Other Constructs and a Learning Progression

Our example is a classroom-intensive context, and formal statistical modeling of this small sample of particular student responses would not be useful. However, the responses of other students involved in learning about data and statistics by inventing displays, measures, and models of variability (Lehrer et al., 2007, 2011) were plotted using a DaD construct map (see Figure 3-13, above), and the results of the analysis of those data are illustrated in Figure 3-17 (Schwartz et al., 2011). In this figure, the left-hand side shows the units of the scale (in logits9_{) and also} the distributions of the students along the DaD construct. The right-hand side shows the locations of the items associated with the levels of the construct—the first column (labeled “NL”) is a set of responses that are pre-Level 1—that is, they are responses that do not yet reach Level 1, but they show some relevancy, even if it is just making appropriate reference to the item. These points (locations of the thresholds) are where a student is estimated to have a probability of 0.50 of

9_{The logit scale is used to locate both examinees and assessment tasks relative to a common,}

underlying (latent) scale of both student proficiency and task difficulty. The difference in logits between an examinee’s proficiency and a task’s difficulty is equal to the logarithm of the odds of a correct response to that task by that examinee, as determined by a statistical model.

R02484 FIG3-17.eps

NL DaD1 DaD2 DaD3 DaD4&DaD5

XXXX | Crab2 Rocket2,Crab2,Headache2 Rocket2 XXXXXXX | Rocket2,Statue6

XXXXX | Rocket2 Statue8 Earthquake XXXXX | Statue8,Statue6 Invest,Earthquake,Statue4,Cherry CandleQ2

XXXXXXX | CandleQ2 Crab1 StateCap2

0 XXXXXXX | Rocket2 Statue4 Headache2 Max5 XXXXXXXXXX | CandleQ2,Crab2 CandleQ1

XXXXXXXXXX | Statue8,Statue6 Applause,EarthQuake

NL DaD1 DaD2 DaD3 DaD4&DaD5

FIGURE 3-17 Wright map of the DaD construct.

SOURCE: Wilson et al. (2013). Copyright by the author; used with permission.

responding at that level or below. Using this figure, one can then construct bands that correspond to levels of the construct and help visualize relations between item difficulties and the ordered levels of the construct. This is a more focused test of construct validity than traditional measures of item fit, such as the mean square or others (Wilson, 2005).

The DaD construct is but one of seven assessed with this sample of students, so BAS was applied to each of the seven constructs: theory of measurement, DaD, meta-representational competence, conceptions of statistics, chance, models of variability, and informal inference (Lehrer et al., 2013); see Figure 3-18.

1. Theory of measurement maps the degree to which students understand the

mathematics of measurement and develop skills in measuring. This construct represents the basic area of knowledge in which the rest of the constructs are played out.

2. DaD traces a progression in learning to construct and read graphical rep-

resentations of the data from an initial emphasis on cases toward reasoning based on properties of the aggregate.

3. Meta-representational competence, which is closely related to DaD, proposes

keystone performances as students learn to harness varied representations for making claims about data and to consider tradeoffs among representations in light of these claims.

FIGURE 3-18 Wright map of the seven dimensions assessed for analyzing and interpreting data.

NOTES: Cha = chance, CoS = conceptions of statistics, DaD = data display, InI = informal inference, MoV = models of variability, MRC = meta-representational competence, ToM = theory of measurement. See text for discussion.

R02484 FIG3-18 convert.eps

4. Conceptions of statistics propose a series of landmarks as students come to

first recognize that statistics measure qualities of the distribution, such as center and spread, and then go on to develop understandings of statistics as generalizable and as subject to sample-to-sample variation.

5. Chance describes the progression of students’ understanding about how

chance and elementary probability operate to produce distributions of outcomes.

6. Models of variability refer to the progression of reasoning about employing

chance to model a distribution of outcomes produced by a process. 7. Informal inference describes a progression in the basis of students’ infer-

ences, beginning with reliance on cases and ultimately culminating in using models of variability to make inferences based on single or multiple samples.

These seven constructs can be plotted as a learning progression that links the theory of measurement, a construct that embodies a core idea, with the other six constructs, which embody practices: see Figure 3-19. In this figure, each verti- cal set of levels is one of the constructs listed above. In addition to the obvious links between the levels within a construct, this figure shows hypothesized links between specific levels of different constructs. These are interpreted as necessary prerequisites: that is, the hypothesis is that a student needs to know the level at the base of the arrow before he or she can succeed on the level indicated at the point of the arrow. The area labeled as “bootstrapping” is a set of levels that require mutual support. Of course, performance on specific items will involve measurement error, so these links need to be investigated using multiple items within tasks.

In document Developing Assessments for the NGSStandards (Page 92-95)