Crossed vs nested random effects: how do they differ and how are they specified correctly in lme4?

Here is how I have understood nested vs. crossed random effects: Nested random effects occur when a lower level factor appears only within a particular level of an upper level factor. For example, pupils within classes at a fixed point in time. In lme4 I thought that we represent the random effects for nested data in either of two equivalent ways: (1|class/pupil) # or (1|class) + (1|class:pupil) Crossed random effects means that a given factor appears in more than one level of the upper level factor. For example, there are pupils within classes measured over several years. In lme4, we would write: (1|class) + (1|pupil) However, when I was looking at a particular nested dataset, I noticed that both model formulas gave identical results (code and output below). However I have seen other datasets where the two formulas produced different results. So what is going on here? mydata <- read.csv("https://web.archive.org/web/20160624172041if_/http://www-personal.umich.edu/~bwest/classroom.csv") # (the data is no longer at `http://www-personal.umich.edu/~bwest/classroom.csv` # hence the link to web.archive.org) # Crossed version: Linear mixed model fit by REML ['lmerMod'] Formula: mathgain ~ (1 | schoolid) + (1 | classid) Data: mydata REML criterion at convergence: 11768.8 Scaled residuals: Min 1Q Median 3Q Max -4.6441 -0.5984 -0.0336 0.5334 5.6335 Random effects: Groups Name Variance Std.Dev. classid (Intercept) 99.23 9.961 schoolid (Intercept) 77.49 8.803 Residual 1028.23 32.066 Number of obs: 1190, groups: classid, 312; schoolid, 107 # Nested version: Formula: mathgain ~ (1 | schoolid/classid) REML criterion at convergence: 11768.8 Scaled residuals: Min 1Q Median 3Q Max -4.6441 -0.5984 -0.0336 0.5334 5.6335 Random effects: Groups Name Variance Std.Dev. classid:schoolid (Intercept) 99.23 9.961 schoolid (Intercept) 77.49 8.803 Residual 1028.23 32.066 Number of obs: 1190, groups: classid:schoolid, 312; schoolid, 107