The use of cross multiplication and other mal–rules in fraction operations by pre-service teachers

Highlights

• This study reports on mal–rules (mainly related to cross-multiplication) used by PST’s in fraction operations..

• The interviews revealed the PST’s strong conviction of whole number operations applied separately to parts of the fraction.

• The use of theorems-in-action provides useful insights into the rules as well as properties about some of the rules.

Abstract

Many studies show that prospective teachers often have misconceptions about fractions. In this case study, we report on some of the mal–rules used by a group of 60 prospective South African primary school teachers. The students’ written responses to two items focusing on addition and multiplication of fractions which formed part of an assessment, were analyzed. Semi-structured interviews were also used to elicit the reasoning used in the students’ calculations. Less than half of the participants completed both items correctly, and many of the other students displayed various mal–rules. To interpret the pre–service teachers’ misconceptions, we studied the rules used by the participants, and expressed them as theorems–in–action. An interesting mal–rule governing the multiplication of fractions was the widespread ‘cross multiplication’ rule which after some mutations led to other mal–rules, illustrating how students’ misconceptions can persist many years after their initial learning.

Introduction

It is well recognized that teachers require a deep and connected understanding of the mathematics concepts that they will be teaching. Consequently, issues about mathematics knowledge for teaching has been a focus of much interest in many different countries (Ball, Thames, & Phelps, 2008; Bansilal, Brijlall, & Mkhwanazi, 2014; Beswick, Callingham, & Watson, 2012; Chinnappan & Forrester, 2014; Kar & Isik, 2014; Venkat & Spaull, 2015). The Conference Board of the Mathematical Sciences (2012), p.23) emphasize that it is not sufficient for “teachers to rely on their past experiences as learners of mathematics” and recommend that mathematics courses for elementary teachers should not only focus on remedying “weaknesses in mathematical knowledge” but should also help teachers develop a deeper understanding of the mathematics they will teach. Mathematics teachers need to be able to explain not only how to compute certain procedures, but why certain procedures make sense in a context, and whether certain procedure result in the same result and why they do so (Lovin, Stevens, Siegfried, Wilkins, & Norton, 2018).

The South African context adds an additional dimension to the concerns articulated about teacher knowledge in mathematics. Since the onset of the democratic era, the higher education sector has expanded rapidly. Statistics shared by the South African Institute of Race Relations (SAIRR) showed that in 2013 approximately 984 000 students were enrolled in higher education (South African Institute of Race Relations, 2016), growing from 495 000 in 1994 (Ramrathan, 2016). These numbers show that the sector has almost doubled in the post-apartheid period; this can be seen as a positive development in trying to overturn inequitable apartheid practices. However this rapid growth and large demand for qualified teachers, has meant that many of the students who enter the teaching field are below average students, and may have weak subject content knowledge (Bowie & Reed, 2016; Deacon, 2016; Taylor, 2014). This poor knowledge background of the future teachers, present a challenge to teacher education institutions who must try to break the “cycle of mediocrity, where school leavers, [high school graduates] who were themselves poorly taught are returned to the schools as poorly prepared teachers” (Deacon, 2016, p.25). Studies focusing on mathematics preservice teachers (PST’s) in South Africa confirm that many PST’s do not have a sufficiently strong background in basic mathematics (Bansilal et al., 2014; Bansilal, Brijlall, & Trigueros, 2017; Ndlovu, Amin, & Samuel, 2017). Although there are many studies in South Africa about pre–service and practicing teachers’ poor knowledge, most have been about high school teachers and few have focused on primary school concepts such as fraction operations.

This case study focuses on a group of primary school PST’s enrolled in a 4–year Bachelor of Education (B.Ed.) programme at a selected South African university. These students, many of whom struggled with mathematics at school, were enrolled in a foundational primary mathematics course in their first year, after which they could go on to study other primary school subject specializations, which may or may not include mathematics. The purpose of this study is to better understand the misconceptions that the PST’s hold about fraction operations. It is hoped that this study can shed light on the problem of under–prepared PST’s in South Africa and, more generally, provide a deeper insight about particular misconceptions held by PST’s about fraction operations.