Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
The goals of the NCTM's reasoning process standard are that "in grades K-4, the study of mathematics should emphasize reasoning so that students can:
(NCTM, Curriculum and Evaluation Standards for School Mathematics, p. 29) Video Overview
- draw logical conclusions about mathematics;
- use models, known facts, properties, and relationships to explain their thinking;
- justify their answers and solution processes;
- use patterns and relationships to analyze mathematical situations;
- believe that mathematics makes sense."
Students in this lesson are asked to estimate the number of listings in the white pages of the metropolitan Milwaukee phone book. This is a Fermi problem because it involves estimating with large numbers and using prior knowledge.
During the lesson, students develop number sense as they explore the magnitudes of large numbers through the investigation and as they discuss the reasonableness of their estimates. Once they complete their first estimates, the teacher draws a number line on the chalkboard, and students discuss how to mark the intervals based on the range of their estimates. Students mark their individual estimates on the number line and then discuss what information would help them make better estimates, such as knowing the city's population and that several people can share a single listing.
Students work in groups to revise their estimates and are given various tools - the white pages, string, ruler, magnifying glass, and calculator - to help determine their estimates. After each group reaches consensus, the new estimates are posted on the number line. Students justify their estimates and explain their reasoning verbally and in writing. The following day, students research the question to arrive at a final class estimate.An Exploration
For Teacher workshops
How Many Listings?
This investigation will familiarize teachers with the mathematical activity that students are working on in this lesson. Each group will need these items:
- white pages of a phone book
- self-stick notes in two colors
- paper and pencil
Follow these steps:
- Ask teachers to estimate how many listings they think are in the white pages of the phone book. Have them write their estimates on self-stick notes of the same color. As a group, decide on appropriate number line intervals and graph each teacher's estimate on a number line.
- Discuss and list known information that will help everyone improve the estimates, such as the population of the area and the average number of individuals who would share one listing. Then have teachers form groups to make a group estimate, record it, and provide reasoning behind it. The new estimates should be written on a self-stick note of a different color and graphed on the original number line. Discuss the display of data and the reasoning behind the estimates. Consider the following questions:
- What happened to the range?
- Are any clusters apparent? Any outlying estimates?
- How did groups arrive at their estimates?
- How do the various approaches help explain the range?
- Give each group a phone book. Have each group develop a strategy for estimating the number of listings that are actually in the phone book and describe its strategy in writing. Have each group report and graph its new estimate and explain its strategy for arriving at this number. How do these estimates compare with the previous estimates? What are some reasons for the differences?
Topics for Discussion
The following areas provide a focus for discussion after you view the video. You may want to customize these areas or focus on your own discussion ideas.
Connecting Estimation and Number Sense
- How did Ms. Chase cultivate the idea of what a good estimate is and how to make a good estimate?
- On the basis of students' estimates, what would you surmise about their sense of large numbers? How do activities like the phone-book estimation develop students' number sense?
- How did Ms. Chase guide students' responses and how did she respond to students' unrealistic estimates, such as 1 million 1 thousand and 10 million? What questions would you ask to help students resolve these very different estimates?
- Discuss how Ms. Chase set the stage for the problem and drew students into the mathematical task. What are the benefits of having students do independent work in addition to group work?
- Ms. Chase did not know the number of listings in the phone book before doing this lesson. How important is it for a teacher to know the actual number of listings in the telephone directory? How can not knowing the exact number be an advantage for a teacher?
Writing in Mathematics
- How was writing a part of this lesson? Comment on students' reaction to writing in mathematics.
- Students were directed to write out their group's thinking and reasoning. How does this written communication support student learning? What are some ways to incorporate student writing into mathematics learning?
- Students reported their strategies for estimating toward the end of the lesson by reading what their groups had written. Comment on the detail and depth of their writing. Then discuss ways to help students improve their written communication of mathematics.
- Discuss the use of student writing for assessment purposes. What criteria would you use to evaluate the writing?
Develop additional activities that involve the phone book, the exploration of mathematics, and connections to social studies and other curricular areas. For each activity, identify the mathematical ideas that might emerge and the mathematical connections that could be emphasized. For example, find the average length of the last names for a particular letter, determine which letter of the alphabet has the longest or shortest average name length, determine which letter has the most last names, or find the most common last name.