YOUNG CHILDREN AND TEACHERS NEGOTIATING MATHEMATICS G.M. BOULTON-LEWIS AND G.S. HALFORD Queensland University of Technology and University of Queensland This is a description of preliminary research carried out to assess and compare the processing loads of typical mathematical representations and strategies used by teachers and young children. The sample consisted of 29 children aged between 5 and 8 years and their teachers. Children were interviewed and videotaped individually before and after teaching of a specified concept. Teachers were asked to describe their teaching strategies. Data from the pre and post test results for subtraction and place value are discussed as an example of the design, analysis and results. Potential for school learning is closely related to processing capacity. Children's capacity to process information as well as their knowledge base develop with age (Carey 1985; Chi and Ceci, 1988; Halford, forthcoming). The difficulty that children (and adults) experience in cognizing many concepts is due to the process of mapping the task into a mental model and the load that it imposes. This has been shown for transitivity (Halford, Maybery and Bain, 1986) and class inclusion (Halford and Leitch, 1989). The cognitive process is often complicated in school learning by the teacher's choice of content and representations and by the limited knowledge of some concepts and strategies that children bring to the situation. STRUCTURE MAPPING THEORY OF COGNITIVE DEVELOPMENT In order to cognize a new concept a child must map the task into a mental model that corresponds to the structure of the task. A familiar situation is often used as mental model for a new concept. However mapping the concept into the mental model (structure mapping) entails processing loads, the size of which depend on the structural complexity of the task. Halford's structure mapping theory of cognitive development (1988; forthcoming) aims to account for cognitive development in terms of this structural complexity factor, together with the factors of capacity and knowledge. He has defined four levels of structure mapping the processing demands of which are known: Element mappings in which individual elements in one structure are mapped into individual elements in another structure, on the basis of similarity or convention; e.g.an image or word representing an object or event. Figure 1A is an example of an element mapping. In this figure a set of two sticks is the mental model which is mapped into the numeral 2. The set of 2 objects is thus being used as a mental model for the meaning of the numeral "2". Relational mappings in which two elements and a relation between them are mapped from one structure to another; e.g. the relation between small numbers as represented by sets of objects. The child can see that a five-object set is greater than a four- object set by putting the sets into correspondence. Therefore a set of five and a set of four objects can serve as a mental model for the mathematical relation 5>4. In Figure 1B the five-object set is mapped into the number five and the four-object set into the number four. The relation greater than between the sets corresponds to the relation greater than between the numbers. This mapping is validated by the fact that the relation "greater than" between the sets corresponds to the relation ">" between the numerals. System mappings in which three elements, with a set of relations between them, are mapped. An example is shown in Figure 1C, where a set of two objects and a disjoint set of three objects are combined to form a set of five objects. When the sets are mapped into numbers the operation on the sets corresponds to the operation on the numbers. The mappings are validated by structural correspondence independent of similarity or convention. Neither the elements nor the operation need be the same in both structures. System mappings are therefore more abstract than element or relational mappings. Multiple system mappings in which sets of four elements, with a set of operations defined on them, are mapped. An example of such a mapping is shown in Figure 1D. In this mapping the mental model consists of numerals and the operations defined on them, multiplication and addition. They provide a concrete example of the distributive law. INSERT FIGURE 1 (available from first author) Each higher level of mapping permits more abstract concepts to be represented but imposes higher processing loads, due to the larger amount of information that is required to validate each mapping decision. Approximate ages can be related to these levels but it is necessary to be cautious in this regard because lack of experience and hence knowledge in a particular domain may prevent a child from cognizing to the upper limit of her processing capacity. REPRESENTATIONS AND PROCESSING CAPACITY Preliminary research in the application of Halford's theory to school learning is described below. It concerns the load and benefits of representations used by teachers and children in the teaching learning process. This is a description of the design of that research and some of the early results for subtraction and place value. Young children are confronted regularly by mathematical representations. Some of these are concrete embodiments of mathematical concepts and processes (e.g. bundles of 10 sticks or the tens rods of the Multibase Arithmetic Blocks) and others are representations inherent in the discipline of mathematics (e.g. number lines and symbols). All these representations are technically analogs, and can be analyzed using analogy theory. According to Gentner's theory, (Gentner, 1983; Gentner and Toupin, 1986) analogies entail a structure-preserving mapping from a source to a target. In teaching mathematics the concrete representation can be the source and the concept to be taught the target. The value of a concrete representation is that it mirrors the structure of the concept and the child should be able to use the structure of the representation to construct a mental model of the concept. We can illustrate this point by considering the addition operation, which is shown in Figure 1C. In elementary addition, a set of two objects and a (disjoint) set of three objects can be combined to form a set of five objects. With the sets mapped into numbers, the operation of union which combines 2 objects with 3 objects to produce 5 objects mirrors the arithmetical operation 2+3=5. In theory, the concrete representation facilitates understanding and retrieval. It has, however, been noted increasingly in recent literature in mathematics education that concrete representations often fail to produce the expected positive outcomes. Lesh et al. (in Janvier, 1987) stated "that concrete problems often produce lower success rates than comparable word problems ..." Lesh et al. in Lesh and Landau (1983) provide "examples of word problems that become more difficult when additional information is given in the form of concrete material ..." Their research also showed that purportedly realistic word problems often differ significantly from their real world analogs. Dufour-Janvier et al. (in Janvier 1987) drew attention to teachers' motives for using representations, the expected outcomes of their use, the related level of achievement by children, the inaccessible or detrimental nature of some representations and the possibility of organising teaching in such a way that use is made of children's representation. Sowell (1989), surveyed research of the 1960s and 1970s, and compared the outcomes of mathematics instruction with concrete or pictorial materials with outcomes of instruction without such materials. The results of these studies were mixed. Early reviews of research on manipulative materials found that they `were beneficial for young children but unnecessary for older children'; that students learn well in a laboratory setting with such materials but other meaningful teaching works as well; and that the use of manipulatives produced better results at all grade levels than not using them at all. Sowell concluded from the review of the results of 60 studies that mathematics achievement is increased through the long-term (as opposed to brief) use of concrete instructional materials and that students' attitudes towards mathematics are improved when they have instruction with such materials by teachers who are knowledgeable about their use. The explanation that we propose for the difficulties experienced with concrete representations is that previous analyses have taken insufficient account of the processing loads that their use entails. The higher levels of mapping, as described earlier, permit more abstract concepts to be represented, but impose higher processing loads, due to the larger amount of information that is required to validate each mapping decision. Applied to mathematics education, Halford's theory implies that children often cannot derive the anticipated benefits of concrete representations because, if the representations are not known well enough to be used automatically, then they make an extra demand on processing capacity and the processing load of the mapping is too high for the resources they can apply to the task. In the system mapping example given above, children must recognise that two objects plus three gives five objects parallels 2+3=5. Our contention is that, although children can physically manipulate the objects, and allocate the appropriate names ("this is two" etc.), they sometimes cannot recognize the structural correspondence between the concrete representation and the mathematical concept it is intended to illustrate because the load is too high. Processing load is increased if children use unfamiliar or inappropriate analogs, or lack declarative or procedural knowledge. For example, the two binary operations, 8-3 and 45-19 are both system mappings according to Halford's theory (1988; in preparation). However the second operation imposes additional loads including the need to consider place value, recomposition, appropriate concrete representations, and perhaps the written algorithm. These loads must be reduced if the underlying subtraction concept is to be understood. In order to reduce the load it is necessary to ensure that the child is able to recall automatically the relations between quantity, numeration and place value, and any symbolic and concrete representations of the task. A detailed discussion of structure mapping, analogy theory, representations and processing load is provided in Halford and Boulton-Lewis (in press). There are descriptions of the representations and strategies that children use to find solutions to mathematical problems. For example Carpenter and Moser (1982) have been concerned with addition and subtraction word problems and Siegler and colleagues (Siegler and Robinson, 1982; Siegler, 1987) with the representations and rules for numbers that young children use when counting, comparing numerical magnitudes and adding. Our intention in recording all verbal and other behaviours, such as use of fingers as representations, was not merely to describe typical developmental patterns but to assess the cognitive processing load of strategies and representations. This allowed us to predict when representations will increase the processing load and, if their use is still considered to be desirable, to make recommendations about reducing the load. In order to make the subtraction task as basic as possible we asked the children to respond to verbal and/or numerical representations of the operations and we assumed (cf. Fischbein, Deri, Nello & Marino, 1985) that the fundamental, intuitive model of subtraction was taking away or the procedure that Carpenter and Moser (1982) defined as separating. We assessed responses to such tasks by children from grade 1 to 3. The place value tasks required the children to read the numbers, represent them with materials, explain how many tens and ones there were and explain why the number was written as it was. Using structure mapping theory we determined the level of complexity of each task. We then determined the additional processing load, if any, of typical mathematical analogs and procedures used by children and teachers . In 1989 we undertook preliminary testing for counting, place value, subtraction, and some aspects of number relations, addition,fractions and money. This paper is a description of the procedure, analysis and results for study 1- subtraction and study 2-place value. STUDY 1 SUBTRACTION Sample This consisted of 29 children, 10 in Year 1, 10 in Year 2, and 9 in Year 3, and their teachers in a suburban Brisbane school in a low to medium socioeconomic area. The mid-year mean ages were respectively 6;1, 7;1 and 8;4. The children were selected to represent a range of mathematical performances at each year level. Method Each teacher was asked to identify in advance those aspects of a concept that she intended to teach in subsequent weeks and, in this case, which aspects of subtraction she intended the children to learn. The year 1 teacher had different objectives for individual children while the other two had more general class objectives. Each child in the sample was interviewed individually before and after instruction to determine knowledge of content, representations and procedures. All pre and post- test interviews with the children were videotaped. The videotaped material was subsequently viewed, transcribed by the interviewer and then categorized and coded in consultation with the authors. Following instruction the teachers were interviewed again and asked to describe the representations and procedures that they had used. Tests The pre and post test procedures are described below. Subtraction tasks included 29 examples which required subtraction of 0, 1, the same number (e.g. 2-2) from numbers to 10, other combinations within 10, subtraction of one digit from two digits with and without regrouping, and two digits from two digits with and without regrouping. Materials Materials were provided for children to use if they chose. These included Multibase Arithmetic Blocks (MAB) in base 10 (tens and units), sticks singly and in bundles, counters, a metre ruler, the symbols " = " and " - " and numerals for all problems written on cards, and paper and pencil for written calculations. Procedure The child and the interviewer discussed the materials and their use. The child was shown the first appropriate subtraction task, asked to read it, to use fingers, materials or paper and pencil to represent the task, find the answer, and talk (if possible) about the process. Each child started at an appropriate level, proceeded as far as possible through the tasks and stopped after difficulty with three tasks in succession. Results The responses of each child on the pre and post tests were transribed from the videotapes and then coded using the following categories; Sucess; The response to each item was coded as Y (Yes) if the child reached a final correct result in some way or N (No) if not. Representations; These were coded as A (Analogs) and then C (Counters), D (Fingers and/or toes) or M (MAB tens or bundles of ten sticks) to represent the analog that was finally used for solution. W (Written algorithm) was used if the child used pencil and paper as well as or without concrete representations. Strategies; These were coded as F (Forward counting), B (Backward counting), R (Recall or number facts), G (Regrouping, recomposition, or place value explanations) or I (Buggy, incorrect algorithms). The data were crosstabulated for combinations of these variables on the pre and post tests. The results show complex interactions of knowledge and use of representations, content and procedures in each child's response to the problems. For example a child's success and strategies often varied from one subtraction problem to another. We also noted that children can demonstrate knowledge in one context or aspect of mathematics but not in another. For example a child may not use knowledge of counting or place value in a subtraction task, despite success on a counting or place value test. Such a lack of connectedness of knowledge has been observed in other research. Lawler (1981) noted that his daughter knew how to do mental calculations with money and also how to do mental arithmetic with numbers by breaking them into multiples of ten. However she did not connect the two techniques. When asked to add 75 and 26 in terms of money she could do so with reference to familiar coins but when she was adding numbers she added tens and counted the remainders e.g. `seventy, ninety, ninety-five, ninety-six, ninety-seven...' Lawler referred to these separate skills as microworlds. They required the same skills but were activated by different conditions. Resnick and Osmanson (1984) found a similar lack of connectedness of knowledge. Children who could write numerals to represent numbers, use correct place- value notation, use concrete analogs such as MAB or coins to construct valid representations, and represent recompositions such as changing 34 from 3 tens and 4 units to 2 tens and 14 units, were not able to relate this understanding to addition and subtraction tasks. An attempt to train the children to map their concrete representations into arithmetical procedures was not very successful. Our contention is that such behaviour occurs because other aspects of the task, which are not well enough known, cause a load in excess of the number of units that the child can process in parallel. Children's representations, strategies and errors Representations that were used successfully and unsuccessfully included fingers (and toes!), counters, MAB 10, symbols and algorithms. The number line was not used by this sample. Knowledge of content (and concepts) that facilitated or impeded the process included forward and backward counting strings, subtraction as an operation, symbols, numeration and place value, and number facts. Procedural knowledge employed successfully and unsuccessfully included the subtraction process, either by taking away or counting back; the use of concrete representations for numbers and operations including regrouping of tens; and the written algorithm. The choice and successful or unsuccessful use of a representation depended on the interaction of the child's knowledge of the representation itself, of content and of appropriate procedures. For example, one child represented 16 - 8 with 1 MAB ten and 6 units for the minuend and 8 units for the subtrahend, did not know how to use them, decided to use 16 fingers and toes instead, tried to count back from 16, lost track of the backward count and obtained 10 as an answer. He did not know the MAB well enough to use it; he did not apply his knowledge of place value (as measured elsewhere); he had been taught how to count backwards as a procedure and therefore did not use the less demanding strategy (as we show below) of forward counting for taking away. All the above in combination apparently imposed an excessive processing and memory load and he did not succeed. Generally speaking processing and memory loads were increased when children had not learned some or all of the necessary content, procedures or representations well enough or when relations between these different kinds of knowledge were not well enough established to be used automatically. Subtractions that caused the most errors were; in year 1, 2-2 (80% on pretest and 50% on posttest); in years 2 and 3, 16-8 (20% pre, 10% post), 25-9 (30% pre, 10% post), 31-6 (37% pre, 21% post); and in year 3, 42-16 (33% pre, 55% post), 44-27 (55% pre, 55% post), and 52-29 (55% pre, 33% post). All of these examples except 2-2 require knowledge of regrouping and place value for efficient solution. Eight of the 10 Year 1 children responded incorrectly to 2-2 on the pretest for a variety of reasons. Five of the 10 were still incorrect on the post test despite the use of counters as analogs. The problem might be specific to this class. Our current research will help to answer that question. We believe that the task causes a processing load greater than other single digit subtractions because the child must keep in mind the following items of information. First he or she must represent the minuend with a set of 2, think of the operation of taking away the whole set, and then quantify the difference, which is difficult, because there is no apparent concrete analog for zero. The most typical incorrect answer for this task was 2. To arrive at this conclusion they put out 2 counters for the minuend and then another 2 for the subtrahend, took away the second lot of 2 and then counted the remaining 2. Children who gave this answer usually succeeded on subsequent tasks which indicated that they had some knowledge of subtraction as an operation. There is a mapping problem here. It consists of deciding which set to map into the difference, the empty set or the extra set of 2 that the child erroneously used to represent the subtrahend. The errors for the other subtractions, 16-8, 25-9, 31-6, 42-16 and like problems, were all similar. Even when children were counting and using analogs, such as counters or fingers and toes, they made errors. We discuss 16-8 in some detail as an example of the difficulties that they encountered. In order to subtract 8 from 16 efficiently a child should know all the relations shown in Figure 2 well enough to use them automatically; that is that 16 equals 10 and 6; that it can be represented as a bundle of 10 and 6 single sticks (or 1 MAB ten and 6 units); that "-" means remove (or difference between); that the bundle of 10 (or the MAB ten rod) must be exchanged for 10 units so 8 can be removed (or taken away); and that if a set of 8 sticks (or units) is left after the subtraction is performed then 8 is written as the answer. If the child does not know all these relations between numerals, operation symbols, quantities and concrete representations well enough then they impose additional loads and he or she may not be able to understand and perform the operation. INSERT FIGURE 2 (available from first author) The procedure that led to the most incorrect responses on the post tests was counting backwards / double counting. It caused difficulties whether it was used with concrete materials such as counters, fingers and toes, or MAB blocks. Higher success rates occured on the pre and post tests when children counted forward or did not count at all. The following protocol examples illustrate some of the problems encountered. e.g. 8 - 3 = 6 (using counters) "I counted backwards 3 places 8, 7, 6." e.g. 16 - 8 = 10 (with fingers and toes and no use of knowledge of place value) "15, 14, 13, 12, 11 (on right hand) 10 (on left hand), 10, I was right." We show in the flowchart in Figure 3 why counting backwards imposes a greater memory and processing load and makes the task more difficult. In counting forwards the processes in boxes B and E can be carried on serially. In counting backwards the processes in boxes B and C must be carried on concurrently. Therefore more information must be processed in parallel in the backward counting procedure. The difficulty occurs because, according to Halford's theory (forthcoming), processing load is a function of the amount of information that must be processed in parallel. Also box E in counting backwards is confusing because the process conflicts with the rule for quantification of sets, which assumes the last number in the count output as the answer. This rule does not hold when subtraction is performed by counting backwards; then it is necessary to count back one more item. The existence of two conflicting rules, one for counting to quantify sets, and another to quantify for subtraction, is an extra source of difficulty. The problem becomes even more difficult when the child does not know sufficient about place value to work with sets of ten and must count backwards across decades. Neuman (1987) found that the most obvious difference between pupils with mathematical difficulties and their classmates was that the former group of children could use practically no problem solving strategies other than double counting. Marton and Neuman (1990) observed that the double counting strategy `seemed to be an effective barrier to developing mental calculation and estimation skills'. We would agree, on the basis of our results, and attribute the difficulty to the extra processing load imposed. INSERT FIGURE 3 (available from first author) Teacher representations and strategies On the basis of interviews with teachers we observed that certain representations and procedures used by them, with the very best of intentions, apparently increased the memory and processing load or caused inefficient behaviour or incorrect responses. Some of these are discussed as examples. The year one children were expected to become very competent at adding two single digits to ten before considering any aspect of subtraction. As a result (it seems) in the pretest half of the year 1 sample automatically added when they were presented with two numbers. At the end of the year, on the post test 2 of the 10 children were still responding in that way. It appears that they had overlearned a rule that `when you see two numbers you add'. This earlier learning was interfering with their ability to understand the relation between addition and subtraction (which would require all their available processing capacity) and the operation of subtraction. Teaching the children to count backwards as a procedure for finding the difference between two numbers as opposed to forward counting of subtrahend and minuend has been discussed in some detail above. This procedure caused problems with both small and large subtrahends but more particularly with subtrahends greater than 3 or 4 and in particular with larger subtrahends (e.g.12-10, 16-8, 25-12, 31-6). Children who attempted more difficult subtraction tasks than these were generally not using backward counting. Some children at all grade levels could not read (verbalize) symbols such as " - " or 13 correctly. Such behaviour probably results from children carrying out mathematical calculations without talking much to each other or the teacher while they are doing them. Many children were not familiar enough with any one concrete analog and the relations between it and quantity, numeration and operations. In fact year 2 and 3 children evidenced better use of material such as MAB on the pre test than they did on the post test. In year 1 the preferred representations were counters, nothing or fingers (and toes) in that order. In years 2 and 3 children used a mix of materials on pre and post tests, with counters being the most popular even for numbers beyond 20, but preferred to use no materials and/or a written algorithm to achieve both correct and incorrect results. The teachers believed that constant use of one kind of material would cause the children to become bored. Intuitively this seems to be a reasonable assumption. However this apparently led to children representing , for example, the multiples of ten in numbers with mixed assortments of counters, sticks and units. One child in year 2 on the post test told us that she was making a `bungle' of ten with as many different things as she could. She had obviously become confused about what a bundle was and what constituted ten as a set or unit. Unless concrete representations are very well mapped into quantities and symbols then they will increase the processing load of a task. Hiebert (1988), discussing developing competence with written mathematical symbols, makes the same kind of assertion when he says; `The process involves building bridges between symbols and referents and crossing over them mentally many times.' In the case described above the child would not have sufficient knowledge of the connection between symbols and referents for tens and units to utilize the concept in two digit subtraction. The practice of teaching subtraction algorithms to 2 or 3 digits without regrouping before those that require regrouping caused some children to use "buggy" algorithms and they were unable to subtract when regrouping was required. In this sample 14% on the pretest and 17% on the post test used a buggy algorithm, where they attempted to take the small number from the larger even when it was in the subtrahend (e.g for 12-9 in year 2 and for 31-20, 42-16, 44-27 ,52-29 in year 3). Others treated each column as a separate single digit subtraction. They did not need to use concrete analogs for subtraction tasks with no regrouping and then were reluctant to use material such as MAB when it would have assisted with subtractions that required regrouping. The protocol of post test behaviour for one child, who had no trouble with the pretest, is an an example of the kind of confusion that occurred. `31 - 20, that's 9, that's 2 tens, you take 2 tens from 30, that leaves 1 ten and you take 1 from 10, that's 9.' It is becoming increasingly clear that many childrens who acquire procedural proficiency in mathematics often do not demonstrate corresponding conceptual knowledge (Cauley 1988; Hiebert, 1988). Cauley (1988), describing performance of 7-10 year olds with subtraction algorithms, maintained that incorrect rationales for judgements that borrowing increased or decreased the size of the minuend indicated a weakness in knowledge of place value; and that the majority of procedurally proficient students did not believe that the value of the minuend was conserved during borrowing. Responses also suggested a poor grasp of place value concepts. She suggested that an understanding of the part/whole structure of number may be necessary to fully understand place value and subtraction. She cited Kamii (1986) who argued that place value conventions will only become meaningful for children if they construct a system of tens on the system of ones through their own mental activity; and that traditional instruction often disregards children's own construction of the base ten system and hence they are forced to learn conventional rules by rote. We suggest below and elsewhere (Boulton-Lewis and Halford, in preparation) that knowledge of place value, based on the part/whole structure of numbers is an additive concept hence requires a binary operation and is therefore a system mapping task as is subtraction. Because both concepts therefore make the same demand on processing capacity, and simultaneously probably make an excessive load unless both are overlearned, we suggest that place value concepts should be taught separately and then applied to support calculation of the most difficult operations of which a child is capable. The present practice of allowing children to perform subtraction up to three digits without any real requirement to consider place value ia probably an intuitive recognition of the difficulties involved but leads often to a limited and erroneous concept of the use of algorithms. The post test results for subtraction indicate that some children learned from and used the teachers' strategies and representations effectively (although not on all tasks). There were fewer correct responses on the post test than the pretest for 16-10, 25-15, 31-6, 42-16, and no improvement for 12-9 and 12-10. Some children had learned algorithmic procedures without understanding them; some had learned buggy algorithms which `worked' sometimes; some were using lower level strategies than they had used previously, whilst other children still continued to use their prior representations and strategies correctly or incorrectly. Persistence in using their own strategies and representations probably occurs, either because the child does not know the relations between representations, quantities, symbols, and operations well enough to use them and therefore the teacher's representation makes an excessive load on processing capacity or the child does not understand the the algorithmic process introduced by the teacher because it does not fit apparently with the process that he or she is currently using. One child described the teacher's procedure as "easy" and his own as "hard", because the teacher had said the child's procedure was hard, but he continued to use the hard one correctly because he thought he knew what he was doing. He either did not understand the teacher's procedure well enough to use it or had not practised it sufficiently to feel confident with it. The challenge lies now in further analysis of particular tasks and the responses of selected children to determine exactly why some children continue to use or invent their own strategies. STUDY 2 PLACE VALUE On the basis of structure mapping theory we identified two levels of cognizing and representing place value in numbers greater than ten. At the first level the number is cognized and represented as a relational mapping. A child who has knowledge of a number at this makes a one-to-one correspondence between single units and each counting number in the set that the number represents. The child would not be able to respond correctly to questions about how many tens and how many units there were in the number. Such behaviour indicates that the child has global knowledge of a number (e.g 16) as a whole only or a collection of ones, rather than as of a whole made up of parts consisting of tens and units. Figure 4A shows examples of relational mappings between numbers and representations. At the second level the number is represented and explained as a system mapping. A child who has knowledge of a number at the system mapping level should represent it correctly with sets of ten and units and be able to answer questions about how many tens and units there are in the number. Understanding at this level also implies that the child knows that a number, such as 16, can be thought of both as 16 units or as a set of 10 plus 6 units. Figure 4B is an example of the mapping between the number and a representation at this level. We believe that in order to understand the part whole relationship inherent in the place value representations of numbers the child must be aware of the additive property at least, if not the multiplicative property of the regrouped tens, hundreds and so on in a number greater than ten. INSERT FIGURE 4A & 4B (available from first author) Children in the early years of school have limited experience of numbers. Hence it is likely that children's responses will vary from one number to another and that they will not always choose to represent a number using the most efficient set of materials. The latter assumption is supported by the categories for responses used by Ross (1986) and Miura et al. (1988), when discussing research in the development of children's concepts of place value numeration. They distinguished three categories as follows; `(a) one-to-one collection - the construction used only unit blocks (e.g. 24 unit blocks for 24); (b) canonical base ten representation - the construction used the correct number of ten blocks and unit blocks, with no more than nine units in the ones position (e.g., 2 ten blocks and 4 unit blocks for 24); and (c) noncanonical base 10 representation - the construction used some other correct number of ten blocks and unit blocks that allowed more than 9 units in the ones position (e.g., 1 ten block and 14 unit blocks for 24).' It was also expected that some children in the age range of this sample would experience difficulty in reading numbers from written numerals and hence representing them with an analog. Numbers in English present problems that do not occur in some other languages (e.g.those rooted in ancient Chinese) because the spoken numerals lack reference to the tens and ones that are contained in them (e.g. eleven, twelve, thirteen, etc. and twenty, thirty ... one hundred). As well numbers such as 14 and 40 are phonetically similar and the order of pronouncing the unit part of the number does not correspond with left/right placement of the place of the digits in numbers such as 14. As a result fourteen is often confused with forty and the numerals for 14 read as forty by young children. Kamii (1986) presented an explanation for the difficulty of place value and its implications for teaching in the primary grades. She cited results of recent research in the US, Canada, and Switzerland which indicated that most children in the 1st and 2nd grades do not understand that the 1 in 12, for example, means 10 and that place value is not understood in 3rd and 4th grades by a large number of children. She described three behaviours that she maintained do not in reality deal with place value. These are firstly putting out the correct number of single units for a number such as 16 or writing the correct numeral for a set of 16 objects. That is because these numerals can designate whole numerical quantities and do not require knowledge of the part whole relationship of the 10 and 6 in 16. Secondly research has shown that even first graders can tell that 61 is greater than 16. They can know this because 61 comes after 16 in the counting or written sequence of numbers and do not need to know that the 6 in 61 represents six tens. Thirdly many first graders can remember that 10 plus 10 is 20 and that 20 and 20 are 40. It is possible for them to operate on whole tens and treat 20 for example as 2 bundles, sets, ten rods or other representations of ten as a unit set. On the basis of her research she maintained, as we do, that the construction of the system of tens on the basis of knowledge of ones is a problem of part whole relationships. As proposed earlier this requires knowledge at the system mapping level. Sample This was the same as for study 1. Method This was the same as study 1. The teachers were asked to identify those aspects of place value that they intended to teach in subsequent weeks. For place value children in year 3 were only interviewed on a post test. Tests The pre and post tests for place value were as follows; Children were tested for knowledge of place value in 8, 10, 12, 13, 16, 18, 20, 30, 40, 50, 80, 33, 27, 36, 41, 56, 92, 108, 150. These numbers were chosen because; 8 could be used as a training item as well as a test of whether the child knew that there were no sets of ten in this number; 10 because it represents a single set of ten or 10 units; 12 to 18 because these are combinations of tens and units and cause difficulties for children because of the way they are spoken in the English language; 20 to 80 because they are multiples of ten with no units and because they are often confused with the teens numbers and vice versa; 41 to 92 because they are combinations of tens and units of increasing size; and 108 and 150 to determine whether any of the children could represent and explain numbers beyond 100. In fact no child in years 1 and 2 attempted these last two numbers. Materials A range of materials was provided for children to use if they chose. These included Multibase Arithmetic Blocks (MAB) in base 10 (tens and units for years 1 and 2 plus some hundred blocks for year 3), matchsticks (or paddlepop sticks) singly and in bundles of ten, counters, a metre ruler, and the numbers, as above written on cards. Procedure The child and the interviewer discussed the materials. The interviewer briefly explained any unfamiliar materials in case the child wanted to use them. The child was shown the first number and asked to read it; then to put materials underneath the number to show how much it was; asked how many tens and how many ones there were in the number; and asked to explain why the number was written in that way. Each child started at a number appropriate to his or her year level, proceeded through the numbers and stopped after difficulty with three tasks in succession. Results The responses of each child for the pre and post tests were transcribed from the videotapes and then coded using four categories. The categories were as follows; Success; The response was coded as Y (Yes) if the child represented the number in any correct way with materials, either canonical or non-canonical, or N (No)if not. Knowledge of place value; This was coded as Z (for no knowledge) if the child could not correctly answer questions about how many tens and units in a number or explain the value of the numerals on the basis of their place in the number. G was the code for global knowledge of a number when the child gave explanations that indicated that e.g 16 was the name for a collection of units only. P was used to code responses that clearly indicated knowledge of a number as consisting of sets of ten and units and hence place value. Errors; These were coded as A (for automatic adding of numbers) when a child responded for example to a number such as `13' with the answer `4'. R was used for reversal or confusion when a child read a number such as 13 as 30. D was the code for a response to a number such as 21 where the child explained and represented the 2 and the 1 in the number as digits symbolizing sets of units. Representations; Five codes were chosen for the different representations that children used. O was used when the child made a one-to-one matching of unit objects to the counting sequence for the number; S when the digits in a number such as 21 were treated as sequential units and represented by 2 units and 1 unit; I when the representation was inconsistent and was made up of a mixture of different objects and/or random use of sets of tens and units (e.g. 2 MAB tens and 13 sticks for 33); F for fingers (and toes); and T for correct use of bundles of ten or Multibase material. Teacher objectives, strategies and representations In year 1 the teacher expected the children to learn about place value (or tens and units in numbers) to about 30. In year 2 they were expected to have knowledge of place value to about 99. Both teachers used a range of materials including single objects such as counters, Unifix cubes, MAB units, straws and sticks. Unifix cubes were also joined together to make rods of ten and sticks were made into bundles of ten. MAB ten rods were used occasionally. Children were asked to make numbers with materials on charts of various kinds to represent given numerals or else to put out materials and then to write the correct numerals. They also counted in tens starting at numbers such as 10 or 2, e.g. 2, 12, 22...92. In year 3 similar procedures were used for numbers into the hundreds and more use was made of MAB materials. Both year 1 and year 2 teachers said that they used a range of materials so that the children would learn that numbers could be represented in a variety of ways and so that the children would not become bored. Error patterns The error pattern for reading and interpreting digits in numbers was interesting. On the pretest the greatest number of errors occurred for 12, 13, 16, and 18 in both years 1 and 2. There were still one or two children who had trouble with reversals or did not know the place value of digits in 12, 13, and 16 on the posttest. Those children who attempted numbers from 20 to 92 on the posttest did not make any of the errors that they had made on the pretest. Pre and post test performance The changes in knowledge of place value, for selected items, from pretest to posttest are shown in Table 1. The majority of the years 1 and 2 children gave place value explanations for these items on the pre and post tests. The figures in the Table show that there was hardly any change in these explanations from pre to post test. For those children who did not understand place value teaching did not make much difference. All the year 3 children gave place value explanations on the post test although they did not all use bundles or MAB 10 to represent tens in all numbers. Of the year 1 and 2 sample 4 children in year 1 and 9 in year 2 gave correct place value explanations on 75% or more of the items that they attempted. Of the children in year 2 and 3 who gave place value explanations for 75% or more of the place value items only 38% could use this knowledge on the majority of appropriate subtraction tasks that they attempted. INSERT TABLE 1 Children's use of representations The most interesting and unexpected result that occurred was in the use of representations. On 50% or more of the items that were attempted by children in year 1 and year 2, 50% only of the sample changed from using less to more efficient materials to represent tens in numbers (viz. from single units or mixed materials to bundles of ten sticks or MAB ten rods). Twenty-five percent of the sample did not improve and, most curious of all, 25% actually changed from using tens to using single units or a mixture of materials. Such behaviour is illustrated by the following protocol for one child. Pretest for 20 I(Interviewer) `What's this number?' C(Child) `Oh twenty, that's easy.Two tens' Placed two MAB tens under 2. I `How many tens do you have?' C `Two tens and no ones.' I `Why do we write the 2 here and the 0 there?' C `Ooh because if you put that number there (0 before 2) it would make 2.' Posttest for 20 C `This is going to be fun, I'll do all different things. Now ten' (one MAB ten). He then put out 6 counters, 2 sticks, and 1 MAB unit. I `What do you have there? C `20... 1,2,3,4,5,6,7,8,9 ( counted all single materials), 9.' I `Nine, so nine and ten make twenty? Are you certain? C `Yes' I `How many tens?' C `One' I `How many ones?' C `Nine' I `What does 20 tell you' C `Twenty' The kind of confusion described above is apparently only temporary in this school. By late in year 3 all the children were using units of ten for tens in numbers up to 50. Two of them used mixed materials for numbers beyond 50 and two more for numbers beyond 100. Table 2 is a summary of the prerequisite knowledge, at the relational and system mapping level, for cognition of place value in numbers greater than 10. The children described above, who changed from using more to less efficient materials have learned that numbers can be represented in a variety ways using ten and unit materials. In the process however they have not made strong mappings between digits in the ten place of a number , the spoken number name, and materials that most effectively represent ten. Consequently they are increasing the load of processing information about such numbers by the need to count or think of all or most of the number in units. We believe that the change from more to less efficient use of materials to represent place value in numbers occurred because teachers in years 1 and 2 encouraged children to use a mixture of materials. INSERT TABLE 2 Representation of specific numbers Generally there was more success on pre and posttests in interpreting and representing numbers such as 20, 30, 40 and even 23 and 27 than 12, 13, 16, 18. This was foreshadowed above because the spoken numbers from eleven to nineteen in English do not correspond with the order in which the numerals are written. The difficulty can be explained on the basis of the flowchart shown in Figure 5, which compares the items of information that the child must take into account to read, understand and represent 13 as compared with 30. There was no use of fingers as representations of numbers except for 10 and 12 on the pretest. INSERT FIGURE 5 (available from first author) Language use Some interesting and incorrect language was used by a few children in describing tens and units. Units or ones were described by one child as numbers `in real time' (perhaps he meant they were real as opposed to sets of ten) and by another child `as casual numbers' (perhaps he meant that they were loose as opposed to bundles or ten rods). Whatever they meant by the words that they used, we believe that by using incorrect terms the children are increasing the difficulty of understanding that numbers are made up of tens and units. A more serious error was made by two year 2 children on pre and posttests. They were describing sets of ten as `bungles'. When queried as to what she meant by `a bungle' one of the children explained that a `bungle of ten' was any collection of ten objects (not necessarily joined in any way), such as counters, MAB units, sticks etc. On the pretest this child had actually used bundles of ten sticks to represent tens despite the fact that she described them as `bungles'. Between the two tests she had become confused about the concept of a bundle as being ten objects joined together in some way to represent a unit of ten. Understanding place value requires knowledge of relational mappings as shown in Table 2 and part whole relationships at a system mapping level. Children who respond in a global or one- to-one way have knowledge of some but not all of the requisite mappings. Therefore, if children are to understand place value as a concept in its own right, and more importantly use that knowledge in operations, teachers should try to ensure that they know all the necessary mappings well enough for them not make any additional processing load. The change from use of tens materials to use of mixed materials that we observed with the year 1 and year 2 sample has implications for teacher strategies and representations. The first is that if the objective is to have children learn about place value then they should practice exchanging unit materials for sets of ten once they have made global one-to-one responses. The second is that if children are to use concrete representations without increasing the processing load they must know the material so well that they can use it automtically. It is probably important for motivational reasons to allow children to use a wide range of materials to represent numbers and it also helps them to construct mental models of numbers which map into a variety of different representations. However they should also be encouraged to use a particular representation of sets of tens and units regularly, on the basis that it is more efficient and so that that material becomes well mapped into the place value of numbers. If such mappings become automatic then children should be able to use them more effectively in algorithms for addition and subtraction and so on. CONCLUSION This research provides preliminary information about the effect of the processing load of representations and procedures on children's learning of some mathematical concepts. Some strategies and representations that children themselves choose to represent mathematical tasks, and some that they are taught, impose unnecessary loads. For example many young children use fingers (and toes) or counters, and counting, to add and subtract numbers beyond the first decade. As we have shown, the need to monitor cognitively fingers, double counting and the operation to be performed, imposes a greater load on capacity than operating on numbers using knowledge of part-whole relationships or even forward counting. It is difficult to know whether to allow children to use such `natural', demanding, inefficient strategies until they develop a more effective strategy or to intervene and try to lead them to learn and adopt less demanding strategies earlier. Some of the representations and procedures that teachers introduce, with the intention of facilitating learning, actually make the task more difficult. For example where teachers use concrete analogs to embody such operations as the algorithmic procedure for subtraction, then the step by step strategies procedures of the algorithm, in conjunction with concrete representations, impose loads that are too great for most young children to process in parallel unless, as we have discussed above, the child knows all the component parts of the task very well. The demands of such procedures are considered in some detail in Halford and Boulton-Lewis (in press). Because of the stripped down fundamental nature of the subtraction tasks used in this study we have not considered the loads made by other representations and procedures that teachers said they used. For example unfamiliar words and situations in word problems, and materials that cause distractions (such as a pictorial representation of a fish eating worms to illustrate subtraction), can all increase the amount of information that children must process to reach a solution. Research describing the effects of teaching inappropriate or novel substitute strategies and methods to low and high ability students is summarized by Clarke, Aster and Hession (1987). They claim that some teaching of this kind can kill learning. The results of our research lead us to suggest that whether or not teacher-determined strategies and representations facilitate or intefere with learning depends on their goodness of fit with the child's existing procedures, how much they increase the processing load for each individual child, how motivated the child is, and how well they are taught. This implies that teachers should, as far as possible, attempt to take what Ramsden (1988) and colleagues describe as a phenomenographic approach to teaching. That is they should attempt to find out as much as possible about each child's declarative and procedural knowledge and help the child to refine and build on that knowledge in such a way as to lead the child to approximate the thinking of an expert in the relevant discipline area. Resnick, Bill, and Lesgold (in press) have described a successful program where children `find' their own mathematical problems and bring them to school to solve cooperatively. The grade 1 children in this program, who were not restricted to a limited range of numbers (e.g. within 10), performed better on standardized tests than a similar group of children taught in a more conventional manner. Perhaps the motivation provided by ownership of real problems allowed children to cope with the additional load of using large numbers and unsophisticated strategies. Tests of short term memory such as digit and word span, and the data from capacity measures indicate that there are limits to the amount of information that all young children can process in reaching a decision or solving a problem. These limits should be considered in conjunction with what is known about the minimum level of cognition required to cognize a particular aspect of the mathematics curriculum. We should also take into account the demand that representations and strategies will make on processing load. And finally we must try to ascertain what prior knowledge a particular child has of requisite concepts, representations and procedures. This paper is a description of two aspects of our current research. We argue on the basis of the results that concrete teaching aids are useful only if children clearly recognize the correspondence between the structure of the material and the structure of the concept (or in terms of analogy theory, if they can map one structure into another). Because seeing this correspondence imposes a processing load success requires that all other processing loads be minimized, to make as much capacity as possible available. In practical terms this means the child must know the representations, concepts and procedures thoroughly, so that recognition of the relevant relations is automatic. Teachers can, with the very best intentions, depending on the strategies and representations that they use, increase the processing load for children who are trying to cognize mathematical concepts and this will intefere with learning rather than facilitate it. It can occur particularly if children are encouraged to use materials, that they do not know well, as concrete analogs for mathematical concepts and operations. The second is that if teachers do not succeed in matching their representations and strategies as closely as possible to those that children are using then temporary confusion can occur which will also interfere, at least in the short term, with learning. We have identified and described the processing loads imposed by some of the representations and procedures that are used in teaching subtraction and place value to young children. If teachers believe that such representations and procedures are necessary or beneficial then they should use them in a way that will minimize memory and processing loads and so produce the expected benefits. REFERENCES Boulton-Lewis, G.M. and Halford, G.S. (in preparation) The processing loads of young children's and teachers' representations of place value and implications for teaching. Brown, J.S. and VanLehn, K. (1982). Towards a generative theory of "bugs". In T.P. Carpenter, J.M. Moser and T.A. Romberg (Eds.), Addition and Subtraction: A Cognitive Perspective. Hillsdale, N.J: Erlbaum. Carey, S. (1985). Conceptual Change in Childhood. Cambridge, MA: MIT Press. Carpenter, T.P. and Moser, J.M. (1982). The Development of addition and subtraction problem solving skills. In T.P. Carpenter, J.M. Moser and T.A. Romberg (Eds.) Addition and Subtraction: A Cognitive Perspective Hillsdale, N.J.: Erlbaum. Cauley, M.D. (1988) Construction of logical knowledge: study of borrowing in subtraction. Journal of Educational Psychology, 80, (2), 202-205. Chi, M.T.H. and Ceci, S.J. Content knowledge: its role, representation and restructuring in memory development. Advances in Child Development and Behaviour, 20, 91-142. Clarke, R.E., Aster, D. and Hession, M. A. (1987). When teaching kills learning: types of mathemathanic effects. Paper presented at the Annual Meeting of the American Educational Research Association, Washington DC, April 20-24. Gentner, D., (1983). Structure-mapping: a theoretical framework for analogy. Cognitive Science 7, 155-170. Gentner, D. and Toupin, C. (1986). Systematicity and surface similarity in the development of analogy. Cognitive Science 10, 277-300. Ginsburg, H.P. (1977). Children's Arithmetic: The Learning Process. New York: D. Van Nostrand. Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S., (1985) The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16, 3-17. Halford, G.S. (forthcoming). Children's Understanding: The Development of Mental Models. Hillsdale, N.J.: Erlbaum. Halford, G.S. (1988). A structure-mapping approach to cognitive development. In A. Demetriou (Ed.) The Neo-Piagetian Theories of Cognitive Development: Toward an Integration. Amsterdam, Netherlands: Elsevier Science Publishers for North-Holland. Halford, G.S. and Leitch, E.(1989) Processing load constraints: a structure-mapping approach. In M.A. Luszcz and T. Nettlebeck (Eds). Psychological Development: Perspectives Across the Lifespan. Amsterdam: North Holland. Halford, G.S., Maybery,M.T. and Bain J.D. (1986) Capacity limitations in children's reasoning: A dual task approach. Child Development, 57, 616 - 627. Halford, G.S. and Boulton-Lewis, G.M. (in press) Value and limitations of analogs in teaching mathematics. In A. Demetriou, A. Efkliades and M. Shayer (Eds.) Modern Theories of Cognitive Development Go To School. London: Routledge and Kegan Paul. Hiebert,J. (1988) A theory of developing competence with written mathematical symbols. Educational Studies in Mathematics, 19, 333-355. Janvier, C. (1987). Problems of Representation in the Teaching and Learning of Mathematics. Hillsdale, N.J.: Erlbaum. Kamii, C. (1986) Place value: an explanation of its difficulty and educational implications for the primary grades. Journal of Research in Childhood Education, 1, 75-86. Lawler, R.W. (1981). The progressive construction of mind. Cognitive Science, 5,1-30. Lesh, R. and Landau, M. (Eds.) (1983). The Acquisition of Mathematics Concepts and Processes. New York: Academic Press. Marton, F. & Neuman, D. (1990) Constructivism, phenomenology, and the origin of arithmetic skills. In L. P. Steffe & T. Wood (Eds.) Transforming Children's Mathematics Education (pp 62-75). Hillsdale, NJ: Erlbaum. Miura, T.I., Chungsoon, C.K., Chi-Mei, C. and Yukari, O. (1988). Effects of language characteristics on children's cognitive representation of number: cross-national comparisons. Child Development, 59, 1445 - 1450. Neuman, D. (1987) The origin of arithmetic skills: A phenomenographic approach. Goteborg: Acta Universitas Gothoburgensis. Cited in Marton, F.and Neuman, D. (1990). Ramsden, P. (Ed.) (1988) Improving Learning: New Perspectives. London: Kogan Page. Resnick, L.B., Bill, V. and Lesgold, S. (in press) Developing thinking abilities in arithmetic class. In A. Demetriou, A. Efkliades and M. Shayer (Eds.) Modern Theories of Cognitive Development Go To School. London: Routledge and Kegan Paul. Resnick, L.B. and Osmanson S.F. (1984) Learning to understand arithmetic. In R. Glasser (Ed.), Advances in Instructional Psychology (Vol. 3). Hillsdale, N.J.: Erlbaum. Ross, S.H. (1986, April). The development of children's place- value numeration concepts in grades two through five. Paper presented at the annual meetings of the American Educational Research Association, San Francisco. Siegler, R.S. (1987). The perils of averaging data over strategies. Journal of Experimental Psychology: General, 3, 250-64. Siegler, R.S. and Robinson, M. (1982). The development of numerical understandings. Advances in Child Development and Behavior, 16, 241-312. Sowell, E. J. (1989) Effects of manipulative materials in mathematics instruction. Journal for research in Mathematics Education , 20(5), 498-505. TABLE 1 Knowledge Of Place Value For Selected Items Pretest Postest Item Z* G** P*** Z G P 10 2 2 15 1 4 15 12 4 4 12 3 3 14 13 4 3 13 2 4 14 20 6 14 2 2 16 30 2 2 15 1 1 17 23 2 16 1 1 17 27 1 2 15 1 1 17 *Z Explanations that indicated no, or very little, knowledge of place value **G Global or one-to-one knowledge of numbers, e.g. 16 explained as sixteen ones or units ***P Part whole or place value explanation of numbers TABLE 2 Requisite relational and system mapping knowledge to cognize place value Relational mappings for one-to-one/global response 1. relation between number names in counting string and objects in order 2. counting rule dependent on relation between quantity in set and last number name said 3. relation between written and spoken number names Relational mappings for place value/system mapping response 4. all preceding mappings 5. relation between unit of ten (e.g. MAB, bundles of ten) and place value of digit System mapping for place value response 6. relational between e.g. 10+2 in 12, i.e. that 12 is ten plus two or 12 units Funded in 1989 by a grant from Brisbane College of Advanced Education (now Queensland University of Technology). Continued in 1990 by a grant from the Australian Research Council. Undertaken with the permission of the Queensland Education Department. .By this we mean an incorrect algorithm that the child has actually learned through practice (cf. Brown and VanLehn, 1982).