 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.
Related Units for Length
 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
 1B
Identify measurable attributes (qualities).
“We could find out how long the caterpillar is or how fat it is.”
 1C
Define the attribute being measured.
“Fat means how far it is around the caterpillar” (analogy to circumference of wrist).
“Big means the one that weighs the most.”
 1D
Distinguish (e.g., equal, not equal) or order (e.g., greater, lesser) magnitudes of an attribute by direct physical comparison.
“This book is taller than that one (Student aligns the books and compares).” (This relation of equality tends to be mastered first.)
“Johnny is tallest, and Sally is in the middle, and Jennifer is the shortest.” (Ordinal relations tend to be more difficult for very young children).
 1E
Distinguish (e.g., equal, not equal) or order (e.g., greater, lesser) magnitudes of an attribute by direct comparison of representations.
“Pumpkin A is taller than pumpkin C” (Student aligns paper strips that stand in for height and notices that A’s strip is longer than C’s strip).
 1F
Develop and use local (classroom) conventions to distinguish or order two or more objects by a single attribute.
“We decided that for height the streamer started at the floor and went straight up until you could see it was level with the top of the pumpkin (not the stem). Then we found that pumpkin A was taller than C.”
 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 paperclips long (the paper clips may not be identical lengths).
Examples
 2B
Tile and explain why (the explanation is required).
“The units should touch so there is no gap. Gaps mean that some of the space is not measured.” The units should not overlap. If they do, you measure some of the space twice.
 2C
Use identical units and explain why.
“It is better to measure with all the same units because then you can just count the number of units.”
“If you use different units, then you have to tell which ones—like, this line is 2 red and 3 green units long.”
 2D
Count with reservoir of identical units to tile a length and represent measure by the total. If units are not identical, distinguish among them.
Tiles 8 red unit lengths, counts all, reports measure as “8 reds.”
Tiles 2 blue, 4 red units. Reports measure as “2 blues, 4 reds.”
 2E
Consider suitability of unit and explain why.
“That (distance) is very long, so using my clipboard (as a unit) works better than my pencil (as a unit.)”
 2F
Qualitatively predict the inverse relation between size of unit and measure.
“If we use small steps, the measure is larger than if we use large steps.”
 Length Level 3
 Iterating Units and Symbolizing Length Measure as Distance Traveled
Performances
 3A
Reuse (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
 3B
Symbolize the starting point of measure as zero (0).
When you haven’t traveled yet you start at the starting point, called zero (the heel of the foot), and the toe of your right foot marks 1 foot,. And the next step with your left foot is 2 feet (see the heel of your left foot touches the toe of your right foot).
 3C
Symbolize measure as whole number unit indicating distance traveled.
“You don’t write a 1 in the middle of the unit like this one…
…because the unit starts at zero and ends at 1 — that’s how far you have traveled! When
you put it in the middle, you don’t see where the unit ends!”
 3D
Use and justify standard (including conventional) unit.
“If we all agree to use Justin’s foot as a unit, then we can compare our measurements of the lengths of different desks.”
 3E
Explain how magnitude of measure is the ratio of the number of units accumulated to the unit length.
“This is 5 Goades long. It is 5 times as long as 1 Goade.”
 3F
Iterate composite unit, reflecting 2 levels of unit structure.
“I lined up the zero on my ruler with the start of the path and traveled to the 3 ft. mark at the end of the ruler. Then I kept my place and moved the ruler to line up my place with zero and then the end of the path was at 2 ft. So the total distance that I traveled was 5 ft.”
(Note: simultaneous marking of 3u and 0)
 Length Level 4
 Partitioning and Symbolizing Partitioned Units (2 splits)
Performances
 4A
Partition and compose equipartitions by factors of 2 (2split), 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
 4B
Symbolize relation between origin and partitioned units on scale.
“You don’t write 1/4 in the middle. . .
…put it at the end of the part of the unit, so you can see how far you have traveled.”
 4C
Coordinate whole and part units
“It was 15 inches or 1 1/4 feet.” Note: These are 2level units, or unitsofunits.
“It’s 6 quarters long, or we can say 1 1/2 or 1 2/4 units long.”
 4D
Anticipate outcomes of more than 2 repetitions of a 2split of a unit.
“If I fold the unit to make of 1/2 the unit, and then fold it in half again, and then again—3 times—the unit will be split into 8 (equal) parts.”
(then folds unit to show this is true)
 4E
Symbolize multiplicative comparisons involving splits of 2 with words and arithmetic operations

 4F
Account for change of origin when measurement does not start at zero (whole numbers).
“I can start to measure from the 3 on the inches ruler, and take off 3 inches from the result.”
(Note: Starting at 2 or 3 is generally more difficult than starting at 1. Starting at a nonwhole number is more difficult still).
 Length Level 5
 Partitioning and Symbolizing Units Involving 3splits and Compositions of 2 and 3Splits
Performances
 5A
Generate a 3split 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
 5B
Symbolize relation between origin and 3 splitpartitioned units on scale.

 5C
Anticipate outcomes of repetitions of 3split.
“If you make thirds, and then split it again into thirds, you get 9 parts.”
 5D
Compose splits of 2 and 3 to generate 1/6 or 1/12.
1/2 of 1/3 ft is 1/6 ft.”
“1/2 of 1/2 of 1/3 ft is 1/12 ft.”
 5E
Interpret markings on a standard foot ruler.
Identify inches as 1/12 ft.
Understand how different vertical lengths convey 2splits of inch, such as 1/2 in, 1/4 in, 1/8 in.
 5F
Account for change of origin when measurement does not start at zero (fractions and whole numbers).
“If I start at 3 and go to 7 1/4, the measure is 4 1/4 .”
“If I travel from 2 1/2 cm to 8 3/4 cm, it’s 6 1/4 cm.”
 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
 6B
Use relations among units to quantify results of changes in unit.
“The measure of the height of the plant is 14 cm. If a cm is 10 times as long as a mm, then the measure is 140 mm.”
“If I change the unit so that it is half as long as the original unit, the measure doubles.”
 6C
Derive relations among units, given expression of the same attribute in different scales of measure.
“If the measure of the height of the plant is about 10 cm or about 4 inches, then an inch is about 2 1/2 cm.”
 6D
Anticipate inverse relation between multiplicative comparison of unit lengths.

 6E
Invent and justify a measure of length.
“We can use the time it takes as a measure of distance, because we can assume a constant rate.”
“This bushiness index (extent of branching) tells me how much the elodea plant (its total length) grew in the water.”