Working memory demands of exact and approximate addition

We compared the working memory requirements of two forms of mental addition: exact calculation (e.g., 63 + 49 = 112) and approximation (e.g., 63 + 49 is about 110). In two experiments, participants solved two-digit addition problems (e.g., 63 + 49) alone and in combination with a working memory task (i.e., remembering four consonants). In Experiment 1, participants chose an answer from two alternatives (e.g., exact: 112 vs. 122; approximate: 110 vs. 140). In Experiment 2, participants responded verbally with exact or approximate answers. In both experiments, the working memory load impaired exact and approximate addition performance, but exact addition was affected more. Load also impaired performance on problems with a carry operation in the units (e.g., 28 + 59 or 76 + 57) more than on problems without a unit carry (e.g., 24 + 53 or 76 + 52). These results identify the carry operation as the source of the working memory demands in multidigit addition.     

Alzheimer's disease disrupts arithmetic fact retrieval processes but not arithmetic strategy selection

Three groups of healthy younger adults, healthy older adults, and probable AD patients, performed an addition/number comparison task. They compared 128 couples of additions and numbers (e.g., 4 + 9 15) and had to identify the largest item for each problem by pressing one of two buttons located under each item. Manipulations of problem characteristics (i.e., problem difficulty and splits between correct sums and proposed numbers) enabled us to examine strategy selection and specific arithmetic fact retrieval processes. Results showed that arithmetic facts retrieval processes, which were spared with aging, were impaired in AD patients. However, AD patients were able to switch between strategies across trials according to problem characteristics as well as healthy older adults, and less systematically than healthy younger adults. We discuss implications of these findings for further understanding AD-related differences in arithmetic in particular, and problem solving in general.     

Arithmetic split effects reflect strategy selection: an adult age comparative study in addition comparison and verification tasks.

We tested whether split effects in arithmetic (i.e., better performance on large-split problems, like 3 + 8 = 16, than on small-split problems, like 3 + 8 = 12) reflect decision processing or strategy selection. To achieve this end, we tested performance of younger and older adults, matched on arithmetic skills, on two arithmetic tasks: the addition/number comparison task (e.g., 4 + 8, 13; which item is the larger?) and in the inequality verification task (e.g., 4 + 8 < 13; Yes/No?). In both tasks, split between additions and proposed numbers were manipulated. We also manipulated the difficulty of the additions, which represents an index of arithmetic fact calculation (i.e., hard problems, like 6 + 8 < 15, are solved more slowly than easy problems, like 2 + 4 < 07, suggesting that calculation takes longer). Analyses of latencies revealed three main results: First, split effects were of smaller magnitude in older adults compared to younger adults, whatever the type of arithmetic task; second, split effects were of smaller magnitude on easy problems; and third, calculation processes were well maintained in older adults with high level of arithmetic skills. This set of results improves our understanding of cognitive aging and strategy selection in arithmetic.     

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.     

Evolution of cognitive processes for solving simple additions during the first three school years

The development of a group of children's cognitive strategies forn solving simple additions was studied by analyzing verbal reports given after each problem (I+J) was solved. The evolution of the cognitive processes involved a gradual shift from more primitive and less demanding strategies (in which, e.g., the child's fingers served as memory aid) to reconstructive memory processes (in which e.g., the answer was derived in a counting process in working memory) to retrieval processes (in which the answer was obtained form long term memory search). During the first semester of the first school year 36 percent of the problems (I⁺J≤13, I≠J, I≠0, I≠1, J≠1,) could not be answered, 40 percent of the solutions were obtained in the most frequent processes utilizing external meory aid and 16 percent in reconstructive memory processes. When in the second semester of the third school year, the same children solved th same problems by utilixing the followitn most frequent strategies; 31 percent long term memory retrieval, 38 percent reconstructive memory processes and 19 percent in processes utilizing external memory aid. If a problem was solved by using a given strategy this strategy was often most likely to have been used bt the child on the occasion before and to be used during the following semester as well. For long-term memory solutions this tendency was strongest and for other strategies it was coupled with a gradual shift towards strategies with increasing sophistication in terms of memory representation.     

Extending task analytic models of graph-based reasoning: A cognitive model of problem solving with Cartesian graphs in ACT-R/PM

Models of graph-based reasoning have typically accounted for the variation in problem solving performance with different graph types in terms of a task analysis of the problem relative to the particular visual properties of each graph type [e.g., Human Computer Interaction 8 (1993) 353; Proceedings of the Twenty-first Annual Conference of the Cognitive Science Society. Lawrence Erlbaum Associates, Mahwah, NJ (1999) 531]. This approach has been used to explain response time and accuracy differences in experimental situations where data are averaged over experimental conditions. An experiment is reported in which participants’ eye movements were recorded while they were solving various problems with different graph types. The eye movement data revealed fine grained fixation patterns that are not captured by current analyses based on optimal fixation sequences. It is argued that these patterns reveal the effects of working memory limitations during the time course of problem solving. An ACT-R/PM model of the experiment is described in which a similar pattern of eye fixations is produced as a natural consequence of the decay in activation of perceptual chunks over time.     

大学生问题发现过程的眼动研究

在潜藏式与矛盾式两类问题发现情境中, 以眼动仪为研究工具, 问题发现能力高与低的大学生各20名为被试, 探讨大学生在问题发现总体和4个兴趣区中的眼动特征及其与发现问题数量和质量之间的关系。研究表明：(1)不同能力大学生在不同情境及其兴趣区中的问题发现差异, 能够体现在眼动指标上。回视是反映问题发现能力的敏感指标。回视次数和发现问题数量与质量之间的正相关, 以及在高能力组学生上的优势, 体现了信息的联系和整合性加工在问题发现中具有积极意义。(2)潜藏式问题发现中, 个体平均注视时间更长, 反映其认知加工难度更大。在提供重要信息的区域, 被试会投入更多精力, 表现为在注视时间、注视次数和瞳孔直径大小等指标的上升。(3)眼睛注视区域与发现问题区域间存在对应关系, 显示出“眼随心动”现象。在问题发现的最初和最终阶段, 被试都会出现跨区信息搜寻行为, 分别代表了对问题线索的寻找和最后的检查评估。高能力被试在每个稳定注视阶段的注视时间更短, 这种信息转换的灵活性体现出其信息加工上的优势。动态眼动轨迹分析揭示了单个静态指标难以反映的新特点。     

Children's understanding of the arithmetic concepts of inversion and associativity

Previous studies have shown that even preschoolers can solve inversion problems of the form a+b-b by using the knowledge that addition and subtraction are inverse operations. In this study, a new type of inversion problem of the form d x e/e was also examined. Grade 6 and 8 students solved inversion problems of both types as well as standard problems of the form a+b-c and d x e/f. Students in both grades used the inversion concept on both types of inversion problems, although older students used inversion more frequently and inversion was used most frequently on the addition/subtraction problems. No transfer effects were found from one type of inversion problem to the other. Students who used the concept of associativity on the addition/subtraction standard problems (e.g., a+b-c=[b-c]+a) were more likely to use the concept of inversion on the inversion problems, although overall implementation of the associativity concept was infrequent. The findings suggest that further study of inversion and associativity is important for understanding conceptual development in arithmetic.     

Sequential or parallel decomposed processing of two-digit numbers? Evidence from eye-tracking

While reaction time data have shown that decomposed processing of two-digit numbers occurs, there is little evidence about how decomposed processing functions. Poltrock and Schwartz (1984) argued that multi-digit numbers are compared in a sequential digit-by-digit fashion starting at the leftmost digit pair. In contrast, Nuerk and Willmes (2005) favoured parallel processing of the digits constituting a number. These models (i.e., sequential decomposition, parallel decomposition) make different predictions regarding the fixation pattern in a two-digit number magnitude comparison task and can therefore be differentiated by eye fixation data. We tested these models by evaluating participants' eye fixation behaviour while selecting the larger of two numbers. The stimulus set consisted of within-decade comparisons (e.g., 53_57) and between-decade comparisons (e.g., 42_57). The between-decade comparisons were further divided into compatible and incompatible trials (cf. Nuerk, Weger, & Willmes, 2001) and trials with different decade and unit distances. The observed fixation pattern implies that the comparison of two-digit numbers is not executed by sequentially comparing decade and unit digits as proposed by Poltrock and Schwartz (1984) but rather in a decomposed but parallel fashion. Moreover, the present fixation data provide first evidence that digit processing in multi-digit numbers is not a pure bottom-up effect, but is also influenced by top-down factors. Finally, implications for multi-digit number processing beyond the range of two-digit numbers are discussed.     

Sequential or parallel decomposed processing of two-digit numbers? Evidence from eye-tracking

While reaction time data have shown that decomposed processing of two-digit numbers occurs, there is little evidence about how decomposed processing functions. Poltrock and Schwartz (1984) argued that multi-digit numbers are compared in a sequential digit-by-digit fashion starting at the leftmost digit pair. In contrast, Nuerk and Willmes (2005) favoured parallel processing of the digits constituting a number. These models (i.e., sequential decomposition, parallel decomposition) make different predictions regarding the fixation pattern in a two-digit number magnitude comparison task and can therefore be differentiated by eye fixation data. We tested these models by evaluating participants' eye fixation behaviour while selecting the larger of two numbers. The stimulus set consisted of within-decade comparisons (e.g., 53_57) and between-decade comparisons (e.g., 42_57). The between-decade comparisons were further divided into compatible and incompatible trials (cf. Nuerk, Weger, & Willmes, 2001) and trials with different decade and unit distances. The observed fixation pattern implies that the comparison of two-digit numbers is not executed by sequentially comparing decade and unit digits as proposed by Poltrock and Schwartz (1984) but rather in a decomposed but parallel fashion. Moreover, the present fixation data provide first evidence that digit processing in multi-digit numbers is not a pure bottom-up effect, but is also influenced by top-down factors. Finally, implications for multi-digit number processing beyond the range of two-digit numbers are discussed.     

Children’s strategies in complex arithmetic

Strategies used to solve two-digit addition problems (e.g., 27 + 48, Experiment 1) and two-digit subtraction problems (e.g., 73 – 59, Experiment 2) were investigated in adults and in children from Grades 3, 5, and 7. Participants were tested in choice and no-choice conditions. Results showed that (a) participants used the full decomposition strategy more often than the partial decomposition strategy to solve addition problems but used both strategies equally often to solve subtraction problems; (b) strategy use and execution were influenced by participants’ age, problem features, relative strategy performance, and whether the problems were displayed horizontally or vertically; and (c) age-related changes in complex arithmetic concern relative strategy use and execution as well as the relative influences of problem characteristics, strategy characteristics, and problem presentation on strategy choices and strategy performance. Implications of these findings for understanding age-related changes in strategic aspects of complex arithmetic performance are discussed.     

Retrieval-induced forgetting of arithmetic facts but not rules

Retrieval of a multiplication fact (2×6 =12) can disrupt retrieval of its addition counterpart (2+6=8). We investigated whether this retrieval-induced forgetting effect applies to rule-governed arithmetic facts (i.e., 0×N=0, 1×N=N). Participants (n=40) practised rule-governed multiplication problems (e.g., 1×4, 0×5) and multiplication facts (e.g., 2×3, 4×5) for four blocks and then were tested on the addition counterparts (e.g., 1+4, 0+5, 2+3, 4+5) and control additions. Increased addition response times and errors relative to controls occurred only for problems corresponding to multiplication facts, with no problem-specific effects on addition counterparts of rule-governed multiplications. In contrast, the rule-governed 0+N problems provided evidence of generalisation of practice across items, whereas the fact-based 1+N problems did not. These findings support the theory that elementary arithmetic rules and facts involve distinct memory processes, and confirmed that previous, seemly inconsistent findings of RIF in arithmetic owed to the inclusion or exclusion of rule-governed problems.     

Evidence for visual persistence during saccadic eye movements

Summary 1. The persistence of visual perception was investigated under conditions of visual fixation as well as eye movement. The Ss' task was to discriminate brief double light impulses; their responses were recorded as a function of the duration of the interstimulus interval. Based on these data the critical interstimulus interval was calculated, which yielded equal response frequencies for the perception of one or two stimuli upon presentation of double light pulses.2. In the condition of visual fixation the two stimuli could not be discriminated until the mean value of interstimulus interval exceeded 73 msec. In the condition with eye movements, when the first stimulus was presented in the parafoveal region of the retina before the beginning of the saccade and the second stimulus in the foveal region just after termination of the eye movement, this duration was shown to be statistically of the same magnitude (76 msec).3. Possible alternative interpretations of this latter result, e.g., that it could be explained in terms of masking or saccadic suppression rather than visual persistence was discussed; it was attempted to invalidate such explanations by means of three control experiments.4. The main result, the persistence of visual perception during voluntary eye movements, was discussed in relation to the problem of spatial and temporal stability of visual perception.I thank Prof. Dr. H.W. Wendt for support in correcting the English translation.     

Health-engagement control strategies and 2-year changes in older adults' physical health

This study examined the associations between older adults' daily physical symptoms (e.g., chest pain or difficulty breathing) and 2-year changes in chronic health problems (e.g., cardiovascular disease or cancer) and in functional problems (e.g., difficulty dressing or moving around at home). We reasoned that these associations depend on a person's active control processes aimed at counteracting physical health problems (i.e., health-engagement control strategies, or HECS). In particular, we hypothesized that high levels of HECS buffer the adverse effect of daily physical symptoms on increases in chronic and functional health problems. We found that daily physical symptoms were associated with declines in chronic and functional health among older adults who were not engaged in addressing their health problems, but not among their counterparts who reported high levels of HECS. These findings suggest that active control strategies play an important role in the maintenance of older adults' physical health.     

The surface form x problem size interaction in cognitive arithmetic: evidence against an encoding locus.

Sixty-four university students received simple addition problems with operands presented as arabic digits (e.g. 2 + 3, 8 + 6) or as English number words (two + three, eight + six). Operands either were displayed simultaneously or sequentially with the left operand appearing 800 ms before the right operand. Consistent with previous findings, word problems were slower than digit problems and this word-format cost was larger for large- than small-number problems. The central question addressed by this experiment concerned whether this Format x Size interaction arises during problem encoding processes or in subsequent retrieval or production processes. In the simultaneous condition, both operands would contribute to format-related differences in encoding, whereas in the sequential condition encoding differences would arise only in connection with the second operand. Critically, however, the Format x Size interaction did not differ between the sequential and simultaneous conditions, although the experiment had ample power to detect such an effect. The results argue that the Format x Size interaction does not arise during encoding, but instead arises during calculation or production processes.     

Eye movements are not a prerequisite for learning movement sequence timing through observation

The present experiment examined learning of a three-segment movement sequence using physical or observational practice, and whether permitting eye movements to be made during observation is a prerequisite for learning such a movement sequence. Specifically, participants were required to move a mouse cursor through a three-segment movement sequence in order to satisfy one of three movement time goals (800, 1000, 1200 ms). A yoked-participant design was used in which a physical practice group acted as a learning model, which was viewed simultaneously by two groups that carried out different observational practice procedures. An observation group was permitted to move their eyes whilst observing the model, whereas the fixation group was instructed to maintain fixation on a central target. The difference between pre-test and post-test data indicated that all the three experimental groups significantly altered their timing accuracy, variability and movement kinematics over practice, while the control group’s behaviour was unchanged. These data indicate that movement time as well as the underlying movement control was learned following observation of a movement with or without an explicit contribution from eye movements, albeit to a lesser extent during the final segment of the sequence when compared to the physical practice group. The implication is that while similar processes might normally be involved in physical and observational practice, information afforded by eye movements during observation (e.g., efference copy and eye proprioception) is not necessary for movement sequence learning.     

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.     

A procedural theory of eye movements in doing arithmetic

A procedural theory of eye movement that accounts for main features of the stochastic behavior of eye-fixation durations and direction of movement of saccades in the process of solving arithmetic exercises of addition and subtraction is presented. The best-fitting distribution of fixation durations with a relatively simple theoretical justification consists of a mixture of an exponential distribution and the convolution of two exponential distributions. The eye movements themselves were found to approximate a random walk that fits rather closely in both adult and juvenile subjects the motion postulated by the normative algorithm ordinarily taught in schools. Certain structural features of addition and subtraction exercises, such as the number of columns, the presence or absence of a carry or a borrow, are well known to affect their difficulty. In this study, regressions on such structural variables were found to account for only a relatively small part of the variation in eye-fixation durations.     

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.     

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.