# Remediation of errors with mathematical algorithms

Manson, AJ ORCID: 0000-0001-7171-990X 2019 , 'Remediation of errors with mathematical algorithms', PhD thesis, University of Tasmania.

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## Abstract

This thesis is concerned with students’ fraction understanding and the effects of remedial instruction on understanding, computational skills, and self-efficacy. The study makes the extent of fraction misconceptions among secondary students visible, supporting the literature that achieving a depth of understanding in fractions is both complex and difficult. Students do not construct meaning in isolation; rather, they try to make sense of new ideas based on what they already know. As students grapple with the conceptual development of fractions from natural numbers, they must also make sense of the complex manipulation of procedures. Students often do not remember which procedural operation to use when doing fraction computation and, coupled with a lack of deep understanding, often do not experience success with fractions. As a result, students become despondent about their ability and achievement in the topic, leading to low self-efficacy.
Past research has revealed that inappropriate application of prior knowledge causes an interference effect, which can result in erroneous procedures. The interference effect, known as proactive inhibition (Underwood, 1957), impacts on learning and memory by conflicting associations of prior learning. This thesis explores the effect proactive inhibition has on the learning of fractions and the effect is used to explain how inappropriate prior knowledge results in the misapplication of fraction procedures. Our knowledge of typical errors in fraction computation enables us to identify students who have difficulties performing standard fraction operations.
This large-scale study was conducted in an authentic school setting with students from years seven, eight, and nine (n=361) participating. Drawing on the literature about fraction misconceptions, an instrument was developed to expose fraction errors and to allow the diagnosis of repeat error patterns. The research confirmed the commonality of certain fraction misconceptions and highlights a lack of conceptual understanding.
Students identified as having misconceptions (n=83) were invited to participate in one of two remediation programs. One program, the Old Way / New Way technique (Lyndon, 1989), designed to counteract the effect of proactive inhibition, brings the learner’s “old way” to a conscious level and exchanges it for a “new way” by means of discrimination learning. The effectiveness of this method was examined, in comparison to a traditional reteaching technique. The programs ran concurrently for five weeks, with two sessions per week. The effectiveness of the two intervention strategies was determined through the analysis of pre-, post- and delayed retention test results. Pre- and post- self-efficacy was also examined, to determine the effect the intervention programs had on students’ confidence in their ability to perform fraction algorithms.
The Old Way / New Way (O/N) intervention students gained significantly better pre-post results; however, the effectiveness of the O/N method was not maintained in the delayed retention test. Although the students verbally reported having more confidence after the intervention, this was not fully reflected in the self-efficacy scale. The three psychological domains of functioning were examined in the self-efficacy scale. Questions related to the affective domain examined students’ internal belief system; awareness of their own mathematical knowledge was examined in the cognitive domain questions; and questions about the conative domain looked at students’ striving and level of focused attention in mathematics. Self-efficacy improved significantly in the conative domain pre-post for both intervention groups; however, there was no change in the cognitive domain and reliability was not able to be achieved for the affective domain scores.
Results of this research highlighted the interference effect of prior knowledge, with students not remembering procedures for specific operations and lacking the conceptual understanding to support their responses. Although students want to apply an operational procedure, they often do not have good recall of appropriate procedures, due to the interference effect. Remediation was shown to have an impact on the learning of fractions and to improve self-efficacy, with the Old Way / New Way method yielding significant short-term gains, but such remediation may need to be conducted regularly for long-term impact. This thesis discusses what can be achieved with fraction remediation and discusses whether techniques such as the Old Way / New Way intervention might be able to be used more widely in mathematics.

Item Type: Thesis - PhD Manson, AJ mathematics, fractions, self-efficiency, proactive inhibition Copyright 2019 the author View statistics for this item