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 Foundation (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
Expertise with Combinations 2
ADAPTIVE EXPERTISE WITH
BASIC ADDITION AND SUBTRACTION COMBINATIONS—
THE NUMBER SENSE VIEW
Mastery—the efficient (quick and accurate) reproduction—of the basic number combinations, which include the single-digit sums to 18 (e.g., 9 + 3 = 12) and related differences (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 education 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).
---------------------------------Insert Figure 1 about here
----------------------------------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 knowledge) can be equated with what Hatano named adaptive expertise—meaningfully
learned knowledge that can be creatively (flexibly) adapted and effectively (appropriately) 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 recently, 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 perspective 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).
Expertise with Combinations 3
Information-processing models of number combination learning build heavily on
Thorndike’s (1922) associative-learning theory and focus on skill development (routine expertise). This perspective constitutes the conventional wisdom among cognitive 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 William 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 attend 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 refers to different constructs. For James, it refers to meaningful assimilation and reflection. For Thorndike, the connection is an unthinking association between a stimulus
and a response.) One embodiment of the number sense perspective is a schemabased 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; Steinberg, 1985):
•Phase 1: Counting strategies—using object or verbal counting to determine answers;
•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).
---------------------------------Insert Table 1 about here
-----------------------------------
Expertise with Combinations 4
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 hindrance. 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 assumptions about the process of mastery and the nature of mental-arithmetic expertise.
1. Basic combinations must be memorized as individual facts by rote, and fluency 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 entails 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 mastering 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 neither 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 (calculation 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.
Expertise with Combinations 5
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, including 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 perspective is based on the following three assumptions, for which there is growing research 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 factual knowledge more likely than memorizing individual facts by rote (Bruner, 1963, 1966;
Katona, 1967; Wertheimer, 1959). As with any worthwhile knowledge, meaningful memorization 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 facilitate 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 translated 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 proficient in their execution, they are more likely to discover the myriad of patterns and relations 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 spontaneously 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 process of mastery is gradual and involves both passive quantitative changes (due to the
strengthening of associated factual knowledge or increasing efficiency of reasoning processes) 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 knowledge—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”) allows 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 application 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.
Expertise with Combinations 6
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 combination mastery with fluency (Baroody, Cibulsksis, Lai, & Li, 2004; see Table 2). For example, children who understand the big idea of composition (a number can be composed 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).
---------------------------------Insert Table 2 about here
----------------------------------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 (LeFevre et al., 2003). Furthermore, an understanding of commutativity not only enables children 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 representation (Butterworth, Marschesini, & Girelli, 2003; Rickard & Bourne, 1996; Rickard, Healy,
& Bourne, 1994; Sokol, McCloskey, Cohen, & Aliminosa, 1991). This view is further supported 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 boundaries among different components of knowledge (Baroody, 2003). It follows from the number sense perspective that other aspects of computational fluency such as estimation ability 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 capacity 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.
Expertise with Combinations 7
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.
---------------------------------Insert Figure 2 about here
----------------------------------The focus of the report is two long-term case studies of a preschooler and a kindergartner. Alice was 20 months old when the 54-month case study started. She is a bilingual 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 experiences 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 kindergartners’ addition-development (see Baroody, 1987a, 1987b, 1995; Baroody & Tiilikainen, 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 underlie 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 appears to be a critical foundation for mastery with fluency of basic addition and subtraction 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, Alice!”]
Expertise with Combinations 8
28 months old: Asked
“Two [raising two crayons]
Asked for four items, Alice
and announced, “Two,” did
“four.”
by her mother how many crayons she had, Alice answered,
and one” [raising one crayon with the other hand]; “three!”
on numerous occasions picked up two items with one hand
the same with the other hand, and then announced the total
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 picture 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 concluded, “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 whispered: “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 picture of five soldiers, Alice noted, “Two, two, and one!”
54 months old. Asked to get 8 crayons, Alice made a collection of 8 crayons consisting 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 concept 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 indicate 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 enumerate 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
Expertise with Combinations 9
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 constructing an understanding of number. In brief, in the skills first view, verbal number recognition initially represents routine expertise, whereas, in the simultaneous view, it embodies 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.
---------------------------------Insert Table 3 about here
----------------------------------2. Because it embodies adaptive expertise, verbal number recognition in everyday situations can serve to promote concepts of composition and decomposition which, in turn, can extend children’s understanding of part-whole relations
and addition ability.
• Because verbal number recognition allowed Alice to repeatedly see (decompose) “two” as “one and one” and “three” as “two and one,” she appeared 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:
Expertise with Combinations 10
“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 important insight into part-whole relations or additive composition.
3. The results just discussed indicate that immediate number recognition (subitizing) even the “intuitive numbers” may actually involve several different processes, including decomposition and re-composition). Immediate number recognition, often called subitizing, entails quickly recognizing the number of items in a collection without counting. Although sometimes viewed as a single process, it probably involves 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 computational fluency. More specifically, the case demonstrates how number sense can help
computational estimation, computational proficiency in the form of counting-based procedures, 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 applied 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 byproduct 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
Expertise with Combinations 11
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].)
---------------------------------Insert Table 4 about here
---------------------------------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 semiflexible 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 stating the number after n. Aaron did not generalize the number-after-n rule to 1 + n combinations 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 Table 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 semiflexible estimation strategy for m + n trials, however, dropped from a high of 60% in Session 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 determine 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 beginning with the larger addend strategy (Procedure 4 in Table 5) for 1 + n items and later an
abstract strategy for larger combinations.
Expertise with Combinations 12
---------------------------------Insert Table 5 about here
---------------------------------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 manifested 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 combinations, which entail minimal keeping track (a single application of number-after knowledge) 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 constructing 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 manner 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 countingon 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 reason 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 countingon (Achievement 6 in Table 5), Procedure 4 in Table 5 (CAL is the most common transition 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
Expertise with Combinations 13
The passive storage view is an adequate model for how basic combinations are often, 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 foster 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 development is concept driven, it makes sense that various (related) aspects of computational
fluency and thinking itself co-evolve and become increasingly interdependent and integrated. 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 direct 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 welldesigned, 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 circumvent 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 constantresponse-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 respond within the prescribed time frame (e.g., Bezuk & Cegelka, 1995; Goldman & Pellegrino, 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 remain. Efforts that focus on memorizing the basic combinations by rote do not guarantee
Expertise with Combinations 14
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, meaningful, and inquiry-based way—that is, fosters all aspects of mathematical proficiency
(e.g., conceptual understanding, logical reasoning, and problem-solving ability) in an integrated 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 benefit 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 instructional 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 combinations (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 students 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 encouraging children to look for patterns and relations, using these discoveries to construct reasoning strategies, and sharing, justifying, and discussing their strategies
(Baroody & Coslick, 1998). It follows from this view that instruction should (a) concentrate 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 worthwhile tasks (games, projects, science experiments, real and imaginary problems)
that create a need to explore and use mathematics. Purposeful practice is more ef-
Expertise with Combinations 15
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 discover 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 efficient strategy and students should be encouraged to flexibly use a variety of strategies.
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 construct number sense could prevent, remedy, or at least minimize many, if not most, difficulties 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
baroody@uiuc.edu
Luisa Rosu
University of Illinois at Urbana-Champaign
217-649-1133
rosu@uiuc.edu
Expertise with Combinations 16
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Expertise with Combinations 22
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
a
cardinality concepts
[7, 9]
d
f
ordinal concepts
(meaningful number
comparisons) [4]
h
counting sequence
skills [3, 12, 21]
b
i
i
g
enumeration
[6, 28]
part-whole relations including composition &
decomposition
e
addition & subtraction
concepts [8]
k
k
i
j
j
verbal set
production
[10, 11]
number after
[13]
e
j
i
h
l
c
b
counting-based
number comparisons
m
k
concrete
counting-all
[16]
Qualitatively reason
about part-whole problems [17]
o
m
n
n
automatic
mental number
comparisons [19, 20]
q
o
o
number-after
rule for n+1/1+n
combinations
r
Different
names for
a number
p
s
s
Complement
principle
w
t
w
v
abstract
counting-on
[32]
t
Other reasoning
strategies for
basic addition
combinations
u
v
p
s
Solving missing
addend problems
exactly
decade
after
abstract
counting-all
[26]
Mastery of basic addition
combinations
v
Mastery of basic
subtraction
comw
binations
Reasoning strategies
for multidigit addition
& subtraction
w
Expertise with Combinations 24
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
Passive quantitative
change (increase) in
associative strength
Passive quantitative and
dynamic qualitative change
(insights ➞ reorganizations
in memory)
Not necessary
Essential
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)
Type of mental changes
Role of conceptual
knowledge
Type of expertise
Expertise with Combinations 25
TABLE 2: An Example of a Big Idea: Composition and Decomposition
Big ideas are overarching concepts that connect numerous topics and applications 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 (composed) 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 decomposed 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).
Expertise with Combinations 26
TABLE 3: Three Views of the Developmental Relations Among Number Recognition, 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 concepts-first view. A third perspective is the simultaneous view.
Piagetian Skills-First View: The simultaneous application of meaningless verbal subitizing 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) regularities view and concepts. Specifically, von Glasersfeld (1982) argued that the early reciting 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, because 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 determining 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 comprised 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 concepts-first view, children’s conceptual understanding enables them to devise meaningful
procedures or skills. According to nativists’ accounts (e.g., Gelman & Gallistel, 1978), children have an innate understanding of number and counting principles, and this prior understanding underlies infants’ ability to nonverbally subitize the number of items in small
collections and toddlers’ capacity to rapidly learn number words and enumeration procedures.
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 conceptual 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-
Expertise with Combinations 27
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 seeing 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 knowledge (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).
Expertise with Combinations 28
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 addend
“4” or “2”
“4” or “2”
“6” or “3”
“6” or “3”
D. Number-after larger-addend
rule
“4”
“4”
“6”
“6”
“6” or “7”
“6” or “7”
“7” or “8”
“7” or “8”
Semi-flexible strategies
E. Extension of number-after
larger-addend rule: state the
number after the larger addend or—for combinations
other n + 1—then some
“4”
“4” or “5”
Flexible strategies
F. Precise (appropriately applied) number-after rule for n
+ 1/1 + n combinations and
true estimation for m + n
items
“4”
“4”
Note. Mechanical strategies are essentially a response bias or an inflexibly applied estimation 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).
Expertise with Combinations 29
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
1. An informal change-add-to concept of addition, previously constructed from verbal subitizing experiences,
enables children to invent or assimilate a direct modeling 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.
2. Integrating knowledge about cardinal representations
of number and addition enables children to shortcut the
laborious concrete counting-all procedure. For example, 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).
3. Some children construct an embedded-addend concept—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 enables them to invent semi-indirect modeling procedures. That is, they no longer have to directly model
(concretely represent) one of the addends. Such procedures are particularly useful for sums greater than
10, where it is difficult to represent both addends on
the fingers of two hands.
Procedure Class
Concrete or Level 1 Procedures
1. Concrete counting-all entails three separate and sequential 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).
2. Concrete counting-all shortcuts involve using number or
finger patterns to short-cut one or more steps in the direct-modeling procedure listed previously—to represent
one or both addends or to determine the sum.
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.
Illustrative Procedure
For example, for 3+5, a child might
(a) successively count and raise three fingers to concretely represent the starting amount 3,
(b) successively count and raise five fingers to concretely represent the added-on amount 5,
(c) and then count all (eight) raised fingers 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 extended fingers. This most advanced concrete counting-all shortcut procedure is first observed with sums
up to five and is sometimes called the “fingerrecognition strategy ” (e.g., Siegler & Jenkins, 1989).
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).
Expertise with Combinations 30
TABLE 5 (Continued)
Conceptual Basis–Achievement
4. Discovery of the embedded-addends concept— that
representing both addends can be done simultaneously
with the sum count—enables children to devise indirect-modeling procedures executing both addend
counts simultaneously with the sum count. This requires a keeping-track process (continuing a verbal
sum count from a number for a specified interval).
Procedure Class
Abstract or Level 2 Procedures
4. Abstract counting-all involves starting one count (Step 1)
and then executing this count and the second count simultaneously (Step 2 next):
Step 1: Begin the sum count by counting up to the cardinal 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.
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).
Illustrative Procedure
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
For 3+5, “counting-all beginning with the larger addend”
labor leads to disregarding addend order. Because of
(CAL) would take the form of
their informal change-add-to concept of addition, chil(a) Note that 5 is more than 3;
dren typically feel compelled to deal with the addends
(b) starting with one, count up to the cardinal value of
in the order they are specified in word problems or
the larger addend (“ … two, three, four, five …”); and
symbolic expressions (Baroody, 1987a, 1987b;
then
Baroody & Ginsburg, 1986; Baroody & Tiilikainen,
(c) continue the count for three more terms: “six [is one
2003; DeCorte & Verschaffel, 1987; Wilkins, Baroody,
more], seven [is two more], eight [is three more].”
& Tiilikainen, 2001). For items such as 3+5, representing the larger addend first can minimize the keepingtrack process and, thus, reduce the load on working
memory.
6. Discovery of the embedded cardinal-count concept (in
6. Abstract counting-on involves the same three steps as
For 3+5, “abstract counting-on from the larger addend ”
the context of computing sums, stating the cardinal
Procedure Class 5 (if Achievement 5 has been obtained),
(abstract COL) might take the form of (a) Identify the
value of an addend is equivalent to counting up to this
except that Step 2 is abbreviated by stating the cardinal
larger addend 5; (b) state “Five” (instead of counting
number) enables children to count-on (Fuson, 1992).
number of the larger addend.
from “one” to “five”); and (c) count on three more
This permits starting the sum count with the cardinal
times: “six [is one more], seven [is two more], eight
value of the addend (counting-on) instead of counting
[is three more].”
from one (counting-all).
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 however. 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.
a
A 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.