The development of learners’ spontaneous multiplicative and additive reasoning in mathematically-neutral schematic problems

http://www.ru.nl/bsi/expert-meeting/programmeWe studied the development of learners’ spontaneous multiplicative and additive Quantitative Analogical (QA) reasoning. We offered 366 third, fourth, fifth and sixth graders in primary education four schematic problems in which three values were given and a fourth one had to be found. All problems had a multiple-choice and multiple-answer format, offering a multiplicative, additive and distractor solution. The schematic problems were all neutral in terms of their mathematical model, but differed with respect to their schematic representation (i.e. twodimensional arrow scheme or onedimensional number line) and the number characteristics of the given numbers in the problem (i.e. integer or non-integer number ratios). Our findings showed that the majority of all answers was additive, but also multiplicative and even multiplicative-and-additive answers were given. Moreover, multiplicative and multiplicative-and-additive answers increased with grade, whereas additive answers decreased with grade. Also the problem characteristics played a role, in the sense that arrow schemes evoked more multiplicative and less additive answers than number line representations, just like problems with integer number ratios compared to problems with non-integer ratios. These findings highlight the importance of spontaneous additive reasoning, next to spontaneous multiplicative reasoning, as another way of QA reasoning. Future research should further study the interplay of spontaneous multiplicative and additive QA reasoning in several kinds of mathematically-neutral problems and across grades.status: publishe