Knowledge and confidence of pre-service mathematics teachers: the case of fraction division

Abstract

To make teacher preparation and professional development effective, it is important to find out possible deficiencies in teachers' knowledge as well as teachers' own perceptions about their needs. By focusing on preservice teachers' knowledge of fraction division in this article, we conceptualize the notion of pre-service teachers' knowledge in mathematics and pedagogy for teaching as containing both teachers' perceptions of their preparation and their mathematics knowledge needed for teaching. With specific assessment instruments developed for pre-service middle school teachers, we focus on both pre-service teachers' own perceptions about their knowledge preparation and the extent of their mathematics knowledge on the topic of fraction division. The results reveal a wide gap between sampled pre-service middle school teachers' general perceptions/confidence and their limited mathematics knowledge needed for teaching fraction division conceptually. The results suggest that these pre-service teachers need to develop a sound and deep understanding of mathematics knowledge for teaching in order to build their confidence for classroom instruction. The study's findings indicate the feasibility and importance of conceptualizing the notion of teachers' knowledge in mathematics and pedagogy for teaching to include teachers' perceptions. The applicability and implications of this expanded notion of teachers' knowledge is then discussed.

A Comparative Analysis of Division in Elementary Mathematics Textbooks in Korea and Japan

“…For instance, Li and Kulm (2008) reported that students had difficulties in learning division of fractions and solved problems just by applying an "invert and multiply" algorithm without understanding its meaning. Regarding the division of decimal numbers, students usually focused on dividing vertically by moving a decimal point without knowing why it works (Sherman et al, 2015).…”

La enseñanza inicial del concepto de fracción por maestros en formación

“…Entre los distintas investigaciones realizadas sobre el conocimiento didáctico del contenido, encontramos diversos estudios sobre fracciones (Charalambous, Hill y Ball, 2011;Domoney, 2001;Fuller, 1996), centrados en las operaciones con fracciones (Li y Kulm, 2008;Isiksal y Cakiroglu, 2011;Charalambous, Hill y Ball, 2011) o en la equivalencia de fracciones (Chick, 2003;Marks, 1990). Algunos de estos estudios (D'Ambrosio y Mendoça, 1992;Domoney, 2001;Marks, 1990) sugieren que es necesario el diseño de instrumentos adecuados para el estudio del conocimiento didáctico del contenido, pues los empleados en estudios previos presentaron algunos inconvenientes.…”

“…Un logro de este estudio radica en que, a través de un instrumento aparentemente sencillo, nos pudimos aproximar a este tipo de conocimiento salvando las dificultades de otros estudios (Charalambous, Hill y Ball, 2011;Li y Kulm, 2008), en los que las carencias en el conocimiento del contenido sobre fracciones incidió en sus resultados. A pesar de que otros estudios utilizan ítems similares "¿Cómo explicarías las fracciones a alguien que no sabe lo que son?"…”

“…Entre las distintas investigaciones realizadas sobre el conocimiento didáctico de las fracciones (Charalambous, Hill y Ball, 2011;Domoney, 2001;Fuller, 1996), encontramos estudios centrados en las operaciones con fracciones (Li y Kulm, 2008;Isiksal y Cakiroglu, 2011) o en la equivalencia de fracciones (Chick, 2003;Marks, 1990). Todos ellos resaltan las carencias que los maestros en formación presentan en sus conocimientos sobre la relación multiplicativa parte-todo y las fracciones y, por consiguiente, en las implicaciones para su enseñanza.…”

Mathematics teacher preparation examined in an international context: learning from the Teacher Education and Development Study in Mathematics (TEDS-M) and beyond

“…Further efforts are needed to examine the nature of teacher expertise valued in different education systems, and to identify common knowledge elements that are important for teachers across education systems as well as those cultural-specific manifestations of common knowledge elements and unique elements. Moreover, examination of pre-service teachers' knowledge needed for teaching may call for assessments different from those of in-service teachers (e.g., Li and Kulm 2008).…”