In this article, I argue that the common practice across many school mathematics curricula of using a variety of different representations of number may diminish the coherence of mathematics for students. Instead, I advocate prioritising a single representation of number (the number line) and applying this repeatedly across diverse content areas.…

A growing emphasis on computational thinking worldwide necessitates student proficiency in creating algorithms. Focusing on the use of counterexamples for developing student-invented algorithms, I reanalyze two pieces of data from previously published research, pertaining to two different cases of students' algorithmatizing activity. In both…

This article introduces unit transformation graphs (UTGs) as a tool for diagramming the ways students use sequences of mental actions to solve mathematical tasks. We report findings from a study in which we identified patterns in the ways preservice elementary school teachers relied on working memory to coordinate mental actions when operating in…

The staircase and fractional part functions are basic examples of real functions. They can be applied in several parts of mathematics, such as analysis, number theory, formulas for primes, and so on; in computer programming, the floor and ceiling functions are provided by a significant number of programming languages--they have some basic uses in…

Despite the importance of teacher fractional knowledge, there are several areas of teacher understanding that are not well understood. The purpose of this study was to characterise profiles of pre-service teachers' (PSTs) mathematical competence on the topic of fraction multiplication by examining PSTs' understanding of multiplication of fractions…

Prior research shows that representational competencies that enable students to use graphical representations to reason and solve tasks is key to learning in many science, technology, engineering, and mathematics domains. We focus on two types of representational competencies: (1) "sense making" of connections by verbally explaining how…

Presents an investigation using data collected by students to explore timelines and pie charts. The activity begins with having students keep a chronological record of their activities for one 24-hour period to create and analyze pie charts that show how they have spent their time. Activity sheet also included. (ASK)

The author suggests that slope be introduced as a topic independent of graphs, and that it be presented in a way that makes some sense of using a fraction to represent it. Many examples are included. (MNS)

Considering a fraction as directions for graphing a line through the origin of the x-y plane is suggested. Strengths and weaknesses of the idea are also discussed. (MNS)

Grid paper activities are discussed as a way to help students conceptualize fractions. (MP)

Students are given the opportunity to deal with many kinds of data as they make graphs and interpret the results of polls. Worksheets are provided on filling in a graph grid and making bar and circle graphs as well as interpreting each. (MNS)

Describes an activity in which students examine bicycle chain-rings, cogs, and gear ratios as a means of exploring algebraic relationships, data collection, scatter plots, and lines of best fit. (KHR)

Simple experiences with probability can extend third graders' understanding of the need for fractions. Three activities are given which build on language-arts experiences and incorporate concepts in multiplication, graphs, and fractions. (MNS)

Discusses computer games (called intrinsic models) focusing on mathematics rather than on unrelated motivations (flashing lights or sounds). Games include "Green Globs," (equations/linear functions), "Darts"/"Torpedo" (fractions), "Escape" (graphing), and "Make-a-Monster" (equivalent fractions and…

Practical tips on the teaching of subtraction, addition and subtraction of fractions, and finding the vertex of a parabola are given. (MK)