Students’ post-study questionnaire items

At the conclusion of the classroom study students’ beliefs in the randomness and confidence in Fathom virtual simulation were examined in the post-study questionnaire

168 in Items 7, 12, 14, 21, and 25, the questionnaire is attached as Appendix A.18, and the data for these items are presented in Table 4.15. Questions are paraphrased for brevity. That students accepted the Fathom die as fair was a demanding criterion and not essential for the study: the study sought only that students’ perception of fairness and legitimacy of Fathom were not a barrier to learning. Students’ acceptance of the simulator was high with 83.3% (15/18) of the male and 88.6% (31/35) of the female students either agreeing or strongly agreeing with Item 7 “The Fathom die was as random as it needed to be for what we were doing.” One student only disagreed, and the remaining students were neutral.

Students’ confidence in the Fathom die may be influenced by whether the die behaved as the students expected, given their experiences using a physical die. This was explored in Item 12 “I was surprised how many times I had to roll the Fathom die before I thought it was random;” 22.2% (4/18) of male and 17.1% (6/35) of female students were surprised at the number of times the Fathom die was rolled before they accepted the die was random. In response to Item 25, “I didn’t have confidence in Fathom because the results were too weird” 16.7% (3/18) of males and 8.6% (3/35) of females thought the data generated by Fathom were peculiar. If students had largely accepted that the Fathom simulation was fair, reservations regarding the random behaviour of the simulation remained amongst some students.

Item 14, “Building the simulation myself gave me confidence in the simulation,” considered if assembling the virtual die had contributed to students’ acceptance of the simulator. Of the male students 72.2% (13/18) and 54.3% (19/35) of females thought that assembling and step-wise checking of the simulation were important in developing their confidence. The assembling and checking process may be important to developing acceptance amongst students generally, but the female students may have been more likely to accept the simulation on trust than the male students.

Students’ acceptance of the Fathom die relative to the familiar physical factory-made die was explored through Item 21, “I had more confidence in a physical die than a Fathom die.” At the conclusion of the classroom study, and despite specifically attending to beliefs, at least one third of both male and female had more confidence in the physical die than the Fathom virtual die.

169 4.3.8 Detailed study workshop

Students in the detailed study were asked their preference for the physical or the Fathom virtual die. The physical die was intuitively more real, more fair or random than the virtual die, and students rarely hesitated when expressing their preference for a physical die.

Physical. [die and coin] [Student R1706D]

I’d probably go for the physical die. Real or not it’s the one that I have a gut- instinct preference. [Student N2610H]

The factory-made die. [Students S1001J & T0612M, simultaneously and without hesitation]

I think the factory-made is more fair, but I’d rather use computer [because] it is quicker. [Student S1001J]

Always thought it [Fathom] wasn’t completely random. [Student Y1504L] Table 4.15.

Post-study Questionnaire Students’ Acceptance of the Fathom Die

Post-study questionnaire item Disagree or Strongly disagree Maybe or neutral Agree or Strongly agree % % %

Students’ perceived fairness of the Fathom die 7. By the end of the unit I was convinced the Fathom coin was “as random as it needed to be for the work we were doing”

M 5.6% 11.1% 83.3% F 0.0% 11.4% 88.6% Fathom die behaved as students expected

12. I was surprised how many times I had to “roll” the Fathom die before I thought the die was random

M 83.3% 0.0% 22.2% F 65.8% 17.1% 17.1% 25. I didn’t have confidence in the Fathom die

because some results were too “weird” M _F 77.8% _{77.1% 14.3%} 5.5% 16.7% _8.6% Assembling and step-wise testing of the

Fathom die developed confidence

14. Building the simulation myself gave me

confidence in the simulation M 5.6% 22.2% 72.2% F 11.4% 34.3% 54.3% Confidence in the Fathom die relative to

confidence in a standard factory-made die 21. I have more confidence in a physical die

170 Students were aware of the efficiency and speed of the Fathom simulation. When given a choice between a physical and virtual die the advantage of using a computer at large sample size when a physical die was impracticable was clear: students did not hesitate in preferring the Fathom die at large sample sizes. The virtual coin and die had the disadvantage that the simulation had to be assembled and that the software was not available outside of school. The advantages of the physical die and coin included the portability and ready availability of a die and coin. Arguably this was intelligent use of the available tools. At a more subtle level, and recalling the class discussion, two students noted the virtual die and coin eliminated the bias potentially introduced in using a physical die.

With a computer die you can do can do a lot more, thousands, it’s quicker, I wouldn’t use a real [physical] die. [in that situation] [Student Y1504L]

The Fathom coin does take out the physical biases. [Student R1706D]

[….] but then again you can take the random elements out of a physical die as well [by] throwing it a certain way. [Student Y1504L]

If it was only ten rolls I use the real [physical] one. [….] If I had the option of computer, the computer, but it is harder to carry around. [Student N2701B]

One student thought that the additional difficulty of assembling a Fathom simulation was an impediment, but once assembled and operating the simulation was quick. Assembling the simulation was one of the schemes of the instrumental genesis.

It was easier to do things physically because you weren’t as good in getting set up in Fathom, but much quicker in rolling the die in Fathom. [Student E2611G]

Part D explored the information – the sample size – students required when they made a decision regarding the fairness of a virtual die. The sample size is clearly related to the level of the bias evident. For subtly biased dice with expected values between 0.48 and 0.53 students were prepared to make a decision on whether the coin was biased using sample sizes between 40 to 140 – sample sizes not dissimilar to a physical simulation. One student pair made a decision on bias at a sample size of 420 (Appendix D.8). 4.3.9 Colleague teacher interview

Both colleague teachers felt that, by the end of the program, the students had accepted the Fathom virtual simulation as legitimate. Both teachers noted the advantage, for students and teachers, of the speed and ease with which the data could be generated by the simulation.

171 The students accepted the simulation, no longer questioned the tool, and they

had faith in its ability to model a reality. The [Fathom] die was more than a pretend die and they tried to transfer the information to other situations. [….] Collecting data from physical systems is very time-consuming and difficult. This time-constraint is eliminated once students developed a belief and a confidence in the random behaviour of the simulator. [Colleague teacher of male class]

I didn’t hear any mumblings or dispute about the Fathom die and coin – they seemed to accept it. They would have used the [conventional physical] die in Years 7 and 8. The students recognised that it made modelling more efficient. [Colleague teacher of female class]

Asked whether the elaborate process of developing confidence in the random behaviour of the Fathom simulation by comparing the random behaviour of home-made, the factory-made and the Fathom die was effective, one colleague teacher thought that students found the activity fabricating and testing their own home-made die as particularly engaging, and the fairness measure dot plot as an effective technique to support the analysis of the data but the activity might be necessary only for younger or lower-ability students. A key decision for the teacher is to select appropriate material for any class. The colleague teacher the importance of students “trying it out,” but one of the study’s intentions’ was to extend an informal approach to a more formal disciplined scientific approach.

It depends on the class. A less able class would need the engagement of making the die, and this class liked it, but it was not essential. The students certainly needed the opportunity to “try it [the dice] out.” [Colleague teacher of female class]