GeoGebra is a dynamic software that is frequently used and of increasing importance in mathematics teaching processes in our digital age. Accordingly, in this study a new perspective has been brought to the proofs of the "two square difference identity" expressed for the square, which is a flat polygon, made with different approaches.…

Generalization is considered to be an essential part of mathematical reasoning and proving, and it has many definitions in mathematics education research. Despite its centrality, teachers often have difficulty identifying and responding to generalization in students' work. In this study, we focus on preservice teacher's (PTs) ability to describe…

This article describes "Build It!--The Rectangle Game" task that uses the context of a game to develop mathematical generalizations based on strategy. The underlying mathematics in this game-based task is for students to discover factors and prime and composite numbers through 100. The playful use of "The Rectangle Game"…

We perform dynamic investigation of two surprising geometrical properties, each of which involves additional properties. In the first task the property belongs to two regular polygons with the same number of sides and with one common vertex. It is found that all the straight lines that connect corresponding vertices of the two polygons intersect…

The purpose of this study was to analyze the prompts that preservice secondary mathematics teachers used for the acquisition of mathematics knowledge in dynamic geometry environment tasks. The participants, four preservice secondary mathematics teachers who were enrolled in a computer-supported mathematics education course, designed a dynamic…

Children's failure to reason often leads to their mathematical performance being shaped by spurious associations from problem input and overgeneralization of inapplicable procedures rather than by whether answers and procedures make sense. In particular, imbalanced distributions of problems, particularly in textbooks, lead children to create…

In the context of the sociocultural approach, the authors studied the problem of the development of the students' conceptual mental structures in the process of geometry teaching. An educational activity for the development of a generalized ability to solve geometry construction problems in electronic educational environment served as a…

Linear pattern is the primary material in learning number patterns in junior high schools, but there are still many students who fail to generalize the linear pattern. The students' failure in generalizing the pattern occurred when the students ended to view the problems globally without breaking them into the constructors' components such as the…

This study aims to analyze how the Mathematics Teacher Candidates (MTCs) define the "prism" and "pyramid" concepts in geometry. The research was carried out in 2017-2018 academic year with 92 MTCs. This test consists of 3 open-ended questions that MTCs can answer conceptually. The first openended question was as "Define a…

Previous research has verified the benefits obtained when learners trace out worked examples with the index finger. Our study conducted two experiments to explore the reasons for this phenomenon and its generalizability. Experiment 1 compared the learning effects among tracing, non-tracing, and cueing methods. The cueing method was included to…

The purpose of this study is to determine the progression of exemplifying and example generalization by students. We investigated whether example generalization occurs by analyzing collected data by identifying whether students recognize, describe, and define general features of geometric examples. We also investigate how example generalization…

The authors who are mathematics teacher educators, have found that classifying, composing, and transforming shapes (in particular, rotations and reflections) are areas of difficulty for adults as well as for children. However, these are also some of the most important geometric ideas. They are fundamental topics in the K-8 Geometry and Measurement…

Generalizing is a foundational mathematical practice for the algebra classroom. It entails an act of abstraction and forms the core of algebraic thinking. Kinach (2014) describes two kinds of generalization--by analogy and by extension. This article illustrates how exploration of fractals provides ample opportunity for generalizations of both…

A well-known result of Feinberg and Shannon states that the tribonacci sequence can be detected by the so-called "Pascal's pyramid." Here we will show that any tribonacci-like sequence can be obtained by the diagonals of the "Feinberg's triangle" associated to a suitable "generalized Pascal's pyramid."…

A key aspect of learning algebra in the middle years of schooling is exploring the functional relationship between two variables: noticing and generalising the relationship, and expressing it mathematically. This article describes research on the professional learning of upper primary school teachers for developing their students' functional…