato sviluppato partendo da quello detto «Jass-differenza», piu difficile, ma altrettanto divertente, motto conosciuto nella Svizzera tedesca. In esso il giocatore può vincere sìa con carte di valore alto sìa con carte di poco valore. Deve esaminare le sue carte e stimare quanti punti potrà raggiungere giocando contro uno, due o tre giocatori (Egg, 1989). II gioco «Dado-differenza» offre meno possibilità di operare scelte tattiche e risulta quindi più facile. Inoltre, difficoltà e complessità matematica possono essere adattate a piacere. Oltre al divertimento, ii gioco è un buon mezzo per esercitare la stima dei numeri e la capacità di prendere decisioni strategiche.

Der kommentierte Bericht gibt Einblick in eine Fallstudie in einer Kindergartenklasse. Es ging um die Frage, wie Lernportfolios die Lernprozesse unterstützen können. In einem Teilprojekt lernten die Schülerinnen und Schüler, wie man mit Pet-Flaschen und Bällen ein Kegel-Spiel konstruieren kann. Gleichzeitig wurden sie aufgefordert, Spielsituationen, Ergebnisse und Zahlenverhältnisse mündlich, enaktiv, ikonisch und symbolisch darzustellen. Die Darstellungen übertrugen sie in Lernportfolios, welche vielfach besprochen wurden. Die Fallstudie zeigte, dass die Portfolioarbeit in Kombination mit einem Spiel sehr motivierend sein kann. Modellieren und Mathematisieren auf der Basis von Spielerfahrungen sind für die Schülerinnen und Schüler, aber auch für die Lehrpersonen herausfordernde und bereichernde Tätigkeiten Schlüsselwörter Portfolioarbeit, Kegel-Spiel, modellieren, mathematisieren, Kindergarten, Aktionsforschung The commented report provides insight into a case study in a kindergarten class. The question was how learning portfolios can support learning processes. In one subproject, students learned how to construct a game of bowling using pet bottles and balls. At the same time, they were asked to represent game situations, outcomes, and number relationships verbally, enactively, ironically, and symbolically. They transferred the representations into learning portfolios, which were discussed many times. The case study showed that portfolio work in combination with a game can be very motivating. Modelling and mathematizing based on play experiences are challenging and enriching activities for the students, but also for the teachers.

Developing games with cultural potential is an omnipresent vision. The Monza parlor game has become an open concept and project, with which you can learn holistically at school and in your free time and share wonderful experiences. The game is related to the ladder game, but much more creative. The mathematization is even more successful if teachers have a clear insight and mastery of mathematical knowledge, the curriculum, the math books, and manipulatives. At the same time, they should use creative methods so that the 'exodus' from transmission pedagogy can start. The text is a translation of the article "Monza Gesellschaftsspiel" and contains links to videos and comments in different languages. Keywords - play as a zone of proximal development, systemic play theory, co-construction, Vygotsky, Piaget, mathematization

Spiele mit kulturellem Potenzial zu entwickeln, ist eine allgegenwärtige Vision. Das Monza-Gesellschaftsspiel ist ein offenes Konzept und Projekt geworden, mit dem in der Schule und in der Freizeit ganzheitlich und autonom gelernt sowie schöne Erfahrungen geteilt werden können. Das Spiel ist verwandt mit dem Leiterspiel, aber viel gestaltreicher. Die Mathematisierung ist erfolgreicher, wenn die Lehrerpersonen über gutes mathematisches Wissen verfügen, sowie den Lehrplan, die Mathematikbücher und die Darstellungsmittel beherrschen. Gleichzeitig sollten sie kreative Methoden anwenden, damit der "Exodus" aus der Transmissionspädagogik beginnen kann. To develop games with cultural potential is an omnipresent vision. The Monza game has become an open concept and project. At school and in leisure time, holistic, autonomous learning and sharing of good experiences is generated. The game is related to the snakes and ladder, but much more creative. The mathematization is even more successful if teachers have a clear insight and mastery of mathematical knowledge, the curriculum, the math books, and manipulatives. At the same time, they should use creative methods so that the 'exodus' from transmission pedagogy can start.

Mental calculation with joy. The flexible interview and mental calculation (FI-B) a method with system in system The lecture presents a method with which mental calculation can be diagnosed and promoted holistically. The participants learn for themselves how a screening with 33 tasks works. Afterwards, they gain insight into various case studies in which mental calculation has been systematically, multi-methodically, relationally and with interest/motivation cultivated at all school levels.

The so called «War» is one of several multiplication games portrayed by Kamii & Anderson (2003). This article outlines rules, levels and complexity of the game from additional to multiplicative thinking. Possibilities of schooling mathematical and social thinking and memorization in a third class of an elementary school are described. La cosiddetta "sfida contadina" è uno dei tanti giochi di moltiplicazione ritratti da Kamii & Anderson (2003). Questo articolo delinea le regole, i livelli e la complessità del gioco, dal pensiero addizionale a quello moltiplicativo. Vengono descritte le possibilità di scolarizzare il pensiero matematico e sociale e la memorizzazione in una terza classe di una scuola elementare.