An Investigation into the ARITHMETICAL COMPETENCE OF STUDENTS entering the professional courses o f training for teaching By A. L. BEHR (B .Sc .Ho n s ., B .A., D .E d .) (Head of the Department of Mathematics and Science at the Johannesburg College of Education.) 1. Introduction Those who are responsible for the training of teachers in the field of arithmetic constantly com­ plain that students entering the training colleges are lacking in basic arithmetical knowledge. The arith­ metical incompetence of their students precludes lec­ turers from carrying out an effective training pro­ gramme, and topics such as how children think and reason in arithmetic, new experiments in the field of the methodology of arithmetic, the use and interpreta­ tion of diagnostic and standardized tests, discussions on contemporary literature on the subject of arithmetic, etc., simply cannot be handled. The little time avail­ able in an overcrowded curriculum for teacher-train­ ing must be utilized mainly to instruct students how to do for themselves the sums which they will in turn be called upon to teach their future pupils in the classrooms. These are not idle assertions. At the beginning of this year (1962), the author initiated a pilot investigation into the arithmetical competence of students entering the professional courses of training at the Johannesburg College of Education. During the period of orientation at the commence­ ment of the academic year, all first year profes­ sional students (men and women) were given a test involving the four basic processes in addition, sub­ traction, multiplication and division of integers, and also simple computations involving common frac­ tions and decimals. The test given is of about Standard V level, i.e. what would be expected of pupils at the end of their primary school course. 2. T he Actual Test Given The actual test given is reproduced here.t1) No time limit was set, but students had to indicate on the answer sheet how long (in minutes) they had taken to complete the test. The test was not a speed test, but rather a diag­ nostic test, designed to assess accurately the actual competence and understanding of the testees of the essential basic processes in arithmetic and their methods of attacking simple computations of a mechanical nature. THE TEST (All working must be done on these sheets. No working on separate paper is allowed.) Section 1: Addition 3 2. 64 3. 16,837 4. 1,000 0 57 84,273 100 2 93 10 8 15 101 1 41 9,789 7 26 5 77 9 — 6 4 5. 8 + 4 + 1 + 9 4 -6 + 5 = 6. 24 + 89 + 708 + 50 = 7. 5,500 + 3,344+5,588+7,711 8. 66 -f- 66 -j- 66 | 6 6 + 6 6 = ( 1) The detailed instructions that accompanied the test are not included. SYMPOSIUM 1962/63 61 Section 2: Subtraction I. 79,865 43,201 2. 45,321 -6 ,8 9 7 3. 100,002 -76,893 4. 777,777 5. 101,010,010 -599,999 -90,000,101 6. 1 0 0 -6 9 = 7. 4 ,3 2 6 - 2,654 = 8. 70,006 - 29,148 = Section 3: Multiplication 1. 537 x 9 2. 478 X 12 3. 8,050 4. 629 X ll x l6 5. 84,196 x23 6. 625 x 84 7. 70 x 707 8. 6 9. 23 x 46,823 x456 10. 5,205 X804 11. 204x139 12. 1 ,212x12= 13. Section 4: Division 456x99 = 1. 10.068-M2 2. 813,627 - 9 3. 2 ,700:54 4. 33,888,921:11 Section 5: Fractions 1 5 1 3 f—l i — & 2. -— 1------------ 41 21 221 3i-75!/e Xl'/» 4. 3 |4 -5> /io f l l/„ 5. After a baker had sold i o f his bread, he still had 875 loaves over. How many loaves had he at first ? Section 6: Decimals 1. 0 -2 x 0 -2 x 0 -2 2. 0-2 x0 -3 xO-4 3. 300x0-003x0-03 4. 6-47 4 5. 0-000127-0-04 6. 27-4,000 7. 6 x 0 -5 xO-04 8. 1 1 1 0-025 0-2 0-4 0-5 (Answer as a common fraction) 9. 14-6x0-29 10. 13-87-4 3. Some Details About the Testees The following details concerning the testees are of importance. No. of students who were tested ............ 231 No. of students in possession of the T.S.S.C. 124 No. of students in possession of the N.S.C. 23 No. of students in possession of the U.E.C. 75 No. of students in possession of other school leaving certificates ..................................... 9 The testees were drawn from 99 schools. The schools included State and private institutions, commercial and technical colleges, and were repre­ sentative of all the provinces of the Republic of South Africa. In addition a few testees came from schools in Southern Rhodesia and Portuguese East Africa. Of the total number of testees, 11 only had taken arithmetic at school to Std. VI level, 13 to Std. VII level, 167 to Std. VIII level, 4 to Std. IX level and 36 to Std. X level. It will thus be seen that 207 out of the 231 testees had studied arithmetic to lunior Certificate level or beyond. A further significant point is that 135 out of 231 testees had taken mathematics up to matriculation level. The testees represented, therefore, a good sample and with their background in arithmetic, one would have expected to find very few unable to cope with the test, but as the subsequent analysis of the results show, this was not the case. 4. A n Analysis of the Test Results 4.1: ADDITION Though the testees on the whole succeeded well in adding columns of integers (both horizontal and ver­ tical), their methods of working are worthy of analysis. (a) Of the 231 testees. 127 or 55%(2) indicated that they added seriatim, i.e. as the numbers appear on the paper. The remainder did the additions by seeking out tens or other combinations. The inference that can be drawn from this is that teachers do stress on the whole the learning of the addition bonds. (b) Of the 231 testees no fewer than 58% stated that they found it necessary at times during the test to count on their fingers or make marks on paper. This may be attributed to the fact that many testees were out of practice as far as rapid addition was concerned, and were, therefore, compelled to resort to more infantile methods. The test does show how deeply ingrained these early methods of learning become. (c) Of the 231 testees, there were at least 87 or 38% of the total number, who made use of carrying figures. This is of considerable interest, if it is borne in mind that most teachers and educational authorities frown on the use of crutches! In fact, (2) All percentages are given in round figures merely for the convenience of the reader. 62 SYMPOSIUM 1962/63 the Education Department of the Cape of Good Hope has gone so far as to forbid the use of crutches from Standard II upwards except in the case of the weakest pupils and those in special classes.(3) Some testees, faced as it were with a situation which in a sense was of a critical nature to them, resorted to a method looked upon with disfavour by their former teachers, but which nevertheless from their point of view brought the desired results. There is, the author believes, an important lesson to be learned from the results in this test. The ability to dispense with crutches must depend upon the pupil himself. Some children (and indeed many adults) may never be able to operate successfully without them. The author is of the opinion that no hard and fast rules ought to be laid down. 4.2: SUBTRACTION The most significant fact illustrated by the results of this test is that no fewer than 159 out of the 231 testees, or 69% of the total made use of the decomposition method of subtraction and employed the word borrowing. (One testee spelt it “borough- ing”!) The word borrowing is a misnomer, for borrowing implies giving back, which is not the case in sub­ traction. One gains the impression, especially after having had discussions with students, that the bor­ rowing of a number in subtraction is learned with­ out any explanation of the logic on which the per­ formance is based. The following are some typical errors in equations made by candidates in a recent matriculation exam­ ination : ( a ) 7r2— 7/-= 15, r2= 15 7 (b) x+2 y = 7, .'. * = ----- (') - 2 y 8 (r) 4a = 8, .'. n = ---- - 4 It is certainly worthwhile investigating whether the incorrect use of the negative sign in algebra at high school level, might not very well have its roots in the rather loose and incorrect meaning attached to the word borrowing in the primary primary school. So ingrained is the word borrowing in the minds of many students, that lecturers give up in frustra- (■!) See The Education Gazette, Vol. LIX, No. 7— 17.3.1960, p. 510. (4) Knobel, J. C .: Pie Betekenis en Hantering van Foute en Leermoelikhede in Algebra. (HAUM Boekhandel, Pretoria, 1962). lion after having tried in vain to eradicate this mis­ nomer in a three year course of training! 4.3: M ULT1PLICATION In this test, as in all others concerned with the fundamental processes, the author did not concern himself with the correctness of the answers, but essentially with the methods of attack, i.e. how the testee approached the solution to each multi­ plication sum. From an analysis of the results it will become clear to the reader that each sum is not treated on its merits. In many instances the testee merely does as he was told by his teacher and through years of practice has established definite habits of reaction in given situations. Arithmetic then becomes the mechanical manipulation of number without any raionalization. The following ought to substantiate what the author has just stated. No fewer than 99 out of the 231 testees (i.e. 43%) did the sum as shown alongside. A few left out the second partial product con­ sisting of noughts. This sum could have, in fact, been done mentally had the testees realized that 70 should have been used as the multiplier instead of the multiplicand. Fortunately only 10 out of the 231 testees (i.e. 4%) did the sum as shown alongside. The fact that the sum was done as indicated shows lack of insight, and far too much dependence on rule of thumb methods. Once again 99 out of the 231 testees (i.e. 43%) did the sum as shown alongside. The principle applied success­ fully in example 8 was not ex­ tended to example 9. Ex. 7: 70 X707 490 000 490 49,490 Ex. 8: 6 X 46,823 240,000 36,000 4,800 120 18 280,938 Ex. 9: 23 X456 138 1,150 9,200 10.488 SYMPOSIUM 1962/63 63 Two of the examples given involve a nought in the multiplier or multiplicand. The solutions given by some testees are as follows:— Ex. 10: 5,205 X804 20,820 00,000 420,000 440,820 Ex. 11: Of the 231 testees, 25 or 11% did the sum as indicated; many of these (and others) had the answer incorrect, because of the omission or addition of noughts. 204 X 139 20,400 6,120 1,836 28,356 Of the 231 testees as many as 151, or 65%, did the sum as shown alongside. The similarity between examples 10 and 11 was not realized. The author was interested to know how the testees would react to multiplication sums involving 11, 12 and 16 as multipliers. In all schools much attention is given to the eleven and twelve times tables, and in many schools the sixteen times table is also taught. When multiplying by 11 and 12, pupils are not expected to multiply by 10 and 1 or 10 and 2 separately, but as one operation. Instructions to this effect have been incorporated in the syllabuses issued to teachers in the Transvaal. Here are the efforts of some of the testees: Ex. 2: Ex. 3: Ex. 4: 478 8,050 629 X 12 x n X16 956 8,050 6,290 4,780 80,500 3,774 5,736 88,550 10,064 Example set out as above by 21% of the testees. Example set out as above by 17% of the testees. Example set out as above by 82% (189 out of 231) of testees. Horizontal multiplication is hardly taught in our schools. Very few testees, indeed, did this sum as follows: 1,212X12=14,544. Much time is devoted in our schools to the teach­ ing of short methods of multiplication, especially with multipliers such as 99, 101, 201, etc. The author was therefore particularly interested in how the testees would react to the following sum: Ex. 13: 456X99. No fewer than 175 out of the 231 testees, i.e. 76%, did the sum by the long method. Of the remainder, who attempted to work the sum by the short method, 13 failed completely. Here are some of the efforts: (i) 456X99=4,560—99=4,461 (ii) 456X99=36,040 + 36,040=72,080 (iii) 456 X 99 =4,140 + 4,104 = 8,244 (iv) 456 X 99 =45,600—99 =45,601 4.4: FRACTIONS The testees’ responses to the questions on frac­ tions were, to say the least, appalling. The following is a summary of the results: No. o f testees Percentage o f out of 231 who testees who Example did the sum did the sum incorrectly incorrectly 1. 3 I - U - * 111 48% 1 5 1 2. 45 21 2 2 5 192 831% 3. 35=5% x l 1/, 109 47-2% 4. 35 = 5% of l>/9 180 77-9% 5. After a baker had sold | of his bread, he still had 875 loaves over. How many had he at first ? 170 73-6% Here is a list of typical errors:— Ex. 2: 3?—1|[ - * 9 18 7 7 = — --------- — 5 4 10 201 100 5 1 5 1 Ex. 2: 1 5 1 45 21 225 45 2] 225 6 1 5 6} 221 (45 + 21 )—225 5 = ------- = etc. -1 5 1 G4 SYMPOSIUM 1962/63 Ex. 2: 1 5 1 4J 25 225 = 45 + (5 x 2 J )-2 2 5 Ex. 2: 1 5 1 45 21 225 1 2 5 4 1 2 = —X —1— X ------- X — 0 9 0 9 0 22 Ex. 5: 00 CA II Xjs t Ex. 5: 1=875 loaves 8 - = A 8 II <_/l •I- 00 = ... loaves 8 7 5 x 8 x 8 5 x 8 3 = 875A Ex. 5: Ct i- i II O O <-* H | = - x 875 loaves J=177 8 8 Originally he had -= 1 ,416 loaves 1,4585 loaves 8 1093x8 Ex. 5: 8 7 5 = | 955 .'. 1 = 8757-5 ---- X3 = 171 8 8 2,865 -= 1 ,368 8 8 3585+875 = 1,2335 loaves Ex. 5: x —-|x=875 x = 875 + §a = 8a= 7,000 + 3a: .-. 5a = 7,000, .’. a = 1,400 From example 5 it is clear that the average pro­ fessional student entering the teaching profession does not know how to set out a simple problem, and that the equality sign ( = ) has no meaning to him. 4.5: DECIMALS In all fields of scientific work, and in the newer branches of computation such as cost accounting and statistics, a knowledge of decimals is indispensable. Much attention will have to be given to a study of decimals at school level and at Colleges of Edu­ cation. The results of the test in decimals indicate that professional students on entering the teachers’ train­ ing course have a very inadequate knowledge of the computational aspects of decimals. The following is a summary of the results. Example No. o f testers out o f 231 who did the sum incorrectly Percentage of testees who did the sum incorrectly 1. 0 -2 x 0 - 2 x 0 -2 = 140 60-6% 2. 0-2 x0- 3 x 0 -4 = 127 55% 3. 300x0 003x0 03 = 173 74-9% 4. 6-47-4= 39 16 9% 5. 0 000I2-X0 04= 144 62-3% 6. 2 - 4,000 = 164 71% 7. 6 x 0 - 5 x 0 0 4 186 80-5% 0 025 8. 1 1 1 0 -2 X0-4 0-5 (Answer as a com­ mon fraction) 205 88-7% 9. 14-6x0-29 142 61 5% 10. 13-87-4 76 32-9% Here is a list of typical errors. Only the incorrect answers are given alongside each sum. Example List o f typical incorrect answers I. 0-2 x 0 -2 x0- 2 (i) 0-8; (ii) 0-12; (iii) 0-6; (iv) •006 2. 0 -2 x 0 -3 x 0 -0 4 (i) -24; (ii) 2-4; (iii) -00024 3. 300x0 003x0-03 (i) 9; (ii) -00009; (iii) -00027; (iv) -02700; (v) 300-009; (vi) 300-0009; (vii) 300-033; (viii) 3-033; (ix) 900-009; (x) 2700000 4. 6-47-4 No significant errors 5. 0-00012 : 0-04 (i) 0-03; (ii) -00000048; (iii) ■00003; (iv) 000 003; (v) •0000003 6. 27-4,000 (i) 2,000; (ii) 8,000; (iii) -002; 1 (iv) 2; (v) ------; (vi) -5 2000 7. 6 x 0 -5 x 0 -0 4 (i) 4J; (ii) 4-08; (iii) 0-041 0-025 8. 1 1 1 0-2 XQ-4 0-5 5 1 25 (i) J; (ii) - ; (iii)------; (iv) — ; 2 040 0 1 1 (v) — ; (vi) — 25 -4 SYMPOSIUM 1962/63 65 9. 14-6x0-29 (i) 42-340; (ii) 4,234; (iii) •42340 10. 13-84-4 (i) 3-405; (ii) -345; (iii) 340-5; (iv) 34-25; (v) -03405; (vi) 34i Why do the testees perform so badly in this test on decimals, particularly if one considers the fact that decimals are taught in all classes from Std. IV upwards? The author is of the opinion that in the teaching at school level too much emphasis is placed on the learning and applying of rules without ensuring that the basic principles on which the rules depend are fully grasped and mastered. Some authorities in fact recommend that the manipulation of decimals should be taught by rule of thumb methods. The children should simply be told that the decimals be ignored until the multipli­ cation is completed. Thereafter the number of places in the multiplicand and the multiplier are counted and the decimal point is inserted.(5) A rule of thumb for the division of decimals is also given. If a pupil is not made to understand the principle, the rule is really meaningless and easily forgotten. This then appears to have been the case with the testees. Many did not have occasion to use deci­ mals for a year or longer and, confronted with examples, they attempted to rely on memory instead of tackling the examples de novo from basic prin­ ciples. 5. Where Does the Rem edy Lie? (a) The primary school syllabuses in arithmetic must be recast to stress an understanding of basic principles, and the present emphasis on soul-destroy­ ing and monotonous drill of processes involving long, complicated and abstruse manipulation should be discarded. Oral work should be the key to all primary school arithmetic, and should, in the opinion of the author, occupy one half of the time of each lesson period. (°) Potter, F. E .: The Teaching of Arithmetic (Pitman & Sons, London, 1932) p. 246. See also: The Education Gazette, Province of the Cape of Good Hope, Vol. LIX, 17.3.1960, p. 511. The new syllabus for arithmetic for primary schools in the Transvaal reflects a new trend and shows a movement in the right direction. It is grati­ fying to read, for example, that in the division of decimals by decimals, teachers are instructed to “stress the fact that the divisor must be converted to a whole number and explain why this is done.”(6) (b) The present curriculum for the training of primary school teachers needs to be amended. While not denying the importance of the humanities in any training course for teachers, ours in this scien­ tific and technological age is far too heavily weighed down by the so-called skills, i.e. Art and Crafts, Music, Physical Education, Writing and Blackboard Work, etc. The general atmosphere is still far too much orientated towards the traditional classical approach. All students following a three-year general pro­ fessional course for the primary school with or without specialization in a particular subject, should be given a much more elaborate and challenging syllabus in arithmetic than is the case at present. To provide them with instruction that is merely an extension and an elaboration of the primary school syllabus will no longer do. Instruction in arithmetic alone is not enough. Our students require a well co-ordinated, adequately planned course in basic modern mathematics that will give them insight into the rigours of mathematical thinking, and will make them aware of the meaningfulness of absolute truths, precision of statement and deductive logic. It is this mental equipment which our future teachers need. This, then, is a challenge we in the teacher train­ ing colleges of South Africa must face. The Johannesburg College of Education is attempting to meet this challenge by introducing, with effect from 1963, a new course in modern mathematics for all first year students. More time is to be devoted to the sub­ ject, and the emphasis will be a distinctly different one. (6) Transvaal Education Department: Syllabus for Arithmetic Grade / —Std. V (1960, p. 41 op. cit.) 66 SYMPOSIUM 1962/63