Negative instances and the acquisition of the mathematical concepts of commutativity and associativity

The effects of negative instances in the acquisition of the mathematical concepts of commutativity and associativity were examined. Two treatment levels for commutativity (positive instances or positive and negative instances) and the same treatment levels for associativity were crossed to form a 2 × 2 factorial design with 21 subjects per cell. Subjects were undergraduate elementary education majors. Criterion variables were number of correct responses, stimulus intervals, and postfeedback intervals. Results supported the contention that negative instances enhance concept acquisition but also appear to require more time during treatments. No evidence for a transfer effect for negative instances from one concept to another was found.     

Negative instances and the acquisition of the mathematical concepts of commutativity and associativity

The effects of negative instances in the acquisition of the mathematical concepts of commutativity and associativity were examined. Two treatment levels for commutativity (positive instances or positive and negative instances) and the same treatment levels for associativity were crossed to form a 2×2 factorial design with 21 subjects per cell. Subjects were undergraduate elementary education majors. Criterion variables were number of correct responses, stimulus intervals, and postfeedback intervals. Results supported the contention that negative instances enhance concept acquisition but also appear to require more time during treatments. No evidence for a transfer effect for negative instances from one concept to another was found.Paper presented at the 51st Annual Meeting of the National Council of Teachers of Mathematics, Houston, Texas, April, 1973.     

Young Children's Understanding of Addition Concepts

Children's knowledge of concrete versions of additive composition, commutativity and associativity was investigated in two studies. In Study 1, 24 four- to five-year-olds and 25 five- to six-year-olds judged the equivalence of conceptually related addition problems presented using groups of objects. In Study 2, 45 five- to six-year-olds judged related problems and solved addition problems. Both studies indicated that concrete versions of principles were salient to most children although associativity was more difficult than commutativity and there were considerable individual differences in children's understanding. Study 1 results indicated that schoolchildren were more accurate at recognising additive composition than preschoolers and Study 2 results suggested that commutativity knowledge was related to using advanced counting strategies for solving addition problems. Overall, the research supports the claim that examining early knowledge of addition principles provides important insights into children's emerging part-whole knowledge and mathematical development.     

The role of number words in preschoolers’ addition concepts and problem‐solving procedures

Preschoolers’ conceptual understanding and procedural skills were examined so as to explore the role of number‐words and concept–procedure interactions in their additional knowledge. Eighteen three‐ to four‐year‐olds and 24 four‐ to five‐year‐olds judged commutativity and associativity principles and solved two‐term problems involving number words and unknown numbers. The older preschoolers outperformed younger preschoolers in judging concepts involving unknown numbers and children made more accurate commutativity than associativity judgements. Children with conceptual profiles indicating a strong understanding of concepts applied to unknown numbers were more accurate at solving number‐word problems than those with a poor conceptual understanding. The findings suggest that an important mathematical development during the preschool years may be learning to appreciate addition concepts as general principles that apply when exact numbers are unknown.     

Children's understanding of the commutativity and complement principles: A latent profile analysis

This study examined patterns of individual differences in the acquisition of the knowledge of the commutativity and complement principles in 115 five-to six-year-old children and explored the role of concrete materials in helping children understand the prinicples. On the basis of latent profile analysis, four groups of children were identified: The first group succeeded in commutativity tasks with concrete materials but in no other tasks; the second succeeded in commutativity tasks in both concrete and abstract conditions, but not in complement tasks; the third group succeeded in all commutativity tasks and in complement tasks with concrete materials, and the final group succeeded in all the tasks. The four groups of children suggest a developmental trend – (1) Knowledge of the commutativity and of the complement principles seems to develop from thinking in the context of specific quantities to thinking about more abstract symbols; (2) There may be an order of understanding of the principles – from the commutativity to the complement principle; (3) Children may acquire the knowledge of the commutativity principle in the more abstract tasks before they start to acquire the knowledge of the complement principle. This study contributes to the literature by showing that assessing additive reasoning in different ways and identifying profiles with classification analyses may be useful for educators to understand more about the developmental stage where each child is placed. It appears that a more fine-grained assessment of additive reasoning can be achieved by incorporating both concrete materials and relatively abstract symbols in the assessment.     

Reinforcing the occurrence or nonoccurrence of interim drinking

Four pairs of rats were studied in a yoked control design intended to determine if an interim activity (schedule-induced drinking) was sensitive to operant contingencies. Food was always presented on a fixed-time 30-sec schedule. Additionally, a positive or negative operant contingency was in effect during the first 6 sec of each interval. The positive (drink/food) contingency produced an extra food presentation at the 6th second of an interval if the lead rat drank at least once in the first 6 sec. The negative (no-drink/food) contingency produced an extra food presentation only if the lead rat did not drink in the first 6 sec. Two pairs of rats were first exposed to the positive contingency and then to the negative contingency. Two pairs received training in the reverse order. In drink/food training, all lead rats developed patterns of drinking that produced extra food presentations in most intervals. There were some indications that the positive contingency facilitated early acquisition of drinking, but the yoked rats eventually developed temporal distributions and asymptotic levels of drinking comparable to those that occurred in lead rats. In no-drink/food training, the two lead rats initially exposed to the positive contingency showed high levels of drinking inappropriate to the negative contingency, but the two lead rats initially exposed to the negative contingency showed appropriately low levels of drinking. The latter effects seem attributable to the no-drink/food contingency.     

Temporal specificity in patterning of the rabbit nictitating membrane response

Rabbits were trained in either positive patterning (AX+, A−, X−) or negative patterning (A+, X+, AX−) using one of four intervals between the onset of A and the onset of X on AX trials. These intervals were 0, 800, 2,400, and 5,600 msec. In each task, all groups acquired an appropriate pattern of discriminative responding. Following acquisition, all rabbits were tested with the four different A-X intervals. All positive patterning groups showed an excitatory gradient, in which the highest level of responding occurred at the interval used in training. Conversely, all but one of the negative patterning groups showed an inhibitory gradient, in which the lowest level of responding occurred at the interval used in training. The one exception was the negative patterning group trained with simultaneous AX stimuli (0 msec), which showed a low, broad gradient, indicating transfer of inhibition across all the intervals. The results are discussed with respect to temporal encoding mechanisms and accounts of conditional discriminations.     

The Peano Axioms: An IBL Unit Constructing the Natural Numbers

Abstract

A problem sequence is presented developing the basic properties of the set of natural numbers (including associativity and commutativity of addition and multiplication, among others) from the Peano axioms, with the last portion using von Neumann’s construction to provide a model satisfying these axioms. This sequence is appropriate for mathematically inclined students in their first or second years, and has regularly required about a week of class time. The sequence could be included as one unit within any of several appropriate courses, and could be an ideal “trial size” first attempt at guiding inquiry-based learning in the tradition of R.L. Moore.     

Summation and configuration in patterning schedules with the rat and rabbit

Discrimination between a tone + light compound and its components in positive and negative patterning schedules was examined. In the positive schedule, reinforced compound presentations (C+) were intermixed with unreinforced component presentations (T?, L?). In the negative schedule, the compound was unreinforced (C?) and the components were reinforced (T+, L+). In Experiment 1, appetitive conditioning of rats’ anticipatory magazine responses was used, and in Experiment 2, aversive conditioning of the rabbit’s nictitating membrane response was used. Both experiments revealed that the positive patterning schedule consistently produced rapid acquisition of appropriate discriminative responding. The results of the negative patterning schedule were more complex. Specifically, the results of Experiment 1 demonstrated that naive rats initially showed rapid acquisition of the negative patterning discrimination. However, schedule reversals revealed that experience with the positive patterning schedule virtually abolished subsequent acquisition of discriminative responding under the negative patterning schedule. The results of Experiment 2 revealed that naive rabbits showed very slow acquisition of discriminative responding under the negative patterning schedule. The results are discussed in relation to the unique-stimulus hypothesis, a contextual encoding hypothesis, and a configural hypothesis.     

Effects of Concept and Vee Mappings Under Three Learning Modes on Students' Attitudes Towards the Mappings,Learning Modes,Ecology and Genetics

This study investigates the effects of concept and vee mappings under cooperative, cooperative-competitive and individualistic learning modes, ability levels and gender on students' attitudes towards concept and vee mappings, the three learning modes, ecology and genetics. A pre-test-post-test experiment, with control groups, involving 808 eleventh grade (fourth form) students, was used for data collection. Students taught the use of concept and vee mappings, under cooperative conditions, developed superior positive attitudes towards the two heuristics, ecology and genetics, compared with those taught under cooperative-competitive and individualistic conditions. Statistically significant differences in their attitudes were due mainly to treatment and ability. High ability students developed the most superior positive attitudes to concept and vee mappings, ecology and genetics. Students' attitudes towards cooperation were mostly positive but totally negative to individualisation and mostly negative to competition. Students' positive attitudes to ecology and genetics correlated negatively with their attitudes towards competition and individualisation. The implications of the study's findings for the teaching of science are addressed.     

论母语与第二语言跨语言间的迁移

迁移是教育心理学中的一个重要概念。文章运用迁移的理论分析了正负迁移在第二语言习得中的作用，并提出了克服负迁移的方法。     

论母语与第二语言跨语言间的迁移

迁移是教育心理学中的一个重要概念。文章运用迁移的理论分析了正负迁移在第二语言习得中的作用,并提出了克服负迁移的方法。     

连通主义与隐喻习得中的母语迁移

以连通主义理论为视角,阐释了二语隐喻习得中的两种母语迁移现象,在隐喻习得过程中,母语既能产生正向迁移,也能产生负向迁移。指出教师在教授隐喻时应持正确的态度,运用强化文化学习观念、加大输入量来抑制负迁移的发生和注重个体差异这三种策略来减少或尽量避免负向迁移的发生。     

Bilingualism in Southwest Louisiana

Abstract

The utility of signal detection theory as a framework for the interpretation of concept acquisition was studied in three experiments involving preschool children. In the first two experiments, one group was taught an unfamiliar line angle concept with an example sequence containing minimally different, juxtaposed positive and negative examples and the other group was taught the same concept with a sequence of positive examples. In both experiments, preschoolers taught with the sequence containing positive and negative examples correctly identified significantly more transfer items than the positive only group. In the third experiment, one group was taught an unfamiliar line angle concept with consistent wording from example to example. The other group was taught with instructions that changed from example to example. Subjects in the consistent group identified significantly more posttraining test items but did no better during transfer testing. The results are discussed in terms of signal strength and the signal-to-noise ratio of the instructional communication.     

《豪斯医生》中的隐喻和影射分析

在认知语言学领域里,建构隐喻概念系统需要结合相关语言背景、社会文化和历史环境。试以美国剧集《豪斯医生》(House M.D.)为例,阐述英语隐喻和影射语言在交流中的认知作用及其对思维方式的培养,研究剧中创作、主题和语言等方面的隐喻和影射,并结合实例,分析了剧中源自圣经、古希腊神话、古典文学等古典文化典故和隐喻的文化影射和源于影视、体育、日常俗语等的社会流行文化的典故和隐喻的文化影射。     

二语习得过程中的母语思维现象

母语思维在二语习得过程中是客观存在的，它对二语习得会产生正面的和负面的影响，即正迁移和负迁移。我们应该引导学生利用语言的共性来促进“正迁移”，借助母语思维来促进二语习得，进而提高目的语学习的效率。     

Increasing Self‐Selection and Self‐Location of Words by Mildly Retarded Children during Story Writing

Sentence‐writing behaviour, including use of words self‐selected and self‐located independently of teacher modelling, was established with eight mildly retarded children in a special class. The procedure comprised three components: 1) provision of word lists; 2) praise for self‐selecting and self‐locating words from supplied word lists; and 3) response cost for responses dependent on teacher modelling. The latter were termed Words Shown and Words Told, while self‐selected and self‐located responses were termed Words Found. In a sequential component design it was found that after baseline, supplying instructions and a word list (Phase 2) and continuing the use of word‐lists together with praise for self‐managed behaviour (Phase 3) decreased the frequency of Words Told, but Words Found and Words Shown were not appreciably affected. Words Shown decreased with the introduction of response cost for responses dependent on teacher modelling (Phase 4). Also, in Phase 4 Words Found increased substantially above levels in all previous phases and Words Shown decreased to zero frequency. These changes within sentence‐writing behaviour are discussed in terms of the need for accurate discrimination of and selective attention to positive and negative instances in the acquisition of self‐regulation.     

二语习得中的母语负迁移现象研究——以汉语为母语的日语学习者为例

语言迁移是语言学习中的重要影响因素。正迁移推动二语习得,负迁移阻碍二语习得。研究和探讨母语负迁移对我们的外语学习必将产生重要的作用。该文结合教学实际,从以汉语为母语的日语学习者常见的语用失误入手,着重探讨和分析了跨文化交际中母语负迁移的现象及成因,并就如何克服母语负迁移,提高学生的语用能力提出了相应的建议及对策。     

Second-order conditioning of the rabbit’s nictitating membrane response as a function of the CS2-CS1 and CS1-US intervals

Three experiments investigated conditioning of the rabbit’s nictitating membrane response (NMR) in a second-order conditioning procedure which intermixed first-order trials (CS1-US) and second-order trials (CS2-CS1) from the outset of training. Experiment 1 provided a controlled demonstration that substantial levels of second-order conditioning can be obtained with the NMR preparation. Experiment 2 showed that the level of CR acquisition to CS2 was an inverse function of the CS2-CS1 interval over the values of 400, 800, and 2,400 msec. Experiment 3 found that CR acquisition to CS2 and CS1 in second-order conditioning varied in a parallel fashion across CS1-US intervals. Similarly, Experiment 3A found that the level of CR acquisition to the two components of a serial compound (CSA-CSB-US) varied in a parallel fashion as a function of the CSB-US interval. The results of the CS2-CS1 and CS1-US interval manipulations were all predictable from the known CS-US interval effects in NMR conditioning with a single CS. The present results are discussed with regard to their implications for accounts of serial compound conditioning and second-order conditioning.