Presents a problem introducing pupils to the ideas of consecutive and integer while practicing addition. Investigates the overall pattern of results with regard to formulating some general results, considers the algebra of the problem, and suggests proofs of why some numbers can and others cannot be formulated in this way. (ASK)

Suggests scales as alternative representations of numerical concepts and operations that can be used as arithmetic-mean machines, adding machines, multiplication machines, and geometric-mean machines. (ASK)

Presents solutions for a problem concerning doubling a recipe from a previous article on number concepts in the October 1997 issue. (ASK)

Presents research on students' use and understanding of algorithms in Poland. Concludes that there are detrimental side effects that stem from the teaching and learning of standard written algorithms. (ASK)

Discusses the role of play in the learning of mathematics. Describes a play for the third grade mathematics classroom that focuses on negative numbers and number concepts, and revisits Vygotsky from this point of view. Concludes that mathematically imaginative play can contribute greatly to mathematics learning in elementary school. (Contains 20…

Giving children time and encouragement to explain methods of calculation is a useful strategy. Uses this strategy with a group of children and discovers surprising thinking by one child on number facts. (ASK)

Discusses ways to improve the understanding of problems involving combination and complementation. Analyzes three children's initial mental representations, procedures, and reflections when presented with four different situations. Concludes with the impacts of the logico-mathematical relation of inclusion in these processes. Contains 31…

Two experiments examined 6-month-olds' ability to discriminate between visual displays of various number of dots varying in size and position, and with controls for other extraneous variables. Findings indicated that infants could discriminate between large sets on the basis of numerosity if they differed by a large ratio (8 versus 16, but not 8…

Presents some results of a study on teaching negative numbers. Focuses on the identification of addition and subtraction, the use of the number line, additive problem solving, and the possibility of following several sequences of numerical extensions. Indicates the importance of previous ideas on positive numbers and how these ideas influence the…

This brief investigation exemplifies such considerations by relating concepts from number theory, set theory, probability, logic, and calculus. Satisfying the call for students to acquire skills in estimation, the following technique allows one to "immediately estimate" whether or not a number is prime. (MM)

Discovers and analyzes the proportions of the Golden Section in the rare stone tablet, Tablet of Sippar, representing the four main feats of the god Marduk. (ASK)

Introduces data collated by Glendon Lean on nearly 900 counting systems in Papua New Guinea, Oceania, and Irian Jaya (West Papua) which came from a questionnaire completed by students and talks with village elders. Lean's thesis on the spontaneous developments of these ancient cultures challenged traditional theories describing the spread of…

For the last several years teachers have been using counters and connected cube "trains" and creating base-10 block models to help students develop number sense and understand number concepts. It is described how ten-frame tiles could be more useful tool for building number understanding for many students.

Many assertions about the occurrence of the golden ratio phi in art, architecture, and nature have been shown to be false, unsupported, or misleading. For instance, we show that the spirals found in sea shells, in particular the "Nautilus pompilius," are not in the shape of the golden ratio, as is often claimed. Some of the most interesting…

Within the set of points in the plane with integer coordinates, one point is said to be visible from another if no other point in the set lies between them. This study of visibility draws in topics from a wide variety of mathematical areas, including geometry, number theory, probability, and combinatorics.