THE DEVELOPMENT OF PROBLEM SOLVING ABILITIES AND STRATEGIES TO FACILITATE THEIR ATTAINMENT

RESEARCH ON PROBLEM SOLVING

Problem solving has a special importance in the study of mathematics. A primary goal of mathematics teaching and learning is to develop the ability to solve a wide variety of complex mathematics problems. During mathematics problem solving, one goal is to find a solution for a given problem. Other goals may be to generate new problems, generate alternative solutions, interpret a result, or generalize. The Curriculum and Evaluation Standards for School Mathematics published by the National Council of Teachers of Mathematics, (1989), advocates that a primary goal for students is "that they become mathematical problem solvers" (p5). Stanic and Kilpatrick (1988) note that sets of word problems have long been a part of the mathematics curriculum, citing examples from a text dated about 1650 B.C.

To many mathematically literate people, mathematics is synonymous with solving problems---doing word problems, creating patterns, interpreting figures, developing geometric constructions, proving theorems, and so forth. On the other hand, persons not enthralled with mathematics may describe any mathematics activity as problem solving.

W. R. Reitman (1965), defined a problem as a situation in which you have been given the description of something but do not yet have anything that satisfies that description. Reitman’s discussion described a problem solver as a person perceiving and accepting a goal without an immediate means of reaching that goal.

Henderson, and Pingry (1953), wrote that to be problem solving there must be a goal, a blocking of that goal for the individual, and an acceptance of that goal by the individual. What is a problem for one student may not be a problem for another--either because there is no blocking or no acceptance of the goal. Alan Schoenfeld (1983) also pointed out that defining what is a problem has always been relative to the individual involved.

The basis for most mathematics problem-solving research for secondary school students since 1960 can be found in the writings of G. Polya (1962, 1965, 1973), in the field of cognitive psychology, and, most specifically, cognitive science. Cognitive psychologists and cognitive scientists seek to develop or validate theories of human learning (Frederiksen,1984), whereas mathematics educators attempt to understand how their students interact with mathematics (Schoenfeld, 1983, 1985).

The area of cognitive science has particularly relied on computer simulations of problem solving (Newell & Simon, 1972). If a computer program generates a sequence of behaviors similar to the sequence for human subjects, then that program is a model or theory of the behavior. Newell and Simon (1972), and Larkin (1980) have provided simulations of mathematical problem solving. These simulations may be used to better understand mathematics problem solving.

THE ROLE OF A FRAMEWORK

To discuss and investigate the processes involved in problem solving, researchers find it useful to develop frameworks. Most formulations of a problem-solving framework attribute some relationship to Polya’s problem-solving stages: understanding the problem, making a plan, carrying out the plan, and looking back. As discussed by Wilson, Fernandez, and Hadaway (1993), United States textbooks often present these "stages" as "steps" in linear problem-solving frameworks (see fig.1). However, the linear nature of the frameworks do not promote the spirit of Polya’s stages and his goal of teaching students to think. They all have the following defects:

1. They depict problem solving as a linear process.
2. They present problem solving as a series of steps.
3. They imply that solving mathematics problems is a procedure to be memorized, practiced, and habituated.
4. They lead to an emphasis on answer getting.

Figure 1.

READ

KNOW

PLAN

SOLVE

CHECK

Wilson and his colleagues have presented a framework that emphasizes the dynamic and cyclic nature of genuine problem solving as an interpretation of Polya’s stages. It has been used in a mathematics problem-solving course at the University of Georgia to aid the discussion of issues involved with teaching mathematics problem solving in elementary and secondary schools (see fig.2).

Figure 2

Understanding the

problem

Problem

posing

Looking Making a

back plan

Carrying out

the plan

EVALUATION OF PROBLEM SOLVING

With the increased emphasis of problem solving in mathematics classrooms, the need for evaluation of progress and instruction in problem solving becomes more pressing. Some research dealing with the evaluation of problem solving involves diagnosing students’ cognitive processes by evaluating the amount and type of help needed by an individual during a problem-solving activity. Campione, Brown, and Connell (1988) term this method of evaluation as dynamic assessment. Students are given mathematics problems to solve. The assessor then begins to provide as little help as necessary to the students throughout their problem-solving activity. The amount and type of help needed can provide good insight into the students’ problem-solving abilities as well as their ability to learn and apply new principles.

W. R. Reitman (1965), reported the use of hints to help diagnose student difficulties in problem solving in high school algebra and plane geometry. Problems were developed such that the methods of solutions were not readily apparent to the students. A sequence of hints was then developed for each item. According to Reitman, "the power of the hint technique seems to lie in its ability to identify those particular students in need of special kinds of help" (p. 371)

PROBLEM SOLVING INVOLVING COOPERATIVE GROUPS

1. Campione and his colleagues (1988) also discussed a method to help monitor and evaluate the progress of a small cooperative group during a problem-solving session. A learning leader (sometimes the teacher, sometimes a student) guides the group in solving the problem through the use of three boards:

1. A Planning Board, where important information and ideas about the problem are recorded,
2. A Representation Board, where diagrams illustrating the problem are drawn,
3. A Doing Board, where appropriate equations are developed and the problem is solved.

Through the use of this method, the students are able to discuss and reflect on their approaches by visually tracing their joint work. Campione and his colleagues indicate that increased student engagement and enthusiasm in problem solving, as well as increased performance, resulted from the use of this method for solving problems.

When students solve problems in small cooperative groups, different students may propose approaches to the problem. These suggestions must then be discussed by the group to make a decision on what action to take. Schoenfeld (1985) indicates that this interaction is "precisely the kind of discussion that the students should be having internally" ( p.375). He has advocated that when students work in small groups or pairs to solve problems, teachers act as a coach, asking questions and providing hints as they deem necessary.

To help students work in groups effectively, advocates of cooperative learning suggest that a task be assigned to each member of a group. Lochhead (1985) offers a structure for pair problem solving in which one student solves the problem and the other student listens and monitors the solver’s actions and asks questions to understand the solver’s thinking.

Artzt and Armour-Thomas (1990) found that when students work in groups, students who always take the role of monitors have difficulty executing their own ideas when they are faced with the task of solving a mathematics problem; thus teachers must insure that students assume different roles during different problem-solving sessions and that each student assumes the role of problem solver during some of the sessions.

THE USE OF COMPUTERS AND GRAPHING CALCULATORSIN SECONDARY MATHEMATICS

COMPUTER PROGRAMMING

One objective for student programming is based on the presumed general cognitive skills the student learns as a result of appropriate kinds of programming activity, for example, planning skills, problem breakdown skills, and so on. Another objective of programming activity is the more domain-specific learning that takes place, as one either attempts to solve, say certain mathematics problems, or uses specifically tailored commands to operate on a specific kind of mathematical object.

Numerous studies have sought to determine the effects of computer-programming instruction on various aspects of mathematics achievement and attitudes. More recent studies have targeted the effects of learning computer programming on particular problem-solving abilities. Liao and Bright (1980) reported preliminary results from a meta-analysis of studies that assessed the relationship between computer-programming and cognitive skills related to problem-solving abilities. They concluded that computer programming had a slightly positive effect on student problem-solving performance.

G. W. Blume (1984) also reviewed the effects of computer-programming on mathematical problem solving. He concluded that the extent to which programming-augmented mathematics instruction influenced student achievement on particular mathematical topics or improved problem-solving performance seemed to depend on the nature of the programming activities. He suggested that programming exercises that required coding routine algorithmic procedures may have little effect on mathematics achievement, and that programming activities that clearly involved problem solving were more likely to have a positive effect on problem-solving performance.

When students learn computer programming as a part of their mathematics instruction, it is common for much time to be spent on computer-specific and language-specific questions rather than on mathematical concerns. Nevertheless, mathematics-related benefits may include increasing student experience with variables, algorithmic thinking, and recursion. The question that arises is whether teaching students to program is worth the time and energy cost within the context of teaching mathematics.

THE COMPUTER AS A TUTOR

Computer tutors have traditionally been used for teaching students to perform paper-and-pencil algorithms, and much of the research on use of the computer as a tutor focuses on these goals. In an analysis of Computer-Assisted Instruction (CAI) used to supplement traditional classroom instruction, Burns and Bozeman (1981) found that students achieved better with the CAI supplement than with a curriculum that used only traditional instructional methods. The researchers made no attempt to categorize or describe the nature of the achievement measures, leaving the impression that the results applied to traditional paper-and-pencil routines.

According to Longstreet and Smith-Gratto (1995), CAI can certainly develop more varied, positive reinforcement of the classroom instruction than is available in the textbook format. However, overuse of the same kinds of reinforcers and inadequate scheduling can be counterproductive. The reinforcement must be continual for new behaviors and then intermittent for the maintenance of learned behaviors, the key to successful CAI being its ability to attract and maintain the students’ interests.

"Intelligent tutors" like Anderson’s Geometry Tutor (1986), BBN’s Algebra Workbench (1988), and McArthur’s Algebra Tutor (1988) monitor students’ problem-solving efforts and diagnose and provide feedback on errors. McArthur’s tutor capitalizes on multiple representations and has an alternate "boxes and weights" representation for solving equations. In addition, students can command the program to complete the symbolic manipulation. Unfortunately, intelligent tutors, have been focused almost exclusively on the development of routine manipulation skills.

THE COMPUTER AS A TOOL

Computers as function and relation graphers seem to offer opportunities for significant changes in the content and processes of school mathematics. With the capability for computer generation of graphs, and automatic symbolic manipulation, students have earlier access to the graphical representation of a broader range of functions. First-year algebra students using a computer-based curriculum for example, study some polynomial, rational, and exponential functions in their introductory algebra course, analyzing these functions for such properties as rates of change, extrema, and asymptotes (Fey, 1991).

When students work with computer graphers for a long period of time, they become better at understanding the conceptions of mathematical objects (Schoenfeld, 1988). However, students are not naturally able to interpret computer-generated graphs. Goldenberg (1988) relates misconceptions exhibited by two "bright, successful" second-year algebra students who were using a function grapher to determine the polynomial rule to accompany a given computer-generated graph. For example, the students started out with a reasonable estimate for the function rule but quickly lost sight of the meaning of the y-intercept. But when students work with a computer-intensive-curriculum over a long period of time, these misconceptions do not seem to dominate students’ work with graphs (Heid, Sheets, Matas, & Menasian, 1988). Hence their initial misconceptions need not be either intractable or long-standing.

Frequently, students will use more than one piece of tool software in a mathematics lesson or course. Sometimes the set of software together forms a "toolkit", allowing the user to choose from among several representations and strategies (e.g., tables, graphs, and function rules) when solving a given problem. One example of this type of curriculum is Computer-Intensive Algebra (Fey, 1991), a beginning algebra curriculum that uses a mathematical toolkit for the generation of numerical, graphical, and symbolic representations of functions, and that shifts attention from manual skills to mathematical modeling concepts. Traditional algebra topics like by-hand production of equivalent expressions, of solution of equations, and of factoring are omitted.

In one study of this curriculum, Matras (1988) found that students in two versions of the experimental curriculum exhibited better problem solving in three out of four comparisons concerning the ability to identify the underlying structure of a problem and the ability to solve problems. In other analyses of this curriculum, Heid and colleagues (1998) suggested that students who studied the experimental curriculum for the first eight months of the school year (and studied traditional algebra topics for the last month) performed much better than students in traditional classes on a range of mathematical modeling tasks, with little loss in traditional skills.

THE GRAPHING CALCULATOR USAGE

The graphing calculator is a wonderful tool that can be used to generate visual representations to help develop a deeper understanding of mathematical relationships. Users can produce a graph on the calculator screens by entering the axes scales and the function rule, and may easily rescale displayed graphs. An important feature of graphing calculators is the "zooming in" facility, a technique for viewing smaller and smaller parts of the graph by using finer and finer axes scales.

Very little research has been completed on the effectiveness of graphing calculators. One study (Dick and Shaughnessy, 1988) analyzed the effects of providing classroom sets of symbolic-manipulation calculators for volunteer high school mathematics teachers. The utility and power of the calculator were more influential in changing the attitudes of male students, while increasing familiarity and comfort with calculator use were more influential in changing the attitudes of female students. Teachers were observed using the calculators more for graphing than for symbolic manipulation. Nevertheless, according to the researchers, "the teachers felt that the use of the calculators brought only minor changes in the dynamics of classroom interaction" ( p. 333)

Another study by Fareell (1989) pointed out a dramatic shift in teacher and student roles when graphing calculators were used in pre-calculus instruction. One important difference was the use of a specially designed calculator-based curriculum in the second study but not in the first. In a study focusing on the mathematical performance of upper secondary students who had regular and prolonged access to graphing calculators, K. Ruthven (1990) observed that the graphing-calculator group outperformed students who did not have such access on tasks that required symbolization, but failed to better them on tasks requiring interpretation of function graphs.

Many of the new textbooks today are integrating the use of the graphing calculators with all topics included. Dugopolski (1995), of Southeastern Louisiana University, integrates the use of the graphing calculator throughout an algebra course. This curriculum typifies a growing number of graphing-calculator use that allows for the exploration of realistic applications with multiple representations of solutions.

In traditional mathematics courses, teachers train students to "find the correct solution." Changes in the mathematics curriculum are encouraging teachers to go further. The ease with which students can learn to use graphing calculators make the calculator an ideal tool for mathematical investigations in the classroom. Hence the graphing calculator may well be a promising and cost-effective tool that will further the reform in both the content and pedagogy of school mathematics.

THE INTERACTION OF PROBLEM SOLVING ABILITIES AND THE USE OF COMPUTERS OR GRAPHING CALCULATORS IN SECONDARY MATHEMATICS

For many people the appropriate use of technology has significant identity with mathematics problem solving. This view emphasizes the use of technology as a tool for mathematics problem solving rather than the use of technologies to deliver instruction or to generate student feedback.

Problem-solving research involving the computer has often dealt with programming as a major focus--but has often provided inconclusive results. Indeed, the development of a computer program to perform a mathematical task can be a challenging mathematical problem that can enhance the programmer’s understanding of the mathematics being used. Too often, however, the focus is on programming skills rather than on using programming to solve mathematics problems. Most assuredly there is a place for programming within mathematics study, but the focus ought to be on the mathematics problems and the use of the computer as a tool for mathematics problem solving.

Technology can be used both to make possible and to enhance exploration of conceptual or problem situations. For example, a function-grapher computer program or a graphics calculator can allow students to explore families of curves such as the quadratic function f(x) = ax²+bx+c for different values of a, b, and c. Similarly, a function-grapher computer program or graphic calculator can accomplish the integration of algebraic and graphical representations of equations and thus allow students to make mathematical connections that will enhance their understanding of both.

SUMMARY OF FINDINGS

The primary goal of most students in mathematics classes is to see an algorithm that will give them the answer quickly. Students struggle with, and at times against, the idea that a mathematics class can and should involve exploration, conjecturing, and thinking. "Thus a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking." Polya (1965, p.5)

The NCTM’s Curriculum and Evaluation Standards for School Mathematics (1989) recommends that students should be exposed to a variety of instructional methods to cultivate their many potentials and abilities, the goal being to enhance their mathematics problem-solving abilities. The instructional methods identified in the research review of this paper agree with these recommendations.

The future for computers and graphing calculators in high school mathematics is promising but we cannot completely or accurately predict or describe their eventual impact. We must not forget that it was only in the late 1980’s that computers became resourceful in the typical school. Thus even though the technology itself is rapidly evolving, the reliable practical experience of computers and graphing calculators in schools is limited. As access to this ever changing new technology increases, implementation decisions by schools will follow. Hopefully, these decisions will be educational decisions guided by educational objectives.

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