Problem Solving and Problem Posing in a Dynamic Geometry Environment

In this study, we considered dynamic geometry software (DGS) as the tool that mediates students’ strategies in solving and posing problems. The purpose of the present study was twofold. First, to understand the way in which students can solve problems in the setting of a dynamic geometry environment, and second, to investigate how DGS provides opportunities for posing new problems. Two mathematical problems were presented to six pre-service teachers with prior experience in dynamic geometry. Each student participated in two interview sessions which were video recorded. The results of the study showed that DGS, as a mediation tool, encouraged students to use in problem solving and posing the processes of modeling, conjecturing, experimenting and generalizing. Furthermore, we found that DGS can play a significant role in engendering problem solving and posing by bringing about surprise and cognitive conflict as students use the dragging and measuring facilities of the software

Developing student spatial ability with 3D software applications

This paper reports on the design of a library of software applications for the teaching and learning of spatial geometry and visual thinking. The core objective of these applications is the development of a set of dynamic microworlds, which enables (i) students to construct, observe and manipulate configurations in space, (ii) students to study different solids and relates them to their corresponding nets, and (iii) students to promote their visualization skills through the process of constructing dynamic visual images. During the developmental process of software applications the key elements of spatial ability and visualization (mental images, external representations, processes, and abilities of visualization) are carefully taken into consideration

Objects, actions, and images: a perspective on early number development

It is the purpose of this article to present a review of research evidence that indicates the existence of qualitatively different thinking in elementary number development. In doing so, the article summarizes empirical evidence obtained over a period of 10 years. This evidence first signaled qualitative differences in numerical processing, and was seminal in the development of the notion of procept. More recently, it examines the role of imagery in elementary number processing. Its conclusions indicate that in the abstraction of numerical concepts from numerical processes qualitatively different outcomes may arise because children concentrate on different objects or different aspects of the objects, which are components of numerical processing

How do first-grade students recognize patterns? An eye-tracking study

Recognizing patterns is an important skill in early mathematics learning. Yet only few studies have investigated how first-grade students recognize patterns. These studies mainly analyzed students’ expressions and drawings in individual interviews. The study presented in this paper used eye tracking in order to explore the processes of 22 first-grade students while they were trying to recognize repeating patterns. In our study, we used numerical and color pattern tasks with three different repeating patterns (i.e., repeating unit is AB, ABC, or AABB). For each repeating pattern task, students were asked to say the following object of the given pattern. For these patterns, we identified four different processes in recognizing repeating patterns. In addition, we report differences in the observed processes between the patterns used in the tasks.This project has received funding by the Erasmus+ grant program of the European Union under grant agreement No 2020-1-DE03-KA201-077597

Symbols and the bifurcation between procedural and conceptual thinking

Symbols occupy a pivotal position between processes to be carried out and concepts to be thought about. They allow us both to d o mathematical problems and to think about mathematical relationships. In this presentation we consider the discontinuities that occur in the learning path taken by different students, leading to a divergence between conceptual and procedural thinking. Evidence will be given from several different contexts in the development of symbols through arithmetic, algebra and calculus, then on to the formalism of axiomatic mathematics. This is taken from a number of research studies recently performed for doctoral dissertations at the University of Warwick by students from the USA, Malaysia, Cyprus and Brazil, with data collected in the USA, Malaysia and the United Kingdom. All the studies form part of a broad investigation into why some students succeed yet others fail

Stereometry activities with DALEST

This book reports on a project to devise and test a teaching programme in 3D geometry for middle school students based on the needs, knowledge and experiences of a range of countries within the European Union. The main objective of the project was the development (and testing) of a dynamic three-dimensional geometry microworld that enabled the students to construct, observe and manipulate geometrical figures in space and which their teachers used to help their students construct an understanding of stereometr

Beyond the obvious : mental representations and elementary arithmetic

This study seeks to answer the question: "What kinds of mental representation do children project and how may these be associated with their level of achievement in elementary arithmetic?". Drawing upon theories offering some explanation for the way in which arithmetical activity is transformed into numerical concepts and those that hypothesise the form and quality of mental representations the study suggests that qualitatively different kinds of mental representation may be associated with qualitatively different kinds of arithmetical behaviour. The evidence is drawn from the classification and categorisation of data from two series of semi-clinical interviews carried out with children aged eight to twelve who were at extremes of numerical achievement. The first, a pilot study, largely concentrated on mental representations associated with numerical concepts and skills. Its results suggest that mental representations projected by children may have a disposition towards different kinds of mental representation which transcends arithmetical and non-arithmetical boundaries. Issues raised by this study, in conjunction with a re-appraisal of the psychological evidence, informed the development of the main study. With a similar sample of children this considered the relationship between children's projections, reports and descriptions of mental representations in numerical and non-numerical contexts and in elementary arithmetic. Words, pictures, icons and symbols stimulated the projection of these representations. The evidence suggests that there is indeed a disposition towards the formation of particular kinds of mental representation. low achievers' projected mental representations which have descriptive emphasis. 'High achievers', whilst able to do the same, also project those with relational characteristics, the frequency of which increases as the stimulus becomes more 'language like'. This provides them with the flexibility to oscillate between descriptive and abstract levels of thought. The study indicates that qualitative different thinking in number processing is closely associated to a disposition towards qualitatively different kinds of mental representation. Its concluding comments suggest that these differences may have some considerable implication for the received belief that active methods may supply all children with a basis for numerical understanding

Developing an active learning environment for the learning of stereometry

This paper reports on the design of a dynamic environment for the learning of stereometry (DALEST) and the teaching of spatial geometry and visual thinking. The development of the software was in the framework of DALEST project which aimed at developing a dynamic three-dimensional geometry microworld that enables students to construct, observe and manipulate geometrical figures in space, and to focus on modelling geometric situations. The environment will also, support teachers in helping their students to construct a suitable understanding of stereometry. During the developmental process of software applications the key elements of spatial ability and visualization were carefully taken into consideration with emphasis on enhancing dynamic visualization as an act of construction of transformations between external media and student’s mind

Connections between algebraic thinking and reasoning processes

International audienceThe aim of the present study is to investigate the relationship of algebraic thinking with different types of reasoning processes. Using regression analyses techniques to analyze data of 348 students between the ages of 10 to 13 years old, this study examined the associations between algebraic thinking and achievement in two tests, the Naglieri Non-Verbal Ability Test and a deductive reasoning test. The data provide support to the hypothesis that a corpus of reasoning processes, such as reasoning by analogy, serial reasoning, and deductive reasoning, significantly predict students' algebraic thinking

The effect of two intervention courses on students' early algebraic thinking

International audienceThe aim of this study is to investigate the nature and content of instruction that may facilitate the development of students' early algebraic thinking. 96 fifth-graders attended two different intervention courses. Both courses approached three basic content strands of algebra: generalized arithmetic, functional thinking, and modeling languages. The courses differed in respect to the characteristics of the tasks that were used. The first intervention included real life scenarios, and semi-structured tasks, with questions which were more exploratory in nature. The second intervention course involved mathematical investigations, and more structured tasks which were guided through supportive questions and scaffolding steps. The findings, yielded from the analysis of pre-test and post-test data, showed that the first course had better learning outcomes compared to the second, while controlling for preliminary differences regarding students' early algebraic thinking