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Description: A wide variety of estimators of the between-study variance are available in random-effects meta-analysis. Many, but not all, of these estimators are based on the method of moments. The DerSimonian-Laird estimator is widely used in applications, but the Paule-Mandel estimator is an alternative that is nowadays recommended to be used. Recently, DerSimonian and Kacker have developed two-step moment based estimators. We extend these two-step estimators so that multiple (more than two) steps are used. We establish the surprising result that multi-step estimators tend towards the Paule-Mandel estimator as the number of steps becomes large. Hence, the iterative scheme underlying our new multi-step estimator provides a hitherto unknown relationship between the two-step estimators and Paule-Mandel estimator. Our analysis suggests that two-step estimators are not necessarily distinct estimators in their own right, instead they are quantities that are closely related to the the usual iterative scheme that is used to calculate the Paule-Mandel estimate. Two-step estimators are perhaps best conceptualized as approximate Paule-Mandel estimators.