THE FIRST FOUR STANDARDS
STANDARD 1 - PROBLEM SOLVING
Problem posing and problem solving involve examining situations that arise in mathematics and other disciplines and in common experiences, describing these situations mathematically, formulating appropriate mathematical questions, and using a variety of strategies to find solutions. By developing their problem-solving skills, students will come to realize the potential usefulness of mathematics in their lives.
Meaning and Importance
Problem solving is a term that often means different things to different people. Sometimes it even means different things at different times for the same people! It may mean solving simple word problems that appear in standard textbooks, applying mathematics to real-world situations, solving nonroutine problems or puzzles, or creating and testing mathematical conjectures that may lead to the study of new concepts. In every case, however, problem solving involves an individual confronting a situation which she has no guaranteed way to resolve. Some tasks are problems for everyone (like finding the volume of a puddle), some are problems for virtually no one (like counting how many eggs are in a dozen), and some are problems for some people but not for others (like finding out how many balloons 4 children have if each has 3 balloons, or finding the area of a circle).
Problem solving involves far more than solving the word problems included in the students' textbooks; it is an approach to learning and doing mathematics that emphasizes questioning and figuring things out. The Curriculum and Evaluation Standards of the National Council of Teachers of Mathematics considers problem solving as the central focus of the mathematics curriculum.
"As such, it is a primary goal of all mathematics instruction and an integral part of all mathematics activity. Problem solving is not a distinct topic but a process that should permeate the entire program and provide the context in which concepts and skills can be learned." (p. 23)
Thus, problem solving involves all students a large part of the time; it is not an incidental topic stuck on at the end of the lesson or chapter, nor is it just for those who are interested in or have already mastered the day's lesson.
Students should have opportunities to pose as well as to solve problems; not all problems considered should be taken from the text or created by the teacher. However, the situations explored must be interesting,engaging, and intellectually stimulating. Worthwhile mathematical tasks are not only interesting to the students, they also develop the students' mathematical understandings and skills, stimulate them to make connections and develop a coherent framework for mathematical ideas, promote communication about mathematics, represent mathematics as an ongoing human activity, draw on their diverse background experiences and inclinations, and promote the development of all students' dispositions to do mathematics (Professional Standards of the National Council of Teachers of Mathematics). As a result of such activities, students come to understand mathematics and use it effectively in a variety of situations.
K-12 Development and Emphases
Much of the work that has been done in connection with problem solving stems from George Polya's book, How to Solve It. Polya describes four types of activities necessary for problem solving: understanding the problem, making a plan, carrying out the plan, and looking back.
The first step in solving a problem is understanding the problem. Suppose that we want to solve the following problem:
A farmer had some pigs and chickens. One day he counted 20 heads and 56 legs. How many pigs and how many chickens did he have?
After reading the problem, we want to be sure we understand it. We might begin by noting that we probably have to use the number of heads and the number of legs in some way. We know that pigs have four legs and chickens have two. We see that there must be 20 animals in all. We might observe that, if the farmer had only chickens, there would be 40 legs. If, on the other hand, he had only pigs, there would be 80 legs.
Some techniques that may help students with this important aspect of problem solving - understanding the problem - include restating the problem in their own words, drawing a picture, or acting out the problem situation. Some teachers have students work in pairs on problems, with one student reading the problem and then, without referring to the written text, explaining what the problem is about to their partner.
A second type of activity relating to problem solving involves making a plan. For our pigs and chickens problem, the plan might be to make a chart that shows various combinations of 20 chickens and pigs and how many legs they have altogether. If we have too many legs, we need fewer pigs, and if we have too few legs, we need more pigs.
In order to be successful problem solvers, students need to become familiar with a variety of strategies that are used in making a plan for solving problems. Some of the strategies that are especially useful are making a list, making a chart or a table, drawing a diagram, making a model, simplifying the problem, looking for a pattern, using manipulatives, working backwards, eliminating possibilities, using a formula or equation, acting out the problem, using logic, using guess and check, using a spreadsheet, using a computer sketching program like Geometer's Sketchpad, The Geometry SuperSupposer, or Cabri, writing a computer program, or using a graphing calculator.
Let's carry out our plan for using a chart to solve the pigs and chickens problem. If we have 10 pigs (that's 40 legs) and 10 chickens (that's 20 legs), then we have 60 legs - that's too many legs. Let's try 9 pigs and 11 chickens - still too many. How about 8 pigs and 12 chickens? That's just right.
Carrying out the plan is sometimes the easiest part of solving a problem. However, many students jump to this step too soon. Others carry out inappropriate plans, or give up too soon and stop halfway through solving the problem. To reinforce the process of making a plan and carrying it out, teachers might use the following technique: Divide a sheet of notebook paper into two columns. On the left side of the page, the student solves the problem. On the right side of the page, the student writes about what is going on in his/her mind concerning the problem. Is the problem hard? How can you get started? What strategy might work? How did you feel about the problem?
Let's look back at the problem we have just finished. The pigs and chickens problem may remind some of you of other problems you have solved; it's a little bit like some of the algebra problems involving the value of coins. Others may be intrigued by the pattern that we seem to have started in the last column of our chart and seek an explanation for this pattern. Still others may have solved this problem a completely different way; we could discuss all of the different strategies the students used and decide which ones seem most effective. One strategy used by young children is to draw a picture. Twenty circles represent the animals' heads. Each animal gets 2 legs. Additional pairs of legs are drawn on animals, starting at the left, until there are 56 legs.
This looking back activity is where students reflect upon the problem. Does the answer make sense? Is the question answered completely? How is the problem like others you have seen? How is it different?
While it might seem most logical to begin problem solving with Polya's first activity and proceed through each activity until the end, not all successful problem solvers do so. Many successful problem solvers begin by understanding the problem and making a plan. But then as they start carrying out their plan, they may find that they have not completely understood the problem, in which case they go back to step one. Or they may find that their original plan is extremely difficult to pursue, so they go back to step two and select another approach. By using these four activities as a general guide, however, students can become more adept at monitoring their own thinking. This "thinking about their thinking" can help them to improve their problem solving skills.
Students move through a continuum of stages in their development as problem solvers (Kantowski, 1980). Initially, they have little or no understanding of what problem solving is, of what a strategy is, or of themathematical structure of a problem. Such students usually do not know where to begin to solve a problem; the teacher must model the problem solving process for these students. At the second level, students are able to follow someone else's solution and may suggest strategies for similar problems. They may participate actively in group problem solving situations but feel insecure about independent activities, requiring the teacher's continued support. At the third level, students begin to be comfortable with solving problems, suggesting strategies different from those they have seen used before. They understand and appreciate that problems may have multiple solutions or perhaps even no solution at all. Finally, at the last level, students are not only adept at solving problems, they are also interested in finding elegant and efficient solutions and in exploring alternate solutions to the same problem. In teaching problem solving, it is important to address the needs of students at each of these levels within the classroom.
In summary, the real test of whether a student knows mathematics is whether she can use it in a problem situation. Students should experience problems as introductions to learning about new topics, as applications of content already studied, as puzzles or non-routine problems that have many solutions, and as situations that have no one best answer. They should not only solve problems but also pose them. They should focus on understanding a problem, making a plan for solving it, carrying out their plan, and then looking back at what they have done.
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