最近在一本外文儿童杂志是看了这篇文章，感觉很有意思，原来人家是这样教孩子的。哈哈，很有启发哦 ——I'm Kelly.

Sharing Cookies

Problem

Aja and her sisters, Nolise and Clarice, arrived home from school one afternoon. Aja was really hungry, so the first thing she did was go to the kitchen to find something to eat. On the kitchen table was a plate of cookies along with this note from their mother : ”I had to go to the store. Please cookies equally with your sisters. Be back soon. Love, Mom.” Aja was so hungry that she quickly ate one-third of the cookies on the plate. A little later, Nolise came into the kitchen, found the cookies and the note, and ate one-third of the cookies on the plate. Finally, Clarice came into the kitchen, found the cookies and the note, and, not realizing that her sisters had already eaten some cookies, ate one-third of the cookies on the plate. When the girls’ mother came home, she found 8 cookies on the plate. How many cookies were originally on the plate? How many cookies did each girl eat? Did they share them equally? Explain how you solved the problem.

Extension: Find another number of cookies that the mother could leave on the plate so that the sisters could share them equally without having to break them into parts.

A Simpler problem: Two sisters, Aja and NOlise, came home from school one day. Aja was really hungry, so she went into the kitchen to find something to eat. On the kitchen table she found a plate of cookies along with this note from her mother : ”I had to go to the store. Please share these cookies equally with your sister. Be back soon. Love, Mom.” Aja ate one-half of the cookies on the plate. A little later, Nolise came into the kitchen, found the cookies and the note, and ate one-half of the cookies left on the plate. When the girls’ mother came home, she saw 3 cookies on the plate. How many cookies did Aja first see on the plate? How many cookies did each girl eat? Did they share them equally? Explain how you solved the problem.

Extension: lf there were 4 cookies left on the plate when the mother came home, how many cookies did Aja first see on the plate?

The goal of the “problem solvers” department is to foster improved communication among teachers by posing one problem each month for teachers of grades K-6 to try with their students. Every teacher can become an author: Pose the problem to your students, reflect on your students’ work, analyze the classroom dialogue, and submit the resulting insights to this department. Through contributions to the journal, every teacher can help us all better understand children’s capabilities and thinking about mathematics. Remember that even student’s misconceptions provide valuable information.

Classroom Setup

Spend time discussing this problem with your students but avoid giving too much guidance. Allow your students to work with a partner or in small groups. Encourage the students to use any problem-solving strategy that they know, pay close attention to their thought processes, record their work, and explain their reasoning. This problem lends itself to a follow-up class discussion in which pairs or groups of students present their various solutions. Have the students the students compare their strategies with those of their classmates. Collect actual students’ work, make notes about interactions and discussions that took place, and document the variety of student approaches that you observed. As you reflect on your experience with the problem, keep in mind the following questions:

· What difficulties did students have in under-standing the problem?

· What strategies did you see students using to solve the problem?

· Were you surprised by any students’ responses or interpretations?

· What methods did students use to record their work?

· Did the students relate this problem to any others that they have investigated?

· What extensions to this problem did you or your students pose?

· What did your students learn from investigating this problem?

Share Your Student Work

We are interested in how your students responded to the problem and how they explained or justified their reasoning. Please send us your thoughts and reflections. Include information about how you posed the problem and samples of student work or even photographs showing your problem solvers in action. Send your results with your name, grade level, and school by May 1,2007, to Shery1 stump, Department of Mathematical sciences, Ball State University, Muncie, IN 47306.Selected submissions will be published in a subsequent issue of teaching Children Mathematics and will be acknowledged by name, grade level, and school unless otherwise indicated.

Reference

P61ya, George. How to solve it: A New Aspect of Math-ematical Method. Princeton, NJ: Princeton University Press, 1945

Where’s the Math?

The solution to both versions of this problem involves several steps. When students are encouraged to find their own strategies, some of them may decide to work backward, starting with the 8 cookies left on the plate. Other students may choose to work forward, starting with some representation of the unknown.

The first version of the problem requires greater logical thinking skills than the simpler version does. For example, the first version requires students to recognize that the 8 cookies left on the plate at the end of the story represent the two-thirds that the third sister does not cat. The problem’s extension challenges students to think about the properties of the number 27 that allow this number of cookies to be shared as described without having to break them into parts. The second problem is easier because it involves halves instead of thirds; thus, the part eaten is equivalent to the part not eaten. The problem’s extension merely provides another opportunity to use the same problem-solving strategies.

This problem also provides the opportunity to introduce some math-ematics history into the class discussion. George Pólya, renowned for his extensive study of problem solving , once said, ”If you can’t solve a problem, then there is an easier problem you can solve: find it.” Born in Hungary in 1887, Pólya immigrated to the United States in 1940 and spent most of his professional career at Stanford University. He is probably best known for his book How to Solve It, which was originally published in 1945 and continues to be a valuable resource for teachers today. In it, P61ya identified the four steps for solving problems, a process that continues to guide students toward becoming good problem solvers: (1) understand the problem; (2) devise a plan; (3) carry out the plan; and (4) look back. Some of the problem-solving strategies that students might use when devising their plans for solving the Sharing Cookies problem include drawing a picture, making an organized list, and creating a table.

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