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.     

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.     

Individual differences in cognitive arithmetic

Unities in the processes involved in solving arithmetic problems of varying operations have been suggested by studies that have used both factor-analytic and information-processing methods. We designed the present study to investigate the convergence of mental processes assessed by paper-and-pencil measures defining the Numerical Facility factor and component processes for cognitive arithmetic identified by using chronometric techniques. A sample of 100 undergraduate students responded to 320 arithmetic problems in a true-false reaction-time (RT) verification paradigm and were administered a battery of ability measures spanning Numerical Facility, Perceptual Speed, and Spatial Relations factors. The 320 cognitive arithmetic problems comprised 80 problems of each of four types: simple addition, complex addition, simple multiplication, and complex multiplication. The information-processing results indicated that regression models that included a structural variable consistent with memory network retrieval of arithmetic facts were the best predictors of RT to each of the four types of arithmetic problems. The results also verified the effects of other elementary processes that are involved in the mental solving of arithmetic problems, including encoding of single digits and carrying to the next column for complex problems. The relation between process components and ability measures was examined by means of structural equation modeling. The final structural model revealed a strong direct relation between a factor subsuming efficiency of retrieval of arithmetic facts and of executing the carry operation and the traditional Numerical Facility factor. Furthermore, a moderate direct relation between a factor subsuming speed of encoding digits and decision and response times and the traditional Perceptual Speed factor was also found. No relation between structural variables representing cognitive arithmetic component processes and ability measures spanning the Spatial Relations factor was found. Results of the structural modeling support the conclusion that information retrieval from a network of arithmetic facts and execution of the carry operation are elementary component processes involved uniquely in the mental solving of arithmetic problems. Furthermore, individual differences in the speed of executing these two elementary component processes appear to underlie individual differences on ability measures that traditionally span the Numerical Facility factor.(ABSTRACT TRUNCATED AT 400 WORDS)     

Strategy choice for arithmetic verification: effects of numerical surface form

Canadian university students (n=48) solved simple addition problems in a true/false verification task with equations in digit format (3+4=8) or written English format (three+four=eight). Participants reported their solution strategy (e.g. retrieval or calculation) after each trial. Reported use of calculation strategies was much greater with word (41%) than digit stimuli (26%), and this difference was exaggerated for numerically larger problems. Word-format costs on reaction time (RT) were correspondingly greater for large than for small problems, but this Format×Size RT effect was bigger for true than for false equations. The results demonstrate that surface format affects central, rather than only peripheral, stages of cognitive arithmetic.     

Influence of the large–small split effect on strategy choice in complex subtraction

Two main theories have been used to explain the arithmetic split effect: decision‐making process theory and strategy choice theory. Using the inequality paradigm, previous studies have confirmed that individuals tend to adopt a plausibility‐checking strategy and a whole‐calculation strategy to solve large and small split problems in complex addition arithmetic, respectively. This supports strategy choice theory, but it is unknown whether this theory also explains performance in solving different split problems in complex subtraction arithmetic. This study used small, intermediate and large split sizes, with each split condition being further divided into problems requiring and not requiring borrowing. The reaction times (RTs) for large and intermediate splits were significantly shorter than those for small splits, while accuracy was significantly higher for large and middle splits than for small splits, reflecting no speed–accuracy trade‐off. Further, RTs and accuracy differed significantly between the borrow and no‐borrow conditions only for small splits. This study indicates that strategy choice theory is suitable to explain the split effect in complex subtraction arithmetic. That is, individuals tend to choose the plausibility‐checking strategy or the whole‐calculation strategy according to the split size.     

Identical elements model of arithmetic memory: Extension to addition and subtraction

The identical elements (IE) model of arithmetic fact representation (Rickard, 2005; Rickard & Bourne, 1996) was developed and tested with multiplication and division. In Experiment 1, we demonstrated that the model also applies to addition and subtraction by examining transfer of response time (RT) savings between prime and probe problems tested in the same block of trials. As is predicted by the IE model, there were equivalent probe RT savings for addition with identical repetition (prime 6 + 9 --> probe 6 + 9) or an order change (9 + 6 --> 6 + 9), but much greater savings for subtraction with identical repetition (15 - 6 --> 15 - 6) than with an order change (15 - 9 --> 15 - 6), and no savings with an operation change (15 - 9 --> 6 + 9 or 6 + 9 --> 4 15 - 6). In Experiment 2, we examined transfer in simple multiplication and division and demonstrated symmetrical transfer between operations. Cross-operation RT savings were eliminated, however, when the RT analysis included only trials on which both the prime and the probe problems were reportedly solved by direct retrieval. An IE model extended to accommodate savings associated with procedural strategies provides a coherent account of facilitative transfer effects in simple arithmetic.     

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.     

加减法心算老化研究

以223名60～85岁老年人及30名大学生作为被试,对加减心算能力的老化过程进行研究。任务为加减法及加减法的组合,包括有：单加、单减、先加后减、先减后加、连加及连减。结果表明,年龄与心算类型存在明显的交互作用。单加及单减两种最基本的心算能力对其它类型心算成绩的作用,在不同年龄段表现不同：对于大学生,单加起主要影响作用,而对于老年人,单减起主要影响作用。     

Priming effects of arithmetic signs in 10- to 15-year-old children

In this research, 10- to 12- and 13- to 15-year-old children were presented with very simple addition and multiplication problems involving operands from 1 to 4. Critically, the arithmetic sign was presented before the operands in half of the trials, whereas it was presented at the same time as the operands in the other half. Our results indicate that presenting the ‘x’ sign before the operands of a multiplication problem does not speed up the solving process, irrespective of the age of children. In contrast, presenting the ‘+’ sign before the operands of an addition problem facilitates the solving process, but only in 13 to 15-year-old children. Such priming effects of the arithmetic sign have been previously interpreted as the result of a pre-activation of an automated counting procedure, which can be applied as soon as the operands are presented. Therefore, our results echo previous conclusions of the literature that simple additions but not multiplications can be solved by fast counting procedures. More importantly, we show here that these procedures are possibly convoked automatically by children after the age of 13 years. At a more theoretical level, our results do not support the theory that simple additions are solved through retrieval of the answers from long-term memory by experts. Rather, the development of expertise for mental addition would consist in an acceleration of procedures until automatization.     

On mental multiplication and age.

In 2 experiments, younger and older adults were presented with simple multiplication problems (e.g., 4 x 7 = 28 and 5 x 3 = 10) for their timed, true or false judgments. All of the effects typically obtained in basic research on mental arithmetic were obtained, that is, reaction time (a) increased with the size of the problem, (b) was slowed for answers deviating only a small amount from the correct value, and (c) was slowed when related (e.g., 7 x 4 = 21) versus unrelated (e.g., 7 x 4 = 18) answers were presented. Older adults were slower in their judgments. Most important, age did not interact significantly with problem size or split size. The authors suggest that elderly adults' central processes, such as memory retrieval and decision making, did not demonstrate the typical age deficit because of the skilled nature of these processes in simple arithmetic.     

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.     

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.     

What response times tell of children's behavior on the balance scale task

Analysis of accuracy of responses to balance scale problems gives a global idea of the cognitive processes that underlie problem-solving behavior on this task. We show that response times (RTs) provide additional detailed information about the kind and duration of these processes. We derive predictions about the RTs from Siegler's (1981) model for the balance scale task, including the counterintuitive prediction that young adults are slower than children in solving particular balance scale problems. The predictions were tested in a study in which 191 6- to 22-year-old participants were presented with a computerized balance scale task. RTs were analyzed with regression models. In addition to qualitative differences between items, we also modeled quantitative differences between items in the regression models. Analyses supported the predictions and provided additional knowledge on the rules. Rule II was reformulated as a rule that always involves the encoding, but not always the correct application of the distance cue. RTs provided evidence for the use of a buggy-rule and not an addition-rule. Finally, a relation between rule inconsistency and increased RT was found.     

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.     

Mental chronometry and individual differences: Modeling reliabilities and correlations of reaction time means and effect sizes

We used a general stage-based model of reaction time (RT) to investigate the psychometric properties of mean RTs and experimental effect sizes (i.e., differences in mean RTs). Using the model, formulas were derived for the reliabilities of mean RTs and RT difference scores, and these formulas provide guidance about the number of trials per participant needed to obtain reliable estimates of these measures. In addition, formulas were derived for various different types of correlations computed in RT research (e.g., correlations between a mean RT and an external non-RT measure, between two mean RTs, between a mean RT and an RT effect size). The analysis revealed that observed RT-based correlations depend on many parameters of the underlying processes contributing to RT. We conclude that these correlations often fail to support the inferences drawn from them and that their proper interpretation is far more complex than is generally acknowledged.     

Evidence for Mixed Processes in Normal/Mirror Discrimination of Rotated Letters: A Bayesian Model Comparison Between Single- and Mixed-Distribution Models1

When determining whether a rotated letter is normal or mirrored, an observer mentally rotates the letter to its canonical orientation. To account for patterns of response times (RTs) for the normal/mirror discrimination of rotated letters, previous research formulated a model that postulated a mixture of trials with and without mental rotation. While this model could explain the curvilinear relationship that has been found between averaged RT and letter orientations, the curved RT function is still open to alternative explanations without assuming mixed processes. To address this issue and test the mixed-process hypothesis more directly, we analyzed trial-by-trial RT data instead of averaged RTs by employing a Bayesian model comparison technique. If rotation and non-rotation trials are mixed, trial-by-trial RTs for letters in a particular orientation should not follow a single distribution but a mixed one formed from the superposition of two separate distributions, one for rotation and one for non-rotation trials. In the present study, we compared single- and mixed-distribution models. Bayes-factor analysis showed decisive support for the mixed-distribution model over the single-distribution model. In addition, using the widely applicable information criterion (WAIC), the predictive accuracy of the mixed-distribution model was found to be as high as that of the single-distribution model. These results indicated the involvement of mixed processes in normal/mirror discrimination of rotated letters. The usefulness of statistical modeling in psychological study and necessary precautions to take in the interpretation of the parameters of unconfirmed models are also discussed.     

Does a time-monitoring deficit influence older adults' delayed retrieval shift during skill acquisition?

The authors evaluated age-related time-monitoring deficits and their contribution to older adults' reluctance to shift to memory retrieval in the noun-pair lookup (NP) task. Older adults (M = 67 years) showed slower rates of response time (RT) improvements than younger adults (M = 19 years), because of a delayed strategy shift. Older adults estimated scanning latencies as being faster than they actually were and showed poor resolution in discriminating short from long RTs early in practice. The difference in estimated RT for retrieval and scanning strategies predicted retrieval use, independent of actual RT differences. Separate scanning and recognition memory tasks revealed larger time-monitoring differences for older adults than in the NP task. Apparently, the context of heterogeneous RTs as a result of strategy use in the NP task improved older adults' accuracy of RT estimates. RT feedback had complex effects on time-monitoring accuracy, although it generally improved absolute and relative accuracy of RT estimates. Feedback caused older adults to shift more rapidly to the retrieval strategy in the NP task. Results suggest that deficient time monitoring plays a role in older adults' delayed retrieval shift, although other factors (e.g., confidence in the retrieval strategy) also play a role.     

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.     

Environmental context-dependent recognition memory effects: An examination of ICE model and cue-overload hypotheses

The ICE model account (e.g., Murnane & Phelps, 1993; Murnane, Phelps, & Malmberg, 1999) of environmental context (EC)-dependent recognition memory is described, and new predictions concerning the effect of multiple presentation ECs are derived. Experiment 1 tested the ICE model predictions in relation to predictions derived from the cue-overload hypothesis (e.g., Watkins, 1979). In addition, the sensitivity of recognition reaction time (RT) as a measure of EC-dependent memory effects was examined. Minimal support was obtained for the ICE model, but greater support was provided for the cue-overload hypothesis. In Experiment 2, further manipulations were employed to test ICE model predictions and the cue-overload hypothesis, with relevance to the mental reinstatement and outshining hypotheses. Again support for the cue-overload hypothesis was obtained, and an EC reinstatement effect with RTs was detected. The ICE model is considered in respect of these findings.