An incomplete July 2010 working paper
Thinking about a simple teaching example on the t-test for comparing the average (mean) for some measurement in a group versus the average in another led me to articulate a sequence of thoughts and questions about the foundations of statistical analysis. In particular, my inquiry explores contrasts between: the statistical emphasis on averages or types around which there is variation or noise; variation as a mixture of types; the dynamics (or heterogeneous mix of dynamics) that generated the data analyzed; and participatory restructuring of these dynamics in the future. Two key issues are: Who is assumed to be able to take action—who are the “agents”—and who are the subjects that follow directions given by others? What can it mean to explain differences among averages? Questions are noted to be addressed in a future supplement.
I googled the question “Why study fractions?” (for reasons I describe later) and found a study (reported in Swanbrow 2012) that invites critical thinking at two levels: 1) the assumptions, evidence, and reasoning warrant scrutiny; and 2) what is it that allows researchers and policy makers to proceed as if there are no alternative interpretations to be drawn from the study?
Untested draft of method that represents
a) an extension of mathematical thinking (provisionally defined here);
b) follows the premise that no teacher would be prepared to guide every student in developing their mathematical thinking in the diverse ways that interest different people and
c) ditto in developing their life long learning to respond to changes in work, technology, commerce, and social life that continue to change our needs and capacities for mathematical thinking. Continue reading →
What constitutes mathematical thinking? I am teaching a course on mathematical thinking without defining the term for the students or myself. I want a working definition to arise from the explorations we undertake during the course. Continue reading →
This paper analyzes strategies of model building in ecology twenty years after Levins advocated the use of simple models to generate supposedly qualitative and general biological insights. Modeling has five aspects: elevation of biological processes, construction and analysis of the model, and of observations, analysis of correspondence between the model and observations, and model-based action. I distinguish four different roles we can assign to models: schemata, exploratory tools, redescriptions, and representations of generative biological relations. In the analysis of correspondence fidelity of fit and accuracy of predictions are insufficient evidence that a model represents the biological processes that generated the observations. In order to confirm the model a variety of “accessory conditions” need to be established; these are often overlooked and difficult to establish, especially in the case of simple models of naturally variable situations. I reinterpret the strategy of using simple models in community ecology as exploratory. The simplicity may ensure mathematical generality but not ecological generality; the models are only suggestive of — not support for — ecological hypotheses. Generality in ecological theory will require much more particularity.