Learning Multiplication Facts: A Study of Children Taught by Discovery Methods in England

Journal of Experimental Child Psychology 79, 37–55 (2001) doi:10.1006/jecp.2000.2579, available online at http://www.idealibrary.com on

Learning Multiplication Facts: A Study of Children Taught by Discovery Methods in England Sylvia Steel and Elaine Funnell Royal Holloway University of London, Egham, Surrey, United Kingdom The development of multiplication skills was examined in a group of children ages 8 to 12 years who were taught by discovery methods. Strategies used by the children included direct retrieval, retrieval + calculation, and counting-in-series. Repeated addition was not observed. Retrieval was the fastest and least error-prone strategy; counting-in-series was the slowest and most error prone. Children ages 8 and 9 years used mainly mixed strategies. Children ages 10 to 12 years used mainly retrieval or retrieval and calculation for low operands, but reverted to back-up strategies for high operands based on the strategies available for low operands. There was a general shift away from less effective strategies across ages 8 to 12 years but, by the end of the primary school (age 11 years), relatively few children used the most effective strategy of retrieval for all operands. The development of effective strategies was related to nonverbal reasoning ability and to working memory capacity. The results are considered with reference to experiential and pedagogical models of multiplication. © 2001 Academic Press Key Words: multiplication; children; teaching methods; strategies; experience; operand size; working memory; nonverbal reasoning.

During development, the learning of multiplication facts is believed to be affected by the frequency with which problems are presented and the solution strategy that is selected (Siegler, 1988). Strong connections between operands (i.e., the numbers multiplied together) develop with repeated practice, leading to a reduction in errors. Reported strategies used by children include retrieval; repeated addition, for example, 4 × 3 = 3 + 3 + 3 + 3; writing down the problem without addition; and making marks that can then be counted (Brownwell & Carper, 1943; Cooney, Swanson, & Ladd, 1988; Lemaire & Siegler, 1995; Siegler, 1988). The use of retrieval is thought to increase over time (Brownwell & Carper, 1943; Cooney et al., 1988). Studies of adults suggest that experience does not necessarily lead to the exclusive use of fact retrieval (Dehaene & Cohen, 1995; LeFevre, Bisanz, Daley,

We gratefully acknowledge the cooperation of local schools and the constructive criticisms of two anonymous referees on earlier drafts of the article. Address correspondence and reprint requests to Elaine Funnell, Department of Psychology, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom. E-mail: e.funnell @rhbnc.ac.uk. 37 0022-0965/01 $35.00 Copyright © 2001 by Academic Press All rights of reproduction in any form reserved.

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Buffone, Greenham, & Sadesky, 1996). Although simple facts such as 2 × 3 = 6 may be stored and retrieved from rote memory, other strategies, such as reversing the operands or calculating the answer from simpler facts, may be used for more difficult problems. In a comparison of Chinese and Canadian adults, LeFevre and Liu (1997) found that Chinese adults were more likely to rely upon retrieval than were Canadian adults, who used a variety of strategies. The authors argued that educational and linguistic factors may have influenced the range of strategies available. In a longitudinal study of French children who are taught multiplication with an emphasis on retrieval methods, strategy choice was found to vary with the difficulty of the problem and with the degree of experience (Lemaire & Siegler, 1995). At the start of the study, retrieval was the preferred strategy for easy problems (with products below 9). Initially, both retrieval and repeated addition were used for slightly harder problems (with products 9 to 18) but retrieval became the preferred strategy later on. With the more difficult problems (with products 20 to 81) an initial preference for back-up strategies was replaced by retrieval by the end of the study. Improvements in speed and accuracy accompanied the changes in strategy choice. Similar results were reported in a study of North American children who were introduced to direct fact retrieval at a later stage (Siegler, 1988). Lemaire and Siegler (1995) argued that the greater use of back-up strategies for solving problems with larger operands, and the increasing use of retrieval with experience, fit the predictions of the adaptive strategy choice model (ASCM), developed originally by Siegler and Shipley (1995) to explain addition. More recently, Shrager and Siegler (1998) have suggested that children discover new strategies for themselves, using multiple approaches for prolonged periods, but adaptively learning to apply more advanced strategies over time. Discovering new strategies and making adaptive choices are thought to be integrated processes in strategy development. Shrager and Siegler (1998) have applied this model successfully to the simulation of multiplication. Poor working memory resources are believed to result in the incomplete representation of number facts in long-term memory (Geary, 1990; Geary & Brown, 1991; Hitch & McAuley, 1991; Siegler & Shrager, 1984). Working memory may be involved in the selection of appropriate strategies and in solving more complex problems (Bull, Johnston, & Roy, 1999; Logie, Gilhooly, & Wynn, 1994). Problems containing larger operands produce slower responses and more errors than those containing smaller numbers, suggesting that the calculations involved with larger operands make more demands on working memory (Hitch, 1978a, 1978b). Structural models of multiplication propose that answers to problems with larger operands are more likely to have similar magnitudes than smaller problems, giving rise to greater levels of mutual inhibition (Campbell, 1995; Campbell & Oliphant, 1992). Thus, larger problems take longer to solve than smaller problems and more errors are made. In contrast, the learning-based model

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of Siegler (1988) explains the problem-size effect on the basis of greater experience with small operands, which are known to appear more frequently in children’s textbooks (Ashcraft & Christy, 1995). As a result of experience, stronger associations are formed between small operands and their correct solutions. The study reported here examines these issues in a different learning environment. In England, over the past 20 years, didactic teaching methods have given way to “discovery methods” in which children find the best methods of calculation for themselves. Although this approach is beginning to change, the National Curriculum (1991; revised 1995) continues to emphasize the role of exploratory activities in the development of multiplication skills. At present, children in School Year Groups 1 and 2 (age range = 5 to 7 years) are provided with opportunities to learn patterns of multiples (for example, 3, 6, 9, and 12) and to use these to make predictions. Patterns involving multiplication and division are then introduced, including a square of multiplication facts up to 100. Multiplication facts relating to multiples of 2, 5, and 10 are taught, and these facts are used to learn other facts, such as using double multiples of 2 to produce multiples of 4 as a basis for finding answers to novel questions. Basic calculators are used for learning operations such as repeated addition. Children in School Year Groups 3 to 7 (age range = 7 to 12 years) are provided with opportunities to learn multiplication facts up to 10 × 10, to develop a range of calculation methods using known facts, and to understand multiplication as repeated addition. Although multiplication facts are once more being tested at school, the National Curriculum does not encourage instruction in particular methods of learning, such as rote memory. This raises interesting questions about the effect of the learning environment provided by the present educational system in England on the development of multiplication strategies, particularly when compared with other countries where more didactic methods are used. Our study investigated the attainment levels and strategies used in multiplication by groups of children ages 7 to 12 years. We presented multiple-choice questions on a computer and measured accuracy and response times. Individual strategies were classified according to the algorithm used by Geary, Bow-Thomas, and Yao (1992) and Lemaire and Siegler (1995). Changes in the distribution of strategy choice and in speed and accuracy across school years were examined, and information was collected concerning the teaching strategies to which individual children were exposed. On the basis of previous studies (e.g., Geary et al., 1992; Lemaire & Siegler, 1995; Siegler, 1988) we predicted that the response times of children would reflect the efficiency of the strategy used. Responses should be faster for children using retrieval compared with those using a mixture of retrieval and derived facts and should be faster and more accurate than those using less efficient strategies such as counting. We were particularly interested to know whether the findings of our study would support the predictions of the ASCM model that, with increasing experience, retrieval would emerge as the main strategy used.

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METHOD Participants Two hundred forty-one children in school years 3 to 7 (age range = 7 to 12 years) were drawn from two primary and two secondary schools in England (Year 3, N = 54; mean age = 8;01; Year 4, N = 50; mean age = 9;03; Year 5, N = 48, mean age = 10;00; Year 6, N = 49, mean age 10;09; Year 7, N = 40, mean age = 12;02). Hereafter, these school year groups are referred to as children ages 8, 9, 10, 11, and 12 years old. There were 116 boys and 125 girls from various socioeconomic and ethnic backgrounds. Children were selected from the whole ability range, although those who were considered by their teachers to have specific learning or behavioral difficulties were excluded. Six subjects ages 10 and 11 years and four children ages 8 and 9 years were given extra coaching outside school from parents and/or tutors. They are referred to as the “tutor group.” Design and Materials All children ages 10 to 12 years took all parts of the test. Children ages 8 and 9 years completed as much of the test as was appropriate for their learning experience. The test contained 144 multiplication questions that used all number combinations from 1 × 2 to 12 × 12. Each question had five response choices. The four incorrect answers accompanying the correct answer were chosen to be possible responses, for example, 5 × 4 = 20, 19, 16, 15, 24; and included likely operand errors, for example, 15, 16, 24; and counting errors, for example, 19. The questions were presented on a computer as 24 individual screens each composed of 6 questions. Twenty-two screens presented questions belonging to the same multiplication series: half used low operands, from 1 to 6 (e.g., 2 × 4, 6 × 4, 3 × 4, 5 × 4, 1 × 4, 4 × 4), and half used high operands, from 7 to 12 (e.g., 9 × 4, 7 × 4, 11 × 4, 12 × 4, 8 × 4, and 10 × 4). Figure 1 shows an example of these “banded” screens. The remaining 2 screens presented mixed sets of 6 questions. One screen presented multiplication problems using only low operands, for example, 2 × 5 and 3 × 4, the other used only high operands, for example, 11 × 8 and 9 × 7. The 2 mixed screens were included to compare the strategies used with mixed- and same-set questions. Each question was presented with the sum to the left and the five alternative choices placed in a line to the right, finishing with a box containing question marks for the child to press if unsure of the correct answer. A “DONE” box, clicked at the end of each screen, brought up the next screen. Accuracy was recorded automatically at the completion of each question and response times at the completion of each screen. The screens were arranged in 4 blocks of 5 screens (30 questions per block) and 1 block of 4 screens (24 questions). Within each block the screens contained a range of easy and harder problems. For children ages 10 to 12 years, easy and harder questions were distributed evenly across the test ending with some of the

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FIG. 1. Diagrammatic representation of screen example.

simplest problems to counterbalance any effects of fatigue. For children ages 8 and 9 years, screens were arranged in order of difficulty in order to allow children to stop if the test became too arduous. Seven “timing screens,” preceded by two practice screens, were used to record the time to carry out the physical operations involved in the task. Each screen contained a list of six target numbers on the left with a line of five numbers beside each to the right. The child was required to click the number in the line to the right that was identical to the target on the left. For each child, the fastest and slowest screen times were excluded and the average speed of the remaining five screens was deducted from the mean response times of screens on the main part of the test. In addition to the multiplication test, the Progressive Matrices (Raven, Court, & Raven, 1990) was given, as a measure of nonverbal reasoning, and the DigitSpan subtest of the WISC-III UK (1992) as a measure of working memory. Procedure Children were tested individually with the tester beside them. The timing screens were undertaken first followed by the multiplication test. The children were shown how to use the mouse, which was positioned beside the dominant hand. They were told that any method of finding an answer was acceptable except guessing and counting in 1s from zero. The questions marks could be clicked if they were unable to find an answer. They were asked to work as quickly as possible, to attempt all the questions, and to be as accurate as possible. They were told to click the “DONE” box when all questions had been completed. It was explained that the tester was interested in how they reached their answers, and children were asked not to hide what they were doing, for example, by sitting on their fingers to conceal their use in counting.

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Two practice screens preceded the main part of the test. The tester observed the child and recorded the strategy used for each question. Initial classifications were based on the following criteria. If the answer was spoken without hesitation and without obvious counting-related movements, the response was classified as “retrieval.” If there was verbal evidence of operations based on number facts, the response was classified as “calculation.” Evidence of counting in 1s from zero was classified as “no strategy.” Other forms of counting were recorded separately. Responses that combined strategies were noted. At the end of each block of five screens the child and tester discussed informally whether the tester’s identification of strategy had been accurate. Questions asked by the tester were individually tailored to elicit as much information as possible. The children were asked whether they had reversed operands and, if so, which ones. They were also asked if they had used the strategies used normally in the classroom. The test continued in this way until all trials had been completed, at which point the children were asked a series of questions concerning methods of learning and practicing multiplication. Discussions of the strategies used generally confirmed the observations of the tester, but some discrepancies were found where children reported using procedural methods (for example, verbalizing in their head and hiding the use of fingers) in situations where the tester had assumed retrieval. The initial records were revised accordingly. Turning round operands was mentioned, although a number of children insisted that they only turned round the ones “where you can.” Most children, in all school years, reported that they found learning multiplication facts difficult, especially those containing higher operands. Primary school children (ages 8 to 11 years) reported practicing multiplication facts in the classroom by coloring in tables squares, investigating patterns in numbers, answering questions in work books, and doing tests. They reported that they learned multiplication by writing down the number series, by “looking at them,” saying them to their parents, and listening to tapes. Only 10 children reported learning multiplication facts by the more traditional way (i.e., chanting “Once two is two, two twos are four. . .”). These children were receiving extra tuition outside school either from tutors or parents or both. They all confirmed that they learned by rote and that they practiced regularly. Secondary school children (age 12 years) reported that they no longer practiced multiplication facts in school. Teachers of the children involved in the research were questioned about teaching methods. As predicted, a variety of methods was mentioned, including coloring tables squares, investigations into number patterns, watching TV programs on this topic, and doing examples from the mathematics scheme. No specific data were available from teachers on the amount of time spent practicing multiplication. In both primary schools, children aged 8 and 9 years worked mostly at their own pace on individual arithmetic projects. Where specific work was set, it tended to vary across groups. Children ages 10 to 11 years received more whole-class teaching, but the content of the lessons varied from class to class and school to school.

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RESULTS Response times (seconds per screen) were shorter for the 22 banded screens than 2 mixed screens for both low operands (1 to 6) [t tests for paired samples, t(136) = −5.51] and high operands (7 to 12) [t(95) = −7.44]. For low operands, banded screens, (M = 24.51, SD = 22.76), mixed screens (M = 32.52, SD = 24.92); for high operands, banded screens, (M = 42.30, SD = 24.29), mixed screens (M = 68.88, SD = 42.75). Children clearly benefited from the fact that, on banded screens, all questions were drawn from the same multiplication series. The analyses of data from the 22 banded screens are reported below. All significance levels have a probability level of .05 (two-tailed) or better, and all ANOVAs were followed by paired comparisons using the Modified LSD Bonferroni Test p < .05. Where reaction time and error data were transformed to reduce differences in variance, log base 10 and square root were used respectively. Strategies Strategy Types and Strategy Groups Although children were urged not to guess, a small amount of guessing was observed. Three main strategy types were noted: (1) direct retrieval from longterm memory, (2) calculation using derived facts, and (3) counting-in-series. Calculation used derived facts to obtain the required answers (e.g., for 6 × 5: if 5 × 5 = 25, then 6 × 5 = 25 + 5 = 30). Counting was used sometimes to complete the answer. Some calculations were quite complex: for example, children attempted to calculate questions involving the operand 8 by multiplying by 10 and then counting backward. Counting-in-series produced a series of numbers, for example, 6 × 11 = 11, 22, 33, 44, 55, 66. Numbers were not overtly added, for example, 4 × 2 = 2 + 2 + 2 + 2, but were simply repeated as a memorized sequence. Fingers were usually used to keep a tally of the level reached in the series. The series was sometimes completed by sequential counting, for example, 4 × 7 = 7, 14, 21, 22, 23, 24, 25, 26, 27, 28. For questions where the operands 2, 5, and 10 were in the first position (for example, 5 × 4), the order was often reversed to facilitate counting (for example, 5, 10, 15, 20 in place of 4, 8, 12, 16, 20). Five strategy groups were formed: (1) retrieval only included a small amount of calculation involving two particular operands, 11 and 12; (2) retrieval + calculation included retrieval of some answers (mainly those involving lower facts and multiples of 10 up to 100 and 11 up to 99) and calculation of the remaining answers from derived facts close to the target solution; (3) mixed strategies included some retrieval, some calculation, and some counting-in-series; (4) counting-in-series only; and (5) no strategy included guessing, counting from zero in 1s, and failures to respond. Distribution of strategy groups: Ages 8 and 9 years. Analysis was confined to questions that ranged from 1 × 2 to 6 × 5. The following distribution of children in each strategy group was observed: retrieval only (11%), retrieval + calculation

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FIG. 2. Distribution of strategy group by ages 8 and 9 years (low facts to 6 × 5). R = retrieval; R + Ca = retrieval + calculation; M = mixed strategies; C = counting-in-series; NS = no strategy.

(12%), mixed strategies (54%), counting-in-series (13%), and no strategy (10%). Figure 2 shows that more children age 9 years (20%) used retrieval than in those age 8 years (4%) but overall there were no significant differences between the observed frequencies in each strategy group across these two age groups, χ2(4) = 7.26. Applying a mixture of strategies was the greatly preferred method in both age groups. Only small percentages of children retrieved all the answers (11%) or had no strategy to apply (10%). Distribution of strategy groups for all number problems: Ages 10 to 12 years. The strategies used were distributed as follows: retrieval only (20%), retrieval + calculation (37%), mixed strategies (23%), counting-in-series (20%), and no strategy (0%). There were significant differences in strategy use across ages 10 to 12 years, χ2(6) = 22.03. Figure 3 shows that children age 11 were significantly more likely to use retrieval and less likely to count in series than were children age 10 years, χ2(3) = 8.75. Nevertheless, by age 11 years, knowledge of multiplication facts was still incomplete for 49% children and not in evidence at all for 20%. That is, by the end of primary school, one-fifth of children had failed to

FIG. 3. Distribution of strategy group by ages 10 to 12 years (all facts to 12 × 12). R = retrieval; R + Ca = retrieval + calculation; M = mixed strategies; C = counting-in-series.

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learn multiplication facts for even the simplest multiplication problems. By age 12 years, the trend away from counting-in-series was almost complete, but the increase in retrieval strategies observed between ages 10 and 11 years was not continued, possibly because practice in retrieval was reduced for children age 12 years by the increased use of calculators. Strategies on low operand problems: Combined ages 10 to 12 years. Strategies used for low operands (1 × 2 to 6 × 6) were as follows: retrieval only (54%), retrieval + calculation (10%), mixed strategies (16%), and counting-inseries (20%). As Fig. 4 shows, there has been a significant shift in the distribution of strategies toward retrieval for low operands, χ2(3) = 56.39, between ages 8 and 12 years. Even so, only half the older children retrieved all facts for these relatively simple multiplication problems and one-fifth still counted-in-series. Strategies for high-operand problems: Combined ages 10 to 12 years. Overall, there was a significant shift away from retrieval strategies for low numbers toward less efficient strategies for high operands, χ2(3) = 46.22 (mixed strategies and counting-in-series combined). As Table 1 shows, children tended to shift from the strategy they used for low operands toward the next most effective strategy for use with high operands. That is, retrieval changed to retrieval + calculation, retrieval + calculation changed to mixed strategies, and mixed strategies changed to counting-in-series. Children who counted in series for low operands continued to do so for high operands. Summary Children used retrieval, retrieval + calculation, and counting-in-series to solve multiplication problems. Some children ages 8 and 9 years had no strategy to apply. Most children in this age range used mixed strategies for solving multiplication problems, while relatively few used retrieval only or had no strategy. Children ages 10 to 12 years shifted toward the use of retrieval for low operands and away from no strategy. For high operands, however, children in this age range tended to shift toward the use of less effective back-up strategies, determined by

FIG. 4. Strategies used for low number problems by combined ages 8 and 9 years compared with combined ages 10 to 12 years. R = retrieval; R + Ca = retrieval + calculation; M = mixed strategies; C = counting-in-series; NS = no strategy.

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STEEL AND FUNNELL TABLE 1 Shift in Strategies for Low to High Operands in Combined Ages 10 to 12 Years High number strategy

Low number strategy

Retrieval

Retrieval + calculation

Mixed strategies

Retrieval Retrieval + calculation Mixed strategies Counting-in-series Total

27

40 11

7 3 8

27

51

Countingin-series

18

14 27 41

Total 74 14 22 27 137

Note. Table shows number of children per strategy group.

the strategies they used for low operands. By the last year of primary school (age 11 years), 20% children had failed to learn multiplication facts for even the simplest operands and only 61% retrieved facts for all problems up to 6 × 6. Response Time and Accuracy Only screens with at least three of six correct answers were included in the analysis. This reduced the chance of including responses based on fortuitous correct guessing (chance level of one of six) and avoided the ceiling effects that would arise from including only the most accurate children. Errors included omitted and incorrect answers. The no-strategy group data was excluded because too few screens were completed for analysis. High variance prevented statistical comparisons across the four strategy groups for children ages 8 and 9 years. For this reason, the retrieval and retrieval + calculation groups were combined and the data were transformed. Ages 8 and 9 Response times. Figure 5 shows that reaction time varied with the strategy used. Factorial ANOVA (school year and strategy group as between-subjects factors) showed a significant effect of school year, F(1, 79) = 9.55, and strategy group, F(2, 79) = 16.45, with no significant interaction. Post hoc paired comparisons revealed significant differences in speed in the following direction: combined retrieval and retrieval + calculation < mixed strategies < counting-in-series. The retrieval and retrieval + calculation groups were then compared separately [t test for independent samples, t (22) = −3.04]. The reaction time of the retrieval group (M = 12.30, SD = 7.47) was found to be significantly faster than that of the retrieval + calculation group (M = 29.96, SD = 11.65). Faster responses were associated with increasing age and with the use of particular strategies. The analysis was repeated excluding the four children in the “tutor group” who had reported tuition using rote learning. The significant effect of strategy on

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FIG. 5. Mean response times and mean percentage error rates for ages 8 and 9 years (low facts to 6 × 5). R/R + Ca = combined retrieval and retrieval + calculation; M = mixed strategies; C = counting-in-series. Small bars = SE.

response times remained, F(2, 75) = 12.84, and equivalent post hoc paired comparisons were found between strategy groups. Accuracy. Figure 5 shows that mean error rates also increased across strategy groups. Factorial ANOVA (school year and strategy group as between-subjects factors) indicated a significant effect of school year, F(1, 90) = 5.47, a significant effect of strategy group, F(2, 90) = 19.53, and a significant interaction between school year and strategy group, F(2, 90) = 3.51. This interaction arose from fewer errors in the combined retrieval and retrieval + calculation group and more errors in the counting-in-series group in children age 8 years compared with those age 9 years. Post hoc paired comparisons indicated that mean errors increased significantly with more ineffective strategies: combined retrieval and retrieval + calculation < mixed strategies < counting-in-series-only. The retrieval and retrieval + calculation groups were then compared separately [t test for independent samples, t (22) = −1.09]. Percentage error rates: retrieval (n = 12, M = 3, SD = 3) < retrieval + calculation (n = 12, M = 6, SD = 11). The accuracy of the two groups was not found to be significantly different. Thus increases in accuracy were also associated with increases in age and with the use of particular strategies. These effects remained when the group of children who had been coached in rote learning methods (the “tutor group”) was excluded from the analysis, F(2, 86) = 17.37. Post hoc paired comparisons indicated equivalent differences in significance between strategy groups. Ages 10–12 Response times: Ages 10 to 12 years, all problems. For the purposes of this analysis, the remaining child in the 12-year-old age group who relied completely on counting-in-series, was included in the “mixed strategy” group. Response times according to school year and strategy group are displayed in Fig. 6. Factorial ANOVA (school year and strategy group as between-subjects factors) revealed a significant effect of strategy group, F(3, 126) = 38.24, but no

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FIG. 6. Mean response times and mean percentage errors for ages 10 to 12 years (all facts to 12 × 12). R = retrieval; R + Ca = retrieval + calculation; M = mixed strategies; C = counting-in-series. Small bars = SE.

significant interaction between strategy group and school year. Post hoc paired comparisons showed that reaction time increased significantly across strategies: retrieval < retrieval + calculation < mixed strategies < counting-in-series. The analysis was repeated excluding the six children from the “tutor group,” who had reported tuition using rote learning in preparation for external examinations. The significant effect of strategy on response times remained, F(3, 120) = 23.24, and equivalent post hoc paired comparisons were found between strategy groups. Accuracy: Ages 10 to 12 years. Accuracy levels according to school year and strategy group are displayed in Fig. 6. A factorial ANOVA (with school year and strategy group as between-subjects factors) revealed a significant effect of strategy group, F(3, 126) = 24.66, no effect of school year, and no significant interaction between school year and strategy group. Paired comparisons revealed significant increases in errors with less mature strategies: retrieval only < retrieval + calculation < mixed strategies < counting-in-series. These effects remained when the “tutor group,” which had been coached in rote learning methods, was excluded from the analysis, F(3, 120) = 18.81. Post hoc paired comparisons indicated equivalent differences in significance between strategy groups. Low and High Operands Compared: Combined Ages 10 to 12 Years Response times and accuracy levels were compared across low (1 to 6) and high operands (7 to 12). These are displayed in Fig. 7. For this analysis, the counting-in-series group, who reached the criterion on relatively few screens for both low and high operands, was combined with the mixed strategy group, and the data were transformed. Response times. Response times were significantly longer for high operands than low operands: two factor mixed ANOVA repeated measures, F(1, 134) = 277.58, and differed significantly for all post hoc paired comparisons between

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FIG. 7. Mean response time and mean percentage errors for low facts (1 × 2 to 6 × 6) compared to high facts (7 × 7 to 12 × 12) for combined ages 10 to 12 years. R = retrieval; R + Ca = retrieval + calculation; M/C = combined mixed strategies and counting-in-series. Small bars = SE.

strategy groups. A significant effect of strategy group × operand size was also indicated, F(2, 134) = 13.05. This interaction disappeared when the counting-inseries group, whose response times were based on easier screens on which they made no errors, was removed from the analysis. The retrieval group was faster than the retrieval + calculation group for all operands. For low operands only, these groups were also faster than the mixed + counting-in-series group. When the analysis was repeated with the group counting-in-series omitted, the significant effect of strategy on speed remained, F(2, 107) = 34.76: that is, retrieval < retrieval + calculation < mixed strategies. Accuracy. Figure 7 shows that the error rates across low and high operands revealed a similar pattern to response times. Paired comparisons (Wilcoxon Signed-Ranks test) revealed a significant overall increase in errors for high operands, Z = −9.89) and significant paired comparisons for all four strategy groups. The effect of number size was apparent for all strategy groups, although, for the retrieval and retrieval + calculation strategy groups, the error rates for the lower operands were very small: retrieval (2%), retrieval + calculation (4%), and mixed strategy and counting combined (10%). The corresponding error rates for the higher operands were 13, 19, and 40% respectively. Summary As children ages 8 and 9 years gained more experience, accuracy levels increased while response time decreased. Accuracy levels, but not response times, also increased for children ages 10 to 12 years. For all age groups, retrieval and retrieval + calculation were the most accurate strategies, mixed strategies were next in accuracy, and counting-in-series was the least accurate. Retrieval was the fastest strategy, retrieval + calculation was faster than mixed strategies, which were faster than counting-in-series. Response times were significantly longer, and error rates significantly greater for high operands compared to low.

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Test Validation Two mathematics tests, carried out by the schools themselves, were used to validate the results of the multiplication test. The first test of mathematics, developed by Brighthouse, Godper, and Patilla (1984), is a standardized test of general mathematical competence that was given to children age 8 years. The test is not timed, although there is a time limit. Scores for this test were available for all children age 8 years and 46% of children ages 9 to 11 years. A significant difference, F(4, 117) = 19.59, between the standardized scores (average = 100) was found across the different multiplication strategy groups: retrieval (n = 11, M = 113.74, SD = 13.74) > retrieval + calculation (n = 24, M = 111.33, SD = 12.96) > mixed strategies (n = 47, M = 101.51, SD = 13.05) > counting-in-series (n = 28, M = 90.57, SD = 11.12) > no strategy (n = 8, M = 76.38, SD = 8.96). Paired comparisons revealed significant differences between all strategy groups, except between the retrieval and the retrieval + calculation groups and between the counting-in-series and no-strategy groups. For all age groups, significant negative Pearson correlations were found between correct performance on the standardized test and error scores on the multiplication test: age 8 years [r (51) = −0.70]; age 9 years [r(26) = −0.70]; age 10 years [r(23) = −0.70]; and age 11 years [r (18) = −0.50]. Error rates from the multiplication test were also validated against results from the Mathematics section of Key Stage 2, National Curriculum Tests, undertaken by 108 of the children in our sample as they reached the end of primary school at age 11 years. A significant difference, F(3, 107) = 8.70, between the mean level reached on the National Curriculum test was found across the different multiplication strategy groups: retrieval (n = 12, mean level 4.5) > retrieval + calculation (n = 31, mean level 4.3) > mixed strategies (n = 37, mean level 4.0) > counting-in-series (n = 28, mean level 3.5). The counting-in-series group achieved a significantly lower level than all other groups. There was a significant negative Spearman rank correlation coefficient (rs = −.60) between the level attained on the National Curriculum mathematics tests and mean error scores on the multiplication test. Cognitive Factors Short-Term Memory: Combined Ages 8 and 9 and 10 to 12 Years Table 2 presents the raw scores on forward and backward digit span (taken from the WISC-III UK, 1992) calculated separately for each strategy group. A one-way ANOVA revealed no significant effects of strategy on working memory in children ages 8 and 9 years, but significant effects in those ages 10 to 12 years of both forward, F(3, 133) = 3.78, and backward span, F(3, 133) = 6.59. Mean forward spans in children ages 10 to 12 years were significantly smaller in the counting-in-series group compared with the retrieval group. Mean backward spans in the counting-in-series group were significantly smaller than those found in any other strategy group.

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MULTIPLICATION AND DISCOVERY METHODS TABLE 2 WISC Digit Span Subtest Raw Scores Forward span Combined ages (years)

Retrieval Retrieval + calculation Mixed strategies Counting-in-series No strategy

8 and 9

Backward span

10 to 12

8 and 9

10 to 12

M

SD

M

SD

M

SD

M

SD

8.2 7.5 7.4 7.3 6.8

(1.8) (1.6) (1.6) (2.0) (1.5)

9.4 8.2 8.8 7.7

(2.2) (1.8) (2.3) (1.5)

4.7 4.3 3.5 3.8 3.4

(1.7) (1.3) (1.3) (1.4) (0.7)

5.4 5.2 5.1 3.8

(1.7) (1.4) (2.0) (1.0)

Nonverbal Reasoning: Combined Ages 8 and 9 and 10 to 12 Years Table 3 displays the mean scores on the Progressive Matrices across strategy groups. A one-way ANOVA indicated significant differences for children ages 8 and 9 years, F(4, 99) = 3.85, and those ages 10 to 12 years, F(3, 133) = 5.65. Children ages 8 and 9 years with no strategy had significantly lower nonverbal reasoning scores than children who retrieved or retrieved with calculation. Children ages 10 to 12 years who counted in series, or used mixed strategies, had significantly smaller nonverbal reasoning scores than children who retrieved. DISCUSSION Previous research has suggested that the educational environment has a strong influence on the range of strategies available for use in arithmetic (Fuson & Kwon, 1991; Fuson, Stigler, & Bartsch, 1988; LeFevre & Liu, 1997). Children are also believed to discover for themselves new strategies and to select the strategy that best suits the problems to be solved (Shrager and Siegler, 1998). According to the adaptive strategy choice model, the strategy selected for solving a problem and the frequency with which problems are experienced are believed to be strong predictors of the effective learning of multiplication facts (Siegler, 1988). In this article, we examined these claims using data from a group of children taught arithmetic by discovery methods. We were particularly interested to know what effect this learning environment had on their range of strategies and the degree to which retrieval emerged as the strategy of choice. A novel test of multiplication, using a multiple-choice procedure, revealed that children ages 8 to 12 years used three main strategies to solve multiplication problems: direct retrieval of the answer; retrieval + calculation, and counting-inseries. Some children ages 8 and 9 years had no strategy at all. In contrast to studies reported by LeFevre et al. (1996) and Lemaire and Siegler (1995), there was no evidence for the use of repeated addition, generally considered to be the most common back-up strategy employed. Since the absence of repeated addition applied to all children in our study, we assume that the learning environment was directly responsible for this absence and for the emergence of

52

STEEL AND FUNNELL TABLE 3 Progressive Matrices Raw Scores Combined ages (years) 8 and 9 Strategy group Retrieval Retrieval + calculation Mixed strategies Counting-in-series No strategy

10 to 12

M

SD

M

SD

31.3 30.0 27.1 26.6 23.4

(3.5) (3.2) (4.4) (7.1) (6.9)

45.9 41.7 40.5 38.5

(5.8) (6.7) (7.1) (8.6)

counting-in-series: a strategy rarely reported hitherto (LeFevre and Liu, 1997). Children noted that writing down multiples of numbers was used as a method of learning multiplication, and teachers confirmed that continuous addition was not encouraged. Between ages 8 and 9 and 10 to 12 years, there was a significant shift away from mixed strategies toward retrieval and calculation, but the proportion of children ages 10 to 12 years using retrieval only reached only 20% overall. The highest percentage of children using retrieval (31%) were age 11 years, but this was boosted by six children who were receiving tuition in rote learning of multiplication tables outside school. Although, by age 10 years, all children had acquired some strategy, there was relatively little increase in the proportion of children who used retrieval as their main strategy over the 5 school years examined. Only 18% of children age 12 years used retrieval for all problems. This poor progress toward fact retrieval stands in marked contrast to studies that demonstrate high proportions of fact retrieval following more didactic methods of teaching (LeFevre et al, 1996; Lemaire & Siegler, 1995). There is no reason to suspect that the children in our study were unrepresentative of the population as a whole. The two primary schools involved in the present study both achieved average marks for mathematics at the expected level of 4 or above in the National Curriculum mathematics tests for 1998 (72 and 68% respectively). Their scores, which were above the average for the county (61%), were also above the average for the country as a whole (53%). Research in recent years has suggested that children doing multiplication questions use a variety of adaptive strategies to solve problems and that the choice of strategy is related to the size of the operands (Lemaire & Siegler, 1995; Siegler, 1988). In our study, children ages 10 to 12 years were more likely to use retrieval (54%) for low number questions compared to children ages 8 and 9 years (11%), who instead used mainly mixed strategies (54%). This indicates a shift to more efficient strategies with increasing experience in line with earlier findings (Brownwell & Carper, 1943; Cooney et al., 1988) and provides some support for the adaptive strategy choice model (Siegler & Shipley, 1995). Presumably,

MULTIPLICATION AND DISCOVERY METHODS

53

increasing experience with the set of low number problems has resulted in learning number facts that can then be retrieved (cf. Siegler, 1988). When higher number problems were presented to children ages 10 to 12 years, 47% shifted to less efficient back-up strategies, again supporting the predictions of the adaptive strategy choice model (Siegler & Shipley, 1995) and supporting the findings of the study conducted with French-speaking children (Lemaire & Siegler, 1995). The back-up strategies selected by the children in our study were closely related to the strategies that were available to them for use with low operands. Response times and accuracy levels varied with the strategy used. Retrieving an answer was consistently faster than all other strategies and counting-in-series was consistently slower. Retrieval and retrieval + calculation were consistently more accurate than mixed strategies and counting-in-series. Counting-in-series was significantly less accurate than any other strategy. However, despite the fact that mixed strategies and counting-in-series are clearly ineffective strategies in both time and accuracy, more than 40% of children age 11 years and more than 30% of those age 12 years continued to use these strategies. Clearly, children using less efficient strategies do not inevitably develop more efficient ones. The adaptive strategy choice model (Siegler & Shipley, 1995) predicted that early use of retrieval should lead to frequent use of fast and accurate retrieval and that the early use of effective back-up strategies should eventually lead to a decrease in their use and an increase in retrieval. Counting-in-series, encouraged as a method of mastering multiplication facts, did not appear to promote the formation of strong, accurate associations between question and answer that are required for retrieval. Instead, a significant number of children continued to rely on counting-in-series, except for the smallest operands. In contrast, Lemaire and Siegler (1995) achieved high levels of retrieval with French children taught by rote learning methods that associate the operands directly with the solution. Some children taught by discovery methods, however, did develop retrieval as their main strategy of use. Since direct retrieval was not taught explicitly in school, and only a minority reported such teaching at home, it must be assumed that these children discovered the retrieval strategy for themselves (Shrager & Siegler, 1998). Children who used more effective strategies for multiplication also performed better on other tests of mathematics, supporting the findings of Ostad (1997) and suggesting that these children were, in some way, more able than those who failed to develop the strategy of retrieval. Working memory has been argued to be involved in the selection of strategies and in the solving of more complex problems (Bull, Johnson, & Roy, 1999; Logie, Gilhooly, & Wynn, 1994). A significant relationship was found between strategy and scores on forward and backward digit span for children ages 10 to 12 years. Children who counted in series had significantly lower forward span than those who retrieved and significantly lower backward span than any other strategy group. Inefficient strategies are believed to make more demands on

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working memory than efficient strategies (Hitch, 1978a, 1978b); nevertheless, children ages 10 to 12 years with weaker working memories were more likely to use less efficient strategies. No relationship was observed between strategy and working memory span in the children ages 8 and 9 years, probably because the range of span scores across the strategy groups was relatively small. Since span continues to increase between 8 and 11 years (Dempster, 1981), a smaller range of span scores in the younger age group is to be expected. Children with high nonverbal reasoning scores on the Progressive Matrices developed more effective strategies for multiplication than did the children with low scores. Children ages 8 and 9 years with no strategy had significantly lower nonverbal reasoning scores than children who retrieved or calculated. Children ages 10 to 12 years who counted in series or used mixed strategies had significantly smaller nonverbal reasoning scores than children who retrieved. Teaching methods, degree of experience, and the level of cognitive ability all appear to have influenced the development of strategies for multiplication in this study. Teaching methods used in the discovery learning program were directly implicated in the development of counting-in-series and in the lack of evidence for repeated addition—a common form of back up strategy in other studies. Experience was related to the gradual development over age of retrieval and calculation strategies for low operands, although the development of these strategies across the whole age group and whole operand range was limited, providing only partial support for the ASCM. Relationships between strategy, working memory capacity, and nonverbal reasoning ability suggest that it was the more able children who discovered the most effective strategies. REFERENCES Ashcraft, M. H., & Christy, K. S. (1995). The frequency of arithmetic facts in elementary texts: Addition and multiplication in grades 1–6. Journal for Research in Mathematics Education, 26, 396–421. Brighthouse, A., Godper, D., & Patilla, P. (1984). Mathematics, 7–11. Windsor: NFER-Nelson. Brownwell, W. A., & Carper, D. V. (1943). Learning multiplication combinations. Durham, NC: Duke Univ. Press. Bull, R., Johnston, R. S., & Roy, J. A. (1999). Exploring the roles of the visual-spatial sketch pad and central executive in children’s arithmetical skills: Views from cognition and developmental neuropsychology. Developmental Neuropsychology, 15, 421–442. Campbell, J. I. D. (1995). Mechanisms of simple addition and multiplication: A modified networkinterference theory and simulation. Mathematical Cognition, 1, 121–164. Campbell, J. I. D., & Oliphant, M. (1992). Representation and retrieval of arithmetic facts: A networkinterference model and simulation. In J. I. D. Campbell (Ed.), The nature and origin of mathematical skills (pp. 331–364). Amsterdam: Elsevier. Cooney, J. B., Swanson, H. L., & Ladd, S. F. (1988). Acquisition of mental multiplication skill: Evidence for the transition between counting and retrieval strategies. Cognition and Instruction, 5, 323–345. Dehaene, S., & Cohen, L. (1995). Towards an anatomical and functional model of number processing. Mathematical Cognition, 1, 83–120. Dempster, F. N. (1981). Memory span: Sources of individual and developmental differences. Psychological Bulletin, 89, 63–100.

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