Miller K F, Paredes D R
Department of Psychology, University of Illinois, Champaign 61820.
Cognition. 1990 Dec;37(3):213-42. doi: 10.1016/0010-0277(90)90046-m.
A major stumbling block in acquiring a new skill can be integrating it with old but related knowledge. Learning multiplication is a case in point, because it involves integrating new relations with previously acquired arithmetic knowledge (in particular, addition). Two studies explored developmental changes in the relations between single-digit addition and multiplication. In the first study, third-graders, fifth-graders, and adults performed simple addition or multiplication in mixed- and blocked-operations formats. Substantial interfering effects from related knowledge were found at all age levels, but were more pronounced for younger subjects. Thus in the early stages of learning multiplication, one consequence of learning a new operation is interference in performance of an earlier, related, but less recently studied skill. Consideration of error patterns supported the view that the problem of integrating operations is a prominent one even in the early stages of mastering multiplication. Patterns of errors were generally consistent across all age groups, and all groups were much more likely to give a correct multiplication response to an addition problem than the reverse. A second, longitudinal study confirmed this finding, showing evidence for impaired performance of addition over time within individual children (second-, third-, and fourth-graders) tested on simple addition and multiplication over a 5-month period. Analysis of reaction times for addition indicated that second-graders in advanced math classes and third-graders in regular math classes tended to slow down over the year in responses to addition problems. Fourth-graders, on the other hand, tended to increase their speed of addition over the course of the year. Multiplication showed a different pattern during this period, with no evidence for slowing among children who were able to perform this task. Disruption of previously learned knowledge in the course of acquiring new skills provides evidence that new knowledge and old knowledge are being integrated. This kind of non-monotonic development may provide an empirical method for determining the functional limits of a domain of knowledge.
掌握一项新技能的一个主要障碍可能是将其与旧的但相关的知识相结合。学习乘法就是一个恰当的例子，因为它涉及将新的关系与先前获得的算术知识（特别是加法）相结合。两项研究探讨了一位数加法和乘法之间关系的发展变化。在第一项研究中，三年级学生、五年级学生和成年人以混合运算和分组运算的形式进行简单加法或乘法运算。在所有年龄组中都发现了来自相关知识的显著干扰效应，但在较年轻的受试者中更为明显。因此，在学习乘法的早期阶段，学习一项新运算的一个后果是对较早的、相关的但最近较少研究的技能的表现产生干扰。对错误模式的考虑支持了这样一种观点，即即使在掌握乘法的早期阶段，整合运算的问题也是一个突出问题。所有年龄组的错误模式总体上是一致的，并且所有组对加法问题给出正确乘法答案的可能性都远高于相反情况。第二项纵向研究证实了这一发现，显示了在为期5个月的简单加法和乘法测试中，个体儿童（二年级、三年级和四年级）随着时间的推移加法表现受损的证据。对加法反应时间的分析表明，高级数学课上的二年级学生和常规数学课上的三年级学生在这一年中对加法问题的反应往往会变慢。另一方面，四年级学生在这一年中加法速度往往会加快。在此期间，乘法呈现出不同的模式，没有证据表明能够完成这项任务的儿童速度会减慢。在获取新技能的过程中对先前所学知识的干扰表明新知识和旧知识正在整合。这种非单调发展可能为确定知识领域的功能极限提供一种实证方法。
