f fh h fi Iii UPA LAW YY University of the State of New York Bulletin Entered as second-class matter August 2, 1913, at the Post Office at Albany, N. Y., under the act of August 24, 1912. Acceptance for mailing at special rate of postage provided for in section 1103, act of October 3, 1917, authorized July 19, 1918 Published Fortnightly No. 865 ALBANY, Ney. December 1, 1926 - a THE LIBRARY {iF THE een C | | N48uT MAY 27 1999 no. 65 UNIVERSITY o¢ ,,, tL ARITHMETIC IN THE RURAL AND VIQ/BAGE SCHOOLS OF NEW YORK STATE LBY JACoB 5S. ORLEANS Research Associate, Educational Measurements Bureau AND EF, Eucene SEYMOUR Supervisor of Mathematics . ALBANY THE UNIVERSITY OF THE STATE OF NEW YORK PRESS 1926 G197r-N26-2000(5183)* THE UNIVERSITY OF THE STATE OF NEW YORK Regents of the University With years when terms expire 1934 CuHeEsTER S. Lorp M.A., LL.D., Chancellor - - Brooklyn 1936 ADELBERT Moot LL.D., Vice Chancellor - - - Buffalo 1927 ALBERT VANDER VEER M.D., M.A., Ph.D.,. LL.D. Albany 1937 CHarLtes B. ALEXANDER M.A., LL.B., LL.DI Litt. D. - -3- - - - - - - =~ <]9SS Tee 1928 WALTER Guest Ketitoce B.A., LL.D.- - - - Ogdensburg 1932 James Byrne B.A., LL.B., LL.D. - - - - - New York 1931 Tuomas J. Mancan M.A., LL.D. - - - - - Binghamton 1933 Witt1aAm J. Wattin M.A. - - - - - - - Yonkers 1935 Witit1am Bonpy M.A., LL.B., Ph.D., D.C.L. - New York 1930 WittiAM P. BAKER B.L., Litt. D. - - - - - Syracuse 1929 Ropert W. Hicpir M.A. - - - - - - - ~- Rochester 1938 RoLanp B. Woopwarp B.A.- - - - - - = Jamaica President of the University and Commissioner of Education Franxk P. Graves Ph:D.,: Litt. D,, Eee Deputy Commissioner Avucustus S. Downinc M.A, Pd_D., CAD Siie Counsel ERNEST EV Gorn icc Assistant Commissioner for Higher and Professional Education JAMES SULLIVAN M.A., Ph.D. Assistant Commissioner for Secondary Education GeEorGE M. Witey M.A., Pd.D., LL.D. Assistant Commissioner for Elementary Education J.-Cayce Morrison M.A., Ph.D. Director of State Library James I. Wer M.L.S., Pd.D. Director of Science and State Museum CHARLES C.. ADAMS M.S’, Ph.D: Disa Directors of Divisions Administration, LLoyp L. CHENEY B.A. | Archives and History, ALEXANDER C. Frick M.A., Litt. D., Ph.D. Attendance, Examinations and Inspections, AVERY W. SKINNER B.A., Pd.D. Finance, CLARK W. HALLIDAY Law, IRwin Esmonp Ph.B., LL.B. Library Extension, ASA Wynkoop M.A., M.L.S. School Buildings and Grounds, FRANK H. Woop M.A. Visual Instruction, ALFRED W. AprAms Ph.B. Vocational and Extension Education, Lewis A. Witson D.Sc. FOREWORD Cooperative research holds the key to much of the progress we expect during the next decade. The educational measurements move- ment has not only given us a new method of working, a new technic, but better than this, it means a new attitude. Under its influence dogmatism, self-assurance and complacency give way to an inquir- ing mind, to an assurance that few questions are ever finally settled and to a willingness and desire continuously to seek new truth. For the past quarter of a century the spirit of scientific inquiry has been gaining an ever deeper hold on the minds of leaders in public | school education. The research workers in public school systems, and more especially in the colleges and universities, have developed a technic of work and methods of procedure. The lifting of the entire mass of public school instruction from the level of mere opinion to a scientifically evaluated procedure depends, however, not so much upon giving teachers and supervisors established and verified facts as it does in developing within them the attitude and spirit of the research movement. ‘This latter is cooperative research. It means much to find 121 rural superintendents, 26 village super- intendents and more than 3500 teachers cooperating with the State Supervisor of Mathematics and the Educational Measurements Bureau in a statewide investigation of achievement and methods of teaching arithmetic that is reported in this bulletin, Arithmetic in the Rural and Village Schools of New York State. It is, and Bauld be, a satisfaction to the school people of the State to know that the achievement of the village and rural school pupils was seven months above the norm in computation and four months above in the solution of reasoning problems. And to those respon- sible for the success of children in the rural schools it will be inter- esting to note that in arithmetic achievement the one-room schools are well above the norm in all grades. In the discussion of “practices in teaching arithmetic” in rural and village schools, teachers will find much to arouse their interest and to encourage them to re-evaluate their whole method of practice in the teaching of arithmetic. In the text of the report the authors have raised many questions. Some of these, teachers and superintendents will want to consider in their conference discussions; other questions raised call for addi- tional research before adequate answers can be given. They point the way toward further cooperative endeavor in the search for truth concerning the content and method of arithmetic instruction in the elementary schools of the State. J. Cayce Morrison Assistant Commissioner for Elementary Education oo ae alle EY yao bed a “ * F rey rel Nat C4 Ea eh cis. i ra ht , —" : ; Sh eee ° * ic ; ‘ + > 4 7 ie —< “ ; ane pele 2 OS at ~ ce j Pe « e 2 fe “ be a | / ae * ~4 << 4 . ’ i - A i ‘ ; ° ; : 4 : ~~ ; é "9 ‘ % nt a x { = oat rapt 3 aoe * «> . is ++: L - -? ; s ; : a4, - > ai % + s 2 4 “ee? OF t \ ‘ae a ‘ shen : bites eat nner oe iS es ’ y , ; : » 4 . a * 5 . ~ 4 % —~ oa » ; ‘ They ‘ aie oe ; f 34 =? . a, £ a be>.-f or. F 4 ‘0 Ae i i-“e oie re & phy e epi Sty. SORT IR e ; 7 oe > Dey. A esx ia att Sean Mitton 4 \ ; : x y> . > 4 : ava ke ra ae { {¥ ec5 . z a Mich le... 9a Ws Te m} ~ Dek oa i r Watt! % 7 = ri i om, . | elt eee 2 “Te a ake fs Bint fe ae uy 3 f 5 Sa at ee SGKETERTE ms ey By TL ee ee Pie . : 1 x ; ‘. Ps ~ ri . * » . » ! ~> 4 Z- 2 - University of the State of New York Bulletin Entered as second-class matter August 2, 1913, at the Post Office at Albany, N. Y., under the act of August 24, 1912. Acceptance for mailing at special rate of postage provided for in section 1103, act of October 3, 1917, authorized July 19, 1918 Published Fortnightly No. 865 AEDANY. N.Y. December 1, 1926 « ARITHMETIC IN THE RURAL AND VILLAGE SCHOOLS OF NEW YORK STATE During the month of April 1926 a statewide survey in arithmetic was conducted in the villages and rural schools of New York State by the Educational Measurements Bureau and the Supervisor of Mathematics of the State Department of Education. As in previous years the school subject surveyed was chosen after consultation with the district superintendents of the State. This survey is a timely one since the latest arithmetic syllabus of the State Department of Education appeared at the beginning of the school year. The survey therefore provides data for comparison of achievement in arithmetic with the syllabus requirements and standards. TNE sty “Wit eet owed BRAS By Three phases of arithmetic are included in the survey: computa- tion, problem solving, and problem analysis. The Stanford Achieve- ment Test, Arithmetic Examination, Form B', was used to measure achievement in both computation and reasoning. All pupils included in the survey took this test. To measure the ability to analyze arith- metical problems the Stevenson Problem Analysis Test? was used. This latter test was taken, however, by the pupils in only a few schools. The test in computation contains 47 examples varying in difficulty Mei 3 Perlite. 6 LOSI 129 coc Mang ort? Ueto The reasoning (problem solving) test contains 40 problems varying in difficulty from “ How many are 5 birds and 4 birds?”’ to “ How many cubic feet are there in a cylindrical smoke stack that is 20 feet in diameter and 100 feet high?” This variation in the difficulty of the material is necessary in order to use the same test for all the grades included in the survey. Some items must be so easy that the second grade pupils will be able to do 1 Published by the World Book Co., Yonkers, N. Y. 2 Published by the Public School Publishing Co., Bloomington, III. 6 THE UNIVERSITY OF THE STATE OF NEW YORK them correctly ; and some of the items must be so difficult that hardly any eighth grade pupils will obtain a perfect score. The use of the same test for the several grades is advisable in order to be able to determine what grade level each pupil or class has attained, and not merely whether each pupil or class can obtain a passing mark in a test covering a particular grade. For each of the tests in computation and reasoning 20 minutes time is allowed. The same time is given for all pupils taking the test regardless of grade. This makes it possible to compare the achieve- ment of pupils in different grades. If two pupils in different grades have the same score on the test they are equal in arithmetic achieve- ment as measured by this test. If, however, they did not have the same time limit such a comparison could not be made. The Stevenson Problem Analysis Test consists of six problems. With each problem four questions are given referring to the four phases of problem solving, namely: what facts are given, what is to be found, the process to be used, and the approximate answer. Four choices are given for each question and the pupil is to indicate in each instance the correct choice. A different form is used for grades 7 and 8 than is used for grades 4, 5 and 6. This makes it somewhat difficult to compare achievement in problem analysis above the sixth grade with achievement in the sixth grade or below. In this test the pupils are given as much time as they need in order to finish the test. ADMINISTRATION OF THE SURVEY Twenty-six villages, including 73 schools, and 121 supervisory districts in 47 counties participated in this survey. The number of villages is approximately the same as in preceding surveys. The number of supervisory districts is more than 25 per cent greater than any preceding survey and includes almost 60 per cent of all the districts in the State. In all approximately 75,000 pupils took the tests. This number is more than 35 per cent greater than in any of the preceding surveys conducted by the Educational Measurements Bureau. The tests were given and scored by the teachers, principals or superintendents and the class record sheets were sent to the Educa- tional Measurements Bureau for statistical treatment. In a number of instances the test was given in the second grade. In other instances eighth grade pupils, particularly those who passed the Regents examination in arithmetic, were not given the test. In general, however, where the test was used it was given to pupils in ARITHMETIC IN RURAL AND VILLAGE SCHOOLS —woa—_wo—weoww——eowwwnhm@wwwOw0138O8§3;ow»nwonmmmMSmMS@SmRTNmuNE em AWA 90+F ZIv O6F a 0zs Z6r ie we: B16 OOM ET Cae es ® eae ev6 as, 6 6B UOSUIAI}C ) ioe : 6vESL 278L POTIT 609€T FISbhl SbZhl P9OZIL 6z~ ccccee ei So? Pi “DpROTMEYS:£ Ee Ba ety Sern cree, wats * Tey TPO. er re ee eS ee Re ae 06+ Eb 962 682 8ze ble ae 8 Se Ak Bale een os ae Sp) WOSTS491G.) oer a SPSSZ = L692 Osre ZESb 806+ L68b 9ZIb CEG ig 8 eae st rteees prozueys § fe Pi cui) Ore 99 9¢ 78 CZ gs eee Sip tae Oe | Bie bee 6 ¢ Wie Sea aT a _WOSUdAd3S ») an H IZb8 $68 00ZT 89ST ¥SZI IT9T eri DS Cah akine Santee PE E* LE i PLOPURIS Peers eared eames es LONE faa Sen ee ee C6r L6 Sot FIT 6IT 09 over oe 2.0 ey Fs sisce are eee eene UOSUIADIC ) c998Z = COTE ZObb Z80S €0ss 7825 STfb ole Arete 3 SAS tt peo puea Gel ss ag. seve Sy a OUR Se Sno be: Os eeee eee see. ec et at esee Ate 0m as ee ee ee. OR, eerceceee WOsUrAdIS ) , ihe Ms 86rS Z9S 126 OSOT 676 ZZ0T $S6 we eee Oe Wel er ee ne oe 2.6 @ © se Be pioyuris cy Ks Oe Te ian JeNUUPIUISS Pe Beet. ve eeoeeer eee eeene ISP A . eeee eeee osee ec ee © pista) SS We aeeiace ici v6 oe *"* WOSU9ADSIS ) | se ta ve be Toe as €@7Z 09s IZOT LLET OZtT SZrl 92 16 kero os FS cer DIO sEIE TS. - rare teh ten) sat ee EE Ee EL Se ee hee oe No oy eee 8 l 9 ¢ b ¢ Zz saaqvuad LSaL TOOHOS AO AdAL iSO] SISATeUY W2]GoIq UOSUsAaZ}g 94} pue ‘UOHeUIUeXY WeUIyWIY Ysey, JUSMIAAZIyYOW PsOJURIG ey) Surye} foods jo od} yoda ut opei3 yora ur sjidnd jo saquinyy [ F1avy, 8 THE UNIVERSITY OF THE STATE OF NEW YORK grades 3 through 7. For purposes of comparison the test records are tabulated under five types of schools' — village annual promotion, village semiannual promotion, four or more teacher rural schools, two or three-teacher rural schools and one-teacher rural schools. The four or more teacher classification includes a rather large number of schools that are of about the same size as are many of the village schools. A separate tabulation was made for these schools and the same statistical treatment was given that was made for the above-mentioned five types. Since these larger rural schools form part of the four or more teacher group they are not treated as a separate type in the main body of this report. A supplement to this report, p. 22, is devoted to a discussion of the outstanding points concerning this type of school. Table 1 shows the number of pupils in each grade of each type of school taking the Stanford Achievement Test in Arithmetic, com- putation and reasoning, and the Stevenson Problem Analysis Test. STATISTICAL TREATMENT OF KRESUIS The scores on the tests make it possible to answer a number of questions, including the following: What is the average grade achievement in computation and reasoning in each type of school in New York State compared with the standards given for the test? How do the ditferent types of schools compare with each other? What variation exists between schools of any one type? Does achievement in computation compare more favorably with the test norms than does achievement in reasoning? How capable are the pupils of New York State in analyzing arithmetical problems? How does their ability in this compare with their ability to solve problems ? To what extent is there overlapping between grades in achievement in arithmetic? How do the separate districts and villages compare with the norms? In order to answer these questions, the test scores were tabulated for each village school, or rural district by grades and type of school. The distributions thus obtained for the same grade and type of school were totalled for the entire State. Medians were computed for these total distributions as well as for the separate villages and districts. It was also necessary to compute measures of overlapping and to tabulate distributions of the median scores for each grade for all the districts and villages. 1 Hereafter these will be referred to as village annual, village semiannual. four-teacher, two or three-teacher, and one-teacher schools. ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 9 Table 2 shows the median scores in computation by grades for the different types of schools for the entire State. Table 3 shows the corresponding data for the reasoning test and table 4 for the test in problem analysis. The total possible score in computation is 188 points, in reasoning 160 points, and in problem analysis 24 points. The last line of each table gives the norm or standard for each grade for the seventh month of the school year, which is about the average time of the giving of the tests. TABLE 2 Median scores in computation by grades and types of schools TYPE OF SCHOOL ; 3 4 rate 6 P 92 Annual ...-...seee eee 47.4 72.9 86.4 106.8 124.5 134.4 141.7 Village ... 4 Semiannual A1......... aresete 7352: aol OeelOS Ot) t246.5 e21315 540014859 perianal B* 8 as cs vigetee 6404) 8470 995S- 114760 123.8. 14355 Hour or more teacher.:.. 44.2. 72.7 ° 88.1 106.5 122.3 132.0 142.0 Rural .... 4 Two or three-teacher.... a>, e0ecta.o. 89 25.510/7.0" 120.4 13220 —14023 pee COACHET “2. Fea ated se 47.0 970.2, 86.9°° 1036 117.9) 127.21 138.7 5 ee Be. ee 76 S60 fit rao sot 1A =upper half of the school year; B =lower half of the school year. 2It should be kept in mind in interpreting these data that many eighth grade pupils who had passed the Regents examination in arithmetic did not take this test and that therefore the eighth grade medians are probably somewhat low. TABLE 3 Median scores in reasoning by grades and types of schools TYPE OF SCHOOL P 4 7 Stes ie 6 7 g2 PRCA So as en giv: «0 40s Bed 2503 5. AS I2- 5° 64 Aa 77 Bre 92.0 2101,9 Willaver... 4 semiannual Al .....s00.6 See COFOL 51 Oe Ooo OU St 86 ee lLOS cS SRL ATNMALS TS? oe cele o a.5'2 peice teOue) 83.00 S57.) 209 ok. 163.4, 101,8 Four or more teacher... 19.9 "320946584" 62.5—9 80.0. -9027, 101.5 pete two or three-teacher.... 20.9 31.0 44.6 62.2 76.5 89.7 96.4 PRO POACHEL 5 oc ae osu a> ewe aero 84s 1o. B O19.) 825.1. 86.2.0 90.9 DG e Fe Fe sine in 08 SF sie weno 8 13 30 42 56 70 84 $555 1A = upper half of the school year; B = lower half of the school year, 2Tt should be kept in mind in interpreting these data that many eighth grade pupils who had passed the Regents examination in arithmetic did not take this test and that therefore the eighth grade medians are probably somewhat low. 10 THE UNIVERSITY OF THE STATE OF NEW YORK TABLE 4 Median scores in problem analysis by grades and types of schools GRADES TYPE OF SCHOOL 4 5 6 7 8 Four ‘ox more. teachers ove. os eve cas Lae tae ese 11.1 Rey 19.5 20.0 20.0 Two..or: three-teacher. 5-55; 5% eevee. One eae es 10.7 12.6 18.1 17.6 ee | One-teacher)), Vn. ta cies ae bare hb oe ae eee anes bl i3cs 17.0 16.7 18.1 Norma i868 kk eee ee ae eS ee 16.6 19.4 18.3. 20.2 The data shown in these tables are presented graphically in figures 1,2 and 3. It should be noted that the Problem Analysis Test was not given at all in the village schools and that there are no data given for the second grade for the village semiannual promotion schools. Because of the small number of cases the records for the second grade in the annual and semiannual promotion schools were combined. It is obvious from the data, particularly from the graphs, that the differences between the types of schools in any one grade are slight in computation and reasoning. The types of schools differ more from each other in the upper grades than in the lower. On the whole, the medians are highest for the village semiannual pro- motion schools and lowest for the one-teacher schools, though this is not uniformly the case. The computation medians are in every instance higher than the grade norms. The reasoning medians are higher than the norms in all but two instances. On the whole, the computation medians are much more above the norms than are the reasoning medians. All the medians for problem analysis, except for grades 6 and 7 of the four or more teacher schools, are below the grade norms. The four or more teacher schools are appreciably superior to the two or three-teacher and the one-teacher schools and are practically at the norm. The latter two types are about equal and average about nine months below the norms. Table 5 shows the average extent to which each type of school and | all the schools together exceed or fall below the norms. The New York State schools are, on the average, from five months to nine months above the nationwide standards in computation, and from one month to seven months above in reasoning. For all types of schools taken together arithmetic achievement in the villages and rural schools of New York State is almost eight months above the nationwide standard in computation, more than four months above in reasoning, and in the rural schools almost six months below in problem analysis, as measured by tests used in this survey. ARITHMETIC IN RURAL AND VILLAGE SCHOOLS Village dem ‘Annual Vi lage Annual Aor more teacher —-— 2-3 Ceacher ame = Lleacher I iti Ww Vv VI WW WL Figure 1—Norms and Median Scores in Computation for the Different Types of Schools, Stanford Achievement Test Arithmetic Examination, April 1926 11 12 THE UNIVERSITY OF THE STATE OF NEW YORK | 10 100 90 80} AO Vell © Semi-A y 30 Ville Ana ae nee 4 ormore leacher —-+— 2-Sleacher 4 —— fteacher TL NE YL VIL VIL Carades Figure 2—Norms and Median Scores in Reasoning for the Different Types of Schools, Stanford Achievement Test, Arithmetic Examination, April 1926 ARITHMETIC IN RURAL AND VILLAGE SCHOOLS Aor more leacher —.— 2-3 leacher — — {leacher oOrms Ww VI Wile e yuk Figure 3— Norms and Median Scores on the Stevenson Problem Analysis Test for the Different Types of Schools (Test I for Grades 4, 5 and 6; Test II for Grades 7 and 8), April 1926 13 14 THE UNIVERSITY OF THE STATE OF NEW YORK TABLE 5 Average grade difference between medians for the different types of schools and the norms, and average grade difference for all types together in computation, reasoning and problem analysis. Differences are shown as months of school achievement TYPE OF SCHOOL VILLAGE RURAL TEST ao H(A SJ XO _s AVERAGE Semi- Four- Two orthree- One- Annual annual teacher teacher teacher Computations wieritske ete 8.6 9.044 8.1 iso 5.5 Fd. Reasonine. 1. wie vel set ote ee 5.6 6.6+4 eal 2.9 alee 4.3 Problemyanaly sist... se Le ie ee ues See aS —8.5 —9I.5 —5.9 1The norms for the test are given only up to grade 10.0. In some instances median scores were higher than this norm and therefore could not be stated exactly in terms of grade standing. The differences based on them are therefore given with a + sign in this and following tables. It is significant that the excess over the norm is almost twice as great for computation as it is for reasoning and that in comparison with these problem analysis shows up very poorly. Ability in com- putation is more mechanical, while ability in problem solving, and even more so in problem analysis, involves intelligence more in the form of reasoning. Probably the poor standing in problem analysis is an explanation of the extent to which problem solving achievement in the State falls below achievement in computation. The extent to which the medians for the State as a whole exceed the norms in both computation and reasoning raises the question as to whether on the whole the stress on arithmetic in the State is not too great in comparison with other subjects, and whether too much time is being spent on the mechanics of arithmetic in comparison with the time spent on reasoning. Data from previous surveys indicate that, at least in the past, achievement in other school subjects in New York State does not compare favorably with achievement in arith- metic as shown by the present survey. It would be wise for super- intendents and principals in those localities where achievement in arithmetic is high to investigate achievement in other subjects in order to determine whether stress on arithmetic is at the expense of other school subjects. It can be seen from table 2 that the grade median for computation for the lower half of the grade in the village semiannual promotion schools is in each grade higher than the norm for the end of the grade. In reasoning (table 3) the medians for the lower half of the grade in the village semiannual promotion schools are in no instance appreciably below the norm for the end of the grade. ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 15 Table 6 shows the difference in computation between the several types of schools in terms of months of the school year. The table shows the extent to which the village annual medians exceed or fall below the medians for the other types of schools. The table reads as follows: In the second grade the village annual median exceeded the four-teacher median by one month, the two or three-teacher median by one month, and was the same as the one-teacher median. In the third grade the village annual median was the same as for the village semiannual, the four-teacher, and the two or three-teacher, but was one month higher than the one-teacher median. In the fourth grade the village annual median was three months lower than the village semiannual median, 1.5 months lower than the four- teacher median, etc. Table 7 gives the corresponding data concerning the medians for reasoning, and table 8 the data for problem analysis. TABLE 6 Differences by grades between types of schools in computation, showing the number of months of school achievement by which the village annual medians exceed or fall below the medians for the other types of schools THE VILLAGE ANNUAL IN GRADE PROMOTION MEDIANS EXCEED THE Z 3 4 5 6 7 8 AVERAGE Village semiannual medians by... 0 —3 —1 0 8 —? —? Four or more teacher medians by.. il Q- > —T1.5 Oe reas 5 3 0 ate Two or three-teacher medians by.. 1 0 —2.5 O45 3 3 Hod One-teacher medians by.......... 0 1 —1 Pp eC 8 5 3.1 TABLE 7 Differences by grades between types of schools in reasoning, showing the number of months of school achievement by which the village annual medians exceed or fall below the medians for the other types of schools THE VILLAGE ANNUAL IN GRADE PROMOTION MEDIANS EXCEED THE 2 3 a 5 6 ih 8 AVERAGE Village semiannual medians by. .... —2 —5 —l 0 3. —3+ —1+ Four or more teacher medians by 1 1 —l1 1.5 —1l 1 1 8 Two or three-teacher medians by 1 3 0 2 1 LA ees, 2:7 One-teacher medians by........ 1 7 Gye! 3 6250 9 4.3 16 THE UNIVERSITY OF THE STATE OF NEW YORK TABLE 8 Differences by grades between four or more teacher, and one-teacher rural schools in problem analysis, showing the number of months of school achievement by which the four-teacher school medians exceed the medians for the two or three-teacher, and one-teacher schools i THE FOUR-TEACHER SCHOOL IN GRADES 4 5 6 7 8 AVERAGE Two or three-teacher medians by..... hi 8 6 12 14 8.2 One-teacher medians by........ecseee 4 7 10 15 11 9.2 These tables show the slight difference between the types of schools in computation and reasoning and the noticeable differences between them in problem analysis. The last column represents an average difference between the types of schools. On the whole, the differ- ences between both types of village schools and the larger rural schools is negligible. The one-teacher schools are lower than the village and four or more teacher schools by three months in computa- tion and more than four months in reasoning. In problem analysis the four or more teacher schools are almost a year above the two or three-teacher schools and the one-teacher schools. Up. to this point a general picture of the conditions for the State as a whole has been portrayed. The fact that the median status for the State is higher than the norms does not necessarily mean, however, that all the villages or districts are above the norms or that all pupils are placed in the grades consonant with their arithmetic ability. Table 9 gives a measure of the extent to which pupils are not properly graded in arithmetic. In order to simplify the presentation only the data for the village annual promotion schools are given. The figures for the other types of schools are very much the same. ‘The table reads as follows: Of the second grade pupils who took the test 3.0 per cent had scores higher than the third grade computation median; 6.1 per cent had scores higher than the third grade reasoning median; O per cent had scores higher than the fourth grade computation median; and .5 per cent had higher scores than the fourth grade reasoning median. Of the third grade pupils who took the test 14.8 per cent had higher scores than the fourth grade computation median; 21.8 per cent than the fourth grade reasoning median; .3 per cent than the fifth grade computation median; and 2.4 per cent than the fifth grade reasoning median; 13.8 per cent had scores lower than the computation median of the second grade; and 18.0 per cent lower than the reasoning median of the second grade, etc. ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 17 TABLE 9 Overlapping between grades in village annual promotion schools in com- putation and reasoning, showing the per cent of pupils in each grade whose scores exceed the median of the next and second higher grades and the per cent whose scores are below the medians of the next and second lower grades PER CENT EXCEEDING PER CENT BELOW MEDIAN OF MEDIAN OF GRADE TEST (a Next higher Second higher Next lower Second lower grade grade grade grade ba (computation. ...... 350 OnOe 1? Meira, ac te aePtaniescuste IGaASONING.. os 6.66 0 6.1 Eom Pieters ol Mei ba Ae: 5 3 Computation. ........ 14.8 0.3 13 S8e5 weeks acceler. REASONING {2 062). o'. 21.8 2.4 LS; Oubet + 2 Sebsgclotones 4 Computation... .... 21.9 0.4 6.3 322 WReasOnInNG.... 2.56% 24.5 3.4 5 a oe 6.3 5 Computation....... Toast =f eiaal 14.9 3.4 REASONING: 5.4.0... 18.5 2.9 ARS ES 6 Computation...... ; 22.0 14.6 2One 3.6 Reasoning: ... 26 so: 28.8 rene! 18.8 6.0 7 Womoutation. ~..... Sf i PRCA se as Boe 9.7 PRECISE foc. crs sie ss ZOSIV AY weet ete 28.0 10.7 8 Mtn ICAeI Otc te wh ec ciclec ale = kaise poe 40.8 302 SENT er NS ba Sues aN whee 34.1 TSO The overlapping in arithmetic seems to be great. In comparison with figures for other school subjects, however, it appears that the overlapping between grades in arithmetic achievement is probably the least of all school subjects. In comparison, for example, with English achievement the overlapping in arithmetic is on the whole less than one-fourth as great.? It is evident from table 9 that the overlapping increases with the grades and is greater for reasoning than it is for computation in the lower grades but that the opposite is true for the upper grades. The overlapping over two grades is very much less than over one grade particularly in the lower grades. Undoubtedly these conditions are due to the emphasis placed on arithmetic, particularly computa- tion, as a basis for grade classification and promotion especially in the lower grades, and to the absence of strict uniformity as to the grade in which the Regents examination is given. Hence in many instances doubling up starts early in many of the upper grades. The one outstanding fact that appears from this table is the lower- ing in the per cents of overlapping in the neighborhood of the fourth and fifth grade in both reasoning and computation. This condition is common to all the types of schools. It probably indicates a change 1See table 6, p. 19, University of the State of New York Bulletin, 846, English in the Rural and Village Schools of New York State. 18 THE UNIVERSITY OF THE STATE OF NEW YORK in policy with regard to increased emphasis on arithmetic achieve- ment, presumably in the sixth grade. It may also be explained by the large per cent of retarded pupils usually found in the fifth and sixth grades and by the fact that there is considerable natural over- lapping through grades 1 through 4 where the four fundamental operations upon integers are taught and reviewed in each grade while in grades 5 and 6 distinctly new and more difficult topics are taken up. TABLE 10 Showing for each grade the range in years of school achievement nec- essary to include the middle 50 per cent of the scores in each grade in computation and reasoning TYPE OF SCHOOL TEST 2 3 4 6 7 8 Village annual .......+. } Gomputaton Fe oe 8 ee Village semiannual ..... ) Rompuratyon:- is 41d 16 ee or Four or more teacher... ] Romemon; 1g 1s 18 18 La) 2L0b Two or three-teacher.....f Romputsvons -f 18 a7 Te is 23) Blab One-teacher +++. s+seeees 1 erations gu }eke Gg ag) 0g ae Average seesseeseeeeess | eens ee ie 4:8 8 | zie The variation between pupils in achievement in the same grade, _ as shown by table 10, is greater in the higher than the lower grades. On the whole, the variation is greater in reasoning than in computa- tion in the lower grades. The opposite is true in the upper grades. The differences in variation between types of schools are negligible. The increase in variability with increase in grade is probably due to the greater emphasis on arithmetic in the lower grades, particularly on computation, as a basis for promotion and classification. The extent to which the individual districts and the village schools compare with the norms and with each other is of greater importance locally than the comparison between the state medians and the norms. The distributions of the medians for the separate localities show that in computation 13.1 per cent of 586 grade medians for one-teacher schools are below the grade norms whereas 26.8 per cent of the 586 grade medians for one-teacher schools are above the norms for the next higher grade. On the other hand, in reasoning 211 of the 587 grade medians for one-teacher schools are below the grade norms ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 19 TABLE 11 Comparison of local medians with grade norms MEDIANS ABOVE MEDIANS BELOW THE NORM FOR THE TYPE OF SCHOOL TEST TOTAL NORMS FOR GRADE NEXT HIGHER GRADE NUMBER OF GRADE MEDIANS Number Percent Number Per cent Village annual ...... eee 189 Spothas ants bab Welaees Village semiannual .. | SomP--- 71 Palle Eye See Four or more teacher. nae pete = on ee pe Two or three-teacher.. | 50mP--- 3a ploy as Sy ee ote One-teacher -+...... SR Resiges 389 aiearicine’s en pore and 27 are above the norms for the next higher grade. The data for the other types of schools are also given in table 11. There- fore, although it may be stated that the status of the State as a whole in arithmetic is very good, it is still true that in many localities it is poor and in a number it may be characterized as too good: that is, it is probable that some schools are stressing arithmetic disproportionately. The variation between localities in arithmetic achievement in the same grade can be shown by the range in years of school achievement necessary to include the middle 50 per cent of the medians for the grade. ARLE AZ Range in years of school achievement necessary to include the middle 50 per cent of the median scores in each grade GRADE TYPE OF SCHOOL? TYPE 2 3 4 5 6 7 8 Village annual .......... § Computation? 2.). .8 Bd ey LS 2s. A EB 9 { Reasoning... é.... wine 1.0 4 ip PS Gale wih Foor orimore-teacher.... ' Computation.... .4 .6 Ws ts Moye, ahs 6+ Reasoning...... oO 375 oe 2 .6 oe LEO Two or three-teacher..... § Computation.... .4 .6 ais .6 5052 Oe pleat l Reasoning...... mae Biel .4 0 FY Ae ee EEK 4 Preteaeher te § Computation.... oa .6 .4 EL Pee nae be ee he es | Reasoning...... 12 G5 a: 2 A 245-120 +The data for the village semiannual schools are omitted because the small number of medians for each half year (all less than 20) would make the results too unreliable. The differences between schools are on the whole much less in the lower than in the upper grades and much less in the rural than in the village schools. The amount of variation is in many instances sur- prisingly great, particularly in the upper grades. For instance, in 20 THE UNIVERSITY OF THE STATE OF NEW YORK the seventh grade in two or three-teacher schools, the computation median for the district ‘that is 25 per cent below the highest is two years higher than the median for the district that is 25 per cent above the lowest. Of course, the difference between the highest and lowest medians is still greater, in this instance being over four and a half years. The highest seventh grade computation median among the two or three-teacher schools (for a district) is above the tenth grade norm. The lowest is below the sixth grade norm. Such variation must be recognized to appreciate the fact that many localities are doing poorly in arithmetic and many are probably spend- ing too much time on arithmetic in comparison with the time spent on other subjects, although from tables 2 and 3 it would seem at a first glance that achievement for the State as a whole in arithmetic is quite satisfactory. SUMMARY 1 Arithmetic achievement in New York State for the entire State is high in comparison with the standards for the country as a whole, as measured by the Stanford Achievement Test, Arithmetic Examina- tion, in both computation and reasoning. 2 On the average the schools of the State are seven months above the norms in computation and four months above in reasoning. 3 The differences between the grade medians and the norms are greater in the lower than in the upper grades. 4 The medians for the lower half of each grade in the village semiannual promotion schools are above the norms for the end of the grade in every instance in computation and either at least equal to or barely below the norms for the end of the year in reasoning. 5 The differences between types of schools in median achievement in arithmetic are not large. On the whole, the village semi- annual promotion schools are highest. The village annual promotion and the four-teacher schools are only a little behind. The largest difference is between the village semiannual promotion schools and the one-teacher schools — an average difference of more than three months in computation and more than five months in reasoning. 6 The one-teacher schools are almost uniformly lowest, although, as indicated, the differences are slight. 7 In problem analysis, a test taken by rural schools only, the achievement is low particularly for the two or three-teacher and the one-teacher schools, which are on the average about a year below the norms. ARITHMETIC IN RURAL AND VILLAGE SCHOOLS Zi 8 The differences between the types of rural schools are more marked in problem analysis than in either computation or problem solving. | 9 There is a great deal of overlapping between grades in arith- metic, the higher the grades the greater the overlapping. 10 The overlapping is greater for reasoning in the lower grades and for computation in the upper grades. 11 The overlapping in arithmetic is much less than it was found to be in English and is probably less than in any other school subject. 12 The variation between scores in each grade is great, the range necessary to include the middle 50 per cent of pupils being from one year to more than two and a half years above the second grade and only slightly less than a year in the second grade: that is, before pupils have had two years of schooling there is a variation in arith- metic ability found to be almost a year for the middle 50 per cent of the pupils. | 13 The variation in pupil achievement is greater in the upper than in the lower grades. i4 The variation in pupil achievement is greater for reasoning in the lower, and for computation in the upper grades. 15 There is no appreciable difference between types of schools in the variation in pupil achievement in either computation or reasoning. 16 Although the median achievement for the State as a whole is high in both computation and reasoning, 12 per cent of the district and village grade medians are below the norms in computation and 25 per cent are below the norms in reasoning. 17 On the other hand, 34 per cent of the district and village grade medians are above the norm of the next higher grade in computation and 13 per cent are above the norm of the next higher grade in reasoning. 18 The variation between grade medians for any one type of school is considerable, as shown by the range including the middle 50 per cent. The range above the fifth grade varies from six months to over two and a half years. Below the fifth grade the variation is from two months to 1.2 years. 19 The variation between districts is least for the one-teacher schools and appreciably more for the two or three-teacher and the four or more teacher schools. The variation for the villages is very much greater on the whole. This may indicate the extent to which the types of schools tend to deviate from the syllabus. 22 THE UNIVERSITY OF THE STATE OF NEW YORK RESULTS FOR THE LARGEST RURAL SCHOOLS The group of rural schools referred to throughout the foregoing as four or more teacher schools includes a number of schools that are under district superintendents but that are about as large as many of the schools in villages having superintendents. It is possible that these should be treated separately from the rest of the four or more teacher group although this has not been done in previous surveys. A separate tabulation of those rural schools having 15% or more teachers shows some rather interesting facts. The accompanying tables show the median scores for the village annual promotion schools, for the largest? of the four or more teacher schools and for all the four or more teacher schools, together with the norms. Table 13 gives the data for computation and table 14 for reasoning. These tables are comparable to tables 2 and 3. TABLETS Median scores in computation for certain types of schools GRADES TYPE OF SCHOOL 2 3 4 5 6 he 8 Village annual promotion........ 47.4 72.9 86.4: 106.8% 124. 5°eeig4seeeeeee All four or more teacher........ 44.2. 7 2a7ee Z88et 106.5 1223 132.0 142.0 Largest of four or more teacher?. a5 74.7 86.9 104.3 12154 13h 141.9 UNO LTS @ accha isc slo te ae eee aaa 24 57 76 96 114 123 132.5 1 See footnote 2 at the bottom of this page. TaBLe 14 Median scores in reasoning for certain types of schools GRADES TYPE OF SCHOOL 2 3 4 : 6 7 8 Village annual promotion........ PV Be ys OM CC: 64.4 Ar eh: 92.0 2 AGTSY All four or more teacher........ 19.9. - 32.9 46.8 62.5 80.0 90.7 | 101.5 Largest of four or more teacher?. L957 OO tee AO 61.5 79.9 89 S82 eab02eT NOYMIS Voom tees. cao Set ee 13 30 42 56 70 84 95.5 1 See footnote 2 at the bottom of this page. In the second and third grades in computation and the third and eighth grades in reasoning the medians for the largest rural schools are slightly higher than the medians for all the four or more teacher 1A few schools having less than 15 teachers were included for special reasons. 2 The group. of schools termed “largest of the four or more teacher schools ” or “largest rural schools” refers to those schools, under district superintendents, that have 15 or more teachers. ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 23 schools. In every other instance the median for the largest rural schools is below the median for all the four or more teacher schools. In no instance, however, is the difference so great as two months of school achievement. The largest schools were included in obtaining the medians for all the four or more teacher schools. The differences then between the largest rural schools and the rest of the four or more teacher group would be almost twice as great. These differ- ences would still, however, be almost negligible. | The differences between the medians for the largest rural schools and for the village annual promotion schools are also very small, the average difference being less than one month of school achievement. Table 15 shows the variability of the same three types of schools. This table is similar to table 12. ABLE LS Range in years of school achievement necessary to include the middle 50 per cent of the median scores in each grade GRADE TYPE OF SCHOOL 2 3 4 5 6 7 8 Computation Village annual promotion..... Saar 8 ns 12 1.4 1.8 9 All four or more teacher..... .4 .6 ny, Aff 6 1.6 6+ Largest four or more teacher?. .4 ais .6 We 8 TO 1 Reasoning Village annual promotion..... egsae a eRe, .4 9 1.1 fo AY 4 All four or more teacher..... 3 75 5 5 .6 won A 0 Largest four or more teacher!?. is 1.0 a 6 6 1.5 1.4 1 See footnote 2 on page 22. The variation between localities in the two types of rural schools is almost the same except in grades 7 and 8 in computation and grades 3, 7 and 8 in reasoning. On the whole there is a slight tendency toward greater variation between localities having larger schools, this tendency becoming appreciable in the upper grades. In other respects the interpretation for this table is the same as that given for table 12. One might conclude from the above that there is no appreciable difference on the whole between the largest schools under district superintendents and those rural schools having from 4 to 15 teachers. This is the fact in arithmetic. It must be kept in mind, however, that the differences between all types of schools appeared to be very small in arithmetic. This, however, might not be true in other school subjects. 24 THE UNIVERSITY OF THE STATE OF NEW YORK PRACTICES IN TEACHING ARITHMETIC IN RURAL AND VILLAGE SCHOOLS | There is much information regarding the teaching of arithmetic in the schools of New York State which can not be gained from study- ing the results of standardized tests given in this subject. This information, however, is extremely valuable in that it offers possible suggestions not only for improving the teaching of arithmetic but indirectly for improving the teaching of high school mathematics. For example, standardized tests will not disclose any uniformity or lack of uniformity in the teaching of certain of the fundamental operations, in the time devoted to the study of arithmetic or in the use of the state syllabus in arithmetic; they will disclose little knowl- edge of classroom technic or of the method of handling many prob- lems that come up daily in the teaching. Sources of Data In order to obtain some of this additional information regarding the teaching of arithmetic a questionnaire on the prevailing practice in teaching arithmetic in the public schools of New York State was sent out to teachers in one-teacher, two or three-teacher, and four or more teacher schools.t. The following pages contain an attempt to summarize the answers to this questionnaire and to make certain deductions and suggestions regarding the teaching of arithmetic in this State. In many instances it was evidently from the answers given that some of the questions were misinterpreted or that the teacher filling out the questionnaire desired to tell a “‘ good story” and did not put down the facts. All such answers were eliminated in the following summary. This fact combined with the fact that more than forty counties and 162 schools are represented in answers to the question- naires make the findings reasonably reliable and representative. Table 16 shows the number of replies received from the different types of schools. In making the tabulations all of these were used. TABLE 16 Number of replies to the questionnaire from the various schools TYPE OF SCHOOL NUMBER OF QUESTIONNAIRES RECEIVED One-teacher 1. 4.5605 «232 Ce RE ee. LG 1S. pee 87 Two or three-teacher.... ..iie wash accede hc tus cn lode eee 16 Four or more teacher...... \ small village 2.2... oo. . 2s Janes 42 Ularge village’ .... 2. b). 17 1A copy of the questionnaire appears on pages 34-36 at the end of this report. ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 25 TABLE 16a Number of classes in each grade and the total registration per grade for all the schools represented by the answers to the questionnaire — GRADE NO. OF CLASSES == TOTAL REGISTRATION — 3 74 343 4 84 657 5 103 767 6 97 886 7 94 891 8. 74 607 } Teaching Load Table 17 shows the minimum, maximum and average enrolments in each of the grades from 3 to 8 for the various types of schools. It was frequently found that the maximum or minimum enrolments (more frequently the maximum enrolment) differed quite materially from the next nearest enrolment. The average, however, is fairly representative of the enrolments for the various grades. TABLE 17 Enrolment by grades in the various schools GRADE 3 GRADE 4 GRADE 5 TYPE OF SCHOOL Min. Max. Av. Min. Max. Av. Min. Max. Av. One-teacher viii soo ee 1 16 3 1 9 3 1 12 3 wo or three-teacher....... 7 16 12 4 15 9 2, ie HH Four or more teacher...... 8 45 22 6 43 26 5 43 26 GRADE 6 GRADE 7 GRADE 8 TYPE OF SCHOOL Min. Max. Av. Min. Max. Av. Min. Max. Av. ROS Tener © asics cco ose 1 8 3 1 7 2 1 5 2 Two or three-teacher....... 5 12 8 1 12 6 1 12 4 Four or more teacher...... 5 46 29 4 48 26 9 39 24 It is interesting to note that the average enrolment of the one- teacher school where the teacher has all grades from 3 to 8 inclusive is considerably less than the average enrolment in any grade of the four or more teacher school. Furthermore, since only about 50 per cent of the one-teacher schools have all grades and about 50 per cent of the teachers-in-the- four or more teacher schools have more than one grade, it would be safe to say that the number of pupils in a one- teacher school is about half the number of pupils under a single teacher in a four or more teacher school. This statement may be verified from the data in table 18. 26 THE UNIVERSITY OF THE STATE OF NEW YORK TABLE 18 The number of grades for each teacher in the various schools represented in the survey NO. OF GRADES TWO OR PER TEACHER ONE-TEACHER THREE-TEACHER SMALL VILLAGE LARGE VILLAGE 5 or more 44 3 1 0 4 20 8 0 0 3 BS 4 5 1 2 9 2 15 3 1 0 0 24 14 This table shows that nearly 73 per cent of the teachers in one- teacher schools teach four or more grades each while in the four or more teacher schools nearly 90 per cent of the teachers have no more than one or two grades each. Time Distribution Table 19 shows the time devoted to arithmetic in minutes a week in the one-teacher and village schools. As in table 17 it was often the case that the maximum (or minimum) time spent in one or more of the grades obtained in but one school and the next nearest time allotment differed quite materially. This fact, com- bined with the fact that in many schools, especially in the one-teacher school, the time allotment was estimated, makes this table less reliable than some of the other tables. Nevertheless the range of 35 to 225 in the one-teacher school and of 75 to 450 in the village (four or more teacher) school, though these figures be only approximate, is far too great and indicates a lack of anything approaching uniformity in the matter of time given to arithmetic in the various grades. The median time allotment given in this table perhaps comes nearer the prevailing practice than would the average. TABLE 19 Minutes a week devoted to arithmetic in the various schools GRADF 3 GRADE 4 GRADE 5 TYPE OF SCHOOL Min. Max. Med. Min. Max. Med. Min. Max. Med. One-teacher RA ects tis 35 225 75 45 225 75 45 225 75 Villaced.a47 0.8 cc tne aes ie CYAGY ol ans ie 75 450 - 150 75° * 450i ea 50 , GRADE 6 GRADE 7 GRADE 8 TYPE OF SCIIOOL Min. Max. Med. Min. Max. Med. Min. Max. Med. i a ee One-teacher’ Sst ies ce? 50 240 75 60 240 80 SOL sees 100 Village 2. ee ae 7597450.) 200 75) 2400-2200 150 600 # 200 nn EE Eee ool ——$—$— “gd ARITHMETIC IN RURAL AND VILLAGE SCHOOLS Zh It is reasonable to suppose that the minimum allotment is fairly accurate — at least not too small. This means in some schools only nine minutes per day is given to arithmetic in grades 4 and 5, 10 minutes in grade 6, 12 minutes in grade 7 and 10 minutes in grade 8. It is interesting to note that according to the medians grade for grade the rural (one-teacher) schools are giving less than one-half of the time for arithmetic that the village (four or more teacher) schools are giving. This fact helps to explain why pupils entering high school from rural schools are generally not as well grounded in arith- metic, particularly in analysis, as are pupils coming from the village elementary schools. Table 20 gives an analysis of how the time, in minutes a week, given to arithmetic, as indicated in table 19, is divided in the various schools and grades between oral arithmetic, problems and drills. Here again some of the data had to be approximated as many teachers did not make clear-cut distinctions between these three phases of the work and accordingly their daily program does not indicate a definite amount of time given to each. The data are sufficiently accurate, however, to show general practices in these phases of the work and to indicate again the unusual range and lack of uniformity among the various schools and grades. TABLE 20 Analysis of time spent in arithmetic in the various schools GRADE 3 GRADE 4 GRADE 5 TYPE OF SCHOOL Min. Max. Med. Min. Max. Med. Min. Max. Med. Oral arithmetic Onre-teacher oo ciss sco eo 5 60 25 10 60 25 5 90 20 VOLE RS 3 5a oe PR sepa, Ree Seo 15 85 50 15 100 30 Problems One-tezcher <........- 5 150 20 10 150 20 10 150 30 MEETS) | 2c otra Sicieds LkAntd < 0 200 50 30 200 80 Drills . One-teacher .......... 5 70 20 5 110 20 5 50 20 PMs a oe, ae walle so Bese aha hate Chie 10 150 50 10 90 30 GRADE 6 GRADE 7 GRADE 8 TYPE OF SCHOOL Min. Max. Med. Min. Max. Med. Min. Max. Med. Oral arithmetic (ne-Feacher 55.3.2. 06s 5 60 20 10 60 25 5 60 25 Wiliam nn ptat. fkts) at home’.:.%. See Subject matter (Fill in the blanks for the grades you are teaching) : 1 Have you a separate book for oral arithmetic? .......... 2 Are pupils ever required to make up problems? .......... 3 Do you make a conscious selection of material for drills and problems to give a maximum of emphasis on the use in later life Cc Aa ee 4 What books on the teaching of arithmetic are available for your USEF vo cele die esas s cles suse ede © alesis sleet on 5 Are you following the course as outlined in the new arithmetic Syllabtsifis iron ae dees If not, note any departure 22a 6 What standardized tests in arithmetic have you given in your ClaSSES 2 ois oye’ seve ave bw we ue oie Bare ale talus wes ale When were they given? .....5.0..0. 000.00. What were your, findings? ........ 9.50.6. os 2 «se ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 55 7 Do you give timed drills periodically? .......... Are they PeCMaeRCLIZEU fir ceili st: Do you use results to find indi- widual difficulties ?.,..3...% 6. Do you follow them up with et Tha a) Coe Are these given to the class or thesindiyidual? io... 2... Method of procedure tn classes (Fill in the blanks for the grades you are teaching) : MN TAel VHCAMENLAIS . oo Lk ks cs ape eee ee ee eee (a) Do you insist on all work being checked? .......... (b) Do you recognize distinct steps in teaching each of the fundamental operations? ........... Do you present these steps one ata time? .......... (c) What step have you found most difficult to get in MM PLOT RMR Seach ctolt poate xls 14.0% wisi o's 0s ES a STB IN diel oh 1. Pk oiedaithi's: wa) st )ks d's! 6) 9s le'e eva BRAN EEN PN ea, Waele A gt os ws Sig iald wee a ev wee OTE Oy gine Sips lh ge en (d) Is there uniformity in all grades of the method used in per- forming each of the fundamental operations? .......... (e) List the three most helpful devices you have used in drills. (f) When pupils work at the board in groups are they called upon to explain their work or just to get the answer? 2 Teaching problems (a) Do you attempt to make problems informational and the EE MCI ENT tet etna en Pe a So F8S vaste a Sin ys Se vO Y vite ot (b) Do you attempt to suit the problems to your particular SAD a SPELT, po UC OSES ee Aico eae (c) Do you drill specifically on problem analysis? ............ (d) Do you correlate problem analysis and silent reading in Ra lt ee ar Ba CEI Go oS 0d av Se Me eds s2 gine Bae ein, (e) Do you insist that all answers be estimated before they are Pires LER reaper ; checked after they are found? ....... (f) List the three most helpful devices you have used in teaching ST re ere es ee i i ey (g) In problem analysis what have you done to eliminate such incorrect labelings of steps as 5 rds & 4 rds equals 20 CG) O16) 6 6 6 08 6 2 Oo 8 Ss 4 6 8k OS © soe C6 Owe eaten ee eo ecete eee eeweeeones Cr 36 THE UNIVERSITY OF THE STATE OF NEW YORK (h) Do you give drills in solving problems without num- pers ct. at ae ; with numbers but without all the 3 Home work (a) Do you give any time to supervised study in arithmetic? .. (b) Are all home work examples put.on the board or otherwise explained by the pupils each day?’ ... oe) ee Do they copy from their papers when putting the work on the bOdrd.:icaereee ere In explaining the problems does the pupil or teacher read through the solution put on the Jaray: ge 8 top oh ae (c) Is the home: work handed’ in daily? eee [igor at what time in the recitation? .... 2... 0) ee (d) Is it understood that the first part of the period is devoted to questions on home work? .......... (e) Do you encourage pupils to come to you for help either before school or after school? ..72. 7233 )3eseee Aims 1 What do you consider the chief aim in teaching arithmetic in prades: 1-0 iv.5. viele hae oe ee 1-3. bs «2s 2 Have you made any effort to determine and correct the common arithmetical weaknesses evidenced by pupils in algebra? Ar nerdy ; in physics? ..........- If so, how aides IS cae eet cee ee tie soe sence isa) c iets te iee an 3 Have you made any effort to eliminate waste in drills by deter- mining specific difficulties through diagnostic tests? ......... eooesvpteeoeveeeeeeeeeneeeeeeeeesesveoeos e€ee¢ 6 6 8 0 8 6S 6 6 © Oe) eee ee eee esevoeseoeoseeewpeeeetmoaoeseeweeenewme7eee eee ee ee 8 6 8 8 hf 8S 6S ee ee ee eee eeoeeeveoeseoeeeoecoesueoeaesvn eee Oseeeveeeeeensmeeageeenges 6 6 48 * 8 Be See ee ARID ELMEDVG ENSRURAL AND: VIEPAGE SCHOOLS aN PammeGRArHy OF BOOKS ON THE TEACHING OF ARITHMETIC Brown and Coffman. How to Teach Arithmetic. Row Peterson Klapper. The Teaching of Arithmetic. Appleton Lennes. The Teaching of Arithmetic. Macmillan Lindquist. Modern Arithmetic Methods. Scott Foresman McLaughlin and Troxell. Number Projects for Beginners. Lippincott McNair. Methods of Teaching Modern Day Arithmetic. Badger Overman. Principles and Methods of Teaching Arithmetic. Lyons & Carnahan Smith, D. E. The Teaching of Arithmetic. Ginn Stone. How to Teach Primary Arithmetic. Sanborn The Teaching of Arithmetic. Sanborn Suzzallo. The Teaching of Primary Arithmetic. Houghton-M/ifflin Thorndike. The New Methods in “Arithmetic. Rand McNally Psychology of Arithmetic. Macmillan é BIBLIOGRAPHY OF STANDARDIZED OBJECTIVE tS rS IN ARITHMETIC This bibliography may be obtained in mimeographed form from the Educational Measurements Bureau of the New York State Department of Education. Reference is made to this bibliography on p. 40 of this bulletin. “4 3 0112 105878653 Bulletins and Pamphlets Prepared by the Educational Measurements Bureau BULLETIN NUMBER 734 Morrison, J. C. Educational Measurements. 1921. Out of print 764 Morrison, J. C. Spelling in Ra | sie Rural Schools. 1922. Out of print 772 Morrison, J.C. 5 ANN nisistraive Uses Made of Standard Tests and Scaleg Wwe ie State ‘ EXPY ork, 1921-22. 1923. Out of print 784 Morrison, J.C. Tl tandard ‘Tests and Scales in the Platts- burg High Sc 1 ee 10 cas {NO 798 Coxe, W. W. Silent Rev in New York Rural Schools. 1924. 5 cents yERe' 806 Morrison, J. a) ornell, W. B. & Coxe, W. W. Survey of the Need for Special Schools and Classes in Westchester County, New York. 1924. 5 cents 814 Coxe, W. W. & Orleans, J. S. One Year’s Reading Progress in New York: Rural Schools. 1925. 10 cents 819 Coxe, W. W. Organization of Special Classes for Subnormal Chil- dren. 1925. 5 cents. Revision of Bulletin 802 835 Orleans, J. S. Survey of Educational Facilities for Crippled Chil- dren in New York State. 1925. 5 cents 839 Coxe, W. W. & Cornell, E. L. A Study of Pupil Achievement and Special Class Needs in Westbury, L. I. 1926. 20 cents 841 Coxe, W. W. Study of Pupil Classification in the Villages of New York State. 1925. 20 cents 843 Gray, E. A. Manual of Suggestions for the Use of the Phonograph in Special Classes. 1926. 10 cents 846 Orleans, J. S. & Richards, E. B. English in the Rural and Village Schools of New York State. 1926. 10 cents 850 Coxe, W. W. & Richards, E. B. Suggestions for Teaching Silent Reading. 1926. 5 cents. Revision of Bulletin 803 851 Coxe, W. W. & others. Outline of a Course in Educational Measure- ments for New York State Normal Schools. 1926. 5 cents 865 Orleans, J. S. & Seymour, F. E. Arithmetic in the Rural and Village Schools of New York State. 10 cents Tether, C. H. Hints for Special Class Gardens. 5 cents Ability Grouping in Junior and Senior High Schools. Mimeographed. Free : Suggestions for Reclassifying Pupils in Small Rural Schools. Mime- ographed. Free Coxe, W. W. Adaptation of Methods and Materials of Instruction to Ability Groups. Mimeographed. Free Bibliography of Objective Tests in Arithmetic. Mimeographed. Free Arithmetic Survey, April 1926. Preliminary Reports and Detailed Summary Tables. Mimeographed. Free Publications of the State Department of Education are free to schools and libraries in New York State. Others may purchase limited quantities at the cost of publication listed above. The above bulletins and pamphlets should be ordered from and checks made pay able to The University of the State of New York, Albany, N. Y.