Gale Russell

The Fallacy of Composition: Prospective Mathematics Teachers’ Use of Logical Fallacies

The purpose of this article is to address the lack of research on teachers’ knowledge of probability. As has been the case in prior research, we asked prospective mathematics teachers to determine which of the presented sequences of coin flips was least likely to occur. However, instead of using the traditional perspectives of heuristic and informal reasoning, we have utilized logical fallacies for our analysis of the results. From this new perspective, we determined that certain individuals’—those who provided normatively incorrect responses—utilized the fallacy of composition when making comparisons of relative likelihood. In addition, we discuss how our findings impact models established in the research literature (e.g., the representativeness heuristic) and, further, we suggest that logical fallacies should supplement heuristic and informal reasoning as potential perspectives for research investigating comparisons of relative likelihood.RésuméLe but de cet article est de combler en partie le manque de recherches sur les connaissances des enseignants dans le domaine des probabilités. Comme cela avait été fait dans des études précédentes, nous avons demandé à des futurs enseignants des mathématiques quelle était la moins probable parmi les séquences présentées de tirs à pile ou face. Toutefois, au lieu d’adopter les perspectives traditionnelles du raisonnement heuristique et informel, nous nous sommes servis de sophismes logiques pour analyser les résultats. Dans cette nouvelle perspective, nous avons constaté que certaines personnes—celles qui ont fourni des réponses incorrectes sur le plan normatif—se sont servies de sophismes de composition lorsqu’elles ont comparé les différents niveaux de probabilité. Par ailleurs, nous examinons les façons dont nos résultats peuvent avoir un impact sur les modèles établis dans la recherche actuelle (par exemple l’heuristique de représentativité), et nous proposons que les sophismes logiques soient utilisés pour complémenter le raisonnement heuristique et informel dans la recherche visant à analyser les comparaisons de probabilités relatives.

Moving Forward with the Transreform Approach

With the proposal of a new theory, the inevitable question of “so what?” arises – what can be done with the theory now that it has been proposed? Clearly, just issuing a mandate to all teachers, students, administrators, and parents, as well as the society as a whole, is not going to work, just as proposals to change how mathematics has been traditionally taught to approaches that actively engage the students in knowledge creation and understanding have not been widely (or in many cases, successfully) implemented and maintained. So, where to go from here?

Methodology and Methods

With my story told and the theoretical worldview framework discussed, I can now move into a discussion of the methodology and methods that I used in my researching of the ways of knowing and kinds of knowledge that are valued within mathematics and the teaching and learning of mathematics.

Risk and Risk Education

The final area of interest that emerged from the grounded theory analysis of my first data set, my story, relates to how mathematics knowledge and ways of knowing are present and represented within curriculum. As this is a very broad topic, I have chosen to focus on one particular area of mathematics content that has only recently begun to appear within mathematics curricula documents, that of risk education.

Philosophies of Mathematics

Many of the epiphanies in my story that were analyzed raise a different wording of my research question, namely, what is mathematics and consequently, how should it be taught and learned? Even the disparity between dialogues amongst my story and the two worldviews within the analysis of my story present two very different conceptions of what mathematics is (could be) and how it should (might) be taught and learned.

The Math Wars

Having explored in depth the notion of what people think mathematics is, I now consider the second area of interest that emerged as important to me through the analysis of my story, namely what people believe about the teaching and learning of mathematics, especially how it should be taught and why. For decades now, North America (and elsewhere) has been officially embroiled in the so-called “math wars,” which are the consequence (at least in the start) of different thinking about the teaching and learning of mathematics.

Reflections on My Choice of Data Sources and Methodologies

At times, as I worked through the sharing and analysis of my data, I questioned whether my choices of sources, methodologies, and methods were ultimately being influenced by where my worldview positioning was located, and of course it was. Worldview influences all of our decisions, as that is its role. However, I also came to realize that by assuming the role of speaking with both the Traditional Western worldview and an Indigenous worldview, I was also allowing myself, and my research, to be open to alternative perspectives that would not have been present, or even considered, without the use of this theoretical framework.

Analysis of My Story

At this point in my analysis, I call upon all three methodologies within my collage. I bring forward auto/ethnography as I read and analyze my personal story for its epiphanies related to the cultures of mathematics and the teaching and learning of mathematics; I engage in Gadamer’s hermeneutics, as I use dialogue between my story and each of the two worldviews to obtain greater understanding of these two cultures; and, I use grounded theory coding and comparisons to label incidents that provide indications of concepts inherent within the story, as illuminated by the worldviews, that speak to the valuing of kinds of knowledge and ways of knowing in mathematics and the teaching and learning of mathematics.

Research Possibilities for My Theoretical Framework

As the theoretical framework that I presented at the end of my story has been foundational to the analysis of my research data, some final words about the potential of the framework in relation to the researching of the teaching and learning of mathematics seems appropriate. Thus, in this section, I briefly describe how I see the framework of the Traditional Western worldview and an Indigenous worldview might be applied within future research.

Contrasting Worldviews and the Emergence of a Theoretical Framework

As noted at the end of my story, Little Bear’s (2000) description of the two different worldviews, which he and other Indigenous people often find themselves both trapped within and torn apart by, was an eye-opening and soul stirring event in my life. Finally, I had found someone who could speak to me about First Nations ways of knowing, and what he said resonated with the many twists, turns, conflicts, and ultimately philosophical changes that I had experienced with respect to my thinking about mathematics and the teaching and learning of mathematics over the years.