The Action, Process, Object, Schema Conception of Sampling/Sampling Distribution

1. The Action, Process, Object, Schema Conception of Sampling/Sampling Distribution • Neil J. Hatfield

3. Overview • Purpose • Sfard’s Constructs • APOS-Theory • Multiplicative Conception of Sampling • APOS-Sampling/Sampling Distribution

4. Purpose • Extend Saldanha and Thompson’s Multiplicative Conception of Sampling by theorizing more details for an individual’s understanding of sampling. • Put forward a perspective that could serve as a starting point for developing a hypothesis for how an individual might develop a particular conception of sampling. • To provide a starting point for further research

5. Other Work with Students’ Conceptions of Sampling • Size and Type of the Sample • Watson and Moritz (2000a, 2000b) • Size of the Sample • Watson and Kelly (2006) • Sampling methods • Jacobs (1997) • Distributional • Saldanha and Thompson (2002)

6. Sfard’s Constructs Interiorization “process performed on already familiar objects” (Sfard, 1992) The observer hypothesizes that an individual carries out a sequence of steps on an entity that the individual views as something the individual may manipulate. Condensation “turning a process into a more compact, self-contained whole” (Sfard, 1992) The observer hypothesizes that an individual envisions a sequence of steps and the individual draws upon his/her experiences working through that sequence to anticipate the result of the sequence without explicitly carrying out the sequence. Reification “transition from an operational to a structural mode of thinking” (Sfard, 1994) The observer hypothesizes that an individual has reflected upon experiences with the sequence of steps, unpacking/repacking the sequence so that this reflection leads to the individual ascribing characteristics of the sequence.

7. APOS-Theory Action Conception A conception of a concept where an individual’s meanings lead him/her to follow an explicit sequences of steps to transform an object. Process Conception A conception of a concept where an individual’s meanings lead the individual to think about performing a sequence of steps, may be not carry out the sequence, and anticipate the end product of the sequence of steps. Object Conception A conception of a concept where an individual’s meanings allow him/her to to describe characteristics/attributes of the sequence in addition the implications of the process conception. Schema The collection of cognitive elements (understandings, meanings) that an individual has for a particular concept.

8. Vantage Points 2 x 2 Inner Vantage Point Outer Vantage Point Distant Outer Vantage Point Square the value Multiply by two Subtract four from the value 4 f(x) f(2) 8 4

9. Three Phases of the Sampling Process Saldanha and Thompson, 2002 • Randomly select n objects from a population of interest and calculate/record the value of a statistic of interest. • Repeat the first phase a number of times (typically a large number of times); calculate/record the values of the statistic of interest. • Develop categorizations of the values of the statistic of interest to assist in making a judgment.

10. Multiplicative Conception of Sampling Saldanha and Thompson, 2002 Multiplicative Reasoning on three levels: L1: Conceiving a quasi-proportional relationship between a sample and a population. L2: Envisioning a proto-sampling distribution of the values of the statistic of interest based on categorizations of the entire collection of values. L3: The image of the sampling distribution that supports the individual in “quantifying the expectation of a particular kind of sampling outcome and thus quantifying one’s confidence in a sampling outcome’s representativeness.” (p. 267) Not developmental in Saldanha and Thompson’s original work.

11. APOS-Sampling/Sampling Distribution

12. Key Assumptions • Conception of Population • a collection of all possible objects/people an individual might wish to know something about. • Conception of Sample • A randomly drawn subset of the population of a particular size that may be potentially representative of the population. • Stochastic Process • A repeatable process with an anticipation that the end result of the process will vary with some bounds each time the process is repeated. • All samples are of the same size.

13. Action Conception of Sampling/Sampling Distribution • The objects/people that comprise the sample/population are already familiar objects to the individual. • Sequence of steps: • Conceive a statistic of interest • Envision/carry out a stochastic process to draw a sample from a population of interest • Find the value of the statistic of interest based on the elements of the sample. • Continue to draw samples and find the values of the statistic of interest. • The notion that the value of the statistic of interest tells us not only something about the sample, but also the population necessitates the individual conceiving a relationship between the sample and the population. If the observer hypothesizes that the individual’s understanding is best described by the individual repeatedly moving through each and every step of the sequence and the individual requiring data values to arrive at a value of the statistic, then the observer could refer to this conception of sampling as the action conception of sampling/sampling distribution. The individual is also hypothesized to have an inner vantage point for the sequence. The focus of the individual is on the elements that comprise the sample in order to arrive a value of the statistic. MCS--L1

14. Example of the Action Conception D: Ok. It’s asking…the question is…like “do you like Garth Brooks?”. You’re gonna go out and ask 30 people, it’s gonna ask 30 people 4500 times if they like Garth Brooks. The uh…(talks to himself) what’s this? Let’s see…the actual…like the amount of people who actually like Garth Brooks are…or 3 out of 10 people actually prefer like Garth Brook’s music. (Saldanha & Thompson, 2002, p. 262)

15. Process Conception of Sampling/Sampling Distribution If an observer hypothesizes that the individual’s understanding of sampling/sampling distribution is best characterized by the individual thinking through the three phases of sampling, anticipating the generation of values of the statistic of interest, and developing a proto-sampling distribution, then the observer would refer to this conception as the process conception of sampling/sampling distribution. • The individual continues to work through the sequence of steps (envisioning a statistic, phases 1 and 2) • The individual begins to anticipate that for any sample from the population a value of the statistic may be calculated. • The collection values of the statistic maybe placed into categories by the individual. • The categorized values of the statistic begin to form a proto-sampling distribution. The individual is also hypothesized to have an outer vantage point. The focus of the individual shifts to the values of the statistic that have been collected; focus is on the samples rather than the elements that make up the sample. MCS--L2

16. Example of the Process Conception D: When you go out and take one sample of 30 people, the cut off fraction means that if you’re gonna count, you’re gonna count that sample, if like 37% of the 30 people preferred Garth Brooks. And then it’s going to tally up how many of the samples had 37% people that preferred Garth Brooks. (Saldanha & Thompson, 2002, p. 262)

17. Object Conception of Sampling/Sampling Distribution If the observer hypothesizes that an individual’s understanding of sampling/sampling distribution is best described by the individual using the three phases of the sampling process to construct the proto-sampling distribution, and then the individual uses characteristics of that proto-sampling distribution to make a judgment about the expectation of a particular outcome, then the observer could refer to this conception as the object conception of sampling/sampling distribution. • The individual continues to work with expanded sequence (envisioning the statistic, phases 1, 2, and 3) • The individual reflects upon his/her experiences with the expanded sequence, packing/unpacking various steps. • This reflection could lead to the individual to describe characteristics of the proto-sampling distribution. • E.g. noticing that the proto-sampling distribution has a category of values with a high/low frequency. • The individual uses these characteristics to make a judgment about a particular kind of sampling outcome. The individual is also hypothesized to have an distant outer vantage point. The focus of the individual shifts from the values of the statistic that have been collected to the proto-sampling distribution and a new sample; focus is now on the categorized collection of sample values of the statistic and a new sample. MCS--L3

18. Example of Progressing Towards the Object Conception D1: If like…if you represent—if you give it like the split of the population and then you run it through the how—number of samples or whatever it’ll give you the same results as if—because in real life the population like of America actually has a split on whatever, on Pepsi, so it’ll give you the same results as if you actually went out, did a survey with people of that split. I: Ok, now. What do you mean by “same results”? On any particular survey at all—you’ll get exactly what it—? D2: No, no. Each sample won’t be the same but it’s a…it’d be…could be close, closer… I: What’s the “it” that would be close? D3: If you get…if you take a sample…then the uh…the number of the like whatever, the number of “yes’s” would be close to the actual population split of what it should be. I: Are you guaranteed that? D4: You’re not guaranteed, but if you do it enough times you can say it’s with like 1 or 2% of error depending upon uh how many times—I think—how many times you did it. (Saldanha & Thompson, 2002, pp. 265–266)

19. Thank You • neil.hatfield@asu.edu • mathpost.asu.edu/~neil