This study examines the conceptual understanding of numbers and operations among 372 fifth-grade students, based on their responses to an online three-tier test designed to assess their number sense, declarative knowledge, explanatory knowledge, and confidence. The results showed that most students had low performance and low number sense, with only approximately 18% demonstrating profound conceptual understanding of numbers and operations based on their high performance in five number sense components and high confidence in their solutions. In addition, the majority of students struggled with judging the reasonableness of computational results and solving context-based problems. Furthermore, approximately 78% of participants had misconceptions, with many expressing high confidence in their answers and reasoning. The findings of this study can serve as critical references when designing teaching content and materials for elementary students by carefully considering their performance, number sense, misconceptions, and confidence. The report concludes with a discussion of the implications of the results and suggestions for future research to address critical concerns in mathematics teaching and learning.

本研究根据 372 名五年级学生对在线三级测试的回答，考察他们对数字和运算的概念性理解，该测试旨在评估他们的数感、陈述性知识、解释性知识和信心。结果表明，大多数学生的表现和数感都很差，只有大约 18% 的学生表现出对数字和运算的深刻概念理解，这是基于他们在五个数感组件中的高性能和对解决方案的高度信心。此外，大多数学生在判断计算结果的合理性和解决基于上下文的问题时遇到了困难。此外，大约 78% 的参与者有误解，许多人对他们的答案和推理表示高度自信。在为小学生设计教学内容和材料时，通过仔细考虑他们的表现、数感、误解和信心，本研究的结果可以作为重要参考。该报告最后讨论了结果的意义和对未来研究的建议，以解决数学教学中的关键问题。