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We use numbers and fractions every day, for example when we are doing our shopping or baking a cake. But mathematics is, of course, much more: it is the language of science, or, to use Galileo's words, “the book of Nature is written in mathematical language” (Galileo, 1623) and some mathematical competencies beyond basic arithmetic are required in most professions. Basic mathematics, i.e., elementary arithmetic, elementary geometry and some elements of calculus, is taught in school, not just for everyday life, but as a tool for many different professions. In school, however, mathematics is either “loved” or “hated”, as Hersh and John-Steiner masterfully describe in their book “Loving and Hating Mathematics” (Hersh and John-Steiner, 2010). Research in mathematics education has definitely contributed to reducing school students' hatred of mathematics and this reduction may be seen as one of its many goals.
In contrast with mathematics, the field of mathematics education is strongly interdisciplinary; the closest field to influence it directly is psychology. In fact, mathematics education is consistently shaped by both behavioral and cognitive perspectives, since so many factors—the power of visualizations, the effect of representation formats, but also factors like gender, self-efficacy, etc.—influence and sometimes determine students' performance.
Our aim for this Research Topic and for the collection of papers we are now publishing has thus been to illustrate the relevance of such various psychological perspectives for mathematics education using the contributions of colleagues from around the world. All the contributions we have collected address these interdisciplinary perspectives explicitly or implicitly.
Research on productive failure suggests that attempting to solve a problem prior to instruction facilitates conceptual understanding compared to receiving instruction prior to problem solving. The assumptions are that during the problem-solving phase, students activate their prior knowledge, become aware of their knowledge gaps, and discover deep features of the target content, which prepares them to better process the subsequent instruction. Unclear is whether this effect results from merely changing the order of the learning phases (i.e., instruction or problem solving first) or from additional features, such as presenting problem-solving material in the form of cases that differ in one feature at a time. Contrasting such cases may highlight the deep features and provide grounded feedback to students’ problem-solving attempts. In addition, the effect of the order of instruction and problem solving on procedural fluency is still unclear. The present experiment (N = 181, mean age = 14.53) investigated in a 2 × 2 design the effects of order (instruction or problem solving first) and of contrasting cases in the problem-solving material (yes/no) on conceptual understanding and procedural fluency. Additionally, the quality and quantity of students’ solution attempts from the problem-solving phase were coded. Regarding the learning outcomes, the ANOVA results suggest that for procedural fluency instruction prior to problem solving was more beneficial than problem solving prior to instruction. Merely delaying instruction did not increase conceptual understanding. The contrasting cases did not affect the quality of solution attempts, nor the posttest results. As expected, students who received instruction first generated fewer, but higher-quality solution attempts.