Private: Learning Math: Measurement
Classroom Case Studies, K-2 Part B: Reasoning About Measurement (40 minutes)
Session 10: K-2, Part B
In This Part
- Exploring Standards
- Analyzing a Case Study
The National Council of Teachers of Mathematics (NCTM, 2000) has identified measurement as a strand in its Principles and Standards for School Mathematics. In grades pre-K-12, instructional programs should enable all students to do the following:
- Understand measurable attributes of objects and the units, systems, and processes of measurement
- Apply appropriate techniques, tools, and formulas to determine measurements
Recognize the attributes of length, volume, weight, area, and time•Compare and order objects according to these attributesIn pre-K-2 classrooms, students are expected to do the following:
- Understand how to measure using nonstandard and standard units•Select an appropriate unit and tool for the attribute being measured
- Measure with multiple copies of units of the same size, such as paper clips laid end to end
- Use repetition of a single unit to measure something larger than the unit; for instance, measuring the length of a room with a single meterstick
- Use tools to measure
- Develop common referents for measures to make comparisons and estimates
The NCTM (2000) Measurement Standards suggest that students will begin to understand attributes of area by “looking at, touching, or directly comparing objects.” Instruction that provides students with opportunities to measure and explain their findings will help teachers further this goal. Furthermore, “although for many measurement tasks students will use nonstandard units, it is appropriate for them to experiment with and use standard measures such as centimeters and meters and inches and feet by the end of grade 2” (NCTM, 2000, p. 105).
Principles and Standards for School Mathematics © 2000 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or redistributed electronically or in other formats without written permission from NCTM. standards.nctm.org
Standards are listed with the permission of the National Council of Teachers of Mathematics (NCTM). NCTM does not endorse the content or validity of these alignments.
Analyzing a Case Study
To begin the exploration of what measurement topics look like in a classroom at your grade level, you will watch a video segment of a teacher who took the Measurement course and then adapted the mathematics to her own teaching situation. We will begin by looking at some of the content addressed in the videotaped lesson.
Watch this video segment from Ms. Guerino’s class and think about how both the lesson and the teacher are assisting students in making sense of some of the measurement concepts described in Part A.
You can find this segment on the session video approximately 11 minutes and 42 seconds after the Annenberg Media logo.
Prior to this lesson, students worked with nonstandard units to measure area. What might students learn from measuring objects using units like circular discs and paper clips?
Before her students determine which rectangle has a larger area, Ms. Guerino asks them to predict which one is larger. Where in the lesson are students using estimating techniques to decide which rectangle is larger? Why is it important to have students make estimates before they actually begin measuring?
Why do you think Ms. Guerino asked students to measure the two shapes using both square inches and square centimeters? How might discussion among students and the teacher contribute to students’ understanding of conservation of area? What other measurement concept was Ms. Guerino exploring when she asked students to cover the shapes with different-sized units?
Is it important to have misconceptions surface in a lesson? If misconceptions do occur, how can they be addressed?
The purpose of the video segments is not to reflect on the teaching style of the teacher portrayed. Instead, look closely at the methods the teacher uses to bring out the ideas of measurement while engaging her students in activities.
Measuring with nonstandard units provides students the opportunity to choose the unit they want and to gain a sense of the physical process of a measuring experience. When students use nonstandard units (e.g., paper clips, playing cards, or buttons) to measure area, they notice that they can’t cover the surface completely. The other difficulty with using nonstandard measures is that students can’t compare the measures; for example, 110 paper clips vs. 12 playing cards.
Students made predictions about whether the rectangle or square was larger at the start of the lesson. Doing so helps them consider which attributes are important as well as visualize how much space the shapes take up. Students may even visualize cutting up one shape to cover the other. In the video, one group of students compares the dimensions of the two shapes. The students in this group notice that the length of the rectangle is longer than that of the square, and then they use finger span to estimate that the heights of both the rectangle and square are the same. They conclude that the rectangle must be larger.
Ms. Guerino asked students to use both square inches and square centimeters to measure the shapes so that she could address the idea of conservation. Students may not realize that the areas will stay the same even if they measure with different-sized units.
To help students understand conservation, ask them to predict which rectangle will have the largest area before they cover the shapes with the new unit. Also, asking students to explain why the larger shape is still larger, and if it will always be larger, can help them reason about conservation.
Another measurement concept that Ms. Guerino explored when she asked students to cover the shapes with different-sized units is the concept that the smaller the unit, the greater the number needed to cover the shapes.
It is not only important, but actually helpful when misconceptions surface in a lesson. Misconceptions often provide the contrast needed for students to understand an idea. One strategy to use when misconceptions arise is to present two opposing viewpoints and ask students which idea they agree with and why. You can also ask students to give a counterexample, or give the students a counterexample yourself and ask them what they think. Another reason why it is important to discuss misconceptions: By confronting a conception that turns out to be incorrect, students become engaged in understanding why it is a misconception.
Session 1 What Does It Mean To Measure?
Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.
Session 2 Fundamentals of Measurement
Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.
Session 3 The Metric System
Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.
Session 4 Angle Measurement
Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.
Session 5 Indirect Measurement and Trigonometry
Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.
Session 6 Area
Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.
Session 7 Circles and Pi (π)
Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.
Session 8 Volume
Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.
Session 9 Measurement Relationships
Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.
Session 12 Classroom Case Studies, 6-8
Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.