Approximate number system discrimination training for 7-8 year olds improves approximate, but not exact, arithmetics, and only in children with low pre-training arithmetic scores

We investigated whether training the Approximate Number System (ANS) would transfer to improved arithmetic performance in 7-8 year olds compared to a control group. All children participated in Pre- and Post-Training assessments of exact symbolic arithmetic (additions and subtractions) and approximate symbolic arithmetic abilities (a novel test). During 3 weeks of training (approximately 25 minutes per day, two days per week), we found that children in the ANS Training group had stable individual differences in ANS efficiency and increased in ANS efficiency, both within and across the training days. We also found that individual differences in ANS efficiency were related to symbolic arithmetic performance. Regarding arithmetic performance, both the ANS training group and the control group improved in all tests (exact and approximate arithmetics tests). Thus, the ANS training did not show a specific effect on arithmetic performance. However, considering the initial arithmetic level of children, we found that the trained children showed a higher improvement on the novel approximate arithmetic test compared to the control group, but only for those children with a low pre-training arithmetic score. Nevertheless, this difference within the low pre-training arithmetic score level was not observed in the exact arithmetic test. The limited benefits observed in these results suggest that this type of ANS discrimination training, through quantity comparison tasks, may not have an impact on symbolic arithmetics overall, although we cautiously propose that it could help with approximate arithmetic abilities for children at this age with below-average arithmetic performance.

Aerobic Fitness Relates to Superior Exact and Approximate Arithmetic Processing in College-Aged Adults

Compelling evidence supports the association between the attribute of aerobic fitness and achievement scores on standardized tests of mathematics, but the underlying reasons for this association remain unclear. The present investigation sought to clarify the nature of the relationship between aerobic fitness and arithmetic processing by examining the extent to which these fitness-related differences in mathematics are attributed to individual differences in more efficient processing (efficiency hypothesis) or enhanced allocation of cognitive resources (resources hypothesis) in a sample of 118 college-aged adults. Combining behavioral measures to examine speed and accuracy of processing with pupillary measures that indicate resource allocation, participants completed a complex arithmetic task prior to performing a maximal graded exercise test to assess their aerobic fitness level. The arithmetic task comprised problems with varying levels of difficulty, requiring participants to determine whether a sum of two numbers was greater than or less than 100, which could be solved using either approximate or exact calculation strategies. Higher aerobic fitness was associated with 1) shorter reaction time across all problems, 2) superior accuracy for difficult problems employing exact arithmetic, and 3) greater task-evoked pupillary reactivity for the difficult problems requiring approximate and exact arithmetic strategies. These results indicate that individuals higher in aerobic fitness have more cognitive resources available to solve difficult problems faster and more accurately. These data provide initial evidence to suggest that fitness-related differences in mathematics achievement may result from modulation of cognitive resources underlying superior execution of procedural strategies during arithmetic performance. Accordingly, higher cardiovascular health may be implicated in superior health literacy (e.g., interpreting blood sugar readings and other clinical data), thus affecting the motivation to take action and engage in health behaviors based on quantitative information.

On QZ steps with perfect shifts and computing the index of a differential-algebraic equation

In this paper we revisit the problem of performing a $QZ$ step with a so-called ‘perfect shift’, which is an ‘exact’ eigenvalue of a given regular pencil $\lambda B-A$ in unreduced Hessenberg triangular form. In exact arithmetic, the $QZ$ step moves that eigenvalue to the bottom of the pencil, while the rest of the pencil is maintained in Hessenberg triangular form, which then yields a deflation of the given eigenvalue. But in finite precision the $QZ$ step gets ‘blurred’ and precludes the deflation of the given eigenvalue. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the $QZ$ step can be constructed using this eigenvector, so that the deflation is also obtained in finite precision. An important application of this technique is the computation of the index of a system of differential algebraic equations, since an exact deflation of the infinite eigenvalues is needed to impose correctly the algebraic constraints of such differential equations.

A Combinatorial Approach for Constructing Lattice Structures

Lattice structures exhibit unique properties including a large surface area and a highly distributed load-path. This makes them very effective in engineering applications where weight reduction, thermal dissipation, and energy absorption are critical. Furthermore, with the advent of additive manufacturing (AM), lattice structures are now easier to fabricate. However, due to inherent surface complexity, their geometric construction can pose significant challenges. A classic strategy for constructing lattice structures exploits analytic surface–surface intersection; this, however, lacks robustness and scalability. An alternate strategy is voxel mesh-based isosurface extraction. While this is robust and scalable, the surface quality is mesh-dependent, and the triangulation will require significant postdecimation. A third strategy relies on explicit geometric stitching where tessellated open cylinders are stitched together through a series of geometric operations. This was demonstrated to be efficient and scalable, requiring no postprocessing. However, it was limited to lattice structures with uniform beam radii. Furthermore, existing algorithms rely on explicit convex-hull construction which is known to be numerically unstable. In this paper, a combinatorial stitching strategy is proposed where tessellated open cylinders of arbitrary radii are stitched together using topological operations. The convex hull construction is handled through a simple and robust projection method, avoiding expensive exact-arithmetic calculations and improving the computational efficiency. This is demonstrated through several examples involving millions of triangles. On a typical eight-core desktop, the proposed algorithm can construct approximately up to a million cylinders per second.

Language and Math: What If We Have Two Separate Naming Systems?

The role of language in numerical processing has traditionally been restricted to counting and exact arithmetic. Nevertheless, the impact that each of a bilinguals’ languages may have in core numerical representations has not been questioned until recently. What if the language in which math has been first acquired (LLmath) had a bigger impact in our math processing? Based on previous studies on language switching we hypothesize that balanced bilinguals would behave like unbalanced bilinguals when switching between the two codes for math. In order to address this question, we measured the brain activity with magneto encephalography (MEG) and source estimation analyses of 12 balanced Basque-Spanish speakers performing a task in which participants were unconscious of the switches between the two codes. The results show an asymmetric switch cost between the two codes for math, and that the brain areas responsible for these switches are similar to those thought to belong to a general task switching mechanism. This implies that the dominances for math and language could run separately from the general language dominance.

Exact Arithmetic of Zero and Infinity Based on the Conventional Division by Zero 0/0 = 1

The chief object of this work is to create an exact and consistent arithmetic of zero, denoted 0, and infinity (zero divisor), written as 1/0 and denoted ∞ , based on the conventional division by zero $$ \dfrac{{0}}{{0}}=1. $$ Manifold and undeniable applications of this arithmetic are given in this work in order to show its usefulness.