i ^^ L747 UC-NRLF B 3 ma 7m V *" ■i ^-^ /- Cf^ (1^ V DIAGNOSTIC TEoTS OF ABILITY TO ADD IIITEGERS. By RUDOLPH, LIiroQUIST THESIS Submitted in partial satisfaction •f the requirements f«r the degree •f llASTER OF ARTS EDUG.VriON in the GRADUATE DIVISION of the HIWIRSITY ( 3F CALIFORNIA Dec« 1922. 9-. !a/' f/^'T Digitized by the Internet Arciiive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/diagonstictestsoOOIindrich PART ONE INTKODIIGTORY STATFl/ff-^NT Tills study la Intended to rer.ult in tho constr-uctlon of a raeaHui'lng devlco by mcfine, of v/hlcVi the teacher will be able to determino the particular .addition habltt, , the dovolopinent of whlcVi reodti to be emphaslzod wltli her cIhsb. The t^rowth of the teat raovt-iiifiit. huti ti'un a ti't)i.u;ndoua Increase in the number of teats, but It aeoins Uiat fiuffjolont ernphat'.j h has not been placed upon tl-ie development oi' tho&e that are of dlatsnoatlc, rather than general value, 'i'he teacher needs to know, not only, how i^i-t)ut la the general ability of hei- class or of a r»r^'l-t;uiMr individual, but altio, in ceat) they are not as goo(] aa they thould be, tVie particular habita t}u> t are undevelop- ed. All testa are in some sense or to aoiae degree diagnos- tic. Even aueyi afl teat as the Courtia L>tandard Researcyi Tests altVioui^h they werti not iaaued aa definitely diagnostic teatfi are Bucli lo the extent to wlil ch they enable the teaclier' to cUaermine whijch iother or not it lb addition corablnatlona, as such, that tl-ie children are hf.iV\nR troiible with or wheUier it is carr-ying, or wViether "440 1* .X. ' ' ■ ' ' - ■ ■• •■ ■ ' " I i ■■ .i/ ■•>. !•, , I I n III n I li I 11, I ■ il L . Page 2. General tests, such as Vioody-McCall Mixed i'und- amentals, while not intended to be diagnostic, have never- theless been interpreted as being such by many teachers. Form One for instance, has ten addition examples scattered throughout the test. These vary from simple examples such as addition of the simple combinations to the addition of a sixteen number colvunn, including decimal points. Others of the ten examples involve the addition of broken columns, the addition of figures written on a line connected by a plus sign and v hich must be re-written before they are add- ed. In fact, there are no two examples that involve exact- ly the same addition habits. Failure to do the examples in addition in this test could not be attributed by the teacher to a single or to even a limited number of causes . Even though the child had a preponderance of erx'ors in addition examples of this test, the teacher v^ould not know whether it \vas ignorance of combination or unfamiliarlty v/ith fuch a fcrm as 2-f 3, or inability to cope with form such as is found in broken column addition, or inaccuracy due to the necessity of re-writing the example. Even though the child should fail in the one or two examples involving one of the above habits, the teacher could not be certain that the Page 3 failure wan clue to a disability that required special drill to overcome it. A test, in order to be diagnostic must not | only include examples limited to a single ability, in so far as it is possible to so limit them, but there must also be sufficient Instances of each type of example so that the child will have an opportunity to demonstrate conclusively his ability or inability to cope with examples involving a particular habit. V/oody in the Vioody ■'arithmetic i^cale has devised a test v;hich is more diagnostic than either of the tv/o that have been discussed. But even it cannot hope to give more than a general measur-e of power to arid, for there are not sufficient instances of each addition habit nor are the few of each habit that do occur conveniently segregated and arranged. It included everything from a simple combination to the addition of mixed numbers, with about one or two in- stances of each addition habit. Even though a teacher could quickly spot the failure to perform the two instances of one operation, she could not be certain that this f£iil\are was due to anything but chance, for t'wo examples involving any parti- cular Aritlvnetic habit are not sufficient to measure con- clusively a child's ability therein. Page 4 Dr. V/alter ^. Monroe in his diagnostic tests in Arithj-netic goes further than anyone else in meeting the requirements of a diagnostic test namely, that, the speci- fic habits be isolated in, so far as po&sible, anc that each be measured under conditions that render comparison of achievements in each test comparable with achievement in any others. Dr. Monroe has sought to meet the first require- ments by providing three tests in the addition of integers: (a) single column addition, three numbers high (b) single column addition, thirteen numbers high (c) foui' column addition, five nur.ibers high The thr^ee testf-; given should show v/hether or not a pupil's addition disability involves inaccuracies dufe to fatigue or inaccuracies due to carrying. Beyond this it does not seem to be diagnostic, '^here may be other and more constantly operative causes for Inability than either of those mentioned. Such a one miglrit be broken column form, or inability to do the combinations themselves even when these are isolated and the additional difficulties incident to the use of them in actual sitviations in-oerative. Concerning children who succeed in column addition (five by nine) we may of course assume that they know the combinations and that they m8.ke no mistakes in carrying but Page 5. concerning children *o fail in this test v- e do rfot know to which of the above causes to ascribe their inabilitj'^ to add. Nor do we knov/ whether blank spaces in some of the columns, such as occur in broken column addition would not be a real obstacle to the pupil. It seems desirable, therefore, to isolate each of these habits even more than Dr. Monroe has done, and to measure performance in each under comparable conditions, so that we may single out the cause of the inaccuracy and remedy it. Another point Dr. Monroe has not taken into account fully is that of reducing achievements in the different habits to comparable units. Grade norms are, of course, in a measure, such a unit. If eight and six examples correct are the norms for a grade in each test, respectively, the teacher can of course compara roughly the achievement of her class in one test with its achievement in the other but to the extent that a pupil deviates from the average it be- comes increasingly difficult to make an accurate comparison between the accomplishments of any one pupil in two tests. If a series of tests, is to be diagnostic in a mot t helpful way, it seems desirable that the teacher should be able to re(iuce the achievements in the several tests, particularly in the case of children that deviate most frojn the average, to a common denominator. Page 6 Standard deviation position in a normal age dis- tribution woulr] of coTarse furnisb sucri a common iinit. The difficulty of using such a measure in the past has been that of providing the teacher with the means whereby raw scores can be readily transmuted into standard deviation values. A part of this study will be the construction of such a table for each of the tests so that scores in each may be quickly transmuted into a common unit of measure and the teacher be enabled to compare directly the achieve- ments of a child In each test with his achievements in any other test and thus locate his particular disability with respect to addition-integers. The proposed tests aim to be diagnostic by virtue of: 1. Isolating, in so far as is possible, and measuring, those habits in addition of integers v;hich seems to be of significance in the contribution which they make to the child's ability to add integers, buch the following s eem to be: (a) "•ddition of combinations (b) Bridging the tens (c) Single colvunn addition (d) Addition with carrying (e) Cjpoken coliunn addition 2. Providing means whereby teachers may compare direct- ly the achievements of a pupil in one test with his achieve- ments in each of the othf;rs. Page 1. OUTLINE OF PI^OCKDURF: I. Selection of examples - By random sampllnc of single digits, and their combinations into examples. This procedure was used in ell tests except the first, in which case the addition combina- tions and their reverses were used. . II. Arrangement of examples in order of difficulty - The relative difficulty of the examples was de- termined by preliminary testing of grades three to six in tv/o schools. Note ; Those examples, in the case of which a high percentage of error was due to blurred numbers, were placed at the end of each test. There were nine such in- stances in the five tests, ("ae' in test II; "h ' in test III; "g", "j" and "r" in test IV; "b", "a", "g," and "j" in test V) pages. III. Giving of tests to 1400 children in grades three to six inclusive in seven schools in Berkeley, California. Instructions to teachers and tabulation of results are con- tained on pages 13-20. IV. Tabulation of data - The zero scores in test II that were due to mis- understanding of instructions were omitted. Also, the re- sults of the sixth grade of one school were omitted because a re-test shov/ed first test to have been erroneovis . V. Re-testing of the high third, high fourth, high fifth, and high sixth grades of one school after an inter- Page 8. val of two weeks to get data on reliability of test. VI. Statistical treatment of results - 1. Inter-correlation of each test v;i th every other test. 2. Correlation of each test a second giving to the same children after an interval of two weeks . 3. Correlation with VJoody-McGall. 4. Means and standard deviations for each dis- tribution. 5. Computation of sigma values for each score in each test. Page 9 EVALUATING RESULTS VALIDITY OF TEST - It has not seemed necessary to gather evidence as to the validity of the test beyond that furnished by the inter-correlations. They seem to indicate that a single type of ability is being measured. They are uniformly high, except in the cases of test II with other tests. Since the correlations in which test II plays a part are consistently lov>fer, and since this test resulted in an undue number of zero scores, and, also, since its self-correlation was ].ower than the other tests, it is assumed that its validity as v/ell as its reliability is lov/ and it is therefore thrown out as of little value. The low correlation of test V with V/oody-McCall is inter- preted as being due to the fact that ^''oody-McCall tests a great many abilities other than addition. RELIABILITY - or the degree to which it is con- sistent in its diagnosis, would have been determined by checking one half of the test against the other half. A simpler met?iod however, though a somewhat less satis factorj^ one, was used, namely that of correlating one giving of the test against another giving of the same test. These corre- lations were uniformly high except in the case of test II. The latter v.as discarded as unreliable. Page 10 OBJECTIVITY - There has been no evaluation made of the objectivity either the giving or the scoring of the test. It was sought to increase the former as much as possible by simplifying the instructions as much as was consistent with clear understanding. Since the instructions are not in any case complex it is not thought that lack of objectivity of giving is a serious factor. Steps taken to insure objectivity of scoring were the preparation of an an£v;er sheet with answers so spaced ths-t they could be placed directly under the examples and answers checked. In future giving of the test it is planned to have children score their ov;n papers since the case of adminis- tration seems of greater importance than the slight sacri- fice in objectivity of scoring due to the fact that the children do it. No check was made on the relative accuracy of teachers and pupils' scoring. SCALING - The examples were arranged in order of difficulty by giving them to 300 children in grades tJiree to six in two schools, each pupil having time to work all. In the final form all examples were included, arranged in order of difficulty without any attempt being made to select only those tlriat represented equal increments of difficulty. Page 11 RANGE OF APPLICABILITY - It was desired to secure a test which could be used in grades three to six inclusive. Reference to graphs on pages 39-43 and tables of means and standard deviation on pages 45-46 will show that test I is valuable in grades three to sijc inclusive though slightly less so in the latter two as shov/n by the niunber of perfect scores made there. Test III and IV easily measure grades three to six. Test V while it measures grades, four, five and six well, does not measure grade three. Ther efore , ex- cept for test I in grade sir. and test V in grade three the tests have a range of applicability suited to these grades. NORlvIS - Norms for each test, in to far as the test- ing of 1400 children in seven schools of one school system, can be said to be a basis for norms, are stated in th( following forms : 1. Means for each grade three to six inclusive. 2. Means for each age eight to twelve inclusive. 3. Each of the above expressed in terms of a Sigma index (score -t- sigma) reduced to a scale of 1-200. Tables 21-30 inclusive give value of accomplishment in each test in terms of sigma. In the last mentioned table a score which gives an equivalent of 100 is the norm for the age or grade classification into v.hich the pupil falls. These norms are accurate only to the extent that the follow- ing conditions hold: Page 12 1. That each dit tribution approximates a normal proba- bility curve. 2. That the reliability of each test approaches unity as measured by r-epeatin^^ the tests; self correlation .8 or over for all except test II. 3. That the ran£,e of the tests is such as to makE them applicable to the group to be measured. Reference to distri- bution tables 7-16 inclusive shows that except for test I in grades six and test V in grade three the tests do measure the children. 4. That the inter-correlation approach ninity. (Accept for correlations with test II these are all plus 0.8 or higher. USE OF TABLC OF SIGlJlA INDEX VALUES 1. Norm for any sge or grade is alvyays such a score as will give a Sigma Index of 100. If a child's accomplishment in each test is such as to give him a Sigma Index of IOC in each, his scores are satisfactory for his age, assuming that he is of average intelligence. If, however, he has a sigma score of 80 in test IV and IOC in test III his accomplishment in the farmer is not as much as we have a right to expect from him. Concerning each of the tests in wlriich a child gets a Sigma Index of less than 100 v;e may say, v/ithin the limits of the accuracy of the test that he is not doing as well as he could do in it. Page 14. So, while this administration of the test is not for the purpose of determininc the abilities of our own pupils in addition but rather as a step iril the construction of t. test, still I believe that it v/ill be possible for you to tell some- thing from the results as to the type of difficulty which should be vorked upon with your own class or with particular pupils. Do not expect too much in this respect, however, from this preliminary test. DIRECTIONS FOR GIVING Give the tests in the order indicated above. 1. Distribute the first test, printed side down on the desk of each child. 2. Read the following directions to the children: "This is a test to see how well you can work addition combinations. V/ork as rapidly as you can until you have fin- ished all of them. I'll give you all time to finish. V/hen you have finished turn your paper over, printed side c'own, and sit quietly until the others have finished. Now, turn your pe.pers over, v/rite your name, grade and date at the top of the paper and go to work." For each of the following tests, say: The next is a test in (insert name), ""rite your name, grade and date at the top of the sheet. V/ork all of the examples and v/hen you have done so turn yoior pape r over and wait quietly for the others to finish. 3. '-'ive no help in the working of the examples except to make clear numbers that may be blurred or to make clear the directions above. 4. Note carefully in minutes and seconds how long it takes each of tlie first five and the last one to finish. Page 13, Thousand Oaks School. Berkeley, California. October 23, 1922. II'iSTRUCTlOHS TO T5ACHERS FOR GIVING PRISLIMIKARY TESTS Teachers of Grades 3,4,5 and 6. Please give the following tests today, preferably during the regular Ai'ithmetic period and in place of the Arithmetic Lesson. 1. Addition Combinations. 2. Bridging the Tens. 3. Column Addition without carrying. 4. Column Addition with carrying. 5. Broken column Addition. This is a preliminary test for the purpose of determin- ing the relative difficulty of the items in each test. There is, therefore, to be no time limit to the test, but each cViild is to work all the examples. When the relative difficulty of the elements has been determined these will be re-arranged and the final test psppared. The purpose of the test, in it's completed form, is to help diagnose audition difficulties. It should be pos?ible by the use of these tests to determine .just why a class or an individual pupil is weak in addition so that steps may be taken to remedy the difficulty. Later this same step may be taicen with respect to difficulties in the other fundamentals. Page 15. ADDITIOri COMINATIOHS Name Grade Date Right: a b c d c f> i-> h i .i k 1 7 ' " 2 8 5 9 s s 4 5 1 1 2 £ 1 1 4 _1 2- 2 4 _5 4 5 ri n P q r o t u V IT X 3 1 1 2 A 1 1 6 5 1 3 2 9 o •1 5 7 6 1 1 8 y z aa ab ac ad ac af a."; ah ai a,-] 1-2 320 4 223 65 2 6 _3 1 1 -l 1 ± Z 1 A ± 9. ak al am an ao ap aq ar as at au av 2 4 3 5 o 7 2 3 7 2 4 S Z i _3 7_ ^ 7 _5 _2 _5 a\,- ax ay az ba bb be bd bo bf br. bh 7 3 8 6 9 8 7 3 2 4 5 ^ ^ ^ _5 8 4 £ ± 7^ 6 bi b.i bk bl bn bn bo bo be, br bfc bt 9 4 8 5 a 2 4 6 6 9 9 7 3 8 _3 _8 4 8 _9 4 S 4 7 bu bv hm bx by bz oa cb cc cd cc cf 4 9 3 8 7 9 V 8 6 7 9 8 7 9 2^ £ _7 8 7 £ _8 _9 Cf. ch ci cj ck cl CI^x en CO cp , ^^ -.-• 9 5 9 8 q 5 5 S 8 7 6 6 5 5 6 8 9 7 9 7 5 8 ^j-'-a 2 Nnno a. 14 + 9 = b. 14 * ^ z c 18 ^ 4 - d 12 + 8 = e 19 * 2 = f 18 + 8 c 5 19 -^ 3 = h 19 + 9 = i 14 * 8 - 3 13 + 9 = k 17 + 7 = 1 14 » 7 - BRIDGING HE tva:s Page 16, ■ade Date y 19 * RightG m 17 -♦- 4 = 7 : n 19 ♦ 4 - z 15 4. 7 = o 18 + 3 = aa 19 X 8 = P 12 ■»• 9 = ab 19 4- 6 s q 16 4- 6 M ac 15 A 6 * r 13 4. 8 = ad 18 * 5 = wr 15 J. 7 r ae IS * 5 a t 18 ♦ f5 = af 17 * 6 » u 17 * 9 = ag 15 J. 8 = V 16 + 9 - ah 17 4. 8 = 17 17 A 5 = ai 18 * 7 m X 15 * 9 = ak 19 18 X 5 s 9 • SINGLE COLUMN ApDITlON Name ^ Grade •, __ Date Right c & b c d o f K h 1 .1 k 1 4 1 8 2 8 1 9 9 4 1 1 4 3 5 6 6 1 4 6 4 S 1 8 5 6 8 6 6 2 7 4 1 6 8 2 5 6 8 3 6 4 Z 1 » 9 8 5 X m n p q r t u ▼ ■17 X X 1 7 2 5 3 ■■■ i 6 6 5 4 4 8 3 6 2 5 7 7 4 8 5 9 1 2 8 9 2 1 7 S 9 2 7 7 5 7 1 9 4 S 7 5 6 5 8 5 6 7 1 6 2 6 7 6 1 2 6 5 3 1 2 7 7 5 7 Xf c 3 Page 17. • - BROKEN COLUT-N /DOIT ION Nane b — «i c u &r-do ' :.i b-- R Right; h i a i. 52 39 83 7 802 "86 8 5 572 St 602 1 983 82 1 8 66 68 73 43: 395 565 79 382 57 794 48 2 830 cei 4- 83 99 372 467 4 987 757 79 39. TOl 580 63 527 44 347 962 69 68 Ti k 1 m n P q r c t 3 60 48 550 140 552 95 65 430 7 977 3 9 88 918 28 379 63 75 84 45 793 217 5 02 752 89 8 8 79 ■ 42 62 29 3 7 72 562 340 65i 33 879 21 3 135 89 210 55 56 _Z_ Name ADDiTIOK 0,'ITH CARRYING) Grade Date Rights a b c d e f K h i J 427 892 803 457' 108 295 432 249 512 717 158 992 247 205 494 857 202 789 875 9§4 173 359 392 289 774 352 359 119 710 200 k 1 m n P q ^ r G t 534 221 101 846 375 855 364 501 898 G3:-i 941 263 136 780 287 147 573 875 511 563 659 589 426 966 959 775 902 365 651 204 -^^'^ p?- - .V Page 18, T4BLE 1. RSSULTS 03? PRELIMIMARY TEST IMG- -AJ^DITION C OMBIH.AT I ONS Ex. Errors "^x. Srrora Ex. Errors Ex. "Plrrors a 1 aa 3 ba 3 ca 19 B ab 4 bb 3 cb 10 2 ac 4 DC 11 cc 13 d 1 ad 4 bd 3 cd 14 t 5 as 2 be 3 CO 15 f 1 af 1 bf 4 cf 20 & 5 as 2 bg 5 eg 11 h ah 3 bh 7 ch i 4 Ai 3 ta 10 ci 10 J 1 aj 7 uj 5 cj 14 k 3 ak 3 bk 4 ck 18 1 6 al 3 bl 3 cl 9 m am 2 bia 4 cm 15 n 5 an ? bn J 1 en 10 2 ao •» o bo 4 CO 14 V 2 aP 3 bp 5 cp 13 Q 3 aQ 1 bq 3 cq 6 r ft ar 2 br 9 S e as 4 b8 6 t at 2 bt 5 u 3 au 2 bu 7 Y 1 av 2 bv 5 W 4 aw 2 bw 8 X 2 ax bx 2 y 3 ay 8 by 9 z 2 az 5 bz 8 Pat: :!e ^^. TAhLE 2 INSULTS G¥ PR-aiMIKARY TEST liiG— BRIDGING TTIT! imiS EX. No. of Terrors FX, Ho. of Errors Ex. No. of Hrrore Hx. No. of "Hlrrors a 30 i£ 19 u 28 ae 38 b 29 1 20 V 30 af 21 c 18 m 18 w 16 e^ 31 d i;l n 17 X 21 ah 22 e 10 d 12 y 21 ni 21 f iS p 26 z ' 25 aj 15 £ 1 r> 4. iil si 24 fxU. 28 ai: 25 h 24 r ?7 sb 21 i s afc 19 J 4L t 50 ad 20 Ex. N . of :"rr or £ ^x. uO T aB" Ko.of "terrors c 1. ■x. SlliGLE COLinai ATriTIOi^ No. of No. of "rrors 'FSc. -Errors a 27 h 70 17 V 31 u 54 i 4y P b3 w 29. c 53 J 43 Q 53 X 52 d 33 k 36 r 39 y 46 e 17 1 25 s 52 f ;i7 m 2 b t 39 . g. 39 n 40 u 57 Page 20 < TABL^i: 4. RBbULTb 01' PRSLlMli4AKY TE^TIMG— APDITIOli vVITK CARKYIIjG BX. Wo. of ifirrors 44 Bx. IJo.of Srrors Ex. Ho .of iSrrors lux. K o . of Srrors 43 k 36 40 ^ 34 35 22 59 m £^0 81 46 25 n 50 36 52 73 TABLE 5. BR0KSI3 CCLUKN /J)i;iTiOH ^:x. No. of "Errors ■^. Ko.of ^x. KG. of I'lrrors ■Rx. NO. Of ^'rrors £.-■ iOl f Te- yr 92 P 58 b 95 e ll? 1 77 1 79 c 96 h 68 m 51 r fiB d bi i y4 n 51 8 uS e 77 J 135 60 t 73 TABLE 6 . i!il>j»..B'".iii OF PUriLS T>J\II>G ~ACH 7S3T GRADE Test I Teet II Teet III Test I\ r Test V TKIRD 78 SO 80 21 21 FOURTH 74 68 74 73 73 FI7Tn 77 74 77 79 80 SIXTH 87 89 88 87 R8 TOTAL 316 311 319 260 262 # # DIAGNOSTIC TESTS IN ADDITION Directions f-or giving. It is very important that the following directions be care- fully followed if we are to get from the results the hel|> we hope to get in guiding children. 1. Distribut§ tests, face downward. Warn children not to turn papers over until told to do so. 2. Have them fill m blanks on the back of the test paper. Give whatever help is necessary to insure an accurate record of each of the items. 3.- Read " Instruct ionsTIQ ^e f^upils" aloud, the children following silently. 4. Note carefully the beginning time. It is well to jot it down on a £iece of paper. 5. Allow time as follows :- Test 1 Addition Combinations 2 minutes Test 2 Bridging the tens 2 " Test 3 Single column addition 2 " Test 4 Addition with carrying 3 " Test 5 Broken column addition 3 " 6. When time is up say, "Stop. Turn papers over." 7. Collect papers. SCORING PAPERS 1. Count "Number Attempted." Jk. . This can be quickly done, since the examples are ar- ranged in rows of 10 each. Take as the number, the last one worked, disregarding those skipped. 2. Check with a t^ all answers that are incorrect. Ansv;er sheets should be cut or folded so th?.t they may be placed on the paper directly under the examples. 3. Enter the "Number CORRECT" in the blank after RIGHTS on the back of the test paper. 4. Arrange the papers of each test in order of the size of score, the hSigVjhettsooDce on top. 5. Fasten the papers of each test together by a clip or a rubber band and turn them in to the Principals office. Page 22. 11^310135759 2 2 5 7 2 51 1 1 5627231212 517 2 339384 3 2 5 4 7 a 2 1 ■ 4 4 7 2 5 3 2 4 2_ 4 I- 1 3 6 5 4 5 3 9 6 6 6 3 4 2 3 7 2 5 7 8 3 6 4 3 2 4 3 4 ■ 8 a 6 4 1 6 1 3 5 2 3 8 1 1 6 4 6 7 9_ 3 • 4 9 2 2 7 8 4 7 9 1 9 5 4 4 8 3 9 9 3 4 3 2 6 7 £ 7 6 9 7 5 9 8 9 6 8 :* 9 S 6 9 3 5 5 9 8 6 6 7 7 8 8 9 5 5 7 8 7 5 9 6 7 8 7 8 8 9 ?f)^e as. 22 (reverse side of page } AGE YSAKS kOHTHS KO. ATTEAIPTKD _110. RIGHT GRADE. NAME^ DATE SCHOOL INSTRUCTIONS TO PUPILS . Today we are to take five short tests to see how well we can work the different forms of addition. This first test is in addition combinations. When I say "BEGlli'' turn your paper over, and work as many as you can, being careful to get them right. Stop work prompt- ly when i tell you to do so. Ready — Begin. TIMB TWO KIHUTES. Test 1. Addition combinations. Page 24. 1^ 4 2 19 * 3 18 f 3 19 I 5 17 + 5 IQ I 4 16 r+ 4 17 I 4 17+7 15 + 6 14 + ■?■ 18+5 12 + 8 15+9 19+7 19 + 6 17 + 6 13+7 l4 + 8 17 + 8 19 + 9 16 + 6 16 + 9 15+7 12+9 16+7 13 + 8 IS + 8 17 + 9 19 + a 14 ^ ^ ■" 14 + 9 '^ 16 + 9 = 19 + 8 = 15 + 8 = 10 + 5 = 13 + 9 i: 8 3 4 1 4 1 2 4- 4 2 1 9 5 6 3 4 5 2 1 6 r^ eJ 4 5 2 6 7 7 1 7 6 4 6 1 3 6 2 5 6 1 6 5 7 4 7 2 3 1 7 7 1 1 1 8 6 6 7 1 4 6 8 5 8 8 4 5 2 1 6 8 3 8 2 4 2 7 X 8 6 7 9 8 3 1 8 6 7 8 9 5 2 4 6 6 5 1 6 9 3 3 5 8 .6 6 8 5 5 7 3 6 2 3 7 9 5 6 5 9 5 7 2 9 4 1 9 3 Pe(T« 25, (reverse side of page 24 ) AGE ^YSARS liCNTHS. GRADS NASitE DATE SCHOOL NO. ATTEMPTED HIGHT IIJSTRJCTIONS TO PUPILS . This is a test to see how well you can work another form of addition. READY— BEGIN . TT5ST 2. BRiDGlKG THE TEUS . AGE Y3 - BS MOiiTIiS_ GRADE KAltS DATE SCHOOL SO ATTEMPTED RIGHT IKSTRUCTIOHS TC PUPILS . This is a test to see how well you c; n work still another form of addition. READY — BEGIN. TEST 3. SIKGLB COLUMH ADDITION TIME TWO lilNJTES Ps&e 26. 803 101 512 632 221 lae 534 855 296 42'^ 247 136 875 563 268 i94 041 14 7 857 158 3 92 426 710 204 589 774 659 775 352 173 564 4&7 8<.6 892 511 24 9 375 432 717 501 573 205 780 992 651 789 S87 202 954 875 902 289 666 359 898 119 959 359 200 356 48 530 552 430 140 £5 5 75 S6 802 9 88 28 75 918 63 68 842 8 1 217 5 752 8 92 8 2 3 794 57 62 29 7 340 2 562 757 656 4 467 21 3 89 56 135 55 69 27 347 44 60 ik^ 7 3 572 83 39 52 8 995 3 95 82 977 73 983 1 602 66 435 793 379 382 4 5 830 79 563 395 48 90 42 89 372 79 79 99 83 4 987 396 379 72 '21Q 527 33 68 63 380 701 962 16 AGS Y^ARS GRADH. DATE SCHOOL Page 27, (reverse side of page 26 ) MONTHS 1^0. ATTBMPTEH NO. RIGHT INSTRUCTIONS TO PUPILS. This is a test to see how well you can worK another form of addition. READY— BEGIN . TO T^ACII'CR Nate Change in time. TBST 4 . ADI'ITION WITH CARRYING Time Tt'iree minutes AGS Y15ARS MONTHS NO. ATTISMPTED NO. RIGHT GR.U)3_ NALE__ DATS SCHOOL INSTRUCTIONS TO PUPILS . THIS is a test to see how well you csn woric still lanotber fiorm of addition. READY— JSITG IN. TaST 5. BROKEN COLUMN ADDITION TIME THREE MINUTES - LIaGImOSTIC 7H ADDITIOInI . pB^;e 28. Tefct i. pombi nations. 4 5 6 8 5 10 8 6 10 10 7 9 9 10 6 10 5 9 6 7 9 7 9 10 10 4 3 5 5 6 7 9 9 9 6 8 10 11 5 • 11 16 11 12 8 4 8 1 7 8 11 11 12 13 3 8 7 12 10 14. 18 4 13 2 7 10 11 8 3 16 14 ■ 9 13 14 12 13 14 15 16 15 13 12 16 14 15 17 12 13 15 17 Test 3. Single Column Addit ion 20 29 19 19 21 17 20 20 20 21 26 20 27 25 25 23 27 28 29 27 31 23 30 28 31 Test 4. Addit ion With 1 Garryin 1078 1376 i 2134 1777 1504 1442 663 2097 1399 758 1839 951 2292 2243 2060 IIST 1 1621 993 1871 1731 Test 5. Broke n Coluran . Addition 357 655 1428 919 1287 753 901 1613 1239 1371 17 1777 845 1370 1137 1622 1307 1066 1754 2071 193: Test 2. Bridg 21 22 22 23 22 21 24 2X infi the Ti 21 23 20 24 .336 23 25 22 25 ens . 28 22 27 22 21 23 21 ■ fee 26 27 20 23 25 26 23 15 22 i:: <■ r ' • r ts Page 29. TABLW. 7. DISTRIBUTION OF SCORES iiY GRAD15— ADDITION COMBINATION No*Correct * 3d (irade 4tVi Grafie 5th Grade 6th Grade U-o a 4-7 2 1 1 8-11 3 12-15 1 ' 16-19 20-23 5 i 2 24-27 14 1 23-31 IB 2 1 32-35 38 6 36-3y 45 18 4 1 40-4'5 74 28 4 2 44-47 36 23 13 % 48-51 41 34 12 4 52-55 28 67 23 12 56-59 15 51 20 11 60-63 7 51 33 24 '54-67 7 3? 37 P4 5S-71 7 29 36 33 72-75 3 ?.5 51 43 76-79 18 38 29 80-83 9 24 40 84-87 8 15 14 88-89 5 17 22 1 90 7 23 40 L 343 417 3&4 5«j3 Page SO, TABLH 8 DISTRIBUTIOH OF SCORES BY GRADE— BRIDGING TII^ TTSNS Iffo. Correct 3rd Grade 4th Grade 5'th Grade 6th Grade 0-1 ?2 39 9 4 2-3 14 8 4-5 10 2 3 1 6-7 26 5 9 1 8-9 29 8 2 10-11 33 8 8 B 12-13 40 20 8 4 14-15 38 35 13 4 16-17 32 37 21 11 18-19 34 45 12 18 2U-21 21 53 26 16 22-23 9 32 28 21 , 24-25 3 27 36 20 £6-27 1 21 31 23 28-29 2 14 34 32 30-^1 13 27 15 32-33 (> 9 8 25 34-35 7 23 18 3e-37 5 15 19 38-39 13 40 43 & ?24 401 351 286 k Pa^je 31, ?ABLE 9 DISTRIBUTION OF SCORES BY GRAD3--SINai^ COLUMN ADDITION No. Correct 3rd Grade 4th Grade 5th Grade 6th Grade 13 1 1 9 1 2 13 1 3 19 4 17 4 5 30 14 1 6 25 5 1 1 7 28 6 9 1 8 33 16 5 2 9 31 30 6 1 10 37 38 13 6 11 29 40 13 13 12 32 59 35 16 13 10 56 ■d'd iikJ 14 11 36 27 24 15 4 33 34 30 16 2 24 27 23 17 1 17 33 30 18 13 35 25 19 1 10 22 26 20 5 23 17 21 2 15 9 22 1 12 13 23 2 8 15 24 3 13 17 25 5 13 12 K 345 422 367 303 'J; -I ■».. - iVl I ^i J, I t Pac« '^2, -TABLE IC DISTRIBUTION OF SCORES BY GRAIR-- ADDITION WITH CARRYING No. correct 3cl Grade 4th Grade 5th Grade ' 6th Grade 26 10 1 1 14 1 2 25 2 3 26 4 2 2 " 4 33 10 7 1 5 63 15 1 4 6 43 33 7 2 7 34 42 15 13 8 37 58 j 23 16 9 24 52 i 32 25 10 8 36 32 28 11 3 45 i 54 36 12 7 38 1 49 38 13 1 1 26 42 39 14 12 19 32 15 1 18 18 24 16 3 17 19 17 4 8 11 18 10 7 19 3 7 11 20 5 5 11 ^1 ^ 344 417 345 319 ^ .:[ CI ex li^ !t: 5 SI Page 33, TABLE i>ISIHI£uTIOB 0? SCCK^a 2Y GH.'O}'^ — BPOKT^N COLUM? ADDITICM Jbio. Correct 34 Grade 4th Gradg 5tb Grsade 6tb Grade 116 la 2 1 48 13 2 2 Z 46 25 8 2 3 45 SO 8 8 4 44 58 25 17 5 24 57 31 27 12 65 ^y 37 1 4 54 54 50 8 43 58 41 9 4i 5k; 10 16 37 32 ii 21 19 12 <0 11 16 13 1 11 9 14 J ^ 4 14 15 1 1 13 16 3 12 17 1 2 ^ 18 1 2 4 19 4 20 3 If 341 415 360 365 Pa^.;-?? M. TABLE DISTRIBUTION OF SCORES BY AOfH— ADDITION CCM.BI1U7I0K 1 Wo.correct . 7 6 ■ 9 10 11 12 13 1 1 1 1-4 2 1 1 5-9 1 1 10-14 1 1 1 15-19 20-24 1 4 1 1 25-29 8 8 2 30-34 3 18 13 8 1 35-39 4 31 ii6 4 6 1 40-44 3 50 1 40 ! 14 7 2 45-49 2 28 29 16 10 o 2 50-54 1 2S 44 55 22 10 1 55-59 1 14 37 34 23 7 1 60-64 5 9 30 3G 37 13 1 65-69 1 5 19 42 41 15 4 70-74 10 22 55 45 21 5 75-79 1 7 17 38 32 17 7 30-84 2 12 18 26 13 9 85-09 • 11 20 23 IS 9 90 7 17 25 25 7 S S2 ^17 319 363 300 145 46 J TABLE 13. Ti « ,^ o '* *% DISTRIBUTION OF SCORTi;S BY AGTS— BRIDGING THE T^.UQ -AGE- Ixo. Correct 7 r 8 9 10 11 12 13 10 24 16 9 2 1 3 3 1 ii-3 8 8 3 4-5 3 5 3 5 e.7 1 13 14 2 3 1 8.9 1 19 6 5 2 1 lu-ll 4 14 15 11 11 1 12-i-? 2 26 17 8 11 1 . 14-15 1 24 21 17 9 4 ' 16^17 4 25 21 27 lb 2 1 itt-iy 21 32 19 ■^l 6 >0-?l 3 19 33 31 10 7 3 22-23 11 17 22 33 10 ' 24-25 1 21 31 18 9 4 iJ6-27 1 3 12 ki5 23 a 3 p.a-29 4 12 25 S9 11 29 •■^0-31 1 5 21 18 6 3 32-33 2 9 12 18 5 2 34-35 1 10 13 15 7 1 ^ 36 2 5 8 5 7 3 37 1 15 34 33 27 12 H 19 211 1 303 333 288 113 i 72 ^"i I / # \^^3 36. TABLTi: 14, UibTRliiUTlOll Oi"- SCORES BY AGE— o.l.GLH COL^r-i: ATT^T.CK - So. Correct 7 b i# xu 11 13 a 5 1 1 to 5 1 2-3 16 tt 3 4*5 k, ii4 ly D <• 6-7 4 50 13 13 2 3-9 2 4«; oy 15 13 4 iU-11 b ^9 (iw 42 23 7 12-13 i .^'J 65 81 44 16 5 ?.4.15 1 17 47 57 51 16 9 ir,-i7 4 >] 48 54 5 16-3 9 ££: 44 38 24 7 20-21 1 9 24 r.l 13 ■» t i:;j-ir3 1 5 11 14 17 7 24 y ^0 8 C 2 2b 1 16 11 7 £i 17 ^j:;6 v.i;to 5 5 296 io5 48 -?) PatTe ^B, TABLE 16. DISTRIBUTION OF SCORES BY AGS— BR0KJ5N COLUMN ADDITION -AG S- No. Correct 7 a 9 10 11 12 iS 5 58 39 11 2 1 4 25 8 7 5 2 2-3 4 58 45 25 14 6 2 4-5 5 44 64 83 46 15 8 6-7 1 20 74 91 75 28 9 8.9 12 49 73 81 30 7 10-11 2 18 43 43 19 7 12-13 1 4 13 13 16 4 14-15 1 i 4 5 12 7 4 16-17 1 6 2 7 4 18-19 2 3 3 1 4 20 \ 2 , 2 H ^- 19 221 308 360 298 134 47 ' ?«ge *S9. ADDITIOM COMBIl^ATIONS. uH ■ 3^ /& 6th Grade. ^S ?i^ /(^ ^^ >^2J ^T- /4 ■ 31 5th Grade. / r 4th Grade. ^ I > 3rd Grade. " '\ J' '» ^C' — ^t" — i ' »^" ""» T' — t" ■■■ ■% ) r- — -1 ]i r r — t h • ■ >»' • ' J rn. if/, /V 'Y X3 Z7 3/ 3s 3 1 f3 *7 -i/ -i'S -ij 6 J 0? ?/ To' "7/ ^3 8-f ■*o 4. 3 J3R1DGOG THE T3NS. Pai;e 40. 6th Grade. J — '- Sth Grade. n 4th Grade. ^^ J?6 3rd Grade. / 3 3r 7 ^ ^7 o O- /> /^ 2/ ^J ;^J' 2/ 2? 3/ 3J' 37 3^ .i> J> 6 7 5^ ? ^o // ^1- '3 'V /J- y^ /> 'f /9 2_ li y/ 'i' /(, '? 'i' ''> yjiJ -^4 91' 1% H^ ^ ra^e 43. Jt 1 BROKEN COLIMN ADDITION. >-J si 6th Grade. f r -^ I — ' >— 1 ^ _n i/y - 36 . ' 1 5th Grade. %1 - WV^I«. /f ' 1 1 r I ' S_-^_ 1 .>^ 1 s-^ tt--^' - Jt • 4th Grade. 7'' - . x /X — . ' f 1 — J L-u-j—i , . p « . , » V 3 ^ ^ s - C 7 J^ ^ / D '■ /- (, ^5 . V 1 y / fe / -> ' > > f i. • 3rd Grade. 1 » r / ;?. 3 y. or (6 7 F f /<• 'f t i, «i t)^ /-J^ /fe /y /* /y ^< Page 45. CO « to CM oo « CM o CM s ^ CO o E-' cq H O I I CO 2; o M I CO CO )Si t CO 00 CO CO C3 in CM 00 C5» to « to o to CM f-4 10 to to CO r-l U3 CM • CO to • CM 10 M H CM CO 1. TABLS 25 SIGaA INET^i VALJt-L: _? SC^: PrS IK ?5ST Y. Score 8 yrfei 9 /rs 10 vrs, 11 yrg, la yrf 2 A 6 8 10 11 12 13 - 14 15 16 le Id 20 M^ 92 Jti'JL ice X'iU 1^8 1&6 164 172 IflO LIS I9e S.QO 65 100 106 liJ 116 124 iro IZG 14; 14 6 It a U5 IVl 1''7 183 5.S82 65 7X 7<> 83 31. 92 9B 103 108 114 iiy li-4 i.'^.y I'b 141 146 IM 157 If.? 168 5.376 57 63 69 74 30 36 38 104 110 llf igs 127 l?,y< \7,9 145 151 157 16? 166 5.882 60 66 71 76 ei S7 QO 97 102 107 113 118 1J:3 129 1.^)4 139 145 l^C •155 • 5.262 Pag4 52, 7ABLB 26. SIGKA INDISX VM-U^'IS FOR SrQRBS IN TCST I. -GHADT5- Score Third Fourth :"ifth i:ixth < 30 20 11 10 39 28 18 15 48 35 26 4 20 57 43 32 12 25 67 50 39 20 30 76 57 46 28 35 85 65 63 36 40 95 72 60 44 45 104 79 66 52 50 113 87 73 60 55 122 94 80 68 60 132 101 87 76 65 141 109 94 84 70 150 116 101 92 75 159 123 108 100 80 169 131 115 108 85 178 138 122 116 90 187 145 129 124 i 1.851 1-470 ■ " •' ■ 1,388 1,6 hOTK - to find Sigma index Values for intermediate points add amount on line (#) to the value oi the m-xt lower score in thft first column above. ICx. ii^ore 27= 67 plus (2 x 1.851) or 70.702 4, --> i c Page 55, TABLE 27, SIGMA INDEX VALUTAS POR SCOR-RS IfJ Tp^K? II. -GRADE- Score Third Fourth Fifth bixth 2 67 63 52 37 4 73 67 56 42 6 80 72 60 47 8 86 76 64 52 10 92 80 68 56 12 99 85 72 61 14 105 89 76 66 16 111 93 80 71 18 118 98 84 76 20 124 102 88 81 22 130 106 92 86 24 137 111 96 91 26 143 115 100 96 28 150 119 104 101 30 156 124 108 106 32 162 128 112 111 34 169 132 116 116 36 175 137 120 121 # -.,.3.i74 iLill^ 2<02 , 2.4«a IilOTS - to find Sigma index Values for intermediate pointe add amount on line (#) to the value of the n-xt lower score in the first column above, ifix. Score 27-143 plus 3.174 or 146.174 Pai^o 54. TaBLB 28. SIGliA IhlQ-'^X VaLuKS FOB SCORi;S IK TKSY IH -GRAX*'^- Score Third yourth Firth Bixtta 1 62 36 '^<■J 15 2 67 44 33 20 3 72 49 38 25 4 77 54 42 30 S 83 59 47 35 6 88 64 51 40 7 93 69 56 45 8 99 74 61 50 ^ 104 7y 66 55 10 - 109 84 70 61 11 114 89 74 66 12 120 94 79 71 13 125 99 83 76 14 130 104 88 81 15 135 110 92 36 16 141 115 97 91 17 146 120 102 96 16 151 125 106 101 19 156 130 111 106 20 162 135 115 111 21 167 140 120 116 5.263 5.076 4.558 5.063 TABLE 29. Page 55, SIGMA IHDEX VA in?;S FOP SCORES IN TI^ST IV. T -GRAD15- Score Third Fourth Fifth Sixth 1 64 50 33 31 2 71 56 39 37 3 79 61 45 43 4 87 67 51 48 5 95 72 57 54 6 102 78 63 60 7 110 84 69 66 6 118 89 75 71 9 125 95 80 77 10 133 101 86 83 11 141 106 92 89 12 149 112 98 94 13 156 118 104 100 14 164 123 110 106 15 172 129 116 112 16 179 134 122 117 17 187 140 128 123 18 195 146 134 129 19 151 139 135 20 157 145 141 , # 7.718 5.632 5.898 5.762 P8m:;s 56. TABLE 30. SIGMA Il'JDBX VALUl^S FOR SCCRSS Hi T^ST V . -GRADE- Scor« Third Fourth i'ifth Sixth 1 85 63 51 55 2 95 70 58 ei 3 105 77 65 66 4 115 84 71 71 5 125 91 78 76 6 135 99 85 32 7 145 106 92 87 8 154 113 99 92 9 164 120 105 97 10 174 127 112 103 11 154 134 119 108 12 194 141 126 113 13 149 133 118 14 157 140 124 15 163 146 1£9 16 170 153 134 17 177 160 139 18 184 166 145 19 191 174 150 20 199 181 155 #< 9.92 7.142 6.826 5.262 ONE MONTH USE PLEASE RETURN TO DESK FROM WHICH BORROWED EDUCATION-PSYCHOLOGY LIBRARY This book is due on the last date stamped below, or on the date to which renewed. l-month loans may be renewed by calling 642-4209 Renewals and recharges may be made 4 days prior to due date. ALL BOOKS ARE SUBJECT TO RECALL 7 DAYS AFTER DATE CHECKED OUT. m LD 21A-30m-5,'75 (S5877L) General Library University of California Berkeley