This article presents an interpretive analysis of three different mathematics teaching cases to establish where the bigger picture should lie in the teaching and learning of mathematics. We use pre-existing data collected through pre-observation and post-observation interviews and passive classroom observation undertaken by the third author in two different Grade 11 classes taught by two different teachers at one high school. Another set of data was collected through participant observation of the second author’s Year 2 University class. We analyse the presence or absence of the bigger picture, especially, in the teachers’ questioning strategies and their approach to content, guided by Tall’s framework of three worlds of mathematics , namely the ‘conceptual-embodied’ world, the ‘proceptual-symbolic’ world and the ‘axiomatic-formal’ world. Within this broad framework we acknowledge Pirie and Kieren’s notion of folding back towards the attainment of an axiomatic-formal world. We argue that the teaching and learning of mathematics should remain anchored in the bigger picture and, in that way, mathematics is meaningful, accessible, expandable and transferable.

中文翻译：

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在数学的教与学中，更大的前景在哪里？

本文对三种不同的数学教学案例进行了解释性分析，以确定在数学的教与学中应将更大的前景放在哪里。我们使用由第三作者在同一所中学的两位不同老师教授的两个不同的11年级班级中进行的观察前和观察后访谈以及第三作者进行的被动课堂观察收集的既有数据。通过第二作者第二年大学课程的参与者观察收集了另一组数据。我们在Tall的三个数学世界框架（即“概念体现的世界”，“ proceptual-symbolic”）的指导下，分析了大图的存在与否，特别是在教师的提问策略及其内容处理方法中世界和“公理形式”世界。在这个广泛的框架内，我们认可皮里（Pirie）和基伦（Kieren）的观点，即朝着公理形式世界的方向退缩。我们认为，数学的教与学应该停留在更大的范围之内，这样一来，数学是有意义的，可访问的，可扩展的和可转移的。