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.     

Stability and change in children's division strategies

Age-related changes in children's performance on simple division problems (e.g., 6/2, 72/9) were investigated by asking children in Grades 4 through 7 to solve 32 simple division problems. Differences in performance were found across grade, with younger children performing more slowly and less accurately than older children. Problem size effects were also found in that children were faster and more accurate on small problems than on large problems. Two strategies changed across age, with children in Grade 4 relying heavily on the strategy of "addition" (adding the divisor until the dividend was reached) to solve the problems and children in Grades 5 through 7 relying primarily on the strategy of "multiplication" (recasting the division problem as a multiplication problem) to solve the problems. Surprisingly, the frequency of direct retrieval (retrieving the answer directly from memory) did not increase across grade and never became the dominant strategy of choice. Reasons for why retrieval use remains infrequent and age invariant are discussed. Overall, the results suggest that division is a unique operation and that the continued study of division may have implications for further understanding of how procedural and conceptual knowledge of arithmetic develops.     

Target strength and retrieval-induced forgetting in semantic recall

Retrieval-induced forgetting (RIF) occurs when practice of a memory item impairs retrieval of related, unpracticed items. Here, we demonstrated that RIF in semantic memory is retrieval dependent. University students either studied (7 × 8 = 56) or retrieved (7 × 8 = ?) the answers to a set of multiplication problems for 40 blocks and then were tested on their addition counterparts (7 + 8 = ?). For the retrieval practice group, but not the study practice group, response time for the multiplication-practiced addition facts was about 100 msec slower, relative to control addition problems, in the first of five postpractice addition blocks. Subsequent blocks of addition were interleaved with retrieval blocks of all the multiplication counterparts, which permitted measurement of RIF for the control addition problems after only a single retrieval of their multiplication counterparts. The control problems presented RIF in excess of 200 msec, much larger than the RIF observed after massive practice. This is consistent with the hypothesis that inhibition of competitors should be weaker when target strength is high than when target strength is only moderate (Anderson, 2003; Norman, Newman, &; Detre, 2007). The evidence that RIF in semantic retrieval is both retrieval dependent and weaker following massive target practice than following moderate target practice provides strong support for inhibition-based theories of RIF.     

Effects of response time deadlines on adults' strategy choices for simple addition

In this investigation of adults' solution strategies for simple arithmetic, participants solved addition problems (e.g., 2 + 3, 8 + 7) under fast and slow response deadlines: The participants were instructed either to respond before a 750-msec warning beep, or to wait for a 2,500-msec beep before responding. After each trial, they indicated whether they had solved the problem by direct memory retrieval or by using a procedural strategy (e.g., counting, transformation). It was predicted that the fast deadline condition should curtail the use of procedural strategies, which generally are slower than direct retrieval. Furthermore, this deadline effect should be exaggerated for numerically larger problems because procedural strategies are especially slow for the larger problems. As predicted, we observed a deadline x size interaction whereby the fast deadline increased reported use of retrieval, especially for large problems. The results confirm that reported use of direct retrieval decreases systematically with elapsed time, and they provide additional evidence that young, educated adults rely substantially on procedural strategies even for simple addition.     

The representation of multiplication and division facts in memory

Recently, using a training paradigm, Campbell and Agnew (2009) observed cross-operation response time savings with nonidentical elements (e.g., practice 3 + 2, test 5 - 2) for addition and subtraction, showing that a single memory representation underlies addition and subtraction performance. Evidence for cross-operation savings between multiplication and division have been described frequently (e.g., Campbell, Fuchs-Lacelle, & Phenix, 2006) but they have always been attributed to a mediation strategy (reformulating a division problem as a multiplication problem, e.g., Campbell et al., 2006). Campbell and Agnew (2009) therefore concluded that there exists a fundamental difference between addition and subtraction on the one hand and multiplication and division on the other hand. However, our results suggest that retrieval savings between inverse multiplication and division problems can be observed. Even for small problems (solved by direct retrieval) practicing a division problem facilitated the corresponding multiplication problem and vice versa. These findings indicate that shared memory representations underlie multiplication and division retrieval. Hence, memory and learning processes do not seem to differ fundamentally between addition-subtraction and multiplication-division.     

Retrieval-induced forgetting of arithmetic facts across cultures

Retrieving a single-digit multiplication fact (3×4 =12) can slow response time (RT) for the corresponding addition fact (3+4=7). The present experiment investigated effects of problem type (i.e., tie addition problems such as 3+3 vs. non-ties such as 3+4) and cultural background on this retrieval-induced forgetting (RIF) phenomenon in young adults. Canadians answering in English (n=36), Chinese adults answering in English (n=36), and Chinese answering in Chinese (n=36) received four blocks of multiplication practice and then two blocks of the addition counterparts and control additions. Tie addition problems presented a robust RIF effect that did not differ between groups, but only the Canadian group showed RIF for non-ties and only for small non-ties with sum≤10 (3+4). The Chinese groups' RIF effect for addition ties, but not small non-ties, converges with recent evidence that ties are solved by direct memory retrieval whereas small non-ties may be solved by highly efficient procedural processes in skilled performers.     

Cognitive addition and multiplication: Evidence for a single memory network

In an experiment using verification task procedures, 100 subjects responded to simple and complex problems of addition and multiplication. Identical structural parameters were found to model reaction time accurately to both addition and multiplication problems. Slope estimates for a memory network parameter did not differ significantly between simple and complex problems within an operation or between addition and multiplication problems. Both complex addition and complex multiplication problems were processed columnwise, with column sums or products being retrieved from an interrated memory network. The two types of complex problems included similar processes for carrying and for encoding of single digits, and both were self-terminated when an error in the units column was encountered. Addition and multiplication facts appear to be retrieved from a single interrelated memory network. A conceptual model for this interrelated network is discussed.     

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.     

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.     

Situating math word problems: The story matters

In two experiments, we explored how the situation model of a math story problem impacts math problem performance. Participants completed multiplication story problems in which a set of objects was associated with or dissociated from a protagonist, making them more or less accessible in memory during answer retrieval. On the basis of previous findings that the sum of two numbers interferes with retrieval of their product, the number of objects in the math problem was either highly interfering (“9” for 4 3 5) or less interfering (“8” for 4 3 5) for multiplication retrieval in the problem. Participants made more errors in problem solving when highly interfering numerical content was associated with the protagonist and, thus, foregrounded. Moreover, the lower one’s working memory, the bigger this effect. In sum, small changes in the situation model of a math story problem can harm performance. These changes shift the balance of factors that influence math performance away from math knowledge and toward individual differences in general cognitive capacity.     

The production and verification tasks in mental addition: An empirical comparison

We respond to A. Baroody's comment (1984, Developmental Review, 4, 148–156) with an empirical comparison of the production and verification tasks. With the exception of performance at the first grade level, the two tasks yield essentially identical conclusions. The results of an adjunct task, in which the rate of mental counting was assessed, suggest that children as young as second grade are relying on memory retrieval to a significant degree. In contrast to Baroody's speculation, there appear to be no widespread difficulties associated with results from the verification task. Furthermore, the task permits a more analytic examination of performance and underlying mental process than is afforded by the production task. We conclude by reiterating the empirical support for a model of fact retrieval, and suggesting that accessibility of the arithmetic facts is the basic factor which underlies both fact retrieval and procedural knowledge performance.     

The roles of the central executive and visuospatial storage in mental arithmetic: A comparison across strategies

Previous research has demonstrated that working memory plays an important role in arithmetic. Different arithmetical strategies rely on working memory to different extents—for example, verbal working memory has been found to be more important for procedural strategies, such as counting and decomposition, than for retrieval strategies. Surprisingly, given the close connection between spatial and mathematical skills, the role of visuospatial working memory has received less attention and is poorly understood. This study used a dual-task methodology to investigate the impact of a dynamic spatial n-back task (Experiment 1) and tasks loading the visuospatial sketchpad and central executive (Experiment 2) on adults' use of counting, decomposition, and direct retrieval strategies for addition. While Experiment 1 suggested that visuospatial working memory plays an important role in arithmetic, especially when counting, the results of Experiment 2 suggested this was primarily due to the domain-general executive demands of the n-back task. Taken together, these results suggest that maintaining visuospatial information in mind is required when adults solve addition arithmetic problems by any strategy but the role of domain-general executive resources is much greater than that of the visuospatial sketchpad.     

Retrieval-induced forgetting of arithmetic facts

Retrieval-induced forgetting (RIF) is a widely studied phenomenon of human memory, but RIF of arithmetic facts remains relatively unexplored. In 2 experiments, we investigated RIF of simple addition facts (2 + 3 = 5) from practice of their multiplication counterparts (2 × 3 = 6). In both experiments, robust RIF expressed in response times occurred only for high-strength small-number addition facts with sums ≤ 10, indicating that RIF from multiplication practice was interference dependent. RIF of addition-fact memory was produced by multiplication retrieval (2 × 3 = ?) but not multiplication study (2 × 3 = 6), supporting an inhibitory mechanism of RIF in arithmetic memory. Finally, RIF occurred with multiplication practiced in word format (three × four) and addition tested later in digit format (3 + 4), which provides evidence that digit and written-word formats for arithmetic accessed a common semantic retrieval network. The results support the view that addition and multiplication facts are stored in an interrelated semantic network and that RIF of competing addition facts is an intrinsic process of multiplication fact retrieval.     

Practice effects on strategy selection and strategy efficiency in simple mental arithmetic

Two experiments were conducted to investigate the effects of practice on strategy selection and strategy efficiency in mental arithmetic. Participants had to solve simple addition or multiplication problems, after having received 0, 3, or 6 practice sessions (Experiment 1), and before and after having received 3 practice sessions (Experiment 2). Strategy selection was measured by means of trial-by-trial strategy reports, whereas strategy efficiency was measured by means of response latencies. Results showed significant practice effects on retrieval frequency, procedural frequency, retrieval efficiency, and procedural efficiency. However, practice effects on strategy efficiency appeared to be both strategy-specific (i.e., only for procedural strategies) and operation-specific (i.e., only for multiplication problems). Implications of the present results for mathematic cognition and its modeling are discussed.     

Inconsistent operations: A weapon of math disruption

Word problems embed a math equation within a short narrative. Due to their structure, both numerical and linguistic factors can contribute to problem difficulty. The present studies explored the role of irrelevant information in word problems, to determine whether its negative impact is due to numerical (foregrounding hypothesis) or linguistic (inconsistent‐operations hypothesis) interference. Across three experiments, participants solved multiplication and division word problems containing irrelevant numerical information, which was either associated or disassociated with the protagonist. Results demonstrated increased solution errors on division problems when irrelevant numbers were disassociated with the protagonist. When memory for numerical information was emphasized, disassociation was specifically impacted low‐working memory individuals. The effect of disassociation on division performance persisted even when irrelevant numbers, but not words, were removed from problems. These results suggest that, even in the presence of numerically interfering information, it is the language of word problems that often drive their difficulty.     

Procedural knowledge versus fact retrieval in mental arithmetic: A reply to Baroody

Based on a review of reaction time studies, a model of mental arithmetic performance which emphasizes the process of fact retrieval from organized memory representations was proposed (M. H. Ashcraft, Developmental Review, 1982, 2, 213–236). In contrast to this view A. J. Baroody (Developmental Review, 1983, 3, 225–230) proposes that most mental arithmetic performance depends on procedural knowledge such as rules, heuristics, and principles. While Baroody's idea is both intriguing and potentially important, its exposition is quite vague and speculative. Without concrete suggestions as to the nature of the proposed rules and heuristics, especially for routine problems like 4 + 3 and 8 × 5, Baroody's proposal appears to be pertinent only to special cases like N + 0 and N + 1. Lacking this sort of elaboration, the alternative does not provide a useful or compelling explanation of the existing Chronometric results, and seems, at best, to be premature.     

Calculation latency: The μ of memory and the τ of transformation

Does numeral format (e.g., 4 + 8 vs. four + eight) affect calculation per se? University students (N=47) solved single-digit addition problems presented as Arabic digits or English words and reported their strategies (memory retrieval or procedures such as counting or transformation). Decomposition of the response time (RT) distributions into μ (reflecting shift) and t (reflecting skew) confirmed that retrieval trials contributed predominantly to μ, whereas procedure trials contributed predominantly to τ. The format × problem size RT interaction (i.e., greater word-format RT costs for large problems than for small problems) was associated entirely with μ and not with τ. Reported use of procedures presented a corresponding format × size interaction. Together, these results indicate that, relative to the well-practiced digit format, the unfamiliar word format disrupts number-fact retrieval and promotes use of procedural strategies.     

Production,verification, and priming of multiplication facts

In the arithmetic-verification procedure, subjects are presented with a simple equation (e.g., 4 × 8 = 24) and must decide quickly whether it is true or false. The prevailing model of arithmetic verification holds that the presented answer (e.g., 24) has no direct effect on the speed and accuracy of retrieving an answer to the problem. It follows that models of the retrieval stage based on verification are also valid models of retrieval in the production task, in which subjects simply retrieve and state the answer to a given problem. Results of two experiments using singledigit multiplication problems challenge these assumptions. It is argued that the presented answer in verification functions as a priming stimulus and that on “true” verification trials the effects of priming are sufficient to distort estimates of problem difficulty and to mask important evidence about the nature of the retrieval process. It is also argued that the priming of false answers that have associative links to a presented problem induces interference that disrupts both speed and accuracy of retrieval. The results raise questions about the interpretation of verification data and offer support for a network-interference theory of the mental processes underlying simple multiplication.     

The involvement of working memory in children's exact and approximate mental addition

The involvement of working memory (WM) was examined in two types of mental calculation tasks: exact and approximate. Specifically, children attending Grades 3 and 4 of primary school were involved in three experiments that examined the role of verbal and visuospatial WM in solving addition problems presented in vertical or horizontal format. For Experiment 1, the children were required to solve addition problems with carrying. For Experiment 2, they were required to solve addition problems without carrying. Then, for Experiment 3, the children needed to solve approximate problems with and without carrying. Results confirmed that different WM components are involved in solving mental addition problems. In Experiment 1, horizontally presented addition problems were more impaired than vertically presented ones, according to a verbal WM load; conversely, vertically presented addition problems were more affected by a visuospatial WM load, especially when the children were required to perform approximate calculations. In Experiment 2, this pattern emerged in neither exact nor approximate calculations. Finally, in Experiment 3, the specific involvement of WM components was observed only in problems with carrying. Overall, these results reveal that both approximate calculation and carrying procedures demand particularly high WM resources that vary according to the task's constraints.