The authors illustrate three basic types of reasoning used in mathematics by showing how they operate in practical and mathematical situations. The importance and function of the different types of reasoning in each situation is outlined. As a consequence the authors note that while introducing new techniques by example is good from a pedagogical…

An important message teachers convey to their students is that mathematics is founded on reasoning and is not just a collection of rules to apply. Many mathematical rules such as area formulas can be derived in ways that are age-appropriate to students, and it is an important part of mathematics education that students engage in these derivations…

Learning about the area formulas provides many opportunities for students even at the beginning of junior secondary school to experience mathematical deduction. For example, in easy cases, students can put two triangles together to make a rectangle, and so deduce that the area of a triangle is half the area of a corresponding rectangle. They can…

Digital photos and digital movies offer an excellent way of bringing real world situations into the mathematics classroom. The technologies surveyed here are feasible for everyday classroom use and inexpensive. Examples are drawn from the teaching of Cartesian coordinates, linear functions, ratio and Pythagoras' theorem using still images, and…

Discusses the benefits of new technologies, especially graphing calculators, in the teaching and learning of mathematics. Presents two activities for teaching algebra and regression using graphing calculators. (ASK)

Introduces two methods of introducing trigonometry: ratio method and unit circle method. Presents a teaching experiment with two groups of students, half of whom were taught with ratio method and the other half with unit circle method. Concludes that ratio method students were better able to master the skills required and made greater improvements…

Outlines the embedded mathematics in which important but hidden calculations are being done for communications in commerce. Explains two key concepts, error detecting codes and error correcting codes. Illustrates these ideas using two familiar examples, barcodes and ISBN numbers. (ASK)

Demonstrates ways students think about decimals and how this changes as students get older. Presents a test teachers can use to help diagnose the mistakes students make. (MKR)

Discusses some of the aspects of problem solving that teachers can observe for assessment purposes in a student's practical and written work. Suggests that these observations can be incorporated into a student's assessment traditionally or using alternative assessment devices. (PK)

Suggests that problems designed for classroom use should include such factors as ways to introduce the problem, the skills to be developed, the mathematical prerequisites for the problem, key questions, extensions, and solutions. An example is provided which illustrates these and other points. (JN)

This article describes the factors to which the classroom teacher needs to attend to enhance the mathematical problem-solving abilities of students. Emphasis is placed upon the means necessary to develop the attribute of being a problem solver, rather than focusing on the goal of becoming a problem solver. (JJK)