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…

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…

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 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…

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…

Almost 20 years ago, Cuoco, Goldenberg, and Mark wrote a seminal paper for the "Journal of Mathematical Behavior" entitled "Habits of Mind: An Organizing Principle for Mathematics Curricula" (Cuoco et al., 1996). The article remains as relevant today as when it was originally published. The premise of their paper is that…

This article re-considers some interrelations among Pythagorean triads and various Fibonacci identities and their generalizations, with some accompanying questions to provoke further development by interested readers or their students. (Contains 3 tables.)

Two thousand four hundred years ago Socrates gave a remarkable lesson of geometry, perhaps the first detailed record of a pedagogical method in vivo in history [Plato. (2008). "Apologia de Socrates. Menon. Cratilo." Madrid: Alianza Editorial]. Socrates asked Meno's slave 50 questions requiring simple additions or multiplications. At the…

Students often view mathematics as a set of unrelated facts and procedures and fail to make the connections between and among related topics. One role of a teacher is to help students understand that mathematics is an interrelated discipline. Another role is to assist students in the scaffolding of their knowledge so that they can make connections…

In experiment 1, novice fourth-grade students (N = 92) who compared multiple examples that separately varied each critical aspect and then simultaneously varied all critical aspects developed better conceptual knowledge about the "altitude of a triangle" than students who compared multiple examples that did not separately vary each critical aspect…

Presents a method that connects the area formulas for triangles, rectangles, parallelograms, and trapezoids by focusing on the relationships between the bases and heights of each figure. Transformations allow figures to be reconceptualized to establish a general concept of area that can be applied to other figures. (MDH)