generalization abilities in mathematics

8/2/2019 Generalization Abilities in Mathematics

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Generalization Abilities in MathematicsAuthor(s): Reuben S. EbertReviewed work(s):Source: The Journal of Educational Research, Vol. 39, No. 9 (May, 1946), pp. 671-681Published by: Taylor & Francis, Ltd.Stable URL: http://www.jstor.org/stable/27528729 .

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GENERALIZATION ABILITIES INMATHEMATICS*Reuben S. Ebert

State Teachers College, Buffalo, New YorkEditor's Note: The building of meanings, concepts, and generalizations isan important function of elementary mathematics. The author presents dataon these for a large group of elementary school pupils and the relations withother abilities.

INTRODUCTIONBuilding concepts, developing meanings, establishing principles, dis

cerning relationships, and formulating and understanding generalizationsare interrelated aims in the teaching of elementary mathematics. The formulation and the understanding of generalizations, sufficiently adequate forintelligent use, may well be considered as culminations of the related learnings implied above. Mathematics is not only computation and manipulationof symbols; it is also understanding of the principles of computation andof the meanings of the symbols. It deals with not only important concretesituations and experience, so very necessary for meaningful learnings, butalso with useful abstractions and generalizations which grow out of concreteexperience.

The extent of achievement in mathematical generalization is not easilyascertained. This investigation makes an effort to procure direct evidence ofsuch achievement among eighth-grade pupils by focusing attention uponpowers of observation, analysis of relationships, formulation of generalstatements, and illustration. It attempts also to discover relationships betweensuch generalization ability and both mental and reading ability. Interestingand stimulating information should result from such an investigation in anarea of recognized importance, which, up to the present time, has receivedlittle attention by research workers.

PURPOSES OF THE STUDYThis investigation proposed to ascertain the extent to which eighth

grade pupils comprehend selected generalizations in the field of elementarymathematics. More specifically, it set out to answer the following questions:* An abstract of a thesis submitted in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in the School of Education of New York Uni

versity, 1944. 671


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672 JOURNAL OF EDUCATIONAL RESEARCH [Vol. 39,No. 91. To what extent can eighth-grade pupils?(a) Present correctly an additional similar illustration of a mathematical

relationship after observation and analysis of several given illustrations ofthe relationship?(b) Write a word statement of the general truth or fact representedby several given illustrations, or specific cases, of amathematical relationship ?(c) Express in written form a specific illustration of a general truthor fact in elementary mathematics which was observed and studied in sentenceform?

2.What relationship exists between the mental ability of eighthgrade pupils and their ability to generalize inmathematics?

3.What relationship exists between the reading ability of eighth-gradepupils and their ability to generalize in mathematics?

4. What is the relative difficulty of some selected generalizations inmathematics?5.What similarities or differences exist in the abilities of eighth-grade

pupils to comprehend generalizations relating to various areas of elementarymathematics ?DEFINITIONS

1.While "ability to generalize" is recognized as a highly complexability, for purposes of this study the term shall be limited to the followingthree aspects:

(a) The ability to write a mathematical illustration exemplifying thesame relationship as in several given illustrations of the relationship.(b) The ability to write a word statement of the general truth or factexemplified by several given illustrations of amathematical relationship.(c) The ability to illustrate aword statement of a general truth or factinmathematics by writing a mathematical relationship.2. "Selected generalizations" shall mean the generalizations to be usedin the generalization test constructed for this study.3. "Comprehension" shall mean the degree of understanding of the

generalization in question, or of the aspect of generalization in question, asevidenced by the evaluation of pupil responses to the generalization-testitems.

4. "Relative difficulty" of a generalization shall mean the difficultyrevealed by the degree of comprehension of that generalization by the pupils.


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May, 1946] GENERALIZATION ABILITIES IN MATHEMATICS 61$5. "Areas of elementary mathematics" shall mean those areas intowhich the generalizations used in the generalization test of this study shallbe divided; such as, operations with integers, operations with common

fractions, measurement, etc.CONSTRUCTION OF TESTS AND SELECTION OF PUPILS

Elementary textbooks, courses of study, professional literature, andassociates of the investigator suggested rules, principles, and other statementsof relationship that could be considered as generalizations in mathematics.Sixty generalizations inword-statement form were thus selected, with specialcare given to clarity and simplicity of statement, accuracy of meaning, andprobable range of vocabulary of eighth-grade pupils. Practically all of thegeneralizations selected were in pure mathematics, in which the relationshipsare relatively certain. These generalizations were sent to twenty-six selectedjudges in various parts of the United States, twenty-one of whom judgedthem as worthy, probably worthy, or of doubtful worth as knowledge foreighth-grade pupils. Only those generalizations judged worthy or probablyworthy by seventy-five per cent or more of the judges were retained fortest construction. Suggestions from judges assisted in the refinement of thestatements and in the addition of a few.

Each retained statement served as the basis for the creation of severalmathematical relationship patterns exemplifying the relationship stated in

sentence form. For example, one statement was worded as follows: "Twoor more numbers have the same sum regardless of the order in which theyare added.'* The relationship patterns constructed to exemplify thisparticular generalization were:

2+3 + 4= 9 2+3+5+6=163+ 2= 5 3+ 2+ 4=9 3+ 2+ 6+5=162+ 3= 5 3+ 4+2 = 9 3+ 5+ 6+2=164+2+3 = 9 6+3+2+5=16Relationship patterns were constructed to exemplify each retained sentence statement, and were sent to selected teachers of mathematics, who had

agreed to judge the patterns. Several professors of psychology, education, andmathematics judged the patterns also. Nineteen judges appraised the patternsof relationship as valid, probably valid, or not valid for purposes of evokingthe thoughts and meanings in the sentence statements. Fifty-four patternswere thus judged valid or probably valid by seventy-five per cent or moreof the judges, and were retained for testing purposes. As a result of criti


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674 JOURNAL OF EDUCATIONAL RESEARCH [Vol. 39,No. 9cisms and suggestions from judges, a few of the patterns were alteredslightly to achieve greater total effectiveness and potency of the patterns.A preliminary study of the generalization test was made by administering it to fifty-five pupils in Bronxville, New York. This preliminarystudy provided an opportunity for consideration of such matters as, timenecessary for the administration of such a test, vocabulary difficulties in thesentence statements, ambiguities or confusions resulting from the relationshippatterns, standards for marking pupils' responses, and possible tabulationforms.

After minor changes were made in the test, its reliability was ascertained by administering it to a group of fifty-five pupils in Buffalo, NewYork upon two different occasions about six weeks apart. The two sets ofscores yielded a reliability coefficient of .94, with a probable error of .01.Thus a test was constructed and refined for the evaluation of threedefined phases of generalization in mathematics. Phase-A and phase-Bevaluation were made possible by two kinds of responses to the fifty-fourrelationship patterns as test items, and phase-C evaluation by responses tofifty-four sentence statements as items. Phase-A responses were the "imitations" of the relationship patterns, and phase-B responses were the writtensentences indicating the general truths or facts exemplified by the relationship patterns. Phase-C responses were the pupils' illustrations of the sentencestatements.

The tests used for ascertaining the mental ability and the reading abilityof the pupils were the Henmon-Nelson Test of Mental Ability?Form A,Elementary School Examination?Grades 3-8, and the Advanced ReadingTest: Form D of the Stanford Achievement Test.

Approximately 900 eighth-grade pupils from five cities and villagesof western New York participated in the study. Complete test data for 674pupils were obtained. The most descriptive statement that can be made regarding the representativeness of this sample is that the participating communities included a variety of social and economic groups, and that thepupils revealed a wide range of mental ability and of reading ability.

ADMINISTRATION OF THE TESTSThe administration of the generalization test to the approximately 900

eighth-grade pupils required five sittings of about forty-five minutes each onfive different days. The first three sittings were on successive days during


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May, 1946] GENERALIZATION ABILITIES IN MATHEMATICS 615one school week inMarch or April of 1943, depending on the convenienceto individual schools. The other two sittings were on successive days abouttwo weeks after the first three sittings. At the first three sittings, pupils weregiven the fifty-four relationship patterns as test items in groups of eighteeneach, and were asked to write like patterns of relationship (phase ?) andcomplete sentences which stated the general truths or facts exemplified bythe patterns (phase B). After the two-week interval, which was regardedas a reasonable time during which pupils might forget the patterns, the fiftyfour sentence statements were presented as test items in two groups oftwenty-seven each, and the pupils were asked to write mathematical illustrations of these statements (phase C).No rigid time limits were imposed. Pupils were allowed to work atconvenient rates of speed and to finish at different times. For purposes ofthe study it was imperative that practically all pupils attempt all items ofthe test, and previous experimentation had indicated that the time allowedfor each of the five sections of the test was sufficient for slowest pupilsto finish.

The test of mental ability and the reading test were administered at theconvenience of the participating schools at about the same time in the schoolyear that the generalization test was administered. Each of these testsrequired a working time of thirty minutes.The mental-ability test and the reading test were marked by the use ofobjective keys submitted by the publishers. All responses to the items of thegeneralization test were marked on a four-point scale as follows:

3 right and adequate2 right, but not adequate1 not right, yet not entirely wrong0? wrong or omittedDescriptive standards for each of the four intervals on this linear scalehad emerged during the preliminary study of the test and had been refinedat the time of the reliability study. It is believed that they were sensibly

adequate in classifying the pupils' responses numerically, as identified above.These descriptive standards were followed closely by the investigator anda trained assistant, as the generalization test papers were marked.RESULTS

The following results are based upon test scores made by the 674 eighthgrade pupils from the fiveWestern New York communities that participatedin this study, Schools A, B, H, K, and L.


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676 JOURNAL OF EDUCATIONAL RESEARCH [Vol. 39,No. 9The sample of pupils ranged in mental age from 9 years 10 monthsto 18 years, with a mean mental age of 14 years 7 months. The range in

reading age was from 9 years 10 months to probably about 19 years, depending on the validity of extrapolation of high reading ages. The mean reading

age was 14 years.The mean mental age for the 324 boys was 14 years 4 months, and forthe 350 girlswas 14 years 9 months. The mean reading age

for the boyswas 13 years 11 months, and for the girls was 14 years 1 month.The mean mental ages for the five different schools ranged from 14years to 15 years 5 months, and the mean reading ages ranged from 13years 4 months to 15 years 6 months. The five schools ranked themselvesin exactly the same order with respect to mean mental ages and the meanreading ages.Total generalization scores ranged from 55 to 470, with a mean scoreof 314.6. A total score of 486 would have been a perfect score. On any oneof the three phases of the generalization test, a perfect score would have

been 162. Phase-A scores ranged from 3 to 161, with amean score of 123.3.Phase-B scores ranged from zero to 155, with a mean score of 82.1, andphase-C scores ranged from 30 to 159, with a mean score of 108.4.The difference between the mean phase-A score and the mean phase-Bscore was 41.2. Between the mean phase-A score and the mean phase-Cscore there was a smaller difference, 14.9, but the difference between themean phase-B score and the mean phase-C score was 26.3. The criticalratios of these differences to their standard errors were greater than three.In the order of differences mentioned above, critical ratios were 23.5, 9.6,and 14.2. All differences between mean phase scores were statisticallysignificant.Intercorrelations for phase scores were .72 for phase A and phase B,.68 for phase A and phase C, and .85 for phase B and phase C.The 674 pupils were divided into five mental-ability groups, each having a range of approximately one-and-one-half years. Also, they were dividedinto five reading-ability groups, each group except the highest having arange of about one and one-half years. Total generalization achievement by

mental-ability groups from M-l to M-5 (highest to lowest) was indicatedby mean scores of 376.7, 339.7, 294.7, 254.3, and 189.3. All differencesbetween the mean total generalization scores of the ten possible pairs ofmental-ability groups were statistically significant. Critical ratios ranged from


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May, 1946] GENERALIZATION ABILITIES IN MATHEMATICS 6113.56 to 12.30. Achievement on the three phases of generalization by mentalability groups revealed differences inmean phase scores of which 27 out of30 were statistically significant. As measured by group means, the five

mental-ability groups were convincingly different with respect to achievementin generalization.The mean total generalization scores for the five reading-ability groupsfrom R-l to R-5 (highest to lowest) were 373.5, 343.0, 315.0, 274.0 and205.8. All differences between the ten possible pairs of mean scores werestatistically significant, with critical ratios ranging from 3.26 to 14.29. Withrespect to achievement on the three phases of generalization by readingability groups, the differences between mean phase scores were such thatonly three out of thirty were not statistically significant. As measured bygroup means, evidence was convincing that the reading-ability groups weredifferent with respect to generalization achievement.

Correlations between mental scores and phase generalization scoreswere .45 for mental and phase A, .51 for mental and phase B, and .52 formental and phase C. Correlation of mental scores with total generalizationscores yielded a coefficient of .54.As compared with correlations between mental scores and generalization

scores, correlations between reading scores and generalization scores wereslightly lower in one case and slightly higher in three cases. The correlationcoefficient was .44 for reading and phase A, .54 for reading and phase B,and .53 for reading and phase C. Reading scores and total generalizationscores produced a correlation coefficient of .56. None of these coefficientsreported is sufficiently high to indicate strong relationships.

Using the mean of 2,022 marks of 3, 2, 1 or 0 assigned to each generalization as an indication of achievement on a particular generalization,achievement scores varied from a high of 2.7 to a low of 1.2. Generalizationsdealing with number relationships in one process were easiest for the pupils,and those dealing with common-fraction relationships were the most difficult. For the total group of pupils, for all mental-ability groups, and readingability groups, phase A was easiest; phase C was next in difficulty; andphase B was the most difficult.The total generalization scores of the thirty-six pupils who made identical high scores of 86 on the mental test ranged from 167 to 470. The totalgeneralization scores of the eighteen pupils who made identical low scoresof 59 on the reading test ranged from 134 to 378. Similar wide variation


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678 JOURNAL OF EDUCATIONAL RESEARCH [Vol. 39,No. 9in generalization ability was revealed in other very narrow mental-ability

groups and reading-ability groups.Only a small difference was revealed between the mean generalizationscores of boys and girls, and this difference was in keeping with the differ

ences in mean mental scores and mean reading scores.Larger differences were revealed among the mean generalization scoresfor the different schools. The high mean score was 344.6 and the low was274.5. Although the five schools ranked themselves in exactly the same orderwith respect to mean mental scores and mean reading scores, they rankedthemselves quite differently with respect to mean generalization scores.

CONCLUSIONSThe degree to which the findings of a particular investigation result inconclusions pertinent or applicable to other situations is an important consideration. It is somewhat difficult to state the degree of representativenesswhich may be assumed in this study. Certainly the eighth-grade pupils who

participated are not representative of the nation at large or of New YorkState. Such factors as methods in teaching and time spent on the study ofmathematics probably affected the generalizations test results of this investigation, but these factors were not evaluated.What may be said is that all teachers of these pupils attempted to follow the same State syllabus in elementary mathematics, and that the pupils,all in a single grade, varied widely in mental ability and in reading ability.All conclusions pertain to the sample of pupils used in this investigation andare limited by the purposes, plans, and methods of the investigation.The following conclusions seem reasonable on the basis of evidenceobtained in this investigation:

1. Generalization ability in mathematics, as defined, identified, andmeasured in this investigation, varies greatly among eighth-grade pupils.2. Eighth-grade pupils differ widely in their ability to achieve on eachof the three phases of generalization studied in this investigation.3. For eighth-grade pupils, the writing of general truths or facts insentence statements is by far the most difficult of the three phases ofgeneralization studied.4. Eighth-grade pupils write mathematical patterns exemplifying thesame relationship as in observed patterns much more easily than they illustratesentence statements by writing mathematical patterns.


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May, 1946] GENERALIZATION ABILITIES IN MATHEMATICS 6795. Among eighth-grade pupils, and as measured by correlations of testscores, there is a very strong relationship between ability to write sentencestatements of the general truths or facts in observed mathematical patterns,and ability to write mathematical illustrations of sentence statements. The

relationships between each of the above stated abilities and the ability towrite amathematical pattern of the same relationship exemplified by severalobserved patterns are strong.6. As measured by mean scores of various groups, generalizationachievement as a composite of the three phases studied in this investigation,or as any single phase achievement, is convincingly different for differentmental-ability groups and different reading-ability groups, always in favorof the higher ability groups.7. Differences among mental-ability groups and reading-ability groupsin the ability to write sentence statements of the general truths or facts inobserved mathematical patterns are much more pronounced than in the othertwo phase abilities studied in this investigation.8. As measured by correlations of test scores, significant relationshipsexist between generalization ability and mental ability and between generalization ability and reading ability.9. Although significant relationships exist between generalization ability and mental ability and between generalization ability and reading ability,and although different mental-ability groups and different reading-ability

groups differ significantly in generalization ability, there is considerableoverlapping among both mental-ability groups and reading-ability groups ingeneralization ability.

10. As measured by mean achievement scores, the generalizations studiedin this investigation differ widely in difficulty.11. As classified and evaluated in this investigation, generalizations

dealing with number relationships in one process are easiest for eighth-gradepupils, and those dealing with common-fraction relationships are the mostdifficult.

GENERALIZATIONS USED IN THE STUDYWord-statement generalizations used as test items which called for

pupil illustration, and used as bases for construction of relationship patternsare appended for reference. They are as follows:1. Two or more numbers have the same sum regardless of the order inwhich they are added.


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680 JOURNAL OF EDUCATIONAL RESEARCH [Vol. 39,No. 92. Common fractions that have the same denominators may be sub

tracted.3. The area of a rectangle equals the product of its length and width.4. Annexing a zero at the right of a whole number multiplies the number by 10.5. 50% of a number is one-half of the number.6. The distance an object travels at a steady rate is the product of therate and the time.7. Zero added to a number gives a sum which is the number itself.8. Common fractions that have the same denominators may be added.9. The perimeter of a square equals four times the length of one side.10. Dropping a zero from the right of a whole number divides thenumber by 10.11. 25% of a number is one-fourth of the number.12. The cost of a number of things, when the price of each is the same,is the price of each multiplied by the number of things.13. Two or more numbers have the same product regardless of theorder used in multiplication.14. Moving the decimal point one place to the left in a number dividesthe number

by10.

15. The value of a common fraction is not changed if the numeratorand the denominator are each divided by the same number.16. The perimeter of a rectangle equals the sum of two lengths andtwo widths.17. Per cent means hundredths.18. The average cost of one thing, when the cost of a group of thingsis known, is the cost of the group divided by the number of things.19. Zero subtracted from a number gives an answer which is thenumber itself.20. The value of a common fraction is not changed if the numeratorand the denominator are each multiplied by the same number.21. The volume of a rectangular solid is equal to the product of itslength, width, and thickness.22. Annexing two zeros at the right of a whole number multiplies thenumber by 100.23. 100% of a number is the number itself.24. The average of a group of numbers is their total sum divided bythe number of numbers.25. 1multiplied by any number gives a product which is the numberitself.26. The value of a common fraction is, in general, changed if the samenumber (except zero) is added to the numerator and to the denominator.27. Dropping two zeros from the right of a whole number dividesthe number by 100.28. The area of a square is equal to the square of its side.


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May, 1946] GENERALIZATION ABILITIES IN MATHEMATICS 68129. 200% of a number is twice the number.30. Zero multiplied by any number gives a product of zero.31. Moving the decimal point one place to the right in a number multiplies the number by 10.32. Per cents may be changed to decimals by dropping the per centsign, %, and moving the decimal point two places to the left.33. The value of a common fraction is, in general, changed if thesame number (except zero) is subtracted from the numerator and from the

denominator.34. The area of a triangle equals one-half of the product of its baseand altitude.35. Multiplying by 100 and then dividing by 2 gives the same answerasmultiplying by 50.36. Decimals may be changed to per cents by moving the decimal pointtwo places to the right and then annexing the per cent sign, %.37. Quotient times divisor equals dividend.38. When dividing by a fraction, the same answer is obtained by

multiplying by the inverted fraction.39. Moving the decimal point two places to the left in a numberdivides the number by 100.40. The sum of the angles of any triangle is 180?.41. In dividing any number by 1, the quotient is the number itself.42. Dividing the numerator by the denominator changes a proper fraction to a decimal fraction.43. Moving the decimal point two places to the right in a numbermultiplies the number by 100.44. The number of decimal places in a product is the sum of thenumber of decimal places in the multiplicand and in the multiplier.45. Multiplying by 100 and then dividing by 4 gives the same answeras multiplying by 25.46. When one number has been subtracted from another number, the

answer added to the subtracted number equals the other number.47. A common fraction is an indicated division.48. Annexing zeros to decimal fractions does not change the value ofthe decimal fractions.49. The base angles of an isosceles triangle are equal.50. The value that a figure in a whole number represents depends uponits position in the number.51. Dividing by 100 and then multiplying by 2 gives the same answeras dividing by 50.52. A common fraction expresses a ratio.53. Dividing by 100 and then multiplying by 4 gives the same answeras dividing by 25.54. The rate at which an object moves is the distance divided by thetime.

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