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."
Cranberries, the largest agricultural product in Massachusetts, are the focus of this estimation lesson. The lesson begins as students are asked to predict how many scoops of cranberries will fit in a jar. The students record their individual estimates and make a class graph of the estimates. After discussing the graph, one student partially fills the jar by adding three scoops of cranberries. Each group is then given its own jar of the same size containing three scoops of cranberries and asked to estimate how many scoops would fill the jar.
After reaching group concensus, students report and discuss the new estimates. Even though the groups have identical-sized containers, the estimates vary. Each group then scoops cranberries into its jar to check its estimate.
After students report their results, they are asked to estimate the number of individual cranberries in their jars. The groups report their results, and as the class discusses the average number of cranberries in a scoop and the average number of scoops per jar the concepts of range, mode, and median emerge. Using their averages, the groups determine an estimate for the number of cranberries in each jar.
During the lesson, students make and discuss a graph of their estimates and use fractions, mental computation, skip counting, and spatial reasoning.An Exploration
For Teacher workshops
The Cranberry Connection
This investigation will familiarize teachers with the mathematical activity that students are working on in this lesson. Each group will need these items:
Follow these steps:
- self-stick note paper (one piece for each person)
- a jar or container (same size for all groups)
- a scoop (same size for all groups)
- cranberries or other small objects
Topics for Discussion
- Show the teachers the jar, the scoop, and the cranberries or whatever items you are using to estimate. Ask teachers how many scoops of cranberries they think can fit in the jar. Tell teachers to write their individual estimates of the number of scoops on self-stick notes and then make a bar graph of their estimates. Discuss their observations of the graph, considering the following questions:
- What is the range?
- Is it a large or small range?
- Why do teachers think this range occurred?
- Identify the mode. What other interesting observations can be made about the data?
- Place a few scoops of cranberries into the jar. (In the video, the teacher places three scoops in each jar before giving them to the students.) Let the teachers know that they will now have a chance to revise their estimates in groups.
- Divide the teachers into groups of four. Give each group a jar, a scoop, and some cranberries. Tell each group to place the same number scoops of cranberries into its jar as you did. The teachers can revise their estimates by reaching group consensus on a new estimate. Have each group report its estimate. Record the group estimates on the chalkboard, and then discuss the new range and other observations. Each group can now scoop cranberries into its jar and determine the total number of scoops. Record the actual number of scoops and discuss reasons for possible discrepancies.
- Now ask teachers how many cranberries they think are in their jars. Have each group estimate the number of cranberries in its jar. Discuss the estimates and teachers' strategies for reaching those estimates.
- It is likely that some groups used the concept of average to make estimates. Explore this idea further with the whole group. What is the average number of scoops in a jar? What is the average number of cranberries in a scoop? Discuss reasons for using the mode, the median, or the mean. Using these measures of the average number, compute an estimate for the number of cranberries in a jar and compare it with the previous estimates.
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.
Developing Estimation Strategies
- Ms. Dufault asked students to estimate the number of scoops before estimating the number of individual cranberries. What are the benefits or drawbacks of this approach?
- Why did Ms. Dufault put three scoops of cranberries in each jar and show this partially filled jar to the students before asking them to revise their estimates? How did this referent affect students' estimates?
- Discuss the value of allowing students to make and then revise their estimates.
- What strategies were used by students to determine the number of scoops and then the number of cranberries needed to fill the jar? How effective were the students' strategies? What is your evidence?
- Discuss chunking as an estimation strategy and how it was used in this lesson.
- The students never actually counted the cranberries in their jars. What are the pros and cons of having students find the actual count after estimating?
Integrating Mathematical Ideas
- List the mathematical ideas that were explored in this lesson.
- Discuss what you learned about the students and their mathematical understanding. Did any of the students' reasoning surprise you? If so, what?
- Comment on the students' number sense. How does estimation promote number sense?
- Ms. Dufault mentioned that this lesson involved spatial reasoning. Identify instances in which this concept was apparent. Discuss the role of spatial sense in estimation.
- Estimation was a focus of the lesson, yet Ms. Dufault integrated many aspects of data analysis and statistics into this lesson. Discuss the advantages and disadvantages of integrating mathematical ideas.
- Comment on Ms. Dufault's use of the words mode and range during the lesson.
- Even though the concept of median was used to determine the average number of scoops and the average number of cranberries in a scoop, Ms. Dufault did not use the term median. Would you have introduced this term to the students? Why or why not?
An Estimation Walk
Students have lots of questions about their world that involve quantities and quantitative comparisons. Take students on a walk around the neighborhood to observe and wonder: How many? How much? How far? How tall? How big? Stop periodically to record their questions. For example, ask "How tall is that big tree?" or "How long is a block?" If time allows, select some of their questions to explore and estimate during this walk, saving the others for future estimation walks.