Making sense of instruction on fractions when a student lacks necessary fractional schemes: The case of Tim

An analysis of elementary school children’s fractional knowledge depicted with circle, rectangle, and number line representations

“…While the teacher flexibly partitioned a unit on the number line (0-1 interval) and did change the location of B1^when necessary, a student understood this use of representation differently and constructed a misconception that Bn/n^and B1^on the number line could be at two different locations (Izsák et al, 2008). In another case study that Olive and Vomvoridi (2006) conducted with a 6th grader, they reported that the child in the study could use fraction symbols to name unit fractional parts of rectangular shapes successfully. However, when it comes to adding two unit fractions and showing the result with rectangular shapes, it was quite different than adults' perspective.…”

Learning about the numerator and denominator in teacher-designed lessons

“…This shows once more that the teaching and learning of fractions are complex enterprises, and that some pupils do not learn even when their teacher has deliberately planned to overcome difficulties. This study adds to previous research by identifying and viewing pupils' difficulties from the point of view of what dimensions of fractions the learners have or have not discerned or differentiated between (cf., Olive and Vomvoridi 2006). Within variation theory learners need to discern critical aspects in order to learn something specific, in this case operating with non-unit fractions of a discrete quantity.…”

“…Another study by Olive and Vomvoridi (2006), investigating a single student's (Tim) learning of fractions in the 6th grade, showed that at the beginning of the intervention he did not differentiate between the fractional whole (6/6) and the unit fraction (1/6). He called a circle partitioned into six parts both one-sixth and six-sixths.…”

“…A teacher can assume that it may be necessary for the learner to discern what the denominator and numerator indicate; what the whole is; that the parts the whole are partitioned into must be equal etc. However, if a pupil says during a lesson or interview that 1/4 is left when 1/5 is taken from a whole, it is likely that this pupil does not discern the meaning of the numerator and denominator, and does not realise that the denominator does not change even if one part is taken away, it is still fifths (Olive and Vomvoridi 2006). It is critical for the pupil to discern this aspect of fractions and differentiate it from other aspects.…”

“…One explanation is that the continuous model of fractions causes problems when pupils meet fractions of discrete quantities since they have difficulty identifying appropriate units (Mack 1993). Approaches to studying pupils' learning of fractions include interviews with pupils (Watanabe 1995), following particular pupils' learning trajectories during instruction (e.g., Hackenberg 2007; Olive and Vomvoridi 2006;Steffe and Olive 2010;Steffe 2004;Tzur 2004), or classroom studies of how pupils understand fractions (e.g., Ball 1993;Empson 1999). Empson (2011) argues that what children learn is sensitive to the context in which they learn, which is constituted of many factors, including most immediately the types of instructional tasks and how teachers organize students' engagement with these tasks (Empson 2011).…”