Fostering Taiwanese preschoolers’ understanding of the addition–subtraction inverse principle

Available online at www.sciencedirect.com

Cognitive Development 23 (2008) 216–235

Fostering Taiwanese preschoolers’ understanding of the addition–subtraction inverse principle Meng-Lung Lai a,∗ , Arthur J. Baroody b , Amanda R. Johnson b b

a National Chiayi University, Taiwan University of Illinois at Urbana-Champaign, USA

Abstract The present research involved gauging preschoolers’ learning potential for a key arithmetic concept, the addition–subtraction inverse principle (e.g., 2 + 1 − 1 = 2). Sixty 4- and 5-year-old Taiwanese children from two public preschools serving low- and middle-income families participated in the training experiment. Half were randomly assigned to an experimental group; half, to a control condition. Participants were tested for an understanding of inversion before and after intervention. One-third of the 5 year olds from both groups performed at the marginally or reliably successful levels before the intervention, and three quarters of them did so in the posttest. Only one of the 4 year olds was marginally successful before the intervention and 4 year olds in the experimental group somewhat benefited from the intervention. Significant social class effect were evident. © 2007 Elsevier Inc. All rights reserved. Keywords: Mathematical development; Early childhood; Arithmetic principles; Learning potential; Algebraic reasoning

A general understanding of the addition–subtraction inverse principle or inversion entails immediately recognizing that adding and then subtracting the same number (or vice versa) leaves any initial number unchanged. This is symbolically represented as a + b − b = a or a − b + b = a. Achieving such knowledge can enhance four key aspects of mathematical proficiency cited by the U.S. National Research Council (Kilpatrick, Swafford, & Findell, 2001), namely conceptual understanding of number and arithmetic, reasoning, problem solving, and computational fluency (the efficient, appropriate, and flexible application of arithmetic skills). Constructing an understanding of inversion has been considered an important conceptual and reasoning milestone. Ginsburg, Cannon, Eisenband, and Pappas (2006) noted that, “according to Piaget (1952), true addition and subtraction knowledge requires an understanding of the inverse ∗

Corresponding author. Tel.: +886 3 3861252. E-mail address: [email protected] (M.-L. Lai).

0885-2014/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.cogdev.2007.06.002

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

217

relation between the two operations” (p. 212). Specifically, understanding that addition and subtraction are interdependent, not independent, operations is necessary for the construction of a relatively complete knowledge of the additive composition of number and arithmetic reasoning involving part-whole relations (Bryant, 1992; Bryant, Christie, & Rendu, 1999; Inhelder & Piaget, 1958; Piaget, 1965; Rasmussen, Ho, & Bisanz, 2003; Vilette, 2002). Recognizing that addition and subtraction can undo each other has even been taken as evidence that a child is capable of reversing a mental action (reversibility), a defining characteristic of operational (logical) thinking (Piaget & Moreau, 2001; Resnick & Ford, 1981). Applying mathematical principles can also facilitate problem solving and routine computation by eliminating computational effort and increasing both solution accuracy and speed (Baroody, Ginsburg, & Waxman, 1983; Wertheimer, 1959). The inverse principle, specifically, can serve as a computational shortcut for relatively novel three-addend problems or computationally challenging problems such as 8 + 27 − 27 (Klein & Bisanz, 2000; Rasmussen et al., 2003). It further provides the conceptual rationale for the useful “add-back” procedure for checking subtraction computations. For example, unlike 51 or 59, 49 is the correct answer to 84–35 because, when added to the amount taken away (35), it equals the starting amount (84). Unlike learning this checking procedure by rote, its meaningful learning can prompt flexible adaptation to checking division by repeated addition or multiplication (Baroody, 2003). Everyday mathematical knowledge is a key basis for learning school mathematics (Ginsburg, Klein, & Starkey, 1998). It is plausible that an initial understanding of the inverse principle emerges from preschoolers’ concrete everyday experiences (e.g., Baroody & Wilkins, 1999). For example, unhappy with only one cookie, a young child grabs a sibling’s cookie, is momentarily happy with the outcome of having more (2) cookies, is required by an irate parent to relinquish the sibling’s cookie, and glumly recognizes that the original unsatisfactory state of affairs (a single cookie) has been re-established. From such experiences, a child might induce that adding one to any (small) collection can be undone by taking one away. This local informal knowledge could provide a basis for later constructing a general understanding of inversion and for assimilating the principle’s algebraic representation (a − b + b = a). 1. Rationale of the present study Relatively little research has been done on preschoolers’ understanding of arithmetic principles in general and the inverse principle in particular (Ginsburg et al., 1998). Some research indicates that preschoolers can construct a local or unreliable understanding of the principle—that is, restricted to adding and subtracting one but otherwise inconsistent (Klein & Bisanz, 2000; Rasmussen et al., 2003; Starkey & Gelman, 1982). The primary aim of the present study was to assess whether preschoolers can be helped to take the first step toward a general and reliable understanding of inversion. Secondary aims were to gauge whether social class affected this learning potential and to further examine when and how a relatively general and reliable understanding of inversion emerges without adult intervention. 1.1. Learning potential Previous research has not entailed directly examining preschoolers’ potential for learning the inverse principle. Put differently, research to date has focused on whether preschoolers understood the principle or not, but not on whether—with minimal adult intervention—they can construct the concept. A testing effect, though, has hinted that the inverse principle is within children’s zone of

218

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

proximal development (Lai & Baroody, 2006). For example, they found that although no 4 year olds and only a few 5 year olds understood the inverse principle at a relatively general reliable level, a half dozen participants were dramatically more successful on the second half of the testing. These results suggest that minimal intervention may draw some preschoolers’ attention to this important mathematical regularity with transformations involving collections somewhat larger than one. The primary aim of the present study was to gauge 4 and 5 year olds’ learning potential for a somewhat general understanding of inversion. 1.2. Difference due to socio-economic status (SES) There is growing evidence that SES-related differences in mathematical knowledge begin in early childhood, because young children from economically disadvantaged families receive less support for mathematical development than their middle-class peers (Baroody, Lai, & Mix, 2006). Low SES can adversely affect early informal knowledge (Case, 1975; Entwisle & Alexander, 1990, 1996; Shaw, Nelsen, & Shen, 2001). Children from low-income families enter school with significantly less number and arithmetic knowledge than do middle- or high-SES students (see reviews by Arnold & Doctoroff, 2003; Kilpatrick et al., 2001, and Secada, 1992). A secondary aim of the present research was to gauge whether differences in SES had an impact on preschoolers’ learning potential for inversion. 1.3. Natural emergence of a general and reliable understanding of inversion Piaget hypothesized that children could not construct an understanding of inversion before 6 or 7 years of age when they achieved operational thinking and its signature capacity of reversible thought (Piaget & Moreau, 2001). The results of a number of recent studies with preschoolers, however, challenge this view (e.g., Bisanz & LeFevre, 1990; Klein & Bisanz, 2000). Gauging a true understanding of inversion, though, is fraught with difficulties; perhaps the most serious is that children may use computation or some means other than this principle to respond. Piaget (1967) distinguished between true reversibility based on logical thinking and empirical reversibility based on successive actions that only appear to undo each other. The former involves what could be called principled inversion, logically viewing addition and subtraction as interrelated operations (e.g., for 3 + 1 − 1, recognizing that adding 1 is undone by subtracting 1), whereas the latter involves what could be called empirical inversion, treating addition and subtraction as separate and unrelated operations (e.g., for 3 + 1 − 1, adding 3 + 1 to get 4 and then subtracting 1 to get 3 and restoring the original amount). Research on the inverse principle has basically involved computational shortcut, possible versus impossible events, or algebraic-reasoning tasks. In order to more directly assess principled inversion and avoid the limitation of previously used tasks, a true algebraic-reasoning task was developed (Baroody & Lai, 2003, 2007; Lai & Baroody, 2006). Unlike previous tasks, which involved two collections, this task entailed comparing the end state with the original state of a single collection after two intermediate transformations had been made. Because it requires reasoning from an unknown starting amount, the true algebraic-reasoning task better controls for a computation confound than other tasks. Even so, this task has a number of limitations. Another secondary aim of the study was to use a revised true algebraic-reasoning task to investigate what is still unclear: when does a relatively general and reliable informal understanding of the inverse principle typically begin to emerge without intervention? The key improvements over previously used true algebraic-reasoning tasks are summarized below.

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

219

1.3.1. An appropriate starting amount A meltdown phenomenon was observed in previous studies (e.g., Baroody & Lai, 2003, 2007). Most 4 year olds appeared to be overwhelmed by the large starting collection (19 poker chips). In order to minimize this problem, preliminary testing identified an appropriate starting amount for each child. That is, it served to identify a child’s zone of proximal development, which lies between the zone of competence and zone of incompetence (Dowker, 1997, 2003). 1.3.2. Trials involving two to five items Although trials involving transformations of one might evoke a fragile understanding of the principle in less mathematically developed children, trials involving transformations of two to five were used in the present study for four reasons. (a) Previous research (Brush, 1978; Cooper, Starkey, Blevins, Goth, & Leitner, 1978; Smedslund, 1966) already indicates that preschoolers have localized knowledge of the inverse principle (i.e., many recognize that adding one is undone by subtracting one). (b) Including trials involving one would have extended the inverse testing considerably and increased the chances of unreliable results such as inattentiveness. (c) The first important step toward a broader understanding is competence with other “intuitive numbers,” namely two and three, numbers that toddlers and even infants can discriminate (Starkey & Cooper, 1995). (d) The second step toward a broader understanding is competence with numbers just bigger than the “intuitive numbers” such as four and five. 1.3.3. Addition- and subtraction-first trials split evenly—controls for the order effect Previous studies (e.g., Baroody & Lai, 2003, 2007) indicated that children performed better on the subtraction-first inverse trials. This result appears to be counterintuitive because addition-first inverse trials lend themselves to the possible application of the negation strategy (e.g., for x + 4 − 4: 4 − 4 = 0 and x − 0 = x), whereas subtraction-first inverse trials do not (Bisanz & LeFevre, 1990). However, three of the four subtraction-first trials in the Baroody and Lai study were administered in the second session and some children (5 year olds particularly) appeared to exhibit a learning effect. Thus, it was unclear whether such items are, in fact, easier than addition-first trials. In this study, the learning effect was controlled by splitting the addition- and subtraction-first trials evenly. 2. Methods 2.1. Participants Thirty 4 year olds (mean = 4 years 8 months; range = 4–5 to 4–11) and thirty 5 years olds (mean = 5–9; range = 5–7 to 5–11) completed the study. There were an equal number of boys and girls at each age level. Half the participants were drawn from a public school (30 children) located in the middle-SES neighborhood in the city of Taoyuan (population of 400,000), and half from the other public school (30 children) located in the lower SES neighborhood in the town of Dayuan (population of 80,000). 2.2. Procedure An overview of the training study is outlined in Table 1. Familiarization, preliminary testing, pretest, training, and posttest sessions were conducted on different days over a period of 8 weeks.

220

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

Table 1 Overview of the study Session

Materials

1.

Familiarization

Familiarization

Animal spots Car race Hidden stars

2.

Preliminary tests

Is this? Game (verbal number recognition); Inverse practice trials

Number card deck

Inverse (algebraic reasoning) task

A mat, toy cars, a box

Experimental group: adapted magic tasks (half boys and girls) Comparison group: composition and decomposition (half boys and girls)

Toy trains

Is this? Game Inverse (algebraic reasoning) task

Number card deck A mat, blue chips, a box

3.

Pretest

4.

Training

5.

Posttests

A mat, cars, and a box

Toy cars

A child was seen individually for each session. The tester suspended the testing if the child became inattentive (e.g., resorting to wild guessing for four trials), exhausted (e.g., saying something like “I am tired of this game” or “can we do something else?”), or distracted and the testing was resumed on another day. In the familiarization session, each participant played three object counting games with a tester for the purpose of building rapport with the researcher. The preliminary session included the number recognition task and preliminary inverse trials. Performance on the former determined a developmentally appropriate starting amount in the inverse tasks for each participant. The inverse practice trials served (a) to familiarize participants with the main (inverse) task used for the pretest and posttest and (b) to check and promote their comprehension of the relational terms same, more, and fewer used in the inverse task. In the pretest, a revised algebraic-reasoning task served as the tool to identify the inverse understanding of thirty 4- and 5-year-old children. Half of these participants were assigned to the experimental group and the other half to the comparison group with SES counterbalanced. In the training sessions, participants in the experimental group received inverse (reversibility) training while those in the comparison group received the number composition/decomposition training. In the posttest, participants’ inversion was again assessed with the revised algebraic-reasoning task. 2.3. Tasks 2.3.1. Number recognition The number recognition task served to identify a starting amount (five or seven) that children cannot immediately recognize but not overwhelmingly large so as to cause a meltdown. In the context of a game (Is This?), participants were asked to indicate, within 3 s, whether a collection of haphazardly arranged dots had one to five dots. Five trials were included in the number recognition task. For number one, the tester uncovered, in turn, six different collections (10, 1, 2, 1, 7, 3) and asked, “Is this one?” The same procedure was repeated with number two (3, 1, 2, 4, 2, 7), three

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

221

(4, 3, 7, 1, 2, 3, 5), four (5, 4, 2, 1, 3, 4, 6, 7), and five (5, 7, 2, 4, 6, 5, 3). The starting amount five in the inverse tasks was administered to those participants who were unsuccessful on this number recognition task or failed the trial number five. In order to check for false positives due to a response bias and to more accurately examine a child’s overall number sense, trials of two, three, and four were given. If participants succeeded on all five trials, seven was then used as the INITIAL starting amount. 2.3.2. Inverse practice Participants were asked to compare the end state with the initial state of the following trials: 4 + 1 − 1 (same), 4 – 1 + 1 (same), 4 + 1 + 1 (more), and 4 − 1 − 1 (fewer). The detailed procedure is delineated in the next section. (Note that all practice trials begin with a number just bigger than intuitive numbers.) The tester provided feedback by announcing correct answers with justifications if participants missed any practice trials. Those incorrect trials were then redone at the end of the session. 2.3.3. Inversion The participants’ understanding of the inverse principle was gauged by using a true algebraicreasoning task (Fig. 1), an adaptation of Huttenlocher, Jordan, & Levine’s (1994) nonverbal addition–subtraction task. The task was introduced by the tester showing a black mat and a decorated box to a child, who sat across the table from the tester. The tester then explained, “This is the parking lot and its canopy.” After briefly shown a collection of five or seven “toy cars” haphazardly arranged on the black mat, children were aware of several items to begin with and children’s were told that counting was unnecessary. The tester then continued, “In the morning, there were this many cars parked in the parking lot. Some cars came in and some cars left in the afternoon. Now let’s watch carefully and decide if the number of cars in the afternoon is more than what we started with, the same number of cars as what we started with, or fewer than what we started with in the morning.” Next, two to five toy cars were either (a) put out to the left of the cover for 3 s and then added to the starting amount (slid under the cover), or (b) withdrawn from the starting amount on the right side of the cover, left for 3 s, and then removed. (Directions specified are from the child’s vantage.) On inverse trials, the first transformation was then undone by the second performed on the other side of the cover (clearly from a different location). The undoing process involved different toy cars and physical arrangements to avoid children’s judgments on perceptual cues. As children could not recognize the number of the original collection, they were asked to determine the relative effects of the transformations (i.e., whether the final collection was more than, the same as, or fewer than the original collection) instead of the exact number of the outcome. Unlike the previously used shortcut tasks, reaction time is not an issue in this (algebraic) task, because it is improbable that children could determine the starting amount and, thus, calculating the exact answer was not possible. As Fig. 2 shows, four addition-first (x + n − n) and four subtraction-first (x − n + n) inverse trials were used. Also, there were two trials each of four types of control items: (a) unknown original amount plus a relatively large number minus a relatively small number (x + L − S) such as x + 4 − 2, (b) x − S + L such as x − 2 + 4, (c) x − L + S such as x − 4 + 2, and (d) x + S −L such as x + 2 − 4. These trials were presented in the fixed semi-random order with no more than two consecutive inverse trials, and subsequent trials had to be of a different type (e.g., an addition-first inverse trial could be followed by a subtraction-first one, and an x + L − S trial could only be followed by another type of control trial).

222

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

Fig. 1. Nonverbal addition–subtraction inverse task (addition-first trials) from the child’s vantage.

2.4. Scoring 2.4.1. Number recognition task Participants were scored as correct if they responded “yes” to the examples and “no” to the nonexamples of the given trial. If children may not have attended to any trials, they were readministered at the end of the session. Slow responses and even counting the collection were scored as incorrect. For each trial, children were considered unsuccessful if they missed any examples.

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

223

Fig. 2. Inverse task test trials (I) and control trials (C) from the child’s perspective.

2.4.2. Inverse tasks Participants were scored as correct on an inverse trial (1 point) if they responded “the same.” Participants were scored as correct on a control item (1 point) if they correctly indicated that the transformations resulted in a different number (i.e., “more” or “fewer”) than the original collection. If children may not have attended to one of the transformations, or otherwise may not have exhibited their true competence, the trial(s) in question were readministered at the end of the session or in a follow-up session. A correct response on such a second try was scored as a partial success (0.5 points). Otherwise, incorrect responses to either inverse or control trials were scored as 0 points. Overall inverse competence was gauged by a response pattern across both types of trials that indicated a participant understood the task and used principled reasoning. Specifically, three criteria were used to conservatively define reliable and general inverse success: 1. Reliable correctness on the inverse items, required scoring at least six of eight trials (with three choices per trial or a 1 in 3 probability of responding correctly by chance, p < .02, Binomial Theorem). Marginally reliable was defined as scoring correct on five trials (p = .088, Binomial Theorem). The marginally reliable criterion makes statistical and theoretical sense because such a result is what might be expected with a partial or inconsistent grasp of the concept. 2. Reliable correctness on the control items, required scoring at least seven of eight control trials (with a 2 in 3 chance of guessing either “more” or “fewer”, p = .04, Binomial Theorem). Marginally reliable was defined as scoring 5 or 6 points—correct on at least five control trials (p = .088, Binomial Theorem). 3. Absence of systematic errors resulting in false positives. The response patterns of participants were checked to see if they responded in a manner consistent with any of the systematic errors listed in Table 2. For this reason, the Binomial Theorem was used to identify whether a child’s

224

Table 2 Response pattern profiles for correct answers (in italic) and key systematic error patterns Response bias observed (number of participants)

Inverse items x+n−n

Posttest 9 2/3

16

0

0

0

1/2

1/2

0

0

3

1

Correct responses E.1: State the “same” E.2a: State “more” E.2b: State “fewer” E.3: Equate any change to the collection with a change E.4: Focus on perceptual cues— length/area E.5a: Compare the size of the first set seen to that of the second E.5b: or vice versa E.6: Focus on the effects of the first transformation E.7: Focus on the effects of the second transformation

Response bias observed (number of participants)

x+3−3

x+4−4

x+5−5

x−2+2

x−3+3

x−4+4

x−5+5

Same Same More fewer

Same Same More fewer

Same Same More fewer

Same Same More fewer

Same Same More fewer

Same Same More fewer

Same Same More fewer

Same Same More fewer

More/fewer

More/fewer

More/fewer

More/fewer

More/fewer

More/fewer

More/fewer

More/fewer

Fewer/fewer

More/more

Fewer/fewer

More/more

More/more

More/more

More/more

More/more

Same

Same

Same

Same

Same

Same

Same

Same

Same More

Same More

Same More

Same More

Same Fewer

Same Fewer

Same Fewer

Same Fewer

Fewer

Fewer

Fewer

Fewer

More

More

More

More

Control items x+L−S

Pretest 14 2/4

Posttest 9 2/3

16

0

0

0

Correct responses E.1: State the “same” E.2a: State “more” E.2b: State “fewer” E.3: Equate any change to the collection with a change E.4: Focus on perceptual cues— length/area

x−S+L

x+S−L

x−L+S

x+3−2

x+4−2

x−2+3

x − 2+4

x+1−2

x+1−3

x−3+2

x−4+2

More Same More fewer

More Same More fewer

More Same More fewer

More Same More fewer

Fewer Same More fewer

Fewer Same More fewer

Fewer Same More fewer

Fewer Same More fewer

More/fewer

More/fewer

More/fewer

More/fewer

More/fewer

More/fewer

More/fewer

More /fewer

Same/more

Same/more

Same/more

Fewer/more

Same/fewer

Same/fewer

Same/fewer

Same/fewer

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

Pretest 14 2/4

x–n+n

x+2−2

Table 2 (Continued ) Control items x+L−S

1/2

1/2

0

0

3

1

E.5a: Compare the size of the first set seen to that of the second E.5b: or vice versa E.6: Focus on the effects of the first transformation E.7: Focus on the effects of the second transformation

x−S+L

x+S−L

x−L+S

x+3−2

x+4−2

x−2+3

x − 2+4

x+1−2

x+1−3

x−3+2

x−4+2

More

More

Fewer

Fewer

Fewer

Fewer

More

More

Fewer More

Fewer More

More Fewer

More Fewer

More More

More More

Fewer Fewer

Fewer Fewer

Fewer

Fewer

More

More

Fewer

Fewer

More

More

Note. A more complete description of the error patterns listed as follows: • Error patterns E.1 and E.2 entail consistently responding with one of the three choices (more, same, fewer). • Error pattern E.3 is, in effect, tantamount to consistently using more or fewer. • Error pattern E.4: A child might respond based on a perceptual cue (e.g., length). For example, they might respond “less” if the second array was shorter. • Error pattern E.5 stems from simply comparing the size of the two collections seen (i.e., treating the inverse and control trials as a number comparison task). • Error pattern E.6 entails mentally representing the first transformation algebraically (e.g., something was added or taken away), if not exactly, recognizing its directional effect (now there are more or less), and simply ignoring the second transformation. • Error pattern E.7 would involve, in effect, ignoring the unknown starting amount and treating the first exact amount as the starting amount. The focus of attention would be on whether the second transformation involved addition or subtraction of items.

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

Response bias observed (number of participants)

225

226

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

responses were reliably consistent with an error strategy across both components. Consider, for example, the state “same” response bias (error pattern E.1). E.1 overall was inferred if the child answered “more” on three quarters (12) of all trials (p < .004, Binomial Theorem) and on over half of each of the inverse and control trials (at least five of the former and the latter). Participants making E.1 were scored as unsuccessful automatically. Overall competence was based on whether a participant was reliably correct on each component and whether false positives due to any systematic errors across both components could be discounted. Specifically, in the absence of systematic error pattern, participants were scored as successful overall if they were reliably correct on both the inverse and control components. These children already developed a robust (general and reliable) understanding of inversion. Participants were scored as marginally successful if they were reliably correct on the inverse component and marginally reliable on the control component, reliably correct on the inverse component and unreliable on the control component if the former was not due to the “state the same” response bias, or were marginally reliable on the inverse component and at least marginally reliable on the control component. These children began to notice the inversion while still unable to apply it flexibly. Otherwise, participants were scored as unsuccessful if they were correct on less than five inverse and control trials (p > .25, Binomial Theorem) or if they exhibited any systematic errors discussed in the error analyses section of the results. 2.5. Training 2.5.1. Experimental group—reversibility In the training phase, an adaptation of the magic task used by Gelman (1972) was used. A child was first shown three or four toy trains for 3 s and then these items were covered with a box. To create the expectation of finding this number, the tester lifted the box and revealed the initial collection for about 2 s and then repeated the process. The tester then surreptitiously added or withdrew from the initial collection one to three items, revealed the reconstituted collection, and asked the child to fix the collection so it looked just like the initial collection. This training phase, 2 weeks for each preschool, involved three sessions: addition-only, subtraction-only, and mixed addition and subtraction, with different materials (cars, boats, and trains) used, respectively. Each training session lasted approximately 10 min and was administered to individual participant on different days with 2 days apart. All training sessions first started with the easiest transformations (e.g., 3 + 1 and 3 − 1) and then proceeded to more challenging ones (e.g., 3 + 4). With the exception of three 4 year olds, participants were not overwhelmed by the relatively large collections used in on the challenging trials and performed successfully on them. Feedback, such as “how many items were there to begin with” or “how many items were either added to or taken away from the INITIAL collection”, was provided to those who were not successful on a trial. This adapted magic task served to concretely introduce an undoing operation (reversibility). By repairing the collection to its original state, participants actually were performing the second transformation on the inverse trials. 2.5.2. Comparison group—number decomposition/composition Two number decomposition/composition games were administered to the comparison group in order to control for the inverse training effect. These games were designed to help children more accurately estimate the large collection of 5 to 11 items. Participants were first instructed to

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

227

decompose a large collection into two or three smaller ones that were within their zone of comfort in terms of number sense (Dowker, 1997, 2003). For example, six items could be decomposed into a group of two items and another group of four items or two groups of three items each (3 + 3), so that these children could apply number combinations they had already mastered (e.g., number doubles). In the same manner, eight items could be decomposed to three groups such as 3, 3, and 2. In brief, participants were encouraged to see the large collections in a way they could easily figure out by means of using the decomposition strategy. In another session, participants were given a chance to exercise heuristics to estimate large collections (i.e., 6, 8, and 10 items). First, the tester showed participants a collection of 6 items arranged haphazardly for 3 s, and then he covered some of these items (e.g., three items) with a box. Participants were asked to use the visible part to guess about how many items there were in the collection. The purpose of this task was to help participants come up with reasonable estimates for large collections. 3. Results The results of the training experiment are summarized in Table 3 below. Overall, 19 of 30 participants from the experimental group showed improvement in the posttest, whereas only 8 of 30 participants from the comparison condition did so (p < .01, 2 × 2 Chi-square). Using inverse competence of the posttest as the dependent measure, a 2 (SES: middle versus low) × 2 (age: 4 years versus 5 years) × 2 (training: inversion versus number sense) univariate ANCOVA with the pretest inverse competence as the covariate revealed main effects on SES (F [1,51] = 6.60, p = .013, ES = .111), age (F [1,51] = 8.98, p < .01, ES = .150), and training (F [1,51] = 6.37, p = .015, ES = .115). The Tukey HSD post-hoc test indicated that middle-SES children did significantly better than the low SES, 5 year olds did significantly better than 4 year olds, and children receiving the inverse training outperformed those who did not (see Fig. 3). Looking carefully at the performance on different groups, 4 year olds receiving the inverse training performed significantly better than their counterpart in the comparison group (p < .01), although the performance of the former was still below the marginal successful level. No significant interactions were observed. The previously noted main effects were verified using more conservative nonparametric tests and the results were similar. Specifically, a Mann–Whitney test revealed significant effects on age Table 3 Participants’ performance in the pretest and posttest by age and group Task

Groups

Competence

Posttest Unsuccessful Age

Experimental (inversion) Pretest

Comparison

Successful Marginally successful Unsuccessful Successful Marginally successful Successful

Marginally successful Age

Successful Age

4

5

4

5

4

5

0 0 6 0 0 13

0 0 1 0 0 6

0 1 5 0 0 2

0 1 8 0 0 0

0 0 3 0 0 0

2 2 1 3 3 3

228

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

Fig. 3. Training effect on age groups and interventions.

(Z = −3.52, p < .01) and SES (Z = −2.23, p = .026) and marginally significant effect on training (z = −1.90, p = .057). 3.1. The effects of training The inverse training was most useful for those unsuccessful participants from the experimental group. Specifically, among the 24 previously unsuccessful participants from the experimental group, 17 (71%) improved to either marginally successful or even reliably successful in the posttest. In contrast, among the 24 previously unsuccessful participants from the comparison group, only five (21%) improved on the posttest. (p < .01, 2 × 2 Chi-square). Surprisingly, for those participants who were already scored as marginally successful in the pretest, the inverse training was not necessary to accelerate the understanding of inversion. Whereas two of four such experimental participants improved in the posttest, all three of the comparison participants did so (p = .29, 2 × 2 Chi-square). It appears that those marginally successful participants from the comparison group improved without the inverse training but merely from a testing effect (exposure to the test itself). 3.2. Inverse and control results in the pretest and posttest Table 4 shown below indicates the number of inverse and control items correct separately for the pretest and posttest by AGE for the whole sample and for those participants who did not make the E. 1 (“state the same”) response bias. Apparently, 5 year olds improved much better than their 4-year-old counterparts (p = .055 for the whole sample and p = .06 for the sample excluding E.1). Table 4 Inverse and control results in the pretest and posttest Age

Pretest

Posttest

Pretest (excluding E.1)

Posttest (excluding E.1)

Inverse

4 5

4.47 4.48

5.12 6.35

2.92 4.17

4.03 6.19

Control

4 5

4.30 6.17

5.25 6.90

6.05 6.74

6.97 7.17

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

229

3.2.1. Improvement on the inverse and control components When the participants with response bias favoring the inverse component were excluded, 4 year olds performed marginally better in the posttest than in the pretest (p = .10) on the inverse component. Interestingly, 4 year olds also performed better in the posttest than in the pretest (p = .05) on the control component. Looking at the performance of 5-year-old participants, their improvement on the inverse component was clearly more pronounced (p < .01) than that on the control component (p = .17). 3.2.2. Intervention effect on the gain scores of inverse component In the experimental group, participants with more gain scores on the inverse component tended to outnumber those participants with more gain scores on control component (a total of 17:8, p = .10, Sign Test). In contrast, there was almost no difference in the comparison group (a total of 9:8). Overall, children’s better performance on the inverse items was much more pronounced in the posttest than in the pretest. 3.3. How SES and AGE affect the learning potential of inversion (posttest) Using the pretest–posttest gain scores on the inverse component as the dependent measure, a 2 (SES: middle versus low) × 2 (age: 4 years versus 5 years) × 2 (training: inversion versus number sense) univariate ANOVA revealed a main effect on training (F [1,52] = 4.50, p = .039, ES = .08), marginally significant effects on age (F [1,52] = 3.85, p = .055, ES = .069), and SES (F [1,52] = 3.17, p = .081, ES = .057). The Tukey HSD post-hoc test indicated that children receiving the inverse training significantly outperformed those who did not, middle-SES children did marginally better than low SES, and 5 year olds did marginally better than 4 year olds (see Fig. 4). 3.4. Error analyses Most children’s error patterns were consistent with E.1 (respond “the same” regularly) or E.3 (equate any change to the make up of a collection with a change). Regularly responding “the same” (pattern E.1) was one of the most prevalent systematic errors. These participants apparently did not understand how to respond to the tester’s question appropriately, and seemed to simply repeat the last choice they heard from the tester’s probe.

Fig. 4. Different SES’s performance in the pretest and posttest.

230

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

Children who believe that any change in the make-up of a collection means that it is not the same might manifest this conception in another way, namely haphazardly responding “more” or “fewer” (error pattern E.3). This phenomenon, however, became less obvious after the inverse intervention. Another error analysis indicated that participants who scored as marginally successful and unsuccessful (Level 1 or 0) overall exhibited a tendency to focus on the exact amounts and the intervening transformation (exhibited error pattern E.7) on the control items. These participants did extremely poorly on the two control items in which the last term was misleading (e.g., the second transformation was taken away while the outcome, overall, was more, such as x + 3 – 2) and the difference of the terms was not obvious (e.g., x + 3 − 2 and x − 3 + 2). In contrast, they did relatively well on the two items in which the last term was helpful and the last two terms were obviously different (x + 1 − 3 and x − 2 + 4). Where the two factors were in conflict, the direction of the last transformation had a slightly greater impact than the closeness of the terms. 4. Discussion and conclusions The purpose of this study was twofold: to examine whether 4- and 5-year-old children could acquire the novel conceptually based inverse principle and to investigate when and how a reliable and general understanding of this principle begins to emerge. 4.1. Learning potential As can be seen in Fig. 3 in the result section, 5 year olds have much more learning potential on this inverse principle than the 4 year olds. A larger portion of 5 year olds were already marginally successful in the pretest. After the inverse training, they reliably succeeded on both the inverse and control trials in the posttest. It is interesting to note that the majority of the 5-year-old children scored as marginally successful in the pretest became reliably successful in both the experimental and comparison group. For these children, the inverse principle was apparently already within their zone of proximal development. Thus, with minimal intervention (exposure to the test itself), they quickly picked up this principle. However, the inverse training was extremely effective for those participants scored as unsuccessful in the pretest. Apparently, it helped draw these children’s attention to this important mathematical regularity. For the 4-year-old participants in the experimental group, although it was obvious that these children benefited from the inverse training to some degree and, thus, performed better than those participants in the comparison group, these children’s understanding of the inverse principle was still far from the general and reliable level. These findings confirm Piaget’s claim that the effect of instruction is most salient when the target to be learned is considered “moderately novel” to children. In the same manner, such an intervention is not likely to benefit for most 4 year olds, because this principle is “completely novel” for them. 4.1.1. Differences between SES This study obtained similar findings that the middle-SES children improved more than their lower SES counterparts after the training (see Fig. 4). Copley (2004) suggests that there is a need to provide greater mathematics enrichment to all preschool children. This would seem to be especially important for low-income children who receive minimal support for mathematical

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

231

development in their learning environments. One approach is to enrich the home or preschool learning environment through a systematic mathematics curriculum. At both SES levels, experimental children’s scores were greater than comparison children’s scores in the posttest. What is also worth noticing fact is that the posttest performance of low SES intervention children was better than the pretest performance of the middle-SES children. It is an interesting finding and suggests that the intervention on the low SES group may eliminate the SES-related gap. 4.1.2. Suggested intervention One objective of this study was to develop, implement, and assess the effectiveness of the inverse intervention. The inverse scores of both low- and middle-income children who received the intervention rose from pretest to posttest, a finding indicative of the acquisition of inverse principle. This fact implies that the adapted magic task is an effective and legitimate intervention to trigger children’s conceptual understanding of the inverse principle. 4.2. When does a reliable and general understanding of the inverse principle begin to emerge? 4.2.1. Typical achievement of competence Whereas three of the 4-year-old participants exhibited a reliable and general understanding of the addition–subtraction inverse principle in the posttest, almost half (7 of 15) of the 5-yearold participants did so. The estimate of inverse understanding for the younger participants is clearly more conservative than Bryant et al.’s (1999) liberal estimates. In the same manner, the performance on our 4-year-old participants was more conservative than what Klein and Bisanz (2000) and Rasmussen et al. (2003) claimed in their study. They analyzed the data by using the number of trials and found that 4 year olds responded correctly on about half of the inverse trials. Specifically, their 5-year-old participants responded correctly overall on just over half of the inverse trials. If a somewhat less conservative criterion for inverse success (successful or marginally successful) is used, the success rate (38%) of our 5-year-old participants in the pretest still does not approximate Bryant et al.’s data and the success rate (77%) for our 5-year-old participants in the posttest is comparable to Bryant et al.’s results. However, as noted earlier, some of the participants’ correct responses in the Bryant et al.’s study were probably due to computation, not principled reasoning. When the highly and moderately conservative estimates of inverse competence derived from this study and the relatively liberal estimate from Bryant et al.’s study are considered together, it appears that a reliable and general understanding of the principle begins to emerge at about 5 years of age. 4.3. How does a reliable and general understanding of the inverse principle begin to emerge? Two lines of evidence from this study suggest that the development of inverse understanding is not an all-or-nothing phenomenon but from incomplete understanding and gradual development. 1. The participants who were unsuccessful or marginally correct on the inverse component but reliably correct on the control component (eight 4-year-old children and five 5-year-old children in the pretest) were probably not inconsistent on the former because of inattentiveness (cf. Smedslund, 1966). The inconsistency of these children, in particular, might well reflect an

232

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

emerging understanding of the inverse principle. Indeed, half of these children’s understanding of inversion became marginally reliable or even reliable in the posttest. 2. The observed marginal size effect for 5 year olds may indicate a partial understanding of inversion. As recognizing collections of two is easier than doing so for three or four and mentally computing 2–2 is easier than doing so for 3–3 and 4–4, it is possible that some children reasoned qualitatively that the original unknown collection had not changed in size. Taken together with previous research, it appears that a nascent (localized and inconsistent) understanding of the inverse principle, represented by a weak schema, may begin at about 3 or 4 years of age (Klein & Bisanz, 2000; Rasmussen et al., 2003; Starkey & Gelman, 1982) and becomes more robust about 2 or 3 years later (Bisanz & LeFevre, 1990; Bryant et al., 1999, Study 2 in particular). This process may begin with concrete situations involving the relatively obvious case of a subitizeable number plus one then minus one (Brush, 1978; Cooper et al., 1978; Smedslund, 1966). A first step toward gradually generalizing the inverse concept may be prompted by computational experiences involving a subitizeable number plus two minus two (e.g., Bryant et al., 1999; Klein & Bisanz, 2000). However, size effect will not appear in an exactly staggered fashion because children would pay much more attention to a relatively large collection (e.g., x − 5 + 5). The increased attention and effort required rendered the inverse relation more salient. The fact that inverse trials involving two and three were still easier for some 5-year-old participants suggests that a general and reliable understanding of inversion does not begin to emerge for some children until several years after the beginning of school, which is consistent with the recent works (Gilmore, 2006) revealing even 9-year-old children do not fully grasp the inversion principle. Typical of a weak schema, children’s early understanding of the inverse principle may be precedent-driven or local in nature and, thus, inconsistently and even illogically applied (Baroody & Ginsburg, 1986; Baroody, Wilkins, & Tiilikainen, 2003). In time, children construct a strong schema for the inverse principle—one that is principle-driven and general in nature and, thus, logically and consistently applied regardless of number size or context (whether problems are presented nonverbally and concretely, as word problems, or written expressions). Children are then more likely to have an explicit knowledge of the principle and be able to justify their answers in terms of the principle. Although a few preschoolers may achieve this depth of understanding (Klein & Bisanz, 2000; Rasmussen et al., 2003), most children may not construct an explicit understanding of the inverse principle until well into elementary school (Bryant et al., 1999; Siegler & Stern, 1998). The disappointing performance by 4-year-old participants suggests that even a reliable understanding of the inverse principle at the nonverbal level with the intuitive numbers two to five emerges relatively late—at about the same time previous research (Bryant et al., 1999) indicates children are becoming reliably successful on verbal inverse tasks with larger numbers. This is consistent with other data that indicate children develop verbal number skills at about the same time they do comparable nonverbal skills (Baroody, Benson, & Lai, 2003; Benson & Baroody, 2003; but cf. Levine, Jordan, & Huttenlocher, 1992). Contrary to the mental models view (Huttenlocher et al., 1994), a verbal representation of number may facilitate exact mental representation and computations even with small numbers (Mix, Sandhofer, & Baroody, 2005) and, thus, facilitate the discovery and generalization of the inverse principle. Indeed, verbal subitizing (immediately recognizing small numbers and labeling them with a number word) and even object enumeration—perhaps in conjunction with the development

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

233

of working memory (Klein & Bisanz, 2000)—may play a key role in discovering the inverse relation. 4.3.1. Reasons for non-success The children who were not reliably accurate on the inverse component and marginally reliable or, especially, not reliably accurate on the control component might have been unsuccessful on the former for a variety of reasons: 1. These participants may not have constructed even a moderately weak inverse schema. In the absence of such a schema, they may have used some other conceptual basis for responding, such as assuming that any change in the makeup of a collection implies the collection is different. 2. They were generally inattentive, perhaps because they did not have the required conceptual understanding or because of disinterest (cf. Smedslund, 1966). 3. Both the inverse and control trials may have been too difficult for some younger children because it required them to reason qualitatively or algebraically (i.e., about the effects of operations on an unspecified number). This type of reasoning may be foreign to many children who, even at an early age, expect to be asked questions that have an exact or quantifiable answer. 4.4. Educational implications The results of the present study indicate that minimal intervention may draw attention to the important mathematical regularity of inversion for children as young as 5 years of age or even some 4-year-old children. Helping children understand inversion early can deepen their number sense in that they view addition and subtraction as related operations and reinforce the belief that looking for patterns and relations is central to mathematics. Both may help students (a) to discover the general computational shortcut inversion affords, (b) to look for other computational shortcuts (e.g., viewing 9 + 8 as the easier problem [9 + 1] + [8 − 1] = 17), and (c) to recognize that addition and subtraction are related in other ways (e.g., viewing 5 − 3 = ? as 3 + ? = 5). In addition, this study indicates that children younger than 7 years of age are capable of algebraic reasoning. Previous research indicates that kindergartners can abstract arithmetic regularities and use the resulting algebraic rules for performing mental arithmetic (Baroody, 1992) and even that children as young as 3 can engage in rule-based algebraic reasoning with simple or one-step subtraction situations such as n − 0 (none) or n – n (Baroody, Lai, Bi, Li, & Baroody, 2004). The present study further indicates that many kindergarten-age children may also reason algebraically about more complex, two-step process (inverse relations). Acknowledgements Support for the research and preparation of this report was provided, in part, by grants from the National Science Foundation (BCS-0111829; “Foundations of Number and Operation Sense”) and the Spencer Foundation (2000400033; “Key Transitions in Preschoolers’ Number and Arithmetic Development: The Psychological Foundations of Early Childhood Mathematics Education). The opinions expressed are solely those of the authors and do not necessarily reflect the position, policy, or endorsement of the National Science Foundation or the Spencer Foundation.

234

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

References Arnold, D. H., & Doctoroff, G. L. (2003). The early education of socioeconomically disadvantaged children. Annual Review of Psychology, 54, 517–545. Baroody, A. J. (1992). The development of kindergartners’ mental-addition strategies. Learning and Individual Differences, 4, 215–235. Baroody, A. J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 1–34). Mahwah, NJ: Erlbaum. Baroody, A. J., Benson, A. P., & Lai, M. L. (2003, April). Early number and arithmetic sense: A summary of three studies. Paper presented at the biennial meeting of the Society for Research in Child Development, Tampa, FL. Baroody, A. J., & Ginsburg, H. P. (1986). The relationship between initial meaningful and mechanical knowledge of arithmetic. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 75–112). Hillsdale, NJ: Erlbaum. Baroody, A. J., Ginsburg, H. P., & Waxman, B. (1983). Children’s use of mathematical structure. Journal for Research in Mathematics Education, 14, 156–168. Baroody, A. J., & Lai, M. L. (2003). Preschoolers’ understanding of the addition subtraction inverse principle. Paper presented at the biennial meeting of the Society for Research in Child Development, Tampa, FL. Baroody, A. J., & Lai, M.-L. (2007). Preschoolers’ understanding of the addition-subtraction inverse principle: A Taiwanese sample. Mathematics Thinking and Learning, 9(2), 131–171. Baroody, A. J., Lai, M. L., Bi, K., Li, X., & Baroody, A. E. (2004, April). Preschoolers’ understanding of subtractionrelated principles. Paper presented at the annual meeting of American Educational Research Association, San Diego, CA. Baroody, A. J., Lai, M.-L., & Mix, K. S. (2006). The development of young children’s number and operation sense and its implications for early childhood education. In B. Spodek & O. Saracho (Eds.), Handbook of research on the education of young children (pp. 187–221). Mahwah, NJ: Erlbaum. Baroody, A. J., & Wilkins, J. L. M. (1999). The development of informal counting, number, and arithmetic skills and concepts. In J. Copley (Ed.), Mathematics in the early years, birth to five (pp. 48–65). Reston, VA: National Council of Teachers of Mathematics. Baroody, A. J., Wilkins, J. L. M., & Tiilikainen, S. (2003). The development of children’s understanding of additive commutativity: From protoquantitative concept to general concept? In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 127–160). Mahwah, NJ: Erlbaum. Benson, A. P., & Baroody, A. J. (2003, April). Where does non-verbal production fit in the emergence of children’s mental models. Paper presented at the annual meeting of the Society for Research in Child Development, Tampa, FL. Bisanz, J., & LeFevre, J. (1990). Strategic and nonstrategic processing in the development of mathematical cognition. In D. F. Bjorkland (Ed.), Children’s strategies: Contemporary views of cognitive development (pp. 213–244). Hillsdale, NJ: Erlbaum. Brush, L. (1978). Preschool children’s knowledge of addition and subtraction. Journal for Research in Mathematics Education, 9, 44–54. Bryant, P. (1992). Arithmetic in the cradle. Nature, 358, 712–713. Bryant, P., Christie, C., & Rendu, A. (1999). Children’s understanding of the relation between addition and subtraction inversion, identify, and decomposition. Journal of Experimental Child Psychology, 74, 194–212. Case, R. (1975). Social class differences in the intellectual development: A Neo-Piagetian investigation. Canadian Journal of Behavioral Sciences/Revue Canadienne des sciences du Comportement, 7, 244–262. Cooper, R. G., Starkey, P., Blevins, B., Goth, P., & Leitner, E. (1978, May). Number development: Addition and subtraction. Paper presented at the meeting of the Jean Piaget Society, Philadelphia. Copley, J. V. (2004). The early childhood collaborative: A professional development model to communicate and implement the standards. In D. H. Clements, J. Sarama, & A. M. DiBiase (Eds.), Engaging young children in mathematics: Findings of the 2000 national conference on standards for preschool and kindergarten mathematics education. Mahwah, NJ: Lawrence Erlbaum Associates, Inc. Dowker, A. D. (1997). Young children‘s addition estimates. Mathematical Cognition, 3, 141–154. Dowker, A. D. (2003). Young children’s estimates for addition: The zone of partial knowledge and understanding. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 243–265). Mahwah, NJ: Erlbaum. Entwisle, D. R., & Alexander, K. L. (1990). Beginning school math competence: Minority and majority comparisons. Child Development, 61, 454–471.

M.-L. Lai et al. / Cognitive Development 23 (2008) 216–235

235

Entwisle, D. R., & Alexander, K. L. (1996). Family type and children’s growth in reading and math other the primary grades. Journal of Marriage and the Family, 58, 341–355. Gelman, R. (1972). Logical capacity of very young children: Number invariance rules. Child Development, 43, 75–90. Gilmore, C. K. (2006). Investigating children’s understanding of inversion using the missing number paradigm. Cognitive Development, 21(3), 301–316. Ginsburg, H. P., Cannon, J., Eisenband, J., & Pappas, S. (2006). Mathematical thinking and learning. In K. McCartney & D. Philips (Eds.), The handbook of early child development. Malden, MA: Blackwell Publishing. Ginsburg, H. P., Klein, A., & Starkey, P. (1998). The development of children’s mathematical knowledge: Connecting research with practice. In I. E. Sigel & K. A. Renninger (Eds.), Handbook of child psychology: Vol. 4. Child psychology in practice (5th ed., pp. 401–476). New York: Wiley & Sons. Huttenlocher, J., Jordan, N. C., & Levine, S. C. (1994). A mental model for early arithmetic. Journal of Experimental Psychology: General, 123, 284–296. Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence. New York: Basic Books. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, D.C.: National Academy Press. Klein, J. S., & Bisanz, J. (2000). Preschoolers doing arithmetic: The concepts are willing but the working memory is weak. Canadian Journal of Experimental Psychology, 54, 105–114. Lai, M. -L., & Baroody, A. J. (2006, April). American preschooler’s understanding of the addition–subtraction inverse principle. Symposium paper presented at the Annual Conference of National Council of Teachers of Mathematics, Saint Louis. Levine, S. C., Jordan, N. C., & Huttenlocher, J. (1992). Development of calculation abilities in young children. Journal of Experimental Child Psychology, 53, 72–103. Mix, K. S., Sandhofer, C. M., & Baroody, A. J. (2005). Number words and number concepts: The interplay of verbal and nonverbal processes in early quantitative development. In R. Kail (Ed.), Advances in child development and behavior, Vol. 33 (pp. 305–346). New York: Academic Press. Piaget, J. (1952). (C. Gattegno & F. M. Hodgson, Trans.). In The child’s conception of number. London: Routledge & Kegan Paul Ltd. Piaget, J. (1965). The child’s conception of number. New York: Norton. Piaget, J. (1967). Cognitions and conservations: Two views. A review of Studies in cognitive growth. Contemporary Psychology, 12, 532–533. Piaget, J., & Moreau, A. (2001). The inversion of arithmetic operations. (R. L. Campbell, Trans.). In J. Piaget (Ed.), Studies in Reflecting Abstraction (pp. 69–86). Hove: Psychology Press. Rasmussen, C., Ho, E., & Bisanz, J. (2003). Use of the mathematical principle of inversion in young children. Journal of Experimental Child Psychology, 85, 89–102. Resnick, L. B., & Ford, W. W. (1981). The psychology of mathematics for instruction. Hillsdale, NJ: Erlbaum. Secada, W. G. (1992). Race, ethnicity, social class, language, and achievement in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 623–660). New York: Macmillan. Shaw, K., Nelsen, E., & Shen, Y. L. (2001, April). Preschool development and subsequent school achievement among Spanish-speaking children from low-income families. Paper presented at the annual meeting of the American Educational Research Association, Seattle, WA. Siegler, R. S., & Stern, E. (1998). Conscious and unconscious strategy discoveries: A microgenetic analysis. Journal of Experimental Psychology: General, 127, 377–397. Smedslund, J. (1966). Microanalysis of concrete reasoning. I. The difficulty of some combinations of addition and subtraction of one unit. Scandinavian Journal of Psychology, 7, 145–156. Starkey, P., & Cooper, R. G. (1995). The development of subitizing in young children. British Journal of Developmental Psychology, 13, 399–420. Starkey, P., & Gelman, R. (1982). The development of addition and subtraction abilities prior to formal schooling in arithmetic. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 96–116). Hillsdale, NJ: Erlbaum. Vilette, B. (2002). Do young children grasp the inverse relationship between addition and subtraction? Evidence against early arithmetic. Cognitive Development, 17, 1365–1383. Wertheimer, M. (1959). Productive thinking. New York: Harper & Row. (Original work published 1945).