# An Inquiry Based Introduction to Binomial Distributions

MSc Thesis defense by Vibeke Vestermarken

Supervisor: Carl Winsløw

Censor: Nicole Koefoed

Abstract:

This thesis describes the design of a teaching sequence on binomial distributions, by the use of Study and Research Paths (SRP). The research question motivating the design is "Can a greater coherence be generated between the subjects regarding binomial distributions by the use of Study and Research Paths?". In order to answer this question, a content analysis has been conduced using the Anthropological Theory of Didactics (ATD) as theoretical framework. The content analysis describes the knowledge to be taught at upper secondary school and how the subject probability and statistics, connects in relation to binomial distributions. Further is deals with the challenges which can present themselves when transposing the knowledge from a scholarly level to the upper secondary school. A Praxeological Reference Model (PRM) has been created based on

the work of the content analysis illustrating the connections found in the subject. Based on the observations from the content analysis and PRM a teaching sequence of three 1.5 hour lessons was designed using the SRP as a design tool. An a priori analysis was when made of the design, describing the possible questions and answers to be developed by the students. The a posteriori analysis of the observed lessons focuses on the questions posed by students, creating a SRP diagram illustrating the students paths when working on the generating question. Further, has the students' praxeologies for the random variable, binomial distribution and statistics been analysed. The study found that the students pose many of the questions from the a priori analysis however no questions regarding mean and standard deviation was posed by the students.

Further it was seen that the students created many techniques for solving the questions themselves creating a connection between the topics covered. However, no technology of the distributions were established properly resulting in the students not connecting the probability and statistics theory. In regards to the normal distribution the students disregarded this result in relation to the condence intervals, as they were using CAS to determine the intervals. In conclusion it was suggested by the study that the concept random variable and related distribution should be paid more attention to, in order to create a more coherent subject across the probability and statistics theory. However, the results of students creating the techniques motivated by self-generated questions did create a coherence within the two topics and the students were able to use previous knowledge to create new techniques by the use of SRP.