(1) See the section on the long tail of under-achievement by English pupils, in comparison with leading Western European countries, in Prais, S.J. (1997), ‘How did English schools and pupils really perform in the 1995 international comparison in mathematics’, National Institute Economic Review, April, 1997; an expanded version is to appear in Oxford Studies in Comparative Education, 1998.

(2) The quotations are from the conclusions (p. 451) and abstract (p. 423) of ‘Within-class grouping: a meta-analysis’, by Yiping Lou of Concordia University, Montreal, plus five co-authors (listed unalphabetically), all Canadian educationists (four from Concordia, one from Alberta), Review of Educational Research (University of Wisconsin-Madison), 1996 (vol. 66, no. 4), pp. 423-58. The first-named author is described as a PhD candidate in the department of the second co- author, Professor P.C. Abrami, Director of the Centre for the Study of Classroom Processes at Concordia. This article considerably extends previous surveys on this topic by Slavin (1987) and by Kulik (1987, 1991; full references are in Lou et al.); it has received recent favourable attention in British official educational circles and, since it may affect English educational policy adversely, it deserves rather fuller examination here than might otherwise seem necessary (on punctuation : ‘effect-size’ has become a technical term among educationalists; for clarity I have written it here with a hyphen throughout).

(3) Instead of the standard deviation of the control group, sometimes an estimate of the pooled standard deviation of the control and experimental groups is taken. If they do not differ significantly, such a pooling may be unobjectionable. But if we suspect, for example, that the experimental group has a higher standard deviation, then such a pooling may amount to ‘throwing away the baby with the bathwater’.

(4) Putting children into groups for only a short period such as a term or semester, and then returning them to their previous organisational arrangements, cannot be expected to leave much effect; this was noted in the course of a recent valuable research survey of wider alternative methods of Setting and Streaming by W. Harlen and H. Malcolm (Scottish Council for Research in Education, Edinburgh, 1997, p. 19)—though based on a study of 12 year-olds in the US as long ago as 1960. That research survey did not, however, consider the Continental organisational strategies mentioned at the end of the present note.

(5) These extreme values are probably not untypical ‘outliers’, as appears from the more detailed analyses quoted below relying on their published 95 per cent confidence interval. Unfortunately, the 95 per cent confidence interval they pub lished for their overall size-effect of +0.17 (of +0.16 to +0.23; p. 439) was subject to a misprint—as evident from its asymmetry—and Professor Abrami has now kindly told me (in response to my query) that it should read +0.14 to +0.21.

(6) Their average was calculated by weighting the estimated effect-size of each comparison by the number of pupils involved; it would have been more efficient to weight by the precision of each estimate (the inverse of the error-variance), which would depend partly on the number of pupils and partly on the closeness of the relationship.

(7) Based on their table 8, p. 443. Their summary gives only the number of comparisons (n), the average effect-size, and the ‘95 per cent CI’, which is the confidence interval for that average; but what the reader really needs is the original range within which 95 per cent of the estimated effect-sizes lie. Elementary statistical theory fortunately allows the reader to deduce this (at least approximately) by multiplying the range of their published confidence interval by √n. For example, the 95 per cent CI calculated by the authors for the effect-size of +0.29 quoted in the text above is +0.24 to +0.35, and is based on 36 comparisons; the 95 per cent range of the original studies is thus √36 (0.35-0.24) = 0.66, leading to a 95 per cent range in the original comparisons from -0.37 to +0.95. The table of the Normal integral function shows that the latter correspond to the 36th and 83rd percentiles, ie to the 9th and 21st pupils in a class of 25.

(8) See, for example, another recent valuable research review by S. Hallam and I. Toutounji, What Do We Know About the Grouping of Pupils by Ability? (University of London Institute of Education, 1996), esp. pp. 8-9.

(9) There are some unfortunate typographical errors in that table, in that the mean effect-size for 5-7 pupils was published as -0.02 instead of +0.02 (the confidence interval published as ‘-0.02 to -0.09’ should read ‘-0.09 to +0.04’; I am grateful to Professor Abrami for responding to my query on this).

(10) See, for example, Luxton, R. and Last, G. (1997), ‘Under-achievement and pedagogy’, National Institute Discussion Paper no. 112, February (forthcoming in Teaching Mathematics and its Applications, 1998); Bierhoff, H. (1996), ‘Laying the foundations of numeracy’, Teaching Mathematics and its Applications; and S.J. Prais (1997), ‘School readiness, whole-class teaching and pupil's mathematical attainments’, Oxford Review of Education.

(11) The current validity of this requirement (which is to be traced to the National Curriculum specification of 1986, and previously to the English legendary ‘seven-year spread of attainments’ at each age) was recently confirmed by an OFSTED report which noted: ‘it is normal for attainment at [age 11] the end of Key Stage 2 to range over three or four National Curriculum leads in the core subjects and to cover work as high as Level 5 and sometimes Level 6’ (Using Subject Specialists to Promote High Standards at Key Stage 2, OFSTED, 1997, p. 3). It will be remembered that a National Curriculum ‘Level’ is defined to correspond to two years of teaching; and that Level 6 is expected for average 15 year- olds.