One promising approach for connecting undergraduate content coursework to secondary teaching is using teacher-created representations of practice. Using these representations effectively requires seeing teachers' use of mathematical knowledge in the work of teaching. We argue that the dimensions of Rowland's (2013) Knowledge Quartet, especially Foundation and Contingency, form a fruitful framework for this purpose. We contribute an analytic framework to characterize the quality of mathematical knowledge observed in the Foundation and Contingency dimensions, developed using a purposive sampling from over 300 representations. These representations all featured geometry teaching. We showcase the framework with examples of "high" and "developing" Foundation and Contingency.Then, we compare our coding along these dimensions with performance on a measure of mathematical knowledge for teaching geometry. Finally, we describe the potential for generalizing this framework to other domains, such as algebra and mathematical modeling.

将本科内容课程作业与中学教学联系起来的一种有前途的方法是使用教师创建的实践表示。有效地使用这些表示需要看到教师在教学工作中对数学知识的使用。我们认为，罗兰 (2013) 的知识四重奏的维度，尤其是基础和应急，为此目的形成了一个富有成效的框架。我们贡献了一个分析框架来描述在基础和应急维度中观察到的数学知识的质量，该框架是使用从 300 多个表示中有目的的抽样开发的。这些表现形式都以几何教学为特色。我们用“高”和“发展中”的基础和应急的例子来展示这个框架。然后，我们将这些维度的编码与几何教学的数学知识衡量指标进行比较。最后，我们描述了将该框架推广到其他领域的潜力，例如代数和数学建模。