Introduction and the four phases of the classroom study

The classroom teaching sequence is presented here in full and in the same order as presented to students, but not all items presented to students were used in the subsequent data analysis. The classroom sequence was grouped in four phases that were linked to the three research questions (Section 2.7). Phase 1 provided pre-study testing and background information on the students. Phase 2 and Phase 3 were teaching sequences where Phase 2 sought to cultivate students’ acceptance of the Fathom simulation as legitimate, and Phase 3 provided students with the opportunity to examine sample size. Phase 4 provided the post-study assessment that examined students’ development for all three research questions.

Phase 1. Establish a baseline of students’ number-sense of fractions and percentages, gain an understanding of students’ intuitive sense of the distribution of the proportion of heads from 50 tosses of a coin and the distribution of the frequency at which faces occur in 30 rolls of a die, and assess students’ ability to analyse and interpret the dot plot graph format used in the study. Fathom was introduced to students using an exploratory data analysis task of New York marathon race times that provided a base-line of students’ ability to interpret a data distribution and examined students’ first use of Fathom’s basic functions, modular structure, and terminology.

100 Phase 2. Develop acceptance of the Fathom simulator. The literature review noted that the beliefs and misconceptions of probability that students bring to the classroom are present across all stages of student development, are difficult to change, and may confound learning (e.g., Batanero & Sanchez, 2005). Statistics education researchers recommend that students are given opportunities to make predictions and challenge their beliefs, to test simulations, and to develop expertise by allowing more robust global resources and principles to out-compete existing local knowledge (e.g., Pratt, 2002). This phase of the study sought to develop students’ confidence in the Fathom die simulator through an objective scientific statistical enquiry that examined the fairness of physical dice and the virtual Fathom die. The three dice examined were identified as a home-made die – a die that students fabricated themselves using SculpeyTM _{modelling clay; a conventional factory-made die – a term}

chosen purposefully in preference to a “real” die that may suggests that the Fathom die is not legitimate; and the Fathom simulation virtual die. Class discussion supported students’ progression from naive and informal perceptions of fairness to a formal measurement of fairness of the dice using an objective fairness measure statistic. Instrumental genesis (Section 2.5.6.1) incorporates the notion of schemes, which are the mental organisation and structure, the skills, and the supporting concepts to use the software in a meaningful way. Schemes are reciprocal (where the tool acts on the user and vice-versa), personal, and evolutionary. Such a definition does not address subjective beliefs of probability or the simulator as a legitimate mathematics tool explicitly, but this study extended the use of schemes to incorporate students’ acceptance and confidence that the data generated by the simulator were legitimate – in the eyes of the students that the simulation was fair and the data generated were genuine. Much of the research of beliefs in probability is based on physical simulation models, and this study extended this earlier research to consider students’ beliefs of virtual simulation. This also increased the task complexity because students considered both physical and virtual dice. Without confidence in the virtual simulation this study speculated that learning was likely to be

101 superficial, any misconceptions were likely to persist, and further learning using the software undermined. The development of schemes, such as changing beliefs, are time-consuming processes, but such an investment is justified by purposefully examining the legitimacy of the simulation through statistical enquiry and meaningful mathematical activity that objectively examined the fairness of the simulation using the fairness measure statistic. This phase served three objectives simultaneously by attending to students’ beliefs of the legitimacy of simulation, developing basic familiarity with Fathom, and modelling the process of scientific enquiry. This phase provided the classroom tuition for Research Question 1 largely in its entirety.

Phase 3. Introduce and apply the large population sample size model (Section 2.4.13) using Fathom. The principal objective of this phase was to introduce and use the sample size model in a way that potentially would convince students of the model’s usefulness. The model was given to students because it was thought unlikely the students could develop the model independently, and the model was presented without derivation or formal proof because the mathematics involved is too complex for high school students. As an alternative students proved the sample model’s utility and legitimacy using a frequentist approach with a Fathom simulation. Such an emphasis on utility and application lay somewhere between an informal approach that was designed to cultivate intuitions and the more formal mathematical approach appropriate at senior school. The real-world scenario chosen was a public opinion survey of a local controversial and well-publicised issue of supporting or opposing the construction of the Mt. Wellington Cable Car. This phase largely provided the classroom tuition for Research Question 2.

Phase 4. Conduct post-study assessment. The post study assessment was conducted in three parts and under traditional examination conditions. The first part considered students’ use of Fathom and their ability to assemble a basic simulation, the second considered students’ development of understanding of re-sampling and the sample size model, and the third was a follow-up test item that assessed students’ long-term

102 retention of the concepts through re-presentation of the national and state election survey item introduced first in the pre-testing in Phase 1.

The classroom work samples in each of the four phases are summarised in Tables 3.3 – 3.6, the work samples are presented separately in a consistent format that identifies the items’ key objectives provides a description of the item, and gives an explanation of the methodology used to analyse students’ responses. The work samples are grouped by the three research questions in Tables 3.8 – 3.10.