The study of how children learn numbers has yielded one of the most productive research programs in cognitive development, spanning empirical and computational methods, as well as nativist and empiricist philosophies. This paper provides a tutorial on how to think computationally about learning models in a domain like number, where learners take…

The same and different routine offers an opportunity to build a strong foundation for future grades while also deepening the experiences of prekindergarten and transitional kindergarten students. Students have opportunities to make sense, share ideas, and engage in the ideas of others. We conjecture this type of experience supports students and…

An elementary method to calculate the area, centroid and volume of rotation of the Koch curve is presented. Classroom extensions are provided to allow students to investigate the method used.

Ben Zunica describes a lesson in which computational thinking has been successful in assisting students to understand the process of simplifying surds. The strengths and limitations of this approach are discussed. The author concludes that computational thinking can assist in solidifying understanding of a range of mathematical processes for…

Every application of integration by parts can be done with a tabular method. The trick is to identify and consider each new integral in the table before deciding how to proceed. This paper supplements a classic introduction to integration by parts with a particular tabular method called Row Integration by Parts (RIP). Approaches to tabular methods…

We propose an algorithm that allows calculating the remainder and the quotient of division between polynomials over commutative coefficient rings, without polynomial long division. We use the previous results to determine the quadratic factors of polynomials over commutative coefficient rings and, in particular, to completely factorize in Z[x] any…

Play allows young children to acquire and practice mathematics skills and concepts while engaging in meaningful and enjoyable activities (Bobis, 2010; Reed & Young, 2018). In particular, open-ended play provides children opportunities to discover materials, explore concepts, and solve problems (Rosli & Lin, 2018) and in the process,…

The article shows that even a small number of iterations give a very large length of the Koch Curve. The overall purpose of this work is to show that even a small number of iterations can give a very complex fractal structure. In this case it is the very big length of Koch's curve. As examples, the distance from the Earth to the Moon and from the…

Visuospatial representations of numbers and their relationships are widely used in mathematics education. These include drawn images, models constructed with concrete manipulatives, enactive/embodied forms, computer graphics, and more. This paper addresses the analytical limitations and ethical implications of methodologies that use broad…

Having students solve shape puzzles, such as with pattern blocks or tangrams, is a popular way to encourage students to compose shapes in different ways and talk about the effects of different compositions (Clements et al., 2004; Clements & Sarama, 2009; Sales, 1994), meeting Common Core geometry standards in kindergarten through second grade…

In this article the author describes an intervention program for students who were approximately six months behind expected standards. The program, "Prepare 2 Learn," had four components: building prior knowledge; developing mental computational fluency; encouraging a growth mindset; and meta-cognitive strategies. The qualitative data…

This article illustrates how teachers can use number lines to support students with or at risk for learning disabilities (LD) in mathematics. Number lines can be strategically used to help students understand relations among numbers, approach number combinations (i.e., basic facts), as well as represent and solve addition and subtraction problems.…

A correlation coefficient taking positive values is introduced. It is more easily understood than other correlation measures especially in social science contexts. Estimation issues are addressed. A data application is given.

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…

The aim of this article is to advise readers that natural numbers may be introduced as ordinal numbers or cardinal numbers and that there is an ongoing discussion about which come first. In addition, through several examples, the authors demonstrate that in the process of answering the question "How many?" one may, if convenient, use…