LB U5Q9 ■ DIAGNOSTIC AND REMEDIAL TREATMENT FOR ERRORS IN ARITHMETICAL' TSA'GONINC By .7. J. Osourn Director oi induce tionai iAeasureuents Issued by State Department of Public Instruction Kadi son, Wisconsin December 1922 01 DIAGNOSTIC AND REMEDIAL TREATMENT FOR ERRORS IN ARITHMETICAL REASONING INTRODUCTION A voluminous literature nas appeared in recent years toased upon trie correct reponses which children make. The last decade will be remen^e- ed in educational history as the time when large numbers of standard tests first began to be published, and when, the rank and file of the teaching profession began to get aqquainted with the application of elementary statistical procedure to the field of education* Splendid as this movement has been it is by no means complete, The purpose of this and other circulars which are To follow is to describe, in some measure. a further line of attack upon our educational problems based oh the study of wrong responses. There is good argument to support the thesis that a study based upon wrong responses is likely to be even more stimulating and helpful than a procedure based upon the results of the child's susseesful efforts. In the business wo aid much study is given to the points at which the out put of the factory Jraila to function as it should. The manufacturers of automobiles, for example, have '■very definite information concerning the weaknesses of their cars. From a study of these weaknesses it is possible to improve the output.. tn general a contact with failures and breakdowns stimulates thinking- Improvements can come only when there is a somewhat definite knowledge of failures. Progress in the study of errors has long been retarded by the idea that there is the utmost aiversity in the mistake of children. The actual evidence is far to the contrary, The same wrong answers occur in all types of schools and over wide areas to such an Sxtent that it js possible to forecast how many pupils in any individual school will make certain mistakes. Errors are like diseases in some respects. There are a great many different diseases, but only a few of them are wide-spread and frequent in occurrence. In like manner the evidence so far available shows that four or five different things are resppnsible for at least three fourths of the trouble which children have. To identify these widespread errors, determine their causes, and plan their cure is clearly an important task in educational research. Such data will lead to more intelligent supervision, more scientific methods of teaching, better organization of curriculum material, and more efficient teacher t ~training. With these facts in mind it is rather surprising to find how little is available in print concerning the nature and cause of fa: lure. This circular is one of a series begun last year and designed to investigate these questions as they apply to the sever:! school subjects, THE IMPROVEMENT 0? ABILITY I N ARITH METICAL REASONING In discussing a remedial program on Arithmetical Reasoning the following items must be known and agreed upon:- 575378 -2- 1. A list of typical problems which have been widely used and on which lists of wrong answers arc available. 2. An identification and classification of the most serious disabili- ties. 3. A list of the wrong answers which are symptomatic of each disability in each type of problem^.. 4. Tre remedy to apply after the specific disabilities are known and indentified. Each of the these will be considered in order. 1. A list of the Problems which were used. In this study the problems in the Buckingham Problem Test, Porm 1 were used. {l) V7e learn 2 words a day in our class. How many do we learn in 8 days? ($}}23 children belong to our class but only 19 are present. How many children are absent? (3) James has 28 marbles. He gives half of them to Charles. How many has he left 9 (4) If you can get 3 ginger-bread dogs for 5 cents, how many can you get for ten cents? (5) A boy owned 3 kites, each of them having 150 feet of string. How many feet of string had he? (5) A baseball team took 12 players on a trip. The trip cost the team $36.00. How many was that for each player? (7) An automobile was run 30 miles every day for a week. How many miles did it go? (8) Henry gathered 5 quarts of nuts. He sold them at 8 cents a quart and spent the money for oranges at 4 cents apiece. How many did he buy? (9) If an electric car runs 9 miles an hour, how many hours will it take to travel from one city to another, 117 miles away? ( 10) Ned sold his rabbit for 30 cents. This was 3/5 of what he paid. What did he pay for the rabbit? (ll)lf a girl had two one-dollar bills, three five-cent pieces, two dimes, and three quarters, how much money did she have? ( 12) now many pencils can you buy for 50 cents at the rate of 2 for 5 cents? (13)A boy had 210 marbles. He lost l/3 of them. How m^.ny were left? (14) Two tubs of maple sugar weighed 42 lbs. One weighed 18 l/4 lbs. How many pounds did the other weigh? Digitized by the Internet Archive in 2008 with funding from Microsoft Corporation http://www.archive.org/details/diagnosticremediOOosburich -3- (16) A store takes in the following suns: $1250; v-00, $175, $16^.25.,^ ^120.50, $32.75, $68.50. It pays out $600, v. by. tcr.d'n to icpr-ir in a. serious .. J. -- j th| ctij.l&Sa self -"confidence in general, arid aav even result in a feerjaanpni feeling of incapacity I. inferiority complex). Tne rer.cval of these errors, therefore , becomes a matter of major importance in -r. . > future development^ of our methods oi teaching, ^3r.;dJ.al Suggest ions For Those V/ho Totally ~Tail t o Comprehend flag Prob lem little is known concerning a child's mental reactions in the fa.ce of a-situation which he cannot understand. The safest assumption at the present stage of development in our educational psychology is that a child acts quite similarly to an animal under similar situations, A hungry rat in a cage trying to get food which is some distance out- side of the cage responds by making all sorts of reactions of which a rat is capable. The rat's problem is to get food. The child feels a need to vrite something on his tablet -- do something that will get a number that looks like the right answer. 3oth rat and child are blind- ly trying to satisfy needs. Neither has much of an idea as to how this may be done. Each tries the thing which is most easy for him to do, — the thing which, in the past, he has learned to do best, and each will- continue to make this seme reaction to sinilar situations unless the reaction proves unsatisfactory, In short, whenever humo>.n bo ings meet a new situation which they cannot analyze, they invariable* fall back upon the animal type of behavior in (-9) such circumstances. In educational psycnology tiria method of meeting new situations is called trial and error or fumbling and success. Our problem' therefore is fco substitute a higher type of learning for a lower one. This is the essence of teaching p'.n-ilo to thins and reason. We must meet this prcblea in some aaanner'. Fre re sai j t^te s _3? or_ yhinkirg^ Specific directions on how to teach children to think are not yet available, hut a few points are clear enough t o be stated somewhat definjtely. They are as follows:* 1. A supply of free ideas or concepts must be provided. 2. The ability to compare, combine, and judge the relative value of these ideas for the particular situations is necessary, 3. All of the relevant ideas must function, but none that are 'irrele- vant can enter into the thinking process. 4. Relevant ideas must be encouraged and rewarded; those that are irrelevant must be discouraged and penalized* 5. Thinking is impossible until adequate experience has been provided in trial and error and imitation methods . The amount which may be called adequate varies according to the intelligence and past experience of tne child. The foregoing may be regarded as well established principles in teaching of thinking. The present problem- of the field worker in education is to develop a technique which will assist in the real ization of the needs as they are now known. Some things to do for children who show a total failure to com- prehend the problem are as follows:* To increase the s upply of free jdeas or concep ts. Find whether or not the child is reading the problem correctly - This can be done in a very satisfactory manner by using the problems f'S silent reading material before any attempt is made at solution. Samples of s i lent r eading exercj ses v'hich are _prer e qui 8 it e to the S. Qluti on of Problems in Arithme tic. To the pupils Can you understand your arithmetic problems when you read them? If so, you can do these e3terci.se correctly. After each problem is a list of things which might be done. You are to show that you understand by drawing a line under the word or words whish tell the right, t hi np; to do . (l) Mary wishes to BS&e 35 cents; if she has already saved 24 cents, how much more must she sa.^e? Add Subtract Multiply Divide -3 0- (2) How many tablets can I buy for 40 cents, if each tablet costs 8 cents? Add Subtract Multiply Djvi.de (3) Fred's toother has 3 revs of fruit Jars with 9 jars in each ror. How many fruit jars has she? Add Subtract Multiply Divide (4) Frank had 2 apples and .his mother gave. him 3 more. How many apples did he then have? Add Subtract Multiply Divide Fill in the blanks and underline the words which tell the right things to do. (5) How much cheaper is it to rent a room for a year at $12.50 a month than it is to rent another foam for the same length of time at &3.50 a week? .. Divide by _ Divide b; Multiply by Multiply by Th*n Add Subtract Multiply Divide (6) A farmer sold 3375 bushels of potatoes at &0.90 a bushel, and wishes to buy j.and at $75 an acre. How m&£y acres can 5te buy? = Divide by Multiply by , Multiply S by Tner. Add Subtract Multiply Divide (7) If a man receives #7. 20 for 8 hrs . work, how much should he get . for 6 1/2 hrs..vork? Add __to Multiply by Subtract frorn Divide by _ Then Add Subtract Multiply Divide Exercises like the above will insure against careless reading. If the child fails tc respond correctly after a short practice upon exercises of this sort, it may be taken as a sure sign that, another t/ je of prerequisite training is necessary. Such children have not had sufficient experience with concrete things. Free ideas are lack*" ing and must be obtained, if at all, through the actual handling of the objects. They wj 11 need repeated practice of this sort which deals at first with one step problems only. -11- In problem I above it will be necessary to provide toy or real money. In problem 2 actual apples «ril3i b3 needed and so on. In" general th3?e children wall be lacking in intelligence also. Much tjme and patience will p ,y b9 needed. When success teOV been attained with tne simple problems more complicated on^s snould be used. S ome things to do for ch ildren who give only partial respon se s to the problem . Most of the vtrouble here is due to a failure to read the problem completely. Exercises like the following will prove neipful. To the pupil: Some boys and girls fail to get their problems be- cause they do not see everything. How sharp are your eyes? You will get caught in the following exercises i^ you are not careful. (1) A boy had 240 marbles and lost l/4 of them. How many had he left? Are ycu a3±ced to find how many marbles the bey lost? (2) John sold his rabbit for 20 cents which was 4/5 of what the rabbit cost him. How much did he pay for the rabbit? Underline the words wnich make the following statment true. John sold his rabbit for (more thai., less than) it cost hime Are you asked to find out how much money John lost on his rabbit? Underline the word which tells the right things to do. Add Subtract Multiply Divide So me thin g s to do for c hildren who fa.-"l to understand ouanti tativ e relation s. Make a list of the important quantitative relations which occur in the problems which ycu assign. The following list is suggestive: (1) The relation between cost, expenditure, income, and selling price. A certain house rents for $960 a year, the taxes and uokeep cost $300 a year. How much can I afford to pay for this house in order to clear 6% a year on my investment? Suppose I have bought the house at this price and am offered ^12,000 for it. Should I sell ? Why? (2) The relation between rate, time, and distance. The distance from New York to Chicago is 980 miles. Ho\v long will it take a train to run this distance <*% the race of 49 miles per hour? (3) The relation between length, breadth, and thickness. -12- I need a bin that will contain