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-
- 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)
This lesson takes place during an extensive unit on plants and seeds and focuses on using estimation to develop number sense. Students are shown three transparent bags filled with beans; each bag is labeled with the amount it contains: 11, 25, and 50. Students are told they can refer to these labeled bags as they work in groups to estimate a bag of uncounted beans. Each group views and estimates the quantities of beans in its bag and discusses the reasoning behind its estimate. By estimating the quantity of beans, the students develop number sense. After deciding on a group estimate, students build place-value ideas by grouping their beans by tens and then counting by tens and ones to find the total number of beans. After each group records its actual quantity, the quantity is recorded on a class chart alongside the group1s estimate. To close the lesson, students reconvene and discuss their estimates, how they arrived at them, and how many beans were actually in their bags.
For Teacher workshops
A Bagful of Beans
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
- a bag of beans or other small items (vary the number of items for each group; place 25 to 50 items in some bags, 50 to 100 or just over 100 items in the other bags)
- small containers for grouping items
- bags of ten, twenty-five, and fifty beans each
Follow these steps:
- Divide teachers into groups and give each group a bag of beans. Ask teachers how many beans they think are in their bags. Have each group make and record an estimate on the basis of group consensus.
- Show everyone the bag that contains ten beans and tell groups to revise their estimates using the bag as a referent. Then present the bag with twenty-five beans and allow groups to revise their estimates again. Present the bag with fifty beans and have groups revise their estimates a final time.
- Discuss the impact that revealing successively larger referents had on teachers' estimates. Consider the following questions:
- How were the referents helpful in giving teachers a sense of the quantity in their own bag of beans?
- Did teachers revise or keep their initial estimates?
- After seeing each referent, did teachers increase or decrease their estimates?
- Would teachers like to see an even larger referent? Why or why not?
- Discuss ways to count the beans, and encourage teachers to suggest strategies they think would be appropriate for young children to use to get an accurate count. Then tell the groups to decide on one of these strategies for counting their beans and to count them. Once the actual count is completed, have teachers compare their counts with their estimates and discuss whether they tended to overestimate or underestimate and possible reasons for this discrepancy.
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
- What is the value of providing referent sets for estimating? How were the referent bags used by students?
- How does estimation promote number sense?
- How did students decide on their group estimates?
- Ms. Mosley commented that her students frequently work on estimation. List ways to include estimation in all (or almost all) lessons, in daily routines, and in special activities.
- Formulate a definition of estimation and devise various estimation strategies.
- What is the value of having children group objects into tens? How does this activity promote number sense and estimation ability? How does this help develop place-value ideas?
- What size of numbers should kindergarten and first-grade children work with? Why?
Working in Small Groups
- Describe the collaboration among the children in the small groups. How did the size of the groups affect the dynamics?
- Outline what you learned about the students from the small-group interactions.
- What do students gain from working in groups? What are the drawbacks?
- Ms. Mosley stresses the importance of each student participating in small-group work. What evidence can be found to document this participation?
- In what ways did Ms. Mosley facilitate collaboration among the children in their small groups? How can further collaboration be encouraged?
- One of the students said she needed help to tally. Describe the help she received, how it was given, and any additional ways she could have been supported.
- What was the role of the recorder in each group? Why did Ms. Mosley want students to record their actual counts? What recording methods did you observe students using?
Consider the notion that some people believe an estimate is only used to get at the exact count. Then organize two lists of real-life situations. The first list should contain situations in which an estimate is appropriate and the second list should contain situations in which an exact count is needed. Think about how you could use the lists to respond to an individual who thinks estimation tasks are incomplete unless an exact answer has been found.