ABSTRACT

Students often show difficulties in understanding rational numbers. Often, these are related to the natural number bias, that is, the tendency to apply the properties of natural numbers to rational number tasks. Although this phenomenon has received a lot of research interest over the last two decades, research on the existence of qualitatively different profiles regarding students’ understanding is scarce. The current study investigated the different ways students reasoned in arithmetic operation items with fractions and decimals. A cross-sectional study with 1,262 participants from 5^th to 10^th grade was performed. A TwoStep Cluster Analysis revealed eight different student reasoning profiles. We found that the natural number bias is first overcome in addition and subtraction, and later in multiplication and division. Moreover, differences regarding representation were only found in addition and subtraction items, indicating that natural numbers interfered more strongly in fractions than in decimal numbers. Finally, results showed that some students’ difficulties with rational number multiplications and divisions had other explanations than the natural number bias.

摘要

学生在理解有理数方面经常表现出困难。通常，这些与自然数偏差有关，即倾向于将自然数的属性应用于有理数任务。尽管这种现象在过去的二十年中引起了很多研究兴趣，但关于学生理解存在质量差异的研究却很少。目前的研究调查了学生在分数和小数算术运算项目中推理的不同方式。^5日至 10^日有 1,262 名参与者的横断面研究等级被执行。双步聚类分析揭示了八种不同的学生推理概况。我们发现自然数偏差首先在加减法中被克服，然后在乘法和除法中被克服。此外，关于表示的差异仅在加减项中发现，表明自然数对分数的干扰比对十进制数的干扰更大。最后，结果表明，一些学生在有理数乘法和除法方面的困难除了自然数偏差之外还有其他解释。