Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

## Buffalo Estimation

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:

• 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."
(NCTM, Curriculum and Evaluation Standards for School Mathematics, p. 29) Video Overview
Mr. Wszalek is a mathematics resource teacher who assists teachers in individual classes. Students in this class have been studying the Oregon Trail and are now using a buffalo theme to study estimation. Having previously estimated the size of one buffalo and how many standing buffalo would fit in their classroom, students now explore how many buffalo would fit on the playground.

Prior to the video, students made individual estimates, and the class now discusses these estimates. A typical-size buffalo is modeled on the floor for reference. Students develop number sense as they think about the reasonableness of their estimates and use a referent to adjust their estimates.

After studying the model, students revise their estimates and then discuss the range between their first and second estimates and why the range became smaller. Students then form into groups to arrive at an estimate they will share with the class. The class brainstorms ways to figure out how many buffalo would fit on the playground and plans to continue its explorations the following day.

An Exploration
For Teacher workshops

A Roomful of Buffalo
This investigation will familiarize teachers with the mathematical activity that students are working on in this lesson. Each group will need these items:

• paper and pencil
• metersticks or yardsticks

1. Because the lesson in the video evolved from a study of westward migration and the Oregon Trail, you might want to begin with a discussion of this historical period and point out that large herds of buffalo roamed the plains.

2. Ask teachers how many standing buffalo could fit in the room. Have teachers make an initial estimate and then discuss their reasoning and strategies. Identify the range of the estimates.

3. Knowing the size of a typical buffalo will help teachers refine their estimates. To determine the size of a typical buffalo, teachers can do research, rely on personal experiences, or use the measurement decided on by Mr. Wszalek's students - 8 feet long by 4 feet wide. Construct a model of a typical buffalo by marking off the floor space with tape, rulers, or other means.

4. Ask teachers to revise and share their estimates individually. Compare the range of these estimates to the range of their initial estimates. Now have teachers work in groups to measure the room and determine the number of standing buffalo that could fit in the room. Compare these results to the estimates.

5. To extend the activity, find an open area, such as a playground, parking lot, or lawn. Have teachers estimate and then determine the number of buffalo that could fit in this larger area. Ultimately, students in the video work toward determining how much area a million buffalo would cover. You may want to incorporate this challenge into your training.

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.

Making Connections

1. How were estimation and number sense interwoven in this lesson?

2. List all the mathematical ideas embedded in this lesson. Highlight ways in which these ideas connect with each other in the lesson.

3. Describe the statistical concepts that emerged during the discussion and how Mr. Wszalek involved students in understanding them.

4. What connections were made to other subject areas?

5. Cite the connections made to real life. What other connections to real life could have been made in this lesson or could be emphasized in follow-up lessons?

6. What other questions could be considered when doing the buffalo estimate? For example, how much grazing space does a buffalo require? What impact would this information have on the estimate?

Facilitating Classroom Discourse

1. Examine both small-group and whole-group discussions in this classroom. What are the advantages and disadvantages of each type?

2. Classify the techniques Mr. Wszalek used to promote mathematical discourse, such as open-ended questioning and nonjudgmental responses. How did these techniques promote the students' thinking?

3. List the steps Mr. Wszalek used to help students improve their estimates.

4. What strategies did students use to come up with an estimate for their groups?

5. What suggestions did students make for figuring out the number of buffalo that would fit on the playground? Outline how they selected a strategy. How else could this discussion have been handled?

Extension
Estimation Opportunities

Think of and describe other interesting situations that would pique students' curiosity about estimating both area and volume. For example, how many students would fit in the classroom or on the playground? How many bicycles would fit into the classroom?

Buffalo Research

Research the history of buffalo in North America and their status today. For example, how did early Americans use all parts of the buffalo? What was the value of the buffalo to early societies? Do buffalo still exist? Why have their numbers declined so dramatically? Also investigate other mathematical ideas related to buffalo. What is the buffalo birth rate? What is their life expectancy? What, if anything, would happen to their numbers with different rates of birth and death?

Use this information to plan a unit on buffaloes, or consider similar questions to plan a unit on an animal native to your region or relevant to a lesson you're teaching.