How do secondary students approach different types of division with remainder situations? Some evidence from Spain

Division-With-Remainder (DWR) problems are particularly complex, as suggested in many studies. The purpose of this work was to establish whether students’ difficulties in DWR problems came from an inadequate initial representation or from an inadequate final interpretation of the numerical answers, and whether remainders could be grouped into two blocks depending on the kind of answer, either directly matching the terms of the division or not. Forty-five Spanish secondary students, aged 12–13, were requested to solve two Types of Division Situations (i.e., Equal Groups and Comparison), each one involving four Types of Remainder (i.e., Remainder-Not-Divisible, Remainder-Divisible, Remainder-as-the-Result, and Readjusted-Quotient-by-Partial-Increments). Our data showed that: (a) the selection of the correct solution procedure depended on the Type of Division Situations, being easier in Equal Groups than in Comparison problems; (b) correct interpretations were higher than the percentages reported in other researches; and (c) success in problems whose answers were the quotient or the remainder was higher than in Readjusted-Quotient-by-Partial-Increments problems. The results obtained suggest that students’ difficulties originate in the initial representation of the DWR problems and that it would be more adequate to refer to the difficulty of Readjusted-Quotient-by-Partial-Increments problems in particular, rather than to the difficulty of DWR problems in general.