Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

## Problem Solving

The goals of the NCTM's problem-solving process standard are that "in grades K-4, the study of mathematics should emphasize problem solving so that students can-

• use problem-solving approaches to investigate and understand mathematical content;
• formulate problems from everyday and mathematical situations;
• develop and apply strategies to solve a wide variety of problems;
• verify and interpret results with respect to the original problem;
• acquire confidence in using mathematics meaningfully." (NCTM, Curriculum and Evaluation Standards for School Mathematics, p. 23)
Video Overview
This video profiles classroom excerpts in which students are investigating and learning mathematics through problem solving. The excerpts illustrate problem-solving approaches to teaching across the content standards and at various grade levels. For example, students are observed in the following contexts:
• determining how much stain to purchase for their pencil boxes
• measuring the speed and distance a balloon travels
• determining the number of wheels on vehicles in a parking lot
• discussing strategies for dividing marbles equally among a group
• working in pairs to decorate a milk-carton house
• estimating the number of people in a phone book
• working in teams to estimate animal populations in Yellowstone National Park
• figuring out how many students can fit in different areas of a school
• finding three dominoes that have a total of eight dots
• analyzing graphs found in a newspaper
• creating methods to find out how many valentines students exchange
• brainstorming strategies for measuring ant tunnels
• building a rocket shape out of geometric shapes

An Exploration
For Teacher Workshops

Solving a Problem You Can't See
This investigation focuses on mathematics as problem solving. Each group will need these items:

• paper and pencil
• some coins

1. Put three coins in your hand. Without revealing the types of coins, tell teachers you have three coins in your hand and ask them how much money they think you have. Discuss and verify some possible amounts with the whole group.

2. Divide teachers into groups of four. Give each group paper and pencil and two, three, or four types of coins. Challenge them to find all the possible amounts. Have the groups of teachers share their solutions and, more importantly, their strategies and reasoning. How can they be sure that they found all the possible amounts?

3. Discuss the problem-solving experience. Was the problem interesting and engaging? In what ways? How did the problem provide a context for understanding and using mathematical concepts and skills? Compare this problem-solving context to counting money with an approach in which students are just given some coins or pictures of coins and told to find the total monetary value.

4. Discuss ways to modify the problem for students at different grade levels.

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.

Learning through Problem Solving

1. In what ways did problem solving provide a context for learning mathematics concepts and skills in the different classroom episodes? Identify the specific concepts and skills that students were learning in some of the lessons.

2. Cite examples of students' enthusiasm and engagement in the various lessons. How does problem solving promote this eagerness and interest in learning mathematics?

3. Compare the emphasis placed on problem-solving processes and reasoning in the lessons with the emphasis placed on finding correct solutions. Was there a reasonable balance? Provide examples from the video to support your response.

4. How does a focus on solving problems promote confidence in students as learners? Cite examples from the classroom episodes in which students displayed confidence.

5. Describe the interaction of students in the classroom episodes as they worked together to solve problems.

Designing Problem-Solving Experiences

1. Select your favorite problem from all those profiled in the video. Why did you select this particular task? How would you modify this problem for students at different grade levels?

2. What is the teacher's role in a classroom that focuses on problem solving? Give examples of the interactions between the teachers and their students in the various lesson episodes.

3. Which problem-solving experiences arose from everyday situations? Which problem-solving experiences arose from mathematical situations? Cite the advantages and disadvantages of using problems from everyday situations and from mathematical situations.

4. Describe ways to allow students the freedom to investigate problems in their own way, such as with the wheel problem, while still providing some structure for the lesson.

5. In one episode, the teacher formally used a five-step problem-solving heuristic with her first-grade students. What are the pros and cons of using such a heuristic with students of various ages?

6. In one of the episodes, students listed such problem-solving strategies as acting it out and drawing a picture. Their teacher stated that she helps her students learn these strategies at the beginning of the year. Describe ways to help students learn to use various problem-solving strategies.

7. Traditionally, students are asked to solve many computation problems or exercises in one lesson. How does this practice compare with the number of problems investigated in the various lessons in the video? What are the issues of time needed for problem solving? What are the issues of time needed for designing problem-solving experiences?

Extension
Examining Mathematics as Problem Solving

Each classroom excerpt profiled in this video is from a featured lesson in TEACHING MATH: A Video Library, K-4. You may want to watch the full version of these lessons to further examine and explore the role of problem solving in learning mathematics. The following list provides information about each full video and the page number of the accompanying print unit. Videos are listed in the order in which their excerpts appear in the Problem Solving video.