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."
Before this lesson, students were introduced to literature about pumpkins and worked on estimation using objects, counting in groups of ten, and graphing. In this lesson, students develop their sense of larger numbers by working with real quantities of pumpkin seeds (approximately 200-600 seeds). Students are asked to estimate how many seeds they think are in a small pumpkin. Students' estimates are marked on a graph, and the pumpkin is cut in half and its inside revealed. On the basis of their observations, students revise their estimates. Working in small groups, students are given their own pumpkins with which to estimate and count the number of seeds. The groups are required to reach consensus both for their group's estimate and for their seed-counting strategy. Students in a whole-class discussion report on - and compare the actual counts with - the original estimates, and discuss the range among the actual counts. Another pumpkin about the same size as the first is held up and students are asked how many seeds they think it holds. To conclude, students are asked to share something they learned about pumpkins. Topics for Discussion
For Teacher workshops
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.
Reaching Group Consensus
- Several opportunities existed in this lesson for students to work together as a group to reach consensus on an issue. Which tasks required students to reach consensus? What are some of the advantages and disadvantages of having students work together to reach consensus?
- In the "Goofy Jack-o-Lanterns" group, one boy thought the estimate should be 80, another student thought it should be 100, and a third student suggested 1000. How did the group members react to this last suggestion? What do the suggestion and the reaction tell you about the students' number sense? As this discussion continued, someone suggested 180. What was the rationale stated for using 180 as the group estimate? What does this rationale indicate about the students' number sense?
- Ms. Richardson intervened and helped the "Goofy Jack-o-Lanterns" reach consensus. How else could the situation be handled? What are some of the advantages and disadvantages of a teacher's intervening in this situation?
- What strategies did you observe students using to determine the number of seeds in their pumpkins? What does this tell you about their number sense?
Exploring Content and Connections
- Identify the mathematical content and mathematical connections in this lesson.
- Why do you think Ms. Richardson cut a pumpkin into halves and showed them to students? How does her approach take into consideration students who have never seen the inside of a pumpkin?
- What was the value in having students revise their initial estimates? What do you think of what Ms. Richardson chose to show her students before their first two estimates? List the pros and cons of her choices.
- The class noted that all the actual counts of the pumpkin seeds were more than the estimates. Ms. Richardson then asked, "Why do you think it was always more?" How did students respond to this question? What did you think of the flow of the discussion and the ideas that emerged?
- Toward the end of the lesson, Ms. Richardson held up a pumpkin and stated, "If this was the first thing this morning and I said, 'how many seeds are in this pumpkin?' would you say 30? 75? 100? 200? 300?" How did students respond to each number? What advantages or disadvantages can be found in Ms. Richardson's approach of giving numbers in comparison to asking for numbers?
- Ms. Richardson commented, "I would guess anywhere between 200 and 500 for a small pumpkin." Where did these numbers come from? What is the value of giving a range rather than a specific number?
- Students were asked to estimate the number of seeds in a pumpkin two sizes larger. What does it mean for a pumpkin to be two sizes larger? What were some of the students' estimates? How would you have answered this question? Explain your reasoning.
ExtensionsAssessing Children's Understanding of Number Representations
Brainstorming Pumpkin Activities Consider this question: What else can you estimate and measure with a pumpkin? Compile a list of ideas. For example, estimate and measure the circumference and weight (or mass) of the pumpkin; count the number of vertical lines; make line graphs showing the relationship between two variables for a sample of pumpkins; and think about correlations between the number of seeds and the weight. Also brainstorm ideas of things to do with a pumpkin that can lead to connections to science, such as studying the life cycle of a pumpkin, or connections to language arts, such as creative writing and drawing.
Develop a set of tasks and questions to assess a child's understanding of the connections among written symbols, quantities, and oral names. For example, to assess the connection between oral name and quantities, ask a child to show three pennies. Be sure also to assess the other direction, from quantities to oral name. For example, display five pennies and ask, "How many pennies do I have?" Develop tasks for all six contexts. Depending on the age of the students, assess their ability to recognize symbols as well as their ability to write them. Then use tasks to assess some children's understanding of the connections among these representations for number.