The double-edged sword of conjecturing

ABSTRACT

Employing a statistical modeling inspired pedagogy is becoming a widespread practice in the statistics education community. Many have incorporated the practice of formulating conjectures in their modeling-enhanced educational designs and have reported on its benefits. We further elucidate the mechanism through which students’ conjecturing may be beneficial, in particular to their emergent reasoning with informal statistical models and modeling, as well as examine what challenges it may entail – the double-edged sword of conjecturing. We introduce a framework to describe young learners’ reasoning with informal statistical models and modeling (RISM), in which students’ conjecturing is represented as one of two parallel planes of model creation and refinement. We offer a case study of a pair of students’ participation in an integrated modeling learning sequence, including both real-world modeling tasks and probability-world modeling tasks. The pair was chosen as both students held strong, opposing real-world conjectures. Our goal is to elucidate the roles these conjectures can play, for better or for worse, to fully harvest the pedagogical potential of conjecturing.

KEYWORDS:

• Conjectures
• informal statistical models and modeling
• reasoning with statistical models and modeling
• integrated modeling approach
• statistics education

Acknowledgments

We thank the University of Haifa, the I-CORE Program of the Planning and Budgeting Committee and the Israel Science Foundation grant 1716/12 for supporting this research, as well as the Connections research team. We are also grateful to the Azrieli Foundation for the award of an Azrieli Fellowship.

Notes

1. Dvir and Ben-Zvi (2018) provide a detailed explanation of the distinction between mathematical and statistical models.

2. Some of the ideas discussed in this Section have been previously published in Dvir and Ben-Zvi (2018, 2019).

3. The students are asked to model their conjecture, as opposed to a null hypothesis, both to maintain a connection between the two worlds of inquiry (real and probabilistic), as well as to allow them to more deeply examine the underlying probabilistic mechanism (e.g., uncertainty related to samples’ behavior) of their conjecture model.

4. A full depiction of the learning sequence can be found at http://connections.edtech.haifa.ac.il.

5. The TinkerPlots Sampler allows students to expand the focus on data and statistics and incorporate probability, by designing and running probability simulations.

6. The Divider tool divides a fully-separated numeric variable graph into sections. It is typically utilized to indicate the center of the distribution with a gray box, but the width and the location of the intervals – the boundaries of the gray box – can be easily changed.

7. The researcher’s goal was to encourage the pair to examine the probabilistic mechanism of the Sampler model the pair had created, and begin to consider the sampling variability associated with their real sample size.

8. The pair drew another sample but did not notice that the representation they were examining was not an updated one, and they were still examining the first simulated sample (Figure 11).

Additional information

Funding

This work was supported by the Azrieli Foundation [non]; Israel Science Foundation [1716/12]; Israeli Centers for Research Excellence [non].

Notes on contributors

Michal Dvir

Michal Dvir is a doctorate student in the Mathematics Education Department at the University of Haifa. Her background is in abstract mathematics. Her thesis on young learners’ statistical reasoning received the University of Haifa Rector award. Based on her achievements in her doctorate research, she is part of the Azrieli Fellowship Program promoting academic excellence and leadership. ORCID ID: https://orcid.org/0000-0003-4777-2457. E-mail: [email protected].

Dani Ben-Zvi

Dani Ben-Zvi is a professor of statistics education and educational technologies in the Faculty of Education at the University of Haifa, Israel. He is the head of the Department of Learning, Instruction & Teacher Education. Dani’s research interests draw upon two central aspects of human life: Statistical thinking and technology. Focusing on these two aspects in his research, he studies (a) students’ statistical learning and the development of their statistical reasoning as it occurs in the social context of the classroom; and (b) the role of innovative technological tools in offering new forms of understanding, learning and communicating. ORCID ID: https://orcid.org/0000-0002-9946-3456. SCOPUS Author ID: 24448178900. E-mail: [email protected].