## The Effect of External Representations on Numeric Tasks

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Citations: | 6 - 0 self |

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@MISC{Zhang_theeffect,

author = {Jiajie Zhang and Hongbin Wang},

title = {The Effect of External Representations on Numeric Tasks},

year = {}

}

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### Abstract

This article explores the effect of external representations on numeric tasks. Through several minor modifications on the previously reported two-digit number comparison task, we obtained different results. Rather than a holistic comparison, we found parallel comparison. We argue that this difference was a reflection of different representational forms: the comparison was based on internal representations in previous studies but on external representations in our present study. This representational effect was discussed under a framework of distributed number representations. We propose that in numerical tasks involving external representations, numbers should be considered as distributed representations and the behavior in these tasks should be considered as the interactive processing of internal and external information through the interplay of perceptual and cognitive processes. We suggest that theories of number representations and process models of numerical tasks should consider external representations as an essential component.

### Citations

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Citation Context ... parameters into account, we used the Akaike Information Criterion 2 (AIC), which is an index of fit that penalizes more heavily models with more parameters as opposed to those with fewer parameters (=-=Akaike, 1973-=-, 1983; see also Myung & Pitt, in press). The smaller the AIC of a model, the better the fit. The AIC values (Table 1) indicate that the parallel model is the best fit for targets below the standard a... |

864 | A feature-~ntegratlon theory of attention - GELADE, G - 1980 |

296 | The processing of information and structure - Garner - 1974 |

223 | The Number Sense: How the Mind Creates Mathematics - Dehaene - 1997 |

179 | Sources of mathematical thinking: Behavioral and brain-imaging evidence - Dehaene, Spelke, et al. - 1999 |

160 |
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Citation Context ...y have a common internal representation, which is similar to a physical continuum such as a format-independent line-like analog representation (e.g., Dehaene, Dupoux, and Mehler, 1990; Dehaene, 1992; =-=Gallistel & Gelman, 1992-=-; Restle, 1970). For multidigit Arabic numerals, there are at least three different models of comparison (Hinrichs, Yurko, Hu, 1981). Let us use the reaction times of comparing a series of target nume... |

139 | Representations in distributed cognitive tasks
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Citation Context ...ptual information that, in conjunction with the memorial information and cognitive processes provided by internalNumber Comparison 4 representations, determine the behavior in numerical tasks (e.g., =-=Zhang & Norman, 1994-=-, 1995; Zhang, in press). In this view, perceptual processes are not the peripheral encoding processes that re-represent external representations into internal representations. Rather, they directly o... |

113 | The nature of external representations in problem solving - Zhang - 1997 |

106 |
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Citation Context ... of numerals may have a common internal representation, which is similar to a physical continuum such as a format-independent line-like analog representation (e.g., Dehaene, Dupoux, and Mehler, 1990; =-=Dehaene, 1992-=-; Gallistel & Gelman, 1992; Restle, 1970). For multidigit Arabic numerals, there are at least three different models of comparison (Hinrichs, Yurko, Hu, 1981). Let us use the reaction times of compari... |

89 | The Priming Method: Imaging Unconscious Repetition Priming Reveals an Abstract Representation - Naccache, Dehaene |

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Citation Context ...esults suggest that different types of numerals may have a common internal representation, which is similar to a physical continuum such as a format-independent line-like analog representation (e.g., =-=Dehaene, Dupoux, and Mehler, 1990-=-; Dehaene, 1992; Gallistel & Gelman, 1992; Restle, 1970). For multidigit Arabic numerals, there are at least three different models of comparison (Hinrichs, Yurko, Hu, 1981). Let us use the reaction t... |

79 | Applying Occamâ€™s razor in modeling cognition: A Bayesian approach - Myung, Pitt - 1997 |

76 |
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Citation Context ...and multidigit Arabic numerals. For one-digit Arabic numerals, the time to make magnitude comparisons (larger or smaller) decreases linearly with the logarithm of the numerical distance between them (=-=Moyer & Landauer, 1967-=-; for reviews, see Banks, 1977; Moyer & Dumais, 1978). For example, it is faster to compare 1 and 9 than 8 and 9. This distance effect remains even when the comparison is based on English written numb... |

70 | Unconscious semantic priming extends to novel unseen stimuli - Naccache, Dehaene |

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52 |
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Citation Context ... numerals (at least one-digit numerals) may have a format independent, line-like analog internal representation. The format-independent view was explicitly formulated by McCloskey and his colleagues (=-=McCloskey, 1992-=-; McCloskey, Caramazza, & Basili, 1985; McCloskey, Sokol, & Goodman, 1986). Based on the studies of the dissociation of different numerical processing stages in dyscalculia, they proposed a model that... |

49 | Information measures and model selection - Akaike - 1983 |

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37 |
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Citation Context ... & Potts, 1984) and Japanese kanji and kana numerals (Takahashi & Green, 1983). In addition, the distance effect resembles that found for physical stimuli such as line lengths and dot patterns (e.g., =-=Buckley & Gillman, 1974-=-). These results suggest that different types of numerals may have a common internal representation, which is similar to a physical continuum such as a format-independent line-like analog representati... |

34 |
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Citation Context ... Arabic numeral: it was converted to a phonological number word for memory retention. Because numerals can be compared only when they are in the same form of representation (e.g., Noel & Seron, 1993; =-=Dehaene & Akhavein, 1995-=-), the target, which was an external Arabic numeral, and the standard, which was an internal number word, must be converted into a common representation. This common representation could be a phonolog... |

27 |
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Citation Context ...two numerals was larger or smaller. This difference in task demands was unlikely the cause of the different results because it has been shown that these two types of tasks are essentially equivalent (=-=Dehaene, 1989-=-). However, to directly test our hypothesis that the different comparison processes were due to different representations, we conducted Experiment 3 as a control experiment. In this experiment, the ta... |

25 | Architectures for numerical cognition - Campbell - 1994 |

24 |
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Citation Context ...epresentation, which is similar to a physical continuum such as a format-independent line-like analog representation (e.g., Dehaene, Dupoux, and Mehler, 1990; Dehaene, 1992; Gallistel & Gelman, 1992; =-=Restle, 1970-=-). For multidigit Arabic numerals, there are at least three different models of comparison (Hinrichs, Yurko, Hu, 1981). Let us use the reaction times of comparing a series of target numerals (e.g., 11... |

23 |
A representational analysis of numeration systems
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(Show Context)
Citation Context ...ersity Many numerical tasks involve external representations. In these tasks, people need to process information distributed across internal and external representations in an interactive manner (see =-=Zhang & Norman, 1995-=-). For example, to multiply 735 by 278 using paper and pencil, we need to process not just the information in internal representations (e.g., the value of each individual symbol, the addition and mult... |

22 | Primed numbers: Exploring the modularity of numerical representations with masked and unmasked semantic priming - Koechlin, Naccache, et al. - 1999 |

21 |
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(Show Context)
Citation Context ...s of external representations in numerical tasks. We only make a few trivial modifications on the two-digit number comparison task that has been previously published (Dehaene, Dupoux, & Mehler, 1990; =-=Hinrichs, Yurko, & Hu, 1981-=-). The results, however, are quite different. They raise concerns over the existing theories of number representations. Their implications will be discussed under a framework of distributed number rep... |

19 | Decade breaks in the mental number line? Putting the tens and units back in different bins - Nuerk, Weger, et al. - 2001 |

19 | Neural correlates of simple and complex mental calculation - Zago, Pesenti, et al. - 2001 |

18 |
Encoding and processing of symbolic information in comparative judgments
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(Show Context)
Citation Context ...igit Arabic numerals, the time to make magnitude comparisons (larger or smaller) decreases linearly with the logarithm of the numerical distance between them (Moyer & Landauer, 1967; for reviews, see =-=Banks, 1977-=-; Moyer & Dumais, 1978). For example, it is faster to compare 1 and 9 than 8 and 9. This distance effect remains even when the comparison is based on English written number words (Foltz, Poltrock, & P... |

15 |
Cognitive processes in verbal-number production: Inferences from the performance of brain-damaged subjects
- McCloskey, Sokol, et al.
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(Show Context)
Citation Context ...at independent, line-like analog internal representation. The format-independent view was explicitly formulated by McCloskey and his colleagues (McCloskey, 1992; McCloskey, Caramazza, & Basili, 1985; =-=McCloskey, Sokol, & Goodman, 1986-=-). Based on the studies of the dissociation of different numerical processing stages in dyscalculia, they proposed a model that assumes a single abstract representation for all types of numerals and n... |

14 | En encoding-complex view of cognitive number processing - Campbell, Clark - 1988 |

14 | Semantic priming in number naming - Reynvoet, Brysbaert, et al. |

13 | Numerical transcoding: A general production model - Deloche, Seron - 1987 |

10 |
Mental comparison of size and magnitude: Size congruity effects
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(Show Context)
Citation Context ...ews, see Banks, 1977; Moyer & Dumais, 1978). For example, it is faster to compare 1 and 9 than 8 and 9. This distance effect remains even when the comparison is based on English written number words (=-=Foltz, Poltrock, & Potts, 1984-=-) and Japanese kanji and kana numerals (Takahashi & Green, 1983). In addition, the distance effect resembles that found for physical stimuli such as line lengths and dot patterns (e.g., Buckley & Gill... |

10 | Comparative Judgments of Multidigit Numbers - Poltrock, Schwartz - 1984 |

9 | Cognitive number processing: An encoding-complex perspective - Campbell, Clark - 1992 |

9 | Mental manipulation of arithmetic symbols - Gonzalez, Kolers - 1982 |

7 |
Numerical Cognition
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- 1993
(Show Context)
Citation Context ...ons. At the end, the products of the numerical processing are externalized into the environment through some decoding processes. This is the view reflected in many studies on numerical cognition (see =-=Dehaene, 1993-=-, for a collection of reviews). A different view is that external representations need not be re-represented as internal representations in order to be involved in numerical tasks: they can directly a... |

7 |
Mental comparison
- Moyer, Dumais
- 1978
(Show Context)
Citation Context ...umerals, the time to make magnitude comparisons (larger or smaller) decreases linearly with the logarithm of the numerical distance between them (Moyer & Landauer, 1967; for reviews, see Banks, 1977; =-=Moyer & Dumais, 1978-=-). For example, it is faster to compare 1 and 9 than 8 and 9. This distance effect remains even when the comparison is based on English written number words (Foltz, Poltrock, & Potts, 1984) and Japane... |

6 |
Integrated versus modular theories of number skills and acalculia
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(Show Context)
Citation Context ...s (e.g., English number words) without the mediation of any central representations. As a representative for the format-specific view, the encodingcomplex model developed by Campbell and Clark (1988; =-=Clark & Campbell, 1991-=-) denies any central abstract representations. In this model, different forms of external number representations activate different forms of internal number representations, which are functionally int... |

6 | Notational constraints on mental operations - Gonzalez, Kolers - 1987 |

6 | Place information in multi-digit number comparison - Hinrichs, L, et al. - 1982 |

5 |
Arabic number reading deficit: A single case study or when 236 is read (2306) and judged superior to 1258." Cognitive Neuropsychology 10(4
- Noel, Seron
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(Show Context)
Citation Context ...ory was no longer an Arabic numeral: it was converted to a phonological number word for memory retention. Because numerals can be compared only when they are in the same form of representation (e.g., =-=Noel & Seron, 1993-=-; Dehaene & Akhavein, 1995), the target, which was an external Arabic numeral, and the standard, which was an internal number word, must be converted into a common representation. This common represen... |

4 | Efficiency of arithmetic operations in bilinguals as a function of language - Marsh, Maki - 1976 |

3 | Notational constraints and number processing: A reappraisal of the Gonzalez and Kolers - Noel, Seron - 1992 |

3 |
Numerical judgments with kanji and kana
- Takahashi, Green
- 1983
(Show Context)
Citation Context ... to compare 1 and 9 than 8 and 9. This distance effect remains even when the comparison is based on English written number words (Foltz, Poltrock, & Potts, 1984) and Japanese kanji and kana numerals (=-=Takahashi & Green, 1983-=-). In addition, the distance effect resembles that found for physical stimuli such as line lengths and dot patterns (e.g., Buckley & Gillman, 1974). These results suggest that different types of numer... |

3 |
Speech of simple arithmetic in bilinguals
- McClain, Huang
- 1982
(Show Context)
Citation Context ...different external number representations. For example, the representational effect caused by different types of number words (preferred and nonpreferred languages) in bilinguals (March & Maki, 1976; =-=McClain & Huang, 1982-=-) can be explained in terms of the different comprehension and production processes for different types of number words. Format-Specific Representations According to the format-specific view, the repr... |