This is the fifth in a series of statistical articles for mathematics teachers. In this article the authors discuss graphs for exploring relationships between one categorical variable and one numerical variable using stemplots and boxplots and between two numerical variables, using scatterplots and time series plots. [For "The Data Files 4:…

Learning curves are an important tool in cognitive diagnostics modeling to help assess how well students acquire new skills, and to refine and improve knowledge component models. Learning curves are typically obtained from a model estimated on real data obtained from a finite, and usually limited, sample of students. As a consequence, there is…

In this design experiment study, we investigated the problem: What conditions might foster students' discernment of a critical aspect--variation in unidirectional change (e.g., discerning a "decreasing" increase)? At a public middle school in a large US city, we led a sequence of three days of whole class lessons, followed by task-based,…

Statistical reasoning focuses on properties that belong not to individual data values but to the entire aggregate. We analyze students' statements from three different sources to explore possible building blocks of the idea of data as aggregate and speculate on how young students go about putting these ideas together. We identify four general…

Boxplots are statistical representations for organizing and displaying data that are relatively easy to create with a five-number summary. However, boxplots are not as easy to understand, interpret, or connect with other statistical representations of the same data. We worked at two different schools with 259 middle school students who constructed…

This paper extends work about quantitative reasoning related to covarying quantities involved in rate of change. It reports a multiple case study of three students' reasoning about quantities involved in rate of change when working on tasks incorporating multiple representations of covarying quantities. When interpreting relationships between…

In the identification process, there may be gifted students who may be unnoticed or students who are misdiagnosed and are disappointed. In this context, this study is a step that may solve these two problems about the identification of mathematically gifted students with the help of data mining, which is data analysis methodology that has been…

In the context of a generic harmonic oscillator, we investigated students' accuracy in determining the period, frequency, and angular frequency from mathematical and graphical representations. In a series of studies including interviews, free response tests, and multiple-choice tests developed in an iterative process, we assessed students in both…

Histograms are adept at revealing the distribution of data values, especially the shape of the distribution and any outlier values. They are included in introductory statistics texts, research methods texts, and in the popular press, yet students often have difficulty interpreting the information conveyed by a histogram. This research identifies…

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,…

The aim of this study is to examine middle school students' attitude towards mathematics in the context of their mathematic learning preferences using data mining which is data analysis methodology that has been successfully used in different areas including educational domains. "How do I actually learn?" questionnaire and attitude scale…

This paper shows how the application of simple statistical methods can reveal to students important insights from climate data. While the popular press is filled with contradictory opinions about climate science, teachers can encourage students to use introductory-level statistics to analyze data for themselves on this important issue in public…

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…

Some statisticians have pointed out that the field of statistics essentially exists to study the variability seen in everyday life. Because variability is so foundational, the "Guidelines for Assessment and Instruction in Statistics Education" (GAISE 2007), published by the American Statistical Association (ASA), recommend that teachers…

Research shows that multiple external representations can significantly enhance students' learning. Most of this research has focused on learning with text and 1 additional graphical representation. However, real instructional materials often employ multiple "graphical" representations (MGRs) in addition to text. An important open…