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| Overview |
 This activity demonstrates the strategy of problem posing (Brown & Walter, 1983/2005). A piece of math-related children’s literature, Sixteen
Cows (Wheeler, 2002), is used as the springboard for this problem-posing.
After hearing the story read aloud, students are invited to brainstorm
some literary and mathematical observations to the story. With the teacher’s
guidance, students then turn those observations into “what-if” mathematical
extensions. These extensions become mathematical problems that students
solve, both individually and as a whole class. Since this strategy highlights
changing attributes of a story, it underscores for children the range of choices
that authors have.
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| From Theory to Practice |
Problem-posing involves taking a “what-if” stance toward a problem, situation,
or story. It consists of describing, modifying and extending the attributes
of a story. As children list these attributes, they see a world of related
stories embedded within the first story (Whitin & Whitin, 2004). It has
been argued that the more learners change a given story/problem, the better
they understand it (Brown & Walter, 1983/2005). As children have the opportunity
to discuss their observations with peers, they are better able to write about
the relationships that they see in the story (Short & Harste, 1996). Sorting
through mathematical attributes supports children “to analyze situations
carefully in mathematical terms and to pose problems based on situations they
see” (NCTM, 2000, p. 19). In this lesson, children note the attributes
of a story and use these as the basis for their own mathematical extensions.
Further Reading:
Whitin, David J. & Phyllis Whitin. New
Visions for Linking Literature and Mathematics. Urbana, IL: National
Council of Teachers of English; and Reston, VA: National Council of Teachers
of Mathematics, 2004
Overview
of Standards for Grades Pre-K-12. Reston, VA: National Council of Teachers
of Mathematics, 2000.
Brown, Stephen. and Marion Walter. The Art of Problem Posing. Hillsdale,
NJ: Lawrence Erlbaum, 1983/2005.
Short, Kathy and Jerome Harste, with Carolyn Burke. Creating Classrooms
for Authors and Inquirers. Portsmouth, NH: Heinemann, 1996.
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| Student Objectives |
Students will
- identify attributes of a story.
- pose a problem based on these attributes.
- use pictures, numbers and words to solve a mathematical extension of the story.
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| Instructional Plan |
Resources
Preparation
Instruction and Activities
Session One
- Introduce Sixteen Cows to the class or other selected book for
problem posing.
- Invite a few students to respond to each page as you read
it aloud.
- After reading the story, invite the class to respond further by asking:
- What did you find interesting about this story?
- What else did you notice?
- As the students are responding to the story, fill in the chart with their observations.
- If the students do not mention some of the mathematical features of the
story, then ask them to do so directly:
- What do you notice about the numbers in this story?
- How do the numbers change?
- Add these observations to the left hand side of the chart as well.
- Encourage many different answers, even though they may seem redundant.
The language of each observation can offer different possibilities for extensions.
Possible observations include the following:
- “Both numbers are 8.”
- “They each have the same number of cows.”
- “They each have an even number of cows.”
- “The cows didn’t look the same but the people each had 8.”
- After the students have shared about 10 observations (including at least
6-7 mathematical ones) and they have been recorded, demonstrate how to create
some problem extensions. If needed, follow the Teacher Talk handout for
problem-posing examples.
- Brainstorm some possible “what-ifs” for four or five of the
students’ observations.
- Decide on one “what-if” extension (based on the various abilities
of your students) and ask each student to solve it, using pictures,
numbers and words to explain their answer.
- After the children have had time to solve this problem, invite different children to share their solutions with the whole class.
- Emphasize the different ways that children used pictures, numbers and
words in their explanations. For instance, you might say, “I appreciate
how you used arrows to show how you added those numbers together” or “I
appreciate how you used the word ‘double’ to explain what you
did first.”
- Begin a class chart that lists effective ways that children use pictures, numbers and words to make their thinking visible.
Session Two
- Return to the chart of observations and “what-if” extensions.
- Read the list over together, emphasizing how the extensions were derived from the initial observations.
- Select two other problems to solve, and invite students to solve at least
one of these. Again, encourage students to use pictures, numbers and words
in their explanation.
- Provide another sharing time when the class can discuss the solutions
as well as their different ways of representing their thinking.
- Invite students to revise and edit one of their solutions to publish.
Some options are as a class book or bulletin board display on problem-posing.
Some of their work might be included in a family newsletter as well.
Extensions
- Invite students to develop an alternative story that is based on some of
their “what-if” extensions.
The following alternative provides an example:
If there are now 4 people sharing
16 cows, how might this mathematical variation change the story line? What
will be the central problem of this story? Will there now be two couples marrying
at the end of the tale?
- Read other books that lend themselves to problem-posing extensions.
Before reading the next book aloud to the children, you might create a chart
yourself (or with other teachers) of your own observations and “what-if” extensions.
In this way, you can envision more of the mathematical possibilities ahead
of time; then, you can share the story with students and brainstorm together
some observations and possible extensions.
Web Resources
- National Council of Teachers of Mathematics Standards
http://illuminations.nctm.org/info/standards.asp
- Overview of the teaching and learning standards from the National Council of Teachers of Mathematics.
- Online Concentration
http://illuminations.nctm.org/tools/tool_detail.aspx?id=73
- Online game of concentration, from the MarcoPolo partner Illuminations,
which asks students to look for multiple ways of representing a number.
- Sixteen Cows
http://www.lisawheelerbooks.com/sixteen_cows.htm
- Web site devoted to the book Sixteen Cows, with additional teaching ideas and extensions.
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| Student Assessment/Reflections |
Feedback for this lesson should be ongoing, integrated with the students' problem-posing and problem-solving process. It's not something that occurs after the project is completed, but while students are working. Since the focus is on the strategies and processes students use, feedback and reflection takes the form of kidwatching and specific commentary that helps students expand and extend their language strategies. Use this information to create formal or informal checklists, which you can refer to as you observe students at work.
- Whole group and small group participation
- In your observations of student groups, these questions can guide your
observations:
- Are students engaged in the group work?
- Do they listen to the suggestions
of others?
- Do students work cooperatively together throughout the lesson?
- Do students provide useful ideas when participating in the group?
- Do the members actively look for solutions for one another’s problem-posing?
- Do students listen to, share with, and support the efforts of others?
- Student explanations of their strategies
- As you read or listen to students’ explanations, these questions can
guide your observations:
- Are the pictures, numbers, and words clear?
- Do they add to the reader’s or listener’s understanding of
the strategies?
- Does the explanation show a complete understanding of the mathematical
concepts used to solve the problem?
- Quality of story problem written extensions and
individual student follow-up work, including clarity of ideas and details in
written work
- As you read students’ story problems, these questions can
guide your observations:
- Is the problem and solution correct? creative?
- Is the solution easy-to-understand and logical?
- Are the details clear and complete?
- Does the story follow a clear sequence? Are the ideas clear and organized?
- Does the story problem include creative
details and/or descriptions?
- Is the story is readable,
clean, neat and attractive?
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1 - Students read a wide range of print and nonprint texts to build an understanding of texts, of themselves, and of the cultures of the United States and the world; to acquire new information; to respond to the needs and demands of society and the workplace; and for personal fulfillment. Among these texts are fiction and nonfiction, classic and contemporary works.
5 - Students employ a wide range of strategies as they write and use different writing process elements appropriately to communicate with different audiences for a variety of purposes.
6 - Students apply knowledge of language structure, language conventions (e.g., spelling and punctuation), media techniques, figurative language, and genre to create, critique, and discuss print and nonprint texts.
11 - Students participate as knowledgeable, reflective, creative, and critical members of a variety of literacy communities.
12 - Students use spoken, written, and visual language to accomplish their own purposes (e.g., for learning, enjoyment, persuasion, and the exchange of information).
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