We introduce the concept of the "sequence of the ratios of convex quadrilaterals," identify some properties of these sequences and use them to provide new characterizations for some classic quadrilateral families. The research involves aspects of geometry, arithmetic and mathematical analysis, which converge to produce the results.

This article describes an instructional sequence used with inservice secondary mathematics teachers as part of a graduate course. The instructional sequence focused on designing tasks using Interactive Geometry Software (IGS). This study answers a call for replication studies to investigate the use of an established instructional sequence. Results…

This study reports the classroom mathematical practices proposed in an instructional sequence grounding on the principles of inquiry-based learning (IBL) through the use of the dynamic geometry software (DGS) in learning to teach geometric transformations. A 5-week instructional sequence consisting of IBL activities that were guided by the…

The foundation of geometric knowledge and understanding starts in primary school, and in contemporary mathematics education there is an emphasis on identifying learning progressions and trajectories as a way of moving students forward. The van Hiele (1986) theory is well-documented and provides insights into the progression and differences of…

Building from the evidence-based learning progression in geometric reasoning from the RMFII project, this paper presents data from students' solutions to three problems in geometry and measurement situations to identify key components needed to nurture reasoning. To show emerging analytical reasoning students must coordinate multiple pieces of…

Learning progressions (LPs) describe the development of domain-specific knowledge, skills, and understanding. Each level of an LP characterizes a phase of student thinking en route to a target performance. The rationale behind LP development is to provide road maps that can be used to guide student thinking from one level to the next. The validity…

In Part 1 of their article, Erin Host, Emily Baynham and Heather McMaster use a combination of digital technology and concrete materials to explore the concept of "corners". They provide a practical, easy to follow sequence of activities that builds on students' understandings. [For "Using Digital Technology to See Angles from…

Geometry is a course that is increasingly required for students to graduate from high school. However, geometric concepts can pose a great challenge to high school youth with math difficulties. Although research on teaching geometry to students with mathematics difficulties is limited, teachers are challenged daily with providing support for youth…

We analyze the interaction of 3 students working on mathematics problems over several days in a virtual math team. Our analysis traces out how successful collaboration in a later session is contingent upon the work of prior sessions and shows how the development of representational practices is an important aspect of these participants' problem…

Measurement is an integral component of the PK-12 mathematics curriculum. At various grade levels, much attention is given to the areas of geometric shapes. However, there is research evidence that measurement is problematic for US students. In this paper, the merits and demerits of two possible sequences (conventional and alternative) in teaching…

Circle Geometry, a senior mathematics topic, is often regarded as time-consuming and associated relational concepts difficult for students to grasp. Units of work that introduce students to circle geometry theorems are frequently described as a string of tedious constructions. This article explores teacher-designed dynamic geometry software…

Engaging students with real-world activities can be effective in supporting the learning of mathematics, "whenever possible, real-world situations will provide a context for both introducing and applying geometric topics" (NCTM 1989). Tool making is a way to bring real-world mathematics into the classroom and using the tool is a way of…

Good early mathematics is broader and deeper than early practice on "school skills." High-quality mathematics should be a joy--not a pressure. It can emerge from children's play, their curiosity, and their natural ability to think. This article describes the areas of mathematics that young children can learn, and encourages elementary teachers to…

A sequence of number pairs can be used to generate many interesting examples in teaching mathematics subjects at various levels. It is often used in elementary or middle school mathematics classes to illustrate the concept of "patterns." In this paper the authors present a few interesting ways of using this sequence to form examples for high…

The objectives and the scope and sequence for the elementary school mathematics program of the Denver Public Schools (Colorado) are presented. The guide begins with a statement of the philosophy and general objectives of the K-12 mathematics program and a discussion of the methods of achieving those objectives. Several important ideas and concepts…