‘Ours is not to reason why, just invert and multiply’: an insight into Irish prospective secondary teachers' conceptual understanding of the division of fractions

Abstract

This paper reports on an investigation into the conceptual knowledge, through the domain of fraction division, of prospective secondary mathematics teachers in their final/fourth year of undergraduate initial teacher education. The aim of the study is to assess if levels of conceptual knowledge in this domain are sufficient in order for these prospective teachers to teach for understanding. Some impact of conceptual understanding on teaching approaches is also explored. The content of the teacher education program and its suitability for preparing prospective mathematics teachers is also considered. The results indicate that in spite of four years of mathematics teacher education, these prospective mathematics teachers hold a fragmented and limited conceptual understanding of dividing fractions. To address this issue and to ensure the effective implementation of Project Maths, it is argued that initial teacher education programs need to be revised to include an emphasis on school mathematics.

Danish Pre-service Teachers’ Mathematical and Didactical Knowledge of Operations with Rational Numbers

“…PSTs' understanding the meaning of a fraction as a part of whole leads them to some difficulties to explain the tasks on HTT 2. Some of them did not realise other interpretations for those task such measurement (or quotitive meaning), partitive meaning, and product and factors (Alenazi, 2016;Ball, 1990;Ma, 1999;Putra, 2016Putra, , 2019Slattery & Fitzmaurice, 2014). Some potential didactical techniques such as τd4, constructed directly from their mathematical knowledge, tend to be neglected because they consider that it is too abstract for the pupils.…”

“…Previous studies on how teachers deal with multiplication and division of rational numbers also show that teachers may succeed to solve operations of fractions themselves but still be unable to explain the reasons behind the used techniques (Alenazi, 2016;Ball, 1990;Erdem, Gökkurt, Sahin, Basibüyük, & Soylu, 2015;Ma, 1999;Putra, 2016Putra, , 2019Slattery & Fitzmaurice, 2014). Ma (1999) found that the U.S. teachers had various misconceptions about the meaning of division by fractions.…”

“…This means that they could not explain the meaning of the task based on several interpretations, such as measurement (or quotitive meaning), partitive meaning, and product and factors. Some of PSTs' mistakes on multiplication and division of fractions come from their perception that multiplication "makes bigger" and division "make smaller" (Slattery & Fitzmaurice, 2014;Tirosh, 2000).…”

Specialised Knowledge Network Activated in Teacher Education to Answer to a Mathematical Why on Fraction Division

“…On the other hand, it is understandable that the situation was more challenging from a mathematical point of view, considering that the literature indicates gaps in teaching knowledge about the division of fractions (Bayoud, 2011;Özel, 2013;Slattery & Fitzmaurice, 2014). Moreover, historically the contents of basic education (and its justifications) have lacked a systematic approach during initial training, with articulations between theory and practice, between school and academic mathematics, productively for teaching demands (Santos, 2005;Fürkotter & Morelatti, 2007;Nacarato & Passos, 2007;Moriel Junior & Cyrino, 2009;Fiorentini et al, 2016;Crecci et al, 2017).…”

“…In these situations, the teacher should not only know the mathematical justification itself but also know how to teach it respecting the student's level (Peterson, 1972;Lorenzato, 1993;Nobre, 1996;Moriel Junior & Wielewski, 2013;Leikin et al, 2017). However, there are indications that there are usually gaps in knowledge and teacher education (Lorenzato, 1993;Fiorentini, 2005;Santos, 2005;Angelo et al, 2009), and often the teachers have the same difficulties as the students (Bayoud, 2011;Özel, 2013;Slattery & Fitzmaurice, 2014).…”