A New Approach to High-Order Electroencephalogram Phase Analysis Details the Mathematical Mechanisms of Central Nervous System Impulse Encoding

This paper presents a new electroencephalogram (EEG) analysis technique which is applied to example EEGs pertaining to nine human subjects and a broad spectrum of clinical scenarios. While focusing on technique physical efficacy, the paper also paves the way for future clinically-focused studies with revelations of several quantified and detailed findings in relation to high-order central nervous system communicative impulse encoding akin to a sophisticated form of phase-shift keying. The fact that fine encoding details are extracted with confidence from a seemingly modest EEG set supports the paper’s position that vast amounts of accessible information currently goes unrecognised by conventional EEG analysis. The technique commences with high resolution Fourier analysis being twice applied to an EEG, providing newly-identified harmonics. Except for deep sleep where harmonic phase, φ, behaviour becomes highly linear, φ transitional values, ∆φ, measured between harmonics of progressively increasing order are found to cluster rather than follow a normal distribution (e.g., χ2 = 303, df = 12, p < 0.001). Clustering is categorised into ten Families for which many separations between ∆φ values are writable in terms of k = j/4 or j/3 (j = 1, 2, 3 ...), with a preference for k = j/2 (χ2 = 77, df = 1, p < 0.001), amounts of a Family-specific quantum increment value, α∆φ. A parabolic relationship (r > 0.9999, p < 0.001) exists between α∆φ (and the parabola minimum associates with an additional inter-Family or universal quantum increment value, αmin). Ratios of α∆φ typically align within ± 0.5% of simple common fractions (95% CI).

Quantum defect and common fractions

The common fractions N1/N2, where N1 and N2 - the small integers, quite often are used at the quantum-mechanical description of microcosm objects (for example, fractional charges of quarks and some quantum characteristics, such as particles spin). Recently the fractional quantum Hall effect was discovered, and common fractions have considerably expanded their presence in microcosm physics. The theory of the fractional quantum Hall effect has appeared nontrivial, so the Nobel Prize on physics in 1998 was awarded not only for discovery of the effect in 1982 (Daniel Tsui and Horst Störmer) but also for the theory creation in 1983 (Robert Laughlin). And now one more sensational discovery: common fractions were «detected» at the analysis of experimental characteristics of «hydrogen-like» atoms and ions (with only one electron on an outer shell). It has appeared, that the effective main quantum number of outer shell electron, that is, subject to quantum defect (Rydberg correction), can be expressed in common fractions.

IN SEARCH OF STRATEGIES USED BY PRIMARY SCHOOL PUPILS FOR DEVELOPING FRACTION SENSE

Purpose – Most literature has focused solely on either knowledge about number sense or understanding of fractions. To fill the research gap, this study examined pupils’ abilities in both number sense and fractions. In particular, it investigated Year 4 and Year 5 pupils’ use of strategies in developing their fraction sense. Methodology – This study adopted a descriptive research design, utilising a mixed approach in data collection. An instrument called the Fraction Sense Test (FST) and a clinical interview were used to collect data. The FST comprised 3 strands: fraction concept, fraction representation and effect of operation. A two-stage cluster sampling method was employed to select 396 Year 4 and Year 5 pupils. The sampling involved random selection of the primary schools in the first stage, followed by pupils within the selected schools in the second stage. In addition to descriptive statistics, content analysis of interview transcripts was conducted to identify the presence of concepts and strategies applied among the pupils. Findings – The study found that the pupils scored lowest in effect of operation. It was also revealed that there were four strategies which helped the pupils to develop fraction sense, namely (1) comparing fractions using benchmark fractions of common fractions such as ½, ¼, zero and 1, (2) understanding denominators to determine the size of equal parts, (3) comparing fractions using unit fraction, and (4) applying the strategies in (1) and (2) to manipulate fractions in effect of operation. Significance – The findings provide useful input to facilitate the development of fraction sense ability.

Mental representation of fractions: It all depends on whether they are common or uncommon

This study examined whether common and uncommon fractions are mentally represented differently and whether common ones are used in accessing the magnitudes of uncommon ones. In Experiments 1 and 2, college education majors, most of whom were female, Caucasian, and in their early 20s, made comparisons involving common and uncommon fractions. In Experiment 3, participants were presented with comparison tasks involving uncommon fractions and asked to describe the strategies which they used in making such comparisons. Analysis of reaction times and error rates support the hypothesis that for common fractions, it is their holistic real value, rather than their individual components, that gets represented. For uncommon fractions, the access of their magnitudes is a process of retrieving and using a known common one having a similar value. Such results suggest that the development of the cognisance of the magnitudes of fractions may be principally a matter of common ones only and that learners’ handling of uncommon fractions may be greatly facilitated through instructions on matching them with common ones having a similar value.

Nifty Nines and Repeating Decimals

The traditional technique for converting repeating decimals to common fractions can be found in nearly every algebra textbook that has been published, as well as in many precalculus texts. However, students generally encounter repeating decimal numerals earlier than high school when they study rational numbers in prealgebra classes. Therefore, how do prealgebra students in the middle grades convert repeating decimals to fractions without using the age-old algebraic process (multiplying and finding the difference of two “stacked” equations) or without applying the precalculus approach of treating repeating decimal digits as an infinite geometric series?.

Why Learning Common Fractions Is Uncommonly Difficult: Unique Challenges Faced by Students With Mathematical Disabilities

In this commentary, I examine some of the distinctive, foundational difficulties in learning fractions and other types of rational numbers encountered by students with a mathematical learning disability and how these differ from the struggles experienced by students classified as low achieving in math. I discuss evidence indicating that students with math disabilities exhibit a significant delay or deficit in the numerical transcoding of decimal fractions, and I further maintain that they may face unique challenges in developing the ability to effectively translate between different types of fractions and other rational number notational formats—what I call conceptual transcoding. I also argue that characterizing this level of comprehensive understanding of rational numbers as rational number sense is irrational, as it misrepresents this flexible and adaptive collection of skills as a biologically based percept rather than a convergence of higher-order competencies that require intensive, formal instruction.