Teachers whose mathematical meanings support understanding across different contexts are likely to convey them in productive ways for coherent student learning. This exploratory study sought to elicit 67 secondary mathematics pre-service teachers’ (PSTs) meanings for quadratics with growing pattern creation and multiple translation tasks. A framework for categorizing their patterns, in terms of structural correspondence to quadratic equation forms, is shared. PSTs’ translations and explanations were analyzed using an existing framework for representational fluency. Evidence was found for some PSTs applying prior productive meanings to an unfamiliar context and others discovering new meanings. It was found that some PSTs were not able to draw on their prior knowledge of quadratic functions successfully, and possible reasons for this are explored. Successful figural pattern creation was found to be associated with higher representational fluency. Seven PSTs were also interviewed to probe their cognitive and affective experiences with a “new” type of quadratics task; several described “aha” moments and used emotional languages like “horror” and “delight”. Implications for learning to teach quadratic functions, creative visualization tasks and future research are discussed.

中文翻译：

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以新的眼光看二次方程：中学数学职前教师对图形生长模式的创造

数学意义支持跨不同背景理解的教师可能会以富有成效的方式传达这些意义，以促进学生的连贯学习。这项探索性研究试图引出 67 位中学数学职前教师 (PST) 对具有不断增长的模式创建和多项翻译任务的二次方程的含义。共享一个用于根据二次方程形式的结构对应对其模式进行分类的框架。PST 的翻译和解释使用现有的表达流畅性框架进行分析。有证据表明，一些 PST 将先前的生产意义应用于不熟悉的上下文，而另一些则发现了新的意义。发现一些 PST 无法成功利用他们对二次函数的先验知识，并探讨了可能的原因。发现成功的图形模式创建与更高的表现流畅性有关。七名 PST 也接受了采访，以探索他们对“新型”二次方任务的认知和情感体验；一些人描述了“啊哈”时刻，并使用了“恐怖”和“喜悦”等情感语言。讨论了学习教授二次函数、创造性可视化任务和未来研究的意义。