Summation of symbols by pigeons (Columba livia): the importance of number and mass of reward items

In each of 3 experiments, different sets of 4 pigeons (Columba livia) were trained to discriminate between 2 visual symbols that covered wells containing food items that varied in number, mass, or both. In Experiment 1, the symbols were associated with 0, 1, 3, 5, 7, or 9 pieces of grain reward. The pigeons learned to choose the symbol corresponding to the larger reward, and on summation tests, they chose the pair of symbols that summed to the larger total reward. When number of food pellets was varied but mass of reward was held constant in Experiment 2, preference for the larger number symbols failed to appear. When number was held constant and mass was varied in Experiment 3, the pigeons showed a clear preference for the larger mass symbols on single-symbol and summation tests. These findings show that pigeons summate the value of symbols and are more likely to represent symbols by mass of food reward than by number of food items.     

4～5岁儿童对书面数符号的表征和理解能力的发展

本研究对61名中班儿童的书面数符号表征和理解能力进行了跟踪调查。一年中三次个别面试的结果表明,约40％的4岁儿童已能运用1—10的书面数符号表征数量。至中班末期,大多数5岁儿童能够较熟练地运用至少1一17的书面数符号来表征数量,尽管其中有人在两位数的表征和数字的正确写法上有困难。儿童的书面数符号表征与他们的基数概念发展水平密切相关。大学附属幼儿园4—5岁儿童的书面数符号表征能力好于为工人家庭服务的幼儿园儿童的能力。     

Inducing ordinal and cardinal representations of the first five natural numbers

The prediction that the ordinal property of natural number symbols (using these symbols to represent the terms in an ordered progression) is more easily learned than the cardinal property of natural number symbols (using these symbols to represent the manyness of collections) was examined in this experiment. Preschoolers who evidenced no proficiency with either the ordinal or cardinal properties of natural number symbols were trained to acquire these properties via simple feedback. Both properties proved to be trainable. The most important findings were that the ordinal property was much easier to train than the cardinal property, ordinal training effects were more durable across a 1-week interval than cardinal training effects, and ordinal training appeared to transfer better than cardinal training.     

5～6岁儿童对书面数符号的表征和理解能力的发展

本研究对54名大班儿童的书面数符号表征和理解能力进行了跟踪调查。一年中三次面试结果表明,约85%的儿童(M=5:10)能运用书面数符号来表征30个以上的物体的数量。至大班末,94%的儿童(M=6:4)能运用书面数符号来表征100个以上物体的数量。大班末儿童对20以内书面加法运算成绩好于20以上书面减法运算,他们在遇到需进位和借数的加减运算时会有困难。6岁儿童在对加减运算式表征中的符号-实物的转换上有困难,儿童书面数符号表征、书面加减运算及对书面加减运算式的表征之间有显著正相关。大学附属幼儿园儿童的书面数符号表征能力在大班上学期好于为一般居民服务的幼儿园儿童,但无显著差异;在大班末期,为一般居民服务的幼儿园儿童的得分反超大学附属幼儿园儿童的得分,但无显著差异。     

Are number gestures easier than number words for preschoolers?

Some researchers have argued that children's earliest symbols are based on their sensorimotor experience and that arbitrary symbol-referent mapping poses a challenge for them. If so, exposure to iconic symbols (such as one-finger-for-one-object manual gestures) might help children in a difficult domain such as number. We assessed 44 preschoolers’ processing of number gestures and number words on two tasks: give-a-number and how-many. Children were generally more accurate when mapping number words than number gestures onto a number of toys, suggesting that the iconicity of number gestures does not help children map symbols to numbers. We argue that children learn number gestures as arbitrary symbols and do not take advantage of their iconicity.     

Information and the memory span

Messages differing in number of symbols and symbol information load were presented tachistoscopically to 4 adult Ss, The messages were constructed by random drawing with replacement from an alphabet of 8 black form symbols and an alphabet of 32 colored form symbols. The number of symbols recalled varied as a function of alphabet; however, the information in recall was constant for all conditions. The number of symbols recalled and the information in recall was independent of message length.     

Information and the memory span

Messages differing in number of symbols and symbol information load were presented tachistoscopically to 4 adult Ss. The messages were constructed by random drawing with replacement from an alphabet of 8 black form symbols and an alphabet of 32 colored form symbols. The number of symbols recalled varied as a function of alphabet; however, the information in recall was constant for all conditions. The number of symbols recalled and the information in recall was independent of message length.     

Identifying rotated letter-like symbols

Previous research has shown that the identification of rotated alphanumeric symbols seems to be performed via the extraction of critical features encoded invariant to the symbol’s orientation. The present research argued that the use of such feature extraction processes might be a function of, first, a subject’s familiarity with the symbols, and second, the number of symbols from which a presented symbol is sampled. Earlier research has used highly over learned alphanumerics, in sets of six symbols; this practice is argued here as being seemingly conducive to feature extraction. In two experiments, the generality of a feature extraction interpretation, in contrast to one of mental rotation, was tested by having subjects previously trained to relative high- vs. low-familiarity criteria identify novel symbols in conditions in which a presented symbol was 1 of either 5 or 20 possibilities. Identification response times were found to be constant across all nonstandard orientations of presented symbols, irrespective of symbol familiarity or symbol set size. The findings support the generalization of a feature extraction interpretation to varying numbers of novel symbols of varying familiarity.     

Pigeons (Columba livia) learn to link numerosities with symbols

After responding to each element in varying, successive numerosity displays, pigeons (Columba livia) had to choose, out of an array of symbols, the symbol designated to correspond to the preceding number of elements. After extensive training, 5 pigeons responded with significant accuracy to the numerosities 1 to 4, and 2 pigeons to the numerosities 1 to 5. Several tests showed that feedback tones accompanying element pecks, the familiarity of element configurations, and the shape of the elements were not crucial to this performance. One test, however, indicated that the number of pecks issued to the elements was important for numerosities above 2. An additional test confirmed that the birds chose the symbol that corresponded to a particular numerosity rather than the positions that the symbols had held during training.     

Beyond quantity: Individual differences in working memory and the ordinal understanding of numerical symbols

In two different contexts, we examined the hypothesis that individual differences in working memory (WM) capacity are related to the tendency to infer complex, ordinal relationships between numerical symbols. In Experiment 1, we assessed whether this tendency arises in a learning context that involves mapping novel symbols to quantities by training adult participants to associate dot-quantities with novel symbols, the overall relative order of which had to be inferred. Performance was best for participants who were higher in WM capacity (HWMs). HWMs also learned ordinal information about the symbols that lower WM individuals (LWMs) did not. In Experiment 2, we examined whether WM relates to performance when participants are explicitly instructed to make numerical order judgments about highly enculturated numerical symbols by having participants indicate whether sets of three Arabic numerals were in increasing order. All participants responded faster when sequential sets (3-4-5) were in order than when they were not. However, only HWMs responded faster when non-sequential, patterned sets (1-3-5) were in order, suggesting they were accessing ordinal associations that LWMs were not. Taken together, these experiments indicate that WM capacity plays a key role in extending symbolic number representations beyond their quantity referents to include symbol-symbol ordinal associations, both in a learning context and in terms of explicitly accessing ordinal relationships in highly enculturated stimuli.     

How do symbols affect 3‐ to 4‐year‐olds' executive function? Evidence from a reverse‐contingency task

In two experiments, 330 3‐ to 4‐year‐olds competed for stickers in a game in which the optimal response strategy was to point to an empty box that their opponent would receive in order to obtain a baited box for themselves. When the baited box contained stickers, children showed a strong tendency to point at the baited box and therefore lose the stickers to their opponent. In Experiment 1 children performed better when the number of stickers to be won was represented with one of five different types of symbol: numerals, number words, dots, a photograph or sweets. In Experiment 2 children transferred their improved performance in symbolic conditions to non‐symbolic conditions. These findings suggest that symbols enable children to formulate an efficient response strategy, and that this effect may be qualitatively different in children from the effect of symbols in non‐human primates.     

The approximate number system is not predictive for symbolic number processing in kindergarteners

The relation between the approximate number system (ANS) and symbolic number processing skills remains unclear. Some theories assume that children acquire the numerical meaning of symbols by mapping them onto the preexisting ANS. Others suggest that in addition to the ANS, children also develop a separate, exact representational system for symbolic number processing. In the current study, we contribute to this debate by investigating whether the nonsymbolic number processing of kindergarteners is predictive for symbolic number processing. Results revealed no association between the accuracy of the kindergarteners on a nonsymbolic number comparison task and their performance on the symbolic comparison task six months later, suggesting that there are two distinct representational systems for the ANS and numerical symbols.     

关于数字卦与六十四卦符号体系之形成问题

将见于商周器物上的数字卦与传逝文献的有关记载结合起来分析,商周时期应已存在用两个符号记写的六十四卦体系;周初陶拍所见易卦与传本《周易》相同的非覆即反的排列方法,也表明当时的筮书应是用两个确定的符号记写的;六十四卦应如文献所记是由八卦重合而成的,而这种八卦形成的前提同样是将其记写符号确定为两个;构成八卦的阴阳爻应是按阴阳观念将偶数记成“一一”,将奇数记戚“—”的产物。     

Numerical predictors of arithmetic success in grades 1–6

Math relies on mastery and integration of a wide range of simpler numerical processes and concepts. Recent work has identified several numerical competencies that predict variation in math ability. We examined the unique relations between eight basic numerical skills and early arithmetic ability in a large sample (N = 1391) of children across grades 1–6. In grades 1–2, children's ability to judge the relative magnitude of numerical symbols was most predictive of early arithmetic skills. The unique contribution of children's ability to assess ordinality in numerical symbols steadily increased across grades, overtaking all other predictors by grade 6. We found no evidence that children's ability to judge the relative magnitude of approximate, nonsymbolic numbers was uniquely predictive of arithmetic ability at any grade. Overall, symbolic number processing was more predictive of arithmetic ability than nonsymbolic number processing, though the relative importance of symbolic number ability appears to shift from cardinal to ordinal processing.     

An experimental approach to the design of cartographic symbols

Graphic designers and an experimental psychologist worked together to improve the design of two map symbols which are frequently confused: the symbols for cuttings and embankments on topographic maps. The problem was analysed in terms of the function of the symbols and their likely cognitive representations. Tests were developed to evaluate alternative designs, including an intervisibility task which requirred users to visualize the landform from the symbols viewed in the context of a map. Tests were given to schoolchildren and to experienced map users in order to compare the standard symbols with five alternative designs. Children's performance was strongly affected by the symbols they used, but experienced users were much less affected. After some refinement of the symbols a further experiment demonstrated the superiority of a number of alternative designs over the existing symbols on a range of test: scores were almost double on the intervisibility task. The paper makes recommendations to cartographers and argues for greater consideration of the inexperienced map user in the design process.     

Espbase: A microsoft access tool for selecting symbol and icon sets for usability

The ESPbase provides a tool for storing symbols and icons along with information about their characteristics. Information about a wide range of symbol characteristics is included on the database to facilitate the selection of symbol sets for research and design. The database includes information about the graphical characteristics and functions of symbols. It also includes ratings of symbol concreteness, complexity, familiarity, and meaningfulness. Symbols and icons can be accessed on the basis of each of these characteristics or any combination of characteristics. This makes it easier to select symbols on the basis of usability and design requirements. It also means that symbols can be easily selected for research while controlling their characteristics on a number of dimensions.     

Pigeons (Columba livia) associate time intervals with symbols in a touch screen task: evidence for ordinality but not summation

The present experiments examined whether pigeons can sum symbols that are associated with various temporal consequences in a touch screen apparatus. Pigeons were trained to discriminate between two visual symbols that were associated with 0, 1, 2, 3, 4, or 5 s either of delay to 4 s of hopper access (delay group) or duration of hopper access (reward group). In Experiment 1, the pigeons in both groups learned to select the symbol associated with the more favorable outcome, and they successfully transferred this discrimination to novel symbol pairs. However, when tested with 2 pairs of symbols associated with different summed durations, they responded on the basis of a simple response rule rather than the sum of the symbol pair. In Experiment 2, the reward group was presented with four symbols at once and was allowed to successively choose one symbol at a time. All pigeons chose the symbols in order from largest to smallest. This indicates that pigeons formed an ordered representation of symbols associated with different time intervals, even though they did not sum the symbols.     

Rapid proportion comparison with spatial arrays of frequently used meaningful visual symbols

It has been shown that when two arrays of Arabic numerals were briefly presented, observers could accurately indicate which array contained the larger number of a target numeral. This study investigated whether this rapid proportion comparison can be extended to other meaningful symbols that share some of notable properties of Arabic numerals. We tested mainly several Japanese Kanji letters, each of which represents a meaning and can work as a word. Using physically identical stimulus sets that could be interpreted as different types of letters, Experiment 1 first confirmed the rapid proportion comparison with Arabic numerals for Japanese participants. Experiment 2 showed that the rapid proportion comparison can be extended to Kanji numerals. Experiment 3 successfully demonstrated that rapid proportion judgments can be found with non-quantitative Kanji letters that are used frequently. Experiment 4 further demonstrated the rapid proportion comparison with frequently used meaningful non-letter symbols (gender icons). The rapid processing cannot be attributed to fluent processing of familiar items, because it was not found with familiar phonograms (Japanese Kana letters). These findings suggest that the rapid proportion comparison can be commonly found with frequently used meaningful symbols, even though their meaning is not relevant to the task.     

Continuous visual properties of number influence the formation of novel symbolic representations

Numerical symbols are thought to be mapped onto preexisting nonsymbolic representations of number. A growing body of evidence suggests that nonsymbolic numerical processing is significantly influenced by the associated visual properties of continuous quantity (e.g., surface area, density), but their role in the acquisition of novel symbols is unknown. Forty undergraduate students were trained to associate novel abstract symbols with numerical magnitudes. Half of the symbols were associated with nonsymbolic arrays in which total surface area and numerosity were correlated (“congruent”), and the other symbols were associated with arrays in which total surface area was equated across numerosities (“incongruent”). As numbers are represented in multiple formats (words, digits, nonsymbolic arrays), we also tested whether providing auditory nonword labels facilitated symbol learning. Following training, participants engaged in speeded comparisons of the newly learnt symbols. Comparisons were affected by the ratio between the numerosities associated with each symbol, a characteristic marker of numerical processing. Furthermore, comparisons were hardest for large-ratio comparisons of symbols associated with incongruent area and numerosity pairing during learning. In turn, these findings call for the further investigation of visual parameters on the development of numerical cognition.