adaptive expertise with basic math

Expertise with Combinations 1

Adaptive Expertise with Basic Addition and Subtraction Combinations—

The Number Sense View

Arthur J. Baroody and Luisa Rosu

University of Illinois at Urbana-Champaign

4/7/06*

Paper presented at the annual meeting of the American Educational Research

Association, San Francisco, CA, April 2004, as part of a symposium titled, “Developing Adaptive Expertise in Elementary School Arithmetic,” and chaired by A. J. Baroody & J. Torbeyns.

This work was supported, in part, by a grant from the National Science Founda-tion (BCS-0111829), the Spencer Foundations (200400033), and the Department of Education (R305K050082). The opinions expressed are solely those of the authors and do not necessarily reflect the position, policy, or endorsement of the National Science Foundation, the Spencer Foundation, or the Department of Education.

Running Head: EXPERTISE WITH COMBINATIONS


ADAPTIVE EXPERTISE WITH

BASIC ADDITION AND SUBTRACTION COMBINATIONS— THE NUMBER SENSE VIEW

Mastery—the efficient (quick and accurate) reproduction—of the basic number com-binations, which include the single-digit sums to 18 (e.g., 9 + 3 = 12) and related differ-ences (e.g., 12 – 9 = 3, has been a central goal of elementary instruction since ancient times (see Figure 1). Despite the sometimes-heated debate about mathematics educa-tion reform (the so-called “math wars”), there is general agreement today that all children need to achieve this goal (National Council of Teachers of Mathematics or NCTM, 2000). The National Research Council (NRC; Kilpatrick, Swafford, & Findell, 2001) upped the ante by recommending that the goal be computational fluency—the efficient, appropriate, and flexible application of basic arithmetic skills—and that fostering such fluency should be intertwined with promoting other aspects of mathematical proficiency (conceptual understanding, problem solving, reasoning, a productive disposition).

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As traditionally conceived, mastery of the basic combinations can be equated with what Hatano (1988, 2003) called routine expertise—knowledge that is learned by rote and that can be applied efficiently to familiar tasks. What we call mastery with fluency (the appropriate and flexible, as well as efficient, application of basic combination knowl-edge) can be equated with what Hatano named adaptive expertise—meaningfully learned knowledge that can be creatively (flexibly) adapted and effectively (appropri-ately) applied to new, as well as familiar, tasks.

Theoretical Framework

Psychologists and educators have long debated how basic number combinations are learned and how best to promote fluency (see reviews by, e.g., Baroody, 1985, 1994; Baroody & Tillikainen, 2003; Cowan, 2003). One perspective on these issues focuses on mastery; a profoundly different perspective focuses on mastery with fluency. Brownell (1935) labeled the first view “drill theory” and the second “meaning theory.” More re-cently, Geary (1996) contrasted information-processing theories and schema-based views. These polar perspectives also could be characterized as the passive storage view and the active construction view. In recent years, the latter has been called the number sense view (e.g., Baroody, in press; Gersten & Chard, 1999).

1. Passive Storage View. Thorndike (1922) summarized the essence of this perspec-

tive in the preface of his classic book The Psychology of Arithmetic: “We now understand that learning is essentially the formation of . . . bonds between situations and responses” (e.g., a bond between an arithmetic expression such as 5 + 3 and the sum 8; p. v).


Information-processing models of number combination learning build heavily on Thorndike’s (1922) associative-learning theory and focus on skill development (rou-tine expertise). This perspective constitutes the conventional wisdom among cogni-tive psychologists, many elementary educators, and the public (Ginsburg, Klein, & Starkey, 1998). Siegler’s (e.g., 1986) strategy-choice model is, perhaps, the most popular of these models.

2. Number Sense View. In his 1892 “Talk to Teachers,” the eminent psychologist Wil-liam James foretold and elegantly summarized the number sense view:

“…the art of remembering is the art of thinking; … when we wish to fix a new thing in either our own mind or a pupil’s, our conscious effort should not be so much to impress and retain it as to connect it with something already there. The connecting is the thinking; and, if we at-tend clearly to the connection, the connected thing will certainly be likely to remain within recall” (James, 1958, pp. 101-102).

The number sense view also has roots in Brownell’s (1935) meaning theory, Piaget’s (1965) constructivist philosophy, and Vygotsky’s (1962) social-learning theory, which is now ‘the central theoretical grounding within the field of early childhood education” (Winsler, 2003, p. 254; see also Bredekamp & Copple, 1997). This perspective, which constitutes the conventional wisdom in the field of mathematics education (Ginsburg et al., 1998), focuses on conceptual development (adaptive expertise). (Note that, although both Thorndike and James both use the term connection, it re-fers to different constructs. For James, it refers to meaningful assimilation and reflec-tion. For Thorndike, the connection is an unthinking association between a stimulus and a response.) One embodiment of the number sense perspective is a schema-based view (e.g., Baroody, 2003; Baroody & Ginsburg, 1986; Baroody & Tiilikainen, 2003; Baroody, Wilkins, & Tiilikainen, 2003).

There is general agreement that children typically progress through three phases in

learning the basic number combinations (Kilpatrick et al., 2001; Rathmell, 1978; Stein-berg, 1985):

•Phase 1: Counting strategies—using object or verbal counting to determine an-

swers; •Phase 2: Reasoning strategies—using known facts and relations to deduce the

answer of an unknown combination; •Phase 3: Retrieval—efficiently producing answers from a memory network.

However, proponents of the passive storage and number sense views differ dramatically about the role of Phases 1 and 2 (counting and reasoning strategies) in achieving phase 3 and about the nature of Phase 3 itself (see Table 1).

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The Passive Storage View

Early proponents of the passive storage view—and many to this day—regard the counting and reasoning phases (phases 1 and 2) as unnecessary or even as a hin-drance. They have described non-retrieval strategies as, for example, “immature,” “crutches,” “a bad habit to avoid the real work of memorizing the basic facts” (Murray, 1941; Smith, 1921; Wheeler, 1939). Some modern proponents of the passive storage perspective at least view these preliminary phases as opportunities to practice basic combinations and to imbue them with meaning before they are memorized (e.g., Siegler, 1986, 1988). Even so, all proponents of the passive storage view agree that Phases 1 and 2 are not necessary for achieving the mental storehouse of facts that is the basis of basic combination mastery. This conclusion is the logical consequence of three assump-tions about the process of mastery and the nature of mental-arithmetic expertise.

1. Basic combinations must be memorized as individual facts by rote, and flu-ency is best achieved through repeated practice. Mastery is achieved through rote memorization—a passive or non-conceptual process of building up specific associations via practice (Siegler & Shipley, 1995). That is, learning a basic number combination simply en-tails forming and strengthening an association or bond between an expression such as 7 + 6 and its answer 13 (and decreasing the strength of associations between the expression and competing incorrect answers such as 6). In the strategy choice model, for example, a trace is laid down in long-term memory (LTM) every time an answer is computed or stated (e.g., Siegler & Jenkins, 1989). As the number of traces builds up in LTM, the association between an answer and a combination is strengthened. In this view, then, the mastery process entails a passive quantitative change in the associative strengths. Furthermore, although existing knowledge can initially facilitate or interfere slightly with the formation of a correct association, the frequency of practicing the correct answer is THE key factor in mas-tering the basic number combinations (e.g., Siegler, 1986, 1988). In other words, forming a bond merely requires extensive practice (e.g., Ashcraft, 1992; Siegler, 1986, 1988; Torgeson & Young, 1983). Thus, mastery can be accomplished directly without counting or reasoning (e.g., through flash card drills and timed tests) in fairly short order. It requires nei-ther conceptual understanding on the part of the student nor taking into account a child’s developmental readiness—their existing everyday (informal) knowledge—on the part of the teacher.

2. Mastery consists of a single process, namely fact recall. Fact recall entails the

automatic retrieval of the associated answer to an expression. Each basic combination is stored discretely in a factual memory network, perhaps as a verbal statement such as, “Five plus three is eight” (Dehaene, 1997).

3. Arithmetic knowledge is modular in nature. In recent years, researchers from a variety of fields have drawn a sharp distinction (a) among procedural knowledge (calcula-tion routines such as counting or reasoning strategies), conceptual knowledge of arithmetic, and factual knowledge (e.g., knowledge of the basic combinations) and (b) between exact calculation processes and inexact estimation processes (e.g., Dehaene, 1997; Delazer, 2003; Donlan, 2003; Dowker, 1997, 2003; Heavey, 2003; LeFevre, Smith-Chant, Hiscock, Daley, & Morris, 2003). Accordingly, fact retrieval replaces conceptually based counting or reasoning strategies as the preferred strategy. Put differently, the fact-retrieval network of the brain is autonomous and, thus, operates independently from the conceptual component and from other components of computational fluency, including estimation.


Number Sense View

According to the number sense view, children should learn the basic combinations as a system of interrelated information, not as numerous separate associations (Baroody, 1985; Olander, 1931). As Brownell (1935) noted, arithmetic knowledge, in-cluding knowledge of the basic combinations, is “a closely knit system of understandable ideas, principles, and processes” (p. 19). Phases 1 and 2 are essential for constructing this web of knowledge and thus necessary for achieving mastery with fluency. This per-spective is based on the following three assumptions, for which there is growing re-search support.

1. Fluency with the basic combinations is a by-product of children’s growing number sense. Mastery with fluency is achieved through meaningful memorization in which relational (conceptual) learning plays a key role. Focusing on structure (underlying patterns and relations) makes the learning, retention, and transfer of any large body of fac-tual knowledge more likely than memorizing individual facts by rote (Bruner, 1963, 1966; Katona, 1967; Wertheimer, 1959). As with any worthwhile knowledge, meaningful memo-rization of basic combinations can reduce the amount of time and practice needed to achieve mastery, maintain efficiency (e.g., reduce forgetting and retrieval errors), and fa-cilitate application of extant knowledge to unknown or unpracticed combinations (Baroody, 1985, 1993, 1999b; Baroody & Ginsburg, 1986; Carpenter, Fennema, Peterson, Chiang, & Loef, 1989). For example, recognizing that addition is commutative can enable a child to equate an unknown combination with a known combination (e.g., 3 + 5 = ? can be trans-lated into 5 + 3, which has the known sum of 8). This can reduce by nearly half the amount of practice needed to master addition combinations (Rathmell, 1978; Trivett, 1980). As children construct relatively efficient counting strategies in Phase 1 and become more pro-ficient in their execution, they are more likely to discover the myriad of patterns and rela-tions necessary to advance to Phase 2. For example, children may master the addition doubles such as 7 + 7 = 14 relatively easily and early because they recognize the sums of consecutive doubles from the well-known count-by-two sequence. They then often spon-taneously use this known knowledge to invent the “doubles + 1” reasoning strategies (e.g., 7 + 8 = 7 + 7 + 1 = 14 + 1). With practice, reasoning strategies become semi-automatic or automatic (Jerman, 1970) and a basis for mastery with fluency (Phase 3). Thus, the proc-ess of mastery is gradual and involves both passive quantitative changes (due to the strengthening of associated factual knowledge or increasing efficiency of reasoning proc-esses) and dynamic qualitative changes (due to insights that can lead to a reorganization of memory).

Research indicates that discovering patterns or relations, indeed, facilitates mastery with fluency. For example, children may first memorize a few n + 1 combinations by rote. Once they recognize that such combinations are related to their existing counting knowl-edge—specifically their (already efficient) number-after knowledge—they do not have to repeatedly practice the remaining n + 1 combinations to learn them. Discovery of the number-after rule (“the sum of n + 1 is the number after n in the counting sequence”) al-lows children to efficiently deduce the sum of any n + 1 combination for which they know the counting sequence, even those not previously practiced (Baroody, 1988, 1989, 1992a). This can include large combinations such as 1,000,128 + 1. (Note that the appli-cation of the number-after rule with multi-digit numbers builds on previously learned and automatic rules for generating the counting sequence.) With practice, the number-after rule for n + 1 combinations becomes automatic and can be applied quickly, efficiently, and without thought.


Constructing an explicit knowledge of big ideas—overarching concepts applicable to

many topics and applications—is particularly important for constructing number sense, the well-connected knowledge that provides the basis for reasoning strategies and com-bination mastery with fluency (Baroody, Cibulsksis, Lai, & Li, 2004; see Table 2). For ex-ample, children who understand the big idea of composition (a number can be com-posed in different ways using different parts; e.g., 1 + 7, 2 + 6, 3 + 5, and 4 + 4 = 8…) and decomposition (a number can be decomposed into constitute parts in different ways; e.g., 8 = 1 + 7, 2 + 6, 3 + 5, 4 + 4) are more likely to invent reasoning strategies, such as the “doubles + 1” (e.g., 7 + 8 = 7 + 7 + 1 = 14 + 1) or “make a ten” (9 + 7 = 9 + 1 + 6 = 10 + 6 = 16).

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2. Mastery with fluency with basic combinations involves multiple (efficient)

processes. The memory network of mental arithmetic experts involves interconnected concepts, facts, and reasoning processes. Recent evidence further indicates that the basic number-combination knowledge of mental-arithmetic experts is not merely a collection of isolated or discrete facts but a web of richly interconnected ideas. For example, evidence indicates that even adults use a variety of methods, including efficient reasoning strategies or fast counting, to accurately and quickly determine answers to basic combinations (Le-Fevre et al., 2003). Furthermore, an understanding of commutativity not only enables chil-dren to learn all basic multiplication combinations by practicing only half of them (Baroody, 1999b) but may also enable our memory to store both combinations as a single represen-tation (Butterworth, Marschesini, & Girelli, 2003; Rickard & Bourne, 1996; Rickard, Healy, & Bourne, 1994; Sokol, McCloskey, Cohen, & Aliminosa, 1991). This view is further sup-ported by the observation that the calculation prowess of arithmetic savants does not stem from a rich store of isolated facts but from a rich number sense (Heavey, 2003). In brief, Phase 1 and 2 are essential for laying the conceptual groundwork (relational knowledge) and providing reasoning strategies that underlie mastery with fluency (Phase 3).

3. The evolution of computational fluency entails the increasing integration of factual, conceptual (relational), and procedural knowledge. Thus Phase 3 does not replace Phases 1 and 2 but represents an efficient elaboration of these earlier phases. Although instruction by rote leads to compartmentalization, meaningful instruction and learning (the growth of interconnected knowledge) can lead to increasingly fuzzy bounda-ries among different components of knowledge (Baroody, 2003). It follows from the num-ber sense perspective that other aspects of computational fluency such as estimation abil-ity with basic combinations, the development of counting-based strategies, the ability to solve non-routine (e.g., missing-addend) problems or expressions (4 + ? = 7), and the ca-pacity to perform multi-digit mental arithmetic (estimation and exact mental calculation) also grow out of a child’s developing number sense and are interconnected with children’s growing fluency with basic addition and subtraction combinations.


Aim and Method

Summarized in Figure 2 is the hypothetical learning trajectory (HLT) for how children

develop adaptive expertise with basic addition and subtraction combinations suggested by the schema-based (number sense) view. The aim of the present paper is to discuss preliminary (case study) evidence regarding the validity of the schema-based (number sense) view and the HLT depicted in Figure 2.

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The focus of the report is two long-term case studies of a preschooler and a kinder-gartner. Alice was 20 months old when the 54-month case study started. She is a bilin-gual child whose speech in both English and her native Romanian became coherent at about 30 months. She started to go to an English-speaking day-care when she was 19 months old. At home, both Alice’s mother (a former teacher with a Master’s degree in Mathematics and currently a doctoral student in education) and father (a professor in Computer Science) encouraged verbal number recognition in Romanian. At day-care, Alice typically was involved in verbal counting by rote, object counting, or recognizing written numbers (all in English). When she was 34 months old, Alice’s day-care experi-ences were supplemented by intensive use of books and applications of numbers (e.g., each child’s name and age were placed on a poster). Alices’s behavior and progress during the case study were monitored and recorded in an electronic diary by the second author (her mother) and discussed with the first author. Data were collected on both adult and child-initiated situations.

Five-year-old Aaron was a participant in an 8-month microgenetic study of kinder-gartners’ addition-development (see Baroody, 1987a, 1987b, 1995; Baroody & Tiilika-inen, 2003). Details about both case studies are available upon request. These results are supplemented by other long-term micro-genetic studies or short-term case studies of 3 to 5 year olds.

Results and Discussion The Case of Alice

The case of Alice underscores the key role verbal number recognition (reliably and

efficiently recognizing the number of items in small collections and labeling them with a number word) plays in initial number sense development. Specifically, it appears to un-derlie several number and arithmetic concepts and skills (paths b, d, e, & h in Figure 2), including a basic understanding of composition and decomposition (path c), which ap-pears to be a critical foundation for mastery with fluency of basic addition and subtrac-tion combinations. The case of Alice illustrates paths c, e, d, and perhaps b and h.

25 months old: Alice noticed stones near a lake. She called her mother’s attention

to the size of the stones, “A big stone!” [Mother, “Yes.”] “A small stone” [Mother, “Yes.”] “Another small stone!” [Mother, “Yes, Alice…”] “Two small stones!” [Mother, “Great, Al-ice!”]


28 months old: Asked by her mother how many crayons she had, Alice answered,

“Two [raising two crayons] and one” [raising one crayon with the other hand]; “three!” Asked for four items, Alice on numerous occasions picked up two items with one hand and announced, “Two,” did the same with the other hand, and then announced the total “four.”

30 months old (case A): While her mother read her a bedtime story, Alice readily

recognized collections of one and two. She was initially confused by a picture of three children but quickly recovered and announced, “Two kids and one, three!” Shown a pic-ture of four puppies, the girl commented on what the dogs were doing and how they looked. Suddenly Alice asked, “How many?” [Mother: “Don’t you know?”]. Alice put two fingers of her left hand on two dogs and said, “Two.” While maintaining this posture, she placed two fingers of her right hand on the other two puppies and said, “Two.” She con-cluded, “Four.”

30 months old (case B): At bedtime, Alice pointed to a picture in the book

her mother is reading and says, “A girl, a boy, another girl...three!” 30 months old (case C): While playing on the sofa with crayons, Alice whis-

pered: “Yellow [labeling one yellow crayon], blue [labeling a blue crayon], yellow [labeling a second yellow crayon]...three.

31 months old. Several weeks after the previous episode, Alice showed her father the book just discussed. For the picture of four dogs, she again engaged in the same decomposition (“two and two”) and composition process (“four”). Going on the next pic-ture of five soldiers, Alice noted, “Two, two, and one!”

54 months old. Asked to get 8 crayons, Alice made a collection of 8 crayons con-

sisting of 3 yellow, 3 red, and 2 green crayons. As she did this out of the view of her mother, it is unclear how Alice arrived at solution. When she how many she would have if there were 3 yellow, 3 red, and 3 green crayon, she immediately and mentally, (without counting) recognized the new total would be 9 crayons. However, when asked how many 4 red crayons and 4 yellow crayons would make altogether, she quickly took 4 red crayons and 4 yellow crayons and counted all the crayons to determine the sum.

The results summarized above are not unique to Alice. We have found that many

children act in a similar manner. For example, on a immediate verbal number recognition task, 4 year-old Ari announced, “That’s three, because two and one is three.”

The previous examples illustrate the following points: 1. Verbal number recognition can enable children to see two as one and one

and three as two and one or as one and one and one—as a collection composed of units or a whole composed of individual parts—and, thus, construct a true con-cept of cardinal number (Baroody, Lai, & Mix, 2006; Freeman, 1912). Piagetians (e.g., von Glasersfeld, 1982) have long argued that this skill is meaningless and does not indi-cate a concept of number because children see a collection as a whole, but not as a whole composed of individual parts. In this skills-first view, children also learn to enu-merate collections as a rote skill, because children focus on the individual items but not the whole. von Glasersfeld argued that both verbal number recognition and enumeration


did not take on meaning (result in a concept of number) until both skills were applied in conjunction. This enabled a child to see a collection as a whole comprised of parts. The vignettes of Alice at 25 and 30 (cases B & C) months of age, in particular, indicate that her verbal number recognition of small collections involved seeing a collection both as a whole and in terms of its constitute parts. In other words, this skill co-evolved with con-structing an understanding of number. In brief, in the skills first view, verbal number rec-ognition initially represents routine expertise, whereas, in the simultaneous view, it em-bodies adaptive expertise. See Table 3 for a more detailed discussion of the skills-first view and the simultaneous (development of skills and concepts) view—as well as the concepts-first view.

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2. Because it embodies adaptive expertise, verbal number recognition in eve-ryday situations can serve to promote concepts of composition and decomposi-tion which, in turn, can extend children’s understanding of part-whole relations and addition ability.

• Because verbal number recognition allowed Alice to repeatedly see (de-compose) “two” as “one and one” and “three” as “two and one,” she ap-peared to learn the basic combinations that “one and another one make two” (1 + 1 = 2) and “two and another one make three” (2 + 1).

• Verbal recognition of two and three also allowed her to decompose larger collections she could not recognize and provided a basis for meaningfully learning larger sums. Based on her behavior described in the 30-month (case A) and 31-month vignettes, it appeared that she could not recognize and label collections larger than four or five items. To quantify these “larger” collections, she decomposed them into collections she could readily recognize (4 ➝ 2 & 2; 5 ➝ 2, 2, & 1). She then used the known relation “2 & 2 makes 4” (learned from her parents) to specify the cardinal value of the collection. In James’ (1958) words, her parents made a conscious effort to connect the new concept of “four” to something Alice already knew. Specifically, they built on her meaningful knowledge of the cardinal number two and decomposition (that a number can be decomposed into its parts; e.g., 2 ➝ 1 + 1) and composition (that parts can be composed into a whole; e.g., 1 + 1 ➝ 2). As a result, Alice was able to assimilate “2 and 2 is 4” in relatively quick manner and retain this factual knowledge. At 31 months, this process was just getting started for collections of five. The process of learning to recognize four and five illustrates how integrated conceptual and procedural knowledge (meaningful verbal number recognition to decompose a collection) used in conjunction with factual knowledge (a known fact such as 2 + 2 = 4) can work together to expand all three (e.g., conceptual knowledge of the cardinal concept of four, the skill of verbally recognizing four, and the factual knowledge that 2 + 2 + 1 = 5 and perhaps ultimately that 2 + 2 + 1 = 2 + (2 + 1) = 2 + 3 = 5).

• At 54 months, Alice recognized immediately that 3 + 3 + (2 + 1) = 8 + 1 = 9—that incrementing one of the parts by 1 also increased the original whole of 8 by 1. The sum 8 + 1 was probably quickly constructed by building on her number after knowledge (i.e., the sum of 8 + 1 is the number after 8 in the counting sequence:


“9”). Put differently, Alice appeared to recognize the co-variation principle: adding items to (or subtracting items from) a part increases (decreases) the whole by the same amount—if Part 1 + Part 2 = Whole, then [Part 1 + a number] + Part 2 = Whole + the number (Irwin, 1996). The co-variation principle represents an im-portant insight into part-whole relations or additive composition.

3. The results just discussed indicate that immediate number recognition (su-

bitizing) even the “intuitive numbers” may actually involve several different proc-esses, including decomposition and re-composition). Immediate number recogni-tion, often called subitizing, entails quickly recognizing the number of items in a collec-tion without counting. Although sometimes viewed as a single process, it probably in-volves a number of skills, including nonverbal number recognition (“nonverbal subitiz-

ing”; e.g., equating • • • with ) and immediate verbal number recognition (“verbal subitizing”; e.g., immediately recognizing or labeling • • • or • • • as “three”). The latter

encompasses recognizing regular patterns (e.g., equating a triangular array such as ••• with “three”) or decomposing collections into smaller recognizable collections and either using addition or multiplication to determine the total (e.g., viewing as 2 and 3 = 5

or viewing as 3 groups of 4 or 12). The Case of Aaron

The case of Aaron illustrates James’ (1958) point that “the art of remembering is the art of thinking”—that an effort to fix a new thing in mind should focus on connecting it to something already known. As such, it illustrates that fluency with basic combinations is a by-product of growing number sense and that mastery of basic combinations entails multiple efficient processes. The case of Aaron further shows how the development of conceptual understanding can promote adaptive expertise in various areas of computa-tional fluency. More specifically, the case demonstrates how number sense can help computational estimation, computational proficiency in the form of counting-based pro-cedures, and mastery of basic number combinations co-evolve in an interrelated fashion.

From the beginning of the study, Aaron seemed to efficiently “retrieve” the sum for

word problems or expressions involving a small number plus 1 (e.g., 3 + 1 or 5 + 1). However, when his responses are considered in the context of his responses to other combinations, it appears that his correct responses to n + 1 combination were due to a conceptually guided reasoning process. Specifically, Aaron apparently understood that addition is an incrementing process and that his counting knowledge—including his number-after knowledge—was useful in indicating the correct direction of sums (e.g., responding to 3 + 1 with an answer of 4). This conceptual knowledge, though, was ap-plied non-selectively to all problems or expressions (i.e., even to those not involving one). In Session 6, for example, the boy responded to m + n or n + m combinations (where 1 < m < n) with the number just after the larger addend (e.g., responded to 3 + 4 with “5”).

Aaron’s consistent response pattern (systematic error) appeared to be the by-

product of a relatively inflexible estimation strategy but one that at least was consistent with his conceptual knowledge that addition is an incrementing process. His mechanical strategy (Strategy D in Table 4) is more advanced than other mechanical estimation strategies that are not entirely consistent with conception (Strategy C) or do not honor it


at all (Strategies A and B) and that also produce systematic errors. (Unlike Aaron, many novices to verbal-addition resort to the highly mechanical and ineffective Strategies A, B and C [Baroody, 1988, 1989, 1992a; Dowker, 2003].)

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In time, Aaron’s inflexible estimation strategy was replaced simultaneously by more and more sophisticated strategies for responding to combinations involving 1 and those involving the addition of 2 or more. In the fourth session when presented 1 + 7, Aaron verbally counted from one to seven and concluded, “I have to do it with the blocks.” In Session 6, he responded to 1 + 3, for instance, with: “Four or something” (see the semi-flexible Strategy E in Table 4). These responses indicated that he did not yet recognize that 1 + n and n + 1 were equivalent and that both could be determined exactly by stat-ing the number after n. Aaron did not generalize the number-after-n rule to 1 + n combi-nations until Session 9 or so, when he first also applied this rule discriminately—that is, to combinations involving 1 but not to m + n combinations (the flexible Strategy F in Ta-ble 4).

During this same time period that he used the semi-flexible Strategy E (in Table 4)

with combinations involving 1, Aaron often responded to m + n combinations with a range of numbers just past n (e.g., responding to 3 + 4 with “5 or 6” or “about 4, 5, or 6”; see again Strategy E in Table 4). In other words, his strategy for larger combinations was still not distinct from that for n + 1 or 1 + n combinations. The use of this semi-flexible estimation strategy for m + n trials, however, dropped from a high of 60% in Ses-sion 5 to 25% in Session 8 and then 0% in Session 9. Paralleling this trend, the use of a more flexible and effective (state a number a few past the larger addend) estimation strategy for m + n trials (e.g., for 4 + 3, guessing 6; again see Strategy F in Table 4)—went from 40% to 50% and then to 100%.

Aaron’s construction of a precisely (appropriately) applied number-after-n rule for n +

1 and 1 + n combinations apparently served as a scaffold for his invention of the abstract counting-on from larger addend procedure (Procedure 6 in Table 5). For a span of 8 sessions (3.5 months), the kindergartner relied on a concrete counting-all strategy to de-termine sums exactly (see Procedure 1 in Table 5). Nevertheless, he sporadically tried throughout this period to compute sums in a more efficient manner. When presented 2 + 3 in Session 4, he commented, “I want to try it without blocks.” He then proceeded to verbally count, “One, two (pause), three,” and noted, “I’m almost there but I ran out of thinking.” He then abandoned the effort and fell back on concrete counting-all. For 2 + 4, his attempt to use an abstract strategy resulted in counting to ten, concluding, “I don’t know,” and finally resorting to concrete counting-all with blocks to determine the sum. For 1 + 3, Aaron counted up to five. After the interviewer commented, “Close,” the boy responded, “Four.” In subsequent sessions, he invented the abstract counting-all begin-ning with the larger addend strategy (Procedure 4 in Table 5) for 1 + n items and later an abstract strategy for larger combinations.


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Aaron’s performance in Session 4 suggests he recognized that the sum for 1 + n and m + n combinations had to be somewhat more than n and could in theory be determined by continuing the counting process past the cardinal value of the larger addend. This understanding (the embedded-addends concept; Achievement 4 in Table 5) was mani-fested as an imperfect but reasonable “counting just past the cardinal number of the larger addend” strategy (e.g., counting up to five for 1 + 3) and his subsequent educated guess of four. Consistent with a concepts-first view of development, what Aaron seemed to lack was the keeping-track procedure necessary to continue the count accurately. (This is what Aaron seems to have meant in Session 4 by “I ran out of thinking.”) Not surprisingly, he first correctly applied the embedded-addends concept with 1 + n combi-nations, which entail minimal keeping track (a single application of number-after knowl-edge) and then did so four months later (Session 13) with larger combinations, which entail a more demanding keeping-track process (e.g., 2 + 3: the sum is three numbers after “one, two”—“so, three is 1 after, four is 2 after, five is 3 after”; Baroody, 1995; Baroody & Ginsburg, 1986).

Children’s experience with n + 1 combinations may provide the impetus for con-

structing an embedded cardinal-count concept (Achievement 6 in Table 5). Aaron began responding to nearly all n + 1 and 1 + n combinations automatically in a consistent man-ner during Session 11 and did so with all such combinations in Session 12. At this same time, he first used counting on from the larger addend (abstract counting-on; Procedure 5 in Table 5) with 2 + 6. In Session 13, he began to use this advanced strategy regularly.

Like Aaron, seven other children in a microgenetic study invented abstract counting-

on shortly after discovering the number-after rule for n + 1 (and 1 + n) combinations: the sum of only such combinations is the number after n in the counting sequence (e.g., the sum of 5 + 1 is the counting number after five: six; Baroody, 1995; see also Bråten, 1996). This refinement in the integration of addition and counting knowledge appears to provide a scaffold for abstract counting-on with larger combinations. A child might rea-son for 5 + 3 that if 5 and 1 more is the number after five in the counting sequence, then the sum of 5 + 3 must be three numbers after five: six, seven, eight. Because children frequently discover the labor-saving device of disregarding addend order (Achievement 5 in Table 2) before constructing the relatively advanced conceptual basis for counting-on (Achievement 6 in Table 5), Procedure 4 in Table 5 (CAL is the most common transi-tion strategy between CAL and abstract counting-on (Procedure 6 in Table 5). This also explains why counting-on from the first addend (COF; e.g., 3 + 5, counting: “3, 4, [is one more], 5 [is two more], 6 [is three more], 7 [is four more], 8 [is five more]”) is so seldom observed (Baroody, 1987a, 1987b; Baroody & Ginsburg, 1986; Baroody & Tiilikainen, 2003).

Conclusions Theoretical Implications


The passive storage view is an adequate model for how basic combinations are of-ten, or even typically, taught. Such non-conceptual models feature the development of combination mastery as independent from that of number sense or thinking. These, then, are models of the development of routine expertise. The existing evidence that a single retrieval strategy replaces less mature strategies and that arithmetic knowledge is modular in nature may be, in part, an artifact of traditional instructional practices that fos-ter routine expertise.

The number sense view models how basic combinations should be taught. Such

models, with their focus on conceptual development, suggest that the development of combination mastery is dependent on the development of number sense and thinking. These, then, are models of the development of adaptive expertise. Because develop-ment is concept driven, it makes sense that various (related) aspects of computational fluency and thinking itself co-evolve and become increasingly interdependent and inte-grated. That is, as a child constructs deeper concepts (more connections), fluency with basic combinations, other related computational skills (e.g., estimation, computational procedures), and thinking itself (reasoning, and problem solving) increases. Educational Significance

Proponents of the passive storage view recommend focusing on a short-term and di-rect approach, whereas supporters of the active construction view suggest a long-term and indirect approach.

The Direct Short-term Approach (Passive Storage View). According to the passive

storage view, the best approach for ensuring mastery of basic combinations is well-designed, extensive drill and practice (e.g., Goldman, Pellegrino, & Mertz, 1988; Thorndike, 1922). Because of supposed learning and memory deficits, children labeled as “learning disabled” (LD) must “over-learn” material—that is, practice skills many times more than required by typically developing children or use substitute strategies that cir-cumvent their deficits (e.g., learn by rote how to use a calculator; e.g., Sliva, 2004).

In recent years, there has been some concern about a brute force approach—

memorizing all the basic combinations of an operation in relatively short order (e.g., Burns, 1995; Gersten & Chard, 1999). Some have recommended focusing on a few combinations at a time, ensuring one set of facts is mastered before introducing a new one (e.g., Cooke, Guzaukas, Pressley, & Kerr, 1993; Hasselbring, Goin, & Bransford, 1988; Stein, Silbert, & Carnine, 1997; cf. Thorndike, 1922). A controlled- or constant-response-time (CRT) procedure entails giving children only a few seconds to answer and providing them the correct answer if they either respond incorrectly or do not re-spond within the prescribed time frame (e.g., Bezuk & Cegelka, 1995; Goldman & Pelle-grino, 1986; Hasselbring et al.; Koscinski & Gast, 1993a, 1993b; Silbert et al., 1990). These procedures are recommended to minimize (a) associative confusions during learning, (b) reinforcing incorrect associations, and (c) reinforcing “immature” (counting and reasoning) strategies. In this updated version of the conventional view, then, Phase 1 and 2 of number-combination development are still seen as largely unnecessary steps for, or even a barrier to, achieving Phase 3.

However, even if a teacher focuses on small groups of combinations at a time and

uses the CRT procedure, the limitations and difficulties of a rote approach largely re-main. Efforts that focus on memorizing the basic combinations by rote do not guarantee


efficiency, let alone appropriate and flexible use (all aspects of computational fluency). Such learning frequently leads to forgetting or confusion, and seldom results in transfer.

The Indirect Long-term Approach (Number Sense View). Mastery with fluency may best be achieved by instruction that promotes number sense in a purposeful, mean-ingful, and inquiry-based way—that is, fosters all aspects of mathematical proficiency (e.g., conceptual understanding, logical reasoning, and problem-solving ability) in an in-tegrated or intertwined manner, as recommended by the NRC (Kilpatrick et al., 2001). Children who learn the basic combinations in such a manner should have the ability to use this basic knowledge accurately and quickly (efficiently), thoughtfully in both familiar and unfamiliar situations (appropriately), and inventively in new situations (flexibly).

There is some evidence that even children with serious learning difficulties can bene-

fit from an approach based on the number sense view (Gersten & Chard, 1999). Such children are capable of benefiting from a purposeful, meaningful, and inquiry-based in-structional approach and achieving at least modest levels of computational fluency (Baroody, 1987a, 1996a, 1999a; Bottge, 1999; Bottge, Heinrichs, Chan, & Serlin, 2001; Bottge, Heinrichs, Mehta, & Hung, 2002). Children labeled LD or mentally retarded can discover arithmetic relations (Baroody, 1987c; Baroody & Snyder, 1983) and construct pattern-based rules for efficiently generating the sum of n + 0 and n + 1 number combi-nations (Baroody, 1987a, 1988, 1995).

Six instructional implications of this recommendation and current research are:

1. Number sense cannot be imposed on children. A teacher must patiently help stu-dents construct it through exploration, inventing informal strategies, and sharing ideas.

2. Teachers should help pupils explicitly construct big ideas, such as composition and

decomposition.

3. Instruction should promote meaningful memorization of basic combinations by en-couraging children to look for patterns and relations, using these discoveries to con-struct reasoning strategies, and sharing, justifying, and discussing their strategies (Baroody & Coslick, 1998). It follows from this view that instruction should (a) con-centrate on “combination families” (related combinations), not on individual facts, and (b) encourage pupils to build on what they already know, which can greatly reduce the amount of practice needed to master new combinations (see, e.g., Baroody, 1993, 1995). For example, mastering subtraction combinations is easier if they are related to complementary and previously learned addition combinations (e.g., 5 – 3 = ? can be thought of as 3 + ? = 5). (c) Research indicates that unguided discovery learning might be appropriate for highly salient patterns or relations, such as additive commutativity, but that more structured discovery learning activities may be needed for less obvious ones, such as the addition-subtraction complement principle (Baroody, Berent, & Packman, 1982; Baroody, Ginsburg, & Waxman, 1983).

4. The learning and practice of combinations should be done purposefully via worth-

while tasks (games, projects, science experiments, real and imaginary problems) that create a need to explore and use mathematics. Purposeful practice is more ef-


fective than drill and practice in promoting mastery (Bottge, 1999; Bottge et al., 2001, 2002; Carpenter et al., 1989).

5. Practice is important but needs to be used wisely, namely as an opportunity to dis-

cover patterns and relations and to make reasoning strategies more automatic.

6. As “experts” use a variety of strategies, including automatic or semi-automatic rules and reasoning processes, proficiency should be defined broadly as including any ef-ficient strategy and students should be encouraged to flexibly use a variety of strate-gies.

Another key implication of the number sense view is that educators must take the

long view if they wish to truly help children, particularly those at risk for school failure, to achieve fluency with basic combinations. Long-term, conceptually based intervention that begins early and gradually helps children at risk for or with learning difficulties con-struct number sense could prevent, remedy, or at least minimize many, if not most, diffi-culties with basic combinations and other aspects of computational fluency. In brief, early childhood and elementary education should focus on fostering number sense and adaptive expertise. A natural outcome of such instruction will be computational fluency with basic combinations and other aspects of thinking.

Author Information: Arthur J. Baroody University of Illinois at Urbana-Champaign 217-333-4791 [email protected] Luisa Rosu University of Illinois at Urbana-Champaign 217-649-1133 [email protected]


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FIGURE 1: Babylonian Clay Tablet (circa 4000 B.C.)

Tablet 1 Tablet 2

Expertise with Combinations 23 FIGURE 2: A Hypothetical Learning Trajectory of How Some Key Number and Arithmetic Skills May Underlie Combination Fluency

verbal number recognition

counting sequence skills [3, 12, 21] enumeration

[6, 28]

ordinal concepts (meaningful number

comparisons) [4]

counting-based number comparisons

automatic mental number

comparisons [19, 20]

part-whole relations in-cluding composition &

decomposition

Qualitatively reason about part-whole prob-

lems [17]

Solving missing addend problems

exactly

Different names for a number

Complement principle

number-after rule for n+1/1+n

combinations

Other reasoning strategies for basic addition combinations

Reasoning strategies for multidigit addition

& subtraction

addition & subtraction concepts [8]

concrete counting-all

[16]

abstract counting-all

[26]

abstract counting-on

[32]

Mastery of ba-sic addition

combinations

Mastery of basic subtraction com-

binations

number after [13]

decade after

c

i

e

e

j

i j l

n

j verbal set production

[10, 11]

k

i

b

i

g

a

f

n

h

d

h

k

b

cardinality concepts [7, 9]

o

q

k

m o

o

p

p

s s

s

w w

m

r

u

v w

w

t

t

v v


TABLE 1: A Comparison of the Passive Storage and Number Sense Views

Passive Storage View Number Sense

(Active Construction) View Unit of learning Association between an

expression and its answer Relational knowledge

(patterns and relations)

Mechanism of learning Memorization by rote—Observing and practicing

(strengthening) an association

Meaningful memorization—Discovering patterns and

relations—e.g., connecting new information to existing

knowledge

Type of mental changes Passive quantitative change (increase) in associative strength

Passive quantitative and dynamic qualitative change (insights ➞ reorganizations

in memory)

Role of conceptual knowledge

Not necessary Essential

Type of expertise Routine expertise Adaptive expertise

Rate of learning Achieved in short order Achieved gradually

Mental Representation of mental arithmetic expert

Autonomous fact-retrieval network

and single (fact) retrieval

process (rigid system)

Integrated web of facts and relations

and multiple retrieval processes

(flexible system)


TABLE 2: An Example of a Big Idea: Composition and Decomposition

Big ideas are overarching concepts that connect numerous topics and applica-tions within and among domains (Baroody et al., 2004). A fundamental idea underlying various aspects or topics of mathematics is that a quantity or whole can consist of parts and be broken apart (decomposed) into them and that the parts can be combined (com-posed) to form the whole:

• Other names for a number (e.g., different parts can compose to make the same whole, such as 1 + 7, 2 + 6, 3 + 5, and 4 + 4 = 8… and a whole can be decom-posed in constitute parts in different ways, such as 8 = 1 + 7, 2 + 6, 3 + 5, 4 + 4…)

• Invention of reasoning strategies and mastery of larger addition combinations

(e.g., 7+ 8 = 7 + [7+ 1] = [7 + 7] + 1 = 14 + 1= 15 or 7+ 8 = [7 – 2] + [8 + 2] = 5 + 10 = 15)

• Part-whole number relations (e.g., one of two or more parts is smaller than its whole and conversely a whole is larger than any one of its multiple parts)

• Missing-addend (part) problems (e.g., in the problem below or in the equation 4 + ?

= 6, the missing part must be smaller than the whole 6 and when added to 4 equals 6)

Georgia had 4 dresses. Her mother bought her some more. Georgia found 6 dresses hanging in her closet. How many new dresses did Georgia's mom buy?

• Invention of reasoning strategies and mastery of subtraction combinations (e.g.,

for 6 – 4 = ?, the whole 6 minus the part 4 is equal to the other part, which when added to 4 makes 6; i.e., 6 – 4 = ? is related to the missing-addend expression 4 + ? = 6 and, because 4 + 2 = 6, the unknown part is 2)

• Renaming (carrying and borrowing) procedures (e.g., for 37 + 28, a child must be able to recognize that 15 (the sum of the ones digits 7 + 8) can be decomposed into a 10 and a 5 and that this 10 must be added to the three tens and two tens shown in the tens place)

• Geometry (e.g., a square can be decomposed into two [right, isosceles] triangles and vice versa).


TABLE 3: Three Views of the Developmental Relations Among Number Recogni-

tion, Number Words, a Number Concept, and Object Counting

Whether immediate nonverbal number recognition (nonverbal subitizing), immediate

verbal number recognition (verbal subitizing), or object counting (enumeration) indicate a conceptual understanding of number is a matter of considerable debate. Proponents of the Piagetian or extremely pessimistic view imply a skills-before-concepts (skills-first) view, whereas nativists or proponents of the extremely optimistic view advocate a con-cepts-first view. A third perspective is the simultaneous view. Piagetian Skills-First View: The simultaneous application of meaningless verbal su-bitizing and object-counting skills leads to a number concept and the meaningful use of these skills. In the skills-first view, number and arithmetic skills are learned by rote memorization through imitation, practice, and reinforcement. The result is that skill learning is piecemeal. Through applying their skills, children discover number (and arithmetic) regu-larities view and concepts. Specifically, von Glasersfeld (1982) argued that the early recit-ing of number words is a meaningless skill. He also hypothesized that the skill of verbal subitizing is initially a perceptual skill that does not imply an understanding of number, be-cause children view collections as perceptual configurations—as a set or whole, not as a set of individual items or units. He further argued that enumeration is also initially a skill learned by rote. Although the counting process forces children to treat counted objects as individual items (units), they do not conceptually understand that counting is a tool for de-termining the total number of units (the cardinal value of the collection or the whole). Thus, the initial use of number words, verbal subitizing, and enumeration do not entail viewing a collection in terms of a true number concept—as a set of individual items. All three skills do not start to become meaningful until children use them in conjunction. By enumerating a collection they also can immediately recognize and verbally label, children can realize that the former results in the total and that the perceptual configuration of the latter is com-prised of units. In brief, the simultaneous application of meaningless verbal subitizing and object-counting skills leads to a number concept and the meaningful use of these skills. Nativists’ Concepts-First View: Innate Counting Principles Underlie Meaningful Nonverbal Number Recognition by Infants and Rapid Acquisition of Meaningful Number Word and Object Counting (Enumeration) Skills by Toddlers. In the con-cepts-first view, children’s conceptual understanding enables them to devise meaningful procedures or skills. According to nativists’ accounts (e.g., Gelman & Gallistel, 1978), chil-dren have an innate understanding of number and counting principles, and this prior un-derstanding underlies infants’ ability to nonverbally subitize the number of items in small collections and toddlers’ capacity to rapidly learn number words and enumeration proce-dures. Simultaneous View: Meaningful Number Recognition Underlies Natural Object Counting (Enumeration). According to the simultaneous view, conceptual knowledge can lead to the invention of procedural knowledge, the application of which can lead to a con-ceptual advance, which in turn can lead to more sophisticated procedural knowledge, and so forth (Baroody, 1992b; Rittle-Johnson & Siegler, 1998). Alternatively, a skill can be learned by rote and its application can lead to the discovery of a concept. This understand-


TABLE 3 (continued) ing can lead to a procedural advance, and reflection of its application can lead to a deeper understanding and so forth. In some cases, concept and skill co-evolve (the simultaneous view; Rittle-Johnson & Siegler).

Children’s initial use of the first few number words may well be meaningless as von Glasersfeld (1982) suggested. However, the use of these words in conjunction with see-ing examples and non-examples of each intuitive number can imbue them with meaning. As children hone the skill of verbal number recognition, they construct a cardinal concept of one, two, three, and four—abstracting oneness, twoness, threeness, and fourness (Baroody, Benson, & Lai, 2003; Benson & Baroody, 2002). In this view, then, children’s initial procedural knowledge (the verbal number recognition skill) and conceptual knowl-edge (cardinal concepts of one to about three) develop simultaneously. Moreover, this co-evolution of number skill and concept may also enable children to see two as one and one and three as two and one or as one and one and one. That is, meaningful verbal number recognition enables children to see a collection composed of units or a whole composed of individual parts (Freeman, 1912).


TABLE 4: Evolution of Mental Arithmetic Strategies from Most Mechanical to Most Adaptive Using 3 + 1, 1 + 3, 5 + 3, and 3 + 5 as Examples

Response Patterns Sample Strategies 3 + 1 1 + 3 5 + 3 3 + 5

Mechanical strategies A. State last addend rule “1” “3” “3” “5”

B. State larger addend rule “3” “3” “5” “5”

C. Global number-after rule:

state the number after an ad-dend

“4” or “2” “4” or “2” “6” or “3” “6” or “3”

D. Number-after larger-addend rule

“4” “4” “6” “6”

Semi-flexible strategies

E. Extension of number-after larger-addend rule: state the number after the larger ad-dend or—for combinations other n + 1—then some

“4” “4” or “5” “6” or “7” “6” or “7”

Flexible strategies F. Precise (appropriately ap-

plied) number-after rule for n + 1/1 + n combinations and true estimation for m + n items

“4” “4” “7” or “8” “7” or “8”

Note. Mechanical strategies are essentially a response bias or an inflexibly applied esti-mation strategy that, like written buggy algorithms, generate a detectable error pattern. The more primitive of these strategies (A, B, and C above) are typically used blindly and, thus, may generate unreasonable estimates (sums that do not make sense).


TABLE 5: The Conceptual Bases and Major Achievements in the Development of Informal Counting-Based Procedures for Computing Sums, Related Procedure Classes, and Illustrative Procedures Using 3+5 as an Example

Conceptual Basis–Achievement Procedure Class Illustrative Procedure Concrete or Level 1 Procedures 1. An informal change-add-to concept of addition, previ-

ously constructed from verbal subitizing experiences, enables children to invent or assimilate a direct model-ing procedure. That is, their preexisting goal sketch empowers them to model the meaning of a problem or expression or comprehend this modeling process by another.

1. Concrete counting-all entails three separate and sequen-tial counts: Step 1: Count out items to represent the first addend (an

addend count). Step 2: Count out items to represent the second addend

(a second addend count). Step 3: Count all the items put out to determine the sum

(a sum count).

For example, for 3+5, a child might (a) successively count and raise three fingers to con-

cretely represent the starting amount 3, (b) successively count and raise five fingers to con-

cretely represent the added-on amount 5, (c) and then count all (eight) raised fingers to determine

the sum. 2. Integrating knowledge about cardinal representations

of number and addition enables children to shortcut the laborious concrete counting-all procedure. For exam-ple, typically developing children learn finger patterns for representing the numbers up to at least five before entering school (Ginsburg & Baroody, 2003), and they often quickly learn to use this number knowledge to immediately represent one or both addends or to quickly recognize a sum without counting (Baroody, 1987b, see Figure 1 on p. 142; Baroody & Gannon, 1984). Even children with severe learning difficulties quickly draw on finger and number patterns to simplify the concrete counting-all procedure (Baroody, 1996b).

2. Concrete counting-all shortcuts involve using number or finger patterns to short-cut one or more steps in the di-rect-modeling procedure listed previously—to represent one or both addends or to determine the sum.

For 3+5, children commonly (a) put up three fingers simultaneously to represent the original amount 3, (b) then raise five fingers simultaneously to represent the amount added (5 more), and (c) finally determine the sum by counting all (eight) of the extended fingers.

Children who can readily use finger patterns up to 10 might further shortcut the computation process for 3+5 by immediately recognizing the eight extended fingers as “eight ”— instead of counting the eight ex-tended fingers. This most advanced concrete count-ing-all shortcut procedure is first observed with sums up to five and is sometimes called the “finger-recognition strategy ” (e.g., Siegler & Jenkins, 1989).

3. Some children construct an embedded-addend con-cept—realize that representing one of the addends (e.g., one addend count) can be embedded in (i.e., done simultaneously with) the sum count, which en-ables them to invent semi-indirect modeling proce-dures. That is, they no longer have to directly model (concretely represent) one of the addends. Such pro-cedures are particularly useful for sums greater than 10, where it is difficult to represent both addends on the fingers of two hands.

3. Concrete counting of the added-on amount requires two separate and sequential counts — one to directly model the added-on amount (Step 1 next) and the second to verbally (indirectly) represent the starting amount (Step 2 next) and to determine the sum (Steps 2 and 3 next): Step 1: Count or put out objects to represent the amount

added-on (an addend count). Step 2: Count up to the cardinal value to represent the

starting amount (first portion of the sum count done in conjunction with a second but embedded addend count).

Step 3: Continue the sum count from this number by counting out the items previously counted or put out.

For example, for 3+5, a child might (a) first extend five fingers to represent the added-on amount, (b) next verbally count up to the cardinal value of the

starting amount (one, two, three), and (c) then continue this count as he or she pointed, in

turn, to each of the previously extended finger (four, five, six, seven, eight).


TABLE 5 (Continued)

Conceptual Basis–Achievement Procedure Class Illustrative Procedure Abstract or Level 2 Procedures 4. Discovery of the embedded-addends concept— that

representing both addends can be done simultaneously with the sum count—enables children to devise indi-rect-modeling procedures executing both addend counts simultaneously with the sum count. This re-quires a keeping-track process (continuing a verbal sum count from a number for a specified interval).

4. Abstract counting-all involves starting one count (Step 1) and then executing this count and the second count si-multaneously (Step 2 next): Step 1: Begin the sum count by counting up to the cardi-

nal value of the first addend (simultaneously represent the first portion of the sum count and the starting amount).

Step 2: To indicate how much more was added, complete the sum count by counting past the cardinal value of the first addend the number of times specified by the second addend. This necessitates a keeping-track process done in tandem (simultaneously) with a sum count.

For 3+5, for instance, a child might “count-all beginning with the first addend” (CAF):

(a) verbally count out the starting amount (“One, two,

three”) and then (b) count, “four [is one more], five [is two more], six [is three more], seven [is four more], eight [is five more].” Note that the portion in brackets is the keeping-track process.

5. Discovery of additive commutativity or the need to save labor leads to disregarding addend order. Because of their informal change-add-to concept of addition, chil-dren typically feel compelled to deal with the addends in the order they are specified in word problems or symbolic expressions (Baroody, 1987a, 1987b; Baroody & Ginsburg, 1986; Baroody & Tiilikainen, 2003; DeCorte & Verschaffel, 1987; Wilkins, Baroody, & Tiilikainen, 2001). For items such as 3+5, represent-ing the larger addend first can minimize the keeping-track process and, thus, reduce the load on working memory.

5. Abstract procedures that disregard addend order entail three steps: Step 1: Choose the larger addend. Step 2: Begin the sum count (and simultaneous addend

count) by counting up to the cardinal value of the larger addend.

Step 3: Complete the sum count by counting beyond the cardinal value of the larger addend the number of times specified by the simultaneous keeping-track process (the addend count for the smaller addend).

For 3+5, “counting-all beginning with the larger addend” (CAL) would take the form of

(a) Note that 5 is more than 3; (b) starting with one, count up to the cardinal value of

the larger addend (“ … two, three, four, five …”); and then

(c) continue the count for three more terms: “six [is one more], seven [is two more], eight [is three more].”

6. Discovery of the embedded cardinal-count concept (in the context of computing sums, stating the cardinal value of an addend is equivalent to counting up to this number) enables children to count-on (Fuson, 1992). This permits starting the sum count with the cardinal value of the addend (counting-on) instead of counting from one (counting-all).

6. Abstract counting-on involves the same three steps as Procedure Class 5 (if Achievement 5 has been obtained), except that Step 2 is abbreviated by stating the cardinal number of the larger addend.

For 3+5, “abstract counting-on from the larger addend ” (abstract COL) might take the form of (a) Identify the larger addend 5; (b) state “Five” (instead of counting from “one” to “five”); and (c) count on three more times: “six [is one more], seven [is two more], eight [is three more].”

Note. Children do not necessarily go through all six phases or invent Class 1 to 6 procedures in the order shown here. Considerable evidence indicates some strong trends how-ever. Children in the United States, in any case, typically use Class 1 or 2 procedures as their initial counting-based strategy. Many next invent CAF (a Class 4 procedure), then CAL (a Class 5 procedure), and finally COL (a Class 6 procedure). Nevertheless, research has indicated many children do not easily or otherwise fit this pattern of development.

aA modified version of CAF is identical to CAF, except that a child would simultaneously represent the first addend with a verbal count and with objects (e.g., raising a finger with

each count). The same would be true for modified versions of the procedures discussed in Classes 5 and 6.

adaptive expertise with basic math

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