on learning problems in mathematics

ON LEARNING PROBLEMS IN MATHEMATICS SUMMER 1988

Volume 10, Number 3 Editorial Board

Mahesh Sharma, Chairman Center for Teaching/Learning Mathematics George W. Bright University of Houston Grace M. Burton University of North Carolina at Wilmington John F Cawley University of New Orleans Robert Underhill Virginia Polytechnic Institute and State University Nancy Wilson Prince George County Public Schools

Staff Mahesh Sharma - Chief Editor George W. Bright - Editor Virginia Usnick - Associate Editor lillian Travaglini - Executive Assistant

Publisher Center for Teaching/Learning of Mathematics

Focus on Learning Problems in Mathematics Summer Edition 1988, Volume 10: Number 3 © Center for TeachinglLearning of Mathematics

Children's Initial Understandings of Ten Paul Cobb

Grayson Wheatley

Purdue Uni.,ersity The research reported ip this paper was supported by the National Science Foundation under grant #MDR 847-0400. All opinions and recommendations expressed are, of course, solely, those of the authors. The initial goal of the study reported in this paper was to assess beginning second graders' concepts of ten as part of a curriculum development project. An analysis of the development of this concept proposed by Steffe (Steffe, 1983; Steffe, Cobb & von Glasersfeld in press) was used for this purpose. The levels identified by Steffe are related to qualitative changes in children's concepts of addition and subtraction and can be accounted for in terms of the construction of increasingly powerful conceptual operations. When the children were interviewed, it became apparent that the majority of them had constructed meanings for "ten" that did not correspond to those identified by Steffe. The children seemed to operate in two separate contexts: (a) pragmatic, relational problem solving and (b) academic, codified school arithmetic. In many respects, they did not construct ten as a structure composed of ones when they operated in the school context - ten was, for them, one thing which was not itself composed of units. Consequently, the meanings the children gave to " ten" and to "one" were unrelated to each other - the tens were not made up of ones and the ones could not be constructed by unpacking a ten for the simple reason that there was no structure to unpack. The differences between the subjects of the current study and those of Steffe's study appear to reflect differences in their prior instructional experiences. Steffe's subjects participated in a two-year teaching experiment and received instruction that encouraged them to reflect on and reorganize their own problem solVing activity. In contrast, our subjects had received typical textbook instruction in which rules were taught for assigning value to

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digits based on their position (e.g., one place, tens place). Whereas Steffe's work with children reflected the Piagetlan notion that the construction of a sophisticated concept of ten Involves a series of reflective abstractions, textbook instruction is based on the assumption that it involves empirical abstraction from groups of objects. For Steffe we see what we understand whereas, for textbook authors, we apparently understand what we see - tens are out there in Instructional representations just waiting to be perceived. In the following sections, we discuss the rationale for conceptual analyses, outline Steffe's analysis, and then report the study that is the focus of this paper. Finally, we discuss instructional implications by conSidering both the contexts Within which children do school arithmetic and the basis that their concepts of ten provide for further learning. Rationale for Conceptual Analyses

A fundamental assumption of conceptual analyses is that children's actions are always rational given their understandings. We have all seen children who, from our adult perspective, do some strange things as they attempt to solve mathematics tasks, One reaction is to wonder how the children could be so stupid or to ask what is wrong with them. This reaction, in our view, reflects the limitations of the observer rather than the child. It reflects the inability of the adult to put aside his or her relatively sophisticated understandings of mathematics and imagine what things might be like from the child's perspective. An alternative approach is to readily admit the inadequacy of adult mathematics for understanding children and for planning instruction. From this perspective, children's apparently strange actions are viewed as problems for the observer to solve. The trick is to develop an understanding of children's mathematics so that their actions can be seen as rational and sensible. The focus of a conceptual analysis is therefore on children's meanings - on how they Interpret and attempt to solve mathematical tasks. This type of analysis differs from a logical task analysis in that it acknowledges that much of children's knowledge is not, for them, an object of reflection and consequently does not correspond to anything the adult can see "out there." ConSider, for example, the case of a child who attempted to solve a miSSing addend task in which six felt squares were viSible, some more were hidden by a cloth, and the child was told that there were eight squares in all. The child made five attempts to solve the task, each time counting the six visible squares starting at "one." These repeated counts of the same six squares certainly seem strange. One approach is to infer that the child is inadequate in some way perhaps because he has a poor memory and repeatedly forgets that there are six squares. Alternatively, we could ask why does the child have to

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count from "one" each time? Why would a sensible person do that? One plausible answer is that the child cannot create numbers such as "six" in a purely conceptual manner. The child actually has to count in order to make the number six and it ceases to exist for the child once the counting episode is completed. Consequently, the child has to count from one" again when he makes another attempt to solve the task. In making this Inference, we have to put aside the adult's taken-tor-granted notion that numbers exist in the world just waiting to be apprehended. In doing so, we strive to imagine the world of the child's mathematics in which numbers are transitory entities that have to be made and remade by actually counting and do not exist Independent to the activity of counting. The value of a conceptual analysis becomes apparent as soon as we use its results as a lens through which to view children's mathematical activity. We find that we understand children better and that much that had previously seemed strange or bizarre now makes sense. We also begin to develop rationally grounded expectations about what they might do in particular situations. For example, if our inferences about the child discussed above are viable, we would be surprised if he could use 4 + 4 = 8 to solve 4 + 5 = _ _ . This is because the strategy of reasoning that five is more than four, so the sum must be one more than eight involves the construction of relationships between numbers. But, there are no numbers out there in the child's mathematical world for him to relate to each other. It is not that the child fails to notice the relationship between the two number sentences. Rather, such relationships do not exist in his world. Equipped with an understanding of children's mathematics, we can also begin to appreciate why textbook based instruction frequ~ntly gives rise to difficulties for both teachers and children. We will draw on the analysis of children's concepts of ten presented in the following pages to argue that textbooks lead teachers to systematically misteach place value. We will further argu~ that as a consequence of limitations of the textbook approach, the profound difficulties that children experience when they are introduced to twodigit addition and subtraction with regrouping and entirely predictable. The source of these difficulties can be squarely located on the failure of the textbook approach to take account of children's mathematics. An analysis of place value that seems reasonable to an adult in terms of his or her own relatively sophisticated understanding of place value is no substitute for a conceptual analysis of children's mathematics.

An Analysis of the Concept of Ten Steffe's analysis extends a model of children's counting types (Steffe et aI. , 1983) and is compatible with an analYSis of concepts of addition and subtraction that are indicated by the use of thinking

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strategies and mature counting by one method to find sums and differences (Cobb, 1983; Steffe et aL, in press). The analysis is subtle in that it requires the reader to suspend his or her'own knowledge of place value numeration. In effect, the reader is asked to make problematic his or her own ability to operate the units of ten and of one and thus reconstruct an arithmetical reality left long ago. It is by questioning the obvious that we gradually become aware of aspects of our own arithmetical knowledge that we typically take for granted. Although a greater effort of decentration is required than to understand other analyses (e.g., Bednarz & Janvier, 1982; Resnick, 1982, 1983; Ross, 1986) the effort is worthwhile in that much of what we take for granted is precisely what children have to construct if they are to develop an adequate understanding of the positional numeration system. As we will see later, we overlook what is obvious in terms of adult mathematics at our peril when we develop instructional materials. For the purposes of the current study, it suffices to consider the concepts of ten only of children who have attained the most advanced stage of counting by one in Steffe et aL's (1983) model of children's counting types, abstract counting. Steffe et aL (in press) identified three increasingly sophisticated concepts of ten that children construct once they have reached this stage in counting by one. These are called ten as a numerical composite, ten as an abstract composite unit, and ten as an iterable unit. It is only with the last of these units that the child increments (or decrements) by ten when counting by ten. Prior to the construction of ten as an iterable unit, the child struggles to resolve difficulties that are beyond the comprehension of adult mathematics.

Ten as a Numerical Composite Ten as a numerical composite is structurally no different from the meaning given to other number words by children when they first attain the abstract stage. In all cases, the children's focus is on the constituent elements of the composite - the individual ones that make it up - rather than on the composite itself as a single entity. Consider, for example, the way that Jason, a second grader, solved the following task. (In the protocol, "strips" refers to eight-inch paper strips on which ten squares were glued.) T: (Places twenty squares under a cloth and three strips by the cloth) There are twenty little squares under the cloth. How many squares are there altogether? J: (After a pause, Jason finally makes two sweeping gestures over the cloth with his index finger. He then sequentially puts up ten fingers) 31 - 32 - 33 - 34 - 35 - 36 - 37 - 38 - 39 - 40 (he again sequentially puts up ten fingers), 41 - 42 - 43 - 44 - 45 46 - 47 - 48 - 49 - 50. (Steffe et aL, in press, p. 209) -4-

Jason re-presented the 20 hidden squares as two strips before he counted as indicated by his two sweeping gestures. However, even though he knew the number word sequences " 10, 20, 30 . . . . 'he had to count the elements of the strips he re-presented by one. It was because he could not take a numerical composite of ten (i.e., a re-presented strip) as a Single, discrete thing itself composed of ones that he could not count the two composites "40 50." From Jason's perspective, the only things available to count were the squares on the strips that he re-presented. Conversely, if children at this level take visible or re-presented material such as a strip of ten squares as one thing, then the slrip loses it composite quality - it is a one rather than a ten even though the children may call it " ten. " T: Shut your eyes (places four strips under a cloth and three visible strips by the cloth). Open them. There are seventy little squares altogether. How many strips are under the cloth? J: Three (looks at the visible strips), 4 - 5 - 6 - 7 (looking at the successive places on the cloth), four. (Steffe et. at. , in press, p.209) Here Jason counted the strips as ones. Chjldren at this level may sometimes count aVallable materials such as strips " 10 20, 30, .. . " This is nothing more than a modified form of counting by ones. The child has merely learned to use a new number word sequence to count things by ones. In summary children for whom ten is a numerical composite are yet to construct ten as a unit of any kind. There are either ten ones or a singly entity sometimes call "ten" but not both Simultaneously. We note in passing that textbook authors assume that children can "see" both ten ones and one ten Simultaneously when they look at colorfully depicted bundles of ten sticks (or whatever). A significant proportion of second graders. perhaps the majority, are unable to do this even though it seems self-evident to the adult. Ten as an Abstract Composite Unit

The first major advance made by Steffe et. a\.'s (in press) subjects was to take numerical composites as single entities while simultaneously maintaining their ten ness. The first true unit of ten is called ten as an abstract composite unit. John, another second grader, was inferred to have constructed this type of unit of ten on the basis of his solution to a task in which four strips of ten squares and tWo individual squares were visible and he was told that 25 squares and two individual squares were visible and he was told that 25 squares were hidden, To find how many in all, he pointed first to each visible strip and then to each visible square as he counted, "35, 45 55, 65 - 66, 67." The crudal features to this solution are that John started counting in the middle of a decade (i.e" he counted-on from "25" rather

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than, say, "20") and that he coordinated counting by tens and by ones in a single episode. Together, these features support the inference that each counting act by ten was a curtailment of performing ten counting acts by one - each counting act by ten signified a unit composed of ten ones. Although children do make a significant advance when they construct ten as an abstract composite unit, their concepts of ten still have important limitations. First, the children do not increment (or decrement) by ten when they count by ten. In other words, each counting by ten act does not mean ten more (or ten less). Instead, each act signifies another ten that is experienced as being next to the ten just counted. In other words, the meaning of counting by ten to them Is one ten, another ten, . . . rather than one ten, another ten makes 29, another makes 30 ... Second, and perhaps more importantly, it is essential that material of some kind (hidden or otherwise) be available that the children can take as abstract composite units of ten. In the tasks used by Steffe et. al. (in press), fot' example, it was crucial that the children believe that strips were hidden. The most sophisticated type of solution produced by children at this level when they count by ones is illustrated by Tyrone's solution to the following task. T: (Places three strips in front of Tyrone and hide two strips and two more squares). How many are there here (the three strips that are visible)? Ty: Thirty T: If I put these (hidden squares) with them, there would be fifty-two. How many would be under here? How many strips? Ty: (Sequentially puts up twenty-two fingers in syncrony with sub-vocal utterances. He then says, after a crucial lengthy pause where he appeared to be deep in thought) there would be two of them (slaps the strips) and two of them (gesturing towards a pile of squares). T: How did you figure that one out so well? Ty: I counted, 10, 20, 30, (in synchrony with moving strips); and then 40, 50; and then 51, 52. (Steffe et. aI., in press, p. 163) Although Tyrone counted by ones to solve the task, he transformed each sequence of ten counting acts that completed two open hands into units of ten as he went along. He then reflected on what he had done after he reached "52" and reorganized it into abstract composite units of ten and abstract units of one (I.e., "I counted ... 40, 50; and then 51, 52"). In contrast to the first of Jason's two solutions that we presented, Tyrone seemed to be aware that his activity carried the significance of counting by ten. However, there was no indications that he developed this awareness until he reviewed what he -6-

had done. 1n short, re-presented modules of ten counting acts rather than re-presented strips served as the material that Tyrone took as abstract composite units of ten. It is possible for children to construct abstract composite units in this way in the absence of suitable materials by keeping track of how many open hands (i.e., a finger pattern for ten) they complete when they count by ones. Because children at this level are dependent on re-presentations of some kind to construct units of ten and because counting by ten does not increment by ten, the children have extreme difficulty in structuring a number such as thirty-nine as three composite units of ten and nine units of one in purely symbolic settings such as using a paper-and-pencil algorithm to add two-digit numbers. Further, even when suitable materials are available to support their construction of abstract composite units of ten a number such as thirty-nine lose.s its thirty-nineness. The children can construct either three abstract composite units of ten and nine units of one or thirty nine as a single entity composed of ones but not both simultaneously. But the ability to simultaneously construct a numerical whole and the units of ten :md one that compose it is precisely what is requIred to understand the conventional paper-and-pencil algorithms in a meaningful way.

len a. an Interable Unit The second major advance made by Steffe et. al. 's (in press) ;ubjects was to anticipate that they could solve tasks by counting by :en and by one in the absence of SUitable materials. They were no onger dependent on re-presentations to create composite units of :en but could take the unit of ten as weU as the unit of one for granted )efore they counted. ConSider, for example Tyrone's solution to a nlssing subtrahend task. T: (Places the sentence "71 - _ _ = 39" in from of Tyrone.) We have seventy-one take away a number and that leaves us with thirty-nine. Ty: (Sequentially puts up three fingers on his left hand) 61 - 51 41. (He then puts up a finger on his right hand and pauses) 41 - 40! - 39. (He then places "32" in the blank space.) . (Steffe et. al., in press, p. 167) ryrone anticipated that he could construct the unknown subtrahend >y iterating a composite unit of ten and then a unit of one until he 'e ached "39." The unknown subrtahend was, for him, a single entity hat could Simultaneously be structured In terms of compOSite units )f ten and units of one before he counted. The term " iterating" is lSed to emphasize that Tyrone constructed composite units of ten as Ie counted - they were not there waiting to be counted. This :ontrasts with the child who creates and then counts abstract comlosite units of ten because he or she believes that strips of ten are lidden beneath a cloth. It is because of this generative quality of

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Tyrone's counting activity that Steffe et. al. (in press) say that he had constructed ten as an iterable unit. Metaphorically, counting so many abstract composite units of ten is like Jumping 'from one lattice of ten rungs to the next lattice, when the lattices are already in place. Iterating a unit of ten so many times is like repeatedly laying down a lattice of ten rungs end-toend, with the intention of finding how many times it can be done. (Steffe et. a1. , in press, pp. 232-233) Finally. we note that Tyrone's counting activity carried the significance of " 61 is ten, 51 is twenty, ... . In other words, counting by ten did not merely juxtapose composite units of ten but repeatedly decremented 71 by ten. Thus, both of the two major limitations of ten as an abstract composite unit are transcended with the construction of ten as an iterable unit. Steffe et al. (in press) concluded that: The construction of the iterable unit of ten was required before the children [who participated in their study] could understand the positional principle of the numeration system. We were surprised at how difficult it was for them to understand that each decade comprises a number sequence of numerosity ten and also that a counting by ten act could increment by ten more ones. (pp. 233234) . Steffe et aJ.'s analysis, as presented above, deals with children's construction of increasingly sophisticated concepts of ten and their growing awareness of their own activity in the course of a teaching experiment. The remainder of the paper focuses on the constructions that children make as they attempt fo make sense of typical textbook-based school instruction. II

Method

Subjects The subjects were 14 second grade children drawn from a single class. A further six children from this class were interviewed but the tasks that focused on concepts of ten were omitted because all six were at the same conceptual level as eight of the 14 children to whom these tasks were administered. Presentation of these tasks would, in all likelihood, have produced redundant information. The 14 children had been introduced to the notion of grouping (e.g., 73 is 7 sets of tens and 3 ones) and had been taught to add and subtract two-digit number without regrouping at the end of their first grade year when they were in five different classes. These topics had been reviewed shortly before the interviews were conducted. Procedure

All 14 children were individually interviewed during October of their second grade year. The interviews, which were conducted in

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two parts of approximately 35 minutes each on different days, were video-taped for later analysis. Tasks Counting by one, thinking strategy, and subtraction tasu. These tasks were presented at the beginning of the first part of the interview to infer the children's concepts of addition and subtraction that are closely related to their concepts of ten (Steffe et aI., in press). A full description of these tasks can be found in Cobb (1987). Horizontal sentences. Each child was asked to solve, in order, the sentences 16 + 9 =__ , 28 + 13 =__. 38 + 24 =__ , and 39 + 53 =__ These sentences were presented in horizontal rather than vertical, column format to investigate whether the children had constructed their own algorithms or whether they could adapt school-taught procedures. Tens tasks. The materials used to present these tasks were cardboard individual squares and strips to which were affixed ten squares. The child was first asked how many squares were on each strip and was allowed to verify that there were ten on each in any way he or she chose. The following tasks were then presented: 1. The interviewer repeatedly put down a strip, each time asking, " How many squares are there altogether?" 2. The interviewer first put down four individual squares and then repeatedly put down a strip. each time asking, " How many squares are there altogether?" If necessary, this task was repeated with intial collections of seven and then three squares rather than four. 3. Two boards to which were affixed sequences of strips and squares were gradually uncovered and the child was asked "How many are there now?" each time the cloth was pulled back to reveal more strips andlor squares. The two sequences of strips and squares and the cumulative sums after each uncovering were: (a) one strip (10), three squares (13), two strips (33) , four squares (37), three squares (40), one strip (50), two squares (52), two strips (72) and (b) four sq uares (4) one strip (14). two strips (34), one strip and two squares (46), two strips and five squares (71) (See Figures 1a and 1b).

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~ Figure la: The first Uncovering Task

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~ Figure lb: The second Uncovering Task

4. A variety of addition and missing addend tasks were presented at the discretion of the interviewer using visible and screened collections of strips and squares. An example of a relatively simple task is one in which four strips are visible, the child is told thirty squares are hidden beneath the cloth and asked to find how many squares there are in all. In the most complex tasks, a collection of strips and squares is visible (e.g., three strips, four squares), a child is told how many squares there are in all (e.g., 67), and asked to find how many squares are hidden. Worksheet task. The children were asked to complete a worksheet of eleven two-digit addition tasks presented in vertical, column format. Two sequen~es of tasks were of particular interest. First, the second through sixth tasks were 22 22 22 22 and 22 +14 +15 +16 + 17 +18 Successive tasks can be solved by using an elementary thinking strategy, the students' use of which was investigated when the thinkIng strategies tasks were administered. If a student solved the first three tasks without relating them, the interviewer said, "Look at YOl.,lr answers, 36, 37, 38. Can that help you do the next, 22 plus 17?" The second sequence of interest was 28 37 and 39 16 + 9 + 13 +24 +53, the eighth through eleve'nth tasks. They are, from the adult perspective, the same addition tasks that were presented earlier in horizontal form. Thus, it was possible to compare the children's performance in these two settings. All remaining tasks on the worksheet were twodigit addition tasks that did not involve regrouping.

Findings On the basis of their performance on the counting-by ones, thinking strategy, and subtraction tasks, the fourteen children were placed at three levels with respect to their addition and subtraction concepts. These levels, which for ease of discussion we will call level one, level two, and level three, are closely related to the three concepts of ten outlined above (Le., ten as a numerical composite, ten as an abstract

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composite unit, and ten as an iterable unit) . It was found that all 14 children interviewed were abstract counters with respect to counting by one as indicated by their ability to solve missing addend tasks (Steffe et al. 1983). Eight were placed at levell, three at level 2, and three at level 3. Conceptual Level One Two

Three

Concept of Ten

General Indicator

Numerical composite

Count individual elements of suitable materials Abstract composite unit Coordinate counting by ten and by one starting in the middle of a decade in presence of visible or hidden materials Iterable unit Coordinate counting by ten and by one in absence of suitable materials

Table 1: Concepts of Ten by Conceptual Level

Horizontal Sentences The only method available to the eight children at level one was to count-on by ones. This is consistent with Steffe et al.'s (in press) contention that children at this level can, at best, construct ten as a numerical composite and then o nly in the presence of appropriate materials. One child said that 16 + 9 =__ was " too big," seven solved this sentence correctly, and two also solved 28 + 13 =_ _ All eight children's difficulties stemmed from their inability to keep track of counting by ones. Of the three children at level two, OM also attempted to count-on by ones but lost track when he attempted to solve 28 + 13 = _ _ . A second child, John, counted-on to solve 16 + 9 = _ _ but then attempted to solve the remaining three sentences by mentally placing the second addend under the first and using the school-taught algorith m. In each case, he failed to carry and produced answers of 28 + 13 = 31,37 + 24 = 5 1, and 39 + 53 = 712 and then 82. There was no indication that he conceputalized the tasks in terms of units of tens and ones. Rather, he seemed to view a two-digit number as two juxtaposed single digit numbers of equal rank. For example, to justify his initial answer of 712 to 39 + 53 =__ , J ohn said, J: Wait, we take off the first number [the "1" of "12" , the sum in the ones place] and then we keep the second number. I: Alright, keep which one? J: Two. I: Now what?

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J: 28. I: How did you get that? J: I mean 82. The third child at level two, Auburn, counted-on by ones but organized her counting activity into modules of ten - ten as an abstract composite unit constructed by reviewing counting by one activity. She produced correct answers to the first three sentences but gave 83 as her answer to 39 + 52 = _ _ . However, she explained, "I counted-on from 40 ... I counted-on five tens and then I did three ones." In other words, she knew how to solve the task but made a tracking error when counting and believed that she had constructed five modules of ten when, in fact, she had only constructed four. In contrast to the children at level one, her difficulties were procedural rather than conceptual. This child's construction of ten as an abstract composite unit and the failure of all three children at level two to coordinate counting by tens and ones in the absence of suitable materials (Le., failure to construct ten as an iterable unit) is consistent with Steffe et al.'s (in press) analysis. The three children at level three gave evidence that they had either constructed or were in the process of constructing their own algorithms. Carrie's algorithm seemed to be her adaption of the standard addition algorithm. The language she used to justify her solutions indicated that she constructed units of ten and of one as arithmetical objects. For example, she justified her answer of 61 to 37 + 24 = - - in the following way. "I knew there were 50 [Le., 30 + 20] you make 60 and you have one left over." Clearly, the tens she constructed were Simultaneously both single entities and composites of ten ones. It will be recalled that ten as an iterable unit is constructed by reorganizing the activity of counting by ones. In contrast, Carrie's algorithm seems to have been constructed as she made sense of school instruction with its emphasis on collections of ten items. She did not attempt to increase 37 by 24 but instead constructed units of ten and of one when she gave meaning to each numeral and then added units of the same rank. Her concept of ten will be called ten as an abstract collectible unit. This unit is, like ten as an abstract composite unit, a single entity that is itself composed of ten ones. Further, the result of adding, say, three tens and two tens is a single entity itself composed of five composite units of ten. For this reason, the child can switch flexibly from adding units of ten to adding units of one. Were this not the case, Carrie's answr to 37 + 24 = might have been fifty-eleven. As with ten as an interable unit, the child who has constructed ten as an abstract collectible unit can take abstract composite units as given in the absence of suitable materials. This unit of ten differs from the iterable unit in the history of its construction and in the way it might be filled out in problematic situations (Le., re-presenting collections rather than counting activity). -12-

Shawn's algorithm involved increasing the first-addend to the next multiple of ten, then adding tens and finally adding on the remainder. For example, he explained his solution to 28 + 13 = _ _ by saying, "Because that equals 30, and that equals to .... and that's 40 so far and then there's one one more." Similarly, for 37 + 24 = - _ ,. "50 and there was one more so - and this is 60 and then there's one more left." This algorithm seems to be an elaboration of the going-through-ten thinking strategy (he solved 16 + 9 = _ _ by adding first four and then five) and seems to reflect counting rather than collections as its underlying source. The manner in which he coordinated increments of ten and one indicates that he conceptualizes, say, 37 + 24 = - - as to increase 37 by 24. We therefore infer that his algorithm is an alternative expression of ten as an iter.· ble unit. The third child inferred to be at level three, Stephanie, also attempted to construct ten as an iterable unit, though with less success. She solved 16 + 9 = _ _ by counting-on by ones and then appeared to construct ten as an abstract composite unit by reviewing counting by one activity when she solved 28 + 13 = _ _ and 37 + 24= _ _ . Finally, she gave 83 as her answer to 39 + 53 = _ _ , explaining "I counted from 39-40-50-60-70-80, and I added the three up with it." This solution is fraught with procedural errors, but represents an initial attempt to coordinate counting by tens and by ones in the absence of suitable materials. The modifications she made in her methods as she attempted to solve the four sentences captures the transition from counting-on by ones to constructing ten as an iterable unit. This was possible because she was already at level three and had constructed the prerequisite conceptual operations for ten as an iterable unit. In summary, we have seen that only one of the 14 children, John, attempted to solve the horizontal sentences by using a school-taught method. This observation will take on greater Significance when the children's performance on the worksheet task is considered. None of the eight children at level one could reorganize their counting-on by ones methods. In contrasts, one child at level two and all three at level three either had or were in the process of going beyond counting by ones. The observations that only the latter three children attempted to solve at least one task by constructing units of ten without first counting by one is consistent with Steffe et al.'s (in press) claim that only children at level three can understand the positional principle of the numeration system.

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Tens Tasks

The first two tens tasks in which the interviewer repeatedly put down strips of squares were designed to investigate whether the children could generate both the number word sequences "10, 20, 30, ... " and sequences of the form "four, fourteen, twenty-four, thirty-four, ... " In other words, to produce a sequence of the latter type, the child only has to abstract a pattern from his or her sequence of answers. Consequently, appropriate responses indicate that the child is reciting as opposed to counting by tens. We found that all the children but one child at level one could immediately recite "10, 20, 30, ... " and all but the same child could abstract sequences of the type "four, fourteen, twenty-four, thirtyfour, ... " from, at most, a sequence of four prior responses. Thus, even children at level one are generally able to cope with the linguistic demands of the two place numeration system. This finding is in concert with those reported by Baroody, Gannon, Berent, and Ginsburg (1983), Ross (1986), Scriven (1968), and Smith (1972, 1973). For example, Ross (1986) reported that "many children in second and third grade are still working to sort out the left-right distinction of 'tens and ones'" (p. 34). When such a child "says the '4' in '48' means 'four tens,' s/he is demonstrating only verbal knowledge based on the left and right positional labels; the child does not recognize that the '4' represents 40 objects" (p. 34). Nonetheless, this verbal knowledge is sufficient to succeed on many standard textbook tasks such as the following: In 27, which digit is in the ones place? How many tens are in 84? 35 =_ tens and _ ones. 7 tens + 5 ones =_. (p. 35) Kamii (1986) also argued that textbook instruction encourages the construction of linguistic rules at the expense of conceptual structure. The contention that successful performance on the first two tens tasks can be accounted for solely in terms of abstractions from verbal activity is confirmed by the attempts the eight children at level one made to solve the remaining tens tasks. None of the eight children attempted to coordinate counting by tens and by ones (Le. ten as an abstract composite unit). Further, there was no indication that any of the children even constructed ten as a numerical composite. On the tasks in which an array of strips and individual squares was gradually uncovered, six of the children at level one always counted-on by ones. For example, having found that four squares and one strip was fourteen, these children counted the individual squares of two strips

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"15, 16, 17, ... , 34" when two more strips were uncovered. The mixing-up of strips and individual squares on these tasks precluded a solution based solely on abstracting a number word pattern. The remaining two children at level one counted all the strips by ten and then all the individual squares by ones each time the cloth was pulled back. For example, when two further strips were uncovered after they had reached the cumulative sum of 52 In the first uncovering task (see Figure 2), these two children first counted all the visible strips "10,20, .. ., 60" and then counted the squares "61 62, ... , 72." Their performance on the remaining tens tasks in which addtion and missing addend tasks were presented by using visible and screened collections of strips and squares aids the interpretation of these solutions.

•• •• •• •• •• ••• •• ••• ••• •• •• •• •• • • ••

~

Figure 2:

•• ••• •• • ••• •• •• •• ••

The First Uncovering Task Before the Final Two Strips are Revealed

Both children succeeded on addition and missing addend tasks if they could first count strips by tens and then squares by ones. Otherwise they failed. The most sophisticated solution either child produced occurred on a task in which four strips and four squares were visible, the child was told three strips (not 30 squares) were hidden and was asked to find how many in all. Initially, the child attempted to count-on from 44 by ones but gave up when he lost track of his counting activity. He did not organize counting by ones into modules of ten (I.e., did not construct ten as an abstract composite unit) or attempt to count the squares of three hidden strips (I.e., did not take a represented strip as a numerical composite of ten). The child then began to count all the visible squares by ones even though he had previously counted the visible strips and squares "10, 20, 30, 40 44." The interviewer intervened when the child reached 24 in his count by ones and said, "Four and ten more?", to which the child replied, ''I'd have to count to add up - it's hard." The interviewer then suggested that he try and find another way to count. After a pause of several seconds, the child counted the visible strips "10, 20,

-15-

30, 40" then the covered strips "50, 60, 70" and finally the visible squares "71, 72, 73, 74." Again, it should be stressed that this was the most sophisticated solution produced by any of the eight children at level one. The child's inability to construct ten as either a numerical composite. or an abstract composite unit strongly indicates that the count "10, 20, 30, ... " was not a curtailment of counting by ones. Instead, each number word uttered referred to a visible or re-presented strip taken as a single object rather than to be ten squares on each strip. In other words, the child's count of the strips carried the significance "1, 2, 3, ... " and the child is said the have constructed ten as an abstract singleton. The fact that the child continued counting "71, 72, 73, 74" indicates that he had differentiated between items that could be counted using the sequence "10, 20, ... " and those that could be coordinated with the standard number word sequence. The child's solution involved coordinating counting tens as abstract singletons and counting ones as abstract units. This coordination was based solely on figural imagery. The child only had to be aware of what type of items he was counting - he re-presented either a strip as a single item or a square. Consequently, it is inferred that the child's answer, ''74'', did not signify a collection of 74 items but seven strips and four squares: In general, the child's solution did not express the idea that one ten is composed of ten ones - the child did not construct a unit of ten of any type but instead constructed ten as a singleton. In all, three of the children at level one coordinated counting tens as abstract singletons and ones as abstract units at least once when the uncovering tasks and tasks involving screened collections were presented. The remaining five children at this level could only count by tens or by ones when they attempted to solve these tasks. They were unsuccessful when either collection was composed of both strips and squares, indicating that they could not coordinate counting tens as abstract singletons and ones as abstract units. Of the three children at level two, one relied on her knowledge of the basic addition facts to add' the strips and then the squares. This separation of tens and ones and her explanations indicated that she also constructed ten as an abstract singleton. For example, she was asked to find how many in all given five strips and three squares visible and thirty squares covered. She answered "83" and explained, "I went five plus three is eight and I knew there was three ones and 80 plus three is "83." We infer that the translation from eight strips to 80 was primarily linguistic, the sort of rule she learned

-16-

as a consequence of textbook-based instruction. This was further indicated by her solution to a subsequent task in which two strips and four squares were visible and she was asked to find out how many squares were covered given that there were 56 in all. She initially answered "nine" and then said "three strips and six little squares" when the interviewer asked, "Nine strips?" In this case, she failed to keep separate the two types of Singleton units with which she was operating. Surprisingly, this was the same child, Auburn, who reviewed counting by one activity and constructed ten as an abstract composite unit when she solved the horizontal sentences. The remaining two children at level two constructed ten as an abstract composite unit when they solved both tasks in which an array was gradually uncovered and on the addition and missing addend tasks involving screened collections. For example, one of these children solved the second of the uncovering tasks by counting, " 4 - 14 - 24,34 - 44, 45, 46 - 56,66,67,68,69, 70,71. " The Same child, John, had attempted to solve the horizontal sentences by applying a school-taught algorithm. . One of the children at level three also counted ten as an abstract composite unit on the two uncovering tasks. The other two children at this level coordinated counting ten as an abstract Singleton and one as an abstract unit on these tasks. However, all three children used the same method when they solved the tasks involving screened collections. For example, one child was asked to find how many in all given three strips and five squares visible and 22 squares hidden. She answered, "57" and explained, "I knew those were two tens under here and three tens were out and these were five and you added two." This explanation suggests that she constructed ten as a unit of some type. In particular, she added first units of ten and then units of one. As all three children used this method, we tend towards the inference that they constructed ten as an abstract collectible unit. However, a case can be made for contention that they constructed ten as an abstract Singleton. The most striking feature of the children's performance on the tens tasks is the lack of conformity with Steffe et al. 's (in press) observations. Not one instance of either ten as a numerical composite or as an iterable unit was identified. This is all the more remarkable because Steffe et al. used the same types of tasks. In place of ten as a numerical composite, we found that the children at level one constructed ten as an abstract Singleton. Their construction of ten did not involve a sense of ten ness, of ten as composed of individual units. At level three, we inferred that all three children constructed ten as an abstract collectible unit rather than as an iterable unit. On a screened collection task corresponding to 34 + 25 =_, for example, they did not attempt to increment 34 by 25 but instead constructed each

-17-

number as so-many tens and so-many ones and then relied on knowledge of basic facts to add units of the same rank. One question that arises is whether the children had a sense of, say, 34 as a single entitv or whether, in fact, they constructed three tens and four ones as two unrelated entities. In other words, did their answers refer to the sum of 34 and 25 as a single entity? Their performance on the worksheet tasks will help us address this question. This issue has considerable instructional relevance given that ten as an abstract Singleton and as an abstract collectible unit seem to reflect a prior instructional emphasis on the value of each digit of a two-digit number. The second interesting aspect of the children's performance was that most of the children used strongly contrasting methods to solve the horizontal sentences and the tens tasks. Further, these differences could not be explained solely by the fact that materials were used to present one set of tasks but not the other. For example, the children at level one all constructed ten as an abstract Singleton on the tens tasks but attempted to count-on by ones to solve the horizontal sentences. Ten as an abstract singleton does not seem to be derived from counting-on by ones but instead appears to be a consequence of school instruction. As it so happened, two of these children spontaneously volunteered that the tens tasks were similar to their school work. One said, "We did almost the same thing in class today. It was just on a piece of paper." The other explained, "It's just like in class. You have the tens place and the ones place." This suggests that ten as an abstract Singleton replaced rather than built upon their counting-based meanings. In other words, the horizontal sentences and tens tasks were separate contexts for the children. The meanings that they gave to two-digit numerals or number words in the two situations were unrelated. The same can be said of the child at level two who constructed ten as an abstract composite unit to solve the horizontal sentences and ten as an abstract Singleton to solve the tens tasks. Further, two of the three children at level three constructed ten as an iterable unit in the sentence setting and ten as an abstract collectible unit on the tens tasks. Analysis of the children's performance on the worksheet tasks also allows us to further investigate the contextuality of the children's meanings .. Worksheet Tasks

Two of the children at level one said they did not know how to solve any of the two-digit addition tasks presented in column format. The remaining twelve children were all able to routinely solve tasks that did not involve regrouping. Significantly, not one of the twelve children related the five tasks in which the second addend succesSively increased by one even after the interviewer intervened.

-18-

22

22

+14,

22

22

+15,

+16

+17

(Le.

22

+18).

The children seemed to view the intervention as a non-sequitor and only one child made any comment at all. When it was brought to his attention that his answers to successive tasks were 37,38, and 39 he explained, "That's the way they print'em." In short, the possibility of relating tasks did not occur to any of the children. In contrast, eleven of the twelve children had used a thinking strategy in the same interview when the thinking strategy tasks were presented using horizontal addition sentences. This contrast in the children's performance strongly indicates that the two situations were separate contexts for them. The horizontal sentence situation was a context in which they incremented one number by the other and their answers referred to a number taken as a single entity. Their failure to relate task in column format indicates that they did not formulate the intention of adding, say, 22 as a single entity to 15 as a single entity when they gave meaning to

22 +15. Rather, they "saw" the task as either two single-digit addition tasks, 2 + 5 =_, or as two separate tens and ones tasks corresponding to 20 + 10 =_ and 2 + 5 =_ and 2 + 1 =_, This was true even of the children at level three. The children's performance on the tasks that involved regrouping is also generally consistent with the contention that they operated in two separate contexts. Of the nine children who were at levels one or two, one used the standard algorithm correctly. She explained that her father had taught her this method. As she could only count-on by ones on the horizontal sentence tasks and constructed ten as an abstract Singleton on the tens tasks, we infer that this was primarily a rote procedure. The remaining eight children all experienced difficulties when they added the ones digits and obtained a two-digit answer. Four wrote down both digits, thus yielding answers in the hundreds, and four omitted the "1" 6f the sum of the ones digits. None of the children gave any indication that they interpreted the "1" as a ten. In the case of

16 +9,9 for example, four produced the answer 15 and four 115. When the sentence 16 + 9 =_ was presented earlier in the interview, all eight counted-on and arrived at the answer of 25. The child who had constructed ten as an abstract composite unit to solve the horizontal sentences was questioned about the appropriateness of her answer of 15. After she had completed the worksheet she was again asked to

-19-

solve 16 + 9 = _ and counted-on appropriately. I: So when we count we get 25 and when we do it this way (points to the worksheet) we get 15. Is that okay to get two answers or do you think there should be only one? A: (Shrugs her shoulders). I: Which do you think is the best answer 15 or 25? A: 25. I: Why? A: I don't know. I: If we had 16 cookies and nine cookies, would we have 15 altogether? A: No. I: Why not? A: Because if you counted them up together, you would get 25. I: But is this (Points to the worksheet answer of 15) right sometimes or is it always wrong? A: It's always right. This exchange indicates that 15 and 25 could both be correct answers for the child, depending on the situation. In particular, we interpret her final comment to mean that 15 is always right in the worksheet setting. This strongly indicates that 16 plus 9 presented horizontally and vertically were different tasks to her. We should stress that this child was one of only four who was able to progress beyond counting-on by ones to solve the horizontal sentences. The classroom teacher considered her to be one of her best three mathematics students. In summary, there is every indication that the children at levels one and two gave incompatible meanings to numerals in the two settings. We infer that these children constructed ten as an abstract singleton in the worksheet setting. This finding is consistent with Steffe et al. 's findings that the first coordination of units of ten and of one in the absence of suitable materials involves the construction of ten as an iterable unit. As this unit is associated with level three, these children's current conceptual levels precluded their construction of a unit of ten itself composed of ten ones in the worksheet setting. By the way of contrast, the three children at level three were able to construct viable units of ten in the worksheet setting. One child used the standard regrouping algorithm and explained, "If the answer [in the ones column] is more - like ten - then you put a '1' above the tens column." This explanation gives some indication that her use of the algorithm was more than a rote procedure. However, we were unable to determine whether her answers referred to the sum of two numbers constructed as single entities. Nonetheless, she seemed to realize that the one she carried signified a unit of ten that could be added with other units of ten. We therefore attribute to her ten as an abstract collectible unit, the same unit she constructed to -20-

solve the horizontal sentences and tens tasks. She was the only child who constructed the same type of unit of ten in all three task settings. The remaining two children at level three interpreted the worksheet task as a problem solving situation and attempted to construct their own algorithms. The first child initially added first the tens column then the ones column. This method resulted in an answer of 15 for 16

+ 9 I: S: I: S: I: S:

Is that okay? (Erases her answer of 15 and writes 115). Okay, read your answer. One-hundred fifteen. Alright, so you add 16 and nine and get 115. (Looks away and reflects) 16 - 17, 18, ... 25. (Erases 115 and writes 25). Similar interventions were unsuccessful with all the children at levels one and two. This child reflected on her activity when the interviewer said, "you add 16 and nine ... " This suggests she reconceptualized the problem as one that involved adding 16 and nine as single entities. As a consequence, she judged that her answer of 115 was too large. In contrast, the children at lower conceptual levels could not assess the appropriateness of their answers because they did not conceptualize the problem in terms of adding two numbers. Their interpretations of worksheet problems were divorced from their selfgenerated counting meanings. The child capitalized on her insight as she worked the remaining three tasks that involved regrouping. Each time she added by columns, but then paused as if to reflect on her answer before revising it if necessary. For example, she initially wrote 82 as her answer to 39

+53. After a considerable pause, the interviewer asked her what she was thinking about. She explained, "I'm trying to count up from 39 by tens five times." In other words, she was in the process of constructing ten as an tterable unit. She then counted subvocally and erased the "8" of "82" and wrote a "9". By way of justification, she said, S: From 39 I counted 40, 50, 60, 70, 80. Then nine and three more makes 92. I: How did your eight get to be a nine? S: 39 and five tens is 90. In other words, she counted-on five units of ten from 30 and added first nine and then three to the result, 80 ("Nine and three more makes 92.")

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The third child at level three initially attempted to use the standard regrouping algorithm. He verbalized a rule about carrying a one when he attempted to solve the first task that involves regrouping, 22

+18. However, he interpreted this literally and carried one unit of one, writing "9" in the ones place and "4" in the tens place. He paused and expressed doubt in his answer, " I don't get this... forty-nine. Is that right?" Nevertheless, he did not change his answer and continued to use the method of writing "9" in the ones place and carrying a unit of one on the next three tasks that involved regrouping. Each time, he seemed dissatisfied with his answer. Finally, after an unprompted struggle of several minutes, he constructed a novel method to solve

39 +53. He used the compensation thinking strategy in the ones column. Sh: I took one of the three's away... and there's ninety. I: I don't see that. Sh: 92. I: What do you mean, taking one of the three's away? Sh: I need one to put up here (points to the "9" of "39" ,. He Increased the nine by one to make a ten and compensated by decreasing the three by one. He then wrote the resulting two as his answer in the ones place and carried the ten. Finally he, we nt back to

37

+24 and resolved it correctly usi ng this method. In summary, one child at level three used the standard algorithm with at least an understanding that she was dealing with units of ten rather than abstract singletons and the other two were able to construct their own algorithms. In contrast to children at lower conceptuallevels, these two children's activity was generally reflective. They were able to assess the appropriateness of their answers because they could construct units of ten in the absence of suitable materials and realized that they were adding numbers as single entities.

Instructional Implications As we said at the outset, the interviews were conducted as part of a curriculum development project. We will therefore outline implications drawn from the analysis of the children's understandings of ten for the development of instructional activities. First, however, we will consider the limitations of the textbook approach and then discuss the general goals of arithmetic instruction, thus placing our subsequent remarks in context.

-22-

Our findings suggest that the textbook approach to place value with its emphasis on assigning values to digits based on their position (ones place, tens place) as a precursor to the introduction of standard two-digit addition and subtraction algorithms is seriously flawed. Even when drawings of bundles of sticks and manipulatives are used, this approach encourages children to construct ten as an abstract singleton rather than as a unit that is itself composed of ten ones. As an activity for the reader, we suggest playing student and completing exercises in any major textbook series from the introduction of place value through two-digit addition and subtraction without regrouping. As you do this, ask yourself what you need to know to produce pages of correct answers. We predict that you will find that a child needs three types of relatively superficial knowledge to be successful. These are purely verbal number word patterns (e.g., seventy-three is seven tens and three ones or, more generally, a rule of the form pty-q is p tens and q ones), numeral patterns (e.g., 4 tens is written as 4 with a zero after it - 40), and ten as an abstract Singleton. The reasons for children's difficulties with algorithms that involve regrouping become readily apparent once we view textbook instruction from the child's point of view. Ten as a composite unit of some type (e.g., ten as an iterable or abstract collectible unit) is required if children are to use an efficient addition or subtraction algorithm in a meaningful way. But the textbook approach does not provide opportunities for children to construct units of this type. From the children's perspective, they are suddenly asked to do things such as regrouping which make absolutely no sense given the concept of ten they have used with success for some time - ten as an abstract Singleton. A Singleton is not made of anything else, so how can it be "unpacked" into ones? This is a complete non sequitor for the children; it flatly contradicts what they have learned as a consequence of textbook place value instruction. In short, instructional materials that seem perfectly reasonable in terms of the adult's mature understanding of place value lead to predictable difficulties when analyzed in terms of children's mathematics. Despite the points made above, it might nevertheless be argued that the construction of ten as an abstract singleton is acceptable because it enables children to produce right answers to tasks they otherwise find impossible. For example, when the tasks were presented in column format nine of the eleven children at levels one and two were able to solve two-digit addition tasks that did not involve regrouping.. ObViously, we eventually want children to construct methods that enable them to produce correct answers. However, a price has to be paid for the children's construction of ten as an abstract Singleton that is, for us, far too high. First, the children's use of methods involving ten as an abstract Singleton were generally situation specific (e.g., used to solve tasks

-23-

The third child at level three initially attempted to use the standard regrouping algorithm. He verbalized a rule about carrying a one when he attempted to solve the first task that involves regrouping, 22

+18. However, he interpreted this literally and carried one unit of one, writing "9" in the ones place and "4" in the tens place. He paused and expressed doubt in his answer, "I don't get this .. .forty-nine. Is that right?" Nevertheless, he did not change his answer and continued to use the method of writing "9" in the ones place and carrying a unit of one on the next three tasks that involved regrouping. Each time, he seemed dissatisfied with his answer. Finally, after an unprompted struggle of several minutes, he constructed a novel method to solve

39 +53. He used the compensation thinking strategy in the ones column. Sh: I took one of the three's away.. .and there's ninety. I: I don't see that. Sh: 92. I: What do you mean, taking one of the three's away? Sh: I need one to put up here (points to the "9" of "39" ). He increased the nine by one to make a ten and compensated by decreasing the three by one. He then wrote the resulting two as his answer in the ones place and carried the ten. Finally he, went back to 37

+24 and resolved it correctly using this method. In summary, one child at level three used the standard algorithm with at least an understanding that she was dealing with units of ten rather than abstract singletons and the other two were able to construct their own algorithms. In contrast to children at lower conceptuallevels, these two children's activity was generally reflective. They were able to assess the appropriateness of their answers because they could construct units of ten in the absence of suitable materials and realized that they were adding numbers as single entities.

Instructional Implications As we said at the outset, the interviews were conducted as part of a curriculum development project. We will therefore outline implications drawn from the analysis of the children's understandings of ten for the development of instructional activities. First, however, we will consider the limitations of the textbook approach and then discuss the general goals of arithmetic instruction, thus placing our subsequent remarks in context.

-22-

Second, the two subjects of a longitudinal teaching experiment who were able to construct ten as an iterable unit were immediately able to generalize their activity to situations involving multiplication and division (Steffe & Cobb, 1984). These two children had not just constructed relatively sophisticated units of ten. They had constructed powerful conceptual operations that enabled them to construct and coordinate two units of different ranks (e.g., seven and one) in a wide variety of situations. In short, when efficient instruction is construed to mean the development of fleXible, generalizable, conceptually based methods and the encouragement of intellectual autonomy, we feel that an approach that acknowledges that substantive mathematical learning is a problem solving process and encourages the construction of self-generated algorithms is more appropriate than one that stresses empirical abstraction from grouped collections. With regard to specific suggestions, we have developed educational activities for all areas of second grade mathematics since completing the interviews discussed in this paper. Twenty-three teachers are currently using these materials in their classrooms. The basic instructional approach in all areas including arithmetical computation is small group problem solving followed by whole class discussion of the children's solutions. This approach has been discussed in more detail elsewhere (Cobb, Wood, & Yackel, in press a, b; Cobb, Yackel & Wood, in press). One of its key features is that we have used our understanding of children's mathematics to develop activities that give rise to genuine mathematical problems for children at a variety of different levels. The arithmetical activities used in the first part of the school year are designed to give children opportunities to construct increasingly sophisticated thinking strategies for finding sums and differences (Cobb & Merkel, in press). Next, activities designed to help children construct composite units are introduced. As an example, our findings indicate that children at level one can construct number word sequences of the type four, fourteen, twenty-four, thirty-four . .. . These sequences facilitate the construction of more sophisticated units of ten (e.g., ten as an abstract composite unit and as an iterable unit) and, more generally, the realization that, say, 46 is ten more than 36. Consequently, we have developed a variety of hundreds board activities to encourage the construction of these sequences. Other activities introduced at this time include those involVing money and a wide variety of story problems. In addition, we derived activities from the covered collections tasks used in this study. For example, three strips and seven squares might be visible and the children are asked to find how many are hidden given that there are 61 in all. Such tasks which require crossing a decade, are impossible to solve if the child constructs ten as an abstract Singleton. This task requires coordination of tens and ones and the child cannot arrive at a correct answer by first finding how many tens are covered and then

-25-

presented in the familiar textbook column format but not to solve horizontal sentences). Second, and more importantly, the situation specific nature of these methods indicates that the children were operating in two separate, unrelated contexts. Ten as an abstract singleton does not involve a conception of ten as ten ones and is divorced from the children's pragmatic problem solving methods. As a consequence, the children were unable to assess the appropriateness of their answers and thus to modify their methods. For example, four of the children thought that their answer of 115 to 16

+ 9

was reasonable despite the fact that they had previously counted-on to solve 16 + 9 = _ _ correctly. The children's construction of two separate arithmetical contexts is, in our opinion, one source of students'· generally debilitating instrumental beliefs about mathematics (e.g. Cobb, 1985, 1987; Confrey, 1983; Peck, 1984; Schoenfeld, 1985). We will just note that the children's inability to assess the appropriateness of their answers means that they can only rely on an authority to know whether or not they are correct. Further, any modifications they might make if they are told they are incorrect or cannot remember a prescribed procedure will be purely syntactic in nature. This is because their activity is divorced from the context of pragmatic, relational problem solVing. The children's increasing reliance on authority also mitigates against the encouragement of intellectual autonomy, for us a major goal of instruction (ct., Kamii, 1985). Third, it should be noted that the encouragement of situation specific methods that express concepts such as ten as an abstract Singleton is completely incompatible with the National Council of Teachers of Mathematics (1980) emphasis on problem solving. Without going through the usual arguments about the advent of the calculator and the computer, we note that such methods stand in stark opposition to the goals of developing mental computation and estimation abilities. These abilities reflect constructions made within the context of relational problem solving. Finally, it might be argued that children who initially construct ten as an abstract Singleton will eventually endow their methods with viable meanings when they reach level three. There are two points to be made in response to this argument. First, we found that children at level three are capable of constructing their own algorithms for adding two-digit numbers. Thus, if one values methods that express a viable underlying conceptual structure, instruction that builds on children's self-generated methods is certainly not less efficient than typical textbook instruction. In fact, we speculate that such instruction is likely more efficient if meaning and understanding are valued. -24-

Cobb, p. (1983). Children's strategies for finding sums and differences. Unpublished doctoral dissertation, University of Georgia. Cobb, P. (1985). Two children's anticipations, beliefs, and motivations. Educational Studies in Mathematics, 16, 111-126. Cobb, P. (1987). An investigation of young children's academic arithmetic contexts. Educational Studies in Mathematics, 18, 109-124. Cobb, P., & Merkel, G. (in press). Thinking strategies as an example of teaching arithmetic through problem solving. In P. Trafton (Ed.), Elementary school mathematics: 1989 yearbook of the National Council of Teachers of Mathematics. Reston, VA: NCTM. Cobb, P., Wood, T. & Yackel, E. (in press al. A constructivistapproach to second grade mathematics. In E. von Glasersfeld (Ed.), Constructluism in Mathematics Education. Dordrecht, Holland: Reidel. Cobb, P. Wood, T., & Yacke.l, E. (In press b). Philosophy of science as a source of analogies for mathematics educators. Syntheses. Cobb, P., Yackel, E., & Wood, I (In press). Young children's emotional acts while doing mathematical problem solving. In DB. McLeod & V. M. Adams (Eds.), Affect and mathematical problem solving: A new perspectiue. New York: Springer-Verlag. Confrey, J., (1984, April). An Examination of the conceptions of mathematics of young women in high school. Paper presented at the annual meeting of the American Educational Research ASSOCiation, New Orleans. Kamii, C. (1985). Young children reinuent arithmetic: Implications of Piaget's theory. New York: Teachers College Press. Kamii, C. (1986). Place value: An explanation of its difficulty and educational implications for the primary grades. Joumal of Research in Early Childhood Education, 1, 75-86. Kouba, C. L. (1986, April). How young children solue multiplication and dluision word problems. Paper presented at the annual meeting of National Council of Teachers of Mathematics, Washington, DC. Lablnowicz, E. (1985). Learning from children. Menlo Park, CA: AddisonWesley. National Council of Teachers of Mathematics (1980). An agenda for action: Recommendations for school mathematics of the 1980's. Reston, VA: National Council of Teachers of Mathematics. Peck, D. M. (1984, October). Barriers to mathematical thinking and problem solulng. Paper presented at the sixth annual meeting of PME-NA, Madison, WI. Resnick, L. B. (1982). Syntax and semantics in learning to subtract. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.). Addition and subtraction: A cognitiue perspectiue (pp. 136-155). Hillsdale, NJ: Lawrence Erlbaum Associates. Resnick, L. B. (1983). A developmental theory of number understanding. In H. P. Ginsburg (Ed.) , The deuelopment of mathematical thinking (pp. 110-151). New York: Academic Press. Ross, S. H. (1986, April). The deuelopment of children's place-ualue numeration concepts in grades two through five. Paper presented at the annual meeting of the American Educational Research ASSOCiation, San Francisco. Schoenfeld, A.H. (1985). Mathematical problem soluing. New York: Academic Press.

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covered as separate tasks. Other activities that how many ~es a~~ttuction and coordination of units of different encou:a~~de ~eJ~~vely simple multiplication and division word probba (1986) has recently demonstrated that second graders ranks lemds: n s~~e cases, first graders are capable of solving the latter types an problems , 1 . d'lrec t'lOStruc ti on. of without pnor Thus far we have focused on activities that encourage the reorgan ization of counting activity. It is also possible that some children might abstract from activities that involve spatial visualization. For example, a chUd could intlaUy re-present two open hands or a specific spatial configuration of ten dots to construct numerical composites or abstract composites units of ten. In contrast to ten as an abstract Singleton, these re-presentations both involve ten individual, potentially countable units. Ten frame activities (Labinowicz 1985) are appropriate in this regard. We delayed the presentation of symbolic two-digit addition and subtraction tasks until children had opportunities to solve the types of activities described above. Even when the symbolic problems are introduced, the children can use any of the available manipulatives as they see fit. Further, these tasks are presented as problems to be solved and discussed rather than as procedures to be mastered. In conclusion, textbook instruction emphasizes the assignment of values to digits in multi-digit numerals on the basis of their position in the symbol string. As Kamli (1986) observed, this approach assumes that learning place value is a matter of figural re-presentation rather than conceptual abstraction. We have seen that children are typically able to create figural re-presentations as a consequence of this instruction. Unfortunately, they are re-presentations of Singletons rather than of composites of ten individllal units. The instructional suggestions we have made above focus on the construction of increaSingly powerful units of ten. Symbolic two-digit addition and subtraction tasks are not introduced until the children have had opportunities to construct and coordinate viable units of ten and one. This approach is, of course, compatible with the generally accepted view that meaning should be devcloped before conventional symbols are introduced. The pay-off is that children can develop increasingly powerful concepts and use them to make meaning in a broad range of situations.

K

References

Baroody, A. J., Gannon, K. E., Berent, R , & Ginsburg, H. P. (1983, April). The deue/opment oj basic Jormal math abilities. Paper presented at the meeting of the Society for Research in Child Development Detroit. Bednarz, N., & Janvier, B. (1982). The understanding of numeration in primary school. Educational Studies in Mathematics, 13, 33-57.

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