In mathematics there is a conceptual shift from additive to multiplicative reasoning that learners in the Intermediate Phase (Grades 4 to 6) need to make to understand more complex mathematical concepts in secondary school. The aim of this study was to explore and describe the multiplicative proficiency of Grade 6 learners with learning difficulties by exploring the current status of their conceptual knowledge of multiplication, as well as their level of procedural fluency and the nature of their strategic competence. A single-case study design was used, with fifteen Grade 6 learners selected from three special needs schools in Pretoria, South Africa, forming together a group as the unit of analysis. During individualised task-based interviews, participants were required to solve ten classes of multiplication problems. They were further required to solve the problems by using different representations, namely abstract, semi-concrete and concrete representations for a multi-dimensional approach that allowed for in-depth understanding of their reasoning.
The findings of this study indicated that only a few participants made the conceptual shift from additive to multiplicative reasoning, mainly when answering the less cognitively complex questions, since they showed conceptual understanding of and procedural fluency in the way they dealt with those questions. However, only two of the participants answered the less cognitively complex questions in a way that demonstrated proficiency in multiplicative reasoning and showed conceptual understanding, procedural fluency and strategic competence. The majority of the participants were not proficient in multiplicative reasoning and did not make the shift from additive to multiplicative reasoning, especially for the more cognitively complex questions. They still thought in additive terms and struggled to solve cognitively complex multiplication problems. However, some of the participants could solve these problems with either semi-concrete or concrete representations, but not with abstract representations. More participants showed procedural fluency in solving classes of problems they had already learned to solve, even if they were cognitively more complex. The three main reasons identified for this lack of proficiency were misconceptions, misrepresentations and calculation errors, which could inform mathematics teachers’ instructional practice to help learners make the shift from additive to multiplicative reasoning.