Ways that Problem Solving Promote Interest in Learning

Solution Preview

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Problem solving promotes eagerness and interest in learning mathematics because you 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)

Problem solving promotes eagerness and interest in learning mathematics because it is a constructivist approach and while solving a problem students can create their own understanding about learning mathematics as they are challenged by the problem solving process.

Problem solving promotes eagerness and interest in learning mathematics because it is a positive challenge, or a "game" to try to beat. This motivates a student to attempt to overcome the hurdle in a creative way.

Problem solving promotes eagerness and interest in learning mathematics because as you can see from the examples below, it is also authentic and students can truly see the purpose for learning the concepts they are being taught while they are being positively challenged. (for example, determining how much stain to purchase for their pencil boxes)

Problem solving promotes eagerness and interest in learning mathematics because you can actually teach strategies to help students solve problems (tricks to help them solve problems) while they are creating their own understanding.

Problem solving promotes eagerness and interest in learning mathematics because students verify and interpret their own results.

Problem solving promotes eagerness and interest in learning mathematics because the entire process builds confidence.
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Problem Solving

http://www.learner.org/channel/schedule/printmat.phtml?printmat_id=58

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

You could go to this website to get more information and to see how to view the video.
Teaching Math: A Video Library, K-4
http://www.learner.org/channel/schedule/printmat.phtml?printmat_id=58
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Here is a research paper:

WHAT RESEARCH TELLS US ABOUT TEACHING MATHEMATICS THROUGH PROBLEM SOLVING

http://tlsilveus.com/Portfolio/Documents/EDCI327_ProblemSolving.pdf

To Appear: In F. Lester (Ed.), Research and issues in teaching mathematics through problem
solving. Reston, VA: National Council of Teachers of Mathematics.

The author thanks Vic Cifarelli, Jim Hiebert, Frank Lester, and Judi Zawojewski for helpful
discussions in the process of preparing this chapter. Preparation of this chapter is partially
supported by a grant from the National Science Foundation (ESI-0114768). However, any
opinions expressed herein are those of the author and do not necessarily represent the views of
NSF.
2

WHAT RESEARCH TELLS US ABOUT TEACHING MATHEMATICS THROUGH PROBLEM SOLVING

The teaching of problem solving has a long history in school mathematics (D'Ambrosio &
Lester, this volume; Stanic & Kilpatrick, 1988). In the past several decades, there have been
significant advances in the understanding of the complex processes involved in problem solving
(Lester, 1994; Schoenfeld, 1992; Silver, 1985). There also has been considerable discussion about
teaching mathematics with a focus on problem solving (e.g., Hembree & Marsh, 1993;
Henningson & Stein, 1997; Hiebert et al., 1997; Kroll & Miller, 1993; Stein, Smith, & Silver,
1999). However, teaching mathematics through problem solving is a relatively new idea in the
history of problem solving in the mathematics curriculum (Lester, 1994). In fact, because teaching
mathematics through problem solving is a rather new conception, it has not been the subject of
much research. Although less is known about the actual mechanisms students use to learn and
make sense of mathematics through problem solving, there is widespread agreement that teaching
through problem solving holds the promise of fostering student learning (Schroeder & Lester,
1989). Many of the ideas typically associated with this approach (e.g., changing the teacher's
roles, designing and selecting problems for instruction, collaborative learning, problematizing the
curriculum) have been studied extensively, and research-based answers to various frequently
asked questions about problem-solving instruction are now available.

Issues and Concerns Related to Teaching Through Problem Solving
This chapter discusses four issues and concerns related to teaching through problem solving.
These four issues are related to four commonly asked questions: (1) Are young children really
able to explore problems on their own and arrive at sensible solutions? (2) How can teachers learn
to teach through problem solving? (3) What are students' beliefs about teaching through problem
solving? (4) Will students sacrifice basic skills if they are taught mathematics through problem
solving? In the discussion of each issue, available research evidence that addresses the issue is
first reviewed, and then research needed to address the issue more completely is suggested.
Issue 1: Are Young Children Really Able to Explore Problems on Their Own and Arrive at
Sensible Solutions?

Teaching through problem solving starts with a problem. Students learn and understand
important aspects of the concept or idea by exploring the problem situation. The problems used
tend to be more open-ended and allow for multiple correct answers and multiple solution
approaches. In teaching through problem solving, problems not only form the organizational
focus and stimulus for students' learning, but they also serve as a vehicle for mathematical
exploration. Students play a very active role in their learning-exploring problem situations with
teacher guidance and "inventing" their own solution strategies. In fact, the students' own
exploration of the problem is an essential component in teaching through problem solving. For
example, in curriculum projects designed to help students in primary grades learn and understand
number concepts and operations with understanding, the learning of number concepts and
operations is perceived as a "conceptual problem-solving activity" in which teachers support
students' efforts to work out their own procedures and rules related to addition and subtraction
(Fuson et al., 1997). However, a fundamental question arises: Are students really capable of
exploring problem situations and inventing strategies to solve the problems?
Many researchers (e.g., Carpenter, Franke, Jacobs, Fennema, &Empson, 1998; Kamii, 1989;
Maher & Martino, 1996; Resnick, 1989) have investigated students' mathematical thinking and
indicated that young children can explore problem situations and "invent" ways to solve the
problems. For example, traditionally, to find the sum 38 + 26, students are expected to add the
ones (8 + 6 = 14), and write down 4 for the unit place of the sum and carry over 1 to the ten's
place. Carpenter et al. (1998) found that many first-, second-, and third-grade students were able to
use the following invented strategies to solve the problem: (1) "Thirty and twenty is fifty and the
eight makes fifty eight. Then six more is sixty-four"; (2) "Thirty and twenty is fifty, and eight and
six is fourteen. The ten from the fourteen makes sixty, so it is sixty four"; (3) "Thirty-eight plus
twenty-six is like forty and twenty-four, which is sixty-four." In their study, Carpenter et al.
(1998) found that 65% of the students in their sample had used an invented strategy before
standard algorithms were taught. By the end of their study, 88% of their sample had used invented
strategies at some point during their first three years of school. They also found that students who
used invented strategies before they learned standard algorithms demonstrated better knowledge of
base-ten number concepts and were more successful in extending their knowledge to new
situations than were students who initially learned standard algorithms.

Recently, some researchers (e.g., Ben-Chaim et al., 1998; Cai, 2000) have also found
evidence that middle school students are able to use invented strategies to solve problems. For
example, when U.S. and Chinese sixth-grade students were asked to determine if each girl or each
boy gets more pizza when seven girls share two pizzas and three boys share one pizza equally,
they used eight different correct ways to justify that each boy gets more than each girl (Cai, 2000).
Collectively, the aforementioned studies not only demonstrate that students are capable of
inventing their own strategies to solve problems, but they also show that it is possible to use the
students' invented strategies to enhance their understanding of mathematics. Thus, it seems clear
that students in elementary and middle schools are capable of inventing their own strategies to
solve problems. However, there are at least two unanswered questions.
Unanswered questions related to issue 1. The first question has to do with students' invented
strategies. In classrooms using problem-based inquiry (e.g., Carpenter et al., 1998; Cobb et al.,
1991), students are given opportunities to use and discuss alternative strategies for solving
problems before being taught any specific strategies. The question is: How do students learn to use
invented strategies in the first place before any instruction takes place? What kinds of experiences
and knowledge do students draw upon to create sensible strategies? Kamii (1989) has argued that
"the procedures children invent are rooted in the depth of their intuition and their natural ways of
thinking" (p. 14). Clearly, we need to learn much more about what students' "natural" ways of
thinking in mathematics are. We also need to determine if these natural ways are content or gradelevel
dependent.

The second unanswered question has to do with the efficiency of students' invented strategies:
When students develop inefficient strategies, how can they be helped to develop more efficient
strategies? Previous research has shown that students are capable of inventing problem-solving
strategies or mathematical procedures, but the research also has shown that invented strategies are
not necessarily efficient strategies (Cai, Moyer, & Grochowski, 1999; Carpenter et al., 1998;
Resnick, 1989). For example, in a study by Cai et al. (1999), a group of middle school students
was asked to solve the following problem involving arithmetic average. One student came up
with an unusual strategy to solve it. In this solution, the student viewed throwing away the top
and bottom scores as taking 15 away from each of the other scores. By inventing this ...