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

Understanding and facilitating the development of intellect

Information flows continuously in the environment. As we attempt to do something, our senses receive large volumes of information. In any conversation, messages are exchanged rapidly. To understand meaning, we have to focus, record, choose and process relevant information at every moment, before it is displaced by other information. Often, information is incomplete or masked by other information or the problems to be solved are new to us. Thus, we must compare different aspects of information or other messages, and use deduction to fill in the gaps in the information, connect it with what we already know or invent solutions to new problems. Children at school learn new concepts every day. Reading, arithmetic or science are very demanding for them. To learn, children must hold information in their heads, use previously acquired concepts to interpret new information and then change their understanding as required. These tasks are possible because we can focus on information and process it before it disappears, alternate between stimuli or concepts according goals, and make decisions based on an understanding and evaluation of information through reasoning. At the same time, we adjust our strategies according to what we already know or depending on our strengths and weaknesses. To understand human intelligence, psychological and cognitive sciences try to specify what cognitive processes are involved in dealing with the above-mentioned tasks, how these processes change during learning, why individuals have different capacities, and how biology and culture may influence them. Any systematic attempt to improve intelligence through education would have to build on the knowledge assembled by research since the end of the nineteenth century. In this booklet we outline how the sciences of the mind view intelligence and suggest a programme for instruction that may uild upon its various processes

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

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

Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis

The tangent line is a central concept in many mathematics and science courses. In this paper we describe a model of students’ thinking – concept images as well as ability in symbolic manipulation – about the tangent line of a curve as it has developed through students’ experiences in Euclidean Geometry and Analysis courses. Data was collected through a questionnaire administered to 196 Year 12 students. Through Latent Class Analysis, the participants were classified in three hierarchical groups representing the transition from a Geometrical Global perspective on the tangent line to an Analytical Local perspective. In the light of this classification, and through qualitative explanations of the students’ responses, we describe students’ thinking about tangents in terms of seven factors. We confirm the model constituted by these seven factors through Confirmatory Factor Analysis

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

Rapid and Sensitive Assessment of Globin Chains for Gene and Cell Therapy of Hemoglobinopathies

The β-hemoglobinopathies sickle cell anemia and β-thalassemia are the focus of many gene-therapy studies. A key disease parameter is the abundance of globin chains because it indicates the level of anemia, likely toxicity of excess or aberrant globins, and therapeutic potential of induced or exogenous β-like globins. Reversed-phase high-performance liquid chromatography (HPLC) allows versatile and inexpensive globin quantification, but commonly applied protocols suffer from long run times, high sample requirements, or inability to separate murine from human β-globin chains. The latter point is problematic for in vivo studies with gene-addition vectors in murine disease models and mouse/human chimeras. This study demonstrates HPLC-based measurements of globin expression (1) after differentiation of the commonly applied human umbilical cord blood-derived erythroid progenitor-2 cell line, (2) in erythroid progeny of CD34+ cells for the analysis of clustered regularly interspaced short palindromic repeats/Cas9-mediated disruption of the globin regulator BCL11A, and (3) of transgenic mice holding the human β-globin locus. At run times of 8 min for separation of murine and human β-globin chains as well as of human γ-globin chains, and with routine measurement of globin-chain ratios for 12 nL of blood (tested for down to 0.75 nL) or of 300,000 in vitro differentiated cells, the methods presented here and any variant-specific adaptations thereof will greatly facilitate evaluation of novel therapy applications for β-hemoglobinopathies

Magnetic Quantum Tunneling: Insights from Simple Molecule-Based Magnets

This article takes a broad view of the understanding of magnetic bistability and magnetic quantum tunneling in single-molecule magnets (SMMs), focusing on three families of relatively simple, low-nuclearity transition metal clusters: spin S = 4 Ni4, Mn(III)3 (S = 2 and 6) and Mn(III)6 (S = 4 and 12). The Mn(III) complexes are related by the fact that they contain triangular Mn3 units in which the exchange may be switched from antiferromagnetic to ferromagnetic without significantly altering the coordination around the Mn(III) centers, thereby leaving the single-ion physics more-or-less unaltered. This allows for a detailed and systematic study of the way in which the individual-ion anisotropies project onto the molecular spin ground state in otherwise identical low- and high-spin molecules, thus providing unique insights into the key factors that control the quantum dynamics of SMMs, namely: (i) the height of the kinetic barrier to magnetization relaxation; and (ii) the transverse interactions that cause tunneling through this barrier. Numerical calculations are supported by an unprecedented experimental data set (17 different compounds), including very detailed spectroscopic information obtained from high-frequency electron paramagnetic resonance and low-temperature hysteresis measurements. Diagonalization of the multi-spin Hamiltonian matrix is necessary in order to fully capture the interplay between exchange and local anisotropy, and the resultant spin-state mixing which ultimately gives rise to the tunneling matrix elements in the high symmetry SMMs (ferromagnetic Mn3 and Ni4). The simplicity (low-nuclearity, high-symmetry, weak disorder, etc..) of the molecules highlighted in this study proves to be of crucial importance.Comment: 32 pages, incl. 6 figure