Moving beyond physical interactions with materials is a significant mathematical step for students that is often difficult to take. Persistent tally-mark use, for example, among older children is a testament to this challenge. For many students, shifting away from tangible tools begins a precarious journey; teachers should support it with…

One of the challenges of learning ratio concepts is that it involves intensive quantities, a type of quantity that is more conceptually demanding than those that are evaluated by counting or measuring (extensive quantities). In this paper, we engage in an exploration of the possibility of developing reasoning about intensive quantities during the…

In this capsule we give an elementary proof of the principal axis theorem within the real field, i.e., without using complex numbers.

"What do you do with the remainder when you divide?" Mrs. Thompson asked her third-grade students. They replied with such comments as, "You can't share that, because they won't be equal!" and "It's not going to come out even because you can't do that!" These answers were consistent with third- and fourth-grade student performance in a pretest and…

We relate the factorization of an integer N in two ways as N = xy = wz with x + y = w - z to the inscribed and escribed circles of a Pythagorean triangle.

Revised guidelines for Swedish early childhood education that emphasize mathematics content and competencies in more detail than before raise the question of the status of pedagogical mathematical awareness among Swedish early childhood teachers. The purpose of this study is to give an overview of teachers' current pedagogical mathematical…

Experts think in patterns and structures using the specific "language" of their domains. For mathematicians, these patterns and structures are represented by numbers, symbols and their relationships (Stokes, 2014a). To determine whether elementary students in the United States could learn to think in mathematical patterns to solve…

Mathematical creativity is rooted in the intellectual abilities and personality traits of each individual, in which the direct influence of education is only moderate. However, education could have more influence on three important components of creativity: expertise, original thinking, and intrinsic motivation, which underlie individual creative…

This research reports on a class of grade one students engaging with number concepts using the iPad application TouchCounts. It is theorized that students develop understandings of number based on a materially engagement with both social and material resources. Rhythm is the fundamental unit of analysis used to attend to student engagement with…

Being able to find the correct answer to a math problem does not always indicate solid mathematics mastery. A student who knows how to apply the basic algorithms can correctly solve problems without understanding the relationships between numbers or why the algorithms work. The Common Core standards require that students actually understand…

For many students, developing mathematical reasoning can prove to be challenging. Such difficulty may be explained by a deficit in the core understanding of many arithmetical concepts taught in early school years. Multiplicative reasoning is one such concept that produces an essential foundation upon which higher-level mathematical thinking skills…

In this paper we report on five Grade 6 students' responses to a proportional reasoning task. We conducted pair interviews within a longitudinal study focused on extending a hypothetical learning trajectory for length measurement. Results suggest that there exists a link between children's level of conceptual and procedural knowledge for length…

The application to rational numbers of the procedures and intuitions proper of natural numbers is known as Natural Number Bias. Research on the cognitive foundations of this bias suggests that it stems not from a lack of understanding of rational numbers, but from the way the human mind represents them. In this work, we presented a fraction…

In this article Jessica Hunt explores the use of clinical interviews to gain a deep understanding of students' knowledge. Examples of clinical interviews are provided and advice for planning, giving and interpreting the results of interviews is also included.

We interviewed 40 students each in grades 7 and 11 to investigate their integer-related reasoning. In one task, the students were asked to write and interpret equations related to a story problem about borrowing money from a friend. All the students solved the story problem correctly. However, they reasoned about the problem in different ways.…