Arthur Bakker

Boundary Crossing and Boundary Objects

Diversity and mobility in education and work present a paramount challenge that needs better conceptualization in educational theory. This challenge has been addressed by educational scholars with the notion of boundaries, particularly by the concepts of boundary crossing and boundary objects. Although studies on boundary crossing and boundary objects emphasize that boundaries carry learning potential, it is not explicated in what way they do so. By reviewing this literature, this article offers an understanding of boundaries as dialogical phenomena. The review of the literature reveals four potential learning mechanisms that can take place at boundaries: identification, coordination, reflection, and transformation. These mechanisms show various ways in which sociocultural differences and resulting discontinuities in action and interaction can come to function as resources for development of intersecting identities and practices.

Learning to reason about distribution

The purpose of this chapter is to explore how informal reasoning about distribution can be developed in a technological learning environment. The development of reasoning about distribution in seventh-grade classes is described in three stages as students reason about different representations. It is shown how specially designed software tools, students’ created graphs, and prediction tasks supported the learning of different aspects of distribution. In this process, several students came to reason about the shape of a distribution using the term bump along with statistical notions such as outliers and sample size. This type of research, referred to as “design research,” was inspired by that of Cobb, Gravemeijer, McClain, and colleagues (see Chapter 16). After exploratory interviews and a small field test, we conducted teaching experiments of 12 to 15 lessons in 4 seventh-grade classes in the Netherlands. The design research cycles consisted of three main phases: design of instructional materials, classroom-based teaching experiments, and retrospective analyses. For the retrospective analysis of the data, we used a constant comparative method similar to the methods of Glaser and Strauss (Strauss & Corbin, 1998) and Cobb and Whitenack (1996) to continually generate and test conjectures about students’ learning processes.

Lessons from Inferentialism for Statistics Education

This theoretical paper relates recent interest in informal statistical inference (ISI) to the semantic theory termed inferentialism, a significant development in contemporary philosophy, which places inference at the heart of human knowing. This theory assists epistemological reflection on challenges in statistics education encountered when designing for the teaching or learning of ISI. We suggest that inferentialism can serve as a valuable theoretical resource for reform efforts that advocate ISI. To illustrate what it means to privilege an inferentialist approach to teaching statistics, we give examples from two sixth-grade classes (age 11) learning to draw informal statistical inferences while developing key concepts such as center, variation, distribution, and sample without losing sight of problem contexts.

Characterizing the Use of Mathematical Knowledge in Boundary-Crossing Situations at Work

The first aim of this article is to present a characterization of the techno-mathematical literacies needed for effective practice in modern, technology-rich workplaces that are both highly automated and increasingly focused on flexible response to customer needs. The second aim is to introduce an epistemological dimension to activity theory, specifically to the notions of boundary object and boundary crossing. We draw on ethnographic research in a pensions company and focus on data derived from detailed analysis of the diverse perspectives that exist with respect to one symbolic artifact, the annual pension statement. This statement is designed to facilitate boundary crossing between company and customers. Our study shows that the statement routinely failed in this communicative role, largely due to the invisible factors of the mathematical-financial models underlying the statement that are not made visible to customers or to the customer Enquiry Team whose task is to communicate with customers. By focusing on this artifact in boundary-crossing situations, we identify and elaborate the nature of the techno-mathematical knowledge required for effective communication between different communities in the pensions company, and suggest the implications of our findings for workplaces more generally.

Technology for enhancing statistical reasoning at the school level

The purpose of this chapter is to provide an updated overview of digital technologies relevant to statistics education, and to summarize what is currently known about how these new technologies can support the development of students’ statistical reasoning at the school level. A brief literature review of trends in statistics education is followed by a section on the history of technologies in statistics and statistics education. Next, an overview of various types of technological tools highlights their benefits, purposes and limitations for developing students’ statistical reasoning. We further discuss different learning environments that capitalize on these tools with examples from research and practice. Dynamic data analysis software applications for secondary students such as Fathom and TinkerPlots are discussed in detail. Examples are provided to illustrate innovative uses of technology. In the future, these uses may also be supported by a wider range of new tools still to be developed. To summarize some of the findings, the role of digital technologies in statistical reasoning is metaphorically compared with travelling between data and conclusions, where these tools represent fast modes of transport. Finally, we suggest future directions for technology in research and practice of developing students’ statistical reasoning in technology-enhanced learning environments.

Improving Work Processes by Making the Invisible Visible.

There is a growing movement for industrial companies to modify their production practices according to methodologies collectively known as process improvement. After World War II, Japanese companies such as Toyota developed new manufacturing paradigms (e.g., lean manufacturing) under the guidance of American experts, particularly W.E. Deming. Since the 1980s, the Japanese methodologies have been spreading to the West in a major way, in the form of programmes such as Total Quality Management (TQM) and Total Productive Maintenance (TPM) (Deming, 1986; Nakajima, 1988).

An Introduction to Design-Based Research with an Example From Statistics Education

This chapter arose from the need to introduce researchers, including Master and PhD students, to design-based research (DBR). In Sect. 16.1 we address key features of DBR and differences from other research approaches. We also describe the meaning of validity and reliability in DBR and discuss how they can be improved. Section 16.2 illustrates DBR with an example from statistics education.

Attributing Meanings to Representations of Data: The Case of Statistical Process Control.

This article is concerned with the meanings that employees in industry attribute to representations of data and the contingencies of these meanings in context. Our primary concern is to more precisely characterize how the context of the industrial process is constitutive of the meaning of graphs of data derived from this process. We draw on data from a variety of sources, including ethnographic studies of workplaces and reflections on the design of prototype learning activities, supplemented by insights obtained from trying out these activities with a range of employees. The core of this article addresses how different groups of employees react to graphs used as part of statistical process control, focusing on the meanings they ascribe to mean, variation, target, specification, trend, and scale as depicted in the graphs. Using the notion of boundary crossing, we try to characterize a method that helps employees to communicate about graphs and come to data-informed decisions.

Diagrammatic reasoning and hypostatic abstraction in statistics education

Abstract Peirces notions of diagrammatic reasoning and hypostatic abstraction are relevant to educational research in areas where diagrams and abstraction play an important role. In this paper, I analyze an example from statistics education in which diagrammatic reasoning created opportunities for hypostatic abstraction. For instance, where students initially characterized data points as being ‘spread out,’ they later said, ‘the spread is large.’ This is a prototypical example of hypostatic abstraction — taking a predicate as a new object that can have predicates itself. More generally, the notion of diagrammatic reasoning proved helpful to identify the key learning processes involved in learning to reason about statistical concepts.

Touchscreen Tablets: Coordinating Action and Perception for Mathematical Cognition

Proportional reasoning is important and yet difficult for many students, who often use additive strategies, where multiplicative strategies are better suited. In our research we explore the potential of an interactive touchscreen tablet application to promote proportional reasoning by creating conditions that steer students toward multiplicative strategies. The design of this application (Mathematical Imagery Trainer) was inspired by arguments from embodied-cognition theory that mathematical understanding is grounded in sensorimotor schemes. This study draws on a corpus of previously treated data of 9–11 year-old students, who participated individually in semi-structured clinical interviews, in which they solved a manipulation task that required moving two vertical bars at a constant ratio of heights (1:2). Qualitative analyses revealed the frequent emergence of visual attention to the screen location halfway along the bar that was twice as high as the short bar. The hypothesis arose that students used so-called “attentional anchors” (AAs)—psychological constructions of new perceptual structures in the environment that people invent spontaneously as their heuristic means of guiding effective manual actions for managing an otherwise overwhelming task, in this case keeping vertical bars at the same proportion while moving them. We assumed that students’ AAs on the mathematically relevant points were crucial in progressing from additive to multiplicative strategies. Here we seek farther to promote this line of research by reanalyzing data from 38 students (aged 9–11). We ask: (1) What quantitative evidence is there for the emergence of AAs?; and (2) How does the transition from additive to multiplicative reasoning take place when solving embodied proportions tasks in interaction with the touchscreen tablet app? We found that: (a) AAs appeared for all students; (b) the AA-types were few across the students; (c) the AAs were mathematically relevant (top of the bars and halfway along the tall bar); (d) interacting with the tablet was crucial for the AAs’ emergence; and (e) the vast majority of students progressed from additive to multiplicative strategies (as corroborated with oral utterances). We conclude that touchscreen applications have the potential to create interaction conditions for coordinating action and perception into mathematical cognition.