The article describes how one fifth-grade teacher helped her students to exercise their Rights of the Learner (e.g., to be confused; to claim a mistake; to speak, listen, and be heard; and to write, do, and represent what makes sense) as they learned to graph and interpret non-linear data.

This work is an exploration of upper elementary students' sense making around four conventional representations of function: equations with algebraic notation, Cartesian graphs, function tables, and natural language. The cornerstone to the empirical work is a task called the Function Puzzle, where students are given 16 cards representing four…

The goal of this study was to document the characteristics of students' dispositions towards mathematics when they engaged in the exploration of parts of unsolved problems: Graceful Tree Conjecture and Collatz Conjecture. Ten students, Grades 4 and 5, from an after-school program in the Midwest participated in the study. I focused on the…

It is now well known that fractions are difficult concepts to learn as well as to teach. Teachers usually use circular pies, rectangular shapes and number lines on the paper as teaching tools for fraction instruction. This article contributes to the field by investigating how the widely used three external graphical representations (i.e., circle,…

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…

This article reports on students' problem-solving approaches across three representations--number lines, coordinate planes, and function graphs--the axes of which conventional mathematics treats in terms of consistent geometric and numeric coordinations. I consider these representations to be a part of a "hierarchical representational…

This paper looks at 21 fifth grade students as they discuss a linear graph in the Cartesian plane. The problem presented to students depicted a graph showing distance as a function of elapsed time for a person walking at a constant rate of 5 miles/h. The question asked students to consider how many more hours, after having already walked 4 h,…

In Singapore, primary school students are taught to use bar diagrams to represent known and unknown values in algebraic word problems. However, little is known about students' understanding of these graphical representations. We investigated whether students use and think of the bar diagrams in a concrete or a more abstract fashion. We also…

This dissertation explores fifth and eighth grade students' interpretations of three kinds of mathematical representations: number lines, the Cartesian plane, and graphs of linear functions. Two studies were conducted. In Study 1, I administered the paper-and-pencil Linear Representations Assessment (LRA) to examine students'…

With regard to transfer, is it better to provide pupils with ready-made representations or is it more effective to scaffold pupils' thinking in the process of generating their own representations with the help of peers and under the guidance of a teacher in a process of guided co-construction? The sample comprises 10 classes and 239 Grade 5…

Every student should understand and use all concepts and skills from the previous grade levels. The standard is designed so that new learning builds on preceding skills. Communications, Problem-solving, Reasoning & Proof, Connections, and Representation are the process standards that are embedded throughout the teaching and learning of all…

A method of graphing multivariate data, called the star plot, is very useful in educational contexts because it provides teachers and guidance counselors with information not present in unsupplemented student achievement data. The utility of the method can be seen in analysis of data from a study of factors related to the problem-solving ability…

This study compared the results of group tests of computational skills and tests of knowledge and skills in mathematics between three groups of first and fifth grade students from the United States, Japan, and Taiwan. The battery of tests were constructed after reviewing the mathematics textbooks used in each location and discussing the major…

This publication contains an updated academic standards of Arizona public schools for grade 5. The contents of this document include the following: (1) The Arts Standard 2006--Grade 5; (2) Comprehensive Health Education/Physical Activity Standards 1997--Essentials (Grades 4-8); (3) Foreign and Native Language Standards 1997--Foundations (Grades…