Verification of Donders' subtraction method

In an experiment on 6 young adults advance (precued) information of the correct choice response was utilized completely: (a) For precue-to-stimulus intervals (PSIs) clearly shorter than the difference between mean choice and simple reaction time (RT), median response latency (L) measured from precue onset was invariant; (b) For clearly longer PSIs, median RT was very near the value for simple RT. This precue-utilization effect would be expected if response actualization had been delayed until the response had been selected and if the requirement for discrimination and selection had no adverse effect on readiness to respond. Donders' (1868/1969) hypothesis in his initial application of the subtraction method, that choice and simple reactions are identical except for the serial insertion of discrimination and selection operations in the former, is thereby strongly supported. If this formulation is accepted, models that hold that response processing can overlap other processing stages may be considered valid only for response selection, not response actualization.     

Elementary realizability

A realizability notion that employs only Kalmar elementary functions is defined, and, relative to it, the soundness of EA-(10-IR), a fragment of Heyting Arithmetic (HA) with names and axioms for all elementary functions and induction rule restricted to 10 formulae, is proved. As a corollary, it is proved that the provably recursive functions of EA-(10-IR) are precisely the elementary functions. Elementary realizability is proposed as a model of strict arithmetic constructivism, which allows only those constructive procedures for which the amount of computational resources required can be bounded in advance.     

Elementary Polyhedral Mereotopology

A region-based model of physical space is one in which the primitive spatial entities are regions, rather than points, and in which the primitive spatial relations take regions, rather than points, as their relata. Historically, the most intensively investigated region-based models are those whose primitive relations are topological in character; and the study of the topology of physical space from a region-based perspective has come to be called mereotopology. This paper concentrates on a mereotopological formalism originally introduced by Whitehead, which employs a single primitive binary relation C(x,y) (read: x is in contact with y). Thus, in this formalism, all topological facts supervene on facts about contact. Because of its potential application to theories of qualitative spatial reasoning, Whitehead's primitive has recently been the subject of scrutiny from within the Artificial Intelligence community. Various results regarding the mereotopology of the Euclidean plane have been obtained, settling such issues as expressive power, axiomatization and the existence of alternative models. The contribution of the present paper is to extend some of these results to the mereotopology of three-dimensional Euclidean space. Specifically, we show that, in a first-order setting where variables range over tame subsets of R ³, Whitehead's primitive is maximally expressive for topological relations; and we deduce a corollary constraining the possible region-based models of the space we inhabit.     

Strategies in subtraction problem solving in children

The aim of this study was to investigate the strategies used by third graders in solving the 81 elementary subtractions that are the inverses of the one-digit additions with addends from 1 to 9 recently studied by Barrouillet and Lépine. Although the pattern of relationship between individual differences in working memory, on the one hand, and strategy choices and response times, on the other, was the same in both operations, subtraction and addition differed in two important ways. First, the strategy of direct retrieval was less frequent in subtraction than in addition and was even less frequent in subtraction solving than the recourse to the corresponding additive fact. Second, contrary to addition, the retrieval of subtractive answers is confined to some peculiar problems involving 1 as the subtrahend or the remainder. The implications of these findings for developmental theories of mental arithmetic are discussed.     

Concept-procedure interactions in children's addition and subtraction

A 3-week problem-solving practice phase was used to investigate concept-procedure interactions in children’s addition and subtraction. A total of 72 7- and 8-year-olds completed a pretest and posttest in which their accuracy and procedures on randomly ordered problems were recorded along with their reports of using concept-based relations in problem solving and their conceptual explanations. The results revealed that conceptual sequencing of practice problems enhances children’s ability to extend their procedural learning to new unpracticed problems. They also showed that well-structured procedural practice leads to improvement in children’s ability to verbalize key concepts. Moreover, children’s conceptual advances were predicted by their initial procedural skills. These results support an iterative account of the development of basic concepts and key skills in children’s addition and subtraction.     

Children's profiles of addition and subtraction understanding

The current research explored children's ability to recognize and explain different concepts both with and without reference to physical objects so as to provide insight into the development of children's addition and subtraction understanding. In Study 1, 72 7- to 9-year-olds judged and explained a puppet's activities involving three conceptual relations: (a) a+b=c, b+a=c; (b) a-b=c, a-c=b; and (c) a+b=c, c-b=a. In Study 2, the self-reports and problem-solving accuracy of 60 5- to 7-year-olds were recorded for three-term inverse problems (i.e., a+b-b=?), pairs of complementary addition and subtraction problems (i.e., a+b=c, c-b=?), and unrelated addition and subtraction problems (e.g., 3-2). Both studies highlighted individual differences in the concepts that children understand and the role of concrete referents in their understanding. These differences were related to using efficient procedures to solve unrelated addition and subtraction problems in Study 2. The results suggest that a key advance in children's conceptual understanding is incorporating subtractive relations into their mental representations of how parts are added to form a whole.     

Errors in children's subtraction

Many of the errors that occur in children' subtraction are due to the use of incorrect strategies rather than to the incorrect recall of number facts. A production system is presented for performing written subtraction which is consistent with an earlier analysis of the nature of such a cognitive skill. Most of the incorrect strategies used by schoolchildren can be accounted for in a principled way by simple changes in the production system, such as the omission of individual rules or the inclusion of rules appropriate to other arithmetical tasks. The production system model is evaluated against a corpus of over 1500 subtraction problems done by 10-year olds and is shown to account for about two-thirds of the (nonnumber fact) errors. It also provides an alternative, simpler interpretation of the subtraction errors analysed by Brown and Burton (1978). Some implications for teaching are discussed.     

How infants process addition and subtraction events

Three experiments are described that assess 5‐month‐old infants’ processing of addition and subtraction events similar to those reported by Wynn (1992a ). In Experiment 1, prior to each test trial, one group of infants was shown an addition event (1 + 1) while another group was shown a subtraction event (2 ? 1). On test trials, all infants were shown outcomes of 0, 1, 2 and 3. The results seemed to require one of two dual‐process models. One such model assumed that the infants could add and subtract but also had a tendency to look longer when more items were on the stage. The other model assumed that infants had a preference for familiarity along with the tendency to look longer when more items were on the stage. Experiments 2 and 3 examined the assumptions made by these two models. In Experiment 2, infants were given only the test trials they had received in Experiment 1. Thus, no addition and subtraction or familiarity was involved. In Experiment 3 infants were familiarized to either one or two items prior to each test trial, but experienced no actual addition or subtraction. The results of these two experiments support the familiarity plus more items to look at model more than the addition and subtraction plus more items to look at model. Taken together, these three experiments shed doubt on Wynn’s (1992a ) assertion that 5‐month‐old infants can add and subtract. Instead they indicate the importance of familiarity preferences and the fact that one should be cautious before assuming that young infants have sophisticated numerical abilities.     

Large-number addition and subtraction by 9-month-old infants

Do genuinely numerical computational abilities exist in infancy? It has recently been argued that previous studies putatively illustrating infants' ability to add and subtract tapped into specialized object-tracking processes that apply only with small numbers. This argument contrasts with the original interpretation that successful performance was achieved via a numerical system for estimating and calculating magnitudes. Here, we report that when continuous variables (such as area and contour length) are controlled, 9-month-old infants successfully add and subtract over numbers of items that exceed object-tracking limits. These results support the theory that infants possess a magnitude-based estimation system for representing numerosities that also supports procedures for numerical computation.     

Realities Confronting Elementary School Guidance

Overlap among 8 pupil personnel services, having implications for elementary school guidance, is illustrated by quotations from Scope of Pupil Personnel Services, written by representatives of the 8 disciplines. The overlap is viewed as advantageous in helping children if pupil personnel services are organized to accommodate flexibility. Guidance personnel are urged to borrow extensively from research in all behavioral sciences in developing an elementary school guidance program catering to known and anticipated requirements of young children for succeeding in a technological society. Other disciplines are cited for their potential contributions to elementary school guidance in some areas not usually emphasized in professional literature.     

Children’s understanding of addition and subtraction concepts

After the onset of formal schooling, little is known about the development of children’s understanding of the arithmetic concepts of inversion and associativity. On problems of the form a + b − b (e.g., 3 + 26 − 26), if children understand the inversion concept (i.e., that addition and subtraction are inverse operations), then no calculations are needed to solve the problem. On problems of the form a + b − c (e.g., 3 + 27 − 23), if children understand the associativity concept (i.e., that the addition and subtraction can be solved in any order), then the second part of the problem can be solved first. Children in Grades 2, 3, and 4 solved both types of problems and then were given a demonstration of how to apply both concepts. Approval of each concept and preference of a conceptual approach versus an algorithmic approach were measured. Few grade differences were found on either task. Conceptual understanding was greater for inversion than for associativity on both tasks. Clusters of participants in all grades showed that some had strong understanding of both concepts, some had strong understanding of the inversion concept only, and others had weak understanding of both concepts. The findings highlight the lack of developmental increases and the large individual differences in conceptual understanding on two arithmetic concepts during the early school years.     

Strategy use in mental subtraction determines central executive load

A dual task method was used to examine the relationship between strategy use and working memory load during subtraction problem solving. Undergraduates mentally solved subtraction problems alone and while performing secondary tasks that involved the central executive of working memory. Analyses revealed that a central executive task involving response selection and input monitoring (CRT-R task) interfered more with subtraction problem solving than a task that involved only input monitoring (SRT-R task). Additional analyses showed that the CRT-R task interfered more when participants used a nonretrieval (counting) strategy than a retrieval strategy. These findings suggest that the response selection subcomponent of the central executive is involved during both retrieval-based and non-retrieval-based simple subtraction problem solving but is involved more during the latter.     

Children's understanding of the relationship between addition and subtraction

In learning mathematics, children must master fundamental logical relationships, including the inverse relationship between addition and subtraction. At the start of elementary school, children lack generalized understanding of this relationship in the context of exact arithmetic problems: they fail to judge, for example, that 12 + 9 − 9 yields 12. Here, we investigate whether preschool children’s approximate number knowledge nevertheless supports understanding of this relationship. Five-year-old children were more accurate on approximate large-number arithmetic problems that involved an inverse transformation than those that did not, when problems were presented in either non-symbolic or symbolic form. In contrast they showed no advantage for problems involving an inverse transformation when exact arithmetic was involved. Prior to formal schooling, children therefore show generalized understanding of at least one logical principle of arithmetic. The teaching of mathematics may be enhanced by building on this understanding.     

神秘现象浅析

人们经常把奇异莫测,异乎寻常的现象称为神秘现象。按照辞海的解释:“神秘:深奥莫测之意。”其实当人们用“神秘”来表述某一种现象时,就隐含着对这一现象的敬畏,隐含着对人的能力的质疑,是原始宗教概念在人身上留下的烙印。因此一切有神论者,就利     

Elementary School Counselors in California

The number, school location, academic preparation and experience, and functions and duties of “elementary school counselors” in California were studied. Questionnaires were answered by 175 counselors, 118 elementary school principals, and 312 teachers in schools employing counselors. The number of children in the participating schools was 133,125, or five per cent of all pupils in grades K-8 in the State. Counselors reported that they spent 50 per cent of their time working with pupils, 17 per cent with teachers, 10 per cent with administrators, 12 per cent with parents, and 11 per cent with probation, welfare, and other officials. The counselors and the principals generally agreed on the rank order of importance of the functions of elementary school counselors and on the most important skills and personality characteristics counselors should have. The teachers reported that counselors gave them most help by testing individual pupils, counseling with pupils, helping them with classroom problems, and participating in parent conferences. Elements of an ideal counseling program at the district and school level as well as deficiencies in present programs were described by the respondents.     

The degree of abstraction in solving addition and subtraction problems

In this study, the incidence of the degree of abstraction in solving addition and subtraction problems with the unknown in the first term and in the result is analyzed. Ninety-six students from first grade to fourth grade in Primary Education (24 students per grade) solved arithmetic problems with objects, drawings, algorithms, and verbal problems. The participants were interviewed individually and all sessions were video-taped. The results indicate a different developmental pattern in achievement for each school grade depending on the levels of abstraction. The influence of the level of abstraction was significant, especially in first graders, and even more so in second graders, that is, at the developmental stage in which they start to learn these arithmetic tasks. Direct modeling strategies are observed more frequently at the concrete and pictorial level, counting strategies occur at all levels of abstraction, whereas numerical fact strategies are found at higher levels of abstraction.