Curriculum Interventions for Area Measurement Instruction to Enhance Children’s Conceptual Understanding

This study examined the effectiveness of 3 curriculum interventions focused on strengthening children’s ability to solve area measurement problems and explored the instructional perspectives of the instructor who implemented the interventions. The interventions involved various degrees of emphasis on area measurement and knowledge of 2-dimensional geometry. Participants were 131 fourth graders, recruited from a city in northern Taiwan, and 1 instructor. The results confirmed the effectiveness of an enriched curriculum integrating knowledge of 2-dimensional geometry in enhancing children’s conceptual understanding of area measurement. The group that received the enriched curriculum outperformed the other groups that received the curricula stressing only 2-dimensional geometry or numerical calculations for area measurement in solving the area problems requiring mathematical judgments and explanations. The curriculum also facilitated children’s reasoning in distinguishing between the perimeter and area of a rectangle, which required higher-order mathematical thinking. Interview data revealed that approximately all children from the 3 intervention groups could identify the mathematical subject-matter components highlighted in the curricula. Interviewees tended to consider the use of formulae to solve area measurement problems to be important, despite some differences in learning gains among the 3 groups. In interviews, the participating instructor revealed a change of perspective regarding the importance of offering students opportunities for manipulation and geometric operations when teaching area measurement, prompted by curriculum enactment.     

Gödel’s proof

In 1931, the great Austrian mathematician Kurt Gödel proved “all consistent axiomatic formulations of number theory include undecidable propositions”. This discovery of Gödel and its proof had enormous repercussions in mathematics and computer science. The proof hinged upon the writing of a self-referential mathematical statement, in the same way as the liar’s paradox — I am lying — is a self-referential statement. In this three-part article, we describe Gödel’s discovery and his famous proof.     

The effect of acceleration signal processing for head impact numeric simulations

Brain injury research in sport employs a variety of physical models equipped with accelerometers. These acceleration signals are commonly processed using filters. The purpose of this research was to determine the effect of applying filters with different cutoff frequencies to the acceleration signals used as input for finite element modeling of the brain. Signals were generated from reconstructions of concussion events from American football and ice hockey in the laboratory using a Hybrid III headform. The resulting acceleration signals were used as input for the University College Dublin Brain Trauma Model after being processed with filters. The results indicated that using a filter with a cutoff of 300 Hz or higher had little effect on the resulting strain measures. In some cases there was some effect of the filters on the peak linear (8–30g) and rotational measures (1000–4000 rad/s²), but little effect on the finite element strain result (approximately 2–6 %). The short duration and high magnitude accelerations, such as the puck impact, were most affected by the cutoff frequency of different filters.