ABSTRACT Data preparation (also called “wrangling” or “cleaning”) – the evaluation and manipulation of data prior to formal analysis – is often dismissed as a precursor to meaningful engagement with a dataset. Here, we re-envision data preparation in light of calls to prepare students for a data-rich world. Traditionally, curricular statistics explorations involve data that are derived from observations that students record themselves or that reflect familiar, relatively closed systems. In contrast, pre-constructed public datasets are much larger in scope and involve temporal, geographic, and other dimensions that complicate inference and blur boundaries between “signal” and “noise.” As a result, students have fewer opportunities to consider sources of variability in such datasets. Due to these constraints, we argue that data preparation becomes an important site for students to reason about variability with public data. Through analyses of repeated task-based interviews with five pairs of adolescent participants, we find that specific actions during data preparation, such as filtering data or calculating new measures, presented opportunities to engage leaners with variability as they prepared and analyzed several public socioscientific datasets. More broadly, our study highlights some changes to theory and curriculum in statistics education that are necessitated by a focus on “big data literacy”.

ABSTRACT Growing scholarship on the pedagogical applications of statistical modeling is currently taking place to create adaptations of this practice to introduce novices to statistics. These are intended to promote novices’ reasoning, and are typically void of formal mathematical procedures and calculations. In this article, we define the potential product that novices can create as they engage in such informal implementations, Informal Statistical Models (ISM), as purposeful representations – with both deterministic and stochastic components – of how observed variability in data is generated. This definition reflects the potential connection between informal constructs that novices and young students can create and the formal view of statistical models, thereby suggesting how informal insights may be later cultivated into more mature understandings. We illustrate the usefulness of this definition by providing an in-depth account of the reasoning that a pair of sixth-grade students employed through their informal statistical modeling process. The findings illustrate a four-episode sequence where students created and developed an initial deterministic model before extending it with an informal stochastic component. This shows how starting with deterministic models can be a valuable precursor to develop initial ISMs.

ABSTRACT Researchers often develop instruments using correctness scores (and a variety of theories and techniques, such as Item Response Theory) for validation and scoring. Less frequently, observations of children’s strategies are incorporated into the design, development, and application of assessments. We conducted individual interviews of 833 prekindergartens to Grade 3 children using 70 length measurement assessment items, recording both correctness and strategy use. We performed qualitative analyses of children’s strategies to validate or refine our hypotheses of how items align with strategies, and the concepts and procedures underlying them. We then used these analyses to refine: (a) our definitions of those strategies, concepts, and procedures; (b) the items in the assessment instrument; and (c) the alignment of both. The results will form the foundation of a Computer-Adaptive Test for length measurement, which we intend to validate in future research.

ABSTRACT Integrating mathematics content into science usually plays a supporting role, where students use their existing mathematical knowledge for solving science tasks without exhibiting any new mathematical meanings during the process. To help students explore the reciprocal relationship between math and science, we designed an instructional module that prompted them to reason covariationally about the quantities involved in the phenomenon of the gravitational force. The results of a whole-class design experiment with sixth-grade students showed that covariational reasoning supported students’ understanding of the phenomenon of gravity. Also, the examination of the phenomenon of gravity provided a constructive space for students to construct meanings about co-varying quantities. Specifically, students reasoned about the change in the magnitudes and values of mass, distance, and gravity as those changed simultaneously as well as the multiplicative change of these quantities as they changed in relation to each other. They also reasoned multivariationally illustrating that they coordinated mass and distance working together to define the gravitational force. Their interactions with the design, which included the tool, tasks, representations, and questioning, showed to be a structuring factor in the formation and reorganization of meanings that students exhibited. Thus, this study illustrates the type of design activity that provided a constructive space for students’ forms of covariational reasoning in the context of gravity. This design can be used to develop other STEM modules that integrate scientific phenomena with covariational reasoning through technology.

ABSTRACTThe goal of our research program is to explicate the learning of mathematical concepts in ways that are useful for instructional design and to develop design principles based on those explications. I review one type of concept and our elaboration of reflective abstraction, coordination of actions (COA) that accounts for its construction. I then postulate a second type of concept and a second type of reflective abstraction that accounts for its construction, the abstraction of commonality (AOC).

ABSTRACT The reversal error is a prevalent phenomenon that consists of reversing the relationship between two variables when writing equations from comparison in verbal language. A study with 169 Basque/Spanish bilingual pre-service teachers examined whether this error is mainly due to the students’ tendency to employ a direct-translation strategy based on the syntax of statements. By using Basque and Spanish the factor of syntax was studied since in Basque, unlike Spanish (or English), the natural way of expressing comparisons prevents students from using direct-translations strategies. Results show a significantly lower number of reversal errors in sentences written in Basque compared with those in Spanish. This finding confirms the importance of syntax statement in the translation from words into equations and suggests that teachers should have awareness of this fact and should design lesson plans devoted to work on word problems accordingly.

As befitting a treatise on design research, this volume steers a course between the pragmatics of formulating and conducting design research, on the one hand, and on the other, the ensemble of theo...

ABSTRACTThis study examined whether secondary students in an urban school district perceived racial differences in opportunity to be successful in mathematics, whether those perceptions differed between students of color and white students, and the relation of those perceptions to teachers’ choice and implementation of mathematical tasks. The results of multi-level regression models based on student survey and teacher observation data revealed two primary findings: (a) students of color were more likely to perceive opportunity differences than were white students; and (b) this difference was greater in classrooms in which teachers attempted to use cognitively demanding tasks but allowed the cognitive demand to decline during the lesson. Implications for both future research and mathematics teacher education are discussed.

ABSTRACTIn this study, we investigate students’ ways of understanding graphing tasks involving quantitative relationships in which time functions as an implicit variable. Through task-based interviews of students ages 14–16 in a summer mathematics program, we observe a variety of ways of understanding, including thematic or visual association, pointwise thinking, and reasoning parametrically about changes in the two variables to be graphed. We argue that, rather than comprising a hierarchy, these ways of understanding complement one another in helping students discover an invariant relationship between two dynamically varying quantities, and develop a graph of the relationship that captures this invariance. From these ways of understanding, we conjecture several mathematical meanings for graphing that may account for students’ behavior when graphing quantitative relationships.

Make sure you read this book to the end. Although an edited book, it is not only cohesive, but it has a plot as well.It emerged out of a research seminar held in 2012 around the Wits Maths Connect ...