Abstract: Numbers can be divided into pure numbers and concretized numbers. Concretized numbers are numbers with measurement units. As for the pure numbers, previous findings indicated that smaller children's number-line estimation results conform to Fechner's law (S=KlgR), which means estimation results increase logarithmically with the increase of the actual number for smaller children, while the estimation increases linearly for older children in the same number range. However, the representation of concretized numbers was rarely discussed in previous studies. As for the generation and propagation of the concept of number, there have been two different views. One is invention-spread hypothesis, the other is module hypothesis. Based on invention-spread hypothesis，concretized numbers are more inclined to be represented linearly in number-line estimation task. In other words, concretized numbers can facilitate the logarithmic-to-linear shift. However, module theory maintains that pure numbers are more suitable to be processed by number module, and easier to be represented linearly than concretized numbers. Hence, compared with pure numbers, concretization might put off the logarithmic-to-linear shift. Three classes of children （47 Second graders，40fourth graders，31sixth graders）attended the experiment. Numbers and number-line estimation task were duplicated those of Siegler and Opfer (2003). Pure number estimation was the same with that of previous studies, while the concretized numbers were numbers that were concretized with concrete meaning. Concretized numbers were divided into two levels, i.e. big image (elephant) concretization and small image (ant) concretization. Specifically, concretizing a number was to paste a small printed picture (elephant or ant) closely behind the number that was 2 cm above the center of the number line. In concretized number-line estimation task, 0 labeled at the left end of each line represented 0 elephant or ant, 1000 on the right end meant 1000 elephants or ants. Based on number line estimation task, we found there exists concretizing effect in number line estimation when numbers are concretized with image. All of the data in three grades showed that within the scale of 0-1000, children’ logarithmic-to-linear representational shift occurred at third grade which was consistent with previous studies, and when the numbers were concretized with images, explanatory power of logarithmic model was increased and that of linear model decreased correspondingly in all of three grades. The regression analysis of estimation median indicated concretizing effect in each grade. For second graders, both R2and β of logarithmic model under concretizing condition are higher than that of unconcretizing condition. For fourth and sixth graders, although the explanatory power of linear had outperformed logarithmic model, the data still indicated obvious concretizing effect--under concretizing condition explanatory power of logarithmic model was increased, meanwhile the explanatory power of linear model decreased. Not only concretizing effect was found in current study but also the image effect emerged. Image effect means the concretizing image size could influence number estimation. We found that the effect of big image was significant but the small image was weak. Elephant and ant are the animals with which most of children familiar, and the former is famous for its big size and the latter small. The current study made use of the characteristics of these two kinds of animals to concretize numbers and found that when concretized with elephant the logarithmic explanatory power for the estimation results increased comparing to non-image concretized number (pure number), and when concretized with ant there was only weak effect. The effect that big image concretizing decreases explanatory power of linear model but enhances that of logarithmic model can be named image effect. In fact, concretizing effect comes mainly from image effect because the data analysis indicated that it was the big image but not concretizing itself that facilitated the change of explanatory power for different models.

Key words: Number representation, pure numbers, image concretized numbers, logarithmic-to-linear shift

摘要： 已有研究中数字线估计任务几乎都使用纯数字。本研究以二到六年级儿童为被试，采用纯数字任务和赋义数字任务来探索赋义表象对数字表征形式的影响。结果表明，对0-1000的数字赋义后，对数模型对赋义数字的解释力上升，而线性模型的解释力下降；表象大小对于赋义数字的估计影响显著，大表象赋义提高了对数模型解释力而降低了数字估计的准确性，小表象的影响比较微弱。

关键词: 数字表征, 纯数字, 表象赋义, 对数-线性表征转换