Abstract:

Not-knowing is an underexplored concept defined by an individual’s ability to be aware of what they do not know as a means to plan and more effectively face complex situations. Mason and Spence (1999) state that “awareness of knowing and of not knowing is crucial to successful mathematical thinking.” However, little research has been done to understand the not-knowing phenomenon. While the importance of not knowing as a step to knowing may be clear, the ability to be aware of the not knowing in the moment may not be an easy task. Therefore in this study, we are examining the following research questions: how do students express what they do not know while solving geometric reasoning tasks? And, what challenges do they face in externalizing the not-knowing?

Ten students were selected for the study at a university in the southwestern border region of the United States. These students were enrolled in a teacher education course on Geometric Reasoning, which focused on problem-solving. Think aloud protocol was used to record student not knowing in the process of problem-solving. Data sources consisted of audio recording, reflections, and semi-structured interviews. Through audio recordings, written reflections, and interviews, we aimed at capturing student externalization of not-knowing in more than one way. In order to analyze the transcriptions, reflections, and interviews, meaning coding, meaning condensation and interpretation techniques were used (Kvale & Brinkmann, 2009) as the main methods of analysis. After an initial reading, we extracted most frequently occurring codes. Then, we categorized the codes. And finally, we grouped categories in emerging themes.

The results clearly demonstrated that the participants had difficulties expressing their not-knowing. The analysis revealed the following four major themes:
1) deflection,
2) pressure,
3) lack of heuristic sense, and
4) fractured prior knowledge.
Deflection of not-knowing was identified as avoidance of challenge when an individual shifts the focus of their not-knowing somewhere else besides themselves. Participants also expressed pressure through direct vocalization, frustration, or sense of urgency while externalizing their not knowing in the process of problem-solving. Concerning the lack of heuristic sense, most of the participant attempted the tasks through trial and error. Students may become “tunnel visioned” in the process of trial and error, clouding their awareness of not-knowing. Whereas lack of heuristic sense deals with strategy, fractured knowledge is present when a participant may have prior knowledge gaps within the given topic, have misunderstandings of the topic, or simply lack the prior knowledge required for the given topic. We observed that if a student has fractured knowledge, it will directly impede her ability to invoke not-knowing.

Understanding these themes can better assist in minimizing the problem while maximizing the potential of not-knowing as a step to knowing and understanding. Students that were better able to express their not knowing ultimately demonstrated a stronger understanding of the concept by the end of the course. These findings might serve as a stepping-stone to further research not-knowing, as it may directly link to more effective and efficient student learning.

Keywords:

Not-knowing, secondary school mathematics teachers, geometric reasoning.