ERRORS IN USING CONVERGENCE TESTS IN INFINITE SERIES - Errors in Using Convergence Tests - Pre-service Elementary Mathematics Teachers

Oct 15, 2026

Uploaded by Aditya

The present study identifies errors made by pre-service elementary mathematics teachers when investigating the convergence of infinite series. Using a qualitative exploratory case-study design with 43 undergraduate participants, the research collects data from a paper-and-pencil test containing seven open-ended questions. Descriptive and content analysis techniques summarize findings, including inappropriate test selection, failure to check criteria, comparison-test and limit-comparison errors, and misconceptions about series behavior. Results also contrast procedural versus conceptual knowledge effects and emphasize balancing both to reduce learning difficulties and support deeper mathematical understanding.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages15

Reading Time38 min

AccessFree to download

Introduction

Errors in Students’ Use of Convergence Tests

Conceptual vs Procedural Knowledge

Related Research and Motivation

ERRORS IN USING CONVERGENCE TESTS IN INFINITE SERIES Murat GENÇ, Mustafa AKINCI Abstract: The present study aimed to identify the errors made by pre-service elementary mathematics teachers while investigating the convergence of infinite series. A qualitative exploratory case study design was used with a total of 43 undergraduate students. Data were obtained from a test administered in a paper-and-pencil form consisting of seven open-ended questions. The data analysis was done using descriptive and content analysis techniques. Findings were presented as follows: inappropriate test selections; failure to check convergence criteria; incorrect use of a comparison test; limit comparison test error; re-test convergence test results; considering ∑ as a multiplicative function; misunderstanding of special series; considering that series has no character when the convergence test is inconclusive; confusing sequences with series; misunderstanding of the nth-term test; misinterpretation of convergence test results. Findings showed that students withinsufficient procedural knowledge had difficulty in solving the given problem even if they understood it, whereas those with insufficient conceptual knowledge could not literally understand what they did even if they solved the problem. Therefore, the establishment of a moderate balance between procedural and conceptual knowledge in the learning of the convergence of series is essential in reducing the errors or learning difficulties for developing deep mathematical understanding. Keywords: infinite series, convergence tests, errors, procedural knowledge, conceptual knowledge 1. Introduction In today’s developing world, mathematics in many areas is becoming increasingly important for individuals, communities, and nations to function across the globe. Because of the importance of mathematics in such a wide range of fields, the difficulties and misunderstandings of students studying mathematics, science, and engineering, especially in calculus courses, have been investigated and reported by several researchers and mathematics educators (Aspinwall & Miller, 2001; Davis & Vinner, 1986; Martin, 2009; Martínez-Planell, Gonzalez, DiCristina & Acevedo, 2012; Monaghan, 2001; Orton, 1983; Tall & Schwarzenberger, 1978) . Some of the studies have shown that students taking calculus courses have difficulty in conceptual understanding because they have a very superficial and insufficient understanding of the basic concepts of mathematical analysis due to the rote-based learning instruction (Steen, 1998; White & Mitchelmore, 1996) . The introduction to calculus begins with the concept of limit, but this concept is inherently difficult, and no matter how it is taught many students have several difficulties in understanding the definition of limit intuitively (Barnes, 1995) . These difficulties mostly stem from the inability to understand the notion of infinity or a fairly complex definition of the limit conceptually (Cornu, 1991, Tall, 1992; Tall & Vinner, 1981) . Many students also experience similar difficulties in understanding the convergence of an infinite series which is one of the important concepts of analysis (Akgün & Duru, 2007) . These difficulties have been said to increase further along with the inadequacy in algebra (Brown, 1996) . Sequences and series are the most fundamental topics in any analysis course and can serve as a basis for the other topics in calculus including limits, continuity, derivatives, and integrals. For example, the topic of series, which is one of the important subjects in the process of calculating the area under a curve, can also offer alternative ways to model some mathematical relations such as the distribution of drugs and populations. (González-Martín, Nardi, & Biza, 2011) . On the other hand, Tall and Schwarzenberger (1978) argued that the subject of the series is one of the significant anomalies of the mathematics curriculum. Many students encountering

FAQs

What is the main purpose of the study?

To identify errors made by pre-service elementary mathematics teachers when investigating the convergence of infinite series.

How was the data collected in the research?

Through a paper-and-pencil test with seven open-ended questions administered to 43 undergraduate participants.

What kinds of errors were reported in using convergence tests?

Errors included inappropriate test selection, not checking convergence criteria, incorrect comparison test use (including limit comparison), and multiple misunderstandings about special series, sequences versus series, and interpreting convergence results.

Oct 15, 2026

Uploaded by Aditya