The use of procedural knowledge in simple addition and subtraction problems

In a first experiment, adults were asked to solve one-digit additions, subtractions and multiplications. When the sign appeared 150 ms before the operands, addition and subtraction were solved faster than when the sign and the operands appeared simultaneously on screen. This priming effect was not observed for multiplication problems. A second experiment replicates these results on addition and multiplication and, moreover, shows that the priming effect in addition is observed for all problems, including very small ones such as 4+3. In fact, the only problems that were not primed by the addition sign were tie problems, which confirms that they have a special status in memory. Taken together, these results suggest that abstract procedures are pre-activated by the addition and subtraction signs and that these procedures are consequently used by adults to solve the problems. No such procedures would be pre-activated for multiplication, which are then most probably solved by retrieval of the result from memory. Moreover, while obviously two different strategies were used by individuals in order to solve addition and multiplication, solution times were similar when the problems were presented in their whole. These results, which question most of the conclusions of the current literature, support Anderson's model (1982) and Baroody's assumptions (1983) on the existence of compacted procedures that could be as fast as retrievals.     

Procedural vs. direct retrieval strategies in arithmetic: A comparison between additive and multiplicative problem solving

The opposition between declarative and procedural knowledge is used to account for the solution by more or less expert and novice arithmeticians of simple additions and multiplications presented either in mixed blocks (Experiments 1 and 3) or unmixed blocks (Experiment 2) in an equation verification task. In the three experiments, presenting the sign (+ vs 2) before the operands had a stronger effect in additions than in multiplications. This priming effect indicates that many participants use a counting procedure for additions that coexists with the declarative knowledge stored in the associative network. In contrast, the small size (and sometimes the absence) of a priming effect for the ''x'' sign, together with the weak effect of size and the frequency of interaction effects, reveals the essentially declarative nature of multiplication solution.     

Operator and operand preview effects in simple addition and multiplication: A comparison of Canadian and Chinese adults

Using an arithmetic-based retrieval-induced forgetting (RIF) paradigm, researchers have found evidence that participants with very high arithmetic proficiency (Chinese adults), but not less-skilled participants (Canadian adults), solved some simple additions (e.g. 3 + 2) using fast procedural skills. Here we sought converging evidence for this using the operator-priming paradigm. Previous research testing simple addition and multiplication found that a 150-ms preview of the operator (+ or ×) facilitated only addition performance. This was taken as evidence that addition, but not multiplication, was solved by procedural algorithms that could be primed by presentation of the plus sign. In the present study, Chinese and Canadian adults (N = 144) were tested in the operator-priming paradigm but, in contrast to the RIF results, there was little evidence that operator-priming effects differed between the groups and robust operator priming was observed in both addition and multiplication. Thus, the operator preview results did not reinforce the results of previous research but the experiment revealed robust group differences in operand preview effects: For the Chinese, but not the Canadians, a preview of the numerical operands produced much greater facilitation for multiplication than addition. The fact that CN obtained a mean 103-ms gain for multiplication from the 150-ms preview of the operands strongly suggests that multiplication was their default operation in this paradigm. This result adds a potentially important new phenomenon to the behavioural distinctions between Chinese and North American adults' arithmetic skills.     

Adults' strategies for simple addition and multiplication: verbal self-reports and the operand recognition paradigm

Accurate measurement of cognitive strategies is important in diverse areas of psychological research. Strategy self-reports are a common measure, but C. Thevenot, M. Fanget, and M. Fayol (2007) proposed a more objective method to distinguish different strategies in the context of mental arithmetic. In their operand recognition paradigm, speed of recognition memory for problem operands after solving a problem indexes strategy (e.g., direct memory retrieval vs. a procedural strategy). Here, in 2 experiments, operand recognition time was the same following simple addition or multiplication, but, consistent with a wide variety of previous research, strategy reports indicated much greater use of procedures (e.g., counting) for addition than multiplication. Operation, problem size (e.g., 2 + 3 vs. 8 + 9), and operand format (digits vs. words) had interactive effects on reported procedure use that were not reflected in recognition performance. Regression analyses suggested that recognition time was influenced at least as much by the relative difficulty of the preceding problem as by the strategy used. The findings indicate that the operand recognition paradigm is not a reliable substitute for strategy reports and highlight the potential impact of difficulty-related carryover effects in sequential cognitive tasks.     

What affects strategy selection in arithmetic? The example of parity and five effects on product verification

The parity effect in arithmetic problem verification tasks refers to faster and more accurate judgments for false equations when the odd/even status of the proposed answer mismatches that of the correct answer. In two experiments, we examined whether the proportion of incorrect answers that violated parity or the number of even operands in the problem affected the magnitude of these effects. Experiment 1 showed larger parity effects for problems with two even operands and larger parity effects during the second half of the experiment. Experiment 2 replicated the results of Experiment 1 and varied the proportion of problems violating parity. Larger parity effects were obtained when more of the false problems violated parity. Moreover, all three effects combined to show the greatest parity effects in conditions with a high proportion of parity violations in problems containing two even operands that were solved during the second half of the experiment. Experiment 3 generalized the findings to the case of five rule (i.e., checking whether a false product ends in 5 or 0), another procedure for solving and verifying multiplication problems quickly. These results (1) delineate further constraints for inclusion in models of arithmetic processing when thinking about how people select among verification strategies, (2) show combined effects of variables that traditionally have been shown to have separate effects on people's strategy selection, and (3) are consistent with a view of strategy selection that suggests a bias either in the allocation of cognitive resources in the execution of strategies or in the order of execution of these strategies; they argue against a simple, unbiased competition among strategies.     

Rote memory and arithmetic fact processing

The goal of the study was to examine the part played by skill in memorizing arbitrary sequences in the efficiency with which normal young adults perform simple arithmetic fact problems. The first experiment showed a clear independent role for sequence memory in all arithmetic fact processing, but a lesser role for semantic retrieval. This result was particularly true for large-answer multiplication problems and subtraction and division problems with large first operands. In a second experiment, which included a visuomotor processing control task, sequence memory predicted processing of all arithmetic problems apart from small additions independently of semantic retrieval, with the most robust independent contribution being to large-answer multiplication problems. The results, which are compatible with Dehaene and colleagues' triple-code model, suggest that rote learning may be a successful way for some people to process arithmetic facts efficiently.     

Interactive effects of numerical surface form and operand parity in cognitive arithmetic

In Experiment 1, adults (n = 48) performed simple addition, multiplication, and parity (i.e., odd-even) comparisons on pairs of Arabic digits or English number words. For addition and comparison, but not multiplication, response time increased with the number of odd operands. For addition, but not comparison, this parity effect was greater for words than for digits. In Experiment 2, adults (n = 50) solved simple addition problems in digit and word format and reported their strategies (i.e., retrieval or procedures). Procedural strategies were used more for odd than even addends and much more for word than digit problems. The results indicate that problem encoding and answer retrieval processes for cognitive arithmetic are interactive rather than strictly additive stages.     

Algorithmic solution of arithmetic problems and operands-answer associations in long-term memory

Many developmental models of arithmetic problem solving assume that any algorithmic solution of a given problem results in an association of the two operands and the answer in memory (Logan & Klapp, 1991; Siegler, 1996). In this experiment, adults had to perform either an operation or a comparison on the same pairs of two-digit numbers and then a recognition task. It is shown that unlike comparisons, the algorithmic solution of operations impairs the recognition of operands in adults. Thus, the postulate of a necessary and automatic storage of operands-answer associations in memory when young children solve additions by algorithmic strategies needs to be qualified.     

Number-word sequence skill and arithmetic performance

In two studies, the role of the number‐word sequence skill for arithmetic performance was investigated. In the first, children between 4 and 8 years of age were asked to count forward and backward on the number‐word sequence and to solve arithmetic problems followed by post‐solution interviews about solution procedures. The results demonstrated that the number‐word sequence skill predicted both number of problems solved and strategy to solve the problems. In Study 2 it was found that solving doubles (e.g., 2 + 2 = ?) problems served as a link between the number‐word sequence skill and the number of arithmetic problems solved. The findings suggest that counting on the number‐word sequence may be an early solution procedure and that, with increasing counting skill, the child may detect regularities in the number‐word sequence that can be used to form new and more accurate strategies for solving arithmetic problems.     

Mental addition in third,fourth, and sixth graders

Research on mental arithmetic has suggested that young children use a counting algorithm for simple mental addition, but that adults use memory retrieval from an organized representation of addition facts. To determine the age at which performance shifts from counting to retrieval, children in grades 3, 4, and 6 were tested in a true/false verification task. Reaction time patterns suggested that third grade is a transitional age with respect to memory structure for addition—half of these children seemed to be counting and half retrieving from memory. Results from fourth and sixth graders implicated retrieval quite strongly, as their results resembled adult RTs very closely. Fourth graders' processing, however, was easily disrupted when false problems were presented. The third graders' difficulties are not due to an inability to form mental representations of number; all three grades demonstrated a strong split effect, indicative of a simpler mental representation of numerical information than is necessary for addition. The results were discussed in the context of memory retrieval versus counting models of mental arithmetic, and the increase across age in automaticity of retrieval processes.     

Adults’ strategy choices for simple addition: Effects of retrieval interference

Simple addition (e.g., 3 + 2, 7 + 9) may be performed by direct memory retrieval or by such procedures as counting or transformation. The distribution of associations (DOA) model of strategy choice (Siegler, 1988) predicts that procedure use should increase as retrieval interference increases. To test this, 100 undergraduates performed simple addition problems, either after blocks of simple multiplication (high-interference context) or after blocks of simple division problems (low-interference context). Addition took longer and was more error prone after multiplication; in particular, there were more multiplication confusion errors on the relatively easy, small-number addition problems (e.g., 3 + 2 = 6, 4 + 3 = 12), but not on the more difficult, large-number additions. Consistent with the DOA, participants reported greater use of procedures for addition after multiplication, but more so for small addition problems. The findings demonstrate that adults’ use of procedural strategies for simple addition is substantially influenced by retrieval interference.     

Additions are biased by operands: evidence from repeated versus different operands

Recent evidence led to the conclusion that addition problems are biased towards overestimation, regardless of whether information is conveyed by symbolic or non-symbolic stimuli (the Operational Momentum effect). The present study focuses on the role of operands in the overestimation of addition problems. Based on the tie effect, and on recent evidence that the nature of operands biases addition problems towards an underestimation when operands are repeated, but towards an overestimation when different, we aim here to further elucidate the contribution of operands to addition problems. Experiment 1 replicates the underestimation of repeated-operand additions and overestimation of different-operand additions, with large numbers (around 50), and explores whether these effects also apply to small operand additions (around 10). Experiment 2 further explores the overestimation of different-operand additions by investigating the roles of operand order and numerical distance between operands. The results show that both factors have an impact on the overestimation size, but are not crucial for overestimation to occur. The results are discussed in terms of arithmetic strategies, spatial organization of numbers and magnitude representation.     

High working memory capacity favours the use of finger counting in six-year-old children

In this study, we show that 6-year-old children with high working memory capacity are more likely to use their fingers in an addition task than children with lower capacity. Moreover and as attested by a strong correlation between finger counting and accuracy in the arithmetic task, finger counting appears to be a very efficient strategy. Therefore, discovering the finger counting strategy seems to require a large amount of working memory resources, which could lack in low-span children. Furthermore, when children with low working memory capacities use their fingers to solve addition problems, they more often use the laborious counting-all strategy than children with higher capacities who use more elaborated procedures such as the Min strategy. Consequently, we suggest that explicit teaching of finger counting during the first years of schooling should be promoted because it could help less gifted children to overcome their difficulties in arithmetic.     

When combined spatial polarities activated through spatio-temporal asynchrony lead to better mathematical reasoning for addition

Several recent studies have supported the existence of a link between spatial processing and some aspects of mathematical reasoning, including mental arithmetic. Some of these studies suggested that people are more accurate when performing arithmetic operations for which the operands appeared in the lower-left and upper-right spaces than in the upper-left and lower-right spaces. However, this cross-over Horizontality × Verticality interaction effect on arithmetic accuracy was only apparent for multiplication, not for addition. In these studies, the authors used a spatio-temporal synchronous operand presentation in which all the operands appeared simultaneously in the same part of space along the horizontal and vertical dimensions. In the present paper, we report studies designed to investigate whether these results can be generalized to mental arithmetic tasks using a spatio-temporal asynchronous operand presentation. We present three studies in which participants had to solve addition (Study 1a), subtraction (Study 1b), and multiplication (Study 2) in which the operands appeared successively at different locations along the horizontal and vertical dimensions. We found that the cross-over Horizontality × Verticality interaction effect on arithmetic accuracy emerged for addition but not for subtraction and multiplication. These results are consistent with our predictions derived from the spatial polarity correspondence account and suggest interesting directions for the study of the link between spatial processing and mental arithmetic performances.     

Inverse reference in subtraction performance: an analysis from arithmetic word problems

Studies of elementary calculation have shown that adults solve basic subtraction problems faster with problems presented in addition format (e.g., 6 ± =?13) than in standard subtraction format (e.g., 13 - 6?=?). Therefore, it is considered that adults solve subtraction problems by reference to the inverse operation (e.g., for 13 - 6?=?7, "I know that 13 is 6?+?7") because presenting the subtraction problem in addition format does not require the mental rearrangement of the problem elements into the addition format. In two experiments, we examine whether adults' use of addition to solve subtractions is modulated by the arrangement of minuend and subtrahend, regardless of format. To this end, we used arithmetic word problems since single-digit problems in subtraction format would not allow the subtrahend to appear before the minuend. In Experiment 1, subtractions were presented by arranging minuend and subtrahend according to previous research. In Experiment 2, operands were reversed. The overall results showed that participants benefited from word problems where the subtrahend appears before the minuend, including subtractions in standard subtraction format. These findings add to a growing body of literature that emphasizes the role of inverse reference in adults' performance on subtractions.     

The tie effect in simple arithmetic: An access-based account

Simple arithmetic problems with repeated operands (i.e., ties such as 4 + 4, 6 x 6, 10 - 5, or 49 / 7) are solved more quickly and accurately than similar nontie problems (e.g., 4 + 5, 6 x 7, 10 - 6, or 48 / 6). Further, as compared with nonties, ties show small or nonexistent problem-size effects (whereby problems with smaller operands such as 2 + 3 are solved more quickly and accurately than problems with larger operands such as 8 + 9). Blankenberger (2001) proposed that the tie advantage occurred because repetition of the same physical stimulus resulted in faster encoding of tie than of nontie problems. Alternatively, ties may be easier to solve than nonties because of differences in accessibility in memory or differences in the solution processes. Adults solved addition and multiplication (Experiment 1) or subtraction and division (Experiment 2) problems in four two pure formats (e.g., 4 + 4, FOUR + FOUR) and two mixed formats (e.g., 4 + FOUR, and FOUR + 4). Tie advantages were reduced in mixed formats, as compared with pure formats, but the tie x problem-size interaction persisted across formats. These findings support the view that tie effects are strongly related to memory access and are influenced only moderately by encoding factors.     

Capacity and contextual constraints on product activation: Evidence from task-irrelevant fact retrieval

Three experiments tested the limiting conditions of multiplication facts retrieval in a number-matching task (LeFevre, Bisanz, & Mrkonjic, 1988). By presenting two digits as cue and by requiring participants to decide whether a subsequent numerical target had been present in the pair, we found interference when the target coincided with the product of the cue digits. This was evidence for obligatory activation of multiplication facts. Also, we showed that multiplication facts retrieval occurred even in the absence of any arithmetic context (i.e., a multiplication sign between the cue digits) and did not require processing resources (i.e., the process met the capacity criterion of automaticity; Jonides, 1981), whereas manipulation of the spatial relation between the two operands (cue digits) negatively affected retrieval. The present work appears to be unique in the context of previous similar studies on mental calculation, which invariably adopted an arithmetic task as the primary demand. We identify this difference as the reason for the failure of all previous studies in revealing independence of multiplication facts from attentional resources. Furthermore, we suggest the application of a contextual definition of automaticity to this kind of retrieval, given the fact that it might depend both on association strength and on contextual setting variables.     

How does a child solve 7 + 8? Decoding brain activity patterns associated with counting and retrieval strategies

Cognitive development and learning are characterized by diminished reliance on effortful procedures and increased use of memory-based problem solving. Here we identify the neural correlates of this strategy shift in 7-9-year-old children at an important developmental period for arithmetic skill acquisition. Univariate and multivariate approaches were used to contrast brain responses between two groups of children who relied primarily on either retrieval or procedural counting strategies. Children who used retrieval strategies showed greater responses in the left ventrolateral prefrontal cortex; notably, this was the only brain region which showed univariate differences in signal intensity between the two groups. In contrast, multivariate analysis revealed distinct multivoxel activity patterns in bilateral hippocampus, posterior parietal cortex and left ventrolateral prefrontal cortex regions between the two groups. These results demonstrate that retrieval and counting strategies during early learning are characterized by distinct patterns of activity in a distributed network of brain regions involved in arithmetic problem solving and controlled retrieval of arithmetic facts. Our findings suggest that the reorganization and refinement of neural activity patterns in multiple brain regions plays a dominant role in the transition to memory-based arithmetic problem solving. Our findings further demonstrate how multivariate approaches can provide novel insights into fine-scale developmental changes in the brain. More generally, our study illustrates how brain imaging and developmental research can be integrated to investigate fundamental aspects of neurocognitive development.     

Cognitive arithmetic: comparison of operations

Adults' performance of simple arithmetic calculations (addition, multiplication, and numerical comparison) was examined to test predictions of digital (counting), analog, and network models. Although all of these models have been supported by studies of mental addition, each leads to a different prediction concerning relations between the times required for addition, multiplication, and numerical comparison. Pairs of single-digit integers were presented and reaction times (RTs) for adding, multiplying, and comparing the stimuli were collected. A high correlation between RT for addition and multiplication of the same digits was obtained. This result is consistent with a network model, but presents difficulties for both analog and counting models. A "ties" effect of no increase in RT with increases in problem size for doubles such as 2 + 2 has been found in previous studies of addition using verification procedures, but was not found with the production task employed in the present study. Instead, a different kind of ties effect was found. Reaction time for both addition and multiplication of ties increased more slowly with problem size than did RT for non-tie problems. This ties effect, and the finding that probability of making errors contributes independently of problem size to RT support a distinction between location and accessibility of information in a network.     

Counting knowledge and skill in cognitive addition: a comparison of normal and mathematically disabled children.

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 13 MD children were administered a series of counting tasks and solved 40 computer-administered 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. Suggestions for future research are presented.