Abstract:

Division-With-Remainder (DWR) problems have caught the attention of researchers because they have proved to be particularly complex problems. One of the peculiarities of DWR problems is that although students do not have difficulties in establishing that DWR problems are division problems or in correctly executing the division, they tend to respond incorrectly because they do not interpret the numerical answer correctly according to the terms of the problem situation. Some authors like Silver and collaborators (Cai & Silver, 1995; Li & Silver, 2000; Silver, Shapiro & Deutsch, 1993) assumed that children´s unsuccessful performance in these problems came from an inadequate final interpretation of the numerical answers. However, this approach could not take into account student´s answers based on the selection of an incorrect algorithm. Furthermore, we think that children may discover the correct solution procedure by several routes which may not necessarily imply adequate initial representation of DWR problems (i.e., based on superficial aspects of the text such as key-words or key-expressions, familiarity…) that led children to construct flawed and/or fragmented initial representations without being hindered from arriving at the appropriate algorithmic solution. Moreover, this theoretical model did not account for the fact that students’ difficulties changed depending on the Types of Remainder. The purpose of this work was, firstly, to check if children´s difficulties stemmed from an inadequate final interpretation of the numerical answers, as Silver and cols. assumed, or they are due to an inadequate initial representation of the problem. Secondly, our study wanted to determine whether types of remainder could be grouped into two blocks: One of them related to those answers directly referred to one of the terms of the division and the other group linked with the answers that did not consist of the quotient or the remainder. We requested forty Spanish secondary students with a mean age of 12;10 years to solve two Types of Division Situations (i.e, Partitive and Measurement Equal-Groups problems), each one involving four Types of Remainder (i.e, Remainder Divisible -RD- , Remainder not Divisible –RND-, Remainder as the Result –RR-, and Readjusted Quotient by Partial Increments –RQPI-). The results showed that: (I) although most students tended to choose division as the resolution procedure in both types of problems, Partitive ones were easier than Measurement ones; and (II) children´s performance was better when they faced answers related the quotient or the remainder (i.e., RD, RND, RR) than when they did RQPI problems. In conclusion, our data suggest that students´ main difficulty arises in the initial representation of the DWR problems as well as that it would be more appropriate to speak of the difficulty of RQPI problems rather than of the difficulty of DWR problems.

Keywords:

Division with remainder problems, partitive division, measurement division.