This paper deals with a set of geometrical problems for mathematical problem-solving at different difficulty levels. All of these are presented as national flags and one has to investigate invariant area proportions when changing the locus of any corner of the flag. This dynamic element suggests the usage of dynamical geometry environments.…

Mathematical problem posing is an important skill for teachers of mathematics, and relates readily to mathematical creativity. This article gives a bit of background information on mathematical problem posing, lists further references to connect problem posing and creativity, and then provides 20 problems based on the multiplication table to be…

This article describes how a sixth-grade teacher's "missing word" task uncovered higher-level thinking and engaged her students in the Standards for Mathematical Practice. The role and selection of the task promotes higher-level thinking and connects to the Common Core State Standards for Mathematics. The task required students to apply…

Not only are the problems teachers pose throughout their teaching of great importance but also the ways in which they use those problems make this a critical component of teaching. A problem-posing episode includes the problem setup, the statement of the problem, and the follow-up questions. Analysis of problem-posing episodes of precalculus…

Many teachers have designed lessons for students who will be working in groups to discuss and solve a problem. After investing time in constructing an interesting problem, creating strategically designed groups, and introducing the problem carefully, teachers may be left wondering how to help students collaborate to make sense of mathematical…

This article presents a problem set which includes a selection of probability problems. Probability theory started essentially as an empirical science and developed on the mathematical side later. The problems featured in this article demonstrate diversity of ideas and different concepts of probability, in particular, they refer to Laplace and…

Too often the discourse of the mathematics classroom is defined as the teacher asking the questions and the students answering them--or trying to. Certainly teachers should not be prohibited from asking questions, but if students are always placed in the position of responding rather than initiating, then one can hardly be surprised if at times…

By working through well-designed tasks, students can expand their thinking about mathematical ideas and their approaches to solving mathematical problems. They can come to see the value of looking at tasks from different perspectives and of using different representations. This article discusses four tasks that encourage high school students and…

The mathematical topic of inverse functions is an important element of algebra courses at the high school and college levels. The inverse function concept is best understood by students when it is presented in a familiar, real-world context. In this article, the authors discuss some misconceptions about inverse functions and suggest some…

This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos--satisfying an axiom sin[superscript 2] + cos[superscript 2] = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two…

In an elementary undergraduate abstract algebra or group theory course, a student is introduced to a variety of methods for constructing and deconstructing groups. What seems to be missing from contemporary texts and syllabi is a theorem, first proved by Edouard Jean-Baptiste Goursat (1858-1936) in 1889, which completely describes the subgroups of…

An undergraduate seminar was designed to help preservice teachers focus on students' learning. Preservice teachers planned and conducted weekly tutoring sessions with fourth graders and discussed their experiences in weekly discussions. The author investigated what preservice teachers learned about teaching mathematics from their focus on…

Several decades ago, V. A. Krutetskii conducted a multiyear study to investigate the various types of thinking that academically advanced, or as he called them, gifted mathematicians used. Following an in-depth look at Krutetskii's nine ways of thinking, a model is proposed that will provide direction for teachers in selecting problems. The model…

We present a general formula for a triple product involving four real numbers. As a particular case, we get the sum of a triple product of four odd integers. Some interesting results are recovered. We derive a general formula for more than four odd numbers.

In early 2008 researchers from the University of Melbourne's "New Technologies for Teaching Mathematics" project created a lesson for the Year 10 students at their Victorian research schools. Two important goals of secondary school mathematics education are to build students' conceptual knowledge and to teach students to think…