This article describes a study of how Chinese preservice teachers unpacked a learning goal pertaining to adding fractions and understanding the concepts underlying the operation. Based on work in the USA by Morris, Hiebert, and Spizter ("Journal for Research in Mathematics Education," 40(5), 491-529, 2009), 50 Chinese preservice teachers…

In this paper, we highlight examples from school mathematics in which invariance did not receive the attention it deserves. We describe how problems related to invariance stimulated the interest of both teachers and students. In school mathematics, invariance is of particular relevance in teaching and learning geometry. When permitted change…

This articles discusses the characteristics of students' mathematical understanding in solving multiple representation tasks. Qualitative explorative methods were used to clarify the characteristics of mathematical understanding. Data were obtained by assigning multiple representation tasks to and interviewing 25 students. It is concluded that…

Myths about early education abound. Many beliefs people hold about early math have a grain of truth in them, but as a whole are not true--they are largely myths. But the myths persist, and many harm children. In this article, we address ubiquitous math myths that may be negatively affecting many young students. We conclude that avoiding the myths…

A second-grade class visits Saint John's Food Pantry each year and spends the morning learning about the needs of the community. Then students pitch in and help the staff for a few hours. When the children return to school, they create signs advertising a canned food drive, and they place collection boxes at various locations around the school.…

We propose that there exists mathematical knowledge for teaching (MKT) specific to visual representations (VRs), abbreviated MKT-VR. We define a VR as a graphic creation, such as a diagram or drawing, which illustrates quantities and shows quantitative relationships or which illustrates geometric properties of figures and shows geometric…

Twenty-four Grade 5 students participated in clinical interviews where they solved integer multiplication number sentences. Drawing on the theoretical perspective of strategies that students use with whole number multiplication and integer addition and subtraction, we describe the strategies that students employ when negative integers are…

In this study we analyze and contrast approaches that future high school math teachers show when solving word problems with the use of pencil and paper and later, with the use of a Dynamic Geometry System (DGS). The resources, representations, strategies and forms of mathematical reasoning that the participants exhibit when they use GeoGebra in…

Background: With the recent pivot to online instruction and/or to mixed online/face-to-face (i.e., hybrid) models of teaching necessitated by the novel coronavirus pandemic, the uses of technology to support instruction would seem to have great importance. Whereas effective integration of technology in mathematics classrooms requires recognition…

The article deals with mathematical literacy in relation to mathematical knowledge and mathematical problems, and presents the Slovenian project NA-MA POTI, which aims to develop mathematical literacy at the national level, from kindergarten to secondary education. All of the topics treated represent starting points for our research, in which we…

Mathematical thinking beliefs are a significant element in supporting teachers' practices of learning and teaching school mathematics. Most prior researches on mathematics beliefs have focused on teachers' beliefs about learning and teaching mathematics in general. This research aimed to validate a Mathematical Thinking Belief instruments (MTB).…

We investigated the effects of proximal grouping of numbers, problem-solving goals to make 100, and prior knowledge on students' solution strategies in an online mathematics game. Logistic regression on 857 problem-level data points from 227 middle-school students showed that students were more likely to use productive solution strategies on…

Understanding the solution of a problem may require the reader to have background knowledge on the subject. For instance, finding an integer which, when divided by a nonzero integer leaves a remainder; but when divided by another nonzero integer may leave a different remainder. To find a smallest positive integer or a set of integers following the…

As a self described lesson collector, author Charles Lovett enjoys gathering "interesting" lessons and teasing them apart to find out what makes them "tick", particularly the pedagogy. He often wonders what decisions the teacher made that generated such an interesting and successful learning environment. Here he describes a…

Around 430 BCE it is reported that a typhoid epidemic carried off about a quarter of the population of Athens in ancient Greece. When the gods were asked what could be done to halt this raging epidemic, they apparently replied that the altar in the Temple of Apollo in Delios would have to be doubled in size. After several attempts to double the…