{"http:\/\/dx.doi.org\/10.14288\/1.0055069":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Education, Faculty of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Minkowitz, Goldie","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2010-02-26T00:50:20Z","type":"literal","lang":"en"},{"value":"1978","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Master of Arts - MA","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"This study was undertaken to compare the effectiveness of two problem-solving strategies at the grade six and seven levels. Both strategies were designed to aid students in associating the verbal statement of a problem with its corresponding\r\nmathematical equation. One approach, the Translation Method, stressed literal, carefully structured translation of word problems, while the second, the Inductive Method, encouraged students to create their own problems, using mathematical\r\nequations given by the teacher. A control group practiced word problems without any instructional guidance.\r\nForty-eight students from the sixth and seventh grades of a private elementary\r\nschool in Vancouver, British Columbia were combined and assigned to the three treatment groups on the basis of their performance on a pretest in translation. For a period of four school days, all subjects used materials prepared by the investigator.\r\nTwo criterion measures were used. Posttest One was composed of traditional\r\nword problems requiring only one mathematical operation for the correct solution. Posttest Two was constructed with novel or challenging word problems requiring more than one operation for the correct solution. Each test contained eight items and was designed for one forty-minute period. Scores of the tests were analyzed using multivariate analysis of variance for the two dependent measures. The three factors considered were Treatment, Sex, and Grade, and a simple main effects analysis was employed to examine male-female differences within each treatment level.\r\nStatistical comparisons among the three groups offered no evidence of superiority\r\nfor one approach over another. In addition, no interaction was found between treatment and sex. Boys were found to be significantly superior to girls in performance\r\non the posttests. Further analysis indicated that Posttest One scores for the Translation Group students differed significantly between boys and girls, with the girls' performance particularly weak for this measure.\r\nSubjective observation revealed differences in attitude. Students found the Translation Method burdensome. Students in the Inductive Group enjoyed that approach,\r\nand students in the Control Group seemed interested in the practice sequence of word problems.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/20988?expand=metadata","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"A COMPARISON OF THREE APPROACHES TO PROBLEM SOLVING IN GRADES SIX A N D SEVEN by GOLDIE MINKOWITZ B . A . , Brooklyn College of the City University of New York, 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS THE FACULTY OF GRADUATE STUDIES Mathematics Department Faculty of Education \u00bbWe accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA -September, 1978 @ Goldie Minkowitz, 1978 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of Brit ish Columbia 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 Date SyU^k+r jt 197? ABSTRACT This study was undertaken to compare the effectiveness of two problem-solving strategies at the grade six and seven levels. Both strategies were designed to aid students in associating the verbal statement of a problem with its correspond-ing mathematical equation. One approach, the Translation Method, stressed literal, carefully structured translation of word problems, while the second, the Inductive Method, encouraged students to create their own problems, using mathe-matical equations given by the teacher. A control group practiced word problems without any instructional guidance. Forty-eight students from the sixth and seventh grades of a private elemen-tary school in Vancouver, British Columbia were combined and assigned to the three treatment groups on the basis of their performance on a pretest in translation. For a period of four school days, all subjects used materials prepared by the i n -vestigator. Two criterion measures were used. Posttest One was composed of tradi-tional word problems requiring only one mathematical operation for the correct solution. Posttest Two was constructed with novel or challenging word problems requiring more than one operation for the correct solution. Each test contained eight items and was designed for one forty-minute period. Scores of the tests were analyzed using multivariate analysis of variance for the two dependent measures. The three factors considered were Treatment, Sex, and Grade, and a simple main i i effects analysis was employed to examine male-female differences within each treatment level. Statistical comparisons among the three groups offered no evidence of superi-ority for one approach over another. In addition, no interaction was found between treatment and sex. Boys were found to be significantly superior to girls in per-formance on the posttests. Further analysis indicated that Posttest One scores for the Translation Group students differed significantly between boys and girls, with the girls' performance particularly weak for this measure. Subjective observation revealed differences in attitude. Students found the Translation Method burdensome. Students in the Inductive Group enjoyed that ap-proach, and students in the Control Group seemed interested in the practice sequence of word problems. i i i TABLE OF CONTENTS Chapter Page 1 THE PROBLEM 1 Learning Assessment Projects 2 Problem-Solving Strategies 4 Sex Differences 9 Definition of Terms 10 Statement of the Problem 11 Treatment-Related Research Questions 11 Sex-Related Research Questions 12 Grade-Related Questions 12 2 REVIEW OF RELATED LITERATURE 13 Research of General Relevance to Translation-oriented Strategies 13 Literature Relevant to the Translation Method 17 Literature Relevant to the Inductive Method 19 Comparisons of Problem-Solving Strategies 24 Literature Relevant to Sex Differences 31 Summary of Review of Related Literature 38 3 DESIGN A N D PROCEDURE 40 Sample 40 DEVELOPMENT OF MEASURING INSTRUMENTS 41 Word Problem Pilot Test 41 Translation Pretest 42 Posttest One 43 Posttest Two 44 DESCRIPTION OF INSTRUCTIONAL TREATMENTS 46 Treatment A : Transiatioha'Metihod 46 Treatment B : Inductive Method 47 Treatment C : Control Method 49 iv TABLE OF CONTENTS - continued Chapter Page DEVELOPMENT OF INSTRUCTIONAL MATERIALS 50 DESIGN 50 CONTROLS 51 Problem-Solving Background 51 Teachers 52 Treatments 52 ASSIGNMENT TO GROUPS 53 PROCEDURE 54 STATISTICAL ANALYSIS 56 Data 56 Analysis of Tests 57 Analysis of Posttest Results 57 Simple Main Effects Model 60 Statistical Assumptions 61 4 RESULTS OF THE STUDY 63 Analysis of Test Reliabilities 63 Analysis ?of Variance for Pretest 64 RESULT OF RESEARCH HYPOTHESES 65 Homogeneity of Variance 65 Hypothesis One : Treatment Differences 68 Hypothesis Two : Treatment-Sex Interaction 71 Hypothesis Three : Sex Differences 71 Hypotheses Four and Five : Sex Differences Within Tests 73 Hypotheses Six, Seven, and Eight : Sex Differences Within Treatment Levels Using the Simple Main Effects Model 74 Hypothesis Nine : Grade Differences 78 v TABLE OF CONTENTS - continued Chapter Page STUDENT REACTION TO TREATMENTS 78 Treatment A : Translation Method 79 Treatment B : Inductive Method 79 Control Group 80 5 SUMMARY, CONCLUSIONS, A N D IMPLICATIONS 81 Summary 81 Conclusions and Discussion 81 Limitations 88 Implications for the Classroom 88 Suggestions for Further Research 89 BIBLIOGRAPHY 92 APPENDIX A - Word Problem Pilot Test and Results 101 APPENDIX B - Translation Pretest 104 APPENDIX C - Posttest One 107 APPENDIX D - Posttest Two 110 APPENDIX E. ~. Treatment A : Translation Method 115 APPENDIX F - Treatment B : Inductive Method 131 APPENDIX G - Treatment C : Control Method 143 APPENDIX H - Item and Test Statistics 149 APPENDIX I - Summary of Data 156 vi LIST OF TABLES Table Page 2.1 Sex Differences in Mathematics Achievement Upper Elementary and High School Years 33 2.2 Sex Differences in Achievement: Grades 4-8 NLSMA: X-Population 36 2.3 Sex Differences in Achievement : Grades 7-10 NLSMA: Y-Population 36 4.1 Analysis of Variance for the Pretest 66 4.2 Summary Data for Three Tests 67 4.3 Homogeneity of Variance 69 4.4 Means of Treatment Groups 69 4.5 Results of Multivariate Analysis of Variance 70 4.6 Means of Treatment-By-Sex Levels 72 4.7 Means; of Male Students Compared to Female Students 72 4.8 Results of Univariate Analysis of Variance by Sex Differences 75 4.9 Comparison of Male-Female Differences for Each Treatment.. 77 4.iT0 Univariate F-Ratio for Treatment A Sex Differences 77 Appendices A . l Scores of Students on Word Problem Pilot Test 102 E.l Translations 117 H.3 Item Reliability Statistics for Word Problem Pilot Test 149 H.2 Item Reliability Statistics for the Translation Pretest 150 H.3 Item Reliability Statistics for Posttest One 153 H. 4 Item Reliability Statistics for Posttest Two 154 I. 1 Summary Data for Three Tests 156 LIST OF FIGURES Figure 3.1 Outline of Study 51 3.2 Treatment Sequence 55 3.4 Procedure 56 v i i ^ACKNOWLEDGEMENTS The author wishes to express gratitude and appreciation to Dr. Douglas Owens, chairman of her committee, for his unfailing guidance and generous personal involvement. Special thanks are also due to the members of the committee, Dr. Ga i l Spitler and Dr. James Sherrill, for their insight and constant encouragement, to Dr. Todd Rogers and Tom O'Shea for their time and effort as statistical consultants, to Ed O'Regan for his advice and help, to Eric Lee and the staff of the Vancouver Talmud Torah for their kind cooperation, and especially to Raphael Minkowitz for his patience and constant support and encouragement. v i i i Chapter 1 THE PROBLEM Problem solving in mathematics in its most advanced form represents the application of mathematical concepts, formulas, and logic within all aspects of l i fe. In fact, the goal of mathematical education has often been viewed as enabling students to use mathematical tools to solve the problems and challenges that they face outside the classroom. Mathematicians of the Cambridge Con-ference on School Mathematics (Goals for School Mathematics, 1963) urged curriculum developers to devote more time and energy to the creation of problem sequences, particularly those which introduce new mathematical ideas. The kind of problem solving which takes place in the schools, however, tends to be more of an extension of arithmetical examples into the domain of word problems. Yet even in this context, the \"simple\" word problems enable students to gain experience with the process of problem solving and to deduce principles which they will later need in coping with more complex problems. Wilson (1967), in his content taxonomy for mathematics, divides the cognitive skills of mathematics into the four hierarchical domains of computation, compre-hension, applications, and analysis. He places the solution of routine problems in the domain of applications, and the solution of nonroutine problems in the domain of analysis. Thus, problem solving demands the highest order cognitive skills that students may develop. These skills are described by Cohen and Johnson (1967) as \"observing, exploring, decision-making, organizing, recogniz--ing, remembering, supplementing, regrouping, isolating, combining, diagramming, 1 2 guessing, classifying, formulating, generalizing, verifying, and applying\" (p. 261). It is hardly surprising that mathematical word problems at all levels frus-trate many students. Current learning assessment projects, discussed below, show that weak-ness in problem solving, particularly solving problems of a more complex nature than the one-step problems using whole numbers, is a widespread characteristic of junior high school and high school students. Learning Assessment Projects One such project, the National Assessment of Educational Progress (NAEP), (Carpenter, Coburn, Reys, and Wilson, 1975) was conducted in 1972-73 with 90 000 individuals participating at four age levels: students aged 9, 13, and 17, and young adults between 26 and 35 years of age. The abilities tested ranged from recall through analysis or problem solving in a variety of content areas. One example was: A sports car owner says he gets 22 miles per gallon of gasoline. How many miles could he go on seven gallons of gasoline? It was found that for this problem, 89 percent of the 17-year-olds and 90 percent of the adults obtained the correct solution. However, an example which was more complex, such as the following, did not produce such encouraging results: Weathermen estimate that the amount of water in nine inches of snow is the same as one inch of rainfall. A certain Arctic island has an annual snowfall of 1.63 inches. Its annual snow-fall is the same as an annual rainfall of how many inches? 3 In this case the correct answer was found by 31 percent of the 13-year-olds, 53 percent of the 17-year-olds, and 58 percent of the adults. When the indi -viduals who attempted to use division but did not obtain the correct solution are included, only about half of the 13-year-olds could analyze the problem suf-ficiently to determine the appropriate operation, and about three-fourths of the 17-year-olds and adults could do so. These results were similar to those found on other exercises. A l l problems were read to students by a tape recording in order to minimize the effect of poor reading abil ity. Another project, the British Columbia Mathematics Assessment project (Robitaille and Sherrill, 1977), was undertaken in 1977 and over 100 000 students from grades 4, 8 and 12 were tested, using the three domains of computation, comprehension, and applications for a number of different content areas. The re-sults for problem solving in grade 12 were found to be \"disappointing^\" indicating that many students are unable to apply the computational skills they have learned to certain types of problems, especially in geometry and measurement. Although certain difficulties were again experienced with more complex, multi-step prob-lems, results for the grade 8 students concerning problem solving were satisfactory overa 11. For both of these assessments, the researchers summarize their findings by urging that greater emphasis be placed on problem solving. The NAEP organizers report, \"As a whole, these age groups need to develop more problem-solving skills\" (Carpenter et a l , p. 470). Robitaille and Sherrill (1977) of the British Columbia Mathematics Assessment recommend: \"Teachers and teacher educators need to stress the overriding importance of problem solving in mathematics, and 4 their students need to learn strategies to use in attempting to solve problems in mathematics\" (p. 33). Problem-Solving Strategies Henderson and Pingry(1970) suggest, \"Unless students study the process of solving problems as an end in itself, there is scant likelihood that they will learn the generalizations which will enable them to transfer their ability to solve problems to new problems as they arise\" (p. 233). Viewed from this perspective it would seem natural that at least a moderate amount of time should be devoted to teaching students some useful method for solving problems. Yet Sti I wei I (1967) found that a relatively small amount of class time was devoted to discussing a problem-solving method for general use: less than three percent of all problem-solving time! Teachers seem to avoid teaching a general strategy which could apply to al l kinds of verbal problems and instead rely on a collection of \"problem-type\" strategies, each pertaining to one particular type of problem only. This method of attack has been criticized for being the least transferable for students of al l popular techniques. Students who are taught rate problems, or interest problems, for example, are limited to solving only those types of problems (Spitler, 1976). One of the most widely known general strategies was developed by Polya (1957). In How to Solve It, Polya suggested a step-by-step solution, basically heuristic in nature, which involves reading to understand the problem and planning for a solution using the following questions as a guideline: 5 1. What is the unknown? 2 . What are the given facts? 3. What condition relates the unknown to the data given? Finding a plan for the solution involves the use of heuristics such as analogies, solving part of the condition, and other strategies, with examining the obtained solution as a final step. This process has become known as the wanted-given approach. Difficulties in attempting to use Polya's checklist to analyze the problem-solving procedures of 56 eighth grade students of above average ability were re-ported by Ki I patrick (1967): \". . .Whatever merits Polya's list has for teaching problem solving, it is of limited usefulness, as it stands, for characterizing the behavior of these subjects. Many of the categories were unoccupied; subjects seemingly did not exhibit behavior even remotely resembling actions sug-gested by the heuristic questions\" (p. 44). Polya's strategies are of unchallenged value in the construction of a general problem-solving model. Yet, to a large extent, this model is intended for the solution of nonroutine and challenging problems, and it may not be as directly useful for students who are having difficulty with simpler routine problems. An alternate problem-solving strategy was suggested by Maurice Dahmus (1970) in a paper presented to the Central Association of Science and Mathematics Teachers at a convention in 1970. This method is directed towards the lower 90 percent of mathematics classes who seem to lack a viable approach to solving word problems. Students are asked to record each phrase of a problem in mathematical terms without first reading through the entire problem. The series of resulting 6 symbols and equations are then combined by substitution methods until only the final equation or system of equations remains; to be solved. Dahmus characterizes his method as \"DPPC\": D-direct: proceed directly from the order of the problem's statement, P-piecemeal: translate piece by piece as you read, P-pure: no operations done before the translation is complete, C-complete: all facts, ideas, and questions used for the solution to be mathematically recorded. This approach has become known as the translation method. Dahmus' method of literal translation is supported by a large body of re -search which links verbal skills to problem-solving abilities. Studies of the language factor in mathematics have been concerned with the possibility that verbal skills have as much influence as\u2014or even more than\u2014computational ability on problem solving (Martin, 1964; Balow, 1964; Knifong & Holton, 1976, among others). Structural analysis of word problems to determine those components which contribute the most to errors have indicated that linguistic factors occupy positions of primary importance in the determination of the difficulty of the problem (Jerman & Rees, 1972; Segal la , 1973; Cook, 1973, among others). Training to increase mathematical vocabulary is one example of a language-based program which has helped students to significantly increase their problem-solving abilities (Dresher, 1834; Johnson, 1944; and Vanderlinde, 1964). 7 Indeed, most of the pages of a booklet prepared by the International Read-ing Association entitled \"Teaching Reading and Mathematics\" (Earle, 1976) are devoted to the steps necessary for students to perceive symbols and attach literal meanings. These steps are seen to be just as crucial to the problem-solving process as is the learning of new vocabulary to a student in literature. Cohen and John-son (1967) expressed the following sentiments in this regard: \"The ability to translate accurately from the written (or spoken) description of the physical situation to an appro-priate mathematical sentence enables a person to cope with a large number of problems in mathematics in an orderly, logical manner. The ability to translate a given situation into mathematical symbolism is considered to be the 'most useful tool in problem solving\"' (p. 262). Yet few studies have examined the potential of using a language-based strategy to approach the solution of word problems. The Dahmus translation method, while providing this direction for problem solving, is quite a rigorous strategy which minimizes the role of sudden insight or inspiration and the type of heuristics recommended by Polya. According to Kilpatrick (1967) and others, however, the assumption that problem solving in reality occurs in well defined, sequential stages is one which should be avoided. Burch (1953) conducted a study of formal analysis or the rigid sequence of ques-tions or steps to be followed before computational work is begun. He found that training elementary school children in formal analysis methods for solving problems was troublesome and confusing for them. Whether the Dahmus translation method is too rigid or too similar to formal analysis is open to question. An attempt to maintain flexibility and creativity on the part of students, while at the same time bridging the gap which exists between the verbal problem 8 and its corresponding mathematical statement, is found in a strategy suggested by Spitler (1976). As one of many suggested techniques for improving problem solving, she proposes the technique of having students create; their own problems from a given mathematical equation. Students are taught to associate certain situ-ational possibilities and different meanings for mathematical operations with a simple equation such as 10 = N 5. This association of real world context with symbolic equations should lead to deeper understanding of verbal problems, thus improving problem-solving abilities. The advantage of this approach, which is inductive rather than deductive, is that students who were directed in creative and divergent thinking patterns have shown increased abilities to accomplish divergent-type tasks. This was demonstrated to be something of a mixed advantage by Richards and Bolton (1971). They found that when a sample of students was tested on \"mechanical\" arithmetic tasks, children taught by discovery methods were significantly lower in performance than students taught by traditional or combined traditional-discovery methods. Yet on a test of divergent-thinking ability, the discovery and balanced methods were superior to the traditional group. The practical usefulness of a creative, inductive method for problem solv-ing, which entails training of students without the use of specific word problems has not been fully investigated. However, Suydam and Weaver (1977), in their summary of research on problem solving, report that creative or divergent thinking is a successful strategy. The available research relating to the testing of different instructional strategies has tended instead to focus on comparisons with some variant of Polya's 9 method against another deductive method. Wilson (1967), Jerman (1973), and Post and Brennan (1976), for example, contrasted some variation of the wanted-given method with some other strategy, obtaining differing results. One study, conducted by Bassler, Beers, and Richardson (1972), contrasted Dahmus1 transla-tion approach with the wanted-given strategy. No differences were found between two groups of ninth-graders on a solution criterion. No further comparisons could be found using the translation method,'and no experimental studies have been con-ducted with Spitler's inductive approach. It remains to be seen whether an inductive, creative method would be effective in the domain of novel problem situations. The question of whether such a method would also improve skills in solving simple word problems should be studied as well . These questions can also be considered for a deductive, formal translation strategy. Wil l literal translation prove useful in the solution of one-step word problems? If so, will it also help students who face complicated prob-lem-solving tasks? The questions indicate that a comparison of these two trans-lation-oriented strategies should be conducted using both a simple problem-solving criterion and a higher-level, more challenging measure of problem-solving skills. Sex Differences The study of Bassler, Beers, and Richardson (1972) did not consider sex differences because it was conducted in an all girls' school. Other researchers who have been able to use larger samples of male-female subjects have usually relied on randomization to remove any possible sex differences, and, in general, Suydam and Weaver (1977) concluded from research that sex is not an important 10 factor in problem solving. However, the report from the British Columbia Mathematics Assessment (Robitaille and Sherrill, 1977) shows that males outperformed females on all the problem-solving objectives, although most differences were small. This is paral-leled by the findings from the NAEP (Carpenter et a l , 1975), in which an analysis by sex of word problem results indicated that males generally did better than fe -males at al l ages. With these recent findings taken into consideration, it seems that the question of sex differences is sufficiently relevant to problem solving to be used as one of the factors in an analysis of the problem-solving strategies. In addi -tion, it was felt that if some interaction between treatment and sex could be found, this would be of particular interest. Definition of Terms For the purposes of this study, the following definitions of terms are out-lined for reference: The Translation Method refers to the strategy proposed by Dahmus which requires literal translation from the verbal statement of a problem to an appropriate mathematical equation. The Inductive Method is a program to aid students' problem solving by hav-ing them create their own word problems from a given mathematical equation. Problem-type Strategies refers to the collection of different methods for solving different kinds of word problems which is currently used by teachers, i . e . , one mode of attack for rate problems, one for work problems, and so forth. 1 One-step Problems are word problems which require only one mathematical operation for the correct solution. Multi-step Problems are word problems which require more than one mathe-matical operation for the correct solution. Statement of the Problem The purpose of this study is to compare the relative effectiveness of two methods for teaching problem solving, using a control group who practice word problems with no instruction as to method. Both strategies are designed to aid students in associating the verbal statement of a problem with its mathematical counterpart. One is a deductive, literal translation method, while the other is an inductive process which encourages creative thinking. At the same time, this study will evaluate the differential effect of the two strategies on male and female students, with the purpose of determining whether there is an interaction between problem-solving method and sex. Two criterion measures will be used. One represents traditional one-step word problems, and the other contains novel or more challenging multi-step prob-lems. Students at the grade 6 and grade 7 levels were selected for the study since it was felt that they would not have been instructed previously in any problem-solving method. Treatment-Related Research Questions Does learning a particular strategy for problem solving improve the per-formance of students more than providing an unstructured practice sequence without 12 instructional guidance? That is, will the groups learning the Translation Method or the Inductive Method do better than the control group on the simple or the complex posttest or both? Wil l one particular strategy for problem solving prove superior to the other on either or both of the posttests? Does an interaction exist between any of the treatment methods and the sex of the students? That is, will boys improve more under one treatment while girls improve more under another method? Sex-Related Research Questions Wil l grades six and seven boys perform better than girls, as research seems to indicate, on a standard problem-solving test? Wil l they prove superior on a non-standard problem-solving test as well? If the boys prove to be better at problem-solving than the girls, is this finding consistent through all treatments or will any one treatment help to equalize the performance of the sexes? Grade-Related Questions If there are significant differences between the achievement of the grade six and grade seven students, are these differences consistent across both sexes and all treatments? Wil l they be demonstrated in both posttests? Xhapter 2 'REVIEW OF RELATED LITERATURE Research relevant to a comparison between problem-solving strategies falls into three general categories. First, research is reported which serves as a general background to the need for linguistic approaches to mathematical problem solving. Next literature relevant to the Translation Method and then the Inductive Method is discussed. Finally, problem-solving comparisons with relevance to the present study are examined. In all categories, literature of general relevance precedes inspection of specifically related research. After literature in these three categories is examined, it will be necessary to examine some research dealing with sex differences. This research will be dealt with only briefly. Research of General Relevance to Translation-briehted Strategies Problem-solving ability has been correlated with verbal skills for several years. Martin (1964) found that the partial correlation between reading com-prehension and problem-solving abilities, with computational ability partialed out, was higher than the partial correlation between computational ability and problem solving with reading comprehension partialed out. Balow (1964), on the other hand, found that computational ability is more influential to successful prob-lem solving than is reading ability, but he suggests that general reading ability may have a greater effect on problem solving than the effect that he found in his study. 13 Harvin and Gilchrist (1970) concluded from their investigations that there exists, a positive relationship between problem-solving ability and reading abil ity. This relationship is not great enough to conclude that the second is a predictor of the first, but it is significant enough to suggest that arithmetic teachers should also teach those reading skills which are peculiar to the nature of mathe-matics. Structural analyses. Manheim (1961) states that \"the word problem remains a generator of fear and frustration for many students... If we ask our students, or ourselves, why such a problem is more difficult than a non-word problem, we are apt to find the difficulty attributed to 'the non-mathematical nature of the problem'\" (p. 234). Attempts to ascertain exactly where the difficulties lie in mathematical word problems have inspired structural analyses of problems in the last decade. Studies have been attempted to isolate those variables, whether linguistic or computational, which contribute most heavily to the errors found among students at different age levels in solving algebra word problems. One study was conducted by Cook (1973), who described 26 variables to which difficulty might be attributable. Some of these were drawn in turn from other studies (Krushinski, 1973, and Suppes, Jerman and Brian, 1968) and they were added to Cook's independently formulated variables. Cook analyzed the variables to determine which accounted for the most variance from the correct solutions to algebra word problems. The following results were reported, with variables listed in order of relative importance as steps along the regression line: 15 1. length of words in the problem statement; 2 . a \"translation\" variable\u2014signifying the number of un-knowns used in solution of other unknowns; 3 . recall of formulas; 4 . number of digits in the quotient of divisions; 5 . number of steps required to isolate the unknown once the equation is found; and 6. the number of operations necessary to solve the problem. The findings in this study suggest the significance of different linguistic factors and skills in the solution of word problems. Although the subjects were college students, the problems chosen are representative of the type of word problems such as distance problems, age problems, encountered by secondary school students. Jerman (1972, 1973, 1974) has been a most prominent figure in the area of structural analyses. One of his most comprehensive studies (Jerman and Mirman, 1974) involved 340 students from grades four to nine, and compared the results for the upper elementary grades of four through six with the results for the junior secondary levels of seven through nine. Seventy-three linguistic variables were isolated for this study, and they were combined with computational variables from previous studies. In the analysis of the data from the lower grades, four through six, Jerman found that three computational variables entered the regression equation. Two of these variables were in the first two steps, accounting for 54 percent of observed variance from the correct solutions. Yet these two variables were no longer significant in the analysis of grades seven through nine. In the higher 16 grades, a linguistic variable occupied the First step in the regression equation. Two computational variables followed in order of importance, but no other com-putational variables entered the regression equation. Jerman concluded that computational variables accounted for most of the variance from the correct solutions for grades four through six, but that linguistic variables had become more significant by grades seven through nine. In a study of college-level students, no computational variables entered in the first twelve steps, thus effectively disappearing as determiners of the difficulty of word problems. Thus, a developmental trend 'for linguistic variables appears to exist, while computational variables progressively decrease in importance. Other studies have noted that the sequencing of the problems has a greater significant effect on their difficulty than other computational-type variables (Suppes, Loftus and Jerman, 1969; Rosenthal and Resnick, 1971). This means that students find a problem much more difficult if it is not similar in type to the problems which preceded i t . The relevance of these studies in the development of problem-solving ap-proaches is strong. For example, it seems quite reasonable to assume that the problem-type strategy used by many teachers is based on the concept of sequencing. Students who are taught to handle one specific type of problem will feel quite comfortable with the presentations found in most secondary school textbooks where examples of that type of problem are all grouped together. While the problem-type strategy effectively eliminates the sequencing variable as an obstacle in the classroom, it loses most of its usefulness when students face the problems in un-familiar contexts. Language-based strategies. The growing realization of the importance of linguistic skills to verbal problem solving has generated attempts to raise problem-solving ability by improving such skills as reading and vocabulary. Henney (1969), for example, compared a large group of fourth graders who were given lessons in reading verbal problems with a second group who studied and solved verbal problems in any way they chose. Results showed that although both groups im-proved significantly, there was no significant difference between the two groups. Other studies on the effect of teaching reading skills in mathematics classes (Lyon, 1975; Parler, 1975) also failed to find significant improvement in mathe-matical achievement or in problem solving. Yet investigations into the benefit of training in vocabulary for increased mathematical achievement have been quite productive. After pupils were given specific training in mathematical vocabulary, gains in problem-solving ability were found by Dresher (1934), Johnson (1944), Lyda and Duncan (1967) (although this study has been shown to be poorly designed (Kane, 1967)) and Vanderlinde (1964). These studies seem to indicate that mathematics teachers have been too casual about introducing new symbols and terms to their classes. Students appear to improve as problem solvers when these terms are clarified and more time is spent on their mathematical vocabulary. Literature Relevant to the Translation Method The general success of mathematical vocabulary training, and the increasing recognition of the language factor as a major cause of difficulty in solving word problems, indicate that a translation approach is in order. General translation strategies have been developed (Maffei, 1973, Taschow, 1969, and Earp, 1970), but Dahmus (1970) has developed the most rigorous and highly detailed method. The Dahmus Translation Method is aimed at the lower 90 percent of mathe-matics students, who are intimidated by long verbal presentations of mathematical information and have no tools for dealing with such problems on a step-by-step basis. For example, students are not permitted to first read through a problem, but are to translate it one phrase at a time. A l l translations and related informa-tion are set down under the caption of Translation, and only then may the student begin to substitute and solve the resulting equation(s). Dahmus claims that he has used this method often with a great deal of success at all age levels. Specific studies which have contrasted his method, or strategies similar to the Translation Method, with other problem-solving approaches, will be reviewed later under \"Comparisons of Problem-Solving Strategies. \" The careful structure of this method drew a protest from Boersig (1970). Her criticism was levelled at two aspects of the Translation Method: First, she notes that implied relationships and recalling formulas form a large part of finding the solution to word problems. Dahmus makes no provision for this. Then, in a reaction to the literal and highly structured nature of the strategy, she asserts: \"A student should not be cheated out of learning problem-solving processes by just teaching him to translate according to verbal word patterns. Rather, he should be given the opportunity to wrestle with a problem and have the satisfac-tion of resolving the conflict\" (p. 643). Wilson and Becker (1970), in a general discussion of problem solving, echo the same sentiments: 19 \"We feel that students come to equate problem solving with \"answer-getting\" and, in doing so, miss the real heart of mathematics. Solving a mathematics problem should involve the systematic application of one's knowledge to the novel situ-ation, lead to the complete understanding of the problem and its solution, and in turn, increase one's knowledge through learning new information (the solution), enhancing problem-solving skills (the process), and discovery of new relationships\" (p. 293). Literature Relevant to the Inductive Method Another approach has been taken by some educators who emphasize the benefits of creative or divergent thinking in the solution of verbal problems. One such attempt is the Inductive Method, suggested by Spitler (1976). Students are en-couraged to generate their own word problems in response to equations given by the teacher, with the goal of enabling students to establish a link between the verbal statement and the mathematical equivalent inductively. Research supporting this kind of creative thinking is difficult to categorize because studies have used the terms \"discovery\" and \"inductive\" approaches inter-changeably with \"creative\" or \"divergent\" approaches. The literature in question has dealt with every aspect of non-traditional teaching. However, it may loosely be characterized as dealing with learning which allows the student to abstract principles from information or experience with a minimum of guidance. (This is usually referred to as inductive or discovery learning.) When the student is en-couraged to generate the experiences from which to draw generalizations, the terms \"creative\" or \"divergent\" thinking are often used. Literature discussing any of these approaches was considered relevant to the Inductive Method, regardless of which particular term was used. 20 Studies of discovery learning. The relationship between mathematical ability in general and discovery teaching methods has been studied increasingly since the 1960's emphasis on \"new math,\" but consistent results have been difficult to find. One extensive study by Worthen (1968) involved 538 fifth-and-sixth-grade pupils. The investigator concluded that expository methods were superior to discovery methods on tests of initial learning, but discovery was significantly superior for concept transfer, transfer of heuristics, and retention. However, a later study (Worthen and Collins, 1971) criticized certain statis-tical methods used by Worthen (1968). Reanalysis of the data with proper statis-t i c a l procedures yielded no significant differences between treatments on any transfer or retention test. Thus the earlier conclusions by Worthen could not be supported. A study which is often cited was conducted by Richards and Bolton (1971), who studied 265 children in their last year at three junior schools. The subjects were matched, while the three schools used different mathematical instruction techniques. One used discovery methods, another used traditional methods, and the third balanced the two approaches. They found that divergent thinking is a minor factor in the determination of general mathematical ability, with general ability being the most important determiner. (Few proponents of creative thinking ap-proaches would question the fact that general ability, as well as verbal ability, are the factors most highly correlated with mathematical abil ity.) A second part of their study found that students at a school which emphasized discovery teaching and divergent thinking were inferior on tests of mechanical performance in mathe-matics, but were superior to students of a traditional school on tests of divergent thinking. 21 Olander and Robertson (1973) used 374 fourth-grade pupils for a comparison of discovery and expository learning on tests of computation, concepts, applica-tions, and attitudes. They found that students under the expository treatment were significantly better in computation on both a posttest and a retention test. How-ever, students in the discovery group better retained their ability to apply mathe-matical knowledge and showed significant improvement in attitudes. Since solving word problems is considered to be an \"application\" sk i l l , this study indirectly sup-ports the possibility that an inductive teaching approach may be helpful in problem solving although it may not be strongly correlated with general mathematical ability or performance on tasks of low cognitive complexity. Studies of divergent thinking. Few studies have specifically related divergent or creative thinking to the process of solving mathematical word problems. Some have attempted to determine whether it is possible to increase divergent thinking in and of itself by means of special treatments. Dirkes' (1974) study is an example of this type of research. Dirkes administered a fourteen-day divergent thinking program in problem solving to fifty-two geometry students and found that they showed significant gains in creative productivity of verbal content when compared to a control group. However, she did not test the students for problem-solving ability either before or after the treatment. Maxwell (1974) categorized 105 students as either divergent or convergent thinkers in a clinical study to better understand problem-solving processes. Her classification was based on a six-problem test which she constructed, using three 22 divergent-type items and three convergent-type items. Forty-nine of the students were then observed as they solved one particular problem, described their methods, and reworked the problem on a second trial . Maxwell found that divergent thinkers used fewer generalizations with more trial and error, and they took more time with the second trial . These results, however, are based on only one actual problem-solving opportunity. In addition, Maxwell does not draw conclusions about the first attempt to solve the problem, but rather concentrates on the students' ability to dissect their methods and rework the problem. Literature of specific relevance to an inductive approach. Wallace (1968) provides one of the few examples of studies which examined the relationship of different factors to the ability of students to solve mathematical word problems using the discovery method. He administered a battery of standardized tests to 548 freshmen at a college in Pennsylvania, and employed regression analysis of the data to draw the following conclusions: 1. The greater the student's mathematical ability, the greater his ability to solve mathematical problems by the discovery method; 2 . The student's ability to solve mathematical problems by the discovery method was dependent to some extent upon his verbal ability; 3 . There was a substantial relationship between a student's mathe-matical achievement and his ability to discover the solution to a mathematical problem; and 4. Female students displayed a slightly greater ability to solve mathe-matical problems by the discovery method than did male students. However, it appears that ho comparison with a control group had been made. Also, the first three conclusions do not seem to establish new relationships which had not already been found to be true of general problem-solving abil ity. Dodson (1970) charred the characteristics of successful problem solvers and found that high scores on divergent thinking were directly related to success with problem solving. In general, although more research supporting these conclusions would be desirable, many mathematics educators would agree with Dirkes (1974) when she assumes that divergent thinking precedes convergent reasoning and evalua-tion in the problem-solving process. As Manheim (1972) suggests: \"One of the big lessons of 'modern' mathematics is that the creation of new mathematics often is inductive rather than deductive. Thus we often try to abstract certain com-monalities from a few cases and then generalize to a very large se t . . . . But students should be encouraged to imag-ine, to postulate, to 'give it a try'. For trying is the essence of induction\" (pp. 235-36). Of particular relevance to the Inductive Method, where students write their own problems, was a study by Keil (1964). The purpose of the study was to de-termine whether students who wrote and solved their own problems in mathematics would prove superior to students who solved textbook problems. Data were ob-tained from test scores of 226 sixth-grade students in eight classrooms of eight schools in a midwestern state. A l l classes were given two standardized mathematics tests and one test of mental abil ity. Students were classified by sex, as well as by three levels of intelligence and two levels of socio-economic status. Four experi-mental and four control classrooms followed their regular textbook program for four days of the week. One day each week was devoted to the investigator's materials. The experimental group wrote and solved problems about a given situation while the control group solved textbook type problems about the same situation. At the end of sixteen weeks, students were given two standardized mathe-matics tests. An analysis of covariance indicated that the experimental group scored higher than the control group on both tests. Further analysis showed that for one test, differences were significant only for low socio-economic pupils. On the other criterion test, subjects in the experimental groups in each of the following categories: boys, girls, high intelligence, average intelligence, and low socio-economic, outperformed their counterparts in control groups. Kiel concluded that, in general, pupils who wrote and solved problems of their own were superior in arithmetic problem-solving ability to pupils who had the usual experiences in problem solving. Comparisons of Problem-Solving Strategies Research of general relevance dealing with problem solving has been plenti-fu l , to say the least. Suydam (1967) conducted an impressive review of all pub-lished research on elementary school mathematics from 1900-1965, and found that problem solving was the most widely researched topic, with 84 of a total of 799 reports. At the same time, she found conflicting results and a generally low quality of research design and reporting to be prevalent. This was especially true of experimental studies. Studies contrasting different problem-solving approaches are typical, and problem-solving studies of this nature were used by Suydam as examples of research yielding inconsistent conclusions. Some of the investigations into the relative effectiveness of problem-solving strategies are presented, despite the fact that they do not deal with either the Translation Method or the Inductive Method. Research of specific relevance to a comparison of these two methods is discussed later. It should be noted that no comparison was ever drawn between the Inductive Method and any other method, and that experiments using the Translation Method are rare. 25 Studies of the effectiveness of formal strategies. Research has indicated that there is some question as to the usefulness of imposing a structure, and especially a highly rigid structure, on problem solving. The classic study of this kind was conducted by Burch(1953) with 305 elementary school children who were trained to solve arithmetic problems using formal analysis ( i . e . , having students answer a specific sequence of questions before beginning computational work). The children attained higher scores on tests which did not require formal analysis than on one which did. Moreover, fifty-one students who were later interviewed i n -dicated that they never used formal analysis unless required to do so, and many became confused when they attempted to do problems in this way. Similarly, Kinsella (1951) compared the effects of a step method against no formal teaching at all using as a criterion the students' success in selecting the correct process for the solution of a problem. His findings showed that success was not dependent on prior success with any certain step or combination of steps, and that these in fact might lower the level of performance on the solution of the whole problem. This again supports the hypothesis that answering any specific set of questions may not produce superior results. More recently, Post (1968) examined the question of structure in general as an aid to problem solving in mathematics. He assembled a list from research of the mental operations underlying the problem-solving process, and referred to this as problem-solving \"structure.\" He divided ten grade 7 classes into experi-mental and control groups, and gave the experimental groups three days of i n -struction in methodology with a six-week reinforcement period, while the control group solved identical problems and then proceeded with other work. He concluded from the lack of significant differences between groups that exposure to the structure of the problem-solving process does not enhance problem solving. Burch's investigations of formal analysis seem more relevant to the question of rigidly-imposed techniques than does Post's analysis of the benefits of a generally defined \"structure.\" Yet the conclusion remains as to the doubtful benefitcof im-posing restrictive guidelines. It seems that a problem-solving strategy should be a guideline which is spe-cific enough to be of some use, and yet not too highly restrictive for students. Kilpatrick (1969) discusses the need for \"finding methods and devices that would improve problem solving without putting the child in the kind of straightjacket provided by formal analysis and other prescriptive techniques\" (pp. 529-30). Research of general relevance to problem-solving strategies. Most of the com-parisons which have been drawn between different methods have used some variation of Polya's method as one of the treatments. Wilson (1967), for example, compared a version of the wanted-given method as suggested by Polya with an action sequence popularly used in the elementary schools. In the action sequence, the student looks for operations suggested by the sequence of actions in a problem (thus actions dealing with \"joining\" would signify addition, and \"separation\" actions would sig-nify subtraction). Results favored the wanted-given approach when the two methods were contrasted with a control group. One of the more recent studies was conducted by Post and Brennan (1976), who compared formal instruction in Polya's strategy with an informal presentation of general heuristics for problem solving and found no significant differences between the two. However, the study did not take teacher effect into consideration, and it used no control group. A modified wanted-given approach was compared with a general problem-solving program in grade 5 by Jerman (1973) and, again, no differences were found in either the posttest or on a follow-up test between the two experimental groups and a control. The wanted-given group did use correct procedures on the posttest more often, but conclusions may be validated by the presence of group differences which were not controlled. One study which seems more relevant to the current study was conducted by Gawronski (1972). Its purpose was to ascertain the existence of deductive or inductive learning styles. This study differs from the general comparison of strategies because the two types of instruction were administered to all subjects, with content matter differing slightly from one group to the next. Thus not only were deductive and inductive learning programs compared, but the hypothesis under investigation was that different students would benefit from each program. Three hundred eighty-one eighth-grade subjects were stratified by sex and mathematical ability. Students who lacked prerequisite skills or who had had previous experience with the content matter were eliminated. The remaining 298 were randomly assigned to classification programs which used programmed texts to present two concepts, one deductively, and the other inductively. Students who scored above the median on the posttest following the deductive concept and below the median on the inductive concept posttest were classified as deductive learners, and students who scored below the median on the deductive posttest and above it on the indue tive posttest were considered inductive learners. Using this method, 32 deductive and 22 28 inductive learners were found. These 54 subjects were then given two additional programs of instruction, one inductive and the other deductive, and posttests were again administered at the close of each program. While it was expected that inductive learners would score higher than deductive learners on the inductive program posttest, with the reverse holding true for the deductive program posttest, this was not the case. Although this study is of particular interest because it involves a deductive-inductive comparison, all posttests used by Gawronski used items which required low cognitive ability. This means that aside from the fact that no significant re-sults were reported, the relevance to the current study is limited by the fact that Gawronski did not consider mathematical word problems in her investigation. Problem-solving comparisons of specific relevance. No comparison similar to that undertaken by Gawronski (1972) could be found in the field of problem solving. However, one study by Shoecraft (1971) contrasted three translation ap-proaches to problem solving, using twelve grade 7 mathematics classes and ten grade 9 mathematics classes. One group, considered the control group, was taught algebra word problems by direct translation (they were called the low imagery or \"LO\" group). Another group was taught by translation with accompanying materials for illustration (the high imagery: materials, or \"HIM\" group). The third group learned translation methods with drawings preceding the translation (the high imagery: drawings, of \"HID\" group). A l l groups were taught number, coin, and age prob-lems for eight days, and work and mixture problems for four days. HIM was found most effective for low achievers, but otherwise students in the LO group generally performed comparably to HIM and better than HID. 29 Shoecrctft concluded that the popular assumption that materials or drawings, in and of themselves, enhance problem-solving achievement is unjustified. This study does provide some support for a translation method, but it is difficult to generalize from it to the study at hand because no group was taught word problems or prac-ticed them without benefit of translation. Thus LO (direct translation) may be better than HIM or HID, but it does not necessarily follow that LO would be su-perior to having students simply practice the problems without instructional guidance. The only study which used Dahmus1 Translation Method in a comparison with another, non-translational strategy, was one conducted by Bassler, Beers and Richardson (1972). Here it was compared with a variation of Polya's method. The experiment was based on a relatively small sample of 48 ninth-grade students in a parochial school for girls, with the sample divided into three ability levels. The instruction was given by videotape presentation to eliminate the factor of teacher effect. Seven forty-minute periods of instruction included teaching the solution of linear equations as well as the particular strategy for problem solving. A posttest of ten problems was given following the treatment, and an unannounced retention test followed four weeks later. Two criteria were used to score the test: the first was a solution criterion and the second' was an \"equation\" criterion in which students were scored on their ability to find a single equation which would solve the problem. Results from the tests were quite low, with means on the solution criterion ranging from 25 percent to 75 percent, and equations-criterion means ranging from two percent to 65 percent for the six cells. There were no significant dif -ferences on the solution criterion scores, but the Polya method students did significantly better on the equations criterion. Both groups improved significantly on the retention tests. The researchers point out that the results should be viewed with some caution in light of the generally low scores. They also note that the instruc-tional medium may not have been as effective as a teacher-based instruction, where open questioning methods in earlier grades favored the Polya method, which might account for some of the differences. One more reservation about the study should be recorded, since the style and design of Bassler et al (1972) is most closely aligned with that of the present study. No control group was used by them for comparison purposes. Therefore, it is not known whether either of the methods used was in fact superior to non-instructional methods. Bidwell (1972) points to other limitations in his abstract of the study. He questions the appropriateness of using an equations criterion, when a careful i n -spection of the Translation Method will show that students are not being trained to direct their efforts toward producing a single equation. Rather, they are taught to use many variables in many minor equations. He also criticized the time length for the tests which, although not explicitly stated in the study, seems to be one forty-minute class period. Ten problems are too many in this short time sequence and do not allow for effective problem solving. In Bidwell's opinion, this would invalidate the research results. Of all the research to date in this area, however, this comparison serves as the most useful precedent. A follow-up study to measure the different effects of two translation-oriented strategies, one deductive and formal, the other inductive and creative, seems in order. Any such study would use teachers as the means of instruction, and would use a lower grade level to eliminate the problems cited by the researchers. It would also have to consider sex differences, which were not relevant to the Bassler, Beers and Richardson study because their subjects were all female. Literature Relevant to Sex Differences As indicated in Chapter 1, the learning assessment projects have found that sex differences, although small, do exist. Robitaille and Sherrill (1977) conclude from the British Columbia Mathematics Assessment project, for example, that female students are more competent at lower level cognitive tasks, while male students obtain higher scores in such high level cognitive tasks as problem solving. Yet Suydam and Weaver (1975) conclude from their review of research that sex differences do not appear to exist in the ability to solve problems. This con-clusion does not seem to be justified upon inspection of related studies and other general reviews of sex differences in mathematics education. One such review of mathematics learning and the sexes was undertaken by Fennema (1974), and it appears from this review that there are conflicting results from studies which, have used sex as a factor in their analyses. In the thirty-six studies which are reported, the total of significant differences which were found depended largely upon the age of subjects used. Of the thirteen investigations conducted with pre-school age children and early elementary levels, nine studies found no significant differences between the sexes. Of the remaining four, three studies found significant differences favoring the girls, and one favored the boys, thus indicating that there are no consistent differences in the early years. The question of sex differences becomes more confused at higher grade levels, as is evident from the twenty studies reviewed in grades 4 through 9. These studies are listed, together with their results, in Table 2 . 1 . This table appeared as Table 3 in the review by Fennema (1974). From this tabulation it may be concluded that no significant differences consistently appear in studies at these age levels, but where significant differences do exist, girls tend to perform better in tests of mathematics computation and boys tend to perform better in tests of mathematical reasoning. The National Longitudinal Study of Mathematical Abil i ty (NLSMA) pro-vides more insight into the sex differences. Data from one group, the X-popula-tion were collected as the students progressed from grade 4 to grade 8 (Carry and Weaver, 1969; Carry, 1970). Data from a Y-population were collected for four years as the group progressed from grade 7 to grade 10. Tables 2.2 and 2 .3 sum-marize the total number of test's for each population, with all grades for the popu-lation represented in the totals, together with the number of significant differences found between the performance of girls and boys at each level of cognitive skills. It seems evident from these tables that boys outperformed girls at al l levels of cognitive complexity, with results being particularly strong at the ninth and tenth grades (although no breakdown by grade is shown in these tables). Also, boys appear to do better than girls at the higher cognitive levels, as noted previously. Several complicating factors appear to exist at the high school level. Lower ability boys tend to dropout more often than low-ability girls, so that the high school boys who are sampled may be of higher ability than the girls. Also, 33 TABLE 2.1 SEX DIFFERENCES IN MATHEMATICS ACHIEVEMENT UPPER ELEMENTARY A N D HIGH SCHOOL YEARS Author D'Augustine (1966) Jarvis (1964) Grade Dependent Variable(s) 5,6,7 Geometry and topology Results Unkel (1966) Cleveland & Bosworth (1967) Zahn (1966) ParsJey, et a I (1963) Parsley, et a I (1964) Standardized achievement test 1-9 Discrepancies between ac-tual achievement as measured by standardized achievement test and an -ticipated achievement as determined by CA, MA & grade placement; 3 SES groups used as independ^ ent variables 6 Standardized achievement test Arithmetic achievement and reasoning (standard-ized test) 2-8 Standardized achievement test 4-8 Standardized achievement test No significant differences found. Boys tended to excel in reason-ing at all IQ levels. Girls per-formed better in fundamentals in 3\/4 IQ levels. Arithmetic reasoning: no signifi -cant differences in discrepancy scores between girls and boys. Arithmetic totals: no significant differences between girls and boys on total, yet at grades 6 , 7 , 8 , girls have significantly higher discrepancy scores with conver-gence again at grade 9. No significant differences found. \"Virtually no differences between the sexes in any aspect of arith-metic achievement.\" On 5 out of 32 subtests boys per-formed significantly better than girls; on 0 out of 32 subtests girls performed significantly better than boys. No significant differences found. Boys with IQ of 125+ outperformed girls with similar IQs on arithmetic reasoning. Girls with IQs of 75-124 outperformed boys with similar IQs on arithmetic fundamentals. The overall differences appear to be nonsignificant. 34 TABLE 2.1 - continued Auth or Grade Singhal & 5-16 years Crago (1971) K - l l grades Dependent Variable(s) Wide range achievement test(Level 1) Results O lander & 4 , 5 , 6 Ehmer (1971) Mathematics; vocabulary of children's contemporary \u2022 mathematics tests Before instruction girls had higher grade-equivalent scores in arith-metic as a total group and at most grade levels. After six weeks (approx.) of instruction boys made gains significantly higher than girls in grades 3, 4, and 9. The dif -ferences in the total gains for boys and girls were nonsignificant. Girls were significantly better at all 3 grades Sowder (1971) 4-7 Wozencraft 6 (1963) Overholt (1965) Discovery of patterns Standardized achievement test Standardized achievement test and conservation-of-substance test No significant differences. No significant differences found in arithmetic reasoning. Girls per-formed significantly better on arith-metic computation. On arithmetic average girls in middle IQ range performed significantly better. When scores were adjusted for dif -ferences in intelligence, boys scored significantly higher than girls on total score, understanding of concepts, and problem-solving abil ity. No significant differences in ability to conserve were found. Alexander 1962 Muscio (1962) 6 Arithmetic reasoning test Quantitative understanding No significant differences found in ability to solve verbal problems. Boys performed significantly better than girls. Sheehan(1968) 9 Abi l i ty to learn to solve Ambivalent results, algebra problems 35 TABLE 2.1 - continued Author Carry (1970); Longitudinal Carry & Weaver 7-10 (1969) Kilpajnck & Longitudinal McLeod (1971); 7-10 McLeod & Kilpa trick (1969,1971) McGuire (1961) Gainer (1962) Grade Dependent Variable(s) Results Computation, comprehen-sion, application, analysis in algebra, geometry, and number systems Computation, compre-hension application, analysis in algebra, geometry, and number systems Standardized achievement test Standardized achievement test Standardized achievement test In 38 out of 75 tests boys performed significantly better than girls. In 16 out of 75 tests girls performed significantly better than boys. In 25 out of 54 tests boys performed significantly better than girls. In 10 out of 54 tests girls performed significantly better than boys. No significant differences found. No significant differences found. No significant differences at grade 5 . At grades 7, 9, and 11, college-bound boys scored significantly higher than college-bound girls. At grade 11, non-college-bound boys scored signi-ficantly higher than non-college-bound girls. Junior High 6-12 years Hilton and Longitudinal Berglund (5,7,9,11) 36 TABLE 2.2 SEX DIFFERENCES IN ACHIEVEMENT : GRADES 4-8 NLSMA: X-POPULATION* Total Tests Significance Found Favoring: Boys Girls Computation 31 6 14 Comprehension 34 22 2 Application 3 3 0 Analysis 7 7 0 A l l tests 75 38 16 * Figures from Car\/y (1970) and Carry and Weaver (1969). TABLE 2.3 SEX DIFFERENCES IN ACHIEVEMENT : GRADES 7-10 NLSMA: Y-POPULATION* Total Tests Significance Boys Found Favoring: Girls Computation 18 5 7 Comprehension 14 6 1 Application 5 3 0 Analysis 14 11 2 A l l tests 51 25 10 *FfomsKiIpatrick and McLeod (1971) and McLeod and Kilpatrick (1969, 1971). for whatever reason, girls do not choose mathematics electives as often as boys, and it is therefore possible that a larger percentage of bright girls than bright boys do not continue in math. This would result in the sample of girls from mathematics classes being of correspondingly lower ability than the boys. On the basis of the research done at the high school level, no conclusions could be drawn. The sex differences favoring boys at the junior high school and upper elementary levels are supported by some of the studies discussed in the preceding pages, although not all used sex as a factor. For example, Kilpatrick (1967), in his study at the grade 8 level, found that girls said \"I don't know\" as a re-sponse more often than boys, and they used equations less often than boys. Maxwell (1974) found that boys were better problem solvers than girls in tenth grade. On the other hand, Post (1968) found that sex was not a factor in his grade 7 study. Other studies either did not take the factor of sex into considera-tion or examined male subjects separately from female subjects (Gawronski, 1972). However, it does seem that the indications from research are sufficiently strong to justify using sex as a factor in the analysis of performance of students in sixth and seventh grades. Specifically of interest is the possibility that of the two teaching strategies being compared, one will serve to benefit one sex more than the other. This is suggested by Wallace (1968), who found that female students displayed a slightly greater ability to solve mathematical ^ problems by the discovery method than did male students. Summary of Review of Related Literature Although many aspects of problem-solving research defy generalizations, some patterns do emerge from the literature. There exists a strong correlation between linguistic ability and problem-solving ability, paralleling and perhaps even exceeding the correlation between computational ability and problem-solving ability. The need to aid students in the language aspect of mathematical word problems was evident from the prominence of linguistic factors in the determina-tion of the difficulty of word problems, particularly at the grade 7 level and beyond. Assorted attempts to teach students methods to improve problem-solving skills, were often successful, although not consistently. The effectiveness of carefully structured problem-solving techniques was questioned by a number of studies. Yet studies which tried to improve problem-solving by divergent or discovery techniques were rare, and no conclusions could be drawn from studies that do exist. Although the ability to use divergent thinking is considered a minor factor in terms of general mathematical ability, it was related to successful problem solving. Comparisons of problem-solving techniques generally used some variation of Polya's method, and tended to be inconclusive or inconsistent. No contrasts could be found which compared detailed, deductive strategies such as the Transla-tion Method with creative, inductive strategies such as the Inductive Method. Sex differences in favor of boys do appear in the research, although not consistently. In general, boys seem superior at higher-level cognitive tasks and girls at lower-level cognitive tasks when differences are found in the upper e le -mentary and lower secondary grades. The sex differences are sufficiently in e v i -dence to justify using sex as a factor in a study of problem-solving performance. Chapter 3 DESIGN A N D PROCEDURE This study was undertaken after a search of literature revealed that no ex-perimental situation had as yet been constructed which would contrast certain translation-based strategies with a noninstructional practice sequence. These strategies, the Translation Method and the Inductive Method, are discussed after a description of the sample involved and the tests used in the study. The actual experimental procedure and the statistical procedures a n t i c -ipated for the later analysis of data are then detailed. Results of the study are examined in subsequent chapters. Sample Grades six and seven were chosen as the level for the treatments, since it was felt that these grades were just beginning to experience a higher concentra-tion of word problems in their texts and curricula and had not yet evolved a con-sistent strategy for dealing with these problems. The study was performed with students in a private Jewish elementary school in Vancouver, British Columbia. This meant that the subjects did not constitute a random sample of the population at large, and significant results of an experiment done with them would indicate that more extensive research should be done before final conclusions are drawn. Two small classes of sixth graders with a total of 26 students between them and one grade seven class consisting of 24 students were collapsed into one sample group of 50 mixed sixth and 40 41 seventh graders for the interim of the treatment. This was done both to enlarge the sample and to eliminate variance which would have resulted from class dif -ferences. The justification for combining these grades was that careful scrutiny of the word problems in textbooks being used by both levels revealed little change in the complexity of grade 7 problems as compared with grade 6 problems. (As a precaution, grade was used as a factor in the analysis of the results of the study.) The original sample, containing 16 students in Group A and 17 in each of the other two groups, was further reduced by two students who missed more than one treatment period. Thus, by the completion of the experiment, the num-ber of subjects was 15 in Group A , 16 in Group B and 17 in Group C, for a total sample size of 48 subjects. DEVELOPMENT OF MEASURING INSTRUMENTS Word Problem Pilot Test A word problem test was constructed to be used as a pilot test. It was originally planned to measure the problem-solving abilities of the students from the sample for later use as a pretest or covariate for the final statistical analysis. Then, when the results were tabulated, it served as a gauge to determine the optimum length df the subsequent posttests as well as to set the level of dif -ficulty for future tests. The pilot test consisted of six word problems which approximated the type of problems found in the grade six and seven textbooks (Investigating School Mathe- matics, Eicholz, O'Daffer and: Fleenor, 1973). A l l items were short and un-complicated, only whole number operations were needed for the solutions, and the computation involved was minimal. The operations required were two division calculations, two subtraction, one addition, and one multiplication. The test was administered in one forty-minute period but many students were finished af -ter 30 minutes. Answers for each item were scored on a scale of 0, 1, 2, with 2 as the maximum for the correct operation with no computational errors. One point was given for the correct operation with computational errors, and no points were allowed for an answer which did not use the correct operation. A l l work and re-sponses were recorded on the test pages so that the paper score could be ascer-tained from the scratch work shown. A copy of the Word Problem Pilot Test may be found in Appendix A , together with scores obtained by students. Because the test proved to be of low reliability (as discussed in Chapter 4), it was not utilized in the final analysis of results. Translation Pretest Because one of the treatments involved the ability to translate directly from an English statement of the problem to the mathematically equivalent sym-bols, it was felt that all three treatment groups should be more or less equivalent in their translation skills when the treatments began. The translation pretest was thus constructed to ascertain the students' translation ability for use when assign-ing students to groups. No standardized test existed for this purpose, so the test was independ-ently constructed by the researcher with English phrases such as \"16 kilometres less than the distance.\" The students were instructed to indicate the proper mathematical phrase using symbols, numbers and letters (in this case, D - 16). The 30 items consisted of six addition-only examples, seven subtraction-only items, four multiplication-only items, seven division-only items, two items re-quiring only equality, one addition with equality, one subtraction with equality, one requiring addition with multiplication, and one multiplication with division. Correct translation was scored as a maximum possible of two points for an item. If the operation was correctly translated but other mistakes were made, or if only one of two required operations was correct, partial credit of one point was given (for example, \"the sum of Jack's score and 25\" written as N + 25 = N) . Appen-dix B contains the translation pretest, which was designed for one 40-minute period. Posttest One Posttest One consisted of eight word problems of varying degrees of dif -ficulty, but involving only one operation for the solution. Problems were drawn from a variety of sources, including Investigating School Mathematics, Books 6 & 7, (Eicholz et a l , 1973), the 1973 editioncof the Canadian Test of Basic Skills, and MP ?-1 Problem Solving (Spitler, 1976), and some were constructed by the i n -vestigator. Items required only whole number operations, with a minimum of dif -ficulty in the computation necessary for reaching the solution. The test contained two addition items, two subtraction items, two multiplication items, and two division items, one of which called for a remainder. The test was administered in one forty-minute period. A l l work was to be shown on the test paper, and on this basis, partial credit was given. Each item was worth three points. One point was deducted if only one of the following mistakes was found: (1) dollar signs and decimal point omitted, (2) remainder not interpreted properly, or (3) a simple careless or computational error. If a student used the figures incorrectly or in an improper order, or had two of the above errors, then two points were deducted from the item, with one point allowed for identification of the proper operation. No credit was given if the student did not choose the correct operation. The first posttest was designed to represent the traditional word problem test taken by students as a standard exercise in the application of whole number operations to the field of problem situations. Typical problems such as rate, distance, money and so forth all appear in the test, which is given in full in Appendix C . Posttest Two Posttest Two was constructed by the experimenter to be on a much more challenging level. The eight items in this test all required a multi-step compu-tational process, and although only whole number operations were required, the solution of each item involved the use of two operations. The; problems were drawn from the \"think\" problems found in Books 6 and 7 of Investigating School Mathematics (Eicholz et a l , 1973), as well as from sources used for the first posttest. The operations required for the test items were as follows: Item 1: addition and multiplication Item 2: subtraction and division Item 3: division and multiplication Item 4: addition and division Item 5: multiplication and subtraction Item 6: addition and division Item 7: division and multiplication Item 8: subtraction and multiplication When the system of scoring was developed, it was apparent immediately that it would not be feasible to mark the papers allowing different degrees of partial credit, since the errors varied so widely from item to item as well as from student to student that it was impossible to delineate in any consistent man-ner how much work merited a certain amount of credit. It was therefore decided to give partial credit only for the same careless errors listed for Posttest One, with no credit for conceptual errors of any kind. Each item was given a total value of four points, and one point was deducted for each of the clerical errors previously noted ( i . e . , (1) omitting dollar signs and decimal points, (2) not using remainders properly, or (3) for a simple careless or computational error). The test was administered in one forty-minute period, with all work shown on the face of the test paper. Ample room was left for these calculations and all test papers were carefully examined to discriminate between computational and conceptual errors. The second posttest, with directions for its administration, can be found in Appendix D. DESCRIPTION OF INSTRUCTIONAL TREATMENTS Treatment A : Translation Method This method, developed by Maurice Dahmus (1970), is a structured treat-ment of word problems based on the assumption that the primary difficulty with problem solving lies in describing English problems in accurate mathematical terms. To meet this difficulty, the Dahmus method trains students to literally translate, phrase by phrase, from the statement of the problem into mathematical symbols. As a first step in the treatment, then, detailed instruction is necessary in the translation of such phrases as \"increased by,\" \"the difference between,\", etc. The actual translation strategy can best be illustrated using a simple example on a grade six level: Jack has 15 more pennies than Betty. If Betty has 48, how many does Jack have? A . Translate: Jack \/ has \/ 15 more pennies than Betty. J = 15+ B If Betty \/ has \/ 48, \/ how many \/ does Jack have? B = 4 8 ? = J B. Relationships or formulas suggested by key words in the problem. (This example needs no outside formulas.) C . Solution - usually by direct substitution into statement of relationship i . e . , J = 15 + B ? = 15 + 48 and solve. Using the Dahmus Method, the student's exercise book should look some-thing like this: Translate Solve J = 15 + B J = 15 + B B = 48 ? = 15 + 48 ? = J ? = 63 The teaching outline for the four days of translation lessons was as follows: Lesson 1 : Familiarized students with common language equivalents of mathe-matical terms (decreased by, a total of, etc. ) ' Lesson 2 : Introduced solving word problems by the translation method. Lesson 3 : Extended the method to include cases requiring outside formulas or relationships not specifically stated in the problem. Lesson 4 : Reviewed method; general practice. The complete set of lesson plans, worksheets, and other materials for Treatment A can be found in Appendix E. Treatment B : Inductive Method This method was suggested by Gai l Spitler (1976). The assumption was that students lack some vital link which they need to make the connection be-tween word phrases and symbols, and that it would be possible to reduce this lack by having students \"create\" word problems to match suggested equations. Students were encouraged to use as many different contexts and equivalent word phrases for the operation as possible. The equation used as an example here is identical to the derived equation in the example for the Dahmus Method: N = 48 + 15 Teacher: \"What kind of problem situation could this mathematical statement be describing? Can you write at least three word or story prob-lems which are as different as possible, which would fit the same equation? \" Examples of Possible Answers: 1. If a man drove from his home a distance of 48 km in the morning and 15 km more in the afternoon to reach a second city, how far apart are the two cities? 2 . Jane is 15. If she is 48 years younger than her grandmother, how old is her grandmother? 3 . If Mrs. Johnson has only $15 left after buying a coat which cost $48, how much money did she have originally? (Students who have difficulty generating problems at first might look through different story problems in the textbook for context ideas.) In the beginning les-sons, students compared answers and each exercise such as the one above was to be followed by a summary list of the different possible solutions or contexts which the students have used in their examples; here they might be categorized as 1. distance 2. age 3 . money and so forth. 49 Later lessons emphasized that equations may correspond to different meanings of the same operation, and that word problems can vary accordingly. For example, in the problem above, the first sample answer corresponds to N = 48 + 15 as a simple example of the sum of two quantities. However, in the third sample answer, the appropriate equation is probably better described as N - 48 = 15, or a word problem using the addition operation as the inverse of subtraction. Variety in this area was also to be encouraged. The teaching outline for the four days of inductive lessons was as follows: Lesson 1: Expressed the same mathematical relationship in very different English terms. Lesson 2: Introduced the concept of creating word problems. Lesson 3: Emphasized different meanings of operations. Lesson 4: General practice. The ;complete set of lesson plans for Treatment B can be found in Appen-dix F. No worksheets were necessary since the work was generated from a series of equations which were written on the board by the teacher* Treatment C : Control Method This method was designed to duplicate as closely as possible a noninstruc-tional sequence of problem solving. The emphasis was on having students solve as many problems as were being done in the first, experimental method, without any guidance as to a general strategy for the solution. When students requested help, they were given the proper equation for the solution with as few words of explanation as possible. 50 Generally the purpose of this method was to determine whether a concen-trated exposure of word problems is all that is really necessary to raise the achievement levels of students in word problems. No lesson plans were required, therefore, and only worksheets were distributed to this group. These worksheets are presented in Appendix G . DEVELOPMENT OF INSTRUCTIONAL MATERIALS \"How to Teach Word Problems\" (Dahmus, 1970) provided the outline for the instruction necessary for Treatment A , and'M P-T Problem Solving (Spitler, 1976) was used as a source for the general introduction of the lesson plans for Treat-ment B. A l l lesson plans, most worksheets, and visual aids were created by the ex-perimenter to correspond to the instructional aims of the treatment methods. Puzzles and teasers included in the worksheets for Group C , and the \"Line Code\" worksheet for Group A's introductory lesson, were drawn from Mathimdgination (Marcy, 1973). DESIGN The study basically follows the outline set by Campbell and Stanley (1963) of the \"true experimental design\" for the posttest-only control group design since the pretest was not used in the final analysis. Here R refers only to the random assignment to groups rather than to general randomization, since the sample was not randomly selected from the population. In order to present a general scheme of the study, the following figure outlines the experimental design: A . Translation \/Group iransianon _ p r o b | e m \u2014 R B. Inductive Posttest .Posttest Pretest Pilot Test ^ Group \u00b0 n e T w o C. Control Group Figure 3.1 Outline of Study CONTROLS Problem-Solving Background Since the same mathematics teacher had taught each of the three original classes during the year, the classes were assumed to have had similar mathematical experience in problem solving, and none of them had been given any concentrated exposure in this f ie ld . The teacher, however, did have a personal preference for teaching problem solving by translating, although he had never taught translation in any prolonged fashion. It was felt that if all three treatment groups were equated on translation skills before the treatments began, any problem-solving superiority that might be later discovered in the translation group could be assumed to be a result of the translation treatment and would not be attributable to having a higher proportion of strong translators in the group. 52 Teachers The teacher referred to above was extremely receptive to the inductive strategy as well as the translation method, and agreed to teach both Treatment A and Treatment B. This effectively eliminated the possibility of variance between different teachers. During the many consultations before and during the actual teaching week, he was fully cooperative, and after observing his lessons on the first and third days of the treatment week, the researcher concluded that he was carefully following the lesson plans for each method. Meanwhile the control group was supervised by a teacher who had little mathematical background and ordinarily taught the students an entirely different subject. They therefore did not press him for explanations of the word problems which they did independently of any instruction and, in this way, the control group did simulate as closely as possible the conditions of pure practice without a specific teaching strategy. Treatments In order to ensure that all treatment groups practiced an equal number of problems of equal difficulty, the lesson plans and worksheets were coordinated in the following way: a set number of problems were planned for each lesson of Treatment A , and the identical problems were used by the control group in their practice worksheets. Since it was anticipated that students of Treatment C would finish the problems in less time, working independently, than students of Treatment A who were using these problems to learn a particular strategy, the worksheets for Treatment C contained puzzles and unrelated games for students who finished earlier. 53 Treatment B did not require any word problems because this method i n -volves students creating problems with equations given by the teacher. For this reason, keeping the difficulty of word problems on a par with the other groups was accomplished by using equations which were derived from word problems being used by those groups. ASSIGNMENT TO GROUPS In order to insure that the treatment groups were equivalent, three things were taken into consideration when the fifty subjects were randomly assigned. First, it was preferable to have an equal number of sixth and seventh graders in each group. Secondly, for reasons cited previously, it was felt that the groups should be roughly matched on their translation skills when the treatments began. Lastly, the three students for whom English was not a native language were ran-domly assigned to three separate groups. Thus, on the basis of performance on the translation pretest, students from grade skvenewerekdiivided intocthree'rgeneraMlevels: superior translators, average translators, and poor translators. Students in the highest level were randomly assigned to Groups A , B, or C . The second level students were then similarly assigned to the groups, and finally the third level. This process was then repeated with students from grade six. . It should be noted here that subjects were not matched with respect to gender, although it became apparent later that this would have been appropriate. As detailed in Chapter 2, however, the significance of sex in relation to problem solving had not been sufficiently clarified, especially at these particular grade levels, to justify random assignment by gender as well as by grade and translation skills, and such an assignment seemed at the time to be unnecessarily contrived. Later, an analysis of variance of results of the translation pretest seemed to sup-port the method chosen for the random assignment to groups. (The reader is re-ferred to Chapter 4 for this analysis.) After the assignment, Group A contained 16 students, and Groups B and C had 17 students in each. At the completion of the experiment, however, the final sample sizes for treatment groups A , B, and C were 15, 16 and 17 respec-tively. PROCEDURE The Word Problem Pilot Test and the Translation Pretest were administered during the regularly scheduled mathematics lessons of the original three classes (two sixth and one seventh grade class), with the translation skills being tested two weeks before the treatments began, and the pilot test given on the last day before the treatments. Students were informed of their assignment to groups on that same day, in their official classes. At this time they were told that they were taking part in a special attempt to improve their problem-solving abilities in preparation for the upcoming administration of the Canadian Test of Basic Skills. The problem-solving lessons involved four consecutive school days, with each lesson lasting forty minutes. On these days, the groups were alternately taught one after the other, with the control group meeting one period later> as summarized in Figure 3 . 2 . Day 1 Day 2 Day 3 Day 4 Period 1 Period 2 Period 3 Group A Group B Group C Group B Group A Group C Group A Group B Group C Group B Group A Group C Figure 3.2 Treatment Sequence After the four days of lessons, students returned to their original classes and took the first posttest on the day immediately following the treatments. This meant that they did not take their posttest in treatment groups, but instead took it within their sixth or seventh grade mathematics classes. A l l students took the test under as similar conditions as possible, with the same mathematics teacher supervisi ng. The second posttest was administered on the third day following the first posttest. This test was also given to students in their original classes, but the second posttest was taken simultaneously by all three classes, with identical test instructions and time limitations provided for each of the three teachers involved as supervisors. Figure 3.3 describes the time element involved in the study. Week 1 Week 2 . Week 3 Week 4 M T W T h F M T W T h F M T W T h F M T W T h F Group A P R E T E S T P Treatment I A L T E S T T E S T Group B O Treatment T B Group C T E S T Treatment C O N E T W O Figure 3.3 Procedure STATISTICAL ANALYSIS General statistical procedures can be divided into two categories: the an -alysis of the measuring instruments themselves for overall reliability, and the analysis of the results of the two posttests. A l l calculations were performed at the University of British Columbia Computing Centre. Data Test papers and scores were obtained from the 48 subjects for each of the following tests: In addition, work and exercises were obtained from each group according to their lessons. Thus, students from Group A handed in each day's worksheet for four days, as did students from Group C . Student-generated problems were 1. 2 . 3 . 4 . Word Problem Pilot Test Translation Pretest Posttest One Posttest Two 57 collected as the result of their Instruction for each of the four days from Group B. Analysis of Tests The Word Problem Pilot Test and the Translation Pretest, as well as the two posttests, were analyzed by the LERTAP item analysis (Nelson, 1974) program to determine the reliability of the tests. An analysis of variance was also per-formed on the results of the translation pretest in order to determine whether the subjects differed significantly by grade and\/or sex, which would justify the assign-ment of students to groups. Analysis of Posttest Results Since more than one dependent variable was involved, a multivariate analysis of variance was performed on results of the posttests by the computer pro-gram MULTIVAR (Multivariance, 197.8). Three factors were being tested for s ig-nificance: sex, grade, and treatment effect. These factors can be pictured as a completely randomized, 2x2x3 fully crossed factorial design, utilizing two post-tests as the dependent measures: M Differences with regard to Sex, Grade, and Treatment are regarded as fixed rather than random effects, and therefore the following multivariate linear model for fixed effects was considered appropriate: (The model should be viewed as two-dimensional in consideration of the fact that two dependent measures were used.) Y i j k m = V + \u2022* i + 3 j + X k + \u00b0e ij + SA jk + B ik + a g X ijk + E ; j k m where Yjj| < m represents the score for a particular subject on the dependent measures: , Vf represents the general mean, a constant for all the subjects,, th cja j represents the effect of the i \u2014 level of fsex, constant for al l subjects in that population, th g j represents the effect of the j \u2014 level of grade, constant for all subjects in that population, XX k represents the treatment effect of level k, constant for all subjects in that population, with the terms j j , 8X j ^ , aX and aBX representing the interaction between the corresponding effects, and e |||