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- Looking Ahead
- Part A: Beliefs About Best Practices for Teaching and Learning Mathematics and Science
- Part B: Beyond Beliefs: The Guiding Principles
- Structure for the Discussion of Guiding Principles
- Guiding Principles
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Throughout history, scholars of science, mathematics and technology have contributed innovative solutions to diverse problems. Students who learn about problem-solving by studying historical problems and their solutions, as well as by engaging in their own interesting problems, can become powerful and effective problem-solvers.
Problem-solving is the process by which students experience the power and usefulness of mathematics, scientific inquiry, and technological design in the world around them. Problem-solving provides a method of inquiry and application that is a consistent context for learning mathematics and science. Learning to solve problems in a variety of contexts, with time for reflection on the experience, results in the development of problem-solving abilities that can be applied in new contexts.
An exemplary curriculum gives students opportunities to solve problems individually and collaboratively, use a variety of tools including calculators and computers, address relevant and interesting mathematical and scientific ideas, understand mathematical and scientific concepts, and experience the power and practicality of the disciplines. Students should have the opportunity to work on practical problems, as well as problems they have generated themselves, to see the connection of mathematics and science to their own lives.
When problem-solving becomes an integral part of classroom instruction and student success, students gain confidence in doing mathematics and applying scientific concepts. They also develop persevering and inquiring minds, grow in their ability to communicate and become more adept in higher-level thinking processes.
Problem-solving involves a cyclical process and can occur in a variety of situations, ranging from conducting surveys and experiments to technological design. Technological design utilizes tools, techniques, and an applied understanding of science to design actual products or solutions to problems. All of these processes should occur across all grade levels, differing only in their levels of sophistication. Students should be doing science and mathematics within their developmental capabilities. They need the experience of sustaining concentration and commitment to problem-solving over varying lengths of time.
Much mathematical and scientific understanding has been developed in response to questions or problems. Thus student exploration of problems in science and mathematics is a way of learning the important ideas in each discipline. For students to become good problem- solvers, they need opportunities and guidance in asking their own questions and in identifying and clarifying problems based on those questions. They should generate criteria for an acceptable solution, suggest possible strategies, and then try one out, making adjustments or starting over with a newly proposed question or strategy. Throughout this process, they should be using a variety of tools and, in some instances, using technology to design and create tools of their own.
Practice-problems that best prepare students to wrestle with ordinary dilemmas of daily life include multidimensional problems with numbers that don't compute as whole numbers, problems with too little or too much information, or problems with multiple solutions with different implications. Technological design problems encourage students to recognize resources and limiting factors. Solutions often create new problems for which new knowledge must be applied and new solutions designed. Problem-solving experience allows students to construct new knowledge and to challenge firmly-held beliefs about the world around them.____________________________________________
Snapshots First-graders are investigating properties and patterns of numbers, using a number of cubes. With numeral cards in front of them, they move cubes to create shapes showing the numbers one to ten. The teacher asks them to share their visuals and make observations first with a partner, then with the whole class. Ways of showing each of the numbers are then displayed so that the class can see the variety of shapes.
With the displays accessible to all, students work with partners to make observations about patterns they see in the shapes. They recreate these patterns with square stickers on paper and share their findings with the class. One group sees that every other number can have pairs of blocks - two has one pair, four has two pairs, and so on. Other children make observations about every other number (even-odds) as well as numbers that can be made into steps (triangles) and squares. A journal entry assignment asks children to use pictures and words to tell what they have noticed about the numbers 1 through 10.
~~~~~ Sixth-grade science lab partners are trying to determine the characteristics of a seesaw and the effects of the change on the fulcrum to the weight needed to balance the seesaw. They have laid a meter stick across a paper towel roll to create a simple model of a seesaw and have taped paper cups to each end of the meter stick. With the fulcrum at the center, the students put various marbles in the cup on one end, adding marbles to the cup at the other end until the lever (seesaw) balances. All attempts with various number of marbles are recorded in a table.
Students make predictions about the placement of the fulcrum so that a certain weight on one side will take double the weight on the other in order to balance. They test out their prediction and then try the same for three and four times the weight working towards a mathematical generalization. Then they share and discuss the findings.
~~~~~ Tenth-grade students work in small groups to construct a pendulum, investigate how it functions and formulate questions that arise. Students must 1) determine all the factors that do and do not affect the period of a pendulum, 2) quantify in general terms the relationship between these factors, and 3) work to find a mathematical model that might explain the relationship of any of these factors. Students work in small groups to construct a pendulum, investigate how it functions, and formulate questions such as: How does the length of a string affect the period? How does the mass of the object affect the period? How does the height from which the pendulum begins its swing affects its period? Each student completes an investigation report using words and visuals (graphs, tables, pictures), then discusses findings with the group and ultimately the class.
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A. Students demonstrate proficiency using a variety of problem-solving strategies.
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Primary
1. Identify and clarify problems by observing, posing questions, communicating prior knowledge and formulating a problem to be solved.
2. Use results in a purposeful way, which includes making predictions based on patterns they have observed. (S-J3)
3. Identify products which were invented to solve a problem. (S-J4)
Intermediate
1. Use results in a purposeful way, which includes making predictions based on observed patterns and interpret data to make further predictions. (S-J3)
2. Demonstrate and explain the problem solving process using appropriate tools and technology and defend the reasonableness of results. (M-B3)
3. Design and build an invention. (S-J4)
Middle
1. Verify and evaluate scientific investigations and use the results in a purposeful way. (S-J3)
2. Design, construct, and test a device (invention) that solves a special problem. (S-J6)
Secondary
1. Verify, evaluate, and use results in a purposeful way. This includes analyzing and interpreting data, making predictions based on observed patterns, testing solutions against the original problem conditions, and formulating additional questions. (S-J2)
2. Demonstrate the ability to use scientific inquiry and the technological method with short term and long term investigations, recognizing that there is more than one way to solve a problem. Demonstrate knowledge of when to try different strategies. (S-J3)
3. Design and construct a device to perform a specific function, then redesign for improvement (e.g., performance, cost). (S-J4)