In situations where independent working time was high (i.e., help was less frequent), high contingent support was more effective than low contingent support in fostering students’ achievement (when correcting for students’ task effort). Low contingent support was more effective in promoting students’ achievement and task effort than high contingent support in situations where independent working time was low (i.e. All teachers taught a five-lesson project on the European Union and the teachers in the scaffolding condition additionally took part in a scaffolding intervention. Thirty social studies teachers of pre-vocational education and 768 students (age 12–15) participated. We investigated both the effects of support quality (i.e., contingency) and the duration of the independent working time of the groups. With the current experimental classroom study we investigated whether scaffolding affects students’ achievement, task effort, and appreciation of teacher support, when students work in small groups. Yet, hardly any evidence from classroom studies exists. Typically these tests are used to determine convergence of series that are similar to geometric series or p-series.Teacher scaffolding, in which teachers support students adaptively or contingently, is assumed to be effective. In this section, we show how to use comparison tests to determine the convergence or divergence of a series by comparing it to a series whose convergence or divergence is known. 9.4: Comparison Tests We have seen that the integral test allows us to determine the convergence or divergence of a series by comparing it to a related improper integral.We will examine several other tests in the rest of this chapter and then summarize how and when to use them. Several tests exist that allow us to determine convergence or divergence for many types of series.Here, we discuss two of these tests: the divergence test and the integral test. In practice, explicitly calculating this limit can be difficult or impossible. 9.3: The Divergence and Integral Tests The convergence or divergence of several series is determined by explicitly calculating the limit of the sequence of partial sums.This process is important because it allows us to evaluate, differentiate, and integrate complicated functions by using polynomials. We will use geometric series in the next chapter to write certain functions as polynomials with an infinite number of terms. We introduce one of the most important types of series: the geometric series. We also define what it means for a series to converge or diverge.
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