Exploring Preservice Mathematics Teachers’ Cognitive Journey In Understanding Convergence Of Infinite Series Using Direct Comparison Test

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Conference Proceeding

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The concept of convergence or divergence of infinite series (IS) is one of the topics in Calculus that most students struggle to understand. Martinez-Planell et al. (2012) found that students have two different constructions of series as an infinite process of adding numbers and series as a sequence of partial sums. As part of an ongoing dissertation project, the current presentation will focus on the preliminary data-gathering activity by the researcher. It aimed to model the cognitive journey of preservice mathematics teachers (PSMTs) as they vertically reorganize their previous constructs into a new mathematical structure while studying the convergence or divergence of IS using the direct comparison test (DCT). Eight PSMTs who had not yet been formally introduced to the topic were given a task to solve, followed by individual interviews. The researcher used a qualitative approach and inductive content analysis based on the RBC-Model for Abstraction in Context (Dreyfus et al., 2015).

Preliminary results showed that the PSMTs had difficulty constructing the notion of convergence using DCT due to their prior constructs in the concept of convergence of an IS. Some PSMTs had a notion that the limit of the partial sums approaches positive infinity when it increases in value, even though the graph of the corresponding partial sums is asymptotic to a horizontal line, or it approaches a finite value. They also had a notion that a sequence of corresponding partial sums should have a defining formula to be considered convergent. Their difficulty in understanding the idea of an IS converging to a limit is due to the subtle shift in thinking about the terms of the series to thinking about the overall sum. Hence, it resulted in the difficulty of using the DCT, even though they had recognized the convergence or divergence of the other given IS used for comparison. The findings of the current study provided basis for reflection by the researcher in conjecturing specific activities that would help students construct a deeper understanding of the convergence or divergence of IS.

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