Design principles receive support from previous research with elementary grade children. These are:

• Measurement as a process of assigning numbers to magnitudes
  Measurement is a process of assigning numbers to magnitudes.
  The fundamental notion is magnitude (e.g., a distance, a space covered, a volume occupied), and we design instruction to build on intuitions of magnitude and relative magnitude to support development of concepts of measure. For example, in the linear measure sequence, measurements with different non-standard units are compared for fixed distances. Every student can see that the distance has not changed; yet the measure has. This invites more careful consideration of the nature of units.
  
  
• Embodied metaphors of measure
  The design relies on two embodied metaphors of measure (See Lakoff & Nunez, 2001 for extended discussion of the role of metaphor in mathematical thinking). By embodied, we mean that mathematics emerges from analogues in everyday bodily activity, so that embodied activity serves as a resource for developing mathematical understanding. The first metaphor is measure-as-motion —motion along a path (a distance), a sweep (an area) or a “pulling through” (volume as moving an area through a length). This helps students conceptualize measured magnitude as a motion between the initiation (zero) and cessation of motion. The second metaphor is the measuring stick . Every person has seen rulers and knows that they, or something like them (e.g., body parts), can be laid end-to-end to span a magnitude. The measuring stick connotes measurement as iterating rigid units.
  
  
• Representational Re-description
  Mathematical concepts develop by representational re-description . Although we rely on the plane of activity for introducing students to measurement, we seek to promote conceptual development by having students re-describe activity symbolically. Re-description often problematizes the nature of units and scale. For example, children may develop a length measure by walking heel-to-toe. One might assume then that they understand that the resulting count was an iteration of their feet. Yet when constructing a foot-strip ruler, the same students often leave spaces or otherwise violate the iterative principles they recently enacted (from a teacher's point-of-view). Representational re-description also includes comparisons of different systems of representation. For example, how should parts of units be registered? Is one system better than another? Why?
  
  
• Guided re-invention of tools
  Measurement is practical, so we attempt to include opportunities for students to create tools and then use them. As students solve the problems that emerge in practice, new opportunities for conceptual development arise. For example, students know that rulers have marks, and students often can name the marks (e.g., “1/4”). But students often do not know how, in principle, these marks were made. Why are marks placed where they are? Attempting to develop their own system of marks is one way to help students grapple with the meaning of units, especially fractional units. Similarly, we introduce students to distribution as an emergent phenomenon—as a way of structuring differences in repeated measures of the same length.
  
  
• Ratio
  We begin with ratio systems of measurement because they afford the greatest number of invariances due to their constraints. For example, ratio measures are additive. This provides a ready model of addition and of multiplication.
  
  
• Measurement models of arithmetic
  We employ measurement as a context for developing greater understanding of arithmetic. In each system of measure, we provide measurement models of arithmetic operations, the meaning of fractions, and properties of the resulting number system, such as additive and multiplicative inverse, additive and multiplicative identity, commutative and associative properties, and distribution of multiplication over addition. We also visit equivalence in various guises (i.e., equivalent fractions as equal measures).
  
  
• Teaching to big ideas
  We have generated a smaller set of "big ideas" about the nature of units and of scale 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.

• 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, color. Children often have difficulty conceiving of objects in this way. They often associate one quality with one object. It is important that they come to see an object as having many potential qualities.
  
  
  
• Suitability of unit
  Different attributes (e.g., length, angle) are measured with units suited to the qualities of the attribute. For young children, the idea might be expressed as "not everything is best measured with a ruler." Angle measure, for instance, requires a different measure (i.e., turns or degrees or radians). For older children, the idea might be expressed as "the unit need not resemble the attribute." For example, time can serve as an indicator of distance if one has knowledge of rate. Similarly, ratios of lengths can measure angles.
  
  
  
• Iteration of units
  Measures are produced by repeatedly tiling a fixed unit, by laying, for example, a measuring stick end-to-end. This may appear obvious, but it is not always transparent to children. For example, young children often lay units together with "spaces" of variable length. Some are confounded when they "run out" of units, not thinking of re-using units. Others believe that units need not be identical , so they will, for example, use both inches and cm. and report the measure as the sum of the units.
  
  
  
• Partitions of units
  A part of a unit can be made by splitting the unit into n congruent parts to achieve 1/n. Reversibly, any unit can be generated by iterating 1/n, n times. Units can be composed. For example, a length measure of 10 is re-scaled as 2, if units are composed into composites of five. Flexible conceptions of unit are anchored in partitioning and composing, and coming to see these as reversible.
  
  
  
• Zero-point
  Any number can serve as the origin of measure. Hence, a ruler with 10 units can be employed to measure a length beginning at zero, or by treating any other number as zero (e.g., 1 or 2). Many children simply read off whatever number aligns with the desired point (e.g., the end of the length of an object), regardless of the number corresponding to the origin. Thus, beginning at 2 and ending at 10, a student declares the measure as 10, not 8. From a stick metaphor perspective, this confusion is very understandable. In collections, zero corresponds to no elements, yet students can see that we are measuring something, so what does zero have to do with it? From a motion metaphor, this is a bit easier to understand. Zero is the beginning.
  
  
  
• Error
  As a practical activity, measure is inherently imprecise. But we can use tools like distribution to think about bounds of uncertainty, rather than simply noticing this quality.