Spontaneous problem posing (SPP) is presented as a phenomenon central to mathematical thinking, wherein learners generate problems without formal prompting, and where posed problems originate from the learner’s motivation to improve their knowledge. Because of this, they may serve as important markers of productive mathematical engagement—in particular, affective engagement—both for problem posers and their classroom communities (D’Mello & Graesser in Learning and Instruction, 22(2), 145–157, 2012). We report on a study which utilized mixed methods to examine SPP and associated affective engagement of students in four early high school mathematics classrooms in two geographic regions of the USA. For each classroom, we used observational and experience sampling methods to examine the patterns of affect problem posers and their peers experienced at individual and group levels, respectively, on a day with at least one instance of SPP observed compared with a day with no observed instances of SPP. Results show evidence of positive affect among problem posers, while their peers reported fewer negative emotions about mathematics tasks on days with SPP than on days without SPP. Moreover, a detailed analysis of two spontaneous problem posers revealed that they showed a desire to extend their own mathematical thinking, expressed dissatisfaction with their current knowledge, and directed their problems primarily towards the teacher. Results are discussed regarding the need to support students’ confidence in their original mathematical ideas in class, and for reducing negative emotional responses to mathematics tasks.

中文翻译：

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高中数学课堂中表现出自发问题的参与和影响模式：一个探索性框架和研究

自发问题提出 (SPP) 是数学思维的核心现象，其中学习者在没有正式提示的情况下产生问题，并且提出的问题源于学习者提高知识的动机。正因为如此，它们可以作为富有成效的数学参与的重要标志——特别是情感参与——无论是对问题提出者还是他们的课堂社区（D'Mello & Graesser in Learning and Instruction, 22(2), 145–157, 2012 ）。我们报告了一项研究，该研究使用混合方法来检查 SPP 和学生在美国两个地理区域的四个早期高中数学课堂中的相关情感参与。每间教室，我们使用观察和经验抽样方法分别检查了在观察到至少一个 SPP 实例的一天与没有观察到 SPP 实例的一天相比，在个人和组级别上遇到的情感问题提出者及其同龄人的模式。结果显示问题提出者之间存在积极影响的证据，而他们的同龄人在有 SPP 的日子里对数学任务的负面情绪比没有 SPP 的日子少。此外，对两名自发问题提出者的详细分析表明，他们表现出扩展自己数学思维的愿望，对现有知识表示不满，并将问题主要指向老师。结果讨论了需要支持学生对课堂上原始数学思想的信心，