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EMBODIMENT AND REASONING IN CHILDREN’S INVENTED CALCULATION STRATEGIES Carol Murphy University of Exeter This paper explores the reasoning of young children in invented calculation strategies. It draws on the non-objectivist philosophies of embodied learning to analyse the use of the Laws of Arithmetic in invented strategies. In doing so it raises questions regarding the ‘a priori’ view of mathematical reasoning and the nature of children’s learning of informal and formal arithmetic. It raises questions regarding the implications of embodied learning in the primary mathematics classroom. INTRODUCTION Through recent developments in cognitive science it is becoming accepted that humans, along with other animals, have an innate, inherited numerosity that may guide the acquisition of mathematics (Butterworth, 1999; Dehaene, 1997). Numerosity is combined with the human ability to use symbols, language and prediction to develop counting and, with the use of numerals, to create mathematics. Lakoff and Nunez (1997, 2000) proposed a theory of embodied learning in mathematics where cognition is situated in the mind and developed through psychological and biological processes. Grounding and linking metaphors support the development of schema. These are influenced both by the body and the environment and develop understanding of mathematical ideas. The theory explores deep issues related to the universal nature of mathematical ideas and the role of culture in shaping the content of mathematics. Epistemologically the theory is non-objectivist. Mathematics is viewed as human imagination where mathematical reasoning is based on bodily experiences (Johnson, 1987). There is much evidence that young children develop their own strategies in arithmetic (Carpenter and Moser, 1984; Steinberg, 1985; Kamii et al., 1993; Foxman and Beishuizen, 1999). The use of invented strategies has been traditionally viewed as evidence of individual construction of mathematics in a Piagetian sense (Steffe, 1983). In this way the coordination of knowledge of numeration and arithmetic operations has been seen as abstract logico-mathematical reasoning and not experimental abstraction (Giroux and Lemoyne, 1998). An embodied view would suggest that the reasoning involved in these strategies is inductive and that abstraction is experimental. This paper draws on the non-objectivist philosophies of embodied learning to analyse the use of the Laws of Arithmetic in invented strategies. It challenges the notion of young children’s mathematical reasoning as deductive and explores the possibility of experiential learning in determining commutativity and associativity. 2006. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds.). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 217-224. Prague: PME. 4 - 217

Murphy A NON-OBJECTIVE VIEW OF MATHEMATICS A Piagetian view of mathematics sees construction of knowledge as a progression from children’s spontaneous concepts and individual, egocentric views to a ‘true’ scientific knowledge (Piaget, 1962). The child’s ultimate intellectual aim is to arrive at a ‘scientific concept’, a detached abstract view of mathematics. Although Piaget recognised mathematics as a “science as a result of man’s mental activity” (Piaget and Beth, 1966, p.305), mathematical reasoning was seen as absolute and ‘a priori’. Cultural studies provide a different ‘world view’ and challenge Piaget’s constructivism (Lerman, 1996). Epsitemologically, Vygotsky suggested a nonobjectivist view of socially constructed, shared knowledge where “concepts are socially determined and thus socially acquired” (p.146) but that cultural tools and concepts exist “outside of the individual’s mind” (p.135). “Objects in mathematics are objective in an intersubjective sense, agreed, useful, long lasting but potentially changeable” but the created reality “takes on a life of its own” (p.146). This interpretation of Vygotsky suggests that there is an externally created reality to be internalised, the implication being that children appropriate the teacher’s (albeit) cultural knowledge (Steffe & Tzur, 1994). In seeing this as the ultimate aim it still suggests an esoteric, expert/novice model of socio-cultural learning that can reinforce the elitist academic view of mathematics. Traditionally cognitive science in the 1970s has supported the objectivist view of mathematics by examining individual reasoning and the manipulation of arbitrary symbols (Nunez et al, 1999). Mathematics was seen as non-corporeal and it did not consider how mind and body worked together. Embodied learning provided an alternative approach in cognitive science that rejected objectivism. Epistemologically “reality is constructed by the observer, based on non-arbitrary culturally determined forms of sense making which are ultimately grounded in bodily experience” where “cognition is about enacting or bringing forth adaptive and effective behavior, not about acquiring information or representing objects in an external world”. (Nunez et al, 1999, p.49) The view of mathematics as an external reality either in an objective or an intersubjective sense may lead it to be taught in an authoritarian way where the mathematics is presented as “fully formed and perfectly finished knowledge” (Ernest, 1994, p.1). In this way academic mathematics may be seen as esoteric, elitist and decontextualised where students acquire very specific meanings to the mathematics taught in schools. The embodied notion of mathematics as internal and ‘mind-based’ may help to remove such an authoritarian view and, in turn, support affective views of mathematics. 4 - 218

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Murphy GROUNDING METAPHORS AND EXPERIENTIAL LEARNING Lakoff and Nunez (1997, 2000) proposed two types of metaphors from which the conceptual structure of mathematics has been developed. The first are termed ‘grounding metaphors’. These allow everyday experiences to project onto abstract concepts. The second are termed ‘linking metaphors’ that yield more sophisticated abstract ideas and allow different branches of mathematics to be linked. In exploring children’s invented strategies the first type is of interest. These are said to be based on commonplace physical activities such as collecting objects into groups, splitting groups of objects and moving objects together and apart. These are linked with basic numerosity skills of subitising and counting. In this way the three basic grounding metaphors are: • • •

Arithmetic is Object Collection Arithmetic is Object Construction Arithmetic is Motion

The first is a “precise mapping from the domain of physical objects to the domain of numbers” (Lakoff and Nunez, 2000, p.55) and is reflected in our language by the word ‘add’ as the physical placing of objects or substances into a container or group of objects. For example: “Add some logs to the fire”. In arithmetic this becomes “If you add 4 apples to 5 apples, how many do you have?”. This metaphor, along with the other two grounding metaphors, can be seen to base arithmetic firmly in experience and as a human construction. There are objects that exist in reality but the idea of a collection or group of objects is a human construction. This does not contradict socio-cultural or constructivist views of mathematics as a human construction. Vygotsky (1978) saw the perception of real objects as a human construction in that we do not just see a world of shape and colour but impose a sense and meaning. Constructivists such as von Glasersfeld (1994) have stated that mathematics would not exist without the notion of ‘unit’ and that this notion is “derived from the construction of objects in our experiential world” and quotes Einstein in referring to the concept of objects as “a free creation of the human … mind” (p.5). INVENTED STRATEGIES AND EXPERIENTIAL LEARNING As stated earlier there is evidence of young children inventing their own calculation strategies. The innate basis for arithmetic would seem to be limited to subitising small numbers of objects (Butterworth, 1999; Dehaene, 1997) but arithmetic is said to exist as a human construction of numbers. For example addition would not be closed under subitising and relies on the human creation of counting and infinity. Although there may be an innate basis for arithmetic, not all arithmetic is innate (Lakoff and Nunez, 2000). With the invention of their own calculation strategies young children often rely on the Laws of Arithmetic. For example when putting a larger number first in countingPME30 — 2006

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Murphy on in addition children will rely on commutativity. When partitioning numbers into tens and units or ‘bridging’ across the decades children will rely on associativity. The question arises ‘How do children learn to use these?’ There is a possibility that social mediation plays a role as children check their answers using these strategies but this does not explain how children develop them in the first instance. Some children will have developed their own strategies before starting school or in a formal mathematics classroom where their use is discouraged. Lakoff and Nunez proposed that the Laws of Arithmetic only exist through human construction of numbers and that the laws are a metaphorical entailment of the Arithmetic Is Object Collection metaphor. Young children can determine commutativity experimentally. In object collections, adding A to B gives the same result as adding B to A. The claim is then that they will find that, for numbers, adding A to B gives the same result as adding B to A. Similarly associativity can be determined experimentally where adding B to C and then adding A to the result is the same as adding A to B and adding C to the result in both collection of objects and numbers. Gelman and Gallistel’s (1978) empirical studies of young children’s counting have found that children develop numerical reasoning as they develop the principles of counting beyond ‘one-one correspondence’. A further principle of counting, ‘orderirrelevence’, requires a more abstract view of number as the child finds out that the order in which you count a set of objects does not affect the number you end up with. This can then extend to addition and the realisation that this can be commutative. “Addition in the child’s view, involves uniting disjoint sets and then counting the elements of the resulting set. According to the order irrelevance principle it does not matter whether in counting the union you first count the elements of one set and then the elements from the other or vice versa” (p.191).

When extended to three sets, associativity is also implicit in the child’s numerical reasoning. In such a way young children may implicitly determine commutativity and associativity as Laws of Arithmetic that tell them how numbers can be manipulated. They have moved beyond the innate numerosities and are beginning to reason with number. INFORMAL AND FORMAL ARITHMETIC By analysing the use of commutativity and associativity in invented strategies in terms of experiential learning it is possible that children develop early reasoning in number inductively but that this reasoning may be intuitive or implicit. Roter (1985) found that children were able to carry out unconscious abstract processes to some degree. By exploring inductive cognitive activities based on sequences of simple geometric shapes determined by complex rules it was found that children could abstract complex knowledge from the environment where the knowledge obtained was tacit. 4 - 220

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Murphy This suggests the development of implicit, informal mathematical practices as used in everyday life. These practices are often intuitive and are used with little or no formal justification. They may be accepted on a pragmatic basis and empirically determined. The notion of embodied learning, grounding metaphors and the deductive determination of the Laws of Arithmetic through the order-irrelevance principle would support this implicit experiential learning of informal mathematics and the unconscious abstraction of knowledge from the environment. Formal mathematics, on the other hand, is based on proofs and axioms where reasoning is deductive. Tall (2001) questioned the founding of formal constructions in mathematics from natural or informal embodied concepts. Formal mathematics is seen to focus on definitions and formal deduction to “avoid any appeal to intuition” (p.203). It is recognised that informal ideas often come before the axiomatic theories and that they may persist after but Tall indicated they could sometimes be contradictory. In this way Tall asserted that not all “thought is related to embodied perception” (p.207) and saw formal and informal mathematics as two distinct perceptions of mathematics. Auslander (2001) also critiqued the role of metaphor in the development of more advanced mathematics and queried how the “spontaneous use of young children meets with the analysis of academic mathematics” (p.2). An embodied view of learning would see all reasoning as imagination based on bodily experiences (Johnson, 1987). Based on experiential philosophy, it looks to the brain and the body to explain all understanding from a naturally based account (Lakoff and Johnson, 1999). Lakoff and Nunez (2000) asserted the role of metaphors as the “basic means by which abstract thought is made possible” (p.39). It is explained that “much of the ‘abstraction’ of higher mathematics is a consequence of the systematic layering of metaphor upon metaphor, often over the course of centuries” (p.47)

It is also however recognised that mathematics viewed as formal and disembodied will look ‘very different’ to embodied mathematics. There is insufficient space in a paper of this length to explore this fully but there is enough to see that a contention exists between the development of natural, informal mathematics and the understanding of formal mathematics and the role that conceptual metaphors may play in this. From a pedagogical perspective this raises questions related to the teaching of arithmetic and the relationship between children’s informal, intuitive arithmetic and the formal mathematics that may be presented to them in the classroom. Figure 1 presents two extreme paths through informal and formal arithmetic. No curriculum would follow just one of these paths. The path that follows through from experiential learning suggests the inductive development of early mathematical reasoning from empirical experience. We know that many children do invent strategies so this grounded route must play a role in many children’s mathematics. It is anticipated that PME30 — 2006

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Murphy Fig. 1: Paths for Informal and Formal Development of Arithmetic Innate – subitising Numerosity Informal

Formal

Experiential

Symbol manipulation Counting as repetition of number names, symbols

Counting as Physical experience. Collection of objects

Order irrelevance Operations as facts

Operations as movement of objects Commutativity, Invented strategies

Associativity

Taught mental calculations

Standard Algorithms children following this route would arrive at the use of the standard algorithms with a greater capacity for reasoning intuitively. If the curriculum includes the teaching of mental calculations that are presented formally it is possible that children will base this on the practical experiences 4 - 222

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Murphy developed from counting rather than the early reasoning based on order-irrelevence and the implicit use of the Laws of Arithmetic. If however children move from a less experiential route and calculations have been learnt as facts with little concrete verification they may learn to use standard algorithm based on limited numerosity skills. The child’s thinking about the algorithms will be limited and learning may be procedural. CONCLUSION The notion of embodied mathematics provides further lenses with which to investigate learning in mathematics. By reviewing children’s learning in mathematics through these lenses we begin to raise further questions related to pedagogy. It has not been possible to explore the contentions between formal and informal mathematics fully but whether the transition from informal mathematics to formal mathematics is seen as bridging a distinct gap or as an evolution from embodied experience to abstract thought, metaphors could be seen as having a key role. A further question would be to consider the different metaphors children use in carrying out informal invented strategies and formal algorithms. This analysis has explored children’s informal use of arithmetic and their unconscious, implicit abstraction of mathematics and begun to consider the pedagogical implications in the teaching of arithmetic from an informal and a formal perspective. It is possible to explain children’s invented strategies from an embodied perspective that challenges the constructivist notion that the mathematical reasoning underpinning these strategies is deductive logico-mathematics in a positivist sense. Empirical studies are needed to determine this so that we can better understand how children’s early reasoning develops into their first mathematical thinking beyond numerosities and how they use this to develop mental calculations and later become proficient at a range of informal and informal strategies. References Auslander, J. (2001). Embodied Mathematics, American Scientist Online, 89(4). http://www.americanscientist.org/template/BookshelfReviews/issue/396 Butterworth, B. (1999). The Mathematical Brain. London: Macmillan Carpenter, T. and Moser, J. (1984). The acquisition of addition and subtraction concepts in grades one through three, Journal for Research in Mathematics Education, 15(3), 179202. Dehaene, S. (1997). The Number Sense: How the mind creates mathematics. New York: Oxford University Press Ernest, P. (1994). Constructing Mathematical Knowledge. The Falmer Press: London. Foxman, D. and Beishuizen, M. (1999) Untaught mental calculation methods used by 11year-olds, Mathematics in School, 28(5), 5-7. Gelman, R. and Gallistel, C. (1978). The Child’s Understanding of Number. Cambridge, Massachusetts: Harvard University Press. PME30 — 2006

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Murphy Giroux, J. and Lemoyne, G. (1998). Coordination of Knowledge of Numeration and Arithmetic Operations on First Grade Students, Educational Studies in Mathematics, 35(3), 283-301. Johnson, M. (1987). The Body in the Mind. Chicago: University of Chicago Press. Kamii, C., Lewis, A. and Jones Livingston, S. (1993). Primary arithmetic: children inventing their own procedures, Arithmetic Teacher, 41(4), 200-203. Lakoff, G. and Johnson, M. (1998). Philosophy in the flesh. New York: Basic Books. Lakoff, G. and Nunez, R. (2000). Where mathematics come from. New York: Basic Books Lakoff, G. and Nunez, R. (1997). The metaphorical structure of mathematics: sketching out cognitive foundations for a mind-based mathematics. In L. English (Ed.) Mathematical reasoning: analogies, metaphors and images. (pp. 21-89). New Jersey: Lawrence Erlbaum Associates. Lerman, S. (1996). Intersubjectivity in mathematics learning: A challenge to the radical constructivist paradigm? Journal for Research in Mathematics Education 27(2), 133-150. Nunez, R., Edwards, L. and Matos, J. (1999). Embodied Cognition, Educational Studies in Mathematics, 39, 45-65. Piaget, J. (1962). Comments on Vygotsky’s critical remarks. Massachusetts: MIT Press Piaget, J. and Beth, E. (1966). Mathematical Epsitemology and Psychology. Dordrecht: D. Reidel Publishing Company. Roter, A. (1985). Unconscious abstraction processes: Can children process as well as adults? Paper presented at the Annual Symposium of the Jean Piaget Society. May 1985. (p.18) Philadelphia. Steffe, L. (1983). Children’s algorithms as schemes, Educational Studies in Mathematics, 14, 109-125. Steffe, L. and Tzur, R. (1994). Interaction and Children’s Mathematics. In In P.Ernest (Ed.), Constructing Mathematical Knowledge. (pp. 8-32). The Falmer Press: London. Steinberg, R. (1985). Instruction on derived facts strategies in addition and subtraction, Journal for Research in Mathematics Education, 16(5), 337-355. Tall, D. (2001). Natural and formal infinities, Educational Studies in Mathematics, 48, 199238. von Glasersfeld, E. (1994) A radical constructivist view of basic mathematical concepts. In P.Ernest (Ed.), Constructing Mathematical Knowledge. (pp. 5-7). The Falmer Press: London. Vygotsky, L. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.

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