The instructional sequence was constructed with several content and pedagogical principles in mind.

• Concepts of distribution as components of data modeling
  Concepts of distribution are introduced as components of data modeling . Data modeling (Lehrer & Romberg, 1996) focuses on concepts of statistics and statistical reasoning as tools for the invention and revision of models. Data modeling always begins with a question and culminates in an inference, which ideally leads to generating a new, more refined question. The process of data modeling includes constructing data (by partitioning objects or events into attributes and their measures), and structuring and displaying data in ways that facilitate inference.
  
• Emergent qualities of collective, embodied activity
  Distribution and related statistics are first encountered as emergent qualities of collective, embodied activity. For example, in the first lessons, distribution emerges as a descriptor of the structure of the measurements made by different measurers. Students often expect that the measures were "random" and (in their view) unstructured. Qualities of distribution are related to activities of measurers. For example, an individual might over- or under-estimate a length, but the collective consequence is a symmetry (the distribution is normal).
  
• Inventing measures and representations
  Statistical concepts are developed by inventing measures and representations, and by comparing them in a way that allows students to evaluate their trade-offs ( meta-measure and meta-representational competencies ). Students invent or appropriate displays that meet some representational purpose and then engage in an analysis of what each representational system makes more visible and what it makes less visible. For example, varying the widths of intervals and then examining the resulting "shape" of the data introduce density of distribution.
  
• Teaching to big ideas
  We have generated a smaller set of "big ideas" about the nature of data modeling and of distribution as guides for teaching and assessment (See Progress Map). These big ideas orient instruction across the lesson sequence, and they guide what we consider worth assessing. These include:

• Question posing.
  Data modeling addresses questions because data are constructed in response to question.
  
• Attributes/ Qualities
  Some quality of the world is measured. This entails re-describing objects and events as bundles of attributes. For instance, a clump of dirt has qualities of weight, volume, density, relative moisture, texture, and color.
  
• Measure
  Measure quantifies quality. Understanding of qualities is enhanced by developing their measures. Statistics can be viewed as measures of qualities of distribution.
  
• Data-Structure and Representation
  Data are constructed, not given. Measures are organized or structured in a way that supports answering questions about the world. Different structures (e.g., lists, tables) afford ready access to different qualities of the data. Data displays make aspects of structure visible, always at a risk of concealing other attributes. Structures and representations are best considered in light of alternatives.
  
• Inference
  Inference rests on sampling distribution-what would happen if a process were repeated? What are the implications of this repetition for contrasts between distributions?

(1) Distribution emerges from repeated measures.

(a) Students measure. Measures vary.

(b) Students invent ways of representing variation and what they expect about the true measure.

(c) Students develop indicators of expected value and spread.

(d) Students test their indicators in light of measurement tools/methods that affect variability.

(2) Sampling distribution emerges from N repetitions of repeated measures.

(a) Students consider what might happen if we "did it again."

(b) We do it again or show another larger group of measurements. We display these measures and see what seems to be the same, different? 

(c) We introduce sampling as a model of repeated process from this larger group of measurements. We conduct experiments with fixed sample sizes and look across samples-we see what happens to our indicators. Do they vary? If so, how?

(3) Repeated measures of the heights attained by rockets, again with multiple measurers. These constitute a reference distribution. We ask what might happen if we kept on launching rockets, again and again. We attempt to establish some expectation about variation and height.

(4) Employing (3) as a reference distribution, we ask students about their expectations of using rockets with pointed nose cones. (Students often think these "cut" through the air). We again conduct repeated measures and trials (the trial is a source of natural variation in this context), lump the measurements together, and compare to the reference distribution-asking about the basis of informal inference.