Figure 1.9. Gen’s reflection on her final card.
shape of their collated distribution. A uniform probability model is used when the values of the distribution are equally likely; and indeed, three was just as likely to be selected as 11 in the children’s initial cards (Figure1.2). They expected numbers to be equally likely. We argue that they were inferring to a uniform probability model, even though they were not explicitly aware of this.
As numbers were drawn, the uniform probability model was challenged. The chil- dren were not surprised by the individual outcomes that were drawn, but by their col- lective distribution. Through their awareness that some numbers appeared multiple times, the importance of attending to frequency (not relevant in a uniform distri- bution) became apparent. Not all children let go of their “lucky numbers”, even at the end of the unit. However even those who did not, found a way to fit their lucky numbers into a distribution that still recognized basic patterns within a triangular probability distribution: a “mountain” in the middle and low frequencies in the tails. The children who were able to capitalize on their awareness of the structure of the theoretical distribution were more successful in the game.
If we do believe that the children were inferring to a model, a statistical struc- ture, we can think more systematically about the models they engaged with, and the benefits that may have been gained. Mulligan and Mitchelmore (2013) argue that considering children’s development of mathematical structure can “provide new in- sights into how young students can abstract and generalize mathematical ideas much earlier, and in more complex ways” (p. 29). We see connections between the desire to understand and develop children’s mathematical-statistical structures and the move- ment to introduce complex concepts at an informal level in earlier years of school- ing. The structures that underpin key concepts in statistics can be introduced through
DISCUSSION AND IMPLICATIONS 21
informal concepts to expose and develop children’s sense of structures within the discipline. These structures enable us to work meaningfully with mathematical and statistical ideas in unfamiliar problems. Models, therefore, provide underlying struc- tures that children can learn to depend on, adapt and make meaning from within a problem.
Children’s search for frequencies of numbers in the game created a number of ex- periences and perspectives of frequency, not all of which were model-based. Hearing particular numbers such as 12 and 15 come up multiple times developed an aware- ness of frequency that may have challenged an equiprobability model, but did not provide a structure from which to anticipate frequencies. Tallying the numbers in the bucket again reinforced the idea that some numbers were more commonly occurring than others but still lacked a sense of pattern that could reveal how the frequencies were structured in the distribution. Several students did recognize patterns once they began listing possible combinations (one way to obtain the sum of two, two ways to obtain the sum of three, etc.) and this initial awareness of a pattern helped them to check their expectations against the frequencies they counted; it was challenged when they realized that there were not 11 ways to make 12, as they anticipated. The sample space supported a way to reliably record all possible outcomes. Students were able to use the sample space to calculate probabilities of outcomes, and make meaning of the small likelihood of even the most frequently occurring numbers. The addition table also revealed patterns in the relationships among the outcomes (e.g., equal sums falling along a diagonal) that enabled the students to be confident in se- lecting numbers for their cards, although these patterns may have contributed to an over-emphasis on the most common numbers. The sample space in the form of an addition table therefore was a model from which they could work, although it lacked the visual opportunity to envision the structure of variability and “modal clump” in a triangular probability distribution. “Paul’s mountain” generated a more visual structure that underpinned the theoretical model.
The theoretical model of a triangular probability distribution represents the vari- ability in the sample space deterministically. The benefit of the theoretical model is in its ability to expose predictable patterns and relationships in the sample space. However, it lacked the sense of randomness that children experienced and came to expect in each game. The familiarity in the mountain that appeared in Paul’s tally-dot plot of Gideon’s winning card (Figure1.6) provided this uncertainty. As an empirical model of a triangular probability distribution, Paul’s second representation embraced both the structure of the expected outcomes and the sense of uncertainty by show- ing variability in its divergence from the theoretical distribution. Models are useful when they incorporate the structures and relationships we understand, expect and are productive-in-use; we believe that the class embraced Paul’s second model more readily because of this. The two models together set up an opportunity to recognize and express the difference between the theoretical frequencies (what was expected) and the empirical frequencies (what occurred). These two concepts and their rela- tionship will not be met formally for several years, yet the structures visible through these models enabled students to make sense of the game and provided a strong sense of utility in the dot plot representations.