A “Coprolitic Vision” for Earth Science Education

William Buckland (1784–1846) first identified and scientifically studied coprolites in the early 1820s. Although some of his contemporaries did not look favorably upon him or his research, Buckland's early experiments advanced paleoecology and taphonomy. Because our informal presentations with coprolites resulted in students' spirited reactions, we investigated whether coprolite introduction, accompanied with its history of science, had potential for meaningful learning in K‐12 Earth Science classrooms. Practicing Earth Science teachers (N = 28) enrolled in an online paleontology course researched coprolites, identified potential student interest, and designed coprolite activities for their individual classrooms. Resulting projects were diverse and creative, and incorporated investigations into fossilization processes, paleoenvironments, food chains, and geologic time. In anonymous surveys, teachers indicated that their students' interest in coprolites is high. We propose inclusion of coprolites and their history in Earth Science classrooms as a portal to hook students' interest and as springboard to additional scientific topics.     

The “Error” in the Indian “Taylor Series Approximation” to the Sine

It has been repeatedly noted, but not discussed in detail, that certain so-called “third-order Taylor series approximations” found in the school of the medieval Keralese mathematician M dhava are inaccurate. That is, these formulas, unlike the other series expansions brilliantly developed by M dhava and his followers, do not correspond exactly to the terms of the power series subsequently discovered in Europe, by whose name they are generally known. We discuss a Sanskrit commentary on these rules that suggests a possible derivation explaining this discrepancy, and in the process re-emphasize that the Keralese work on such series was rooted in geometric approximation rather than in analysis per se. © 2001 Elsevier Science (USA).Es ist mehrfach festgestellt bisher aber nicht ausführlich diskutiert worden, daß einige sogenannte Taylor-reihennäherungswerte dritter Ordnung, die in der mittelalterlichen Schule keralesischen M dhava gefunden werden, ungenau sind. Das heißt, diesc Formeln sind den Termen der Potenzreihe, die später in Europa entwickelt wurde und unter dem Namen Taylorreihe bekannt ist, nicht äquivalent, im Gegensatz zu den anderen Entwicklungen von Reihen, die glänzend von M dhava und seinen Nachfolgern entwickelt werden. Wir behandeln einen Sanskritkommentar zu den Regeln, der eine mögliche Herleitung suggeriert, die diese Diskrepanz erklärt. Dabei betonen wir nochmals, daß die keralesische Arbeit über solche Reihen eher in geometrischen Näherungen als in der Analysis an sich ihre Wurzeln hat. © 2001 Elsevier Science (USA).MSC subject classification: 01A32.     

The “golden” algebraic equations

The special case of the (p + 1)th degree algebraic equations of the kind x^p+1 = x^p + 1 (p = 1, 2, 3, …) is researched in the present article. For the case p = 1, the given equation is reduced to the well-known Golden Proportion equation x² = x + 1. These equations are called the golden algebraic equations because the golden p-proportions τ_p, special irrational numbers that follow from Pascal’s triangle, are their roots. A research on the general properties of the roots of the golden algebraic equations is carried out in this article. In particular, formulas are derived for the golden algebraic equations that have degree greater than p + 1. There is reason to suppose that algebraic equations derived by the authors in the present article will interest theoretical physicists. For example, these algebraic equations could be found in the research of the energy relationships within the structures of many compounds and physical particles. For the case of butadiene (C₄H₆), this fact is proved by the famous physicist Richard Feynman.