It's harder to find research on mathematics in free play and in informal everyday situations. ... Seo & Ginsburg (2004) perform a similar study, but include parents' ...
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HOW AND WHY DO CHILDREN CLASSIFY OBJECTS IN FREE PLAY? A CASESTUDY Vigdis Flottorp Oslo University College I analyze an episode from a field work in a multilingual kindergarten Oslo, using semiotic theory. I examine verbal and non-verbal expressions of two boys playing in a sandpit. A key part of their play is creation of structure. My findings indicate that this structuring become conscious experiences to the children. I would argue that we cannot know about the children’s mathematical and communicative competence without knowing the physical context, the play in the sandpit, and the friendship between the boys. BACKGROUND In Norway we got a new Framework Plan for the Content and Tasks of Kindergartens in 2006, which had a separate chapter devoted to mathematics. It says: (…) the staff must listen and pay attention to the mathematical ideas that children express through play, conversation and everyday activities (…) and support the mathematical development of children on the basis of their interests and modes of expression. (Kunnskapsdepartementet, 2006:42)
So the Norwegian Framework stresses that mathematics have an intrinsic value for children, not only a value for the future. This concerns the justification for mathematics in kindergarten. According to some political signals, mathematics is important first of all for school readiness. A white paper brings out: In school many pupils struggle with doing arithmetic as a basic skill. There is a need for creating more positive attitudes to the subject. The general work in kindergartens and especially with the learning area Numbers, spaces and shapes is important in this context. (Kunnskapsdepartementet, 2009:77)
In the last years learning has been stressed, focusing on basic skills. In Norway the kindergarten became a part of the educational system with the transition from Ministry of Children and Family to Ministry of Education, and mathematics became a separate learning area in the Framework. This has caused a revived debate concerning the concepts of play and learning. Most people agree that kindergarten shall be a learning arena. The question is how this can be realized in mathematics, and what the concept of learning implies. Much research on mathematics in the early years is based on experimental situations, often with single children. It’s harder to find research on mathematics in free play and in informal everyday situations. Some studies though, I have found. Björklund (2007) analyses toddlers in order to discover how they come to understand different aspects
2 of mathematics like parts and whole, similarities and differences. Twenty-three children were videographically observed. Her findings indicate that the children become aware of how things can be seen as a group of items with similar qualities, by first experiencing units as being separate from other units. She finds that the children tend to focus on visual characteristics of items especially when they differ from others. To me, her study is interesting not so much because of her findings, as her focus on very small children in everyday activities. Her attempt to study this reveals methodological problems with interpreting the actions of small children. Fauskanger (1998) analyzes an incident from her own childhood, playing pirates. Measuring, counting and making record of prisoners are central for participating in the pirate play. As one of the youngest, she demonstrates how she learned from the more capable participants. Her study focuses on how the children construct mathematical knowledge in social context meaningful to them. Playing pirates for weeks on a farm in the countryside is a much less institutionalized activity than those going on in kindergarten where children have less time and space to continue and develop their play activities. These bounds are important related to mathematical experiences, and they are of current interest for my field work. The episode I analyze is similar to Fauskanger’s in the sense that the children are constructing knowledge relevant to them. While the episode I am analyzing is lasting for some minutes, the children in Fauskanger’s study are occupied with the same activity for weeks. Tudge & Doucet (2004) investigate children’s engagement in mathematical activities of a complete day, comparing white and black children. The variation among them cannot be explained by ethnicity or class. Seo & Ginsburg (2004) perform a similar study, but include parents’ income. They find that the low-income children seem to possess the intellectual ability to engage in advanced mathematical explorations. The relevance of the latter study is the finding indicating that enumeration is a relatively small part of the children’s mathematical activities compared to shapes and patterns. My interest from the beginning was on geometrical phenomena more than enumeration. The research on small children’s number sense is waste, while studies on their geometrical understanding are relatively sparse (Clements, 2003). When findings indicates there is less enumeration than other types of mathematics in kindergarten, but more research on enumeration, then my interest coincides with what seemed important to focus on. My main aim for my field work was to study the mathematics in children’s everyday activities, focusing especially on geometrical concepts and on how children express them. I needed a tool for analysing all kinds of expressions - actions, gestures, body language and verbal utterance, and that’s why I chose a semiotic approach.
3 THEORY Semiotics is the study of culture as signs, where signs incorporate all kinds of tools used in communication, from linguistic to physical tools. Since these tools are human made, all concepts can be regarded as historically created. This also applies to mathematical concepts. Ideas and mathematical objects (.) are conceptual forms of historically, socially and culturally embodied reflective, mediated activity. (Radford, 2006:42)
Mathematical concepts are like «lighthouses that orient navigators' sailing boats» (ibid.), but they are not ideas separated from our world. Their abstract and general aspects are results of human activity, and new constructions can arise in another context. Consequently, knowledge is created and recreated in every situation. Traditionally, thinking is regarded as a mental activity. Radford (2009) advocates a multimodal perspective, where language, gestures and tools are considered as “genuine constituents” of cognitive activity. “Thinking does not occur solely in the head, but in and through language, body and tools. (Radford, 2009:113). Accordingly, mathematics can become manifest in many ways. Concept formation is closely related to the context. This does not imply that knowledge can be reduced to individual constructions, because we use tools in the concept formation, tools which already have content. The relation between the subjective and the cultural content are like two sides of the same coin (Radford, (2006). One side is the subjective comprehension, intimately related to the person’s experience. On the other side is the cultural content, transferred through the cultural tools in the act of meaning making (ibid., 52). This is why participation is regarded as crucial for learning, rather than acquisition. Sociocultural psychologists prefer to view learning as becoming a participant in certain distinct activities rather than as becoming a possessor of generalized, contextindependent conceptual schemes. (Sfard, 2001, p. 23)
Mathematics can be conceived in many ways. As a subject matter in school, three conceptions can be distinguished (van Oers, 2001). According to the first one, mathematics is synonymous with arithmetical operations. The second conception says that mathematics is about abstract structures applied to concrete situations, and the last one advocates that mathematics is about problem solving with symbolic tools. The second conception presupposes that structures are stable and a priori. This is not consistent with a sociocultural view, whereas the third conception is. According to this view, mathematical activity organizes human experience in a systematic way, also called mathematising. I myself insist on including in this one term the entire organizing activity of the mathematician, whether it affects mathematical content and expression, or more naïve, intuitive, say lived experience, expressed in everyday language. (Freudenthal, 1991:30)
4 What make Sfards distinct activities and Freudenthals lived experiences mathematical? According to Mason and Johnston-Wilder (2004) desirable mathematical activity consist of some ways of acting; stressing and ignoring, specializing and generalizing, distinguishing and connecting, imagining and expressing, conjecturing and convincing, organizing and characterizing. These ways of acting are independent of activity or object. They can be considered more general than Bishops (1988) six mathematical activities - counting, measuring, locating, designing, playing and explaining. Bishop argue that mathematics exist in both literal and illiteral cultures, - the latter the culture of small children. METHOD I visited a kindergarten in Oslo weakly for one year. The kindergarten is a preschool for practicum, located in an area with a high amount of minority speaking people. The section I followed, had 17 children aged 2-6 years. Five of the children had Norwegian as first language. In order to find out what was going on in the children’s activities, I became participating observer. I could not plan the children’s play or give them instructions, but followed their daily activities. Since I wanted to study what children do alone, I tried not to give them too many suggestions. I was interested in the social aspects of meaning making, and consequently I focused on situations with interaction. I was also curious about the role of the verbal language in a multilingual kindergarten. Hence, I concentrated on children with verbal language. I looked for situations with mathematical potential, like block building, drawing, games and conversation. Quite soon I distinguished some children because of their concentration and creativity. These children seemed to participate in the most interesting mathematical episodes. I do not infer that there is coherence between concentration and mathematics. My data selection does not tell anything about what is common or typical, but they can suggest how children express mathematical ideas. My data consist of notes, photos and videos. I classified the material, using Bishops categories. First I transcribed the videos roughly, dividing them in episodes after distinct activities. Then I chose the most interesting ones and transcribed them more thoroughly. The interesting episodes were rich, with regard to meaning and expression. Here the mathematics was developed over some time, for example because a problem arose. It was not always easy to pick the rich episodes. Another selection criterion was that the episode surprised me, like the one I analyze here. ANALYZE The episode lasts for only 3-4 minutes. When I start recording, the boys have placed different toys on the edge of the sandpit. It ends when another child enters the scene.
5 One of the boys, Mohammed [1], excels with his interest in systems and numbers. The other one, Waqas, is a quiet and concentrated boy. They speak Norwegian with each other as their first languages are different. The sandpit toys constitute an essential part of the activity. They are artifacts with cultural meaning, which in this case is ambiguous. They are toys for children, but at the same time they resemble articles for daily use, - some look like kitchen equipment, other like miniature garden utilities. The cones are special since they are neither kitchen nor garden utilities. They are the only items which I have given a mathematical name. The boys never mention any of the toys by name, but their actions reveal what kind of meaning they give to them. They pretend to drink from the cups and stir in the pans. From earlier observations I know that the children often use the toys for making food. Sometimes they expand this activity by making a restaurant, - pretending to serve and sell food. It looks like the play is about doing the same, because when one of the boys is drinking or stiring, the other one is mimicking. Consequently, it is essential to have the same types and the same number of sandpit toys, which they do not have. Mohammed
Waqas
cup
cup
cone
cone
2 pans
1 pan
sea star
sieve
Fig 1: The distribution of the sandpit toys in the beginning
The problem with Waqas having only one pan, is solved easily. He throws the sieve away and finds a frying pan. This item he bangs in the frame of the sandpit, declaring: “Look what I’ve found!” A frying pan is without holes and can have the same function as a cooking pan. This common feature becomes pronounced, when Waqas suggest that they put sand in “everything”. Mohammed responds that he will put sand “only in two”. Then both put sand in their pans, including the frying pan, showing that they agree on what “everything” shall mean. The difference between sea star and sieve, is more difficult to solve, since there is no extra sea star around. Mohammed watches the toys, turns around, grasps the sieve and places it on Waqas’ side, second to the edge. At the same time he points at his own sea star, saying: “Look, sea star!” At the same time he turns the sieve bottom up. Then he controls the system by making one-to-one correspondence: He touches Waqas’ toys one by one while he follows his own toys with the eyes, saying “putting it there” for every item. In the end he declares: “Now everything is alright.”
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Fig 2: Mohammed pointing at the cones while he declares:” Now everything is alright.” The sea star is hidden second from the end to the right.
Mohammed creates a double similarity between the sea star and the sieve. First, he makes up a new criterion for classification – bottom up – which distinguishes the sea star and sieve from the other toys. Secondly, he creates a common feature by reflective symmetry, next further out. Waqas pan
sieve
pan
Mohammed cup
cone
cone
pan
pan
sea star
cup
Fig 2: The categories and the sequence of the toys on the edge of the sandpit.
The symmetry is not incidental, because short afterwards the spades are placed in the sequence. Mohammed places his spade second to the end with the handle turned to the middle. Waqas asks for help, consequently he must have understood that the placement is special and important. Mohammed puts his spade on his knee, handle turned to the middle, saying: “It’s like this.” Then Waqas places his spade on his side, the handle turned to the middle. The symmetry is not perfect, but Mohammed still declares that everything is alright. The purpose of the symmetry is to create a similarity between the sea star and the sieve. The boys are making categories and symmetry because they need. The essential is what is right in the play, which is to do the same with the same number and kind of toys. This can be interpreted like a token of friendship where the boys mimicking each other. Friendship is an essential part of socialization. In this episode Mohammed is the one in command, and many would argue that reciprocity and balance are necessary for a friendship. According to Greve (2009) every friendship has its own style and history. What matters, is that the children have experiences which give them a common we. It
7 looks like this is the case in this episode. They have a common project that they communicate to each other without misunderstandings. DISCUSSION The play in the sandpit has many similarities with desirable mathematical activity described by Mason and Johnston-Wilder (2004). The boys are creating structure and order of the artifacts by stressing and ignoring, distinguishing and connecting. The toys can be classified in many ways, for example after size, color, form or function. The boys distinguish the pans from the rest by size. The pans and sieve are separated from each other by stressing topological aspects. By looking at form, the boys distinguish the cones from the rest. The sea star and the sieve are connected by placement and symmetry, ignoring all other aspects. The boys cognitive activity becomes manifest first of all through their communicative actions. They show, throw, bang and fill sand in the toys. Sometimes the artifacts themselves are mathematical expressions, for example when the spades are placed symmetric. The boys have few verbal utterances, and most of the utterances are deictic: “I will get one like that,” says Waqas, searching for an equivalent to a pan. “It is like this,” demonstrates Mohamed to his mate demonstrating how to place the spade. These words do only have meaning in the context, which is common to the boys. They talk well Norwegian, but still they communicate without many word because they do not need it. Their actions and deictic signals are sufficient. Accordingly it is irrelevant that they are bilingual. They do not use any mathematical words, yet they have made a structure of classification and symmetry. This can be compared to ethno mathematical studies, for example of children’s geometrical patterns (Gerdes, 2007). These are describes in a mathematical language which is as strange to the children of Angola, as it is to these boys. Neither the patterns, nor the boys’ system are coincidental. While the boys’ structure is ad-hoc, the patterns of the Angolan children are stored knowledge, supported by tradition. The boys are making their own unique genre; they do not repeat a pattern. Most of the studies of children’s mathematical classification concerns geometrical forms, based on features with unambiguous definitions. Classification focus usually on to what degree children are able to separate the defining features from the irrelevant ones (Clements, 2003). While all mathematical concepts have a clear definition, the daily life concepts are often ambiguous. What makes a triangle to a triangle, are the human made definitions which have to be learned. Without them, a form with tree corners and curved lines could be a “triangle”. In the boys’ classification, nothing is defined beforehand. They
8 make up their own criteria, depending on the play and the feature of the toys. Their play reflects the adult’s word, but at the same time it is different because they make the rules. They are in our world and in a make-believe world at the same time. Structures are just temporarily stabilized ways of approaching a problem. Mathematical activity in school – in order to be realistic – should focus above all on the processes of structuring instead of the mastery of fixed and prescribed structures. (van Oers, 2001, p. 63)
In this episode the boys solve a problem, but do not use any traditional mathematical concepts or tools. Their actions, gestures and toys are semiotic signs which create a structure. CONCLUSION One single case cannot prove anything about the mathematics of small children. The analyses show that physical experiences and actions are fundamental in classification. It looks like all the communicative signs are intended. Hence, the children’s mathematical experience cannot be unconscious. The children communicate in the most efficient way in the situation. We emphasize the importance of verbalization, but in efficient communication we avoid unnecessary information. It is a challenge for the staff to create situations where verbalization is necessary and meaningful. The staff can learn a lot by observing children in their play. This is nothing new: Adult would do better to learn to observe and appreciate children’s everyday mathematical activities than to provide fancy “educational” toys. (Seo & Ginsberg, 2004:97).
My study shows how mathematics can be useful, and can provide children with joy. Besides, it can give them confidence in their logical ability, which will be beneficial for future schooling. NOTES 1.
All the names are anonymous. They have been translated culturally, - for example, children with Urdu names have got usual Pakistani names.
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9 Clements, D. H. (2003). Teaching and Learning Geometry. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), Research Companion to Principles and Standards for School Mathematics. Reston, VA: NCTM. Clements, D. H., Sarama, J. A., & DiBiase, A.-M. (2004). Engaging young children in mathematics: standards for early childhood mathematics education. London: Lawrence Earlbaum. Fauskanger, J. (1998). Lek i seksåringenes matematikkundervisning. Thesis. Oslo: Universitetet i Oslo. Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Boston: Kluwer Academic Publishers. Gerdes, P. (2007). Drawings from Angola: living mathematics. Maputo, Moxambique: Research Centre for Mathematics, Culture and Education. Ginsburg, H. P., Inoue, N., & Seo, K. H. (1999). Young Children Doing Mathematics. Observations of Everyday Activities. In J. V. Copley (Ed.), Mathematics in Early Ears: National Association for the Education of Young Children, Washington D. C., NCTM, Inc Reston V. A. Greve, A. (2009). Vennskap mellom små barn i barnehagen. Oslo: Pedagogisk forum. Kunnskapsdepartementet. (2006). Rammeplan for barnehagens innhold og oppgaver. Framework Plan for Content and Tasks of Kindergartens. Oslo: Regjeringen. Kunnskapsdepartementet. (2009). Kvalitet i barnehagen. Kindergartens. (Vol. nr. 41 (2008-2009)). Oslo: Regjeringen.
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10 Sfard, A. (2001). There Is More to Discourse Than Meets the Ears: Looking at Thinking as Communicating To Learn More about Mathematical Learning. Educational Studies in Mathematics, 46(1-3), 13-57. Tudge, J. R. H., & Doucet, F. (2004). Early Mathematical Experiences: Observing Young Black and White Children's Everyday Activities. Early Childhood Research Quarterly, 19(1), 21-39. van Oers, B. (2001). Educational Forms of Initiation in Mathematical Culture. Educational Studies in Mathematics, 46, 59-85.
