TEACHER CLASSROOM BEHAVIOR AND STUDENT SCHOOL ANXIETY LEVELS 93

FLANDERS, N. A. Interaction analysis in the classroom: A manual for observers. Ann Arbor: University of Michigan Press, 1960.

FLANDERS, N. A. Some relationships between teacher influence, pupil attitudes, and achievement. In B. J. Biddle & W. J. Ellena (Eds.), Contemporary research on teacher effectiveness. New York: Holt, Rinehart, &Winston, 1964. Pp. 196-231.

GARRETT, H. E. Statistics in psychology and education. New York: David McKay Co., 1966. HO~GH, J. B. Classroom interaction and the facilitation of learning: The source of instructional

theory. In E. Amidon & J. Hough (Eds.), Interaction analysis: Theory, research, and application. Reading, Mass. : Addison-Wesley, 1967. Pp. 375-387.

LIPSET, S. M., LAZARSFIELD, P. F., BARTON, A. H., & LINZ, J. The psychology of voting; an analysis of political behavior. In G. Lindzey (Ed.), Handbook of social psychology. Cambridge, Mass.: Addison-Wesley, 1954. Pp. 1124-1175.

MANN, J. A comparison of the effects of direct versus vicarious individual and group desensitization of test-anxious students. Unpublished master’s thesis, University of Arizona, 1969.

SAMPH T. Observer effects on teacher behavior. Unpublished doctoral dissertation, University of M’ichigan, 1968.

SCHELKUN, R. F., & DUNN, J. A. School anziety and fhe facilitation of perfonncinee. Paper presented at the meeting of the Midwest Psychological Association, May 1967.

SCOTT, W. A. Reliability of content analysis: The case of nominal coding. Public Opinion Quarterly,

SOAR, R. S. Pupil needs and teacher- upil relationships: Experience needed for comprehending reading. In E. Amidon & J. Houg! (Eds.), Interaction analysis: Themy, research, application. Reading, Mass.: Addison-Wesley, 1967. Pp. 243-250.

SUINN, R. M. The desensitization of test anxiety by group and individual therapy. Behaviour Re- search and Therapy, in press.

WOLPE, J. Psychotherapy by reciprocal inhibition. Stanford, Cdif.: Stanford University Press, 1958.

1955, 19, 321-325.

ERROR PATTERNS I N PROBLEM SOLVING FORMULATIONS GEORGE KENNEDY, JOHN ELIOT, GILBERT KRULEE

Northwestern University

High school teachers frequently have noted that many “competent” students find algebraic word problems very difficult to solve. Presumably, such problems are difficult because they require students to combine natural language and mathemati- cal symbols for their solution. Computer scientists, interested in designing programs which will “converse” with students in a step-by-step solution of algebraic word problems, have found that the translation from natural language into symbolic formulation is indeed a complicated process. Their efforts to design such dialog programs, however, have yielded information about the formulation process which can be helpful to teachers and students alike.

One overriding implication to be drawn from these efforts a t computer simula- tion is that the translation from natural language into symbolic formulation is a most important first step in the successful solution of word problems. In this study, attention is focused upon the formulation processes used by students who are asked to solve two sets of algebra problems : the first set being word problem with informa- tion presented in English, the second set being number problems with information presented strictly in terms of mathematical symbols. The purposes of this study were (a) to determine if students differ with respect to their solution patterns for algebraic word problems; and (b ) to determine how students use the information given to them in problem statements.

94 ERROR PATTERNS IN PROBLEM SOLVING FORMULATIONS

The two hypotheses of this study will be tested by analyzing the verbal and written protocols of students in keeping with the formalisms described by Kuck and Krulee (1964), Bobrow (1964), Paige and Simon (1966) and others. Formulation, or the act of solving algebraic word problems, can be described in terms of five admittedly overlapping, steps. First, the student reads the problem statement and forms a rough hypothesis about the kind of problem he is facing. Second, the student looks for information requiring translation into mathematical symbols. Third, the student ascertains what kinds of relationships are needed to form an appropriate equation. Fourth, the student ascertains whether he has identified the physical or logical inferences needed to solve the problem. Finally, the student is ready to solve the equation and obtain a solution.

While studies of computer simulation indicate that translation from natural language into symbolic formulation is an important first step, it may not be the only step which causes difficulty for students. Besides restructuring natural lan- guage and determining the relationships needed to form appropriate equations, students must also be able to identify needed logical or physical inferences. For example, the problem statement “a given mixture contains two ingredients, one of which is 35%” requires the student to infer that the other ingredient is 65y0 of the mixture. Similarly, in the statement “one race driver’s speed is 150 miles per hour while another’s is 120 miles per hour,” the student must infer that the driver with the greater speed will cover more distance in the same amount of time.

Although not all algebraic word problems require these five steps for solution, and although not all students solve problems in this step-by-step sequence, the oral and written protocols of solution patterns from earlier studies indicate that our five-step sequence holds for a high proportion of students and, therefore, is a useful one.

The two hypotheses of this study were prompted by interest in the ways in which students restructure the information given to them in problem statements. The first hypothesis was tested as two sub-hypotheses: (a) that the difference be- tween less and more able students is a function of their ability to recognize the relationships needed to form an appropriate equation, and (b) that the difference between less and more able students is a function of their ability to add or to identify the logical or physical inferences needed to solve the problem.

The second hypothesis is based upon the assumption that a student’s formula- tion pattern may be strongly influenced by the way in which the problem is stated or presented. If the tasks of combining natural language and mathematical symbols are indeed the point of difficulty for less able students, then these students would be more likely to respond to information as it appears sequentially in the problem statement rather than attempt to initiate a formulation by assimilating the problem statement as a whole.

DESIGN AND PROCEDURES Sample

Twenty-eight high school ‘uniors from the Evanston Townshi High School in Evanston, Illinois, were the subjects for this study. All subjects were drawn from t%e same teacher’s average and ad- vanced classes.

The average and advanced students were initially placed in the regular or advanced cl-s respectively, based upon their aptitude scores obtamed from grammar school and during their early

GEORGE KENNEDY, JOHN ELIOT, GILBERT KRULEE 95

years of high school. The Preliminary Scholastic Aptitude Test (offered in the fall of their Junior year) further demonstrated a marked differeuce in their performance. The average students scored 52 and 59 on their verbal and quantitative scores respectfully (78th percentile) while the advanced students averaged 65 and 68 (95th percentile). The twenty-eight students were selected in order to have an equal number of average and advanced students and also to have an equal number of boys and girls in each group.

All of the students live in Evanston, Illinois. A survey of their backgrounds indicates that the majority come from families with professional backgrounds. All of the subjects took the College Entrance Examination and, when questioned after the study, all indicated a strong desire to con- tinue on to college. They also expressed interest in entering professional fields, ranging from engineer- ing and medicine to teaching. In general, these students well reflect the upper to middle class spectrum of a city which ranks in the top ten percent of cities having high per capita income. Lzst of Problems Used In This Study

comparable difficulty as a consequence of student performance in earlier studies. Table 1 contains a list of the problems used in this study. These problems were judged to be of

No. 1

No. 2

No. 3

No. 4

No. 5

No. 6

TABLE 1 3Y - 4 = 4Y + 8

8 4 --

Y = ? A man is three times as old as his son. Eleven years from now he will be only twice as old as his son. How old is the son at present? B(X-B) = X-(2-B) X = ? Mary can wash the dishes in a half hour, and Tracey can wash them in twenty-five minutes. After Mary has worked for ten minutes, Tracey begins to help her. How long will it take both girls to finish the dishes? A-X2 = 4X-21 -X-56 + 9X = 4X X = ? An automobile radiator contains four gallons of a mixture of water and antifreeze. If the mixture is now twenty percent antifreeze, how much of the mixture must be drawn off and replaced by pure antifreeze to get a mixture containing thirty percent antifreeze?

Each of these problems was chosen for particular reasons. Among the number roblems, for example, problem no. 1 is a routine numerical problem. In problem no. 3, however, tBe number of variables, n, exceeds the number of equations, m. The subjects must recall that one variable has to be held constant in order to solve for all the other variables. Similarly, problem no. 5 contains a set of simultaneous equations that must be solved.

Among the word problems, problem no. 2 contains problem elements with mathematical relation- ships between problem elements, all of which are contained in the problem statement. Problems no. 4 and no. 6, however, require all three steps for solving: that is, mathematical interpretation, recognition of the relationships among problem elements, and identification of the needed logical or physical inferences.

Deswiptum of Experimental Procedures A soundproof recording room was used for the study. After each subject was seated, the following

instructions were read: “We are interested in the different ways in which people go about solving algebra problems.

W e want you to solve a series of problems for us-problems which you already know how to solve. I will hand you each problem one at a time. Please read it aloud to me, and then write down the

step-by-step solution of it. As you solve the problem, say aloud anything which comes to mind, regardless of how trivial it may seem to you. I will record your comments about each problem.

Remember, our interest is in the way you solve the problem-not whether you are able to get the right answer for it. Do you have any questions?”

96 ERROR PATTERNS IN PROBLEM SOLVING FORMULATIONS

Subjects were permitted up to twelve minutes for each word problem and six minutes for each numerical roblem. If a subject completed a problem within the time allotted, the experimenter made the iblowing statement:

“DO you have any additions or corrections? I do not mean to imply that your answer is either right or wrong. Rather, I would like you to work on the problem until you me satisfied with your solution.”

The infrequency of silence following the first problem indicated that.most subjects readily adapted themselves to the thinkmg-aloud process. If students were silent wlth subsequent problems, the experimenter asked “are you thinking anything now?” If a subject proceeded with an incorrect etep or hesitated in taking a next step, he was asked to “explain that part of your solution.” Subjects were discouraged from continuing with a problem if they were unable to explain an earlier step of the solution procedure.

The customary approach by information rocessors has been to flow chart the protocols of sub- jects. Because of the number of subjects a n t t h e number of problems given, however, tables were constructed for recording the sequence of steps employed by the subject rather than constructing a flow chart for each problem and each individual. Table 2 provides an analysis of word problem no. 6. The numbers indicate the sequence of steps taken by three subjects.

1 2

1

1 5

TABLE 2.

4 5 6 7 8 9 1 0

3 4

4 3 6 7 5 8 9 ANALYSES OF PROBLEM NO. 6

1 2

11 12 I

3 4

13 14

5 6%

RESULTS OF THIS STUDY In general, the numerical problems offered little difficulty for the subjects of

this study. The word problems, however, were considerably more difficult for the less able student. Specifically, there were 15 out of 28 correct solutions for prob1c:ms no. 4 and no. 6 among the honor students as against 5 out of 28 solutions among the regular students for the same problems.

The first hypothesis of this study was that the difference between less and more able students is a function of their ability to restructure information given to them in the problem statement. This hypothesis was tested in terms of two sub-hy- potheses. The fist sub-hypothesis was that the difference between the two groups of students would be a function of their ability to recognize the relationships needed to form an appropriate equation. An examination of data from this study indicates that both groups of students recognized the relationships needed for equations equally well and that this sub-hypothesis was not supported. The analysis of numerical and word problem data follows.

Among the algebraic numerical problems, problems no. 3 and no. 5 required the subjects to perceive relationships between problem elements. The data for problem no. 3 offered some support for the first sub-hypothesis. In particular, this

GEORGE KENNEDY, JOHN ELIOT, GILBERT KRULEE 97

problem requires variable “B” to be held constant and variable “X” to be factored out of the “XB” term. All subjects recognized the status of variable “B,” and those who failed to solve the problem did not factor out the “X” from the “XB” term. Consequently, the results of this problem are deceiving. It was not the failure to recognize the relationship, but the lack of ability to perform a mathematical opera- tion similar to adding, subtracting, etc.

Problem no. 5 is an example of a problem containing simultaneous equations. Since the problem requires the problem solver to solve for “X” and the second equation only contains the “X” term, the first equation is superfluous. Three girls and three boys from the advanced group attempted to solve the first equation before solving the second equation. Five girls and three boys from the average group also solved the first equation prior to recognizing that the second equation was necessary and sufficient by itself.

The analysis of the solution processes of the algebraic word problems leads us to conclude that there is no difference between average and advanced students in perceiving relationships between problem elements. Up to four relationships existed in problem no. 2, and all but one student in the advanced group solved the problem.

All the subjects were able to convert a “half hour” into “30 minutes” on prob- lem no. 4. All the subjects, with the exception of one girl from the average group, recognized that “1/3 of the job was done” and “2/3 of the job was left to go.” Three girls from the advanced group attempted to solve for the time to finish the dishes by taking the average time of each girl doing the remainder of. the dishes alone. One girl and one boy from the average group attempted the same. Only one subject thought that the number of dishes to be washed was needed in order to solve the problem.

In problem no. 6, two advanced students did not specify all the relationships between problem elements. Three average students did not include all the relation- ships in their protocols. The recognition of the relationship between percentages was demonstrated either by stating the percentages of the ingredients or by actually computing the amount of antifreeze and/or water now and after the alteration of the contents of the mixture.

These results do not indicate differences in the problem-solving abilities of average and advanced students with respect to recognizing needed relationships. The difficulties encountered by average students were also encountered by the advanced students. In both cases, only a few students made mistakes.

The second sub-hypothesis was that differences between average and advanced students are a function of their ability to identify or to add the logical or physical inferences needed to solve the problem. This sub-hypothesis was supported by the data of this study. Differences between average and able students are a function of their ability to identify needed logical or physical inferences. When we compared the cumulative frequency for each step of the problem solutions, the greatest dis- crepancy occurred not at the step requiring students to restructure information or to recognize relationships, but rather a t the step which required students to identify logical or physical inferences in the problem statement.

The second major hypothesis of this study was prompted by the possibility that a student’s formulation pattern could be strongly influenced by the way in

98 ERROR PATTERNS I N PROBLEM SOLVING FORMULATIONS

which the problem is stated or presented, Specifically, it was argued that less able students would formulate word problems sequentially as facts appeared. An analysis of data from this study partially supports this hypothesis. In testing this hypothesis, it was first necessary to devise some form of scoring procedure. Since most students achieved successful solutions for problem 2, this problem was omitted from the analysis. In problems 4 and 6, the problem statement was broken down into five elementary statements which appeared in sequence. This order was labelled 1-2-3-4-5 and used as a basis for comparison with the actual sequence of each subject. From the protocols, the actual sequence in which the elements were processed can be inferred. Let ri be the expected order in which the i t h element will be read, and by definition, ri = i. Let Ai be the actual order in which the i t h element was read by a particular subject. Then a deviation score summarizing the departure of the j t h

subject from the standard order can be.defined as

5

That is, it is the sum of the absolute values of the differences between the actual snd expected order for each of the i elements. Using this scoring procedure, deviation scores for those students who correctly solved a problem and for those who failed to solve a problem were obtained. The results were in the expected direction (al- though in a statistical sense, they were marginally significant) and the hypothesis was partially supported.

SUMMARY The purpose of this study was two-fold: to determine how students process

information given to them in algebraic word problems and to determine if there are differences in the ways in which able and less able students formulate word prob- lems. Specifically, two hypotheses were tested. The first hypothesis was that the difference between less and more able students is a function of their ability to re- structure the information given to them in word problems. To test this hypothesis, problem solving was defined in terms of five steps-three of which required re- structuring of information. Two of these restructurings-that of mathematical interpretation of problem elements and the identification of relationships among elements-involve a rather direct translation from words into symbols. The data from this study indicate that the two groups of students do not differ in their ability to carry out these two types of restructurings, but do differ in their ability to add or to identify inferences about logical or physical assumptions. It is of interest to note that this latter type of restructuring “goes beyond the information given” in that the inferential process is much less directly related to specific words in the problem statement. To some extent, this finding confirms the results obtained by Paige and Simon (1966)) who used somewhat similar problems.

By way of control, students were given three numerically stated problems such that no restructuring of information was needed in order to solve the problems. The able and less able students performed equally well with this set of problems.

GEORGE KENNEDY, JOHN ELIOT, GILBERT KRULEE 99

The second hypothesis stated that students who are unable to solve algebraic word problems formulate the problems sequentially as facts appear, and con- sequently tend to choose a solution pattern based upon information given early in the problem statement. The rationale for this hypothesis was that the successful students will first try to understand the problem, and that the ability to understand does not depend upon a sequential processing of the information. On the other hand, the unsuccessful students would not develop an understanding of the prob- lem, partly by processing the information sequentially. This hypothesis was partially confirmed by the data.

The results of this study indicate that teachers should be less concerned with teaching students to define the relationships between problem elements and more concerned with helping them to identify the logical and physical assumptions made in the problem statement. Further, these results provide some useful guidelines for computer programmers interested in designing dialog programs capable of adapting themselves to “converse” with poor algebra students one way and with able students in a different way.

REFERENCES BOBROW, D. G. STUDENT, a question-answering system for high school algebra word problems.

AFZPS Conference Proceedings. Vol. 26. Fall Joint Computer Conference, Baltimore, 1964. KUCK, D. J., & KRULEE, G. K. A problem solver with formal descriptive inputs. In J. Tou & R.

Wilcox (Eds.), Computer and information sciences. Washington: Spartan Books, 1963. PAIGE, J. M., & SIMON, H. A. Cognitive processes in solving algebra word problems. In B. Klein-

munts (Ed.), Problem solving research, methods and theory. New York: Wiley, 1966.