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.     

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.     

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.     

Modeling Children's Understanding of Quantitative Relations in Texts: A Developmental Perspective

This article describes developmental models of word problem solving that are grounded in the notion of general developmental constraints of the mind. These models were constructed based on the assumption that differences in children's word problem-solving performance are due, at least in part, to developmental differences in their conceptual structures in the quantitative domain. Three levels of knowledge were identified and modeled. The simplest model represents quantitative relations as an ordered array of mental objects. The next level of the model represents numbers as objects of manipulations open two mental number lines that are coordinated in a tentative fashion. The most complex model represents numerical operations as objects of manipulations on two mental number lines that are well-coordinated with explicit, functional rules. These models were implemented as production systems. The accuracy of the predictions resulting from the simulations of the models was tested in an empirical study. Global tests of the models found a good fit of the data to the models. The results wee consistent with the theoretical analysis that the three levels of knowledge were internally coherent and qualitatively different from each other, and that the models could predict children's performance differences to a satisfactory degree.     

Extending exploratory diagnostic classification models: Inferring the effect of covariates

Diagnostic models provide a statistical framework for designing formative assessments by classifying student knowledge profiles according to a collection of fine-grained attributes. The context and ecosystem in which students learn may play an important role in skill mastery, and it is therefore important to develop methods for incorporating student covariates into diagnostic models. Including covariates may provide researchers and practitioners with the ability to evaluate novel interventions or understand the role of background knowledge in attribute mastery. Existing research is designed to include covariates in confirmatory diagnostic models, which are also known as restricted latent class models. We propose new methods for including covariates in exploratory RLCMs that jointly infer the latent structure and evaluate the role of covariates on performance and skill mastery. We present a novel Bayesian formulation and report a Markov chain Monte Carlo algorithm using a Metropolis-within-Gibbs algorithm for approximating the model parameter posterior distribution. We report Monte Carlo simulation evidence regarding the accuracy of our new methods and present results from an application that examines the role of student background knowledge on the mastery of a probability data set.     

Indexed Natural Numbers in Mind: A Formal Model of the Basic Mature Number Competence

The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in the mind is based on the linear number representation. This classical conception is rejected and a competitive hypothesis is formulated according to which the basic mature representational system of cognitive arithmetic is a structure composed of many numerical axes which possess a common constituent, namely, the numeral zero. Arithmetic of indexed numbers is just a formal tool for modelling the basic mature arithmetic competence. The third task is to develop a standpoint called temporal pluralism, which is motivated by neo-Kantian philosophy of arithmetic.     

Representation of Similar Well-Learned Cognitive Procedures

Continued practice on a task is characterized by several quantitative and qualitative changes in performance. The most salient is the speed-up in the time to execute the task. To account for these effects, some models of skilled performance have proposed automatic mechanisms that merge knowlege structures associated with the task into fewer, larger structures. The present study investigated how the representation of similar cognitive procedures might interact with the success of such automatic mechanisms. In five experiments, subjects learned complex, multistep mental arithmetic procedures. These procedures included two types of knowledge thought to characterize most cognitive procedures: “component” knowledge for achieving intermediate results and “integrative” knowledge for organizing and integrating intermediate results. Subjects simultaneously practiced two procedures that had either the same component steps or the same integrative structure. Practiceeffect models supported a procedure-independent representation for common component steps. The availability of these common steps for use in a new procedure was also measured. Steps practiced in the context of two procedures were expected to show greater transfer to a new procedure than steps learned in the context of a single procedure. This did not always occur. A model using component/integrative knowledge distinction reconciled these results by proposing that integrative knowledge operated on all steps of the procedure: An integral part of the knowledge associated with achieving an intermediate result or state includes how it contributes to later task demands. These results are discussed in the context of automatic mechanisms for skill acquisition.     

Attending to feelings and cognitive performance

The effect on cognitive performance of attending to internally generated stimuli was examined. Specifically, the effect of examining feelings and sensations upon performance in the Stroop test and upon the performance of mental addition was measured. In two initial experiments subjects more quickly named, in the Stroop test, the color of the ink (“red”) in which a color word (“green”) was printed following instructions to attend to their feelings. It was found that this facilitation was due in part to the slowing of the automatic, interfering habit of reading for word meanings. In a third experiment, the effect of attending to physical sensations on the back of the hand was found to have a similar facilitating effect and was seen as resulting from the allocation of attention to internally generated stimuli. In a final experiment, attending to feelings and to the physical sensations on the back of the hand was found to increase speed of mental addition. This effect was attributed to increased attention to internally generated stimuli (here knowledge about the arithmetic performance at the moment).     

Mental multiplication: Nothing but the facts?

Current models of adult arithmetic performance assume that representation includes only facts and procedures. However, other kinds of representations such as an analog scale or sets of number multiples might be useful in a variety of multiplication-related tasks. Introducing the practice transfer paradigm, we demonstrate that associations between distinct representational structures can be detected via cross-task transfer, provided that initial performance is retrieval based. Results support the predictions of the integrated-structures model of multiplication knowledge. Implications for well-established item differences such as the problem-size effect are addressed, and the question of how integration occurs is considered.     

Menatal addition: A test of three verification models

Three explanations of adults’ mental addition performance, a counting-based model, a direct-access model with a backup counting procedure, and a network retrieval model, were tested. Whereas important predictions of the two counting models were not upheld, reaction times (RTs) to simple addition problems were consistent with the network retrieval model. RT both increased with problem size and was progressively attenuated to false stimuli as the split (numerical difference between the false and correct sums increased. For large problems, the extreme level of split (13) yielded an RT advantage for false over true problems, suggestive of a global evaluation process operating in parallel with retrieval. RTs to the more complex addition problems in Experiment 2 exhibited a similar pattern of significance and, in regression analyses, demonstrated that complex addition (e.g., 14+12=26) involves retrieval of the simple addition components (4+2=6). The network retrieval/decision model is discussed in terms of its fit to the present data, and predictions concerning priming facilitation and inhibition are specified. The similarities between mental arithmetic results and the areas of semantic memory and mental comparisons indicate both the usefulness of the network approach to mental arithmetic and the usefulness of mental arithmetic to cognitive psychology.     

元认知监测与算术知识制约小学儿童心算策略运用能力的发展:一年纵向考察

为探究元认知监测与算术知识对儿童心算策略运用能力的影响如何随个体发展而变化，采用计算机任务与纸笔测量的方法，对85名小学三、五年级儿童进行了历时一年的纵向追踪研究。研究发现：（1）两组儿童的元认知监测和算术知识均呈增长趋势，算术知识的增长速度五年级显著快于三年级，且元认知监测增长速度与算术知识增长速度显著相关；（2）两组儿童中，元认知监测与算术知识增长速度更快的个体策略执行反应时与错误率的减少速度也更快；（3）五年级儿童的算术知识在元认知监测影响策略选择发展中起着完全中介作用。     

算术策略运用能力的年龄差异：元认知监测与算术知识的作用

采用选择/无选法,以估算与精确心算为研究任务,考察了元认知监测与算术知识影响个体算术计算策略选择与执行的年龄相关差异。129名不同年龄儿童与成人被试参加实验。结果发现：（1）算术知识对儿童及成人的估算复杂策略有促进作用,并对提升他们心算策略运用的速度和减少错误有作用;（2）元认知监测显著影响儿童的估算策略选择,能够促进使用最佳策略;（3）算术知识在估算及心算策略执行的年龄差异方面起完全中介作用,元认知监测则在估算策略选择的年龄发展中起部分中介作用;（4）算术知识对元认知监测在估算及心算策略执行上的作用起完全中介作用,而对估算的策略选择则不存在中介作用,这表明元认知监测在估算策略选择上具有举足轻重的地位。     

Model Evaluation and Multiple Strategies in Cognitive Diagnosis: An Analysis of Fraction Subtraction Data

This paper studies three models for cognitive diagnosis, each illustrated with an application to fraction subtraction data. The objective of each of these models is to classify examinees according to their mastery of skills assumed to be required for fraction subtraction. We consider the DINA model, the NIDA model, and a new model that extends the DINA model to allow for multiple strategies of problem solving. For each of these models the joint distribution of the indicators of skill mastery is modeled using a single continuous higher-order latent trait, to explain the dependence in the mastery of distinct skills. This approach stems from viewing the skills as the specific states of knowledge required for exam performance, and viewing these skills as arising from a broadly defined latent trait resembling the θ of item response models. We discuss several techniques for comparing models and assessing goodness of fit. We then implement these methods using the fraction subtraction data with the aim of selecting the best of the three models for this application. We employ Markov chain Monte Carlo algorithms to fit the models, and we present simulation results to examine the performance of these algorithms. The work reported here was performed under the auspices of the External Diagnostic Research Team funded by Educational Testing Service. Views expressed in this paper does not necessarily represent the views of Educational Testing Service.     

Acquisition and use of mental operators: the influence of natural order of events

The present article reports two experiments investigating the influence of natural order of events on the acquisition and use of knowledge about operations, in short mental operators. The principle of use specificity states that task performance depends directly on the similarity between acquisition context and the present situation. In contrast, the principle of natural order proposes that knowledge about operations can always be applied easier (faster) if reasoning follows the natural order of events. In Experiment 1, participants had to apply alphabet-arithmetic operators and LISP functions in a prognosis task (A + 2 = ?) or a retrognosis task (? - 2 = A). In alphabet-arithmetic, an advantage for the first kind of task at the beginning of training decreased with increasing practice. In LISP, however, a preference for this task (corresponding with a prospective knowledge use) emerged with increasing practice. In Experiment 2, arithmetic relations between digit pairs had to be induced. In a causal context condition, relations were described as input and output of electric circuits, in a neutral context the relations were described as arithmetic dependencies. A preference for the prognosis task was found for the causal context condition (corresponding with a prospective knowledge use) but not for the neutral one. The findings suggest that the natural order of events moderates the acquisition and use of mental operators. Further research is required to clarify the bases for this moderation.     

Do children understand fraction addition?

Many children fail to master fraction arithmetic even after years of instruction. A recent theory of fraction arithmetic (Braithwaite, Pyke, & Siegler, 2017) hypothesized that this poor learning of fraction arithmetic procedures reflects poor conceptual understanding of them. To test this hypothesis, we performed three experiments examining fourth to eighth graders' estimates of fraction sums. We found that roughly half of estimates of sums were smaller than the same child's estimate of one of the two addends in the problem. Moreover, children's estimates of fraction sums were no more accurate than if they had estimated each sum as the average of the smallest and largest possible response. This weak performance could not be attributed to poor mastery of arithmetic procedures, poor knowledge of individual fraction magnitudes, or general inability to estimate sums. These results suggest that a major source of difficulty in this domain is that many children's learning of fraction arithmetic procedures develops unconstrained by conceptual understanding of the procedures. Implications for education are discussed.     

Inductive reasoning about causally transmitted properties

Different intuitive theories constrain and guide inferences in different contexts. Formalizing simple intuitive theories as probabilistic processes operating over structured representations, we present a new computational model of category-based induction about causally transmitted properties. A first experiment demonstrates undergraduates’ context-sensitive use of taxonomic and food web knowledge to guide reasoning about causal transmission and shows good qualitative agreement between model predictions and human inferences. A second experiment demonstrates strong quantitative and qualitative fits to inferences about a more complex artificial food web. A third experiment investigates human reasoning about complex novel food webs where species have known taxonomic relations. Results demonstrate a double-dissociation between the predictions of our causal model and a related taxonomic model [Kemp, C., & Tenenbaum, J. B. (2003). Learning domain structures. In Proceedings of the 25th annual conference of the cognitive science society]: the causal model predicts human inferences about diseases but not genes, while the taxonomic model predicts human inferences about genes but not diseases. We contrast our framework with previous models of category-based induction and previous formal instantiations of intuitive theories, and outline challenges in developing a complete model of context-sensitive reasoning.     

心算形式对不同近似数量系统敏锐度儿童心算表现的影响

选取112名二年级小学生,以点阵比较任务测量近似数量系统敏锐度,同时以工作记忆测验成绩为协变量,探究了不同心算形式（视算、读算）对不同近似数量系统敏锐度儿童心算表现的潜在影响。结果显示：（1）心算形式显著影响心算的正确率,读算形式下儿童的心算表现最好;（2）控制工作记忆影响后,心算形式与近似数量系统敏锐分组均对心算正确率影响显著。总体来讲,读算可能是提高小学儿童简单心算表现的有效形式,并能提高低近似数量系统敏锐度儿童的心算表现。     

Cognitive arithmetic: a review of data and theory.

The area of cognitive arithmetic is concerned with the mental representation of number and arithmetic, and the processes and procedures that access and use this knowledge. In this article, I provide a tutorial review of the area, first discussing the four basic empirical effects that characterize the evidence on cognitive arithmetic: the effects of problem size or difficulty, errors, relatedness, and strategies of processing. I then review three current models of simple arithmetic processing and the empirical reports that support or challenge their explanations. The third section of the review discusses the relationship between basic fact retrieval and a rule-based component or system, and considers current evidence and proposals on the overall architecture of the cognitive arithmetic system. The review concludes with a final set of speculations about the all-pervasive problem difficulty effect, still a central puzzle in the field.     

EEG and fMRI agree: Mental arithmetic is the easiest form of imagery to detect

fMRI and EEG during mental imagery provide alternative methods of detecting awareness in patients with disorders of consciousness (DOC) without reliance on behaviour. Because using fMRI in patients with DOC is difficult, studies increasingly employ EEG. However, there has been no verification that these modalities provide converging information at the individual subject level. The present study examined simultaneous EEG and fMRI in healthy volunteers during six mental imagery tasks to determine whether one mental imagery task generates more robust activation across subjects; whether activation can be predicted from familiarity with the imagined activity; and whether EEG and fMRI converge upon the same conclusions about individual imagery performance. Mental arithmetic generated the most robust activation in the majority of subjects for both EEG and fMRI, and level of activation could not be predicted from familiarity, with either modality. We conclude that overall, EEG and fMRI agree regarding individual mental imagery performance.     

Performance implications of leader briefings and team-interaction training for team adaptation to novel environments

The authors examined how leader briefings and team-interaction training influence team members' knowledge structures concerning processes related to effective performance in both routine and novel environments. Two-hundred thirty-seven undergraduates from a large mid-Atlantic university formed 79 three-member tank platoon teams and participated in a low-fidelity tank simulation. Team-interaction training, leader briefings, and novelty of performance environment were manipulated. Findings indicated that both leader briefings and team-interaction training affected the development of mental models, which in turn positively influenced team communication processes and team performance. Mental models and communication processes predicted performance more strongly in novel than in routine environments. Implications for the role of team-interaction training, leader briefings, and mental models as mechanisms for team adaptation are discussed.