Abstracting by constructing and revising a ‘partially correct construct’ : a case study

This study draws on data from a broader video-stimulated interview study of the role of optimism in collaborative problem solving. It examines the activity of a Grade 5 student, Tom, whose initial constructing activity resulted in a &lsquo;Partially Correct Construct&rsquo;. Insistent questioning from another group member pressuring for clarification led to Tom developing a &lsquo;more correct construct&rsquo; with further potential for revision. This paper raises questions about influences that can stimulate or inhibit construct refinement.<br /

Empirical generalisation as an inadequate cognitive scaffold to theoretical generalisation of a more complex concept

The impact of prior learning on new learning is highlighted by the case of Dean, a Year 8 student who developed his own method to find the sum of the interior angles of a polygon without knowing why his method worked. Enriched transcripts and visual displays of the cognitive, social (Dreyfus, Hershkowitz, &amp; Schwarz, 2001) and affective elements (Williams, 2002) of Dean\u27s interrupted abstraction process informed the identification of factors that inhibited Dean\u27s constructing process. It was found Dean possessed an empirical, not theoretical, generalization (Davydov, 1990) about sums of interior angles of triangles that was an inadequate cognitive artifact for constructing the new more complex theoretical generalization. The study suggests use of tasks designed with the opportunity develop assumed knowledge in conjunction with new concepts.<br /

Consolidating one novel structure whilst constructing two more

This study reports the cognitive processing of a Year 8 female student (Kerri) during a test, and during her subsequent homework as she consolidated as part of abstracting; a topic of recent research interest. This case adds to the body of knowledge about how constructing and consolidating can occur simultaneously. The analysis captured the complexity of the cognitive processing, and their intertwined nature. Data was captured through lesson video, and post-lesson, video-stimulated reconstructive student interviews. It was found that Kerri&rsquo;s constructing and consolidating included characteristics previously identified by others, and an additional feature. Her constructing included &ldquo;branching&rdquo; (Kidron &amp; Dreyfus, 2004, p. 159); but unlike the case cited, one of these branches related to a new goal.<br /

Student-created tasks inform conceptual task design

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How group composition can influence opportunities for spontaneous learning

Classroom video, and video-stimulated interviews of small group work, in a Grade 5/6 classroom are used to show ways group composition can influence learning opportunities. Vygotsky&rsquo;s (1933/1966; 1978) learning theory on the spontaneous creation of knowledge as compared to the guidance of an expert other frames this group analysis. Illustrations from two groups show how opportunities to spontaneously create new knowledge can be limited or enhanced by psychological factors associated with the inclination to explore that have been linked to resilience in the form of optimism (Seligman, 1995, Williams, 2003). This study contributes to our knowledge on forming groups to promote deep learning. It raises questions about other ways in which learning may be influenced by optimistic orientation and about building this personal characteristic to enable deep learning.<br /

The nature of spontaneity in high quality mathematics learning experiences

Spontaneity has been linked to high quality learning experiences in mathematics (Csikszentmihalyi &amp; Csikszentmihalyi, 1992; Williams, 2002).This paper shows how spontaneity can be identified by attending to the nature of social elements in the process of abstracting (Dreyfus, Hershkowitz, &amp; Schwarz, 2001). This process is elaborated through an illustrative example&mdash;a Year 8 Australian male student who scaffolded his learning by attending to images in the classroom that were intended for other purposes. Leon&rsquo;s cognitive processing was not &lsquo;observable&rsquo; (Dreyfus et al., 2001) in classroom dialogue because Leon &lsquo;thought alone&rsquo;. Post-lesson videostimulated reconstructive interviews facilitated study of Leon&rsquo;s thought processes and extended methodological techniques available to study thinking in classrooms.<br /

Impetus to explore: approaching operational deficiency optimistically

&lsquo;Impetus to explore&rsquo; is studied through post-lesson video-stimulated student interviews. This impetus focuses &lsquo;spontaneous&rsquo; decisions to explore unfamiliar mathematics. Uncertainty, quests for elegance and curiosity have been found to contribute to this impetus. This study identified other contributing factors: inability to undertake the teacher&rsquo;s task, &lsquo;optimistic explanatory style&rsquo; (Seligman, 1995), and identifying relevant complexities (Williams, 2002). Explanatory style affects how a person perceives successes and failures. Enactment of optimism in this study illuminated its role in creating an impetus to explore. Re-conceptualising &lsquo;operational deficiency&rsquo; (Chen &amp; Siegler, 2000) to include absence of resources to undertake a task illuminated a task design feature that can increase exploration.<br /