- Summary

The Theory of Measurement: Length (ToML) construct describes how children come to understand foundations of length measurement, such as the nature of units and how measurement scales are constructed to aid measurement. The construct focuses on length measure, although the foundations of linear measure, such as the role of identical units, have counterparts in other forms of measurement as well. Learning about measurement involves a fusion of practical activity (e.g., how to use tools) and the conceptual underpinnings of unit and scale (e.g., units should be identical, the origin of the scale is labeled as zero). ToML is not intended to portray every nuance of learning about the mathematics of length measurement, but instead, to highlight critical conceptual attainments that would bootstrap learning other forms of measurement as well.

- Length Level 1
- Identify the Event/Object to be Measured and Make Direct Comparisons.

#### Performances

- 1A
Pose a question or make statements about a potentially measurable object of interest.

“How big is the pumpkin?” “This pumpkin is big.”

“Which rocket flies the best?”

“Which pumpkin is tallest?”

#### Examples

- Length Level 2
- Explaining Properties of Units and Their Role in Accumulation

#### Performances

- 2A
Associate measure with count.

This book is 4 (Student reads number off a ruler.)” “The pencil is 5 paper-clips long (the paper clips may not be identical lengths).

#### Examples

- Length Level 3
- Iterating Units and Symbolizing Length Measure as Distance Traveled

#### Performances

- 3A
Re-use (iterate) a unit to measure.

I just had one unit so I marked its end and then used it again, marked its end again, and kept doing that. It’s 8 paper clips long”

Note: Iteration includes both the concepts of translation and accumulating count, and the procedural competence involved in keeping track of the translated unit.

#### Examples

- Length Level 4
- Partitioning and Symbolizing Partitioned Units (2- splits)

#### Performances

- 4A
Partition and compose equipartitions by factors of 2 (2-split), and use to measure a length.

“It takes 2 and a half units to measure this notebook.”

“If you split the unit by 2 and then by 2 again, you get 4 equal parts.”

#### Examples

- Length Level 5
- Partitioning and Symbolizing Units Involving 3-splits and Compositions of 2- and 3-Splits

#### Performances

- 5A
Generate a 3-split of a unit and label it as 1/3 u.

“I split it into 3 parts that are exactly the same. Drags finger, saying, “From here (0) to here (first crease of the split unit) is 1/3

*Goade*. And from 0 to here (second crease) is 2/3*Goade*.”

#### Examples

- Length Level 6
- Generalizing Relations Among Units and Measures

#### Performances

- 6A
Anticipate that n iterations of 1/n unit generate a unit length.

“This length is 1/5

*Goade*, so it takes 5 of them to make 1*Goade*unit.” (Some students at this level may need to literally iterate to establish this before being able to anticipate it.)