Of the "big four", division is likely to regarded by many learners as "the odd one out", "the difficult one", "the one that is complicated", or "the scary one". It seems to have been that way "for ever", in the perception of many who have trodden the learning pathways through the world of…

Somewhere in a programme of study for mathematics in Key Stages 3 and 4 you would expect to find the terms mean, mode, median, and range. You might even find the terms central tendency, and variation. Many students following such a programme will be able to calculate the mean, most will be able to quote the mode, and some students will be able to…

Ofsted was asked to provide evidence of effective practice in the teaching of early arithmetic, its report entitled; "Good practice in primary mathematics: evidence from 20 successful schools," was published in November 2011. Did this report pass you by? Unfortunately, as with many other reports not blessed with a "high…

Although not all children may enjoy eating them, it is fairly safe to assume that most students will at least know what ring doughnuts are, and what they look like. Familiarity with these intriguingly-shaped items of patisserie means that students will not immediately be daunted at the thought of doing some mathematics with the relatively…

Multiplication facts are difficult to teach. Therefore many researchers have put a great deal of effort into finding multiplication strategies. Sherin and Fuson (2005) provided a good survey paper on the multiplication strategies research area. Kolpas (2002), Rendtorff (1908), Dabell (2001), Musser (1966) and Markarian (2009) proposed the finger…

The multiplicative principle is the tool allowing the counting of groups that can be described by a sequence of events. An event is a subset of sample space, i.e. a collection of possible outcomes, which may be equal to or smaller than the sample space as a whole. It is important that students understand this basic principle early on and know how…

This is an account of events in a real classroom when the notion that there is a distinction between calculation by counting, and calculation by structuring is a focus for learning. Pupils have "hands-on" experiences that support their understanding of "number". The teaching "aid" might be unfashionable in…

This is the second of three articles that draw on findings from Nunes, Watson and Bryant (2009): "Key understandings in school mathematics: a report to the Nuffield Foundation". The report was soundly based on research about how children learn mathematics, much of it done in the UK in the '80s. Most of the findings about algebra are very…

In "MT218" the author looked at the possibility of basing a classroom activity on a simple, though not totally transparent, number-theoretic result. In this article he considers another relatively straightforward idea from number theory that could be used either as a lesson starter or as the basis of a more substantial task, requiring students to…

The Williams Review (Dept. for Children, Schools and Families, 2008) is significantly important to all early years and primary educationalists, including those who are involved in teaching mathematics to the next generation of primary school teachers. The report, compiled as a result of this review, sets out the ways that children in primary…

The supposed pre-eminence of an external examination can exert a disproportionate influence on a curriculum and the associated learning and teaching. Teaching can easily subordinate learning and understanding to curriculum coverage if the society develops a culture that appears to make such demands. This study focuses on Tanzania and provides the…

Why do definitions play such a central role in mathematics? It may seem obvious that precision about the terms one uses is necessary in order to use those terms reasonably (while reasoning). Definitions are chosen so as to be definite about the terms one uses, but also to make both the statement of, and the reasoning to justify, theorems as…

This article is the first in a series which draws on findings from Nunes, Watson and Bryant (2009): "Key understandings in school mathematics: a report to the Nuffield Foundation." The Nuffield report is soundly based on research about how children learn some of the concepts involved in mathematics. In this series of articles the author takes key…

Repeatedly taking the square root with a calculator yields numbers with logarithmic-type properties, which can be used to define the logarithm functions. (MP)

More mental computation and organized and enjoyable activities to practice such skills are advocated. Suggestions for brief sessions involving challenging questions and quick recall of number facts are presented. The author presents and comments on several sample lessons designed for a wide range of student abilities. (MP)