Helen M. Doerr’s research while affiliated with Syracuse University and other places

Making inferences about unknown populations is central in statistical reasoning. However, little attention has been paid to empirical investigations of how and why students develop sampling models when investigating a categorical variable whose values are nominal. This paper reports on an intervention that draws on the models and modeling perspective, where 25 pre-service teachers were asked to develop a sampling model that could be used to make inferences about the number of different colored beads and the distribution of different colored beads in different sized populations. Using a thematic analysis, three main results about the characteristics of the students' models of sampling and inference with nominal variables were identified: to catch all the low frequency colors in the population; not to overestimate the low frequency colors in the population; and, to formalize the relationships used in making inferences. These results highlight several issues about students' understanding of the relationship between sample representativeness and sample variability and its consequences for making inferences. Exploring students' models of sampling and inference… 159 Quadrante 30(1) 158-177 Resumo. Fazer inferências sobre populações desconhecidas é fundamental no raciocínio estatístico. No entanto, pouca atenção tem sido dada às investigações empíricas sobre como e com que objetivo os alunos desenvolvem modelos de amostragem ao investigarem uma variável categórica cujos valores são nominais. Tendo por base a perspetiva de modelos e modelação, este artigo relata uma intervenção, durante a qual 25 professores em formação inicial foram solicitados a desenvolver um modelo de amostragem que pudesse ser usado para fazer inferências sobre o número de contas de cores diferentes e sobre a distribuição de contas de cores diferentes em populações de tamanhos diferentes. Por meio de uma análise temática, foram identificados três resultados principais sobre as características dos modelos dos estudantes acerca de amostragem e inferência com variáveis nominais: capturar todas as cores de baixa frequência na população; não sobrevalorizar as cores de baixa frequência na população; e formalizar as relações encontradas para fazer inferências. Os resultados põem em evidência várias questões sobre a compreensão dos alunos acerca da relação entre a representatividade da amostra e a variabilidade da amostra e suas consequências na produção de inferências. Palavras-chave: variáveis categóricas; suscitar as ideias dos alunos; perspetiva de modelos e modelação; categorias nominais; formação inicial de professores; amostragem.

In this paper, we draw on a models and modeling perspective to describe the design of a sequence of tasks, known as a model development sequence, that has been used to research the teaching and learning of mathematics. A central research goal of a models and modeling perspective is the development of principles for the design of sequences of modeling tasks and for the teaching of such sequences. We extend our earlier research by elaborating how a model development sequence can be used to support students in developing models that are not only descriptive but also have explanatory power when connected to existing mathematical models. In so doing, we elaborate language issues about representations and context as well as the implementation strategies used by the teacher.

Statistical modeling allows students to construct and improve representations based on their experiences. A model development sequence is used with fifth graders to build models for comparing two distributions of the flight durations of paper helicopters. Emphasis is on the role of the teacher and using models as evidence.

Contributing to a growing body of research on undergraduate students’ quantitative reasoning, the study reported in this article used task-based interviews to investigate business calculus students’ quantitative reasoning when solving two optimization tasks situated in the context of revenue and profit maximization. Analysis of verbal responses and work written by 12 pairs of students during the task-based interviews revealed that nearly all pairs of students created new quantities (e.g., diminishing marginal returns). Students used these new quantities to reason about relationships among computer sales, sales discount, and total revenue in a revenue maximization task. The creation of these quantities helped the students to solve the problem posed in the task. Ten pairs of students interpreted marginal cost as total cost and marginal revenue as total revenue in a profit maximization task. Implications for business calculus instruction and directions for future research are discussed.

This commentary describes some disciplinary literacy practices observed in mathematics classrooms, discusses factors that might influence teachers’ decisions to integrate literacy with content instruction, and proposes that collaboration with disciplinary experts, in addition to literacy experts, may help reorient teachers’ practice toward a disciplinary take on the literacy demands of their subject area.

Mathematical models have a substantial impact at all levels of society, and hence mathematical modelling stands as an important topic in mathematics education. Mathematical modelling has a particular pedagogical/didactical discourse as modelling continues to garner attention in educational research. Diagrammatic representations of mathematical modelling processes are increasingly being used in curriculum documents on national and transnational levels. In this chapter, we critically discuss one of the most frequently used representations of modelling processes in the literature, namely, that of the modelling cycle, and offer alternative representations to more fully capture multiple aspects of modelling in mathematics education.

Teaching from an informal statistical inference perspective can address the challenge of teaching statistics in a coherent way. We argue that activities that promote model-based reasoning address two additional challenges: providing a coherent sequence of topics and promoting the application of knowledge to novel situations. We take a models and modeling perspective as a framework for designing and implementing an instructional sequence of model development tasks focused on developing primary students' generalized models for drawing informal inferences when comparing two sets of data. This study was conducted with 26 Year 5 students (ages 10-11). Our study provides empirical evidence for how a modeling perspective can bring together lines of research that hold potential for the teaching and learning of inferential reasoning. © International Association for Statistical Education (IASE/ISI), November, 2017.

In this article, we examine how a sequence of modeling activities supported the development of students’ interpretations and reasoning about phenomena with negative average rates of change in different physical phenomena. Research has shown that creating and interpreting models of changing physical phenomena is difficult, even for university level students. Furthermore, students’ reasoning about models of phenomena with negative rates of change has received little attention in the research literature. In this study, 35 students preparing to study engineering participated in a 6-week instructional unit on average rate of change that used a sequence of modeling activities. Using an analysis of the students’ work, our results show that the sequence of modeling activities was effective for nearly all students in reasoning about motion with negative rates along a straight path. Almost all students were successful in constructing graphs of changing phenomena and their associated rate graphs in the contexts of motion, light dispersion and a discharging capacitor. Some students encountered new difficulties in interpreting and reasoning with negative rates in the contexts of light dispersion, and new graphical representations emerged in students’ work in the context of the discharging capacitor with its underlying exponential structure. The results suggest that sequences of modeling activities offer a structured approach for the instruction of advanced mathematical content.

The purpose of this study was to investigate students' quantitative reasoning when solving a multivariable problem in a revenue maximization context. We conducted task-based interviews with 12 pairs of business calculus students. Analysis of verbal responses and work written by the students revealed that in reasoning about the relationships among the quantities (sales, discount, and total revenue) in the problem, nearly all the pairs of students created new quantities. The creation of these quantities helped the students to reason about the effect of the discount on sales and total revenue. An important finding of this study is that the students took different approaches to the meaning of the discount and only five pairs of the students interpreted the discount as intended in the design of the problem. Directions for future research are discussed. This study used Thompson's (1993) definition of quantitative reasoning: analyzing a problem situation in terms of the quantities and relationships among the quantities involved in the situation. According to Thompson, what is important in quantitative reasoning is not assigning numeric measures to quantities but rather reasoning about the relationships between or among quantities. Quantitative reasoning, as used in this study, refers to how students described and represented relationships between or among quantities and how they created and used new quantities to solve the problem they were given. The term quantity has been defined and used in similar ways by several researchers (e.g.

Repeated sampling approaches to inference that rely on simulations have recently gained prominence in statistics education, and probabilistic concepts are at the core of this approach. In this approach, learners need to develop a mapping among the problem situation, a physical enactment, computer representations, and the underlying randomization and sampling processes. We explicate the role of probability in this approach and draw upon a models and modeling perspective to support the development of teachers’ models for using a repeated sampling approach for inference. We explicate the model development task sequence and examine the teachers’ representations of their conceptualizations of a repeated sampling approach for inference. We propose key conceptualizations that can guide instruction when using simulations and repeated sampling for drawing inferences.