JOURNAL
OF EXPERIMENTAL
CHILD
PSYCHOLOGY
54, 3722391(1992)
Counting Knowledge and Skill in Cognitive Addition: A Comparison of Normal and Mathematically Disabled Children DAVID
C.
GEARY.
C.
CHRISTINE
BOW-THOMAS,
AND
YUHONG
YAO
The relationship between counting knowledge and computational skills (i.e., skill at counting to solve addition problems) was assessed for groups of first-grade normal and mathematically disabled (MD) children. Twenty-four normal and I3 MD children were administered a series of counting tasks and solved 40 computeradministered addition problems. For the addition task, problem-solving strategies were recorded on a trial-by-trial basis. Performance on the counting tasks suggested that the MD children were developmentally delayed in the understanding of essential and unessential features of counting and were relatively unskilled in the detection of certain forms of counting error. On the addition task, the MD children committed many more computational errors and tended to use developmentally immature counting procedures. The immature counting knowledge of the MD children. combined with their relatively poor skills at detecting counting errors. appeared to underlie their poor computational skills on the addition task. cl IYY? Acadcmlc Prcw. Inc Suggestions for future research arc presented.
The development of arithmetic skills involves the acquisition of problem-solving procedures and the development of memory representations for basic numerical facts (Ashcraft, 1992; Siegler. 1986; Temple, 1991). For the domain of arithmetic, procedures would include the use of counting algorithms, rules (N + 0 = N), carrying, and so on, to complete problem solving (e.g., Baroody, 1983; Hamann & Ashcraft, 1985; Widaman, Geary, Cormier, & Little, 1989). The development of memory representations for basic facts leads to the use of direct memory retrieval We would like to thank the students, teachers, and principals at Parkade and Blue Ridge elementary schools for their support and cooperation. with special thanks to Linda Coutts, Supervisor of Elementary Mathematics. Columbia Public Schools. We would also like to thank Michelle Owen and Mary O’Brien for their assistance with data collection: Tim Cross for programming one of the experimental tasks; and Mark Ashcraft, Peter Frensch. and an anonymous reviewer for comments on earlier drafts. Yuhong Yao is now at Stanford University. Correspondence and requests for reprints should be sent to David C. Geary. Department of Psychology, 210 McAlester Hall. University of Missouri, Columbia. MO 6.521 I. 372 0022~0965/92
$5.00
CopyrIght 0 lYY2 by Academic Preu. Inc All nyhts of rrproduction m any form revnu!
COUNTING
KNOWLEDGE
373
for the solution of simple problems (e.g., 3 + 5 = 8; Siegler, 1986). The relatively poor skills of mathematically disabled (MD) children: that is, children who show a delay in the acquisition of basic mathematical skills, appear to involve both procedural and memory retrieval deficits (Geary, 1990; Geary, Widaman, Little, & Cormier, 1987; Svenson & Broquist, 1975). The memory retrieval deficits of MD children are reflected in less use of direct retrieval, relative to normal children, to solve simple arithmetic problems and a high proportion of errors when an answer is retrieved (Fleischner, Garnett, & Shepherd, 1982; Garnett & Fleischner, 1983; Goldman, Pellegrino, & Mertz, 1988). Moreover, when an answer is correctly retrieved from long-term memory, the solution times are highly unsystematic, unlike those associated with direct memory retrieval in normal children (Geary & Brown, 1991; Geary, Brown, & Samaranayake, 1991). Mathematically disabled children are also less skilled than normal children in the use of counting, or computational, procedures to solve arithmetic problems, especially in Grade 1.’ when using a computational procedure to solve an arithmetic problem, MD children commit many more errors than normal children (Geary, 1990; Geary et al., 1991) and often use developmentally immature strategies. The high frequency of computational errors and use of immature computational algorithms largely disappears by Grade 2 (Geary & Brown, 1991; Geary et al., 1991). while the memory retrieval deficits are still evident well into the elementary school years (Garnett & Fleischner, 1983; Geary et al., 1991; Goldman et al. 1988). These findings indicate that the computational and memory retrieval deficits of MD children show divergent developmental patterns. The finding that the computational skills of MD children quickly approach the skill level of normal children, while retrieval skills do not, suggests that different mechanisms might contribute to the computational and retrieval deficits of MD children. In fact, this pattern suggests that the factors underlying the poor computational skills of MD children might involve a developmental delay, whereas the retrieval deficits might involve a developmental difference or a more fundamental deficit (Ashcraft, Yamashita, & Aram, 1992; Goldman et al., 1988). Given the apparently different mechanisms contributing to the computational skill and longterm memory representation deficits of MD children, it seems likely that the computational skills of MD children can be fruitfully studied independent of any more fundamental memory retrieval deficit. Thus, the ’ In this article, counting knowledge refers to knowledge of how to count objects and counting refers to the act of counting (unless referenced with min or sum counting) these objects. whereas computational skill refers to skill in the use of counting procedures to solve arithmetic oroblems.
374
GEARY,
BOW-THOMAS,
AND
YAO
present experiment focused only on the computational skills of first-grade MD children. In particular, the experiment tested the hypothesis that developmentally immature counting knowledge contributes to the relatively poor computational skills of MD children. A proposal regarding the potential relationship between counting knowledge and computational skills follows a brief overview of conceptual and empirical research on the development of addition skills and the acquisition of counting knowledge. Skill Development
in Addition
Skill development in arithmetic proceeds on at least two dimensions, strategic and speed-of-processing. When solving an addition problem, children typically attempt to retrieve the answer directly from long-term memory (see Siegler, 1986; Siegler & Shrager, 1984). If a satisfactory answer cannot be retrieved, then the child might invoke a second retrieval attempt, or simply guess, by stating any number in working memory. More typically, though, when a satisfactory answer cannot be retrieved, the child resorts to a backup strategy to complete problem solving (Siegler, 1986). Here, first-grade children most frequently count on their fingers or count verbally(Geary, 1990; Siegler, 1987). If counting, whether on fingers or verbally, is required to solve the problem, then children typically use the min or counting-on procedure (Carpenter & Moser, 1984; Groen & Parkman, 1972). With the min procedure, the solution of a problem, such as 5 + 4, begins with stating the cardinal value of the larger valued integer (i.e., 5) and then counting in a unit-by-unit fashion a number of times equal to the value of the min, for minimum, integer (i.e., 4) until the sum is obtained (Groen & Parkman, 1972). Other forms of backup strategy might involve looking at their fingers, but not counting them, to help to remember the answer, or decomposing the problem into simpler problems (Siegler, 1987). Developmentally less mature computational strategies involve counting both integers (the sum or counting-all algorithm) or counting the larger valued integer (the max algorithm). When they count, first-grade MD children tend to use either the min or sum algorithm, whereas normal first-grade children almost always use the min algorithm (Geary. 1990; Geary et al., 1991). When first-grade MD children do use min counting to solve an addition problem it is usually for problems with smaller valued integers (e.g., 3 + 2), whereas the sum algorithm tends to be employed to solve more difficult problems (e.g., 5 + 7). The difficulty of a problem can be indexed by the problem’s sum (Ashcraft, 1992). For the MD children described by Geary et al. (1991), for instance, the frequency of sum counting was positively correlated with the problem’s sum (r(38) = .35, p .25. Mean national percentile rankings across the Applied, Calculation, and Composite indexes for the MD and normal groups are displayed in Table 1. Inspection of Table 1 reveals that the MD children showed a lower mean percentile ranking than the normal children on each of the three measures (ps < .OOl). Experimental
Tasks
Cuurzting tasks. Each child was administered 45 counting trials. The goal was to assess the child’s skill at detecting violations of each of the three how to count rules (Gelman & Meek, 1983), although the tasks also enabled a determination of the child’s understanding of the essential and some of the unessential features of counting (Briars & Siegler, 1984). For each trial, the subject was presented with a row of 8, 12, or 16 alternating red and blue chips. The chips were then counted by a puppet who was learning to count (Briars & Siegler, 1984; Gelman & Meek, 1983). The child’s task was to determine if the puppet’s count was OK or not OK and wrong. The one-to-one correspondence task consisted of 18 trials, 6 trials for 8, 12, and 16). For each set size, the 6 each of the three set sizes (i.e., trials included two correct counts, two pseudo errors, and two incorrect counts. One pseudo error involved counting the red chips first and then counting the blue chips, whereas the other pseudo error involved counting the blue chips first and then the red chips. The pseudo error counts also enabled a determination of whether the child understood that adjacency was an unessential feature of correct counting (see Briars & Siegler, 1984). One of the incorrect counts involved double-counting the first chip, whereas the second error involved double-counting the last chip. The stable order task consisted of 15 trials, 5 trials for each of the three set sizes. Two of the trials were correct counts, whereas the 3 remaining trials involved reversing a word tag (e.g., 4. 6, 5), skipping a word tag (e.g.. 4. 6. 7). or counting with random word tags (e.g., 2, 6. 1).
COUNTING
KNOWLEDGE
379
The cardinality task consisted of 12 trials, 4 trials for each of the three set sizes. Two of the trials were correct counts, whereas the 2 remaining trials involved the puppet stating a word tag one more than the: correct count, or one less than the correct count. To illustrate, for the set size of 8, the puppet counted, “1, 2, 3, 4, 5, 6, 7, 8,” and then stated. “There are 9 chips.” Finally, if the child were to guess on all trials, then the chance level of performance for all three tasks would be .50. On measures combining across all types, if the child understood one of the counting types, for example correct counts, and guessed for all other counting types, then chance level of performance would be greater than SO. Addition tusk. The addition stimuli consisted of 40 pairs of vertically placed single-digit integers (e.g., 5 + 2). Stimuli were constructed from the 56 possible nontie pairwise combinations of the integers 2 to 9 (a tie problem is, e.g., 2 + 2). The frequency and placement of all integers were counterbalanced. That is, each integer appeared 5 times as the augend and 5 times as the addend, and the smaller valued integer appeared 20 times as the augend and 20 times as the addend. No repetition of either the augend or the addend was allowed across consecutive problems. The addition problems were presented at the center of a 30 >( 30-cm CRT controlled by a PC-XT microcomputer. A Cognitive Testing, Station clocking mechanism ensured the collection of reaction times (RTs) with ? 1 ms accuracy. The timing mechanism was initiated with the presentation of the problem on the CRT and was terminated via a Gerbrands G1341T voice-operated relay. The voice-operated relay was triggered when the subject spoke the answer into a microphone connected to the relay. For each problem, a READY prompt appeared at the center of the CRT for a lOOO-ms duration, followed by a lOOO-ms blank screen. Then, an addition problem appeared on the screen and remained until the subject responded. The experimenter initiated each problem presentation sequence via a control key. Procedure
All subjects participated in three experimental sessions. The three counting tasks were administered during one of the sessions. The WRAT and cognitive addition task were administered during the second session, while the WJ-R was administered during the final session. The order of participation in sessions one and two was counterbalanced across subjects (all testing for sessions one and two was completed between November 12 and March 26). For session one, the order of administration of the counting tasks was also counterbalanced across subjects. Within each counting task, the order of trial administration was randomly determined. All subjects were administered the WJ-R after completion of the counting
380
GEARY.
BOW-THOMAS,
AND
YAO
and cognitive addition tasks. Each subject was tested individually and in a quiet room at the school site. The procedures for the three counting tasks followed Gelman and Meek (1983), although, as noted above, the procedures also enabled an assessment of the child’s understanding of some of the essential and unessential features of counting described by Briars and Siegler (1984). Here, the subjects were asked to detect violations of a counting principle (i.e., one-to-one correspondence, stable order, and cardinality), while monitoring a puppet who was learning to count. The instructions were the same as those described by Gelman and Meek, except that the gender of the puppet was always the same as that of the child. For each task, the subject was presented with a single row of 8. 12, or 16 alternating red and blue chips. The child’s task was to monitor the puppet’s count and determine whether the count was OK or not OK and wrong. During the count, if the child looked away from the puppet or if the child did not appear to be attending to the count, then the experimenter would stop and state, “Be sure to pay attention,” and then readminister that trial. Before the first task was administered, each child monitored two correct practice counts. For the addition task, the subjects were asked to solve the 40 addition problems, preceded by 8 practice problems, presented one at a time on the CRT. Subjects were encouraged to use whatever strategy made it easiest for them to obtain the answer, although equal emphasis was placed on speed and accuracy of responding. During the addition task, the answer and strategy used to solve each problem were recorded by the experimenter and classified as one of the earlier described strategies: (a) counting fingers, (b) fingers (i.e., looking at their fingers to help to remember the answer), (c) verbal counting, (d) decomposition, or (e) memory retrieval. After each trial, the subjects were asked to describe how they arrived at the answer. Several previous studies have demonstrated that children can accurately describe problem-solving strategies in arithmetic, if they are asked immediately after the problem is solved (Siegler, 1987, 1989). Based on subject descriptions, the counting tingcrs and verbal counting trials were further classified in accordance with the specific algorithm used for problem solving. That is, the trials were classified as min, based on counting only the smaller valued integer, or sum/max, based on counting both integers or the larger valued integer. Finally. the child’s description was compared to the experimenter’s initial classification (e.g., verbal counting or retrieval) and indicated agreement between the experimenter and the subjects on 96% of the trials. Disagreements typically occurred for trials on which there was no indication of verbal counting, e.g., no lip movements. For these trials, the experimenter scored the trial as retrieval but the subject described a counting or decomposition process.
COUNTING
MEAN
TABLE 2 PERCENTAGE OF CORRECT IDENTIFICATIONS One-to-one correspondence
Group Math disabled Normal Note.
381
KNOWLEDGE
The values
Stable
ON COUNTING
TASKS
order
Cardinality
8
12
16
8
12
16
8
1’-
16
76 89
73 92
69 90
95 97
91 96
91 Y4
YX 99
96 ‘)‘J
x7 97
8. 12, and 16 refer
to set size.
For those trials on which the experimenter and the subject disagreed, the strategy was classified based on the child’s description. RESULTS
For clarity of presentation, the results with brief discussion are presented in three major sections, followed by a more general discussion of the results and their implications. In the first section, analyses of group differences for performance on the counting tasks are presented, followed by a presentation of the results for the addition task. The final section presents results for the relationship between variables representing counting knowledge and computational skills. Counting
Tasks
For the counting tasks, the dependent measure was the number of trials correctly identified, that is, the number of correct and pseudo-error counts identified as correct and the number of incorrect counts identified as incorrect. The percentage of correct identifications across groups, tasks, and set size are displayed in Table 2. The data displayed in Table 2 were analyzed by means of a 2 (group; MD and normal) x 3 (task; one-toone correspondence, stable order, and cardinality) x 3 (set size; 8, 12, and 16) mixed-design analysis of variance (ANOVA), with group as a between-subjects factor and task and set size as within-subjects factors. The results indicated reliable main effects for group, F( 1, 35) =z 12.15, p < .Ol, task, F(2, 70) = 33.59, p < .OOl, and size, F(2, 70) = 3.69, p < .05, as well as a reliable task x group interaction, F (2, 70) = 7.13, p < .Ol. The size x group, task x size, and task x size x group interactions were not reliable (ps > .lO).’ ’ The data displayed in Table 2 were positively skewed. To ensure that the skewed distributions did not affect our results. following Stevens (1986), the square roots of the proportions displayed in Table 2 were submitted to an arcsine transformation to normalize the distributions. The transformed data were then submitted to a mixed-design ANOVA
382
GEARY,
BOW-THOMAS, TABLE
MEAN
PERCENTAGE
OF CORRECT
IDENTIFICATIONS CORRESPONDENCE
AND 3
ACROSS TRIAL TASK
One-to-one Group
Correct
Math disabled Normal Note.
Correct,
and error
refer
49 x4 to trial
TYPES ON ONE-TO-ONE
correspondence Pseudo
96 YX pseudo,
YAO
Error 73
9n
types.
The main effect for size was due to slightly more correct identifications for the set size of 8 (92%) than for the set size of 16 (89%). The reliable task x group interaction was due to fewer correct identifications by the MD children relative to the normal children, on the one-to-one correspondence task, F(1, 35) = 14.13, p < .OOl, but no group difference for performance on the stable order, F( 1, 35) < 1, or cardinality, F(1, 35) = 3.28, p > .05, tasks. For the one-to-one correspondence task, the group difference in number of correct identifications was reliable for each of the three set sizes (ps < .0.5). The performance levels of the MD and normal children for each of the count types (i.e., correct, pseudo error, and error), across set size, for the one-to-one correspondence task is displayed in Table 3. The frequency of correct identifications for this task was analyzed by means of a 2 (group) x 3 (type; correct, pseudo error, and error) mixed-design ANOVA. The results indicated reliable main effects for group, F(1, 35) = 14.47, p < ,001, and type, F(2, 70) = 16.42, p < .OOl, as well as a reliable group x type interaction, F(2, 70) = 4.93, p < .Ol. The interaction was due to group differences in the number of correct identifications for the pseudo error, F(1, 35) = 9.61, p < .Ol, and error, F(1, 35) = 6.17, p < .Ol, trials, but no reliable group difference for correct trials, F (1, 35) < 1. Individualprotocols. To explore further the relatively poor performance of the MD group on the one-to-one correspondence task, individual protocols for the pseudo-error and error trials for each of the MD children were examined. Examination of these protocols indicated that 5 of the 13 MD children appeared to believe that adjacency was a necessary feature of counting, in that they stated that five or six of the six pseudo-error trials were incorrect. Four of the remaining children appeared to be unsure of the status of pseudo-error trials, as indicated by stating that about l/2 (i.e., two, three, or four trials) of the pseudo-error trials were correct and yielded the same pattern of results that was found for the nontransformed data, that is. reliable main effects for group, task, and size. as well as a reliable group x task interaction (ps > .OS). The remaining interactions were not reliable (ps > .05).
COUNTING
TABLE CHARACTERISTICS
Mean % of trials in which strategy was used
383
KNOWLEDGE 4
OF ADDITION ..__~
Mean reaction time in seconds
STRATEGIES
Percentage of errors
Mean % of trials in which min strategy was used
Strategy
MD
Normal
MD
Normal
MD
Normal
MD
Normal
Counting fingers Fingers Verbal counting Retrieval
57 Oh 33 9
44 2 37 17
7.4
8.0 5.8 1.4 3.7
56
23 9 24 19
31
69 91 -
6.2 -‘.
57 66
62
Note. MD, Math disabled. Mean reaction time (RT) excluded error and spoiled RTs. For the counting fingers and verbal counting trials. mean RT was based on min trials, to make the solution times comparable across groups. Columnar totals for the strategy percentages may not sum to 100 due to rounding. ’ For the MD and normal groups, 0.4 and 0.2%, respectively, of the trials were classified as decomposition. h The actual percentage was 0.4. ’ There were not enough correct retrieval trials to produce a meaningful estimate.
and l/2 were incorrect. The 4 remaining children always indicated that the pseudo-error trials were correct. As for the error trials, 76% of the errors occurred when the first chip was double-counted. Summary. The results for the one-to-one correspondence task suggest that MD children as a group show an immature understanding of some basic number concepts, relative to their academically normal peers. The results also suggest that the group difference in counting knowledge does not reside in the understanding of a particular principle (Gelman & Meek, 1983) but rather in the understanding of essential and unessential features of counting (Briars & Siegler, 1984) and in skill in detecting doublecounting errors. Specifically, the counting knowledge of MD children appears to be rather rigid and developmentally immature, as reflected in the tendency of many of the MD children to behave as if adjacency was an essential feature of correct counting. Moreover, MD children appear to be less skilled than normal children in detecting certain forms of counting error, particularly if the error is committed at the beginning of the count. The high error rate at the beginning, rather than the end, of the count suggests that the failure to detect the double-counting errors might have been due to a memory failure rather than to poor conceptual knowledge. Addition Task
Table 4 presents group-level characteristics of addition strategies. Consistent with previous studies (Geary, 1990; Geary et al., 1991; Siegler,
384
GEARY.
BOW-THOMAS,
AND
YAO
1987), both the normal and MD children relied primarily on counting to solve the addition problems. In fact, univariate ANOVAs indicated that the normal and MD groups did not differ reliably in the frequency with which the counting fingers, F(1, 35) < 1, or verbal counting, F(1, 35) < 1, strategies were used for problem solving. Nevertheless, the normal children did rely on direct retrieval for problem solving more frequently than the MD children, F( 1, 35) = 4.13, p < .05. The normal children also committed fewer errors than the MD children for the counting fingers, F(1, 29) = 9.77, p < .Ol, verbal counting, F( 1. 28) = 7.68, p < .Ol. and retrieval, F(1, 19) = 9.19, p < .Ol, strategies. Finally, the normal children used the min algorithm more frequently than the MD children for both the counting fingers, F( 1, 27) = 11.64, p < .Ol, and verbal counting, F(1, 20) = 4.23, p = ,053, strategies. Problem dificulty and use of counting (computational) strategies. Previous research has shown that the use of direct retrieval decreases in frequency as the difficulty of the problem increases (e.g., Siegler & Shrager, 1984). This is so because answers are less likely to be retrieved from long-term memory for difficult problems than for less difficult problems. As noted earlier, for addition. the difficulty of the problem can be indexed by the sum of the augend and addend (Ashcraft. 1992). Thus, the frequency with which direct retrieval is used for problem solving should decrease with an increase in the value of the sum of the problem. In the present study, the frequency of retrieval trials decreased with an increase in the value of the sum for the normal group, r(38) = - .70, p < .OOl, but not for the MD group, r(38) = - .14, p < .2S. The difference in the value of these two coefficients was marginally reliably, z = - 1.89, p < .06. Thus, consistent with many previous studies (e.g., Geary et al.. 1991; Siegler & Shrager, 1984), for the normal group, direct retrieval appeared to be used to solve simpler problems and backup computational (i.e., finger counting and verbal counting) strategies to solve more difficult problems. For the MD group, the failure to find a relationship between the frequency of retrieval trials and problem difficulty, combined with the high proportion of retrieval errors, suggests that the MD children were frequently guessing when they stated a retrieved answer. In other words, rather than resort to a backup computational strategy if they were not certain of the accuracy of a retrieved answer, the MD children appeared to have stated any answer that might have been retrieved, regardless of the potential accuracy of that answer (see Siegler, 1988). Summary. Of particular interest in the current study was group differences in computational skills. Although the normal and MD children did not differ in the frequency with which the counting fingers and verbal counting strategies were used for problem solving, they did differ in skill at executing these strategies. Consistent with previous studies of first-
COUNTING
KNOWLEDGE
385
grade MD children (Geary, 1990; Geary et al., 1991), the MD children in the present study committed more than twice the computational errors of the normal children and used the min algorithm less frequently. Relationship between Counting Knowledge and Computational Skills The question addressed in this final section is whether the immature knowledge of counting features of MD children contributed to their relatively poor computational skills evident on the addition task. Recall that the analyses of the performance on the counting tasks suggested that the MD children as a group were developmentally delayed in their understanding of counting features and were relatively poor at detecting counting errors when these errors occurred at the beginning of the count. A rigid conceptual understanding of counting might impede the adoption of counting algorithms that deviate from the standard count-all, or sum, strategy. If so, then an index of immature counting knowledge (i.e., the frequency of incorrect pseudo-error trials in this study) should be correlated with the frequency with which the min algorithm was used to solve addition problems. If skill at detecting counting errors and developmentally mature counting knowledge contribute to skill at repairing incorrectly executed computational procedures, as suggested by Ohlsson and Rees (1991). then the frequency of incorrect pseudo-error trials and frequency of doublecounting errors should be correlated with the frequency of computational errors on the addition task. To test these hypotheses, the frequency of correct identifications for pseudo-error trials was correlated with the frequency of min counting (across the counting fingers and verbal {counting strategies), and a composite index (the sum of the frequency of correct double-counting and pseudo-error trials) was correlated with the frequency of computational errors (again, across the counting-fingers and1 verbalcounting strategies). The results indicated that, across groups, the frequency of correct identifications for pseudo-error trials was reliably correlated with use of the min algorithm, r(35) = .47, p < .Ol. Moreover, the score for the composite index was reliably correlated with the frequency of computational errors, r(3.5) = - .44, p < .05, but was unrelated to the frequency of retrieval errors, r(35) .X3, p > .50. A t test for dependent correlations indicated that the value of these two coefficients (i.e., - .44 and .18) differed reliably, t(34) = 2.57, p < .05. Nevertheless, the coefficient for the latter correlation should be interpreted with caution, given ,the relatively small number of retrieval trials. Another potential indicator of counting knowledge is the number of correct counts identified as incorrect. Briars and Siegler (1984), however, found that 3-, 4-, and 5-year-old children only rarely identified correct counts as incorrect. This result suggests that errors on correct counts do
386
GEARY,
BOW-THOMAS.
AND
YAO
not occur often and therefore are not likely to be a sensitive measure of individual differences in knowledge of counting features. Indeed, in this study, correct counts were rarely identified as incorrect by the normal and MD children. The frequency of correct counts that were identified as incorrect was not correlated with either the frequency of computational errors (r(36) = - .25, p > .lO) or with the use of the min algorithm (r(34)
= .02, p > .50).
Finally, Analysis of Covariance (ANCOVA) procedures indicated that controlling for group differences in skill at detecting double-counting errors and frequency of pseudo-error trials eliminated the group difference in the frequency of computational errors, F( 1, 33) = 1.21, p > .25. Controlling for the group difference in number of correct identifications on pseudo-error trials also eliminated the group difference in the frequency with which the min algorithm was used to solve the addition problems, F(1, 32) = 3.42, p > .OS. Summary. Consistent with the conceptual presentation of Ohlsson and Rees (1991), the results suggest that a mature understanding of essential and unessential features of counting is related to the adoption of the developmentally mature min algorithm for solving addition problems and. along with skill at detecting counting errors, the accuracy with which computational procedures can be executed. Moreover, these analyses support the argument that for MD children a developmental delay in the understanding of counting features and relatively poor skills at detecting certain forms of counting error contribute to the poor computational skills of these children in the domain of arithmetic. DISCUSSION
The present study enabled an assessment of the relationship between counting knowledge and computational skills for the domain of addition (i.e., skill at using counting procedures to solve addition problems). Ohlsson and Rees (1991) argued that counting knowledge provides a standard against which the execution of computational procedures is evaluated (see also Siegler & Jenkins, 1989). If the execution of a computational procedure violates principled knowledge, then the procedure is modified so that with subsequent executions it conforms to counting knowledge. The conceptual formulation presented by Ohlsson and Rees implies that computational skills should be related to the developmental maturity of the child’s counting knowledge and to the child’s skill at detecting when counting has violated this principled knowledge. The finding that the frequency of computational errors in the domain of addition was correlated with an index representing skill at detecting counting errors and the maturity of the child’s knowledge of counting features provides empirical support of this position. This argument is bolstered by the finding that this same index was not correlated with the frequency of retrieval errors.
COUNTING
KNOWLEDGE
387
The analyses of the relationship between variables representing counting knowledge and computational skills also suggests a relationship between knowledge of counting features and use of the min, or counting-on, procedure (Carpenter & Moser, 1984; Groen & Parkman, 1972). More precisely, the analyses suggested that the adoption of the min procedure might require an understanding of the essential and unessential features of counting (Briars & Siegler, 1984). If so, then why did the MD children sometimes use the sum procedure and at other times use the min procedure to solve addition problems? As noted earlier, the child’s performance and presumably counting knowledge varies with the size of the counted set. So, the understanding of the cardinality of small set sizes does not necessarily indicate that the child will understand the concept of cardinality when applied to larger set sizes. Similarly, the min procedure tends to be employed to solve problems with smaller valued integers and the sum procedure tends to be used to solve problems with larger valued integers (Geary, 1990; Geary et al., 1991). Thus, if counting knowledge varies with set size, then the expectation would be that the min procedure would first be adopted for the solution of easy problems (e.g., 2 + 1) and only later for the solution of more difficult problems (e.g., 8 + 9; Geary, 1990). This argument must be considered a working hypothesis, however, because the experimental tasks used in the current study did not allow for a determination of the child’s knowledge of, for instance, the unessential features of counting for relatively small set sizes (e.g.. 3). Thus, it might be the case that MD children understand that adjacency is unessential when counting small set sizes. Moreover, the counting tasks did not enable a complete assessment of the understanding of all of the unessential counting features described by Briars and Siegler (1984). Finally, the work of Siegler and Jenkins (1989) suggests that if knowledge of counting features is related to the adoption of the min algorithm, then this knowledge is more likely necessary but not sufficient for the abandonment of the sum procedure. The current study also enabled a comparison of the counting knowledge of MD and normal children and allowed for an assessment of the relationship between the counting knowledge of MD children and their relatively poor computational skills. Recall that MD children commit many more computational errors than normal children and show a delay in the adoption of the min procedure (Geary, 1990; Geary et al., 1987; Geary et al., 1991; Goldman et al., 1988; Svenson & Broquist, 1975). As a group, the MD children in the current study appeared to be developmentally delayed in the understanding of essential and unessential features of counting. Many of the 7-year-old MD children assessed in this study appeared to believe that adjacency was an essential feature of correct counting, much like the 5-year-old children described by Briars and Siegler (1984). The MD children were also less skilled than the normal children
388
GEARY.
BOW-THOMAS,
AND
YAO
in detecting certain types of counting errors, in particular, double-counting errors that occurred at the beginning of the count. Failure to detect this type of counting error can be especially detrimental to modifying incorrect min counts. because the most common computational error during min counting involves double-counting the larger valued integer (Geary, 1990; Siegler & Shrager, 1984). In fact, a developmental delay in the understanding of counting features and relatively poor skills at detecting doublecounting errors appeared to mediate the poor, relative to their normal peers, computational skills of the MD children. Nevertheless, this too should be considered a working hypothesis, as the entire range of unessential counting features described by Briars and Siegler (1984) was not assessed in the current study. Nevertheless, the current study can be used to address the issue of whether the functional counting skills of children are better represented by Gelman and Gallistel’s (1978) principles-first model or by the induction model proposed by Briars and Siegler (1984). Recall that Gelman and Gallistel argued that the counting behavior of children is governed by implicit knowledge of the one-to-one correspondence, stable order, and cardinality principles. Briars and Siegler, on the other hand, argued that children first count by rote and then gradually induce the essential and unessential features of counting. Within the framework provided by Gelman and Gallistel, performance on the cardinality task should be dependent upon knowledge of the principles of one-to-one correspondence and stable order. Thus, the poor performance on the one-to-one correspondence task found in the current study should have affected performance on the cardinahty task. This was not the case. This result combined with the finding that a feature of Briar and Siegler’s induction model (i.e., adjacency) most sharply differentiated the counting skills of MD and normal children indicates that the current results are more consistent with the induction than the principles-first model. Regardless, other evidence supports the position that human infants have a basic ability to detect the numerosity of arrays. That is, infants appear to be able to extract quantity information from the environment for small set sizes (Starkey. Spelke, & Gelman, 1990). The findings of Starkey et al. are consistent with Gelman and Gallistel’s (1978) view that young children have an implicit understanding of basic counting principles. Perhaps the numerical abstraction skills of infants reflect an innate bias to attend to quantitative features in the environment. It is not at all clear, however, how such an innate orientation to quantitative features influences more functional skills, such as those reflected in counting behavior, though it is reasonable to suspect such a link (Siegler, 1991). The numerical abstraction skills of infants may in fact provide the basis for early counting skills, but our data suggest that the concepts associated with the functional skills of children are, for the most part, induced.
COUNTING
389
KNOWLEDGE
In summary, the current study provided empirical support for the hypothesis that the link between counting knowledge and computational skills (Ohlsson & Rees, 1991) resides in the developmental maturity of the child’s principled knowledge and the child’s skill at detecting when the execution of computational procedures violates this knowledge. The results also suggest that the poor computational skills of first-grade MD children are related to a developmental delay in the understanding of essential and unessential features of counting (Briars & Siegler, 1984) and relatively poor skills at detecting certain forms of counting error. Finally, the present study suggests that a comparison of normal and MD children across the entire range of essential and unessential counting features described by Briars and Siegler( 1984) and across small and large set sizes as related to computational skills would provide additional, and perhaps more, valuable information on the relationship between principled knowledge and procedural skills. Such a study might also provide more explicit information on the development of conceptual knowledge in MD children. REFERENCES Ashcraft.
M.
H. (1992).
Cognitive
arithmetic:
A review
of data
and theory.
Cognition.
in
pre.S.
Ashcraft. M. H., Yamashita. T. S.. & Aram, D. M. (1992). Mathematics performance in left and right brain-lesioned children. Brain and Cognition, 19, 208-252. Baroody, A. J. (1983). The development of procedural knowledge: An alternative explanation for chronometric trends of mental arithmetic. Developmenfal Review. 3, 225230. Baroody, A. J. (1984). The case of Fehcia: A young child’s strategies for reducing memory demands during mental addition. Cognition and Instruction. 1, 109Y116. Briars, D., & Siegler, R. S. (1984). A featural analysis of preschoolers’ counting knowledge. Developmental Psychology, 20, 607-618. Carpenter, T. P., & Moser. J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15, 179202. Fleischner, J. E., Garnett, K., & Shepherd. M. J. (1982). Proficiency in arithmletic basic fact computation of learning disabled and nondisabled children. Focus on Learning Problems in Mathematics, 4, 47-56. Fuson, K. C. (1988). Children’s counting and concepts of number. New York: SpringerVerlag. Garnett, K., & Fleischner, J. E. (1983). Automatization and basic fact performance of normal and learning disabled children. Learning Disabilities Quarterly, 6, 2213-230. Geary. D. C. (1990). A componential analysis of an early learning deficit in mathematics. Journal of Experimental Child Psychology, 49, 363-383. Geary, D. C., & Brown. S. C (1991). Cognitive addition: Strategy choice and speed-ofprocessing differences in gifted. normal, and mathematically disabled children. Developmental Psychology, 27, 398-406. Geary, D. C., Brown, S. C, & Samaranayake, V. A. (1YYl). Cognitive addition: A short longitudinal study of strategy choice and speed-of-processing differences in normal and mathematically disabled children. Developmental Psychology, 27, 787-797. Geary, D. C.. Widaman. K. F., Little, T. D.. & Cormier, P. (1987). Cognitive addition:
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Comparison of learning disabled and academically normal elementary school children. Cognitive Development, 2, 249-269. Gelman, R.. & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard Univ. Press. Gelman, R.. & Meek, E. (1983). Preschoolers’ counting: Principles before skill. Cogrz:nition, 13, 343-359. Goldman, S. R., Pellegrino, J. W., & Mertz, D. L. (1988). Extended practice of basic addition facts: Strategy changes in learning disabled students, Cognition and Instruction, 5, 223-265. Greeno, J. Cl.. Riley, M. S., & Gelman, R. (1984). Conceptual competence and children’s counting. Cognitive Psychology. 16, 94-193. Groen. G. J.. & Parkman, J. M. (1972). A chronometric analysis of simple addition. Psychological Review, 79, 329-343. Hamann, M. S., & Ashcraft. M. H. (1985). Simple and complex mental addition across development. Journal of Experimental Child Psychology. 40, 49-72. Jastak, J. F., & Jastak. S. (1978). Manual for the Wide Range Achievement Test: 1978 Revised Edition. Wilmington, DE: Jastak Associates, Inc. Ohlsson, S.. & Rees. E. (1991). The function of conceptual understanding in the learning of arithmetic procedures. Cognition and Instruction, 8, 1033179. Siegler, R. S. (1986). Unities across domains in children’s strategy choices. In M. Perlmutter (Ed.), Perspectives for intellectual development: Minnesota symposia on child psychology (Vol. 19. pp. 1-48). Hillsdale, NJ: Erlbaum. Siegler. R. S. (1987). The perils of averaging data over strategies: An example from children’s addition. Journal of Experimental Psychology: General, 116, 250-264. Siegler, R. S. (1988). Individual differences in strategy choices: Good students, not-so-good students. and perfectionists. Child Development. 59, 833-851. Siegler, R. S. (1989). Hazards of mental chronometry: An example from children’s subtraction. Journal of Educational Psychology, 81, 497-506. Siegler, R. S. (1991). In young children’s counting. procedures precede principles. Edurational Psychology Review, 3, l2?- 135, Siegler, R. S., & Jenkins, E. (1989). How children discover new strategies. Hillsdale. NJ: Erlbaum. Siegler, R. S.. & Robinson, M. (1982). The development of numerical understandings. In H. Reese & L. P. Lipsitt (Eds.), Advances in child development and behavior (Vol. 16, pp. 241-312). New York: Academic Press. Siegler, R. S., & Shrager, J. (1984). Strategy choice in addition and subtraction: How do children know what to do? In C. Sophian (Ed.). Origins of cognitive skills (pp. 229293). Hillsdale, NJ: Erlbaum. Starkey. P., Spelke. E. S., & Gelman, R. (1990). Numerical abstraction by human infants. Cognition, 36, 97-127. Stevens. J. (1986). Applied multivariate statistics for fhe social sciences. Hillsdale. NJ: Erlbaum. Svenson. 0.. & Broquist, S. (1975). Strategies for solving simple addition problems: A comparison of normal and subnormal children. Scandinavian Journal of Psychology, 16, 143-151. Temple, C. M. (1991). Procedural dyscalculia and number fact dyscalculia: Double dissociation in developmental dyscalculia. Cognitive Nemopsychology, 8, 155-176. Widaman, K. F., Geary. D. C., Cormier, P.. & Little, T. D. (1989). A componential model for mental addition. Journal of Experimental Psychology: Learning, Memory, and Cognition, 15, 898-919.
COUNTING Woodcock,
R. W.,
Revised. Allen, Wynn, RECEIVED:
K. (1990). November
& Johnson,
M. B. (1989). Woodcock-Johnson TX: DLM Teaching Resources. Children’s understanding of counting. Cognition, 20, 1991;
REVISED
391
KNOWLEDGE
May
14, 1992
Tests of Ach,ievement36, 155-193.
OF EXPERIMENTAL
CHILD
PSYCHOLOGY
54, 3722391(1992)
Counting Knowledge and Skill in Cognitive Addition: A Comparison of Normal and Mathematically Disabled Children DAVID
C.
GEARY.
C.
CHRISTINE
BOW-THOMAS,
AND
YUHONG
YAO
The relationship between counting knowledge and computational skills (i.e., skill at counting to solve addition problems) was assessed for groups of first-grade normal and mathematically disabled (MD) children. Twenty-four normal and I3 MD children were administered a series of counting tasks and solved 40 computeradministered addition problems. For the addition task, problem-solving strategies were recorded on a trial-by-trial basis. Performance on the counting tasks suggested that the MD children were developmentally delayed in the understanding of essential and unessential features of counting and were relatively unskilled in the detection of certain forms of counting error. On the addition task, the MD children committed many more computational errors and tended to use developmentally immature counting procedures. The immature counting knowledge of the MD children. combined with their relatively poor skills at detecting counting errors. appeared to underlie their poor computational skills on the addition task. cl IYY? Acadcmlc Prcw. Inc Suggestions for future research arc presented.
The development of arithmetic skills involves the acquisition of problem-solving procedures and the development of memory representations for basic numerical facts (Ashcraft, 1992; Siegler. 1986; Temple, 1991). For the domain of arithmetic, procedures would include the use of counting algorithms, rules (N + 0 = N), carrying, and so on, to complete problem solving (e.g., Baroody, 1983; Hamann & Ashcraft, 1985; Widaman, Geary, Cormier, & Little, 1989). The development of memory representations for basic facts leads to the use of direct memory retrieval We would like to thank the students, teachers, and principals at Parkade and Blue Ridge elementary schools for their support and cooperation. with special thanks to Linda Coutts, Supervisor of Elementary Mathematics. Columbia Public Schools. We would also like to thank Michelle Owen and Mary O’Brien for their assistance with data collection: Tim Cross for programming one of the experimental tasks; and Mark Ashcraft, Peter Frensch. and an anonymous reviewer for comments on earlier drafts. Yuhong Yao is now at Stanford University. Correspondence and requests for reprints should be sent to David C. Geary. Department of Psychology, 210 McAlester Hall. University of Missouri, Columbia. MO 6.521 I. 372 0022~0965/92
$5.00
CopyrIght 0 lYY2 by Academic Preu. Inc All nyhts of rrproduction m any form revnu!
COUNTING
KNOWLEDGE
373
for the solution of simple problems (e.g., 3 + 5 = 8; Siegler, 1986). The relatively poor skills of mathematically disabled (MD) children: that is, children who show a delay in the acquisition of basic mathematical skills, appear to involve both procedural and memory retrieval deficits (Geary, 1990; Geary, Widaman, Little, & Cormier, 1987; Svenson & Broquist, 1975). The memory retrieval deficits of MD children are reflected in less use of direct retrieval, relative to normal children, to solve simple arithmetic problems and a high proportion of errors when an answer is retrieved (Fleischner, Garnett, & Shepherd, 1982; Garnett & Fleischner, 1983; Goldman, Pellegrino, & Mertz, 1988). Moreover, when an answer is correctly retrieved from long-term memory, the solution times are highly unsystematic, unlike those associated with direct memory retrieval in normal children (Geary & Brown, 1991; Geary, Brown, & Samaranayake, 1991). Mathematically disabled children are also less skilled than normal children in the use of counting, or computational, procedures to solve arithmetic problems, especially in Grade 1.’ when using a computational procedure to solve an arithmetic problem, MD children commit many more errors than normal children (Geary, 1990; Geary et al., 1991) and often use developmentally immature strategies. The high frequency of computational errors and use of immature computational algorithms largely disappears by Grade 2 (Geary & Brown, 1991; Geary et al., 1991). while the memory retrieval deficits are still evident well into the elementary school years (Garnett & Fleischner, 1983; Geary et al., 1991; Goldman et al. 1988). These findings indicate that the computational and memory retrieval deficits of MD children show divergent developmental patterns. The finding that the computational skills of MD children quickly approach the skill level of normal children, while retrieval skills do not, suggests that different mechanisms might contribute to the computational and retrieval deficits of MD children. In fact, this pattern suggests that the factors underlying the poor computational skills of MD children might involve a developmental delay, whereas the retrieval deficits might involve a developmental difference or a more fundamental deficit (Ashcraft, Yamashita, & Aram, 1992; Goldman et al., 1988). Given the apparently different mechanisms contributing to the computational skill and longterm memory representation deficits of MD children, it seems likely that the computational skills of MD children can be fruitfully studied independent of any more fundamental memory retrieval deficit. Thus, the ’ In this article, counting knowledge refers to knowledge of how to count objects and counting refers to the act of counting (unless referenced with min or sum counting) these objects. whereas computational skill refers to skill in the use of counting procedures to solve arithmetic oroblems.
374
GEARY,
BOW-THOMAS,
AND
YAO
present experiment focused only on the computational skills of first-grade MD children. In particular, the experiment tested the hypothesis that developmentally immature counting knowledge contributes to the relatively poor computational skills of MD children. A proposal regarding the potential relationship between counting knowledge and computational skills follows a brief overview of conceptual and empirical research on the development of addition skills and the acquisition of counting knowledge. Skill Development
in Addition
Skill development in arithmetic proceeds on at least two dimensions, strategic and speed-of-processing. When solving an addition problem, children typically attempt to retrieve the answer directly from long-term memory (see Siegler, 1986; Siegler & Shrager, 1984). If a satisfactory answer cannot be retrieved, then the child might invoke a second retrieval attempt, or simply guess, by stating any number in working memory. More typically, though, when a satisfactory answer cannot be retrieved, the child resorts to a backup strategy to complete problem solving (Siegler, 1986). Here, first-grade children most frequently count on their fingers or count verbally(Geary, 1990; Siegler, 1987). If counting, whether on fingers or verbally, is required to solve the problem, then children typically use the min or counting-on procedure (Carpenter & Moser, 1984; Groen & Parkman, 1972). With the min procedure, the solution of a problem, such as 5 + 4, begins with stating the cardinal value of the larger valued integer (i.e., 5) and then counting in a unit-by-unit fashion a number of times equal to the value of the min, for minimum, integer (i.e., 4) until the sum is obtained (Groen & Parkman, 1972). Other forms of backup strategy might involve looking at their fingers, but not counting them, to help to remember the answer, or decomposing the problem into simpler problems (Siegler, 1987). Developmentally less mature computational strategies involve counting both integers (the sum or counting-all algorithm) or counting the larger valued integer (the max algorithm). When they count, first-grade MD children tend to use either the min or sum algorithm, whereas normal first-grade children almost always use the min algorithm (Geary. 1990; Geary et al., 1991). When first-grade MD children do use min counting to solve an addition problem it is usually for problems with smaller valued integers (e.g., 3 + 2), whereas the sum algorithm tends to be employed to solve more difficult problems (e.g., 5 + 7). The difficulty of a problem can be indexed by the problem’s sum (Ashcraft, 1992). For the MD children described by Geary et al. (1991), for instance, the frequency of sum counting was positively correlated with the problem’s sum (r(38) = .35, p .25. Mean national percentile rankings across the Applied, Calculation, and Composite indexes for the MD and normal groups are displayed in Table 1. Inspection of Table 1 reveals that the MD children showed a lower mean percentile ranking than the normal children on each of the three measures (ps < .OOl). Experimental
Tasks
Cuurzting tasks. Each child was administered 45 counting trials. The goal was to assess the child’s skill at detecting violations of each of the three how to count rules (Gelman & Meek, 1983), although the tasks also enabled a determination of the child’s understanding of the essential and some of the unessential features of counting (Briars & Siegler, 1984). For each trial, the subject was presented with a row of 8, 12, or 16 alternating red and blue chips. The chips were then counted by a puppet who was learning to count (Briars & Siegler, 1984; Gelman & Meek, 1983). The child’s task was to determine if the puppet’s count was OK or not OK and wrong. The one-to-one correspondence task consisted of 18 trials, 6 trials for 8, 12, and 16). For each set size, the 6 each of the three set sizes (i.e., trials included two correct counts, two pseudo errors, and two incorrect counts. One pseudo error involved counting the red chips first and then counting the blue chips, whereas the other pseudo error involved counting the blue chips first and then the red chips. The pseudo error counts also enabled a determination of whether the child understood that adjacency was an unessential feature of correct counting (see Briars & Siegler, 1984). One of the incorrect counts involved double-counting the first chip, whereas the second error involved double-counting the last chip. The stable order task consisted of 15 trials, 5 trials for each of the three set sizes. Two of the trials were correct counts, whereas the 3 remaining trials involved reversing a word tag (e.g., 4. 6, 5), skipping a word tag (e.g.. 4. 6. 7). or counting with random word tags (e.g., 2, 6. 1).
COUNTING
KNOWLEDGE
379
The cardinality task consisted of 12 trials, 4 trials for each of the three set sizes. Two of the trials were correct counts, whereas the 2 remaining trials involved the puppet stating a word tag one more than the: correct count, or one less than the correct count. To illustrate, for the set size of 8, the puppet counted, “1, 2, 3, 4, 5, 6, 7, 8,” and then stated. “There are 9 chips.” Finally, if the child were to guess on all trials, then the chance level of performance for all three tasks would be .50. On measures combining across all types, if the child understood one of the counting types, for example correct counts, and guessed for all other counting types, then chance level of performance would be greater than SO. Addition tusk. The addition stimuli consisted of 40 pairs of vertically placed single-digit integers (e.g., 5 + 2). Stimuli were constructed from the 56 possible nontie pairwise combinations of the integers 2 to 9 (a tie problem is, e.g., 2 + 2). The frequency and placement of all integers were counterbalanced. That is, each integer appeared 5 times as the augend and 5 times as the addend, and the smaller valued integer appeared 20 times as the augend and 20 times as the addend. No repetition of either the augend or the addend was allowed across consecutive problems. The addition problems were presented at the center of a 30 >( 30-cm CRT controlled by a PC-XT microcomputer. A Cognitive Testing, Station clocking mechanism ensured the collection of reaction times (RTs) with ? 1 ms accuracy. The timing mechanism was initiated with the presentation of the problem on the CRT and was terminated via a Gerbrands G1341T voice-operated relay. The voice-operated relay was triggered when the subject spoke the answer into a microphone connected to the relay. For each problem, a READY prompt appeared at the center of the CRT for a lOOO-ms duration, followed by a lOOO-ms blank screen. Then, an addition problem appeared on the screen and remained until the subject responded. The experimenter initiated each problem presentation sequence via a control key. Procedure
All subjects participated in three experimental sessions. The three counting tasks were administered during one of the sessions. The WRAT and cognitive addition task were administered during the second session, while the WJ-R was administered during the final session. The order of participation in sessions one and two was counterbalanced across subjects (all testing for sessions one and two was completed between November 12 and March 26). For session one, the order of administration of the counting tasks was also counterbalanced across subjects. Within each counting task, the order of trial administration was randomly determined. All subjects were administered the WJ-R after completion of the counting
380
GEARY.
BOW-THOMAS,
AND
YAO
and cognitive addition tasks. Each subject was tested individually and in a quiet room at the school site. The procedures for the three counting tasks followed Gelman and Meek (1983), although, as noted above, the procedures also enabled an assessment of the child’s understanding of some of the essential and unessential features of counting described by Briars and Siegler (1984). Here, the subjects were asked to detect violations of a counting principle (i.e., one-to-one correspondence, stable order, and cardinality), while monitoring a puppet who was learning to count. The instructions were the same as those described by Gelman and Meek, except that the gender of the puppet was always the same as that of the child. For each task, the subject was presented with a single row of 8. 12, or 16 alternating red and blue chips. The child’s task was to monitor the puppet’s count and determine whether the count was OK or not OK and wrong. During the count, if the child looked away from the puppet or if the child did not appear to be attending to the count, then the experimenter would stop and state, “Be sure to pay attention,” and then readminister that trial. Before the first task was administered, each child monitored two correct practice counts. For the addition task, the subjects were asked to solve the 40 addition problems, preceded by 8 practice problems, presented one at a time on the CRT. Subjects were encouraged to use whatever strategy made it easiest for them to obtain the answer, although equal emphasis was placed on speed and accuracy of responding. During the addition task, the answer and strategy used to solve each problem were recorded by the experimenter and classified as one of the earlier described strategies: (a) counting fingers, (b) fingers (i.e., looking at their fingers to help to remember the answer), (c) verbal counting, (d) decomposition, or (e) memory retrieval. After each trial, the subjects were asked to describe how they arrived at the answer. Several previous studies have demonstrated that children can accurately describe problem-solving strategies in arithmetic, if they are asked immediately after the problem is solved (Siegler, 1987, 1989). Based on subject descriptions, the counting tingcrs and verbal counting trials were further classified in accordance with the specific algorithm used for problem solving. That is, the trials were classified as min, based on counting only the smaller valued integer, or sum/max, based on counting both integers or the larger valued integer. Finally. the child’s description was compared to the experimenter’s initial classification (e.g., verbal counting or retrieval) and indicated agreement between the experimenter and the subjects on 96% of the trials. Disagreements typically occurred for trials on which there was no indication of verbal counting, e.g., no lip movements. For these trials, the experimenter scored the trial as retrieval but the subject described a counting or decomposition process.
COUNTING
MEAN
TABLE 2 PERCENTAGE OF CORRECT IDENTIFICATIONS One-to-one correspondence
Group Math disabled Normal Note.
381
KNOWLEDGE
The values
Stable
ON COUNTING
TASKS
order
Cardinality
8
12
16
8
12
16
8
1’-
16
76 89
73 92
69 90
95 97
91 96
91 Y4
YX 99
96 ‘)‘J
x7 97
8. 12, and 16 refer
to set size.
For those trials on which the experimenter and the subject disagreed, the strategy was classified based on the child’s description. RESULTS
For clarity of presentation, the results with brief discussion are presented in three major sections, followed by a more general discussion of the results and their implications. In the first section, analyses of group differences for performance on the counting tasks are presented, followed by a presentation of the results for the addition task. The final section presents results for the relationship between variables representing counting knowledge and computational skills. Counting
Tasks
For the counting tasks, the dependent measure was the number of trials correctly identified, that is, the number of correct and pseudo-error counts identified as correct and the number of incorrect counts identified as incorrect. The percentage of correct identifications across groups, tasks, and set size are displayed in Table 2. The data displayed in Table 2 were analyzed by means of a 2 (group; MD and normal) x 3 (task; one-toone correspondence, stable order, and cardinality) x 3 (set size; 8, 12, and 16) mixed-design analysis of variance (ANOVA), with group as a between-subjects factor and task and set size as within-subjects factors. The results indicated reliable main effects for group, F( 1, 35) =z 12.15, p < .Ol, task, F(2, 70) = 33.59, p < .OOl, and size, F(2, 70) = 3.69, p < .05, as well as a reliable task x group interaction, F (2, 70) = 7.13, p < .Ol. The size x group, task x size, and task x size x group interactions were not reliable (ps > .lO).’ ’ The data displayed in Table 2 were positively skewed. To ensure that the skewed distributions did not affect our results. following Stevens (1986), the square roots of the proportions displayed in Table 2 were submitted to an arcsine transformation to normalize the distributions. The transformed data were then submitted to a mixed-design ANOVA
382
GEARY,
BOW-THOMAS, TABLE
MEAN
PERCENTAGE
OF CORRECT
IDENTIFICATIONS CORRESPONDENCE
AND 3
ACROSS TRIAL TASK
One-to-one Group
Correct
Math disabled Normal Note.
Correct,
and error
refer
49 x4 to trial
TYPES ON ONE-TO-ONE
correspondence Pseudo
96 YX pseudo,
YAO
Error 73
9n
types.
The main effect for size was due to slightly more correct identifications for the set size of 8 (92%) than for the set size of 16 (89%). The reliable task x group interaction was due to fewer correct identifications by the MD children relative to the normal children, on the one-to-one correspondence task, F(1, 35) = 14.13, p < .OOl, but no group difference for performance on the stable order, F( 1, 35) < 1, or cardinality, F(1, 35) = 3.28, p > .05, tasks. For the one-to-one correspondence task, the group difference in number of correct identifications was reliable for each of the three set sizes (ps < .0.5). The performance levels of the MD and normal children for each of the count types (i.e., correct, pseudo error, and error), across set size, for the one-to-one correspondence task is displayed in Table 3. The frequency of correct identifications for this task was analyzed by means of a 2 (group) x 3 (type; correct, pseudo error, and error) mixed-design ANOVA. The results indicated reliable main effects for group, F(1, 35) = 14.47, p < ,001, and type, F(2, 70) = 16.42, p < .OOl, as well as a reliable group x type interaction, F(2, 70) = 4.93, p < .Ol. The interaction was due to group differences in the number of correct identifications for the pseudo error, F(1, 35) = 9.61, p < .Ol, and error, F(1, 35) = 6.17, p < .Ol, trials, but no reliable group difference for correct trials, F (1, 35) < 1. Individualprotocols. To explore further the relatively poor performance of the MD group on the one-to-one correspondence task, individual protocols for the pseudo-error and error trials for each of the MD children were examined. Examination of these protocols indicated that 5 of the 13 MD children appeared to believe that adjacency was a necessary feature of counting, in that they stated that five or six of the six pseudo-error trials were incorrect. Four of the remaining children appeared to be unsure of the status of pseudo-error trials, as indicated by stating that about l/2 (i.e., two, three, or four trials) of the pseudo-error trials were correct and yielded the same pattern of results that was found for the nontransformed data, that is. reliable main effects for group, task, and size. as well as a reliable group x task interaction (ps > .OS). The remaining interactions were not reliable (ps > .05).
COUNTING
TABLE CHARACTERISTICS
Mean % of trials in which strategy was used
383
KNOWLEDGE 4
OF ADDITION ..__~
Mean reaction time in seconds
STRATEGIES
Percentage of errors
Mean % of trials in which min strategy was used
Strategy
MD
Normal
MD
Normal
MD
Normal
MD
Normal
Counting fingers Fingers Verbal counting Retrieval
57 Oh 33 9
44 2 37 17
7.4
8.0 5.8 1.4 3.7
56
23 9 24 19
31
69 91 -
6.2 -‘.
57 66
62
Note. MD, Math disabled. Mean reaction time (RT) excluded error and spoiled RTs. For the counting fingers and verbal counting trials. mean RT was based on min trials, to make the solution times comparable across groups. Columnar totals for the strategy percentages may not sum to 100 due to rounding. ’ For the MD and normal groups, 0.4 and 0.2%, respectively, of the trials were classified as decomposition. h The actual percentage was 0.4. ’ There were not enough correct retrieval trials to produce a meaningful estimate.
and l/2 were incorrect. The 4 remaining children always indicated that the pseudo-error trials were correct. As for the error trials, 76% of the errors occurred when the first chip was double-counted. Summary. The results for the one-to-one correspondence task suggest that MD children as a group show an immature understanding of some basic number concepts, relative to their academically normal peers. The results also suggest that the group difference in counting knowledge does not reside in the understanding of a particular principle (Gelman & Meek, 1983) but rather in the understanding of essential and unessential features of counting (Briars & Siegler, 1984) and in skill in detecting doublecounting errors. Specifically, the counting knowledge of MD children appears to be rather rigid and developmentally immature, as reflected in the tendency of many of the MD children to behave as if adjacency was an essential feature of correct counting. Moreover, MD children appear to be less skilled than normal children in detecting certain forms of counting error, particularly if the error is committed at the beginning of the count. The high error rate at the beginning, rather than the end, of the count suggests that the failure to detect the double-counting errors might have been due to a memory failure rather than to poor conceptual knowledge. Addition Task
Table 4 presents group-level characteristics of addition strategies. Consistent with previous studies (Geary, 1990; Geary et al., 1991; Siegler,
384
GEARY.
BOW-THOMAS,
AND
YAO
1987), both the normal and MD children relied primarily on counting to solve the addition problems. In fact, univariate ANOVAs indicated that the normal and MD groups did not differ reliably in the frequency with which the counting fingers, F(1, 35) < 1, or verbal counting, F(1, 35) < 1, strategies were used for problem solving. Nevertheless, the normal children did rely on direct retrieval for problem solving more frequently than the MD children, F( 1, 35) = 4.13, p < .05. The normal children also committed fewer errors than the MD children for the counting fingers, F(1, 29) = 9.77, p < .Ol, verbal counting, F( 1. 28) = 7.68, p < .Ol. and retrieval, F(1, 19) = 9.19, p < .Ol, strategies. Finally, the normal children used the min algorithm more frequently than the MD children for both the counting fingers, F( 1, 27) = 11.64, p < .Ol, and verbal counting, F(1, 20) = 4.23, p = ,053, strategies. Problem dificulty and use of counting (computational) strategies. Previous research has shown that the use of direct retrieval decreases in frequency as the difficulty of the problem increases (e.g., Siegler & Shrager, 1984). This is so because answers are less likely to be retrieved from long-term memory for difficult problems than for less difficult problems. As noted earlier, for addition. the difficulty of the problem can be indexed by the sum of the augend and addend (Ashcraft. 1992). Thus, the frequency with which direct retrieval is used for problem solving should decrease with an increase in the value of the sum of the problem. In the present study, the frequency of retrieval trials decreased with an increase in the value of the sum for the normal group, r(38) = - .70, p < .OOl, but not for the MD group, r(38) = - .14, p < .2S. The difference in the value of these two coefficients was marginally reliably, z = - 1.89, p < .06. Thus, consistent with many previous studies (e.g., Geary et al.. 1991; Siegler & Shrager, 1984), for the normal group, direct retrieval appeared to be used to solve simpler problems and backup computational (i.e., finger counting and verbal counting) strategies to solve more difficult problems. For the MD group, the failure to find a relationship between the frequency of retrieval trials and problem difficulty, combined with the high proportion of retrieval errors, suggests that the MD children were frequently guessing when they stated a retrieved answer. In other words, rather than resort to a backup computational strategy if they were not certain of the accuracy of a retrieved answer, the MD children appeared to have stated any answer that might have been retrieved, regardless of the potential accuracy of that answer (see Siegler, 1988). Summary. Of particular interest in the current study was group differences in computational skills. Although the normal and MD children did not differ in the frequency with which the counting fingers and verbal counting strategies were used for problem solving, they did differ in skill at executing these strategies. Consistent with previous studies of first-
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grade MD children (Geary, 1990; Geary et al., 1991), the MD children in the present study committed more than twice the computational errors of the normal children and used the min algorithm less frequently. Relationship between Counting Knowledge and Computational Skills The question addressed in this final section is whether the immature knowledge of counting features of MD children contributed to their relatively poor computational skills evident on the addition task. Recall that the analyses of the performance on the counting tasks suggested that the MD children as a group were developmentally delayed in their understanding of counting features and were relatively poor at detecting counting errors when these errors occurred at the beginning of the count. A rigid conceptual understanding of counting might impede the adoption of counting algorithms that deviate from the standard count-all, or sum, strategy. If so, then an index of immature counting knowledge (i.e., the frequency of incorrect pseudo-error trials in this study) should be correlated with the frequency with which the min algorithm was used to solve addition problems. If skill at detecting counting errors and developmentally mature counting knowledge contribute to skill at repairing incorrectly executed computational procedures, as suggested by Ohlsson and Rees (1991). then the frequency of incorrect pseudo-error trials and frequency of doublecounting errors should be correlated with the frequency of computational errors on the addition task. To test these hypotheses, the frequency of correct identifications for pseudo-error trials was correlated with the frequency of min counting (across the counting fingers and verbal {counting strategies), and a composite index (the sum of the frequency of correct double-counting and pseudo-error trials) was correlated with the frequency of computational errors (again, across the counting-fingers and1 verbalcounting strategies). The results indicated that, across groups, the frequency of correct identifications for pseudo-error trials was reliably correlated with use of the min algorithm, r(35) = .47, p < .Ol. Moreover, the score for the composite index was reliably correlated with the frequency of computational errors, r(3.5) = - .44, p < .05, but was unrelated to the frequency of retrieval errors, r(35) .X3, p > .50. A t test for dependent correlations indicated that the value of these two coefficients (i.e., - .44 and .18) differed reliably, t(34) = 2.57, p < .05. Nevertheless, the coefficient for the latter correlation should be interpreted with caution, given ,the relatively small number of retrieval trials. Another potential indicator of counting knowledge is the number of correct counts identified as incorrect. Briars and Siegler (1984), however, found that 3-, 4-, and 5-year-old children only rarely identified correct counts as incorrect. This result suggests that errors on correct counts do
386
GEARY,
BOW-THOMAS.
AND
YAO
not occur often and therefore are not likely to be a sensitive measure of individual differences in knowledge of counting features. Indeed, in this study, correct counts were rarely identified as incorrect by the normal and MD children. The frequency of correct counts that were identified as incorrect was not correlated with either the frequency of computational errors (r(36) = - .25, p > .lO) or with the use of the min algorithm (r(34)
= .02, p > .50).
Finally, Analysis of Covariance (ANCOVA) procedures indicated that controlling for group differences in skill at detecting double-counting errors and frequency of pseudo-error trials eliminated the group difference in the frequency of computational errors, F( 1, 33) = 1.21, p > .25. Controlling for the group difference in number of correct identifications on pseudo-error trials also eliminated the group difference in the frequency with which the min algorithm was used to solve the addition problems, F(1, 32) = 3.42, p > .OS. Summary. Consistent with the conceptual presentation of Ohlsson and Rees (1991), the results suggest that a mature understanding of essential and unessential features of counting is related to the adoption of the developmentally mature min algorithm for solving addition problems and. along with skill at detecting counting errors, the accuracy with which computational procedures can be executed. Moreover, these analyses support the argument that for MD children a developmental delay in the understanding of counting features and relatively poor skills at detecting certain forms of counting error contribute to the poor computational skills of these children in the domain of arithmetic. DISCUSSION
The present study enabled an assessment of the relationship between counting knowledge and computational skills for the domain of addition (i.e., skill at using counting procedures to solve addition problems). Ohlsson and Rees (1991) argued that counting knowledge provides a standard against which the execution of computational procedures is evaluated (see also Siegler & Jenkins, 1989). If the execution of a computational procedure violates principled knowledge, then the procedure is modified so that with subsequent executions it conforms to counting knowledge. The conceptual formulation presented by Ohlsson and Rees implies that computational skills should be related to the developmental maturity of the child’s counting knowledge and to the child’s skill at detecting when counting has violated this principled knowledge. The finding that the frequency of computational errors in the domain of addition was correlated with an index representing skill at detecting counting errors and the maturity of the child’s knowledge of counting features provides empirical support of this position. This argument is bolstered by the finding that this same index was not correlated with the frequency of retrieval errors.
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KNOWLEDGE
387
The analyses of the relationship between variables representing counting knowledge and computational skills also suggests a relationship between knowledge of counting features and use of the min, or counting-on, procedure (Carpenter & Moser, 1984; Groen & Parkman, 1972). More precisely, the analyses suggested that the adoption of the min procedure might require an understanding of the essential and unessential features of counting (Briars & Siegler, 1984). If so, then why did the MD children sometimes use the sum procedure and at other times use the min procedure to solve addition problems? As noted earlier, the child’s performance and presumably counting knowledge varies with the size of the counted set. So, the understanding of the cardinality of small set sizes does not necessarily indicate that the child will understand the concept of cardinality when applied to larger set sizes. Similarly, the min procedure tends to be employed to solve problems with smaller valued integers and the sum procedure tends to be used to solve problems with larger valued integers (Geary, 1990; Geary et al., 1991). Thus, if counting knowledge varies with set size, then the expectation would be that the min procedure would first be adopted for the solution of easy problems (e.g., 2 + 1) and only later for the solution of more difficult problems (e.g., 8 + 9; Geary, 1990). This argument must be considered a working hypothesis, however, because the experimental tasks used in the current study did not allow for a determination of the child’s knowledge of, for instance, the unessential features of counting for relatively small set sizes (e.g.. 3). Thus, it might be the case that MD children understand that adjacency is unessential when counting small set sizes. Moreover, the counting tasks did not enable a complete assessment of the understanding of all of the unessential counting features described by Briars and Siegler (1984). Finally, the work of Siegler and Jenkins (1989) suggests that if knowledge of counting features is related to the adoption of the min algorithm, then this knowledge is more likely necessary but not sufficient for the abandonment of the sum procedure. The current study also enabled a comparison of the counting knowledge of MD and normal children and allowed for an assessment of the relationship between the counting knowledge of MD children and their relatively poor computational skills. Recall that MD children commit many more computational errors than normal children and show a delay in the adoption of the min procedure (Geary, 1990; Geary et al., 1987; Geary et al., 1991; Goldman et al., 1988; Svenson & Broquist, 1975). As a group, the MD children in the current study appeared to be developmentally delayed in the understanding of essential and unessential features of counting. Many of the 7-year-old MD children assessed in this study appeared to believe that adjacency was an essential feature of correct counting, much like the 5-year-old children described by Briars and Siegler (1984). The MD children were also less skilled than the normal children
388
GEARY.
BOW-THOMAS,
AND
YAO
in detecting certain types of counting errors, in particular, double-counting errors that occurred at the beginning of the count. Failure to detect this type of counting error can be especially detrimental to modifying incorrect min counts. because the most common computational error during min counting involves double-counting the larger valued integer (Geary, 1990; Siegler & Shrager, 1984). In fact, a developmental delay in the understanding of counting features and relatively poor skills at detecting doublecounting errors appeared to mediate the poor, relative to their normal peers, computational skills of the MD children. Nevertheless, this too should be considered a working hypothesis, as the entire range of unessential counting features described by Briars and Siegler (1984) was not assessed in the current study. Nevertheless, the current study can be used to address the issue of whether the functional counting skills of children are better represented by Gelman and Gallistel’s (1978) principles-first model or by the induction model proposed by Briars and Siegler (1984). Recall that Gelman and Gallistel argued that the counting behavior of children is governed by implicit knowledge of the one-to-one correspondence, stable order, and cardinality principles. Briars and Siegler, on the other hand, argued that children first count by rote and then gradually induce the essential and unessential features of counting. Within the framework provided by Gelman and Gallistel, performance on the cardinality task should be dependent upon knowledge of the principles of one-to-one correspondence and stable order. Thus, the poor performance on the one-to-one correspondence task found in the current study should have affected performance on the cardinahty task. This was not the case. This result combined with the finding that a feature of Briar and Siegler’s induction model (i.e., adjacency) most sharply differentiated the counting skills of MD and normal children indicates that the current results are more consistent with the induction than the principles-first model. Regardless, other evidence supports the position that human infants have a basic ability to detect the numerosity of arrays. That is, infants appear to be able to extract quantity information from the environment for small set sizes (Starkey. Spelke, & Gelman, 1990). The findings of Starkey et al. are consistent with Gelman and Gallistel’s (1978) view that young children have an implicit understanding of basic counting principles. Perhaps the numerical abstraction skills of infants reflect an innate bias to attend to quantitative features in the environment. It is not at all clear, however, how such an innate orientation to quantitative features influences more functional skills, such as those reflected in counting behavior, though it is reasonable to suspect such a link (Siegler, 1991). The numerical abstraction skills of infants may in fact provide the basis for early counting skills, but our data suggest that the concepts associated with the functional skills of children are, for the most part, induced.
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KNOWLEDGE
In summary, the current study provided empirical support for the hypothesis that the link between counting knowledge and computational skills (Ohlsson & Rees, 1991) resides in the developmental maturity of the child’s principled knowledge and the child’s skill at detecting when the execution of computational procedures violates this knowledge. The results also suggest that the poor computational skills of first-grade MD children are related to a developmental delay in the understanding of essential and unessential features of counting (Briars & Siegler, 1984) and relatively poor skills at detecting certain forms of counting error. Finally, the present study suggests that a comparison of normal and MD children across the entire range of essential and unessential counting features described by Briars and Siegler( 1984) and across small and large set sizes as related to computational skills would provide additional, and perhaps more, valuable information on the relationship between principled knowledge and procedural skills. Such a study might also provide more explicit information on the development of conceptual knowledge in MD children. REFERENCES Ashcraft.
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