On the relations between historical epistemology and students’ conceptual developments in mathematics

There is an ongoing discussion within the research field of mathematics education regarding the utilization of the history of mathematics within mathematics education. In this paper we consider problems that may emerge when the historical epistemology of mathematics is paralleled to students’ conceptual developments in mathematics. We problematize this attempt to link the two fields on the basis of Grattan-Guinness’ distinction between “history” and “heritage”. We argue that when parallelism claims are made, history and heritage are often mixed up, which is problematic since historical mathematical definitions must be interpreted in its proper historical context and conceptual framework. Furthermore, we argue that cultural and local ideas vary at different time periods, influencing conceptual developments in different directions regardless of whether historical or individual developments are considered, and thus it may be problematic to uncritically assume a platonic perspective. Also, we have to take into consideration that an average student of today and great mathematicians of the past are at different cognitive levels

Cauchy's infinitesimals, his sum theorem, and foundational paradigms

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy's proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy's proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy's proof closely and show that it finds closer proxies in a different modern framework. Keywords: Cauchy's infinitesimal; sum theorem; quantifier alternation; uniform convergence; foundational paradigms.Comment: 42 pages; to appear in Foundations of Scienc

Development of a consensus operational definition of child assent for research

Abstract Background There is currently no consensus from the relevant stakeholders regarding the operational and construct definitions of child assent for research. As such, the requirements for assent are often construed in different ways, institutionally disparate, and often conflated with those of parental consent. Development of a standardized operational definition of assent would thus be important to ensure that investigators, institutional review boards, and policy makers consider the assent process in the same way. To this end, we describe a Delphi study that provided consensus from a panel of expert stakeholders regarding the definitions of child assent for research. Methods Based on current guidelines, a preliminary definition of assent was generated and sent out for review to a Delphi panel including pediatric bioethicists and researchers, Institutional Review Board members, parents, and individuals with regulatory/legal expertise. For each subsequent review, the process of summarizing and revising responses was repeated until consensus was achieved. Panelists were also required to rank order elements of assent that they believed were most important in defining the underlying constructs of the assent process (e.g., capacity for assent, disclosure). In providing these rankings, panelists were asked to frame their responses in the contexts of younger (≤ 11 yrs) and adolescents/older children (12-17 yrs) in non-therapeutic and therapeutic trials. Summary rankings of the most important identified elements were then used to generate written construct definitions which were sent out for iterative reviews by the expert panel. Results Consensus regarding the operational definition was reached by 14/18 (78%) of the panel members. Seventeen (94%) panelists agreed with the definitions of capacity for assent, elements of disclosure for younger children, and the requirements for meaningful assent, respectively. Fifteen (83%) members agreed with the elements of disclosure for adolescents/older children. Conclusions It is hoped that this study will positively inform and effect change in the way investigators, regulators, and IRBs operationalize the assent process, respect children’s developing autonomy, and in concert with parental permission, ensure the protection of children who participate in research.https://deepblue.lib.umich.edu/bitstream/2027.42/136915/1/12910_2017_Article_199.pd

Teen Health Care Decisions : How Maturity and Social Policy Affect Four Hard Cases

34 pagesLegal standards that allow teens to make health care decisions, or any important decisions, must account for the contingency and variability of minors’ capacity. Traditional law denied minors’ legal authority to make any decisions, giving all power to parents. This rule goes too far; the Supreme Court has held that minors have constitutionally protected autonomy-based rights, and modern views about adolescence are inconsistent with the rule. The question is how and where to draw lines. Legal standards are based on minors’ evolving maturity, policy favoring decisions that follow medical advice, and policy supporting parental authority. This paper uses four hard cases to show how these considerations factor into legal rules