Introducing Students to Data Representation and Statistics (original) (raw)

Supporting the Development of Conceptions of Statistics by Engaging Students in Measuring and Modeling Variability

International Journal of Computers for Mathematical Learning, 2007

New capabilities in TinkerPlots 2.0 supported the conceptual development of fifth-and sixth-grade students as they pursued several weeks of instruction that emphasized data modeling. The instruction highlighted links between data analysis, chance, and modeling in the context of describing and explaining the distributions of measures that result from repeatedly measuring multiple objects (i.e., the height of the school's flagpole, a teacher's head circumference, the arm-span of a peer). We describe the variety of data representations, statistics, and models that students invented and how these inscriptions were grounded both in their personal experience as measurers and in the affordances of TinkerPlots, which assisted them in quantifying what they could readily display with the computer tool. By inventing statistics, students explored the relation between qualities of distribution and methods for expressing these qualities as a quantity. Attention to different aspects of distribution resulted in the invention of different statistics. Variable invention invited attention to the qualities of ''good'' measures (statistics), thus meshing conceptual and procedural knowledge. Students used chance simulations, built into TinkerPlots, to generate models that explained variability in a sample of measurements as a composition of true value and chance error. Error was, in turn, decomposed into a variety of sources and associated magnitudes-a form of analysis of variance for children. The dynamic notations of TinkerPlots altered the conceptual landscape of modeling, placing simulation and world on more equal footing, as first suggested by Kaput (Journal of Mathematical Behavior, 17(2), 265-281, 1998). Keywords Statistics education Á Modeling Á Learning The discipline of statistics originated in problems of modeling variability (Stigler, 1986). History has not changed all that much: Professional practices of statisticians invariably include efforts to model variability (Wild and Pfannkuch 1999). It is through the contest among alternative models that statistical concepts become more widespread and stable

Three paradigms in developing students' statistical reasoning

2016

This article is a reflection on more-than-a-decade research in the area of statistics education in upper primary school (grades 4-6, 10-12 years old). The goal of these studies was to better understand young students' statistical reasoning as they were involved in authentic data investigations and simulations in a technology-enhanced learning environment entitled Connections. The article describes three main paradigms that guided our educational and academic efforts: EDA, ISI, and Modeling. The first, EDA, refers to Exploratory Data Analysis-children investigate sample data they collected without making explicit inferences to a larger population. The second, ISI, refers to Informal Statistical Inference-children make inferences informally about a larger population than the sample they have at hand. The third, Modeling-children use computerized tools to model the phenomenon they study, and simulate many random samples from that model to study statistical ideas. In each of these three paradigms, we provide a short rationale, an example of students' products, and learned lessons. To conclude, current challenges in statistics education are discussed in light of these paradigms. Statistical reasoning Although statistics is now viewed as a unique discipline, statistical content is most often taught worldwide in the mathematics curriculum (K-12) and in departments of mathematics (college level). This has led to exhortations by leading statisticians, such as Moore (1998), about the differences between statistics and mathematics. These arguments challenge statisticians and statistics educators to carefully define the unique characteristics of statistics 13

Using students' statistical thinking to inform instruction

The Journal of Mathematical Behavior, 2001

This study designed and evaluated a teaching experiment in data exploration for second grade students. A research-based framework that incorporated a description of elementary students' statistical thinking on four constructs and across four levels of thinking informed the hypothetical learning trajectory for the teaching experiment. Qualitative evidence from four target students revealed that: (a) experiences with different kinds of data reduced children's idiosyncratic descriptions; (b) categorical data were more problematic than numerical data; (c) technology stimulated and influenced children's thinking in relation to organizing and representing data; (d) children displayed multifaceted conceptual knowledge of center and spread; and (e) children's contextual knowledge was a key factor in being able to analyze and interpret data. By the end of the intervention, the class data as a whole showed that at least 84% of the students was exhibiting Level 2 thinking or better on all constructs, i.e., they were no longer exhibiting idiosyncratic thinking. D

An analysis of students’ initial statistical understandings: developing a conjectured learning trajectory

The Journal of Mathematical Behavior, 2002

This paper reports the analysis of performance assessment tasks administered in a seventh-grade classroom. The purpose of the assessments was to obtain data on students' current statistical understandings that would then inform future instructional design decisions in a classroom teaching experiment that focused on statistical data analysis. The tasks were designed to provide information about students' current understandings of creating data, organizing data, and assessing the center and "spreadoutness" of data. In considering the analysis, we found that the students typically viewed the mean as a procedure that was to be used to summarize a group of numbers regardless of the task situation. Data analysis for these students meant "doing something with the numbers." Based on this analysis, a goal that emerged as significant for the classroom teaching experiment was to support a shift in students' reasoning towards data analysis as inquiry rather than procedure. The influence of the students' prior experiences of doing mathematics in school was also apparent when they developed graphs. They were primarily concerned with school-taught graphical conventions rather than with what the graphs signified. In the course of the analysis we distinguished between additive and multiplicative reasoning about data. This distinction is significant given that the transition from additive to multiplicative reasoning constitutes the overriding goal of statistics instruction at the middle-school level.

Towards a characterization and understanding of students' learning in an interactive statistics environment

… of the Fifth International Conference on …, 1998

We shall describe episodes of middle school students working on Exploratory Data Analysis (EDA) developed within an innovative curriculum. We outline the program and its rationale, analyze the design of the tasks, present extracts from students' activities and speculate about their learning processes. Finally, from our observations, we propose a new construct --learning arena, which is suggested as a curriculum design principle, which may also facilitate research.

Implementing a pedagogical cycle to support data modelling and statistical reasoning in years 1 and 2 through the Interdisciplinary Mathematics and Science (IMS) project

Mathematics Education Research Journal, 2023

This paper illustrates how years 1 and 2 students were guided to engage in data modelling and statistical reasoning through interdisciplinary mathematics and science investigations drawn from an Australian 3-year longitudinal study: Interdisciplinary Mathematics and Science Learning (https:// imsle arning. org/). The project developed learning sequences for 12 inquiry-based investigations involving 35 teachers and cohorts of between 25 and 70 students across years 1 through 6. The research used a design-based methodology to develop, implement, and refine a 4-stage pedagogical cycle based on students' problem posing, data generation, organisation, interpretation, and reasoning about data. Across the stages of the IMS cycle, students generated increasingly sophisticated representations of data and made decisions about whether these supported their explanations, claims about, and solutions to scientific problems. The teacher's role in supporting students' statistical reasoning was analysed across two learning sequences: Ecology in year 1 and Paper Helicopters in year 2 involving the same cohort of students. An explicit focus on data modelling and meta-representational practices enabled the year 1 students to form statistical ideas, such as distribution, sampling, and aggregation, and to construct a range of data representations. In year 2, students engaged in tasks that focused on ordering and aggregating data, measures of central tendency, inferential reasoning, and, in some cases, informal ideas of variability. The study explores how a representationfocused interdisciplinary pedagogy can support the development of data modelling and statistical thinking from an early age.

Preparing school teachers to develop students’ statistical reasoning

2008

In this paper we discuss how two different types of professional development projects for school teachers are based on the same framework and are used to prepare knowledgeable and effective teachers of statistics. The first example involves a graduate course for masters' students in elementary mathematics education at the University of Haifa, Israel. The second example is a graduate course for in-service secondary mathematics teachers, at the University of Minnesota, USA. The framework used is based on six instructional design principles described by Cobb and McClain (2004). Our view of such a classroom is a learning environment for developing a deep and meaningful understanding of statistics and helping students develop their ability to think and reason statistically "Statistical Reasoning Learning Environment" (SRLE).

Necessary Knowledge for Teaching Statistics: Example of the Concept of Variability

ICME-13 Monographs, 2018

This chapter explores teachers' statistical knowledge in relation to the concept of variability. Twelve high school mathematics teachers were asked to respond to scenarios describing students' strategies, solutions, and misconceptions when presented with a task based on the concept of variability. The teachers' responses primarily helped us analyze their comprehension and practices associated with the concept of variability and gain insight into how to teach this concept. Secondly, the study shows that students and high school teachers share the same conceptions on this subject. Keywords Professional knowledge • Statistics • Teacher's knowledge Teaching practices • Variability 10.1 Context The importance of statistics in our lives is such that data management has become a major key in the education of responsible citizens (Baillargeon 2005; Konold and Higgins 2003). The abundance of statistical data available on the internet, the studies reported on television news, or the studies and survey results published in newspapers and magazines all show that nowadays, citizens must have analytical skills to develop critical judgment and a personal assessment of the data they are confronted with daily. This role of statistics in our current society makes it necessary to consider teaching this discipline to train our students to be citizens of tomorrow. If the goal is to encourage statistical thinking in students as future citizens, then not only do we need to teach basic statistical data interpretation skills, but it is also essential to teach the

Developing students’ reasoning about samples and sampling variability as a path to expert statistical thinking

Educational Studies in Mathematics, 2014

ABSTRACT This paper describes the importance of developing students’ reasoning about samples and sampling variability as a foundation for statistical thinking. Research on expert–novice thinking as well as statistical thinking is reviewed and compared. A case is made that statistical thinking is a type of expert thinking, and as such, research comparing novice and expert thinking can inform the research on developing statistical thinking in students. It is also posited that developing students’ informal inferential reasoning, akin to novice thinking, can help build the foundations of experts’ statistical thinking.