The Sleep of Reason Produces Monsters: How and When Biased Input Shapes Mathematics Learning

Children's failure to reason often leads to their mathematical performance being shaped by spurious associations from problem input and overgeneralization of inapplicable procedures rather than by whether answers and procedures make sense. In particular, imbalanced distributions of problems, particularly in textbooks, lead children to create spurious associations between arithmetic operations and the numbers they combine; when conceptual knowledge is absent, these spurious associations contribute to the implausible answers, flawed strategies, and violations of principles characteristic of children's mathematics in many areas. To illustrate mechanisms that create flawed strategies in some areas but not others, we contrast computer simulations of fraction and whole number arithmetic. Most of their mechanisms are similar, but the model of fraction arithmetic lacks conceptual knowledge that precludes strategies that violate basic mathematical principles. Presentingbalanced problem distributions and inculcating conceptual knowledge for distinguishing flawed from legitimate strategies are promising means for improving children's learning.

Knowledge of mathematical symbols goes beyond numbers

The written language of mathematics is dense with symbols and with conventions for combining those symbols to express mathematical ideas. For example, reading a factored polynomial function such as f(x) = x²(2x + 15) requires the knowledge that parenthesis can be used to signify function notation in one context and multiplication in another. Mathematical orthography is defined as orthographic knowledge of symbolic mathematics. It entails both knowledge of discrete mathematical symbols and the conventions for combining those symbols into expressions and equations. The ability to read text written in the base-ten system, comprised of digits and conventions for combining digits to express whole and rational quantities, is an important aspect of mathematical orthography. However, success in secondary and post-secondary programs requires more advanced mathematical orthography. The goal of this research was to determine if a simple and novel measure of mathematical orthography captures individual differences in adults’ mathematical skills. Mathematical orthography was measured with a timed dichotomous symbol decision task. Adults (N = 58) discriminated between conventional and non-conventional combinations of mathematical symbols (e.g., x² vs. ²x; |y| vs. ||y). The mathematical symbol decision task uniquely predicted individual differences in whole-number arithmetic, fraction/algebra procedures, and word problem solving. These findings suggest that the symbol decision task is a useful index of symbol associations in mathematical development and, thus, provides a tool for understanding the role of mathematical orthography in individual differences in adults’ mathematical skills.

Understanding students’ game experiences throughout the developmental process of the number navigation game

Serious games for learning have received increased attention in recent years. However, empirical studies on students’ gaming experiences throughout the developmental process of serious games and discussions regarding game design are missing. The aims of the present study were to analyze students’ gaming experiences while playing four consecutive versions of the Number Navigation Game (NNG)—a mathematical game-based learning environment focusing on flexibility and adaptivity with whole-number arithmetic; and to provide an extensive review of the NNG developmental and design process over 3 years with focus on how and why the design decisions were made, and how those choices affected students’ gaming experiences. The study employed a mixed-methods design of quantitative and qualitative research. The Game Experience Questionnaire about eight core game experience dimensions was answered by different groups of students at primary schools in Finland in three different experiments after students played four versions of the NNG from 2014 to 2016. Six semi structured interviews related to students’ game experiences, preferences and game features of the latest version of NNG were conducted. Overall, results indicate that improvement in game’s usability and clarity in the user interface has positive impacts on students’ game experiences. Furthermore, there seems to be a clear advantage in having better aesthetics and value in improving extrinsic elements that could contribute to maintain players’ enthusiasm and situational interest in serious games.