Chapter II



    There is a great deal of research on the use of computers in education. This review is concentrated upon the portion of that research which deals with (a) problem solving, (b) the use of computers in mathematics, and (c) the use of computers to provide concrete experience as an aid to problem solving.

    Many teachers, mathematics educators and researchers (Ediger, 1989; Hamm & Adams, 1988; Hatfield & Kieren, 1972; Kinnaman, 1993; Kulik & Kulik, 1987; Roblyer, Castine & King, 1993) have asserted that computers have undeniable value and an important instructional role in mathematics classrooms. others argue that there is no overwhelming evidence of the worth of computers as a learning tool. Cannings and Finkel (1993) explained:

There is some evidence; there are research results; there are anecdotal remarks by myriad teachers; there are hundreds of articles about thousands of studies; there are stories of improved attitude, improved attendance, and "saving" of large numbers of students at risk. But for every positive item you read, you can find a contrarian who has found the opposite effect. (pp. 51-52)
    For these reasons, research results must be viewed with caution; still, many reports provide progressive evidence that computers are useful educational tools.

Problem Solving

    The development of problem solving skills is an important objective in today's education. The National Council of Teachers of Mathematics (NCTM), in its Agenda for Action, recommended that "problem solving be the focus of school mathematics in the 1980's" (NCTM 1980, 1). Many attempts were made to teach problem solving skills effectively. A considered amount of research has been done in this area.

    Problem solving, viewed as a skill, is a difficult objective to achieve for students and for teachers to teach. Branca (1980) presents the idea that considering problem solving as a basic skill can help teachers and mathematics educators to organize the specifics of daily teaching of skills, concepts and problem solving.
Recent research demonstrates that to improve students' problem solving skills, specific concentration must be given to teaching problem solving strategies. According to LeBlanc, Proudfit, and Putt (1980), "Success in solving the problem does not depend on the application of specific mathematical concepts, formulas, or algorithms; rather, the solution requires the use of one or more strategies" (p. 31). Studies suggest that "students need to have explicit instructions in how to use strategies, to see examples of strategies being applied to solve problems, and to have opportunities to practice problem solving" (Truckson,
1982/3, p.7).

    Teachers, researchers, and mathematics educators have found that students' success at solving problems will
improve if they solve many problems and are encouraged to use general problem guidelines (Suydam, 1982). George
Polya, whose book, How to Solve It (1985), strongly influenced mathematics teachers and students of the 1990ís,
claimed that a better understanding of problem solving strategies [known also as heuristic] "could exert some good
influence on... the teaching of mathematics" (p. 118). Schoenfeld (1980) claims more: "Under appropriate circumstances, many students can learn to use heuristics, with the results being a demonstrable improvement in their problem-solving performance" (p. 9). For Polya (1980):

The first duty of a teacher of mathematics is to use this great opportunity: He should do everything in his power to develop his students' ability to solve problems .... if the teacher helps his students just enough and unobtrusively, leaving them some independence or at least some illusion of independence, they may experience the tension and enjoy the triumph of discovery. (p. 2)
    Research studies of problem solving have changed in the last three decades. The Kilpatrick (1969) report, a comprehensive review of problem solving research written just prior to the beginning of the 70s, indicates that much of the research and development in problem solving focused on the actual solution to the problems or the answers to the exercises. Kilpatrick (cited in Kantowski, 1981) noted that "Since the solution of a problem--a mathematics problem in particular--is typically a poor index of the processes used to arrive at the solution, problem solving processes must be studied by getting subjects to generate observable sequences of behavior" (p. 111). Research by mathematics educators has examined problem solving processes (Kantowski, 1977; Schoenfeld, 1985b; Smith, 1973). Skill in computational processes is necessary for solving problems (Knifong and Holton, 1976; Meyer, 1978); however, having these skills does not guarantee success in the process of problem solving.

    Another change since 1970 has been the development of the meaning of problem solving. At the beginning, problem solving was related to verbal or word problems. In these days, when referring to a problem, it means a "situation demanding resolution for which there is no immediate or apparent solution" (Wiebe, 1993, p. 13), and problem solving "is the process of searching for and finding solutions to problems" (Wiebe). Mathematical problem solving involves mental processes that could be classified as complex. A 1942 work by Brownell cited in Schoen and Oehmke (1980) said that a task must be complex if it is to be referred as a problem. In a more detailed form Schoen and Oehmke defined a problem as:
            A task is a problem to a person if--

    1. the task calls for a solution under certain specified conditions;
    2. the person understands the task but does not see an immediate strategy for solution;
    3. the person is motivated to search for the solution. (p. 216)

Nonroutine and Real Problems

    For this study, the definition of nonroutine and real problems is essential. Those given by Kantowsky (1981) are appropriate for the activities presented in this project:

    Nonroutine and real problems are important for many particular reasons. Kantowsky (1981) alerted that when students experience solving nonroutine problems, they can transfer methods of problem solving to new situations. These experiences can also help students understand the meaning of mathematical structure and acquire the ability to see mathematics in a given situation.

    Several recent studies provide information about the status of solving nonroutine problems. First, at the elementary, secondary, and postsecondary levels most students do not know how to approach such problems and do not appear to use strategies in their solution without specific instruction in techniques for solving nonroutine problems (Lester, 1975; Kantowski, 1974, 1980; Schoenfeld, 1979). Webb (1979) found that conceptual knowledge and heuristic strategy components, among other factors, interact in successful problem solving. For Kantowski (1981) this means that:

    It is not simply computational skill and the knowledge of how to apply algorithms that are important in problem solving; it is also important for a student to be able to plan effectively and to use other heuristics such as organizing data into tables and drawing effective diagrams. (p. 115)

    One of the best known studies in the area of Real Problems was that of the Unified Science and Mathematics for the Elementary School program (Shann, 1976). In this study, students work together in small groups to solve real problems. The students became involved in activities where computational skills were used in application situations so that the use of the skills became meaningful to them. As a result of the study students spent larger amounts of time in more active, self-directed, and creative behavior. Another finding was that these students had higher means on basic skill tests, and significant positive attitudes toward mathematics.

Heuristic Training as a Variable
 in Problem Solving Performance
    Gimmestad (1976) studied the processes used by sixty students of a community college in mathematical problem solving. The most popular processes were found to be deduction, trial and error, and equations. The number of processes used by community college students could be increased by instruction in the use of successive approximation, checking, analogy, specialization, generalization, algorithm, reduction-combination, and working backwards. In a similar study with fourth grade students Sanders (1972) found that the most widely-used strategy was Logical Analysis [use of equations or algorithms]. The findings suggested that Blind Guessing should be discouraged as a means of solving arithmetic verbal problems and Trial and Error, unless it is systematic, also should be discouraged. Logical Analysis, which proved to be successful irrespective of the children's intelligence and other characteristics, was strongly recommended.  The high degree of accuracy obtained by the
strategy Creative or Divergent Thinking would show that teachers should encourage children to find original ways of
solving problems.

    Glage's 1980 study of background, attitude and problem solving characteristics of college students whose lowest ACT scores were in mathematics, indicated that poor problem solvers used trial and error frequently and equations infrequently. For the most part, the subjects could read and understand the problem, but were unable to use analytical and mathematical techniques to solve them. Looking Back problem solving behavior was used rarely by subjects in the study. There was some evidence that looking back behavior was related to the student's attitude toward word problems.

    In a study conducted by Gliner (1989), he found that college students were involved in activities to find out whether success on relating problems in terms of mathematical structure corresponds to success on solving problems. The study showed that the more successful problem solvers recognized underlying mathematical relationships among the problems and were immediately able to organize the problems based on structural criteria. Less successful problem solvers did not see these underlying structural characteristics and, instead organized the problems based on criteria such as question form, context, and common units of measurements.

    A study by Schoenfeld (1985a) is related directly to the present study. Schoenfeld (1982b) studied whether students who received explicit training in the use of particular strategies could use those strategies to solve posttest problems. He also wanted to know if explicit instruction in heuristics makes a difference. Schoenfeld found that problem solving practice is not enough, explicit training is required. The data showed that with explicit instruction about heuristic strategies, students can learn to use them. An interesting detail was that the participants chose the strategies used more often during instruction. For example, for the complex strategy Try to Establish a Subgoal it was not expected that the students would learn enough about it to use it to solve posttest problems; however as a result of working five practice problems, the students learned enough to use it.

Evaluation of Progress in Problem Solving
    An important part of the processes of teaching and learning is evaluation. Teachers are constantly evaluating in several informal or formal ways. With the increase of the attention given to problem solving, there is a need to develop new techniques for evaluating the effectiveness of instruction. However, the complex process of problem solving is more difficult to evaluate. Charles, Lester, and O'Daffer (1987), in the book How to Evaluate Progress in Problem Solving, describe several classroom evaluation techniques and illustrate how these techniques might be used in practice. The techniques mentioned in the book are: observing and Questioning [informal and structured interviews]; Self-Assessment Data [student reports, inventories]; Holistic Scoring [analytic scoring, focused holistic scoring, and general impression scoring]; Multiple-Choice and Completition Tests.
Schoen and Oehmke (1980) described the rationale, development, and potential of a testing approach used in the Iowa Problem Solving Project [IPSP was a three-year project directed by George Immerzeel of the University of Northern Iowa and funded under USEA-Tittle IV, C.]. Their goal was to produce an easily administered test that provides information about the problem solving process. The results indicated that it was possible to construct a psychometrically sound test based on the three steps from the problem solving model [understand the problem, apply the solution strategies chosen, and look back at the solution].

    Marshall (1988) presented an approach that makes a shift from assessment procedures based upon statistical or psychometric model to procedures based upon cognitive models of learning and memory-schema assessment. For Marshall "test development needs to reflect the definition and organization of the knowledge base to be tested." It was recommended that tests' items be constructed to measure declarative [facts and concepts], procedural (skills or techniques], and/or schematic knowledge [relation between declarative facts and procedural rules]. For Marshall,

Once these items are incorporated into a test, we can begin to look at student performance in terms of how much a student knows and in terms of how well the student has organized the information in his or her long-term memory. Having this information about a student, allow us to modify existing instruction and create new instruction with the objective of helping the student to learn more efficiently. (p. 176) Wiebe (1993) provided another explanation of how
evaluation should take place when measuring problem solving skills:
Children should be given credit for using correct problem-solving strategies, finding the correct resources, selecting appropriate media (e.g., the computer or calculator), determining whether or not their answer makes sense, partial results, and answers that are reasonably close to the 'true answer'. Also, when true problem solving is taking place, one cannot expect the same level of accuracy as with practice activities-70% or 80% is an unreasonably high expectation. (p. 13)
    An important source for this project was a second study by Schoenfeld (1985c) where he examined the effects of a college-level course in problem solving. For this purpose, three pairs of tests and associated grading procedures were developed. These instruments were related to the ideas presented by Marshall (1988). Measure 1 consists of a pair of matched tests (pretest and posttest). The tests were compared in terms of solution methods, not by problem type. measure 2 is a qualitative companion to Measure 1. This measure examined the students' subjective assessments of their problem solving behavior. Measure 3 was a test of heuristic transfer. The three measures were used to observe the students' performance before and after problem solving instruction. Schoenfeld explained that most educational research in the area of problem solving was based on extensive protocol analyses, which he considered an incredibly time-consuming task. He added that Measure 1 and Measure 2 provide straightforward and easy-to-gather assessment of heuristic fluency and transfer. The results indicated that student in a problem-solving course can learn to employ a variety of heuristic strategies. There was clear evidence not only of heuristic mastery but also of transfer.

    According to Kantowski (1981), teaching problem solving is one of the most difficult tasks facing the teacher at any level. Describing what the curriculum holds for the 80s, and probably for several others decades, she stated reasons for this difficulty:

  1. The object of instruction is to have students put together knowledge they have already acquired to solve the given problem, there are no new concepts to introduce or algorithmic skills to teach;
  2. Students in a given group are not familiar with the necessary content or algorithms needed to solve some of the problems encountered, there is a diversity of backgrounds;
  3. Students work at different rates;
  4. There are many problem solving styles resulting in different paths to the solution of problems, particularly those that are nonroutine;
  5. Teachers are faced with increasing requirements in the curriculum and are pressured to emphasize computational skills and so have little time to assist students who are having difficulty in problem solving experiences;
  6. There is a lack of good sequences of related problems to use in instruction. (pp. 122-123)
The Use of Computers in Mathematics
    Kantowski (1981) noted that many of the problems cited above could be resolved partially by computers. Computers can provide support in many problem areas. Students working at a computer can work their own rates. The diversity of possibilities, when using a computer is perhaps the single aspect of this machine that makes it an invaluable tool in teaching problem solving. Kantowsky said that "The capacity of the computer to provide for differences in educational backgrounds and preferred styles enables the teacher to deal effectively with what could otherwise be an unmanageable situation" (p. 123).
Heid and Baylor (1993) mentioned four different ways to use computing technology in the teaching of the concepts and skills of mathematics. These are: tutor [drill and practice]; tool [graphing, symbolic-manipulation, and geometric construction]; tutee [student]; and catalyst [motivator]. Each suggests a different perspective on teaching and learning. Heid and Baylor emphasized that "what seems to make the difference is not the fact that teachers use computer ... but rather how they use them" (p. 203).
Research studies have examined the impact and effects of computers used for education and instruction at almost every grade level and for almost every subject area. Becker (cited in Bracey, 1993) pointed out that most studies in the past are in many ways not relevant to us now. Many early studies used mainframes and mini computers. The control of the computers and the software used was in the hands of the researchers, not classroom teachers. There is now a great difference in terms of the software and the instructional environment.

    Research studies in the use of the computer in the teaching of mathematics have been reviewed by several authors (Bennet, 1992; DeVault, 1981; Kaput & Thompson, 1994; Kulik & Kulik, 1987). In the review conducted by Kaput and Thompson (1994), two major types of studies were identified. The first type, called "wave-level" studies, were those that used the calculator or the computers as a complement to existing curricula and instruction. Most of the emphasis in these studies was on computation. A second type, called studies at the "swell" level of change, involved a closer look at the role of the technology in the teaching and learning of mathematics, or in cognition (Blume & Schoen, 1988; Heid, 1988: Kraus, 1982; Szetela, 1982; Szetela & Super, 1987; Wheatley, 1980).

Bennet's (1992) review of recent findings included the following suggestion:

  1. Computer-assisted lessons should require students to work cooperatively rather than competitively or individually.
  2. The commonly referred to sex bias regarding the use of computers in mathematics may be disappearing or perhaps        never existed. This is contrary to past studies/opinions which have tended to support the belief that males are more successful with computers and mathematics than females.
  3. The use of computers in elementary mathematics classes may widen the gap between advantaged and disadvantaged and high-and low-aptitude students.
  4. Students who use computers successfully are probably going to be the same ones who are successful in other areas--the self-assured, independent, and intelligent.
  5. Although computers are not a panacea, it does appear that they should be one of the tools used to improve students' problem solving skills.
  6. A positive teacher using computers appears to be the best teaching combination with the lower remedial mathematics students; however, use of the computer will not overcome the effect of a teacher. (p. 39)
    As a final observation about the last conclusion, Bennet suggested that the computer is a successful teaching tool when it is used by good teachers using appropriate teaching methods. Bennet quoted an anonymous cite "Teachers who are afraid they will be replaced by a computer probably should bell p. 39.

    Kulik and Kulik (1987) applied Glass's (1976) methodology in four separate meta-analyses (Bandgert-Drowns, Kulik, & Kulik, 1985; Kulik & Kulik, 1986; Kulik, Kulik & Bangert-Drowns, 1985; Kulik, Kulik & Shwalb, 1986). The analyses covered a total of 199 comparative studies: 32 in elementary schools; 42 in high schools; 101 in universities and colleges; and 24 in adult education settings. The meta-analyses examined the use of computers in (a) computer-assisted instruction, or CAI, including drill-and-practice and tutorial instruction; (b) computer-managed instruction, or CMI; and (c) computer-enriched instruction, or CEI, including the use of the computer as a calculating device, programming tool, and simulator. Each of the 199 studies included in the meta-analyses was a controlled, quantitative study that met predefined standards for methodological adequacy. Kulik and Kulik concluded that the major finding from the meta-analyses was that computer-based instruction raised student achievement in many studies. They added that it has given students a new appreciation for technology and has had positive effects on students' attitudes toward schools and teaching. Another benefit found was that computers have helped teachers save instructional time.

    Several other studies presented positive results toward the use of computers. These studies have found that computer experience was effective when working with: low-achieving students (Lawson, 1989); graphing abilities (Fredrick, 1989); reflective abstractions (Ayers, Dubinsky & Lewin, 1988); and drill, games and problem solving (Ediger, 1989). In a conference presented at the 1989 annual meeting of the American Educational Research Association, "A Review of Research Issues in the Use of Computer-Related Technologies for Instruction: An Agenda for Research," researchers Williams and Brown (cited in Kinnaman, 1993) drew the following generalizations:

    Williams and Brown concluded that while there is "some evidence that well-designed computer-assisted instruction
can be more effective than traditional instruction, the findings to date can be described only as moderately positive" (p. 55).
Factors Related to the Effectiveness
           of Computers Use
    There are other factors that are important in the effectiveness of the use of computers in the teaching and learning processes. one of these factors is the development of instructional activities and strategies for a specific computer program. Nizamudin (1989) found that an intelligent computer aided instruction program, MicroSEARCH, when used with designed instructional strategies improved student performance. The same results were not found when using the program without the designed instructional strategies.

    The time the students work with computers and the experiences they get also were variables of interest in research studies. DeVault (1981) found that there was consistent evidence to support the claim that achievement gains were related to the amount of time students spend in CAI. Demana and Waits (1992) reported that students must use computers on a regular basis for both in-class and for homework if significant changes are to be made in the mathematics they learn in the 1990s. Wiebe (1993) said that if students are to learn to use a particular program, teachers will have to use computers with problems that could just as easily be solved with other media.

    Another factor that can influence problem solving ability is mathematics anxiety and computer attitudes.  Studies of mathematics anxiety (Castro, 1984; Fenemma & Sherman, 1977; Nolasco, 1988; Sherman, 1979) have shown significant relations between attitudes toward mathematics and achievement of mathematical skills. Some teachers fear that the use of computers during instruction may be affected by a similar anxiety. No empirical research was found, only anecdotal observations (Fisher, 1984; Schubert & Bakke, 1984). In a study of the effects of mathematics anxiety and gender on three computer attitudes [computer anxiety, computer confidence, and computer liking], Gressard and Lloyd (1987) found that mathematics anxiety may be a small but important factor in the high computer anxiety and low computer confidence. The results suggested that "studies of mathematics anxiety may provide insight into computer anxiety, and techniques that have been successful in the treatment of mathematics anxiety may also help to relieve computer anxiety" (p. 134).

The Use of Computers to Provide Concrete Experience
                 as an Aid to Problem Solving
    Most of the current interest in mathematics education is focused on computers and problem solving. Many attempts have been made to use computer programs, programming languages, spreadsheets, graphing software, calculators, and others when working with problem solving. Technology can play a valuable role in developing problem solving and mathematical reasoning skills by freeing the students from tedious computations. As pointed out by Hoeffner, Kendall, Stellenwerf, Thames and Williams (1993), "an electronic spreadsheet can calculate rapidly, generate data from which patterns can be found, show relationships between two or more variables, and investigate 'what if?' questions with ease" p. 51.
Programming and Spreadsheets as
        Problems Solving Aids
    Problem solving research, involving computers, often has dealt with programming languages and spreadsheets. The development of a computer program or a spreadsheet to solve a problem should be a challenging activity that can enhance the students' understanding of the mathematics being used. Wilson, Fernández, and Hadaway (1993) indicated that too often the focus is on programming skills rather on using programming to solve mathematics problems. The same situation could occur with the use of spreadsheets. The focus of teachers who decide to use these tools should be on the mathematics problems and the use of the computer as a tool for problem solving.

    Programming as a problem-solving aid has been discussed extensively (Blume & Schoen, 1988; Hatfield & Kieren, 1972; McCoy & Burton, 1988; McCoy & Dodl, 1989; Stannard, 1984) with different conclusions. Hatfield and Kieren (1972) described a computer program as a dynamic problem solving tool. McCoy and Burton (1988) from their study of the
relationship of computer programming and secondary mathematics, found that "after programming instruction, both ability to use mathematics variables and mathematical problem solving ability scores were significantly improved" (p. 165).
McCoy and Dodl (1989) found that experience with computer programming increased problem solving achievement in mathematics; however, another finding in that study was that the ability of the students has the greatest causal effect on mathematical problem solving achievement. The variables, computer programming, mathematics experience, and gender, each had a moderating effect. This corresponded to the findings by Lutz, Durham, and Coble (1988) that successful computers users were much like students successful in any academic area. They suggested that these might be students who are self-assured, independent, and intelligent.

    Damarin, Dziac, Stull, and Whiteman (1988) suggested significant increases in students' abilities to solve estimation problems because of using computer based instructional materials. Stannard (1984) concluded that the microcomputer has vast potential as a teaching tool in two critical areas: development of problem solving skills and estimation.
Perhaps even more important is the availability of the computer allowing teachers to include a variety of realistic problems with large amounts of data and complex computations. Wiebe (1993) said that the ability to solve a simple word problem from a textbook has little relationship with the ability to solve real-word mathematics problems.  Arganbright (1984) pointed out some advantages of spreadsheet use in a mathematics classroom:

Its operation is easy to learn and requires no previous programming knowledge; it provides a natural way to implement algorithms on a computer that allows for interactive experimentation and modeling; and it furnished an intuitive, concrete means to visualize mathematical algorithms and manipulations. (p. 193)
    This idea corresponded to that presented by Verderbe (1990), "The spreadsheet will allow us to provide computer
experience for those who cannot grasp the fundamentals of writing a computer program" (p. 45). Examples of the use of
spreadsheets can be found in Bright, 1989; Hoeffner, Kendall, Stellenwerf, Thames, and Williams, 1993; Hunt, 1995; Kari and Dubreuil, 1977-88; McDonald, 1988; Sgroi, 1992; Troutner, 1988; Wiebe, 1993.

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