In this article, some prescriptions to define a distribution on the set Q0 of all rational numbers in 0,1 are outlined. We explored a few properties of these distributions and the possibility of ...making these rational numbers asymptotically equiprobable in a suitable sense. In particular, it will be shown that in the said limit—albeit no absolutely continuous uniform distribution can be properly defined in Q0—the probability allotted to every single q∈Q0 asymptotically vanishes, while that of the subset of Q0 falling in an interval a,b⊆Q0 goes to b−a. We finally present some hints to complete sequencing without repeating the numbers in Q0 as a prerequisite to laying down more distributions on it.
In this study, we consider an equivalence test on rational numbers in a scenario with K+2 distributed parties, Alice, Bob, P1,P2,…,PK, where Alice has a private rational number xa, Bob has a private ...rational number xb, and party Pi has a secret si, where i∈{1,2,…,K},K ≥ 2. The parties want to cooperatively detect whether xa=xb without revealing any information about their secrets. This problem has many applications in online collaboration, such as e-voting, which requires public verifiability. First, we develop a provably secure threshold cryptosystem for rational numbers. Next, based on the proposed threshold scheme, we construct a distributed plaintext equivalence test protocol in an honest majority environment. We prove that the proposed protocol is secure and robust in the standard (ideal/real) model.
This article investigates the adaptive neural network (NN) finite-time output tracking control problem for a class of multi-input and multi-output (MIMO) uncertain nonlinear systems whose powers are ...positive odd rational numbers. Such designs adopt NNs to approximate unknown continuous system functions, and a controller is constructed by combining backstepping design and adding a power integrator technique. By constructing new iterative Lyapunov functions and using finite-time stability theory, the closed-loop stability has been achieved, which further verifies that the entire system possesses semiglobal practical finite-time stability (SGPFS), and the tracking errors converge to a small neighborhood of the origin within finite time. Finally, a simulation example is given to elaborate the effectiveness and superiority of the developed.
A critical difference between decimal and whole numbers is that among whole numbers the number of digits provides reliable information about the size of the number, e.g., double-digit numbers are ...larger than single-digit numbers. However, for decimals, fewer digits can sometimes denote a larger number (i.e., 0.8 > 0.27). Accordingly, children and adults perform worse when comparing such Inconsistent decimal pairs relative to Consistent pairs, where the larger number also has more digits (i.e., 0.87 > 0.2). Two explanations have been posited for this effect. The string length congruity account proposes that participants compare each position in the place value system, and they additionally compare the number of digits. The semantic interference account suggests that participants additionally activate the whole number referents of numbers – the numbers unadorned with decimal points (e.g., 8 < 27) – and compare these. The semantic interference account uniquely predicts that for Inconsistent problems with the same actual rational distance, those with larger whole number distances should be harder, e.g., 0.9 vs. 0.81 should be harder than 0.3 vs. 0.21 because 9 < < 81 whereas 3 < 21. Here we test this prediction in two experiments with college students (Study 1: n = 58 participants, Study 2: n = 78). Across both, we find a main effect of consistency, demonstrating string length effects, and also that whole number distance interferes with processing conflicting decimals, demonstrating semantic interference effects. Evidence for both effects supports the semantic interference account, highlighting that decimal comparison difficulties arise from multiple competing numerical codes. Finally, for accuracy we found no relationship between whole number distance sensitivity and math achievement, indicating that whole number magnitude interference affects participants similarly across the spectrum of math achievement.
It was argued recently that number line based training supports the development of conceptual rational number knowledge. To test this hypothesis, we evaluated training effects of a digital game based ...on the measurement interpretation of rational numbers. Ninety-five fourth graders were assigned to either a game-based training group (n = 54) who played a digital rational number game for five 30-min sessions or a control group (n = 41) who attended regular math curriculum. Conceptual rational number knowledge was assessed in a pre- and posttest session. Additionally, the game groups' playing behavior was evaluated. Results indicated that the game-based training group improved their conceptual rational number knowledge significantly more strongly than the control group. In particular, improvement of the game-based training group was driven by significant performance increases in number magnitude estimation and ordering tasks. Moreover, results revealed that in-game metrics, such as overall game performance and maximum level achieved provided valid information about students’ conceptual rational number knowledge at posttest. Therefore, results of the current study not only suggest that aspects of conceptual rational number knowledge can be improved by a game-based training but also that in-game metrics provide crucial indicators for learning.
•We developed a number line based game on fractions and decimals.•The game improved aspects of fourth graders conceptual rational number knowledge.•In-game metrics reflected children's conceptual rational number knowledge.•In-game metrics predicted learning gains through the training.•These learning analytics may support teachers'/students' learning/teaching goals.
•Findings from psychological science often do not translate to school settings.•Centering teachers’ voices in study design improved students’ math scores.•Students’ fraction knowledge improved in ...school settings through play.•We provide a model for combining community knowledge with psychological science.
We combine design-based implementation research with a pre-registered RCT to address a long-standing challenge in psychological science: How to use psychological principles to address real-world problems while designing and implementing interventions in the field. We posit this as a design methodology for optimizing the translation between psychological science and real-world applications. We tested the efficacy of an extensively co-designed version of a game-based rational number intervention, Fraction Ball, versus “business-as-usual” math instruction and physical education in a sample of 4th/5th grade Latine students (N = 360). Insights from nine co-design sessions with 20 teachers informed revisions and additions to a previous version of Fraction Ball, strengthening impacts across 10 of 12 rational number subtests. This methodology provides insights for translating psychological science research and scaling it to address real-world educational needs, such as play-based interventions that improve rational number understanding in authentic contexts.
Abstract
The aim intimately pertinent to this research program is to use constructive mathematical notions to analyze how constructive real numbers and rational numbers are likely to have an ...influence on the outcomes involving economic problems. Researchers endeavor to find a computer program to determine the optimal location of a production plant and to find out that provided that the distances from the production plant to the sales location are rational numbers, then the problem is algorithmically solvable, though, if the distances are constructive real numbers, then the existence of the computer program that solves all such problems will not be possible.
The foundations for more advanced mathematics involve a good sense of rational numbers. However, research in cognitive psychology and mathematics education has repeatedly shown that children and even ...adults struggle with understanding different aspects of rational numbers. One frequently raised explanation for these difficulties relates to the natural number bias, i.e., the tendency to inappropriately apply natural number properties to rational number tasks. This contribution reviews the four main areas where systematic errors due to the natural number bias can be found, i.e., their size, operations, representations and density. Next, we discuss the major theoretical frameworks from which rational number understanding is currently investigated. Finally, an overview of the various papers is provided.
Rational number learning can cause frustration and negative emotions for elementary school students. Fraction Ball, a play-based math intervention, allows students to actively learn rational numbers ...through engaging and interactive activities. Based on a cluster-randomized trial with 16 teachers and 360 students, our pre-registered analyses showed moderate positive impacts of Fraction Ball on overall students' self-reported math-related emotions. Exploratory analyses indicated that students with higher negative emotions at pretest showed larger experimental impacts on decreasing negative emotions at posttest. Finally, we found that Fraction Ball evidenced no trade-off between rational number learning and emotional outcomes at the classroom block level, indicating that positive learning gains in rational number skills were associated with increases in positive emotions and decreases in negative emotions.
•We tested the effectiveness of a playful rational number intervention for improving students' math-related emotions.•We found the intervention improved students' positive math-related emotions.•Students with higher negative emotions at pretest exhibited larger treatment impacts on their negative emotions at posttest.•The cluster-randomized trial allowed a block-level test showing positive relations between impacts on learning and emotions.
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