Mathematical representations are 'visible productions' that embody mathematical ideas. The author guides teachers with suggestions for the use of representations in mathematical problem solving with students. Representations can be used to help make abstract mathematical ideas accessible, allowing teachers and students to interpret, communicate,…

This article draws on the simple context of a two animal farm to create an engaging and mathematically meaningful problem solving task for students. It aims to encourage students to think systematically and to present their findings methodically. The task is structured sequentially, making it suitable for primary school students from Year 1 to…

Several studies related to mathematics understanding found that many undergraduate students lack some basic knowledge of algebra. They memorized only a few topics, formulas, and algorithms without understanding them conceptually, even though they could manipulate those limited number of points correctly or incorrectly. In comparison, most…

In this study, we aim to explain choices that groups of preservice teachers made to implement modelling tasks in primary school. We found that the preservice teachers decided to facilitate everyday-inspired contexts for the tasks and that this could be explained by their reasoning that modelling is supposed to be an activity close to reality.…

Whole-class discussion of open mathematics tasks is an instructional practice K-12 mathematics educators report has the potential to engage all learners. Because this practice has not been extensively and systematically researched, this study aims to describe and analyze the engagement and experience of learners in open mathematics tasks. Drawing…

This article explores a conceptual relationship between learner choice and mathematical sense-making. It argues that when learners can exercise choice in their mathematical activities, mathematical sense-making can be enhanced. The literature around mathematical modelling suggests a link between sense-making and learner choice. A three-tiered…

Ever try to captivate students with real-world applications only to discover that they live in a very different world? Assigning an Extreme Word Problem activity to students alleviates this dilemma by placing the authorship of a problem squarely into the hands of learners. Rather than trying to be hip and finding word problems that resonate with…

Developing textbooks of optimal quality is crucial for enriching the students' learning and understanding. This study examined fraction addition and subtraction problems in the U.S. and South Korean mathematics textbooks according to the types of denominators. In particular, we investigated Everyday Mathematics (EM) and South Korean mathematics…

Every day, 27,000 trees are used to make bathroom tissue. Americans use an average of 23.6 rolls per person per year, and more than 7 billion rolls of toilet paper are sold yearly in the United States alone. Perhaps the amount of bathroom tissue used can be reduced by changing the dimensions of the paper or the core. This brief article presents…

The first author's initial attempts at a whole-class mathematics discussion were woefully dysfunctional. Ten second graders showing their answers to a missing addend question about the number of children left riding a school bus did not result in the mathematical epiphany that the textbook promised. She did not give up on using mathematical…

Although some students might struggle with problem posing, the positive effects on student learning and abilities may be far reaching for those who engage in this activity. Problem posing requires students to create their own problems rather than to solve problems posed by others. Problem posing is not regularly taught; however, reform proponent…

Originally written 30 years ago, this paper is an analysis of the central challenge of schooling--that of engaging fully the powers of students' minds in classroom learning. This challenge maintains its relevance today. The work of engaging what John Dewey referred to as students' "inner attention" becomes the focus of an investigation…

This paper uses multimodal discourse analysis to show that discursive form, in addition to words, gestures and sound dimensions of speech, is an important linguistic resource for expressing mathematical meaning. During a collaborative task, a student spoke an insight about an algebraic property five times over a few minutes, in slightly different…

The goals of problem-solving activities in the elementary grades often include making sense of story problems, developing a range of strategies, and reaching accurate solutions. These are important mathematical aims, but they do not fully address the demands of modeling with mathematics as described in the fourth of the Common Core's eight…

This study examined pre-service primary school teachers' performance in posing problems that require knowledge of problem-solving strategies. Quantitative and qualitative methods were combined. The 120 participants were asked to pose a problem that could be solved by using the find-a-pattern a particular problem-solving strategy. After that,…