Using equivalence‐based instruction to teach piano skills to college students

The purpose of the current study was to evaluate the effects of equivalence‐based instruction (EBI) on the emergence of basic music reading and piano playing skills. Six female college students learned to identify three musical chord notations given their respective dictated names. Participants also learned to play chords on the piano following the dictated name of the chord, and to play the chords to a song on a keyboard. Results are consistent with past research, in that stimuli became substitutable for each other and acquired a common behavioral function. Data suggest that EBI was an effective and efficient procedure to teach adults to read musical notation, as well as play chords and a song on a piano keyboard.     

Stimulus display geometry and colour discrimination learning by pigeons

Pigeons learned a successive conditional visual discrimination on vertically and horizontally arranged pairs of stimulus-response keys. When the discriminanda were two similar hues the pigeons' performance was significantly better on the vertical than on the horizontal task. This was also found in an experiment in which the subjects could see only monocularly. When, however, the discriminative stimuli were patterns of different orientation or markedly dissimilar hues then the performance on the horizontal task had an advantage over that on the vertical one. A horizontal advantage also obtained when similar hues were discriminated on keys clustered closely together. Pigeons thus seem more adept at solving successive conditional discriminations on horizontal than on vertical pairs of keys except when similar hues are displayed on widely separated keys where the reverse is true. It is hypothesized that colour vision inequalities due to regional retinal differentiations are responsible for this latter effect.     

Target detection performance in helmet-mounted and conventional dome displays

An experiment was conducted to assess visual target detection performance using a helmet-mounted display (HMD) and a conventional flight simulation dome display. Measures of workload and mood were also obtained. Participants in both viewing conditions scanned an area 120 degrees vertical by 240 degrees horizontal while attempting to locate targets that appeared to be approaching them from one of a possible 18 locations. Results indicated significantly superior performance in the conventional dome display. Workload and mood measures also showed a significant advantage for the dome display. Results are discussed in terms of their implications for the design and use of HMD systems as components of airborne virtual environment interfaces.     

Asymmetries in perception of tachistoscopically presented horizontal and vertical random-letter strings

Two studies examined perception of briefly presented (100-msec.) strings of letters. In Study One, 20 subjects were presented horizontal 5-letter strings in the left, central, and right visual fields. These were compared with 5-letter vertical strings presented centrally in the lower, central, and upper visual fields. Similar within-string patterns were found for all presentations. Between strings there was a typical right over left visual-field advantage in accuracy of report for horizontal strings. There was no equivalent lower over upper visual-field advantage for vertical strings. In Study Two, 24 subjects were presented vertical strings in the right and left visual fields, vertical strings in the upper and lower visual fields, and horizontal strings in the right and left visual fields. A post-stimulus cueing technique for single letters was used. Between-strings, the same right over left visual-field advantage for horizontal strings was noted but not for vertical strings. Between strings no advantage for lower over upper visual fields was found. An interaction for within-string patterns and visual field was found for vertical strings presented in the upper and lower visual fields. These results are explained in terms of an interaction between scanning and masking effects depending upon orientation and visual field.     

Notational strategies for problem solving in 5- to 7-year-olds

The study focuses on the production and use of notations as a means to solve a memory problem. It reports results from two experiments. In the first, a memory task, 5- to 7-year-old children were asked to notate the contents of three identical boxes in order to later determine the identity and number of the objects inside the boxes once they had been closed and moved to a different position. Statistical analysis of the data revealed developmental trends in subjects' notational performance: prior to age seven their notations were generally ineffective in helping them to later determine the exact contents of the boxes. In order to rule out the possibility that their low performance on the memory task was due to their inability to represent information by means of a notation, we designed a second experiment, a representation task, in which subjects produced a notation to represent the same set of objects as in experiment 1. Results comparing performance on the two experiments (the memory and representation tasks) showed no differences at age five, statistical differences for identity at age six, and only marginal statistical differences for quantity at age seven, mainly due to the statistically significant age-trend in performance. Seven-year-olds' performance improved in both conditions, weakening the differences between the two conditions. These results suggest a metacognitive hypothesis that it is not until age seven that children begin to make the prospective analysis needed to solve the memory task.     

Across-notation automatic numerical processing

In this article, the authors explored the existence of across-notation automatic numerical processing using size comparison and same-different paradigms. Participants were Arabic speakers, who used 2 sets of numerical symbols -- Arabic and Indian. They were presented with number pairs in the same notation (Arabic or Indian) or in different ones (Arabic and Indian). In the size comparison paradigm, 2 digits differing both numerically and physically were compared on the physical dimension. Nevertheless, there was evidence that participants automatically processed the irrelevant numerical dimension in different notation pairs. In the same-different paradigm, 2 digits were presented either in the same or in different notations. Participants had to indicate whether the 2 digits were physically the same. The results again showed evidence for the automatic processing of numerical magnitude for pairs in different notations. Findings of both experiments suggest that numbers in different notations are automatically translated into a common representation of magnitude, in line with M. McCloskey's (1992) abstract representation model.     

Visual perception of musical notation: Registering pitch symbols in memory

Music bears formal relations to language which suggest that perceptual processes in the two modes may also be similar. These experiments examined the way in which experienced musicians differed from non-musicians in their recognition of briefly exposed pitch notation. Experiments I and II together demonstrated that musicians are superior to non-musicians in their immediate written recall of stimuli containing more than three notes, but only when the stimulus is available to them for 150 ms or more. These results are accounted for well by a model proposed by Coltheart (1972) for letter perception under conditions of brief exposure. In this model, two coding processes act simultaneously on the stimulus, one a fast visual coding, and the other a slower, but more permanent abstract (or name) coding. In this case non-musicians appear to be lacking a second, abstract, coding which musicians possess. Experiments III and IV attempted to investigate the nature of the abstract code for musicians by presenting various types of interference in the linguistic or musical mode. Neither concurrent letter naming nor concurrent memorization of pitches appeared to cause a decrement in the original visual task, suggesting that musicians may not have been using simple naming or pitching transformations in coding the visual input.     

No prevalence of right-left over top-bottom spatial codes

Reaction time is known to depend on spatial stimulus-response compatibility in both the horizontal and the vertical dimensions. However, if both dimensions are varied in the same task, horizontal but not vertical compatibility affects performance, even if subjects are instructed to attend to the vertical dimension only (Nicoletti & Umiltà, 1984). Experiment 1 compared the effect of horizontal and vertical instructions in a task with left- versus right-handed key responses placed at a top versus bottom location. A horizontal-prevalence effect was observed only with a horizontal, and not with a vertical, instruction. This suggests that subjects might not have heeded the vertical instruction in the Nicoletti and Umiltà study but instead attended to the horizontal dimension. Experiment 2 did not yield any horizontal prevalence with one-handed responses (joystick movements). This indicates that top-bottom codes become ineffective only if the responses suggest an exclusively horizontal response coding. In sum, results demonstrated multiple spatial coding of stimuli and responses under appropriate conditions and suggest that the right-left codes do not dominate the top-bottom spatial codes.     

Letter string processing and visual short-term memory

The present study investigated whether expertise with letter string processing influences visual short-term memory capacity. Specifically, we examined whether performance in a change-detection task would vary as a function of stimulus type (letters vs. symbols) and type of display (horizontal, vertical, and circular). Participants were asked to detect a one-character change in a briefly presented character array following a delay of 900?ms. Concurrent articulation was used to limit effects of rehearsal. Type of display significantly affected performance with letters, but not with symbols, with a selective increase in change-detection accuracy for horizontally presented letter arrays compared with vertical and circular arrays. These findings confirm the standard limits of storage in visual short-term memory, but critically reveal a selective advantage for letter arrays over symbol arrays when presented horizontally. Such an advantage is probably due to the utilization of a specialized encoding mechanism built up over years of reading experience.     

The relation between angle of display and performance in a prolonged visual task

An experiment is described to test the relation between angle of display and performance in prolonged visual tasks, using the Clock Test in three positions of display surface, vertical, at an angle of 45 degrees to the horizontal, and horizontal. In every case the line of regard was kept at right-angles to the display surface. The results indicate that significantly fewer stimuli are missed in the vertical position of the display surface than in the other two.

The deterioration observed in the second half-hour of the original Clock Test experiments was not found in the present experiment, and two possible reasons for this are discussed briefly.     

不同注意条件下大数与小数的加工差异

考察在注意（注视点）与非注意（非注视点）条件下数字加工的距离效应和符号效应。采用小数（1－4）和大数（6－9）的中文与阿拉伯数字为材料,以判断数字是否大于5为任务。实验结果表明：⑴ 在注意条件下,大小数都出现了距离效应;而在非注意条件下,只有小数出现距离效应;⑵ 在注意条件下,大小数都没有出现符号效应;而在非注意条件下,只有小数出现符号效应,中文数字绩效显著好于阿拉伯数字。     

Vertical versus horizontal spatial compatibility: Right-left prevalence with bimanual responses

For two-choice tasks in which stimulus and response locations vary along horizontal and vertical dimensions, the spatial compatibility effect is often stronger on the horizontal than vertical dimension. Umiltà and Nicoletti [(1990) Spatial stimulus-response compatibility (pp. 89–116). Amsterdam: North-Holland] attributed this right-left prevalence effect to an inability to code vertical location when horizontal codes are present simultaneously. Hommel [(1996) Perception & Psychophysics, 43, 102–110] suggested instead that it reflects a voluntary strategy. This study reports four experiments that examine this issue. Experiment 1 was a conceptual replication of Hommel's Experiment 1, with responses made on a numeric keypad and subjects instructed in terms of the vertical or horizontal dimension. The results replicated Hommel's findings that showed a right-left advantage with horizontal instructions; however, with vertical instructions, we found a benefit of vertical compatibility alone that he did not. This benefit for vertical compatibility alone was eliminated in Experiment 2 using a varied practice schedule similar to that used by Hommel. Experiment 3 showed right-left prevalence and a benefit of vertical compatibility alone, even with varied practice and vertical instructions, when subjects responded on perpendicularly arranged handgrips. These benefits were eliminated in Experiment 4 using Hommel's method of urging subjects to respond only in terms of the instructed dimension. With bimanual responses, right-left prevalence is a robust phenomenon that is evident when comparing across vertical and horizontal instructions and, when the right-left distinction is relatively salient, within the vertical instructions condition alone. Received: 4 February 2000 / Accepted: 5 May 2000     

Decrement in the horizontal-vertical illusion: are subjects aware of their increased accuracy?

47 subjects adjusted the extended vertical lines of 20 inverted-T figures to make them appear equal to a horizontal line and rated the confidence in their accuracy after each trial. One group viewed figures of varying sizes, a second group viewed figures of standard size, and a third group viewed figures of standard size but received feedback on their accuracy immediately after completing Trial 5. Except for a significant increase in accuracy on Trial 6 for the Feedback Group, there were no differences in performance among groups. Subjects made the vertical line significantly shorter than the horizontal line on initial trials but their accuracy improved over trials. In contrast, there was no consistent increase in confidence, and several analyses indicated that confidence ratings were unrelated to accuracy. These results suggest that the subjects were unaware of the decrement in illusion that occurred over trials.     

Right—left prevalence effect with horizontal and vertical effectors

We investigated the right-left prevalence effect in spatial compatibility tasks by assessing subjects’ performance on a two-dimensional task in which both the horizontal and vertical spatial dimensions were task relevant. Two experiments were performed, in which stimulus-response mappings were one-dimensional (Experiment 1) and two-dimensional (Experiment 2). The subjects responded by using either horizontal or vertical effectors to stimuli appearing in four possible locations. With the one-dimensional mapping, the spatial compatibility effect was present only in the dimension relevant to the mapping. With the two-dimensional mapping, the horizontal compatibility effect was always present, whereas the vertical compatibility effect was present only when vertical effectors were used. This pattern of results indicates that horizontal coding takes place with either horizontal or vertical effectors, whereas vertical coding takes place only when vertical effectors are used.     

Learning to ignore irrelevant stimuli: Variations within and between displays

Pigeons were exposed to stimuli differing in orientation. Some birds received both horizontal and vertical on every trial; others received two horizontals on some trials and two verticals on other trials. These stimuli were irrelevant, i.e., did not predict reliably the availability of reinforcement. When required to learn a discrimination between horizontal and vertical, the birds that had experienced the stimulus variation within each display learned more slowly. This was true whether the test problem was a simultaneous or a successive discrimination. It is argued that this result implies that animals will actively learn to ignore stimuli that are irrelevant.     

The Role of Notations in Mathematics

The terms of a mathematical problem become precise and concise if they are expressed in an appropriate notation, therefore notations are useful to mathematics. But are notations only useful, or also essential? According to prevailing view, they are not essential. Contrary to this view, this paper argues that notations are essential to mathematics, because they may play a crucial role in mathematical discovery. Specifically, since notations may consist of symbolic notations, diagrammatic notations, or a mix of symbolic and diagrammatic notations, notations may play a crucial role in mathematical discovery in different ways. Some examples are given to illustrate these ways.

     

Children's note taking as a mnemonic tool

When given the opportunity to take notes in memory tasks, children sometimes make notes that are not useful. The current study examined the role that task constraints might play in the production of nonmnemonic notes. In Experiment 1, children played one easy and one difficult memory game twice, once with the opportunity to make notes and once without that opportunity. More children produced functional notations for the easier task than for the more difficult task, and their notations were beneficial to memory performance. Experiment 2 found that the majority of children who at first made nonmnemonic notations were able to produce functional notations with minimal training, and there was no significant difference in notation quality or memory performance between spontaneous and trained note takers. Experiment 3 revealed that the majority of children could transfer their training to a novel task. The results suggest that children’s production of nonmnemonic notes may be due in part to a lack of knowledge regarding what task information is important to represent or how to represent it in their notes rather than to an inability to make functional notes in general.     

The horizontal-vertical illusion and knowledge of results

The Horizontal-Vertical (HV) Illusion was examined in two studies in which subjects adjusted the vertical line in L-shaped and inverted-T figures or produced lines in the vertical and horizontal planes. On the adjustment tasks, vertical lines were made significantly shorter than horizontal comparison lines, especially for the inverted-T figure. On the production tasks, lines drawn in the vertical plane were significantly shorter than lines drawn in the horizontal plane. The adjusted and created lines of subjects receiving intertrial feedback on illusion magnitude were significantly more accurate and less variable than the estimations of control subjects. Performance on either task or figure type did not differ as a function of sex of subject. The present results show that the HV illusion exists in the absence of line bisection or a comparison line and results from the overestimation of vertical lines. These findings further clarify the relative contributions of the structural and strategy mechanisms in the formation of the Horizontal-Vertical Illusion.     

Identification of symmetry: Effects of stimulus orientation and head position

It has been argued that the perceptual advantage of symmetry depends upon the essentially symmetrical properties of the visual system. According to this explanation, the ease of identification of symmetries about different axes of orientation should decrease with increasing distance from the vertical: Reaction times to vertical symmetry should be faster than those to diagonal symmetry, which in turn should be faster than those to horizontal symmetry. Previous research demonstrating this pattern of responding employed stimuli with linear axes. In the present study, the subjects viewed tachistoscopically presented symmetrical and asymmetrical dot patterns (which had no explicit axes) in one of three head positions: upright, 45 deg left, and 45 deg right. The subjects’ performance failed to support the structural explanation: Identification of symmetry is equivalently fast for vertical and horizontal; vertical and horizontal show strong advantages over obliques, and this general advantage follows retinal coordinates. Findings are discussed in light of alternative theories of symmetry processing.     

Decoding Gentzen's Notation

In this note we consider Gentzen's first ordinal notation, used in his first published proof of the consistency of Peano Arithmetic (1936). It is a decimal notation, quite different from our current notations. We give a rule to translate this notation into our usual set-theoretic notation and we show some of its peculiarities. Then we indicate how to decode Gentzen's assignment of ordinal notations to derivations and give some examples. Finally, we go through his proof of their decrease after the application of his reduction procedure, giving further examples.