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.

Download ToML Construct Map

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.)

Examples