ABSTRACT We explore prospective elementary-school teachers’ attempts to provide signs and symbols with mathematical meaning in a case involving non-integers as numerator or denominator. The data comprises 33 responses to a task inquiring about the existence of numbers between 1/6 and 1/7, in which the participants were asked to compose a hypothetical classroom dialogue addressing this issue, and provide explanations on the mathematics involved. The analysis utilised the theoretical frameworks of “semiotic representation” and “concept image”. The findings indicate a strong reliance on prevalent fraction images and difficulties in assigning meaning to “6½” when it appears in the numerator or denominator of a fraction representation. These difficulties are examined via three approaches demonstrated in the scripts, relating to the symbolic representation of fractions, visual representation of fractions, and permissible operations on fractions. We suggest teacher-education courses may put further emphasis on the set model of fractions and atypical semiotic representations.

中文翻译：

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分数图像：六岁半的情况

摘要 我们探讨了未来的小学教师在涉及非整数作为分子或分母的情况下提供具有数学意义的符号和符号的尝试。数据包括对询问 1/6 和 1/7 之间数字是否存在的任务的 33 个回答，其中要求参与者编写一个针对这个问题的假设课堂对话，并提供有关数学的解释。该分析利用了“符号表示”和“概念图像”的理论框架。研究结果表明，当“6½”出现在分数表示的分子或分母中时，它强烈依赖普遍的分数图像，并且难以为“6½”赋予意义。通过脚本中演示的三种方法检查了这些困难，与分数的符号表示、分数的可视化表示以及对分数的允许操作有关。我们建议教师教育课程可以进一步强调分数和非典型符号表示的集合模型。