20,657 research outputs found
Understanding changes in teacher beliefs and identity formation: A case study of three novice teachers in Hong Kong
Novice teachers often undergo an identity shift from learner to teacher. Along this process, their instructional beliefs change considerably which in turn affect their teacher identity formation. Drawing on
data collected mainly through interviews with three novice English teachers formore than one year, the present study examines their firstyear teaching experience in Hong Kong secondary schools, focusing
on changes of their English teaching beliefs and the impact of these changes on their identity construction. Findings reveal that while the teachers’ initial teaching beliefs were largely shaped in their prior school learning and learning-to-teach experience, these beliefs changed and were reshaped a great deal when encountering various contextual realities, and these changes further influenced their views on their teacher identity establishment, unfortunately in a more negative
than positive direction. The study sheds light on the importance of institutional support in affording opportunities for novice teachers’ workplace learning and professional development
Two asymptotic expansions for gamma function developed by Windschitl's formula
In this paper, we develop Windschitl's approximation formula for the gamma
function to two asymptotic expansions by using a little known power series. In
particular, for with , we have \begin{equation*}
\Gamma \left( x+1\right) =\sqrt{2\pi x}\left( \tfrac{x}{e}\right) ^{x}\left(
x\sinh \tfrac{1}{x}\right) ^{x/2}\exp \left( \sum_{k=3}^{n-1}\tfrac{\left(
2k\left( 2k-2\right) !-2^{2k-1}\right) B_{2k}}{2k\left( 2k\right) !x^{2k-1}}
+R_{n}\left( x\right) \right) \end{equation*} with \begin{equation*} \left|
R_{n}\left( x\right) \right| \leq \frac{\left| B_{2n}\right| }{2n\left(
2n-1\right) }\frac{1}{x^{2n-1}} \end{equation*} for all , where
is the Bernoulli number. Moreover, we present some approximation formulas for
gamma function related to Windschitl's approximation one, which have higher
accuracy.Comment: 14 page
An accurate approximation formula for gamma function
In this paper, we present a very accurate approximation for gamma function:
\begin{equation*} \Gamma \left( x+1\right) \thicksim \sqrt{2\pi x}\left(
\dfrac{x}{e}\right) ^{x}\left( x\sinh \frac{1}{x}\right) ^{x/2}\exp \left(
\frac{7}{324}\frac{1}{ x^{3}\left( 35x^{2}+33\right) }\right) =W_{2}\left(
x\right) \end{equation*} as , and prove that the function
is strictly
decreasing and convex from onto , where \begin{equation*} \beta =\frac{22\,025}{22\,032}-\ln
\sqrt{2\pi \sinh 1}\approx 0.00002407. \end{equation*}Comment: 9 page
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