Mark W. Ellis, Ph.D., NBCT
Data collected from over 750,000 students using i-Ready® Diagnostic & Instruction show that measurement is one of the most challenging math domains for students. According to the authors of Mathematics Learning in Early Childhood, measurement is difficult because it involves three skills:
- Identifying what is to be measured
- Choosing a unit of measure
- Determining the number of units
Six New Concepts for Students
In order to perform these actions proficiently and meaningfully, students must understand six new concepts:
- Conservation of length or other measurable quality—understanding that an object’s length does not change when moved
- Transitivity of measurement—understanding that the measures of two objects can be compared to a third object
- Equal partitioning—understanding that an object can be sliced into segments of the same size or unit without changing its overall length
- Unit and unit iteration—understanding that measurement requires using identical copies of a standard length
- Origin—understanding that a measurement of length is a distance from a fixed starting point, including from zero
- Relationship between abstract number (quantity) and physical measurement characteristic (e.g., length)—understanding that a numerical measurement is a representation of an object’s length
As Amy MacDonald and Tim Lowrie explain in Developing Measurement Concepts within Context students’ proficiency with linear measurement typically follows a, “Progression from identification of the attribute and use of informal measurement through to the use of formal units, with application to contexts being the final stage.”
We see this progression in the Common Core Standards for mathematics:
- Kindergarteners learn to describe measurable attributes of objects and compare measurable attributes of two objects, addressing the concept of conservation.
- First graders learn to compare two objects to a third to determine relative length (transitivity) and begin to use unit iteration to determine lengths.
- Second graders learn to use standard units and tools of linear measurement.
Learning progressions are important in every topic that students find challenging, since the standards were intentionally developed with coherence within and across grade levels. One program designed to build knowledge based on learning progressions is Ready® Mathematics, since each Teacher Resource Book includes a visual map of key progressions and lesson-specific connections to the progressions.
Measurement should not be taught simply as a skill to be mimicked. Teachers must build understanding of measurement concepts starting with familiar contexts and experiences. For example, they can move from visually comparing two lengths, perhaps two pencils, to deciding “how much” longer one is than the other using a third object. In other words, non-standard units can be used to show the need for standard units.
Building the Foundation for Later Grades
Students’ early learning about linear measurement in K–2 sets the foundation for later work with two-dimensional measurement of area in grades 3–5 and three-dimensional measurement in grades 5–7. Students in the upper grades are expected to develop a flexible approach to solving measurement problems, including decomposing figures, choosing appropriate units, and making reasonable estimates. Each of the six key concepts of measurement is essential to students’ success with the later standards.
Thinking about my own teaching, when my sixth graders struggled with making sense of volume, I realized they had only learned rules and procedures to find perimeter and area and did not see any conceptual connections. I first asked students to explain to a partner (then as a whole class) the difference between perimeter and area. After a three-minute discussion, I gave them a task that asked them to find all rectangles with a given area, compare the perimeters of these, and write down their observations.
Students worked in pairs and came to understand why perimeter was a measure of distance in linear units and area was a measure of two-dimensional space having square units. Afterward, they were much more ready and willing to make sense of measuring volume with cube units. The CPALMS website run by Florida State University has a similar activity that’s worth checking out.
In the CCSS, the culmination for measurement comes in grade 8 when students learn about geometric relationships on the coordinate plane using transformations. Much of what students need to know about measurement in order to engage productively with standard 8.G.A.2 goes back to their early learning about measuring length. To be ready for this standard, students must understand concepts of origin, unit, conservation of length, and transitivity. After all, the coordinate plane formed by x and y axes is, at its foundation, two perpendicular number lines. So early learning about measurement matters.
Once students start using standard units, you can stretch their understanding of origin and conservation by requiring them to measure beyond the end of the ruler or other tool or to begin measuring at a non-zero location. This will help them understand that the measure of a specific length is a characteristic of the object itself, not simply a number on a measurement tool.
As you design early measurement activities, build on students’ everyday experiences and interests and make their thinking the center of classroom discussions. For example, ask students to measure various parts of their body with yarn and find the lengths in centimeters. Then ask groups of students to compare the measurements of two students and make posters for the class math wall.
To continually reinforce measurement concepts and skills in an informal but productive way, try occasional estimation tasks that get students thinking and talking. For older students who are struggling with measurement, it may be helpful to assess and revisit their understanding of the six key measurement concepts. Both Constance Kamii’s article “Measurement of Length: How Can We Teach it Better?” and Heidi Bush’s article, “Assessing Children’s Understanding of Length Measurement” offer assessment items that target these concepts.
Grades K–2 Measurement Lesson Ideas:
- Concepts of Measurement (chapter from Mathematics Learning in Early Childhood)
- The Length of My Feet (NCTM Illuminations)
- How Long is a Foot? (activity based on How Long is a Foot? children’s book)
- Measuring Madness (lesson based on Measuring Penny children’s book)
- Measuring Penny (another lesson based on Measuring Penny children’s book)
- Approaching Measurement as Scientists
- Measure Me booklet (requires free registration)
Grades 3–8 Measurement Lesson Ideas:
- Estimation 180 (estimation tasks for grades 5–8)
- Motivating decomposing when determining area (standard 3.MD)
- Investigating Area and Perimeter (investigative lesson with technology option)
- Designing Packages (investigating volume and surface area)
- Finding Surface Area and Volume (technology-enhanced lesson)
- Practicing Surface Area and Volume (web-based activity)
This blog is the second in a series from Curriculum Associates about the most challenging math standards that complements they’re recently released white paper – “Mastering the Most Challenging Math Standards with Rigorous Instruction“.
For more see:
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards – Modeling
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards – Fractions
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards – Statistics
Mark Ellis is a National Board Certified Teacher and professor of education at California State University at Fullerton. He is an author of Curriculum Associates’ Ready® Mathematics program.