HANDS-ON PROBLEM-SOLVING ACTIVITY PERFORMANCE IN THE ELEMENTARY GRADES A Master's Thesis Presented to The School of Graduate Studies Department of Industrial Technology Education Indiana State University Terre Haute, Indiana In Partial Fulfillment of the Requirements for the Master of Arts Degree by Christopher L. Droessler August 1991 ---------------- APPROVAL SHEET The thesis of Christopher L. Droessler, Contribution to the School of Graduate Studies, Indiana State University, Series I, Number 1675, under the title Hands-On Problem-Solving Activity Performance in the Elementary Grades is approved as counting toward the completion of the Master of Arts Degree in the amount of six semester hours of graduate credit. Committee Chairperson Committee Member Committee Member For the School of Graduate Studies ------------ ABSTRACT This study was designed to investigate the performance of elementary-school students on a particular hands-on problem-solving activity (HOPSA), the Tower Problem. It was anticipated that the performance would increase as the grade level increased. The HOPSA used in this study involved giving each student a limited amount of materials to build the tallest possible structure within a given period of time. The following null hypotheses were tested: Ho 1. There is no significant relationship between the grade level of the student and the maximum height of the structure. Ho 2. There is no significant relationship between the grade level of the student and the length of the initial-analysis phase. Ho 3. There is no significant relationship between the grade level of the student and the frequency of occurrence of a single geometric shape used in the structure. The students studied included 8 each from the first-, second-, fourth-, and sixth-grade levels, for a total of 32, from the Indiana State University School. The students were selected using a stratified random sample to insure a representative sample of academic-achievement levels as well as an even gender ratio. The results of this study were as follows: 1. As the grade level increased, there was a significant increase in the maximum structure height. 2. As the grade level increased, there was no significant change in the length of the initial-analysis phase. 3. As the grade level increased, there was no significant change in the frequency of occurrence of a single geometric shape. It was concluded that the ability of elementary-school students in this study to construct a tall structure with limited resources increased with an increase in grade level. It was also concluded that at the grade levels studied, both the length of the initial-analysis phase and the frequency of occurrence of a single geometric shape used in the structure had no significant relationship with the grade level. ------------ ACKNOWLEDGMENTS I wish to thank the following: My Mother and Father, the best parents in the world. My committee Dr. Peter H. Wright, Chair, Dr. Lowell D. Anderson, and Dr. Alvin R. Putnam. Dr. Gregory Ulm, Director; Lynne W. Thomas, Director of Guidance; and the teachers at the Indiana State University School, which sadly no longer exists. Caryn, Dave, Jeff, Jim, John, Julie, Larry, and Mac. Dr. David Dillon. Dr. Eldon Rebhorn. Mary, Maureen, and Jim. ------------ TABLE OF CONTENTS Page LIST OF TABLES . . . . . . . . . . . . . . . . . . . viii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . ix Chapter 1. INTRODUCTION . . . . . . . . . . . . . . . . 1 Statement of the Problem . . . . . . . . . 3 Research Hypotheses . . . . . . . . . . . 4 Design of the Study . . . . . . . . . . . 4 Delimitations . . . . . . . . . . . . . . 6 Assumptions . . . . . . . . . . . . . . . 6 Definition of Terms . . . . . . . . . . . 7 2. REVIEW OF RELATED LITERATURE AND RESEARCH . . 9 Increased Achievement . . . . . . . . . . 9 Analysis Phase . . . . . . . . . . . . . . 10 Previous Knowledge . . . . . . . . . . . . 13 Summary . . . . . . . . . . . . . . . . . 15 3. PROCEDURE . . . . . . . . . . . . . . . . . . 17 Description of Sample . . . . . . . . . . 17 Description of Instrument Used . . . . . . 18 Validity of the Instrument . . . . . . . . 19 Pilot Test . . . . . . . . . . . . . . . . 20 Null Hypotheses . . . . . . . . . . . . . 21 Data Collection . . . . . . . . . . . . . 22 Chapter Page Data Treatment . . . . . . . . . . . . . . 24 4. ANALYSIS OF DATA . . . . . . . . . . . . . . 25 Null Hypothesis One . . . . . . . . . . . 25 Null Hypothesis Two . . . . . . . . . . . 27 Null Hypothesis Three . . . . . . . . . . 30 5. SUMMARY . . . . . . . . . . . . . . . . . . . 33 Conclusions . . . . . . . . . . . . . . . 35 Discussion . . . . . . . . . . . . . . . . 36 Recommendations for Future Research . . . 39 REFERENCES CITED . . . . . . . . . . . . . . . . . . 43 APPENDIXES . . . . . . . . . . . . . . . . . . . . . 46 A. INSTRUCTIONS TO THE STUDENT . . . . . . . . . 46 B. INSTRUCTIONS TO THE OBSERVERS . . . . . . . . 47 C. OBSERVATION SHEET . . . . . . . . . . . . . . 50 D. RAW SCORES FOR THE LENGTH OF THE INITIAL- ANALYSIS PHASE . . . . . . . . . . . . . . . 51 E. MAXIMUM STRUCTURE HEIGHTS AND ANOVA CALCULATION . . . . . . . . . . . . . . . . . 52 F. LENGTHS OF THE INITIAL-ANALYSIS PHASE AND ANOVA CALCULATION . . . . . . . . . . . . . . 53 G. FREQUENCIES OF OCCURRENCE OF A SINGLE GEOMETRIC SHAPE AND ANOVA CALCULATION . . . . 54 --------------- LIST OF TABLES Table Page 1. Mean Maximum Structure Heights, F value, and Probability at the Four Grade Levels . . . . 27 2. Mean Lengths of the Initial-Analysis Phase, F value, and Probability at the Four Grade Levels . . . . . . . . . . . . . . . . . . . 28 3. Mean Frequencies of Occurrence of a Single Geometric Shape, F value, and Probability at the Four Grade Levels . . . . . . . . . . 32 ---------------- LIST OF FIGURES Figure Page 1. Design of the Study . . . . . . . . . . . . . 5 2. Comparison of the Maximum Structure Heights . 26 3. Comparison of the Lengths of the Initial- Analysis Phase . . . . . . . . . . . . . . . 29 4. Comparison of the Frequencies of Occurrence of a Single Geometric Shape . . . . . . . . . 31 ---------------- ABSTRACT Droessler, Christopher L. Thesis: Hands-On Problem-Solving Activity Performance in the Elementary Grades. Indiana State University, Terre Haute, Indiana, August, 1991. Pp. ix + 54. Series I, Number 1675. Committee: Dr. Peter H. Wright, Chairperson Dr. Lowell D. Anderson Dr. Alvin R. Putnam ----------------- Chapter 1 INTRODUCTION People encounter mechanical problems every day that need some sort of action taken in order to solve or eliminate the problem. Because of this, hands-on problem-solving activities (HOPSAs) are an integral part of the technology curriculum at the middle- and high-school-grade levels. In fact, "technologies are, in many ways, a product of problem-solving" (DeLuca, 1991, p. 5). Students in technology classes learn proper problem-solving procedures which will assist them throughout their entire lifetimes. John Dewey (1933) proposed a three-step-problem-solving procedure that is used in technology education. This procedure included a diagnosis phase, an analysis phase, and a solution phase. Sellwood (1989) stated that "practical problem solving is the marriage of an investigative approach to learning through the manipulation and use of materials" (p. 4). Todd & Hutchinson (1991) addressed the appropriateness of HOPSAs in the elementary-school grades by suggesting that "even at five years of age, children have enough experience to engage in design and technology activities that are developmentally appropriate" (p. 5). Bruner (1960) added that "any subject can be taught effectively in some intellectually honest form to any child at any stage of development provided that [the subject is] divorced from its mathematical expression and studied through materials that the child can handle himself" (p. 33). Davis (1983) researched Piaget's stages of cognitive development that present some insight into the development of logical thought. Piaget described a change in the cognitive activity in children at the age of 7 years, which is approximately the second-grade level. At this time, a child goes from the preoperational stage to the concrete-operational stage. This change in a child is signaled by an ability to think through a problem before actually attempting to physically solve it. A concrete-operational child is, however, limited to concrete objects in problem solving and cannot deal with abstract objects. The concrete-operational stage continues until a child is about 11 years of age, which is approximately the sixth-grade level. At this time, a child enters the stage of formal operations. This stage is marked by the emergence of abstract thinking (Davis, 1983). The performance on a HOPSA might show evidence of these stages of development proposed by Piaget. There should be a marked change in performance at approximately the second- and sixth-grade levels due to the student's change in ability to deal with concrete objects. Statement of the Problem Most studies dealing with hands-on problem-solving activities (HOPSAs) have dealt with only simple problems consisting of a manipulation of concrete objects. In those studies, there was usually only one right way to solve the problem, such as arranging blocks in a pattern or getting through a maze. There was a need for research to examine performance on more complex HOPSAs involving abstract objects. Most problem-solving research has been limited to high-school or college-age students. Elementary students, though able to perform HOPSAs, have rarely been researched in that regard. There was also a need to look at the development of problem-solving skills over several years of cognitive development rather than investigating the performance at only a single-grade level. This research was designed to ascertain whether or not there was a significant relationship between the performance on a particular HOPSA and grade level. The students observed in this study should have exhibited some traits that were characteristic of one of Piaget's stages of cognitive development. The problem of this study was to investigate the performance of elementary-school students on a particular hands-on problem-solving activity (HOPSA), the Tower Problem. Research Hypotheses This study was designed to investigate the performance on a particular hands-on problem-solving activity (HOPSA). The activity involved giving each student a limited amount of materials to build the tallest possible structure within a given period of time. The following research hypotheses were proposed: 1. The height of the structure will increase as the grade level increases. 2. The length of the initial-analysis phase will increase as the grade level increases. 3. Due to the students' increase in previous knowledge, the frequency of occurrence of a single geometric shape used in the structure will increase as the grade level increases. Design of the Study This was a descriptive research study. This study involved an analysis of quantitative data obtained from an observation of students engaged in an activity compared with similar data obtained from observations of students engaged in the same activity at other grade levels. A stratified random sample was used to choose 8 individuals in each of four grade levels for a total study of 32 individuals (see Figure 1). ___________________________________________________________ Stratified Stratified Stratified Stratified random sample random sample random sample random sample utilizing utilizing utilizing utilizing gender & gender & gender & gender & ISTEP scores ISTEP scores ISTEP scores ISTEP scores ________ ________ ________ ________ n = 8 n = 8 n = 8 n = 8 ________ ________ ________ ________ Grade 1 Grade 2 Grade 4 Grade 6 ________ ________ ________ ________ Build Build Build Build structure structure structure structure ________ ________ ________ ________ Observation Observation Observation Observation ___________________________________________________________ N = 32 ___________________________________________________________ Figure 1. Design of the Study. The design of the study includes the (a) method for choosing subjects, (b) size of the group, (c) grade level of the subjects, (d) type of activity, (e) method of assessment, and (f) size of the sample. Delimitations This research was delimited to one particular hands-on problem-solving activity (HOPSA), the Tower Problem. This activity is described in detail in this paper under the heading "Description of Instrument Used." Findings from this study should not be generalized to other problem-solving activities. This investigation was delimited to three measurements for each student. They were the (a) length of the initial-analysis phase, (b) maximum height of the structure, and (c) repetition of a single geometric shape. The sample used in this study was delimited to first-, second-, fourth-, and sixth-grade students at the University School located on the campus of Indiana State University in Terre Haute, Indiana. Assumptions The researcher regularly taught a class of ninth-grade students at the University School. The researcher had, on one occasion during the school year of the research, and prior to the research, been a substitute teacher for a sixth-grade-technology class. The researcher had, on three occasions that school year, been a substitute teacher for seventh- through ninth-grade technology classes. The researcher also met occasionally with the school's chess club, composed of students in the seventh through ninth grades. It was assumed that these factors would not have an effect on the results of this study. Definition of Terms An abstract object was something that could not be perceived by the senses. A concrete object was something real or actual that could be perceived by the senses. Elementary school included students in kindergarten through sixth grade. This represented students from approximately age 4 through 11. Geometric shapes included, but were not limited to rectangular prisms, hexahedrons (cubes), triangular prisms, triangular pyramids, square pyramids, cylinders, and cones. A hands-on problem-solving activity (HOPSA) is an activity that poses a challenge needing to be overcome by manipulation of physical materials. The initial-analysis phase was the time the student spent thinking about the problem before starting construction. The initial-analysis phase started at the conclusion of the reading of the instructions. The initial-analysis phase ended if the student cut or folded a piece of paper or used a piece of tape in an attempt to start construction. ISTEP was the Indiana Statewide Testing for Educational Progress, a standardized test used to measure academic-achievement levels. Maximum structure height was the highest achieved height of the student's structure. To ensure stability, the structure had to stand for 15 seconds before it was measured. The structure was measured vertically from the floor to the highest point on the structure. Performance was measured by the length of the initial-analysis phase, the maximum height of the structure, and the frequency of occurrence of a single geometric shape in the HOPSA. --------------- Chapter 2 REVIEW OF RELATED LITERATURE AND RESEARCH Three areas of research related to this study are presented here. All three areas deal with an aspect of solving problems and the correlation with age. The three areas are (a) increased achievement in solving problems, (b) the analysis phase of a problem, and (c) the use of previous knowledge to solve a problem. Increased Achievement Alpert (1928), Harter (1930), Matheson (1931), and Roberts (1933) studied problem solving in preschool children. These studies concluded that there was some relationship between age and achievement. As age increased, so did the achievement level of the students. Heidbreder (1928) studied elementary-age children and concluded that "the general ability to solve problems increased with age [and] the problems [that were used] represented degrees of difficulty which retained the same rank order at the different age levels" (p. 542). A definite form of procedure was exhibited as the age of the student increased. Müller (cited in Gibson & McGarvey, 1937) studied children between the ages of 7 and 17. The study showed a positive correlation between performance on syllogistic-reasoning tests with age. The greatest increases were between the first- and second-, and the second- and third-grade levels. Glaser (1941) claimed that critical-thinking skills could be successfully taught to students as early as 7 years of age. Analysis Phase In order to keep from jumping at the first occurring solution, Dewey (1933) described a three-step "systematized method" (p. 166) for solving problems. The first step was the diagnosis phase which involved the collection and understanding of the facts surrounding the problem. The second phase was the analysis phase. In this phase, potential solutions were generated and hypotheses had to be made. The third phase was the solution phase during which the hypotheses were tested. Glaser (1941) asserted that the knowledge of the methods of problem-solving procedures was necessary to be able to think critically. Dewey's second step, the analysis phase, is the subject of this section of the literature review. Bloom's (1956) taxonomy of the cognitive domain described analysis as involving a thorough examination of the facts surrounding a situation and the use of inductive or deductive reasoning to come to a solution or a hypothesis. Dewey (1933) described this analysis phase as follows: Given a difficulty, the next step is suggestion of some way out - the formation of some tentative plan or project, the entertaining of some theory that will account for the peculiarities in question, the consideration of some solution for the problem. The data on hand cannot supply the solution; they can only suggest it. (p. 15) Dewey (1933) described the term data as the observed facts surrounding a situation that needed to be "managed and utilized" (p. 104). The possible solutions suggested by an examination of the data formed ideas. Dewey suggested that data and ideas formed the two most important factors in all reflective activity. The hypotheses, or ideas, had to be judged to determine which would give the best possible solution to the problem. Possible solutions lead to other possible solutions (Dewey, 1933). Brightman (1980) encouraged generating as many hypotheses as possible. More hypotheses increased the likelihood of solving the problem. The number of hypotheses, of course, could be limited by the amount of available time. During Piaget's concrete-operational stage (ages 7 to 11), behavior sequences could be internalized and thought through in the head before having to manipulate the actual physical objects. The child at this stage developed logical-thought processes that enabled the solving of concrete problems (Wadsworth, 1979). Heidbreder (1928) studied preschool children, elementary-school children, and adults. Reaction time was measured between the time the subjects were presented the data and the time a decision was made. The elementary-age children had a quicker reaction time than both the pre-school students and the adults. The preschool students were often distracted by the equipment used in the experiment or by other objects in the testing room. Discounting the preschool students, there was an increase in reaction time as the age increased. Much research had been done concerning the interpretation of ten ink blots drawn by Rorschach. Researchers agree that an increased number of M responses on the test indicated "inner creation," "artistic inspiration" (Rorschach, 1964, p. 65), creativity (Levitt & Truumaa, 1972), an "ability to delay response," and "accessibility to inner life" (Dana, cited in Levitt & Truumaa, 1972). Rorschach (1964) concluded that the number of M responses decreased with age, thus creative-thinking ability also decreased with age. Levitt and Truumaa (1972), however, found that M responses increased with age. They concluded that creativity increased with age. Levitt and Truumaa (1972) reported the experts agreed that an increase in Sum C responses to the ink blots indicated emotional responsiveness, and impulsiveness. There was a significant-negative relationship between Sum C and age for the children from age 5 through 14. The Sum C decreased from age 5 through age 14. Rorschach defined an individual who had a high number of M responses and a low number of Sum C responses as "introversive" (p. 82). An individual who had a low number of M responses and a high number of Sum C responses was defined as "extratensive" (Levitt and Truumaa, 1972, p. 83). Orme (1986) researched the differences introversive and extratensive children had with respect to problem-solving styles. It was hypothesized that introversive children internalized more of their thinking and thus would make fewer trial-and-error mistakes. They would work the problem through in their head before physically manipulating the materials. It was hypothesized that extratensive children internalized less of their thinking and thus would make more trial-and-error mistakes. They would start manipulating the materials before thinking through the solution. Orme conducted three experiments in which only one showed a significant relation to the hypothesis. Previous Knowledge Past experiences and knowledge played an important role in the solving of any problem. Dewey (1933) noted that the solution to a problem depended upon previous experiences. It was important to recall solutions made previously by the person solving the problem and by others who had solved similar problems. Glaser (1941) conducted research on critical thinking. It was concluded that "thinking cannot be carried on in a vacuum" (p. 9). Past knowledge was very important if critical thinking was to be "fruitful" (p. 9). "The search for alternatives is strongly influenced by past behavior" (Brightman, 1980, p. 187). There was a need to search for similar decisions that had already been made. Previously successful solutions were recycled to solve new problems. Fales, Kuetemeyer, and Brusic (1988) emphasized "designing with nature" (p. 393). "The triangle is a simple basic shape that occurs in nature. It is very strong" (p. 393). Steel beams used in construction came in many shapes. Some had cross sections that looked like letters of the alphabet such as H, I, L, T, and U. The different shapes had different strengths in different directions depending upon the application. The children's activities "must utilize facts and skills gained in other phases of the elementary curriculum" (Miller & Boyd, 1970, p. 9). Sellwood (1989) noted that "outcomes and solutions can be reasoned by an understanding and knowledge built upon firsthand experiences" (p. 6). An activity was developed involving the use of sheets of paper to construct a bridge capable of supporting a weight. The students developed an "understanding of how Œflimsy' sheet materials can be folded, formed, rolled and concertina'd [sic] into strong shapes that will support, package, structure, and adapt to many manufacturing needs" (p. 7). The students were asked if the bridges they made reminded them of bridges they had seen before. The students were encouraged to try out different shapes and different types of bridges. Winter (1990) developed a HOPSA which involved assembling a polypopagon, a discrete three-dimensional geometric shape. Several polypopagons were combined to form structures of varying shapes and sizes. In order to assemble individual geometric structures, the students had to understand how to connect them together. "The synergistic results when one unit, by mere multiplication and manipulation of patterns of assembly becomes a system of disparate and unexpected character" (p. 22). Winter pointed out that "ideas reside in the prepared mind, coming from learned information encoded in some discipline" (p. 17). The impulse to make and possess more than one unit and to use this collection for building things is typical in young children. But it is also clearly an ancient inclination of the human species, in as much as stone and brick structures are found in widely dispersed pre-historic civilizations. These characteristic patterns of construction, the post and beam, the arch, the barrel vault, the dome all illustrate the general responsiveness to modular manipulation and the growing complexity it quickly assumes. (Winter, 1990, p. 20) Summary All research seemed to show an increase in achievement in problem solving or critical thinking with an increase in age. The analysis phase was an important part of the problem-solving procedure. Some research pointed to an increase in the length of the analysis phase with age; however, other research pointed to a decrease in the length of the analysis phase. All researchers seemed to agree that past experiences were necessary to solve a problem. Recalling solutions to similar problems could lead to a more successful solution to the present problem. ----------- Chapter 3 PROCEDURE Description of Sample The students in this study came from the Indiana State University School located in Terre Haute, Indiana. There was a total of 32 students including 8 each from the first-, second-, fourth-, and sixth-grade levels. The students were selected using a stratified random sample to ensure a representative sample of academic-achievement levels as well as an even gender ratio. There were two sections of students at the school for each grade level. Students in this study came from only one section at each grade level. The section used in this study was determined by a flip of a coin. This was done four times, once for each grade level studied. To determine a representative sample of academic-achievement levels, ISTEP (Indiana Statewide Testing for Educational Progress) scores were used. The median ISTEP score for each section was determined. The students whose ISTEP scores were higher than the median were classified as having high ISTEP scores. The students whose ISTEP scores were lower than or equal to the median were classified as having low ISTEP scores. This was done four times, once for each grade level studied. Each grade-level group included 2 students from each of the following classifications: 1. Male & high ISTEP score. 2. Female & high ISTEP score. 3. Male & low ISTEP score. 4. Female & low ISTEP score. The 2 students in each group were determined by utilizing a computer program that generated a random number between one and the size of the stratified group. Description of Instrument Used The instrument used in this study was the Tower Problem, number M 1.1 in Module 1, found in the TE 510 course curriculum at North Carolina Agricultural and Technical State University. This hands-on problem-solving activity (HOPSA) was traditionally used in the classroom as an introduction to construction technology. This HOPSA was generally administered as a group activity. In this study, however, the students worked independently in order to measure individual cognitive performance. This particular HOPSA was chosen for this study because it would be easy for students of all ages to understand, yet it would still provide enough of a challenge for older students. Each student was given the following materials: 1. Four sheets of plain white paper, each measuring 8 1/2 in. (215.9 mm) by 11 in. (279.4 mm). The paper was Champion Bond, White 8 1/2 x 11 - 10M sub 20 long, manufactured by the Champion International Corporation. 2. Clear adhesive tape measuring 18 in. (457.2 mm) long by 1/2 in. (12.7 mm) wide. The tape was Scotch Brand Transparent Tape, 1/2 inch wide, catalog number 174, manufactured by the Minnesota Mining and Manufacturing Company (3M). 3. A pair of safe scissors with rounded points (students could choose left or right handed, but not both). Each student was instructed to make the tallest possible vertical structure using only the given materials. The structure could only touch the floor. Each student had a maximum of 20 minutes to complete the structure. The exact instructions to the student are given in APPENDIX A. Each student worked independently and was placed in such a location so they could not view the work of any other student. Four students from each grade level worked at the same time. Then the other four from the same grade level were immediately tested so as to eliminate any sharing of information between test sessions. Validity of the Instrument This particular hands-on problem-solving activity (HOPSA) had generally been used as a group activity in an introductory construction technology class to examine and evaluate structure design. This HOPSA had never, to the researcher's knowledge, been used as it was in this study to measure individual cognitive performance in the elementary grades. Therefore, in order to ensure instrument validity, a panel of experts who had experience working with HOPSAs were asked to review the instrument. This panel of experts included: (a) L. Helphinstine (personal communication, April 8, 1991), a technology education teacher and Indiana's Affiliate Representative to the International Technology Education Association (ITEA); (b) D. Dillon (personal communication, April 8, 1991), a professor of technology education at North Carolina Agricultural and Technical State University; and (c) E. Rebhorn (personal communication, April 10, 1991), a professor of technology education at Indiana State University and President of the Indiana Industrial Technology Education Association (IITEA). All of these educators agreed that the instrument described in this study was a valid example of a HOPSA that could be typically found in the technology curriculum. They also agreed that this instrument could be used to measure the HOPSA performance of elementary-school students. Pilot Test A pilot test of this study was run on a group of 4 students from several grade levels at the same school. The students did not come from the grade levels used in the study. The same testing site to be used in the study was utilized for this pilot test. This pilot test was conducted to check the reliability of the raters and the rating procedure. Two research assistants examined all of the observation sheets from the pilot test and independently determined the ending point for the initial-analysis phase for each student. The two research assistants identified the same ending point for each student. After the pilot test, a few minor changes were made to the procedure. These changes included: 1. The amount of time given to the students to build their structures was reduced from 30 minutes to 20 minutes. 2. There was a slight change in the configuration of the room. 3. The instructions given to the students and to the observers were clarified. These changes are reflected in the current APPENDIXES A and B, respectively. Null Hypotheses Three null hypotheses were formulated to test the performance of elementary school students on a particular hands-on problem-solving activity (HOPSA). The following null hypotheses were tested: Null Hypothesis 1: µ1 = µ2 = µ3 = µ4 There is no significant relationship between the grade level of the student and the maximum height of the structure. Alternative 1: µ1 < µ2 < µ3 < µ4 The maximum height of the structure will increase as the grade level increases. Null Hypothesis 2: µ1 = µ2 = µ3 = µ4 There is no significant relationship between the grade level of the student and the length of the initial-analysis phase. Alternative 2: µ1 < µ2 < µ3 < µ4 The length of the initial-analysis phase will increase as the grade level increases. Null Hypothesis 3: µ1 = µ2 = µ3 = µ4 There is no significant relationship between the grade level of the student and the frequency of occurrence of a single geometric shape used in the structure. Alternative 3: µ1 < µ2 < µ3 < µ4 The frequency of occurrence of a single geometric shape used in the structure will increase as the grade level increases. Data Collection Trained observers recorded the activities of the students using the procedure outlined in APPENDIX B. The observers recorded all of the student's activities and the times of their occurrence on the observation sheets (see APPENDIX C). Each student's performance was recorded as follows: 1. Any activity by the student, whether it was a manipulation of the materials or a distraction by other objects in the room, was recorded along with the time, in minutes and seconds, of its occurrence. 2. The structure height was recorded in inches rounded to the nearest 1/4 in. (6.35 mm). 3. The frequency and type of geometric shapes used in the construction of the structure were recorded. The researcher utilized two research assistants as raters to examine all of the observation sheets and independently determine the ending point for the initial-analysis phase for each student. Once the ending point was established, the length of the initial-analysis phase was calculated by subtracting the time at the starting point from the time at the ending point. This time was recorded in the form of minutes and seconds. In order to determine the reliability of the raters, a Pearson product-moment coefficient of correlation was calculated using all score pairs from the two raters. The data for each student is presented in APPENDIX D. Both of the raters recorded the same time for each of the 32 students. Therefore, the resulting coefficient was r = 1.00. The mean was determined for each student from the two times calculated by the research assistants. This mean was recorded as the length of the initial-analysis phase for the student. The frequency of occurrence of the geometric shape that was most evident in the student's structure was the score that was recorded. The highest of all of the measured heights of the student's structure was recorded as the maximum structure height. Data Treatment The independent variable in this study was the student's grade level. The dependent variables in this study were the (a) length of the initial-analysis phase, (b) maximum structure height, and (c) frequency of occurrence of a single geometric shape. An analysis of variance (ANOVA) was calculated to determine if there was a significant difference in the performance of the students at the four grade levels for each of the three measurements. A level of .05 significance was necessary in order to reject each null hypothesis. --------------- Chapter 4 ANALYSIS OF DATA Elementary school students were engaged in a hands-on problem-solving activity (HOPSA) that involved giving each student a limited amount of materials to build the tallest possible structure within a given period of time. Following are the results of the statistical analysis of the data for each of the three null hypotheses tested. Null Hypothesis One The first null hypothesis stated that there would be no significant relationship between the grade level of the student and the maximum height of the structure. Maximum structure heights were measured for each student. The heights recorded for the students were compared across the four grade levels studied (see Figure 2). The data for each student are presented in APPENDIX E. An analysis of variance (ANOVA) was calculated to determine if there was a significant difference across the four grade levels. The result of this ANOVA is presented in Table 1. The result of this ANOVA was F = 6.98. Through the use of a table that shows the critical values of the F distribution (Best & Kahn, 1989), the significance level was found to be less than the .01 level. This level showed a greater significance than the .05 level required to reject the null hypothesis. Table 1 Mean Maximum Structure Heights, F value, and Probability at the Four Grade Levels Grade Mean Height F value Probability First 5.94 Second 6.88 6.98* <.01* Fourth 13.69 Sixth 26.19 Note. The mean heights are in decimal inches. * significant. There was a significant relationship between the grade level of the student and the maximum height of the structure. Specifically, the maximum structure height increased significantly as the grade level increased. Null Hypothesis Two The second null hypothesis stated that there would be no significant relationship between the grade level of the student and the length of the initial-analysis phase. The length of the initial-analysis phase was measured for each student. The times recorded for the students were compared across the four grade levels studied (see Figure 3). The data for each student are presented in APPENDIX F. An analysis of variance (ANOVA) was calculated to determine if there was a significant difference across the four grade levels. The result of this ANOVA is presented in Table 2. Table 2 Mean Lengths of the Initial-Analysis Phase, F value, and Probability at the Four Grade Levels Grade Mean Length F value Probability First 0.55 Second 0.64 1.10* >.10* Fourth 0.26 Sixth 0.40 Note. The mean lengths are in decimal minutes. * not significant. The result of this ANOVA was F = 1.10. Through the use of a table that shows the critical values of the F distribution (Best & Kahn, 1989), the significance level was found to be greater than the .10 level. This level did not show a greater significance than the .05 level required to reject the null hypothesis. There was no significant relationship between the grade level of the student and the length of the initial-analysis phase. Null Hypothesis Three The third null hypothesis stated that there would be no significant relationship between the grade level of the student and the frequency of occurrence of a single geometric shape used in the structure. The frequency of occurrence of a single geometric shape was determined for each student. The frequencies recorded for the students were compared across the four grade levels studied (see Figure 4). The data for each student are presented in APPENDIX G. An analysis of variance (ANOVA) was calculated to determine if there was a significant difference across the four grade levels. The result of this ANOVA is presented in Table 3. The result of this ANOVA was F = 0.54. Through the use of a table that shows the critical values of the F distribution (Best & Kahn, 1989), the significance level was found to be greater than the .10 level. This level did not show a greater significance than the .05 level required to reject the null hypothesis. There was no significant relationship between the grade level of the student and the frequency of occurrence of a single geometric shape used in the structure. Table 3 Mean Frequencies of Occurrence of a Single Geometric Shape, F value, and Probability at the Four Grade Levels Grade Mean Frequency F value Probability First 3.13 Second 6.38 0.54* >.10* Fourth 2.88 Sixth 3.38 * not significant. ------------ 60 50 40 30 20 10 0 First Second Fourth Sixth Grade Level Figure 2. Comparison of the Maximum Structure Heights. Each black dot represents each student's maximum structure height. The gray dots represent the mean maximum structure height for each grade level. The lines connect the mean values. ------------ Chapter 5 SUMMARY This study was designed to investigate the performance of elementary-school students on a particular hands-on problem-solving activity (HOPSA), the Tower Problem. It was anticipated that the performance would increase as the grade level increased. It was also anticipated that the students observed in this study would exhibit some traits that were characteristic of one of Piaget's stages of cognitive development. The HOPSA used in this study was the Tower Problem, number M 1.1 in Module 1, found in the TE 510 course curriculum at North Carolina Agricultural and Technical State University. This particular HOPSA was chosen for this study because it was easy for students of all ages to understand, yet it still provided enough of a challenge for older students. The activity involved giving each student a limited amount of materials to build the tallest possible structure within a given period of time. The following null hypotheses were tested: Ho 1. There is no significant relationship between the grade level of the student and the maximum height of the structure. Ho 2. There is no significant relationship between the grade level of the student and the length of the initial-analysis phase. Ho 3. There is no significant relationship between the grade level of the student and the frequency of occurrence of a single geometric shape used in the structure. The students studied included 8 each from the first-, second-, fourth-, and sixth-grade levels, for a total of 32, from the Indiana State University School. The students were selected using a stratified random sample to insure a representative sample of academic-achievement levels as well as an even gender ratio. Three measurements were made for each student. They were the (a) length of the initial-analysis phase, (b) maximum structure height, and (c) frequency of occurrence of a single geometric shape. An analysis of variance (ANOVA) was calculated to determine if there was a significant difference in the performance of the students at the four grade levels for each of the three measurements. The results of this statistical analysis were as follows: 1. As the grade level increased, there was a significant increase in the maximum structure height. 2. As the grade level increased, there was no significant change in the length of the initial-analysis phase. 3. As the grade level increased, there was no significant change in the frequency of occurrence of a single geometric shape used in the structure. Conclusions Several conclusions were made regarding this study. They included conclusions pertaining to each of the three null hypotheses and conclusions regarding the results of this study appearing to show evidence of characteristics of Piaget's stages of cognitive development. Null Hypothesis One The ability of elementary-school students in this study to construct a tall structure with limited resources increased with an increase in grade level. The null hypothesis was rejected. Null Hypothesis Two The length of the initial-analysis phase of elementary-school students in this study constructing a tall structure with limited resources had no significant relationship with the grade level. The null hypothesis was accepted. Null Hypothesis Three The frequency of occurrence of a single geometric shape in a tall structure constructed with limited resources by elementary-school students in this study had no significant relationship with the grade level. The null hypothesis was accepted. Characteristics of Piaget's Stages It appeared that 1/2 of the first-grade students were able to perform concrete-operational tasks as evidenced by their performance in manipulating concrete materials. According to their age, however, they should have been in the final stages of the preoperational stage. The other 1/2 of the first graders, who appeared to be performing at the preoperational level, simply could not handle the concrete operations necessary to build a structure. Simply crumpling a single piece of paper into a ball would have resulted in a higher structure than they produced (Davis, 1983). All of the second- and fourth-grade students appeared to be performing at the concrete-operational level as evidenced by their performance in manipulating the concrete materials. This was the stage they should have been in at this age (Davis, 1983). The sixth graders had, according to their age, just entered the formal-operations stage of cognitive development. It was assumed that their significantly taller structures were evidence of an ability to conceive of the many possible ways to proceed in this activity and an ability to use abstract reasoning that did not limit the possible uses of the materials (Davis, 1983). Discussion Two of the three null hypotheses were accepted as a result of the statistical analysis of the data. That was contrary to the anticipated results of the study. Some possible explanations are discussed here. Null Hypothesis Two The lengths of the initial-analysis phase for all of the students were surprisingly short. Some possible explanations follow. The observers commented to the investigator that giving the students a pair of scissors seemed to signal a need to start cutting the paper without any thought as to what they were doing. The observers said that the students seemed to be analyzing the problem while they were cutting the paper rather than analyzing first before cutting. In this study, the initial-analysis phase ended if they started cutting the paper. The students may not have realized that 20 minutes was more than enough time to plan and build a tall structure. They may have rushed the initial-analysis phase by thinking that they could not afford the time to stop and analyze the situation. In most of the computer and video games that were popular at the time of this study, there was never a chance to stop and analyze the game in order to plan the best move. The game player had to react within seconds to keep from bringing the game to an end. It was of no consequence, however, to end the game since it could easily be restarted. High scores to many games did not require an ability to analyze a situation, only an ability to remember a correct sequence of moves learned from many incorrect trials. The students needed to learn that this random trial and error method of solving problems would not always work in the real world. Null Hypothesis Three Some of the students made towers which consisted of many different geometric shapes which were stacked on top of each other. Due to the procedure used of measuring a repetition of only a single geometric shape, this multiple-shape approach caused some students to have lower scores on that measure than they might have had if some other procedure was used to count the shapes. One structure included over thirty small squares of paper stacked on top of each other as part of the structure. Even though this instance skewed the results at that grade level for this measure, the elimination of this student's score would not have significantly altered the results. Some of the students seemed to be preoccupied with assembling creative shapes or shapes that actually represented real objects such as helicopters, airplanes, castles, and fish. These students spent their time making sure their structure represented something real rather than trying for the absolute tallest structure they could build irrespective of form. Recommendations for Future Research Recommendation One The length of the initial-analysis phase and the frequency of occurrence of a single geometric shape did not show a significant change as the grade level increased when this activity was performed by elementary-school students. Therefore, it is recommended that middle- and high-school students be observed performing this same activity to see if there is a significant change in the length of the initial-analysis phase and the frequency of occurrence of a single geometric shape among students older than those in this study. Recommendation Two This study was a only a descriptive study of students engaged in an activity at four grade levels. It was not used to measure any learning which might have occurred as a result of some treatment. Therefore, it is recommended that the same activity be performed as an introduction to teaching the scientific method of solving problems. The lesson could then be concluded with the same or similar HOPSA, and the results from the two activities could be compared. Recommendation Three It is possible that the presence of a pair of scissors caused the students to begin cutting the paper before they had initially analyzed the problem. Therefore, it is recommended that the same activity be performed without giving the students a pair of scissors but allowing them to tear the paper instead. A straightedge could be provided without explanation to assist in making straight tears. Recommendation Four This study measured a repetition of only a single geometric shape and did not take into account students who successfully used a collection of different geometric shapes in their structure. Therefore, it is recommended that a different procedure be used to measure shapes which would allow for a use of multiple types of shapes. Recommendation Five Some of the lower-achievement-level students, as evidenced by their low ISTEP scores, had structures that were higher than those of the students in the higher-achievement level. The four lower-achievement-level students at the fourth-grade level scored higher than 3/4 of the higher-achievement level students at the same grade level. Therefore, it is recommended that a study be conducted to compare the performance of students of lower-achievement levels, as evidenced by low standardized-testing scores, with students of higher-achievement levels on similar HOPSAs which do not require the use of reading, writing, or arithmetic. ------------------ REFERENCES CITED REFERENCES CITED Alpert, A. (1928). The solving of problem situations by preschool children: An analysis. Teachers College Contributions to Education, 32, 69. Best, J. W., & Kahn, J. V. (1989). Research in education. Englewood Cliffs, NJ: Prentice Hall. Bloom, B. S. (Ed.). (1956). Taxonomy of educational objectives: The classification of educational goals, by a committee of college and university examiners. New York: Longmans, Green. Brightman, H. J. (1980). Problem solving: A logical and creative approach. Atlanta: Georgia State University. Bruner, J. S. (1960). The process of education. Cambridge: Harvard University Press. Davis, G. A. (1983). Educational psychology: Theory and practice. Reading, MA: Addison-Wesley. DeLuca, V. W. (1991). Implementing technology education problem-solving activities. Journal of Technology Education, 2(2), 5-15. Dewey, J. (1933). How we think: A restatement of the relation of reflective thinking to the educative process. Boston: D. C. Heath. Fales, J. F., Kuetemeyer, V. F., & Brusic, S. A. (1988). Technology: Today and tomorrow. Mission Hills CA: Glencoe. Gibson, E. J., & McGarvey, H. R. (1937). Experimental studies of thought and reasoning. The Psychological Bulletin, 34(6), 327-350. Glaser, E. M. (1941). An experiment in the development of critical thinking. New York: J. J. Little & Ives. Harter, G. L. (1930). Overt trial and error in the problem solving of preschool children. Journal of Genetic Psychology, 38, 361-372. Heidbreder, E. F. (1928). Problem solving in children and adults. Journal of Genetic Psychology, 35(4), 522-545. Levitt, E. E., & Truumaa, A. (1972). The Rorschach technique with children and adolescents. New York: Grune & Stratton. Matheson, E. (1931). A study of problem solving in pre-school children. Child Development, 2, 242-262. Miller, W. R., & Boyd, G. (1970). Teaching elementary industrial arts. South Holland IL: Goodheart-Willcox. Orme, D. R. (1986). Rorschach erlebnistypus and problem-solving styles in children. Unpublished doctoral dissertation, Indiana State University, Terre Haute. Roberts, K. E. (1933). Learning in preschool and orphanage children. University of Iowa Studies, Studies in Child Welfare, 7(3), 94. Rorschach, H. (1964). Psychodiagnostics: A diagnostic test based on perception. New York: Grune & Stratton. Sellwood, P. (1989). The role of problem solving in developing thinking skills. The Technology Teacher, 49(3), 3-10. Todd, R., & Hutchinson, P. (1991, January/February). Design & technology: Good practice and a new paradigm. Ties Magazine, pp. 4-11. Wadsworth, B. J. (1979). Piaget's theory of cognitive development: An introduction for students of psychology and education. New York: Longman. Winter, P. (1990, November/December). Polypopagons. Ties Magazine, pp. 15-22. ------------ APPENDIXES APPENDIX A INSTRUCTIONS TO THE STUDENT You have been given four sheets of paper, some tape and a pair of scissors. Using only the materials I have given you, you are to make the tallest vertical structure possible. No part of the structure may touch the ceiling, furniture, chairs, walls, or yourself. The structure may only touch the floor. You will have 20 minutes in which to work. You do not have to use the entire 20 minutes. Let us know if you finish early. The structure must stand by itself for 15 seconds before it can be measured. You can have your structure measured as many times as you would like. You must stay in your own work area. You may not look at the other students work. I will tell you when you have 10, 5, and 2 minutes left to work. Do you have any questions? You may go to your work area and begin working. --------- APPENDIX B INSTRUCTIONS TO THE OBSERVERS Materials Needed by Each Observer 1. A tape measure that is graduated in inches. 2. A clock or watch that measures minutes and seconds. 3. An observation sheet. 4. A pen to record observations. 5. A copy of the "Instructions to the Student." Procedure 1. The researcher will assign an observer to each individual work area. 2. The researcher will assign an individual work area to each student. 3. The researcher will ask each student if they are left or right handed in order to provide the proper type of scissors. 4. The paper, tape, and scissors will be placed in the students' work areas by the researcher. 5. The instructions will be read to the students by the researcher. 6. The students will proceed to their individual work areas. 7. The observer will record the time, in minutes and seconds, when the student they are observing enters the assigned work area. This will occur after the researcher says "You may go to your work area and begin working." 8. The observer will record all of the student's activities and the time, in minutes and seconds, of their occurrence on the observation sheet. Activities will include folding or cutting a piece of paper, cutting or using a piece of tape, placing a paper shape on the floor in order to begin construction, or connecting one shape to another. The first activity to be recorded is the end of the instructions as the student enters the work area. The last activity to be recorded will be when the height of the structure is measured for the last time. The observer will also record activities involving distractions with other objects in the testing area. 9. The observer will record the structure height in inches rounded to the nearest 1/4 inch. The measurement will be made perpendicular to the floor after the student's structure has stood for 15 seconds to insure stability. The structure may be measured more than once during the activity. 10. The observer will list the number of and type of geometric shapes used in the structure each time its height is measured. Such geometric shapes include, but are not limited to, rectangular prisms, hexahedrons (cubes), triangular prisms, triangular pyramids, square pyramids, cylinders, and cones. Shapes that are the same except for the physical size, are to be considered the same shape. A separate piece of paper constitutes a separate shape, unless the separate piece of paper is not a regular shape and was combined with one or more pieces of paper to form a shape. 11. The observer will reinforce the rules to the student when deemed necessary (e.g., the structure may only touch the floor, the student must stay in the work area). ------------ APPENDIX C OBSERVATION SHEET Student's Name page Classification Time Observed Activity ------------ APPENDIX D RAW SCORES FOR THE LENGTH OF THE INITIAL-ANALYSIS PHASE Rater Rater Student One Two Student One Two 1A1 0:50 0:50 4A1 0:15 0:15 1A2 0:25 0:25 4A2 0:00 0:00 1B1 0:58 0:58 4B1 0:52 0:52 1B2 0:20 0:20 4B2 0:10 0:10 1C1 0:52 0:52 4C1 0:16 0:16 1C2 0:00 0:00 4C2 0:15 0:15 1D1 0:00 0:00 4D1 0:19 0:19 1D2 1:00 1:00 4D2 0:00 0:00 2A1 0:16 0:16 6A1 0:10 0:10 2A2 0:00 0:00 6A2 0:30 0:30 2B1 0:30 0:30 6B1 1:35 1:35 2B2 0:30 0:30 6B2 0:00 0:00 2C1 0:10 0:10 6C1 0:14 0:14 2C2 1:00 1:00 6C2 0:10 0:10 2D1 1:10 1:10 6D1 0:31 0:31 2D2 1:30 1:30 6D2 0:00 0:00 ----------- APPENDIX E MAXIMUM STRUCTURE HEIGHTS AND ANOVA CALCULATION First grade Second grade Fourth grade Sixth grade X1 X12 X2 X22 X3 X32 X4 X42 A1 5.00 25.00 12.50 156.25 11.00 121.00 40.50 1640.25 A2 23.00 529.00 5.25 27.56 9.50 90.25 24.00 576.00 B1 1.00 1.00 5.50 30.25 23.00 529.00 25.75 663.06 B2 2.00 4.00 8.50 72.25 3.25 10.56 52.50 2756.25 C1 0.50 0.25 2.50 6.25 16.00 256.00 10.00 100.00 C2 4.00 16.00 9.50 90.25 11.75 138.06 14.25 203.06 D1 0.00 0.00 6.50 42.25 15.00 225.00 3.00 9.00 D2 12.00 144.00 4.75 22.56 20.00 400.00 39.50 1560.25 … 47.50 719.25 55.00 447.63 109.50 1769.88 209.50 7507.88 X 5.94 6.88 13.69 26.19 X = 13.17 …X = 421.50 …X2 = 10444.63 SSt = 4892.68 SSb = 2093.27 dfb = 3 SSw = 2799.41 dfw = 28 F = 6.98 The heights are shown in decimal inches. A1,2 are male students with high ISTEP scores. B1,2 are female students with high ISTEP scores. C1,2 are male students with low ISTEP scores. D1,2 are female students with low ISTEP scores. ------------- APPENDIX F LENGTHS OF THE INITIAL-ANALYSIS PHASE AND ANOVA CALCULATION First grade Second grade Fourth grade Sixth grade X1 X12 X2 X22 X3 X32 X4 X42 A1 0.833 0.69 0.267 0.07 0.250 0.06 0.167 0.03 A2 0.417 0.17 0.000 0.00 0.000 0.00 0.500 0.25 B1 0.967 0.94 0.500 0.25 0.867 0.75 1.583 2.51 B2 0.333 0.11 0.500 0.25 0.167 0.03 0.000 0.00 C1 0.867 0.75 0.167 0.03 0.267 0.07 0.233 0.05 C2 0.000 0.00 1.000 1.00 0.250 0.06 0.167 0.03 D1 0.000 0.00 1.167 1.36 0.317 0.10 0.517 0.27 D2 1.000 1.00 1.500 2.25 0.000 0.00 0.000 0.00 … 4.417 3.67 5.101 5.21 2.118 1.08 3.167 3.13 X 0.55 0.64 0.26 0.40 X = 0.46 …X = 14.80 …X2 = 13.09 SSt = 6.24 SSb = 0.66 dfb = 3 SSw = 5.58 dfw = 28 F = 1.10 The times are shown in decimal minutes. A1,2 are male students with high ISTEP scores. B1,2 are female students with high ISTEP scores. C1,2 are male students with low ISTEP scores. D1,2 are female students with low ISTEP scores. ------------ APPENDIX G FREQUENCIES OF OCCURRENCE OF A SINGLE GEOMETRIC SHAPE AND ANOVA CALCULATION First grade Second grade Fourth grade Sixth grade X1 X12 X2 X22 X3 X32 X4 X42 A1 1 1 1 1 1 1 4 16 A2 5 25 2 4 3 9 3 9 B1 1 1 36 1296 3 9 3 9 B2 8 64 1 1 4 16 5 25 C1 1 1 5 25 3 9 1 1 C2 7 49 4 16 3 9 2 4 D1 0 0 1 1 4 16 5 25 D2 2 4 1 1 2 4 4 16 … 25 145 51 1345 23 73 27 105 X 3.13 6.38 2.88 3.38 X = 3.94 …X = 126 …X2 = 1668 SSt = 1172 SSb = 64.4 dfb = 3 SSw = 1108 dfw = 28 F = 0.54 A1,2 are male students with high ISTEP scores. B1,2 are female students with high ISTEP scores. C1,2 are male students with low ISTEP scores. D1,2 are female students with low ISTEP scores.