Young children solving additive structure problems

This paper describes a study to analyse how 4-6-year-olds (N=45) children solve different types of additive reasoning problems. Individual interviews were conducted on kindergarten children when solving the problems. Their performance as well as their explanations were analysed when solving additive reasoning problems. The additive reasoning problems comprised simple, inverse and comparative problems. Results suggested that Portuguese kindergarten children have some informal knowledge that allowed them to solve additive structure problems with understanding. Children performed better in the simple additive problems and found the comparative problems more difficult.CIEC – Research Centre on Child studies, UM (FCT R&D 317

Mental calculation : its place in the development of numeracy

The current concerns about the standards of numeracy in primary schools, as these are manifest in different official reports (HMI, 1997; DfEE, 1998), have given a revised emphasis to mental calculation. While not completely discounting the wider aspects of mathematical achievement, the topics of space and shape, data handling and measurement are being de-emphasised (Brown et al, 2000) and mental calculation is being emphasised, with there being daily opportunities for children to develop efficient and flexible mental methods of calculating (QCA, 1999; Wilson, 1999). However, the term, mental calculation is not clearly defined (Harries and Spooner, 2000) and without conceptual clarity it may be very difficult for us to recognise, let alone understand, what pedagogical practices are needed to support the objective of increased emphasis on mental calculation. What follows is some consideration of what is meant by the term mental calculation and what this meaning implies for practice

Making Algebra More Accessible: How Steep Can it be for Teachers?

Teacher educators need to support middle grades teachers in developing mathematical knowledge for teaching algebraic concepts. In particular, teachers should become familiar with possible introductions and sequencing to the concept of slope, and common middle school students’ limited conceptions about measuring the steepness of an incline. Steepness can be expressed directly in terms of an angle or indirectly as a slope. Encouraging middle school students to find a measure of steepness using a ratio may help support students’ transition to multiplicative thinking. This mixed – methods study explores middle school students’ responses in solving a comparison problem involving the steepness of two inclines, in order to gain insight into common student strategies. The quantitative portion of the study involved written surveys distributed to 256 Grade 7 participants in the United States. We examined the frequency and types of solutions offered by these participants. We found that 27% of the participants provided an incorrect solution which was consistent with additive reasoning. The qualitative portion of this study consisted of small group interviews of 19 Grade 7 participants, who were conflicted in the different solutions they produced from using additive reasoning and their geometric knowledge

Distinguishing schemes and tasks in children's development of multiplicative reasoning

We present a synthesis of findings from constructivist teaching experiments regarding six schemes children construct for reasoning multiplicatively and tasks to promote them. We provide a task-generating platform game, depictions of each scheme, and supporting tasks. Tasks must be distinguished from children’s thinking, and learning situations must be organized to (a) build on children’s available schemes, (b) promote the next scheme in the sequence, and (c) link to intended mathematical concepts

Mathematics

This chapter discusses mathematics. It is part of a collection which examines educational practice and professional thinking from pre-school and primary, through secondary, further and higher education; and locates Scottish education within its social, cultural and political context

Teaching mathematics : self-knowledge, pupil knowledge and content knowledge

Mathematical learning is significantly influenced by the quality of mathematics teaching (Hiebert and Grouws 2007). In spite of the evidence for teachers seeking to do what they believe to be in the best interests of their learners (Schuck 2009; Gholami and Husu 2010), research and policy reports (within the UK and beyond) draw attention to insufficient mathematical attainment (Williams 2008; Eurydice 2011). Why is there this discrepancy? On the one hand, teachers are open to improving their professional practices (Escudero and S´anchez 2007), and on the other, the findings of mathematical education research make little or no impact on teachers’ practice (Wiliam 2003), even although teachers themselves think that they are enacting new or revised practices (Speer 2005)

Mapping kindergartners’ quantitative competence

In this study we investigated the structure of quantitative competence of kindergartners by testing a hypothesized four-factor model of quantitative competence consisting of the components counting, subitizing, additive reasoning and multiplicative reasoning. Data were collected from kindergartners in the Netherlands (n = 334) and in Cyprus (n = 304). A confirmatory factor analysis showed that the four-factor structure fitted the empirical data from the Netherlands. For the Cyprus data a one-factor structure was found to have a more adequate fit. Regarding the effect of country on performance, a comparison at item level showed that the kindergartners in the Netherlands outperformed those in Cyprus in the majority of quantitative competence items. Analyses of variance revealed for each country a significant effect of kindergarten year on performance, with children in K2 (second kindergarten year) outperforming those in K1 (first kindergarten year). A statistical implicative analysis at item level revealed that in both countries the relevant implicative chain, showing what successful solving of an item implies for correct solving of another item, reflects by and large the sequential steps mostly followed in teaching kindergartners early number. This sequence starts with counting and subitizing, then continues with additive reasoning and finally multiplicative reasoning. These implicative chains also clearly show that the development of early quantitative competence is not linear. There are many parallel processes and cross-connections between the components of quantitative competence.publishedVersionPaid Open Acces

Focusing on young chindren’s additive and multiplicative reasoning

This paper describes a brief study to analyse how 4-6-years-old children solve different types of additive and multiplicative reasoning problems. Individual interviews were conducted on kindergarten children when solving the problems. Their performance as well as their explanations were analysed when solving additive and multiplicative reasoning problems. The additive reasoning problems comprised simple, inverse and comparative problems; the multiplicative ones comprised simples and inverse problems. Results suggested that Portuguese kindergarten children have some informal knowledge that allowed them to solve additive and multiplicative reasoning problems with understanding.CIEC – Research Centre on Child studies, UM (FCT R&D 317

Teachers’ actions in classroom and the development of quantitative reasoning

International audienceThe main goal of this paper is to understand how teachers’ actions in classroom influence the way students solve a task involving quantitative additive reasoning. The study used a qualitative methodology and a teaching experiment was carried out. One task was proposed to two different classes (2nd and 3rd graders), of the same public school, with two different teachers. The results show that 2nd grade students solved the task with ease, but 3rd graders had difficulties. Teachers’ actions of guiding and challenging and how teachers developed the process of communication in classroom do influence students’ performance