John Threlfall

Flexible Mental Calculation

The ability to be flexible in mental calculation by using a method that is efficient for calculating the particular problem being faced is an important aim of teaching in this area. Flexibility is commonly seen as arising from a rational choice between mental calculation ‘strategies’, based on the characteristics of the problem faced. In this article it is argued that strategy choice is a misleading characterisation of efficient mental calculation, and that teaching mental calculation methods as wholes is not conducive to flexibility. An alternative is proposed in which calculation is thought of as an interaction between noticing and knowledge, and an associated teaching approach to promote flexibility is described.

Computer or Paper? That Is the Question: Does the Medium in Which Assessment Questions Are Presented Affect Children's Performance in Mathematics?.

This paper reports on the results of a study of English children’s performance on a computer mathematics assessment compared with a pencil-and-paper assessment. Two matched samples of children were each assessed on one of two mathematics pencil-and-paper tests and assessed a month later on a cloned computer test. The performance scores were better on the computer tests than on the pencil-and-paper tests, although this was not statistically significant in every case. The paper goes on to discuss some of the differences between performance and approach to the question between the two media, and explores possible reasons for these differences. In conclusion, the computer tests were found to have an overall positive effect on children’s performance, although not for every child and, in some instances, the computer assessment limited the way in which a question could be answered.

THE FORMATIVE USE OF ASSESSMENT INFORMATION IN PLANNING – THE NOTION OF CONTINGENT PLANNING

ABSTRACT:  This article is concerned with the relationship between assessment information and teacher planning. In the UK, although planning used to be central to characterisations of formative assessment, the most recent government proclamations under the ‘Assessment for Learning’ banner offer no clear role for teachers making decisions about what to do based on assessment information. In this article, the reasons behind the shift will be examined. ‘Contingent planning’ will be proposed as a mechanism for using assessment information in a planning context.

Children's Strategies with Number Patterns

Summary Pattern and generalisation are both thought to be fundamental to mathematics and are therefore important in mathematics education. This study investigates how children throughout the junior age range generalise about number patterns and the cognitive processes involved in this. The data were collected from 315 children aged between 7 and 11 years, by means of a workbook which gave children the opportunity to work with different types of pattern within different tasks. Analysis of the data revealed the different types of generalisation made by the children and the different cognitive processes which seemed to be involved. This paper examines specifically the strategies children use when they make generalisations about individual number patterns and briefly discusses how these vary across year groups.

The Problem-Solving Methods of Mathematically Gifted and Older Average-Attaining Students.

This paper examines some of the ways gifted students are said to be different from non‐gifted students by comparing the responses of 475, 9‐year‐old “gifted” students with those of 230 average‐attaining 13‐year‐old students on the same mathematical problem‐solving questions. The questions were specifically written for mathematically gifted 9 year olds as part of the World Class Tests project. The performance and approaches used by students in the two samples were found to be very similar, as was the frequency of different responses to the questions, suggesting that many of the mathematically “gifted” are not qualitatively different in their problem‐solving approaches from students of average ability, but are merely precocious.

Uncertainty in mathematics teaching: the National Curriculum experiment in teaching probability to primary pupils

During the 1980s and 1990s, the quantity of probability being taught in mathematics lessons in primary schools in England and Wales went from minimal to extensive and back again. A great deal of effort was invested in a project to incorporate probability into the primary curriculum that has now almost been abandoned. In this article I will attempt to explain this by considering the probability experiences that primary children had, in relation to the hopes for learning that the curriculum contained. I shall argue that what was done did not lead to mathematical learning because in most cases it was not mathematical experience. One consequence of this was that primary aged children did not learn anything about probability that could be reliably assessed, and so probability as a curriculum component did not contribute to some of the purposes of the National Curriculum.During the 1980s and 1990s, the quantity of probability being taught in mathematics lessons in primary schools in England and Wales went from minimal to extensive and back again. A great deal of effort was invested in a project to incorporate probability into the primary curriculum that has now almost been abandoned. In this article I will attempt to explain this by considering the probability experiences that primary children had, in relation to the hopes for learning that the curriculum contained. I shall argue that what was done did not lead to mathematical learning because in most cases it was not mathematical experience. One consequence of this was that primary aged children did not learn anything about probability that could be reliably assessed, and so probability as a curriculum component did not contribute to some of the purposes of the National Curriculum.

MENTAL CALCULATION STRATEGIES

Some of the approaches used by children when calculating mentally involve a flexible and inferential use of number knowledge. These approaches are often called ‘strategies’, and it is a reasonable ambition for teaching to enhance childrens skill in this area. But what exactly does effective performance of this kind involve? This paper considers to what extent childrens mental calculation ‘strategies’ are-in any meaningful sense-strategic, and explores the implications for teaching mental calculation, including the viability of direct instruction in ‘strategies’.

FUNCTIONAL MATHEMATICS: WHAT IS IT?

The authors of this paper are part of one of the teams funded by The Qualifications and Curriculum Authority (QCA) to research pathways in mathematics for the 14–19 age group. Apart of their work has been to examine the concept of functional mathematics and attempt to define it. The paper sets out the origins of the phrase, the problems that the concept is called upon to address and previous solutions to these problems. Through the lessons drawn from these previous solutions the paper attempts to give meaning to the concept of functional mathematics.

“Just” counting: Young children's oral counting and enumeration

SUMMARY This article focuses on the specific skills and abilities of young children in oral counting and enumeration. Responses to an oral counting task and an enumeration task by a sample (n=93) of 3- and 4-year old children attending a range of pre-five establishments in an urban district of northern England are described. The findings, whilst providing partial support to previous work, have features that differ from the results in earlier studies, and offer some additional insights into young childrens performance in oral counting and enumeration. The implications for those working with young children, as well as for the direction of further research, are discussed.

The Role of Practical Apparatus in the Teaching and Learning of Arithmetic

Abstract The use of practical apparatus to help children to learn about arithmetic has a long history, but many teachers have found that it is not as useful as theorists claim and in the current review of practice occasioned by the National Curriculum its use is being questioned. This article reviews the arguments for the value of practical number apparatus and examines the practical activities for which the apparatus is used, to try to explain why the theoretical benefits do not seem to be translating into practice. A number of possible reasons for the divergence are found, most notably the widespread unhelpful use of the apparatus as an aid to calculation. A change of emphasis in the use of number apparatus is recommended, rather than its rejection.