Syntax, Vocabulary, and theVerbal Arithmetic Problem

William J. LinvilleDepartment of EducationIndiana State University

Terre Haute, Indiana 47809

A basic purpose of mathematics taught in the elementary schoolis that such learning should serve as a foundation of knowledge uponwhich the individual may draw for further study of the subject, oruse to solve quantitative problems which arise later in life. Variousexperimental programs have been developed in efforts to make thestudy of mathematics more enjoyable and more meaningful for pupilsin the elementary grades. Yet, with all of the progress which hasbeen made in the areas of methodology and materials used in theclassroom, it would appear that the desired end result of elementarymathematics programs, the ability to apply knowledge to uniqueproblems, has not been reached.The literature related to elementary school mathematics abounds

with evidence that pupils in the elementary grades still experienceconsiderable difficulty in the solution of verbal problems, even thoughthey may give evidence of an understanding of the particular funda-mental operation or operations required to perform the necessarycomputations. Numerous research findings serve to illustrate theimportance of reading skills as they are related to facility in solvingverbal problems. While not all agree as to the specific kinds of readingskills which are essential, agreement is evident as to the vital roleof reading competence in the ability of elementary students to solveverbal problems.

PURPOSE

The purpose of this investigation was to study whether the degreeof syntax used in the sentences which state verbal arithmetic problemsand/or the level of vocabulary used in the statement of the problemsare factors which contribute significantly to the degree of difficultyof the problems when the computational operations are held constant.

HYPOTHESES

The major null hypothesis to be tested in this study was: thereis no difference among children in verbal arithmetic problem testscores when syntax and vocabulary are varied from easy to difficult.

In order to further delineate relationships, the following sub-hypoth-eses were tested:

152

The Verbal Arithmetic Problem 153

1. There is no difference between sexes of children in verbal arithmetic problemtest scores when syntax and vocabulary are varied from easy to difficult.

2. There is no difference between children of high intelligence and children oflow intelligence in verbal arithmetic problem test scores when syntax andvocabulary are varied from easy to difficult.

3. There is no difference between children who scored high and children who scoredlow in reading achievement in verbal arithmetic problem test scores when syntaxand vocabulary are varied from easy to difficult.

4. There no significant interactions among the above noted variables.

MATERIALS

Since no instruments were available which could adequately testthe hypotheses of this study, a series of four tests was preparedby the investigator. Each of the four tests consisted of 10 verbalarithmetic problems. The syntax and vocabulary in the test problemswere allowed to vary in the following manner.

Test I was termed "Easy Syntax; Easy Vocabulary." The syntaxof the sentences stating these problems (excluding the question itself)was constructed so that each of the sentences resembled a kernelsentence as nearly as possible without losing meaning in the problem.A minimum of 70 per cent of the words used in the statement ofthe problems (excluding the question) were taken from the Thorndikeand Lorge listing of the first 1,000 words.

Test II was termed "Easy Syntax; Difficult Vocabulary." The syntaxof the sentences stating these problems (excluding the question) wasconstructed so that each of the sentences resembled a kernel sentenceas nearly as possible without losing meaning in the problem situation.A minimum of 70 per cent of the words used in the statement ofthe problems (excluding the question) were exclusive of the Thorndikeand Lorge first 1,000 words.

Test III was termed "Difficult Syntax; Easy Vocabulary." Theseitems were constructed so that the sentences which stated the problems(excluding the question) each contained a main clause and a subordi-nating clause. A minimum of 70 per cent of the words used in thestatement of the problems (excluding the question) were taken fromthe Thorndike and Lorge listing of the first 1,000 words.

Test IV was termed "Difficult Syntax; Difficult Vocabulary." Thesentences which stated the problems (excluding the question) eachcontained a main clause and a subordinating clause. A minimum of70 per cent of the words used in the statement of problems (excludingthe question) were exclusive of the Thorndike and Lorge first 1,000words.The same computational operations were used in each of the four

tests. Five of the problems required the addition of two three-digitnumbers. The remaining five required the subtraction of one three-

154 School Science and Mathematics

digit number from another three-digit number. The two fundamentaloperations of addition and subtraction were chosen because of theirclose relationship to each other and because they represent operationsto which beginning fourth-grade students have normally been exposed,thereby lessening the chance of incorrect solutions to problems dueto unfamiliarity with the operations involved.Examples of test items constructed are:Test I: "A tree was 295 inches tall. It grew 314 inches. How tall

is the tree now?"Test II: "A rancher herded 295 steers into a corral. There were

314 steers previously in the corral. How many steers are in the Corralnow?"

Test III: "If you drove a care 295 miles in one day and had driven314 miles the day before, how many miles would you have drivenin both days?"Test IV: "When 295 marbles are poured into a container already

containing 314 marbles, how many marbles are in the container?"Validity estimates of the test items with respect to syntax were

obtained by submitting the completed test items to a panel of fourfaculty members from three universities. Three of the four panelmembers work primarily in the area of elementary school mathematics;the fourth in elementary language arts. In addition to validity estimates,the panel reacted with respect to clarity and appropriateness ofcomputational operations selected.A pilot study involving 52 fourth-grade students was conducted

to determine (1) the time period needed for the administration ofthe tests; (2) the adequacy of the test directions; (3) whether thesubjects could indeed handle the arithmetic computations necessaryto affect correct solutions; and (4) reliability estimates for the fourtests. Internal consistency reliability coefficients, using the Kuder-Richardson Reliability Coefficient Twenty, for Tests I, II, III, andIV were .81, .79, .88, and .64 respectively.

PROCEDURES

A random assignment of the four tests was made among 422fourth-grade pupils from 18 fourth-grade classes in 12 six-gradeelementary schools in West Central Indiana. Of the 422 pupils amongwhom the random assignments had been made, 14 were absent atthe time of the testing. This resulted in a total of 104 subjects eachtaking Tests I, II, or III. The remaining 96 subjects of the 408 totalwho were present for testing, took Test IV.

Since the data collected from the test administrations was to beanalyzed with respect to differences between sex, I.Q., and reading

The Verbal Arithmetic Problem 155

achievement, these data were recorded for each of the subjects tested.From the cumulative record of each subject the most recent intelligencetest score, as measured by The SRA Primary Mental Abilities Test,and the reading achievement test score from the previous year, asmeasured by the reading comprehension subtest, Iowa Tests of BasicSkills, were recorded.Following the 18 separate administrations of the tests, the tests

were scored by the investigator with the number of correct responsesfor each subject taken as the dependent variable.

FINDINGS

The research design used to analyze the data from the 18 testingsessions was a2x2x2x4 unweighted means analysis of variance.This method was chosen because of the concern for possible interac-tions among the variables of sex, intelligence, reading achievement,and levels of difficulty in syntax and vocabulary as they were relatedto solving verbal arithmetic problems. It was also decided to use theF-test to determine the significance of the observed differences. Asignificant difference was to be accepted when the probability TypeI error was beyond the .05 level of significance.

In addition to the above, the results of Scheffe Tests of Contrastare reported to further examine significant differences found betweenthe main effects.

Table 1 presents the results of the data analysis for the test ofthe major hypothesis and the subsequent sub-hypotheses. The tableincludes the sums of squares, degrees of freedom, mean squares,and F-ratios for sex, intelligence, reading achievement, and the fourtreatments. The same data is included for all of the tests for interactionbetween variables.Those F-ratios presented in the table represent significance at the

.05 level are so indicated by an asterisk. An additional superscriptis placed beside those F-ratios which also represent significance atthe .01 level.The major null hypothesis tested was:There is no difference among children in verbal arithmetic problem

test scores when syntax and vocabulary are varied from easy todifficult.The analysis of variance applied to the test scores produced an

F-ratio of 9.06 between the means of treatments. From an F-table,using 3 and 400 degrees of freedom, the resulting ratio was significant(.05 level is 2.62). On the basis of these data it was found that therewas a significant difference between the four treatments. Therefore,the null hypothesis was rejected.

156School Science and Mathematics

TABLE 1: ANALYSIS OF VARIANCE AND F-VALUES FOR THE TESTS OF VERBAL PROBLEMSOLVING

Sums of Degrees of MeanSource Squares Freedom Square F-ratio

Sex(A)0.7810.780.19I.Q. (B)64.57164.5715.44*3Reading achievement (C) 327.511327.5178.30^Treatment (D)113.75337.929.06*13A xB0.0310.030.01A xC14.06114.063.36A xD3.0431.010.24B xC4.6914.691.12B xD7.1232.370.57C xD18.0936.031.44A x B x C9.8119.812.34A x B x D6.3132.100.50A x C x D3.0731.020.24BxCxD16.5635.521.32A x B x C x D 12.4434.150.99Error1572.703764.18

^(1,350; .01) = 6.72.^^O; .01) = 3.84.

The first sub-hypothesis tested was:There is no difference between sexes of children in verbal arithmetic

problem test scores when syntax and vocabulary are varied fromeasy to difficult.The analysis of variance applied to the test scores produced an

F-ratio of 0.19 between the means of boys and girls. From an F-table,using 1 and 400 degrees of freedom, the resulting ratio was notsignificant (.05 level is 3.86). On the basis of these data it was foundthat there was no significant difference between the means of boysand girls. Therefore, the first sub-hypothesis was accepted.The second sub-hypothesis tested was:There is no difference between children of high intelligence and

children of low intelligence in verbal arithmetic problem test scoreswhen syntax and vocabulary are varied from easy to difficult.The analysis of variance applied to the test scores produced an

F-ratio of 15.44 between the means for high intelligence and lowintelligence. From an F-table, using 1 and 400 degrees of freedom,the resulting ratio was significant (.05 level is 3.86). On the basisof these data it was found that there was a significant differencebetween means of the two intelligence levels with respect to theirproficiency in solving the verbal arithmetic problems. Therefore, thesecond sub-hypothesis was rejected.The third sub-hypothesis tested was:

The Verbal Arithmetic Problem 157

There is no difference between children who scored high and childrenwho scored low in reading achievement in verbal arithmetic problemtest scores when syntax and vocabulary are varied from easy todifficult.The analysis of variance applied to the test scores produced an

F-ratio of 78.30 between the means for high reading achievementand low reading achievement. From an F-table, using 1 and 400 degreesof freedom, the resulting ratio was significant (.05 level is 3.86). Onthe basis of these data it was found that there was a significantdifference between means of the two reading achievement levels withrespect to their proficiency in solving the verbal arithmetic problems.Therefore, the third sub-hypothesis was rejected.The fourth sub-hypothesis tested was:There are no significant interactions among the above noted varia-

bles.None of the F-ratios obtained for the interactions among variables

from the analysis were significant. Therefore, the fourth sub-hypothesiswas accepted.While there was a significant difference found between the four

treatments and the major null hypothesis was rejected, the analysisof variance did not reveal the points of difference. Scheffe Testsof Contrast were used, therefore, for testing differences betweenpairs of means. These tests revealed that:

1. Easy vocabulary scores were significantly higher than difficult vocabulary scoresacross difficulty levels of syntax at the .01 level.

2. Easy syntax scores were significantly higher than difficult syntax scores acrossdifficulty levels of vocabulary at the .05 level.

CONCLUSIONSAn analysis of the data for this study suggests the following

conclusions:

1. Syntax and vocabulary level can both be determiners of difficulty in verbalarithmetic problems. Vocabulary level could be more crucial in determining successthan syntax.

2. Boys and girls appear to do equally well in solving verbal arithmetic problems.3. Pupils of higher ability can be expected to meet with considerably greater success

in solving verbal arithmetic problems than pupils having less ability.4. Pupils who have scored high in reading achievement can be expected to experience

greater success in solving verbal arithmetic problems than pupils who scoredlow in reading achievement.

REFERENCES

BALOW, IRVING H., "Reading and Computation Ability as Determinants of ProblemSolving," The Arithmetic Teacher, XI (January, 1964), 18-22.

CHASE, CLINTON I., "The Position of Certain Variables in the Prediction of Problem-Solv-ing in Arithmetic," Journal of Educational Research, LIV (September, 1960), 9-14.

158 School Science and Mathematics

FAY, LEO C., "The Relationship Between Specific Reading Skills and Selected Areasof Sixth Grade Achievement," Journal of Educational Research, XLIII (March,1950). 541-547.

GILMARY, SISTER, "Transfer Effects of Reading Remediation to Arithmetic ComputationWhen Intelligence is Controlled and All Other Factors are Eliminated," 77ie ArithmeticTeacher, XIV (January, 1967), 17-20.

HANSEN, CARL W., * ’Factors Associated With Successful Achievement in Problem Solvingin Sixth Grade Arithmetic," Journal of Educational Research, XXXVIII (October,1944), 111-118.

HUNT, KELLOGG W., Grammatical Structures Written at Tl^ree Grade Levels, NationalCouncil of Teachers of English, Champaign, Illinois, 1965, 159 pp.

LOBAN, WALTER, The Language of Elementary School Children, National Council ofTeachers of English, Champaign, Illinois, 1963, 92 pp.

THORNDIKE, EDWARD L. and LORGE, IRVING, The Teacher’s Word Book of 30,000 Words,Teachers College, Columbia University, New York, 1944, 274 pp.

TREACY, JOHN P., "The Relationship of Reading Skills to the Ability to Solve ArithmeticProblems," Journal of Educational Research XXXVIII (October, 1944), 86-96.

PARALLELS BETWEEN MODERN SCIENCE, MEDIEVAL ALCHEMY

A chemist, internationally known in scientific circles for his discoveryof radiocarbon-14, believes there are parallels between modern-day scienceand medieval alchemy.

Martin D. Kamen, this year’s Patten Lecturer at Indiana University, saidalchemy, a science which tried to fuse religion, ethics, and natural philosophy,held sway over the civilized world for almost 2,000 years. Alchemists triedto have a control over nature in a time when most people believed in witches,demons, and angels.He attributes the decline of alchemists in about 1700 A.D. to their attitude

of secrecy and keeping a distance from the everyday man. Alchemy wasshrouded in mysticism and the desire for secrecy. Alchemists wrote textsunder false names, used indefineable terms, and either spent their time readingthe works of the ancients in Latin and Greek or bungled in the laboratoriestrying to change base metals into gold.

But, contrary to popular belief, alchemists were the major intellects oftheir time. Alchemy claimed the attention of Newton, Descartes, Goethe,and others. Those who worked in laboratories, "peering into crucibles andhot stoves," created the basis for modern chemistry. Alchemists must notbe denigrated, even though they didn’t know what they were doing.What caused the decline of alchemists?They suffered from an image problem, just as scientists do today. As

revealed in Geoffrey Chaucer’s "The Canterbury Tales," alchemists wereconsidered "disreputable unprincipled charlatans."Quoting from the 14th Century work. Dr. Kamen said alchemists were

described as "smelling like a goat in a threadbare coat."The alchemist was cast as a character who didn’t care about his appearance,

spending all his time and money trying to convert metal into gold. The alchemist,in Chaucer’s tale, even admitted that all his work was a lie, that he triedto delude everybody and take their money.