Every educator knows the sinking feeling of a lesson gone wrong. As teachers look around the room and realize that many of their students are just not getting it, they often feel like failures. However, the struggle students experience as they persevere through high-quality challenging tasks is not a sign of failure, but rather a key aspect of…

This paper analyses how pre-service teachers approach the notion of the infinite limit of a sequence from two perspectives: Specialized Content Knowledge and Advanced Mathematical Thinking. The aim of this study is to identify the difficulties associated with this notion and to classify them. In order to achieve this, an exploratory qualitative…

We report on an innovative design of algorithmic analysis that supports automatic online assessment of students' exploration of geometry propositions in a dynamic geometry environment. We hypothesized that difficulties with and misuse of terms or logic in conjectures are rooted in the early exploration stages of inquiry. We developed a generic…

This paper demonstrated how to apply Machine Learning (ML) techniques to analyze student interaction data collected in an online mathematics game. Using a data-driven approach, we examined 1) how different ML algorithms influenced the precision of middle-school students' (N = 359) performance (i.e. posttest math knowledge scores) prediction and 2)…

This paper demonstrated how to apply Machine Learning (ML) techniques to analyze student interaction data collected in an online mathematics game. Using a data-driven approach, we examined: (1) how different ML algorithms influenced the precision of middle-school students' (N = 359) performance (i.e. posttest math knowledge scores) prediction; and…

This paper demonstrates how to apply Machine Learning (ML) techniques to analyze student interaction data collected in an online mathematics game. We examined: (1) how different ML algorithms influenced the precision of middle-school students' (N = 359) performance prediction; and (2) what types of in-game features were associated with student…

Presents recently developed generalizations to the sieve of Eratosthenes, showing the principles underlying these improvements, which increase its efficiency without changing too much of its simplicity. Offers several possibilities to propose good investigations for students to explore, find patterns, and make generalizations. (JRH)

Attempts to describe a notion parallel to number sense, called symbol sense, incorporating the following components: making friends with symbols, reading through symbols, engineering symbolic expressions, equivalent expressions for non-equivalent meanings, choice of symbols, flexible manipulation skills, symbols in retrospect, and symbols in…

The article presents a general test for divisibility that includes composite numbers and shows that such a test can be used to determine divisibility by several numbers simultaneously. (MK)

Suggestions for teaching the concept of division by zero are given. (MK)

Compares the approaches to teaching division in Britain and in Holland where different emphasis is placed on the development of mental and written methods. Describes how it is common for pupils in Britain to work from an early stage with pencil and paper rather than mentally whereas early emphasis is placed on mental strategies in Holland. (ASK)

Develops a division algorithm in terms of familiar manipulations of concrete objects and presents it with a series of questions for diagnosis of students' understanding of the algorithm in terms of the concrete model utilized. Also offers general guidelines for using concrete illustrations to explain algorithms and other mathematical principles.…

An unusual algorithm for approximating square roots is presented and investigated using techniques common in algebra. The material is presented as a tool to interest high school students in the logic behind mathematics. (MP)

Describes the development of a method of generating problems that are easy to present in classroom settings because all the important points to be graphed are single-digit integers. Uses an algorithm that generates intersection problems that fit the criteria. A proof of the algorithm is included. (DDR)

PRODUCED AS PART OF A PROJECT FOR THE IMPROVEMENT OF CLASSROOM TEACHING, THIS BOOK DISCUSSES THE BASIC CONCEPTS AND OPERATIONS OF ARITHMETIC WHICH ARE TAUGHT TO ELEMENTARY SCHOOL STUDENTS. THE CONCEPT OF A SET IS USED THROUGHOUT AS A BASIS FOR EXPLANATION, AND THE DISCUSSION STRESSES WAYS BY WHICH STUDENTS CAN BE BROUGHT TO UNDERSTAND THE…