How many people have ever lived?

Lasse Damgaard Christensen defends his MSc thesis in mathematics

Supervisor: Carl Winsløw

Censor: Morten Misfeldt, AAU

Abstract:

This thesis describes the testing and analysis of a Study and Research Path (SRP)
on a classical question from mathematical demography. The hypothesis of the study
has been that an SRP on demography can give high school students an insight into
how mathematical methods can be used to answer questions in the social sciences.
Specifically the SRP concerned the generating question: "How many people have
ever lived?".

The Anthropological Theory of Didactics (ATD) has been used as a theoretical
framework for the study and an epistemological reference model (ERM) has been
developed. We have rst performed our own study and research of the question
and found that it can be answered in multiple ways. The most prominent methods
focused either on modelling yearly births as a function of time in order to calculate
a definite integral or on modelling the population size as a function of time in
order to calculate the integral and divide this with an estimate of an average life
expectancy. We have then designed a teaching sequence and made an a priori
analysis of it. In the a posteriori analysis we have focused on which questions the
students developed, how students used media and students' praxeologies regarding
integration and interpolation.

We have found that students developed questions similar to those in the a priori
analysis which could be divided into three branches: counting deaths, counting
births and counting "person-years". The students mainly studied media to gain
information on the origins of man and to nd demographic data. Only little media
on the methods for calculating an answer was found and it was not studied in
depth. A better study of such media might have given a greater variety in successful
methods used by students. Regarding both interpolation and integration students
almost exclusively used techniques involving CAS. We have found that students
know the praxis parts well but that their knowledge of the theory on both regression
and integration is lacking. Other studies have suggested that more focus on giving
students an interpretation of the definite integral as a Riemann sum may alleviate
the problem.