Educational Psychology, Vol. 23, No. 5, December 2003
Effective Strategies for Mental and Written Arithmetic Calculation from the Third to the Fifth Grade
DANIELA LUCANGELI, Department of Developmental Psychology, University of Padova, Italy PATRIZIO E. TRESSOLDI, MONICA BENDOTTI, MICHELA BONANOMI, Department of General Psychology, University of Padova, Italy LINDA S. SIEGEL, Department of Educational and Counseling Psychology and Special Education, University of British Columbia, Canada
The strategies used to solve mental and written multidigit arithmetical addition, subtraction, multiplication and division were observed in 200 third, fourth and fifth grade children. A strategy was classified as effective if it resulted in the correct solution at least 75% of the time. For mental addition and subtraction, primitive strategies such as counting on fingers and counting on (mental counting from a specific point), and the more sophisticated strategy 1010 (solution of the calculation problem using tens and units separately) were more effective than the strategies learned at school. In written addition, subtraction and multiplication there was a shift from the CAR ⫹ to the CAR- strategy (tabulating with, or without, a carried amount) from the third to the later grades. Results show that typical strategies taught at school progressively substitute every other strategy both in mental and written calculation, but without reaching the criterion of effectiveness. The implications for maths curricula are discussed. ABSTRACT
Although the mental and written execution of simple arithmetic operations is part of our daily activity, there have been relatively few studies related to the acquisition of strategies to solve them. The present study investigates the strategies necessary to arrive at a correct solution of multidigit arithmetical problems by schoolchildren of different ages. Several models of learning and development of arithmetic calculation problems have been proposed. Groen and Parkman (1972) hypothesised that the solution of a calculation problem is achieved with counting processes, using the bigger addendum as ISSN 0144-3410 print; ISSN 1469-046X online/03/050507-14 2003 Taylor & Francis Ltd DOI: 10.1080/0144341032000123769
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a counter and adding the smaller addendum one unit after the other, until the answer is attained. This procedure is often called counting on, and implies a mechanism for recording the number of units gradually added to the counter and for ending the counting when the quantity of the smaller addendum is reached. In support of their models, Groen and Parkman found that reaction times increased linearly as the size of the addendum increased. Subsequent research verified that the counting on model was reflective of children’s learning processes, but that it did not represent the processes in adults. Analysis of adults’ reaction times in mental calculation showed a linear increase too small to indicate that counting processes were used, suggesting a mnemonic retrieval of the result by direct access. Winkelman and Schmidt (1974) hypothesised that the results of different operations could be represented in the memory system and that errors could represent failures in mnemonic retrieval. Most models share the assumption that the solution of arithmetic calculations may be characterised by direct retrieval of the results. The cognitive representation of arithmetic calculation procedures may be organized and structured in terms of connections between the memorised elements and the strength of storage. Thus the probability and the speed of retrieval may depend on experience and amount of practice. Some of the models based on the hypothesis of direct retrieval of results from memory are the retrieval from network model (Ashcraft, 1982), the associative distribution model (Siegler, 1988; Siegler & Jenkins, 1989), and the network interference model (Campbell, 1987, 1994a, 1994b; Campbell & Clark, 1988). Considering these models and summarising the results of research, Ashcraft (1992, 1994) noted that although all they differed in specific aspects (the focus on cognitive characteristics, the modality of the interpretation of errors, and so on) these models share some similar features. They all hypothesise that the resolution of the problem is based on the retrieval of results from memory, through the use of backup strategies of retrieval facts. A different perspective on the cognitive structures needed for calculation is described in the model of McCloskey and colleagues (McCloskey, Caramazza, & Basili, 1985; McCloskey, Macaruso, & Whetstone, 1992). This model assumes that numerical information processing is due to the activity of three modules, or systems, that are functionally distinct components: the system of number comprehension, the system of number production, and the system of calculation. The comprehension and production modules are mechanisms of number elaboration (or processing) that allow the input and output procedures in the system of calculation. Comprehension makes possible the transformation of the superficial number structure, either alphabetically or numerically expressed, into an abstract representation of the quantities, on which the other modules can operate. Production allows the translation of the abstract information into a specific superficial code (words or numbers), then giving the answers of the various arithmetic processes. Finally, calculation operates on the abstract representation of the quantities using the retrieval of arithmetic facts, the knowledge of arithmetic signs and calculation. Thus there are several models of cognitive structures of number elaboration, and several hypotheses. Each of them supposes different cognitive functions involved in calculation, but all of them hypothesise that this system is based on retrieval and procedural processes. All these models offer different hypotheses concerning the development of strategies used to solve arithmetic calculation from early attempts to the skilled performance. A progressive evolution of strategies has been hypothesised from the initial counting
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strategies to automatic processes for the retrieval of results (Ashcraft, 1982, 1994; Baroody, 1987; Siegler, 1988). Analysing strategies for solving simple addition, Svenson and Broquist (1975) found three different levels of strategies: adding one digit at time (1 ⫹ 1), adding more units (2 ⫹ 2, 3 ⫹ 2, and so on) and rounding up to numbers in the teens. Baroody (1987) found similar results and suggested that the development of different kinds of strategies by children depended on the level of instruction and the difficulty of the task. The effectiveness of calculation may depend on the retrieval of arithmetic facts, and also on the development of procedural knowledge. The simplest strategies are those recognized in the rules of the operations involving the number 1 (n ⫹ 1; n x 1; n - 1; n ⫼ 1) and the one involving the number 0 (n ⫹ 0; n x 0; n - 0; n ⫼ 0). The retrieval of this procedural knowledge may facilitate the execution of mental and written calculation (Passolunghi, Domenis-Czerwinskj, & Bortolotti, 1996). Analysing the responses of children aged five to addition problems, Siegler and Robinson (1982) found different kinds of strategies: counting on fingers, verbal enumeration, and mental retrieval. The choice of strategy depended on the level of confidence in the results obtained. After presentation of the problem, children at first attempted to retrieve the answer and subsequently, if they were not confident, they used a backup strategy such as counting on their fingers. Carpenter and Moser’s (1984) and Fuson’s (1988) findings were similar. In mental addition, children use different levels of strategies. Counting all was the first and the least developed. The second-level strategy was based on the adding of the smaller addend one unit after the other to the larger addend until the result is achieved. At the third level, children were able to retrieve the results or the arithmetic fact in question directly from memory. Strategies based on the manipulation of arithmetic facts or of frequent combinations of numbers appear to be characteristic of the performance of expert solvers, both children and adults. Studies on the addition of two numbers found that two different combined strategies are used (Beishuizen, 1993; Beishuizen, Van Putten, & Van Mulken, 1997): the so-called “1010” strategy [87 ⫹ 39 ⫽ (80 ⫹ 30) ⫹ (7 ⫹ 9)] and the “N10” strategy [27 ⫹ 29 ⫽ (27 ⫹ 10 ⫹ 10) ⫹ 9]. In the 1010 strategy (or decomposition strategy) both the numbers are split into units and tens for summing or subtracting separately, and finally the result is reassembled. In the N10 strategy only the second operator is split into units and tens that are subsequently added or subtracted. Beishuizen (1993) showed that the N10 strategy was a more effective and developed strategy than the 1010, which was used more frequently by those subjects who were less accurate. Passolunghi et al. (1996), examining strategies used for mental addition in expert subjects, verified the use of the N10 strategy and, also, a kind of mental algorithm of the addenda previously decomposed into units and tens. These authors also observed the so-called “rounding” strategy, where the first operator is retained, while the second is rounded up to the following ten and then added to the first one and, finally, the difference is eliminated. The aim of this study was to analyse the strategies used to solve arithmetic tasks using the four operations with mental and written calculation. Even if disentangling development from ongoing instructional processes is difficult, it is possible to devise research to assess some specific effects of scholastic instruction on the spontaneous strategies of children. Specifically, we expect to find an increased use of strategies taught directly at school, a decreased use of spontaneous strategies
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derived from the development of number knowledge, and an appearance of mixed strategies derived from both instruction and more sophisticated development of number knowledge. More precisely, regarding strategies taught at school, we refer to strategies requiring the child to follow fixed sequential steps to solve each arithmetic operation. For example, in the case of written addition, children must first add units, then tens and so on, using carrying when necessary. Developmental strategies usually characterise the beginning of calculation knowledge. Counting on fingers and counting on are some specific examples. Regarding mixed strategies, we refer to strategies such as 1010 and N10 that require both a good level of instruction and the development of a more sophisticated knowledge of number composition and decomposition. The above strategies have been studied primarily using single digit arithmetic problems, whereas the focus of our investigation is on strategies necessary for successful performance on multidigit problems. Children between 8 and 11 were included in the study to ensure the possibility that they could use both automatic retrieval of results and procedural knowledge. The four arithmetic operations, both mental and written, were studied using multidigit problems. Furthermore, we were interested in studying the degree of effectiveness of each strategy, that is, the level of correct solutions obtained when using each of them.
Method Subjects The sample included 200 students from third to fifth grade (65 in the third grade, mean age 8.3; 65 in the fourth grade, mean age 9.4; 70 in the fifth grade, mean age 10.2), attending different schools in several cities in northern Italy. They were recruited after an informal request to their teachers. The ratio male:female was approximately 1⬊1. All came from a medium socio-economic background and were free from medical and psychological disabilities.
Materials and Procedure Children were tested using the test of evaluation of calculation abilities (Lucangeli, Tressoldi, & Fiore, 1998). The test is based on the theoretical model of McCloskey et al. (1985) and on a study by Shalev, Weirtman, and Amir (1988). It consists of three parts corresponding to the three modules postulated by McCloskey: (1) mental and written calculation, (2) comprehension and (3) production of the numerical system. We used only the items from the calculation part which consists of addition, subtraction, multiplication and division, both mental and written, for a total of 24 problems for each age group (see Appendix). Every child was tested individually. While they were engaged in each arithmetic operation, the children were asked: “Please, tell me how you solved the problem.” This information was recorded on tape, and together with all the nonverbal behavior the examiner was able to observe during the task, enabled the classification of strategies according to the following system.
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Strategies for Mental Addition and Subtraction COF ⫽ Counting on fingers. Starting from a specific operator, children typically start with the larger one, and change it by the number of units indicated by the smaller number, counting on their fingers one unit after the other. In third grade students, this procedure was often characterized by counting aloud and a rhythmical movement of the head. CON ⫽ Mental counting from a specific point (counting on). This is a slightly higher level strategy, when the child is able to operate with numbers at a representative level, without overt bodily movements although rhythmical body movements can occasionally occur. 1010 or decomposition strategy. The subjects split both the numbers into units and tens for summing or subtracting them separately, the result was reassembled [77 ⫹ 49 ⫽ (70 ⫹ 40) ⫹ (7 ⫹ 9); 77 ⫺ 42 ⫽ (70 ⫺ 40) ⫹ (7 ⫺ 2)]. N10. Only the second operator was split into units and tens, that were subsequently added or subtracted [77 ⫹ 49 ⫽ (77 ⫹ 10 ⫹ 10 ⫹ 10 ⫹ 10) ⫹ 9; 52—28 ⫽ (52— 10 ⫺ 10)—8]. MA ⫽ Mental algorithm. The children often indicated that they simply formed a mental representation of the calculation in a written form and afterwards operated on a mental algorithm, moving from right to left. C10 ⫽ Formation of units of 10. The children formed multiples of ten on which it was possible to operate more easily [43 ⫹ 6 ⫽ (43 ⫹ 7) ⫺ 1; 43—7 ⫽ (43 ⫺ 3)—4]. AUTO ⫽ Automatic calculation (retrieval of the results). At higher grade levels, more children said that they automatically solved the operation through a process of retrieving the answer from memory. Strategies for Mental Multiplication DO ⫽ Different operation. Transformation into addition: for the most simple multiplication, child arrived at the solution by adding the multiplicand several times as indicated by the multiplier [31 ⫻ 3 ⫽ 31 ⫹ 31 ⫹ 31; 31 ⫻ 3 ⫽ (30 ⫹ 30 ⫹ 30) ⫹ 3]. AUTO ⫽ Multiplicative arithmetic fact. The children indicated that they solved the multiplication automatically, through a direct mnemonic retrieval of the results. MA ⫽ Mental algorithm. In a manner similar to mental addition and subtraction, the children indicated that they simply formed a mental representation of the calculation in a written form and afterwards operated a mental calculation, moving from right to left. Strategies for Mental Division DO ⫽ Different operation. Transformation into multiplication: children indicated that they solved the division by considering the dividend as result of a multiplication having as the multiplier the divisor (120 ⫼ 4 ⫽ ? → 4 x ? ⫽ 120). MA ⫽ Mental algorithm. Children indicated that they imagined a division in the same structural form used for a written execution.
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AUTO ⫽ Automatic calculation. The children indicated that they retrieved the answer automatically. Strategies for Written Addition, Subtraction and Multiplication CAR ⫹ ⫽ Tabulating with a written carried amount. In the execution of these arithmetic calculations, the children often needed to write the carried numbers. CAR- ⫽ Tabulating without a written carried amount. The tabulate amount was not written, although it was often kept on fingers. AUTO ⫽ Automatic calculation. The children indicated that they retrieved the answer automatically. Strategies for Written Division DIV ⫽ This strategy involved the application of the specific algorithm of division, without using a transformation into the processes of multiplication. DO ⫽ Different operation. Transformation into multiplication problems: some children needed to transform the division into the corresponding multiplication, both during the execution and afterwards as a final verification of their answer. AUTO ⫽ Automatic calculation. For the simplest division problems, the solution was obtained by automatically retrieving the results. Results For each grade and for each type of calculation, the different strategies were ordered in descending order with respect to frequency of use. As an arbitrary criterion to define a strategy as effective, we chose the 75% (highest quartile) of correct answers for the whole sample of subjects of each different grade. We used this criterion because a lower percentage of correct answers does not qualify a strategy as really effective in an educational context. Thus, it was also possible to observe changes in the use and effectiveness of different strategies from the third to the fifth grade. Using a Tukey-type statistic for independent proportions (Williams, 1992) we compared the percentages of use of the different strategies among the three grades adopting an ⬍ 0.01. This statistic, similar to the analogue post-hoc used to compare means, provides protection from the experiment-wise error. Statistically significant comparisons between grades appear in the figures that follow with the ⬍ or ⬎ sign. Furthermore, for each statistically significant difference between proportions, we calculated the effect size d using the formula (P1 ⫺ P2)/Sp1p2 as suggested by Hedges and Olkin (1985) (Sp1p2 ⫽ composite standard deviation). Mental Addition and Subtraction With mental addition (see Fig. 1a and Fig. 1b), it is interesting to note the statistically significant increase in the use of 1010 strategy from the third to the fifth grade (d ⫽ 0.34) always reaching the criterion of effectiveness, contrary to the second most used strategy MA.
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FIG. 1a. Percentages of use of the different strategies for mental addition
FIG. 1b. Percentage of correct solutions using the different strategies for mental addition
FIG. 2a. Percentages of use of the different strategies for mental subtraction
FIG. 2b. Percentage of correct solutions using the different strategies for mental subtraction
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The more sophisticated strategies N10 and C10, even if generally effective, were used by no more than 10% of the children. With mental subtraction (see Fig. 2a and Fig. 2b), we observed the same strategies as with addition and in the same order of use. However, very different from what was observed with addition, with the exception of CON in the fifth grade and COF in the third grade, the strategies did not reach the effectiveness criterion. The COF strategy decreased statistically in the comparison between the third and the fifth grade (d ⫽ 0.29). Even if less used, it is interesting to observe the increment in the 1010 strategy from the third to the fifth grade (d ⫽ 0.44) that, however, never arrived at the effectiveness criterion. Written Addition, Subtraction and Multiplication The strategies observed with written addition (see Fig. 3a) were very simple and the changes from the third to the fifth grade were as expected. The statistically significant increment of CAR- from the third to the fourth (d ⫽ 0.51) and from the fourth to the fifth grade (d ⫽ 0.83), and the parallel decrement of CAR ⫹ strategy from the third to the fourth (d ⫽ 0.79) and from the fourth to the fifth grade (d ⫽ 0.47), is probably a consequence of the expected increased calculation ability in addition. The pattern of the AUTO strategy is less clear even if it is probable that most of the calculations solved with CAR- in the fifth grade were partially carried out with an AUTO-type strategy. It is also interesting to note that, except for the AUTO strategy in the third grade, all strategies reached the effectiveness criterion (see Fig. 3b). The strategies observed with written subtraction were the same as those observed with written addition (see Fig. 4a). In this case however, there is not the progressive decrement in CAR ⫹ but only a statistically progressive increase in CAR- strategy (d ⫽ 0.24 from the third and fourth grade to the fifth grade). Another difference is in the level of effectiveness obtained only bu the CAR ⫹ in the fourth grade and the CAR ⫺ and AUTO strategies in the fifth grade (see Figure 4b).
FIG. 3a. Percentages of use of the different strategies for written addition
FIG. 3b. Percentage of correct solutions using the different strategies for written addition
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FIG. 4a. Percentages of use of the different strategies for written subtraction
FIG. 4b. Percentage of correct solutions using the different strategies for written subtraction
FIG. 5a. Percentages of use of the different strategies for mental multiplication
FIG. 5b. Percentage of correct solutions using the different strategies for mental multiplication
The strategies observed with written multiplication are the same as those observed with written addition and subtraction (see Figure 5a). The trend is very similar to written addition with a statistically significant decrement of the CAR ⫹ strategy (d ⫽ 0.52 from the third to the fourth grade, and d ⫽ 0.46 from the fourth to the fifth grade) and a parallel increase of CAR ⫺ strategy from the third to the fourth grade (d ⫽ 0.85) and from the fourth to the fifth grade (d ⫽ 0.47), indicating a linear development of this strategy that also reaches levels of effectiveness similar to the AUTO strategy (see Fig. 5b). Mental Multiplication and Division Similar to what was observed with mental addition and subtraction, with mental multiplication most of the calculations are solved using the MA strategy with a significant increase from the third to the fourth grade (d ⫽ 0.52). This strategy did not reach the
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FIG. 6a. Percentages of use of the different strategies for written multiplication
FIG. 6b. Percentage of correct solutions using the different strategies for written multiplication
FIG. 7a. Percentages of use of the different strategies for mental division
FIG. 7b. Percentage of correct solutions using the different strategies for mental division
effectiveness criterion until the fifth grade (see Fig. 6a and Fig. 6b). The DO and the AUTO strategy were far less frequently used. In the other mental calculations, even division, there was a significant use of MA strategy, but in this case it reached the effectiveness criterion from the third grade on, similar to the other strategies (see Fig. 7a and Fig. 7b). The use of DO strategy served to transform the complexity of division into a more manageable and familiar multiplication. Written Division As expected, the most used strategy was the DIV with a statistically significant increase from the third to the fourth grade (d ⫽ 0.63) and from the fourth to the fifth grade (0.60). It almost reached the effectiveness criterion in the fifth grade (see Fig. 8a and Fig. 8b).
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FIG. 8a. Percentages of use of the different strategies for written division
FIG. 8b. Percentage of correct solutions using the different strategies for written division
Discussion In the literature on arithmetic calculation there are several hypotheses regarding the cognitive functioning and development of arithmetic competence. Each assumes different cognitive functions involved in calculation, but all hypothesise that complex calculation is based on retrieval and procedural processes. The literature considers the retrieval of arithmetical facts as the most complex and developed of the calculation strategies. There is progressive development of strategies, from the initial counting to automatic processes for the retrieval of the results. Our study was designed to analyse the strategies used by children of different ages solving multidigit arithmetic problems involving the four basic operations, in both mental and written calculation. In particular, we analysed the evolution and efficacy of the strategies used.
Addition and Subtraction Concerning the effectiveness of strategies, it is interesting to observe the differences between written and mental calculations. In the written calculations the algorithms learned at school (CAR ⫹ , CAR-, and so on) were effective from the third grade for addition and from the fourth grade for subtraction. In mental calculation the same algorithms are not successful even if they were used by more than 30% of subjects. On the contrary, both the elementary strategies of counting on fingers and counting on, and the more complex strategies like 1010, were more effective. This finding may be explained by the fact that mental use of the algorithms learned for written calculation requires a significant cognitive load (both visual-spatial and verbal working memory) compared to the strategies for mental calculation.
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From a developmental point of view, in mental calculation there is a decrease in the use of the elementary strategies (counting on fingers) compared to the more sophisticated strategies requiring knowledge of the composition and decomposition properties of numbers (1010 and C10). It is interesting to note that there is not a progressive increase in the mental algorithm strategy even if the algorithm is the most frequently used strategy in written calculation. Except for addition, this effect may be a consequence of the low level of effectiveness of this strategy. In written calculation, a progressive decrement in the CAR ⫹ strategy and a progressive increment in the CAR- strategy was observed. The latter reduces the need to write the carry, possibly a sign of development of the capacity to maintain the partial results of the calculation in short term memory. Multiplication and Division Both in written and mental calculation, the only strategies observed were specific to the procedural algorithm. Concerning efficacy, these strategies are effective in written multiplication and in mental division. In written division the specific algorithm did not reach the efficacy criteria before the fifth grade; this may be due to the complexity of the items used for written versus mental calculation. The written multiplication problems administered were more complex than the mental calculations; the same effect for division was not observed. With multiplication, the strategies learned for written calculation were effective. This finding provides further evidence that the algorithm of division is the more complex. From a developmental point of view, similar changes for multiplication and addition were observed, that is, a reduction in the need to write the amount to be carried and a shift from the CAR ⫹ to the CAR ⫺ strategy. In general, as expected, there was a progressive increase in the use of algorithms specific to multiplication and division. Based on the results of this study, there is obviously a specific role exerted by the scholastic instruction in the development of strategies used for arithmetic calculation. Educational Implications To summarise the results of this research, we observed that the algorithms learned at school became dominant and more effective (except for written division) from the third to the fifth grade as a consequence of formal instruction and improvement in the retrieval of numerical facts. There was continued use of developmentally earlier strategies that seem to survive along with more complex strategies only for mental addition and subtraction. The choice of a strategy appears to be more a result of experience with it and the frequency of its use than its level of effectiveness—in fact, subjects used less effective strategies like MA more frequently. From a theoretical perspective, it appears that if strategy use is due to the effects of experience, even if children may learn from previous failures they remain fixated on taught algorithms. It appears that algorithms learned at school are not really strategic procedures but rather fixed and not flexible competencies. From the data it seems that children show overconfidence in the effectiveness of strategies learned at school, at least with respect to those strategies classified as developmental and mixed. The importance of a curriculum that emphasizes flexibility is recognized by different authors (Blote, Klein, & Beishuizen, 2000; Deboys & Pitt, 1995). Teachers and curriculum designers should promote “strategy learning” as a core module, transversal to the different units of the maths curriculum.
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Acknowledgment: We thank the two anonymous referees for their precise and helpful suggestions to improve the clarity of the paper. Correspondence: Daniela Lucangeli, Dipartimento di Psicologia dello Sviluppo, Universita` di Padova, Via Venezia 8, 35131 Padova, Italy (e-mail:
[email protected]). REFERENCES Ashcraft, M. H. (1982). The development of mental arithmetic: A chronometric approach. Developmental Review, 2, 213–236. Ashcraft, M. H. (1992). Cognitive arithmetic: A review of data and theory. Cognition, 44, 75–106. Ashcraft, M. H. (1994). Model of mental calculation. Paper presented at the Concepts of Number and Simple Arithmetic Workshop, SISSA-ISAS, Trieste. Baroody, A. J. (1987). Development of counting strategies for single-digit addition. Journal for Education in Mathematics Education, 18, 141–157. Beishuizen, M. (1993). Mental strategies and materials or models for addition and subtraction up to 100 in Dutch second grades. Journal for Research in Mathematics Education, 24, 294–323. Beishuizen, M., Van Putten, C. M., & Van Mulken, F. (1997). Mental arithmetic strategy use with indirect number problems up to one hundred. Learning and Instruction, 7, 87–106. Blote, A. W., Klein, A. S., & Beishuizen, M. (2000). Mental computation and conceptual understanding. Learning and Instruction, 10, 221–247. Campbell, J. I. D. (1987). Network interference and mental multiplication. Journal of Experimental Psychology: Learning Memory and Cognition, 13, 109–23. Campbell, J. I. D. (1994a). Architectures for numerical cognition. Cognition, 53, 1–44. Campbell, J. I. D. (1994b). Mechanism of number-fact retrieval: A modified network interference theory and simulation. Paper presented at the Concepts of Number and Simple Arithmetic Workshop, SISSAISAS, Trieste. Campbell, J. I. D., & Clark, J. M. (1988). An encoding complex view of cognitive number processing: Comments on McCloskey, Sokol, Goodman. Journal of Experimental Psychology: General, 117, 204–214. 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, 3, 179–202. Deboys, M., & Pitt, E. (1995). Lines of development in primary mathematics. Open University set book. Belfast: Blackstaff Press. Fuson, K. C. (1988). Children’s counting and concepts of numbers. Springer: New York. Groen, G. J., & Parkman, J. M. (1972). A chronometric analysis of simple addition. Psychological Review, 79, 329–343. Hedges, L. V., & Olkin, I. (1985). Statistical methods for meta-analysis. New York: Academic Press. Lucangeli, D., Tressoldi, P., & Fiore, C. (1998). ABCA-Test. Trento: Edizioni Erikson. McCloskey, M., Caramazza, A., & Basili, A. (1985). Cognitive mechanism in number processing and calculation: Evidence from dyscalculia. Brain and Cognition, 4, 171–196. McCloskey, M., Macaruso, P., & Whetstone, T. (1992). The functional architecture of numerical processing mechanism: Defending the modular model. In J. I. D. Campbell (Ed.), The nature of the origins of mathematical skills (pp. 493–537). Amsterdam: Elsevier. Passolunghi, M. C., Domenis-Czerwinskj, L., & Bortolotti, E. (1996). Strategie esecutive nelle addizioni mentali entro il centinaio [Executive strategies in mental addition below one hundred]. Studi di Psicologia dell’Educazione, 1–2, 101–115. Shalev, R. S., Weirtman, R., & Amir, N. (1988). Developmental dyscalculia, Cortex, 24, 555–561. 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., & Robinson, M. (1982). The development of numerical understanding. Advances in Child Development and Behavior, 16, 241–312. Siegler, R. S., & Jenkins, E. R. (1989). How children discover new strategies. Hillsdale NJ: Erlbaum.
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Svenson, O., & Broquist, S. (1975). Strategies of solving simple addition problems. Scandinavian Journal of Psychology, 16, 143–149. Winkelman, J. H., & Schmidt, J. (1974). Associative confusion in mental arithmetic. Journal of Experimental Psychology, 102, 734–736. Williams, R. H. (1992). Tukey-like pairwise comparisons among proportions. Educational and Psychological Measurement, 52, 913–914.
Appendix Mental Calculations Addition 43 ⫹ 6 ⫽ 55 ⫹ 7 ⫽ 76 ⫹ 49 ⫽ Subtraction 43 ⫺ 7 ⫽ 52 ⫺ 28 ⫽ 51 ⫺ 16 ⫽ Multiplication 18 ⫻ 2 ⫽ 31 ⫻ 3 ⫽ 57 ⫻ 5 ⫽ Division 66 ⫼ 3 ⫽ 120 ⫼ 4 ⫽ 81 ⫼ 9 ⫽ Written Calculations Addition 47 ⫹ 15 ⫽ 239 ⫹ 106 ⫽ 4920 ⫹ 345 ⫽ Subtraction 80 ⫺ 26 ⫽ 104 ⫺ 28 ⫽ 4329 ⫺ 3783 ⫽ Multiplication 255 ⫻ 18 ⫽ 492 ⫻ 7 ⫽ 134 ⫻ 9 ⫽ Division 2050 ⫼ 45 ⫽ 288 ⫼ 12 ⫽ 7054 ⫼ 9 ⫽
