The ideas of measurement and measurement comparisons (e.g., fractions, ratios, quotients) are introduced to students in elementary school. However, studies report that students of all ages have difficulty comparing two quantities in terms of their relative size. Students often understand fractions such as 3/7 as part-whole relationships or…

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…

We introduce a methodology for diagramming the ways students use sequences of mental actions to solve mathematical tasks. We studied 12 pre-service teachers as they solved a set of fractions tasks, ranked by cognitive demand. We present the unit transformation graphs for one of those pre-service teachers, to illustrate how she experienced and met…

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…

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 (STEM) domains. We focus on two types of representational competencies: (1) "sense making" of connections by verbally…

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)

Students use a spinner device to generate numbers, which they then insert into given open sentences and search for true statements. (MM)

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)

This guide is a continuation of the series of guides starting with SE 012 723. The entire course is ungraded, but the material referred to in this guide would probably be covered in grade five or six. The concept of place value is studied more deeply using Dienes' Multiple Attribute Blocks, and used in the further development of formal algorithms.…

This guide is the seventh of a series of seven curriculum guides included in SE 012 723, SE 012 724, and SE 012 725, and concludes the primary school mathematics course (grades one through six in Victorian schools, Australia). It is expected that "a substantial majority of children leaving grade six will have completed Section H. Some…