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…

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…

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…

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…

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…

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…

One debate that periodically arises in mathematics education is the issue of how to teach calculation more effectively. "Modern" approaches seem to initially favour mental calculation, informal methods, and the development of understanding before introducing written forms, while traditionalists tend to champion particular algorithms. The debate is…

Ever since "mental arithmetic" was updated to "mental calculation," official documents have succeeded in perpetuating several basic misconceptions and misunderstandings about this topic. This situation does not augur well for the successful teaching of mental calculation strategies to young children. In this article, the author…

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…

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)

Procedures using calculators are described for determining the recurring sequence of digits in the decimal representation of numbers of the form 1/n. (MN)

In this article, the author discusses the Soroban (Japanese abacus) in the age of computers and its structure. Since the advent of computers, the Soroban has started to shift from being used purely as a calculating device to being a useful tool in general mathematics education. The beauty of the Soroban is that it represents numbers exactly as you…

While attending a Postgraduate Certificate in Education (PGCE) course, this author was provided with the time and encouragement to try out different activities, styles, and strategies of teaching. She used this time to develop numeracy starter lessons with Y7 students, spending two weeks with each of four sets of students. She decided from the…