ED 038 270 A THOR TITLE PUB DATE NOTE AVAILABLE FROM ED RS PRICE DESCRIPTORS DOCUMENT RESUME SE 007 645 Aziz, M. A. An Improved Method fpr Teaching Basic Addition in Elementary ",chools. 69 16p. Author, 6429 LivingstOn Road, Oxon Hill, Maryland 20021 ($2.00) EDES Price 1F-$0.25 EC Not Available from EDES. *Addition, *Elementary School Mathematics, Grade 1, Grade 2, *Instruction, *Mathematics Education, Teacher Education, *Teaching Techniques ABSTRACT The author claims the development of an improved method for teaching basic additicn in the elementary schools. Tvo advantages of the method are (1) more effective groiping of basic addition facts, and systematic and consistent use of reasoning in their derivaticn, and (2) use of a special classroom technique to improve the proficiency of a child in the application of basic arithmetic facts. An analysis is presented to show how the organization of various methods for teaching addition came into use, and compares their advantages and disadvantages. (RP)
Transcript
ED 038 270
A THORTITLE
PUB DATENOTEAVAILABLE FROM
ED RS PRICEDESCRIPTORS
DOCUMENT RESUME
SE 007 645
Aziz, M. A.An Improved Method fpr Teaching Basic Addition inElementary ",chools.6916p.Author, 6429 LivingstOn Road, Oxon Hill, Maryland20021 ($2.00)
EDES Price 1F-$0.25 EC Not Available from EDES.*Addition, *Elementary School Mathematics, Grade 1,Grade 2, *Instruction, *Mathematics Education,Teacher Education, *Teaching Techniques
ABSTRACTThe author claims the development of an improved
method for teaching basic additicn in the elementary schools. Tvoadvantages of the method are (1) more effective groiping of basicaddition facts, and systematic and consistent use of reasoning intheir derivaticn, and (2) use of a special classroom technique toimprove the proficiency of a child in the application of basicarithmetic facts. An analysis is presented to show how theorganization of various methods for teaching addition came into use,and compares their advantages and disadvantages. (RP)
$2.00 PER COPY$1.10 F014 'TEACHERS
AN IMPROVED METHOD FOR TEACHING
BASIC ADDITION IN ELEMENTARY SCHOOLS
BY M. A. AzuMimeo Meowed Semi ler, Nave Research Laboratory, WASHINGTON, D. C.
(A short trographice sketch of the author appears on pale 1 of this paper)
DISCUSSION BY
ALVIN W. SCHINDLER, AL D.Professor of Educatir nUniversity of Maryland
DOUGLAS P. McNUIT, Ph. D.PhysicistNaval Research Laboratory
KENNETH R. BERG, Ph. D.Assistant Preessm of MathematicsUniversity of Maryland
RUDOLPH E ELLING, Ph. D.rkssociate Professor of Civil Engineeringaemson Ugversity
ABIITItACT
The pape. reports the development of an im-proved method for teaching basic addition inelementary s-,-nools. This improved method canbe used iii the? arithmetic books for first and sec-ond grades. The adoption of the method will notrequire any change in the arithmetic books forupper grades, unless it is desired to upgrade, thecurriculum and the standard of instruction, nor,will it require new or special training for teachers.Teachers slnald understand the method.
The paper presents the development of themethod step by step. It analyzes how the organi-zation of the various methods came into use, andcompares their advantages and disadvantages.The analyses show that the author's method addsseveral advantages to the present method withoutlosing any beneficial aspects of the latter or any
new math incepts. Two of the more significantadvantages of the mi,thod are:
More effective grouping of basic additionfacts, and systematic and consistent useof reasoning in their derivation.
Use -4 a special class-room technique toimprove the proficiency of a child in theapplication of basic addition facts.
Tests conducted by the author on the basis ofindividual instruction showed that by this method,a child learned the basic addition facts moreeasily and became proficient in their applicationin less time. With a few minutes of practice a dayhe achieved, in the fir,t grade, a speed and skillin basic addition which otherwise he might nothave achieved before the seventh or the eighthgrade. Similar results could also be expected onthe basis of class-r nom instruction.
"PERMISSION TO REPRODUCE THIS COPYRIGHTEDMEWL BYAAIICR1ILIff ONLY HAS REDA GRANTEDBy sm. et az,TO ERIC AND ORGANIZATIONS OPERATING UNDERAGREEMENTS WITH THE U. S. OFFICE OF EDUCATION.
FURTHER REPROMTION OUTSIDE THE ERIC SYSTEM
REWIRES PERMISSION OF 111 COPYRIGHT MIER."
itioVdsit 1969 by the Author. Al Elites neessed wider the Istewart Pae-Ameleas Copydght Comovalies. pert of fhb paper
nay he sejesdeesd he ew tarK witiont=eltitlee permiedes from theMahar. RIMINI ill the Wise States et
2
4.i 4 +I +I .ft, +1 A
II , 10 g II8
1 2 0 41 6. 7 4i 1 1 4 i +11 9
+4 tti i +1 I 1 19 , A 4 i 4.1 +i 4
+i +a 45 It kJ
+.39
+5I2 6 9 I 9
8 5+7 +1 41 4 +I 47.9 4-2 4i 1I 8 3 n 17 7
4 7 , T+4 + 'ii 8
4+1
. 13
4 I.4
41 +4 +iID
+8 +2 +7g i
13
+47
7+ 2g
9+2
4+1IT 7
4 4ts fi 1iff
+.1 +I fri6 6 9
+49
+7
41
5 8
118 4
+1 +24 T
96 4- I
1.3 lb-
9+2
7.4111,
TABLE 1. ADDITION FACTSwith NO PARTICULAR ORDER
IN1RODUCTIONTo compute addition like the ones shown in Figure 1, a
child must know the elementary or basic addition of twodigits, such as 5 + 4 = 9, 4 + 6 = 10, etc. All such basic
546793+469831
78649 741+ 6943 + 69
41111111
FIGURE 1. Examples of addition in practice
addition, or basic addition facts, as they are more com-mon!y referred to, of two digits from 1 to 9 are shown,with no particu'ar order, in Table 1. One could conceive ofa time in the state of the art when addition facts were notarranged in order. Clearly then, man's first attempt was toarrange them in scme order. Having known no other way,or having found no reason to arrange differently, he ar-ranged them according to his very old, but not obsolete,natural sense of arranging objects, that is, in rows andcolumns in ascending order of the natural numbers. Thisarrangement is shown in Table 2.t
It is obvious that a child's proficiency in addition de-pends upon his proficiency in basic addition facts. By theauthor's method, a child could learn the basic additionfacts more easily and could become proficient in theirapplication in less time.
TABLE 2. THE BASIC ADDITION BLOCK(ADDITION FACTS ORDERLY ARRANGED)
Adler [I] enlarges the basic addition block of Table 2by including the zero addition facts in it. Referring to theenlarged addition table (page 112), he asks a child, at thecompletion of his third-grade year, to "know these 100addition facts by heart." Upton and others [2] ask a childto "tell or write the answers as quickly as you can" to theaddition problems of Table 1, but scrambled in a differentmanner (page 209). (An addition fact without the sumwill be called an addition problem.)
In the present work, a method is developed in which wemay expect the child to know the basic addition facts at thecompletion of his first-grade year, and in which we shallhave reduced the size of the addition table and refer thechild to a smaner table. We shall not expect the child toknow all the addition facts by heart, but only some of themby heart, and to derive the rest from the ones he wouldknow by heart. Further, we shall expect him to know onlythose facts by heart which are easier than he others. Andfor the others, we shall improve the present method, sothat he can derive them more quickly.
DESCRIPTION OF METHODSWe see in the Introduction that the addition facts of
Table 1, having no particular order, are arranged in Table2, in ascending order of the natural numbers. After this isdone, we must ask, "In which order do we introduce theaddition facts to a child, since we cannot introduce themall at the same time?" For example, do we introduce them " 1
t Table 2 has been labelled as the Basic Addition Block and will 111 "MathematicsGrade 3," Irving Adler, Ph.D., Golden Pressbe referred to as such throughout the paper. [21 "Learning About Numbers," C. B. Upton, K. G. Fuller and
G. H. McMeen, American Book Company
3
TABLE 3. THE ADDITION TABLE TABLE 3A. THE DIRECTION NETACCORDING TO THE OLD METHOD
the order 1 1, 2 + 1, 3 + 1, . . . , in the order 1 + 1,1 2, 1 3, . . . , or in some o t h e r o r d e r ? If w e i n t r o -
d u c e than in the o r d e r 1 + 1, 2 + 1, 3 -I- 1, . . . , then achild travels the basic addition block alor; rows from leftto right. If we introduce them in the order 1 1, 1 + 2,1 3, . , then a Odd travels the basic addition blockalong columns from top to bottom, and so on. After wehave decided how we shall introduce the addition facts toa child, the next question we must ask, "How do we teachhim an individual addition fact?"
We shall review the various methods presented in thispaper in the light of these two questions. Tne reader should,however, keep, in mind that in this section of the paper weshall primarily be describing the various methods, but notanalyzing or comparing them; analysis and comparison willbe found in the next section.
THE OLD METHODIn the old method, strictly speaking in one of the old
methods, the basic addition facts were introduced to a childin t h e o r d e r 1 + 1, 2 + 1, 3 + 1, . . . So, a child trav-eled the basic addition block along rows from left to right,top to bottom. This direction of travel is shown in Table 3.A reader may have observed that the basic addition blockhas not been altered; only a direction, shown by the direc-tion net of Table 3A, has been superimposed on it. At onetime in the old method, still used in some places, a childlearned and performed the basic addition facts by finger-
for THE OLD METHOD
counting; that is, lie counted fiugers eacE time he com-puted the sum of a basic addition fact. For long addition,too, he depended upon finger-counting. The process oflearning was very slow. (The term "long addition" hasbeen used to differentiate between basic addition and ad-dition of two numbers explicitly, each number contziningmore than one digii)
The direction nets need some explanation. Light linesseparate one addition fact from another, and heavy linesseparate one group of addition facts from another. Arrowspoint the direction in which a child travels in a group, whilethe numbers by the arrows indicate how the groups areintroduced to the child.
THE FLASH METHODIn the so-called flash method, addition facts of the basic
addition block were flashed before a child, using flashcards. This was done with the hope that after a time, whenaddition facts were shown to him without the sum, the sumwould appear in his memory. This process facilitated drill-ing and improved a child's speed of learning. But the childstill depended upon his memory, for, the sum must beretrieved from his memory. He still followed rows from leftto right; that is, he still followed the direction of the oldmethod. It may be mentioned here that the flash method isnot a basic method. It is a system of training in which flashcards are employed to improve the spePd and skill of achild in acquiring and in applying the basic addition facts.The system does not have a direction tf its own. It tees I
4
the direction of the method which is being employed at aparticular time.
Some of the flash cards that are available are flash cardsNG. 7020, published by the Milton Bradley Company, flashcards No. 96700, published by the McGraw-Hill BookCompany, and flash cards No. 4570:39, published by theWestern Publishing Company. All of then: flash cards usethe direction of travel of the old method. Figure 2, whichis a summary card of the.Milton Bradley flash cards No.7020, shows this direction of travel, for example.
THE CONSTANT -.;UM METHODThe order in which the basic addition facts are intro-
duced to a child in the method currently in use in theUnited States is shown in Table 4. The method currentlyin use in the United States is referred to as the constant-sum method. The reason for the name is discussed shortly.It is seen that, in this method, a child travels the basicaddition block diagonally. The reader may have observedagain that the basic addition block of Table 2 has not beenaltered; only a new direction, shown by the direction netof Table 4A, has been superimposed on it. This directionof travel has been used by authors of first and second gradearithmetic books in various modifications. For the benefitof those who may not be fa.-niliar witt the present systemof elementary education, the following books may be citedas examples.
I. -Ekmentary Mathematics: Second Edition, Grade I,Harcoun. Bract & World. Inc
2. "Aethmetic Workshop.- Second Ediii,n. Book I.American Book Company
3. -Elerraentary School Mathematics," Second Edition, Booki. Addison-Wesley Publishing Company
It is seen that the sums of the addition facts of a step inthis method remain constant for all the facts of the samestep. For this reason, this method has been referred to asthe constant-sum method.
To make the direction of travel of the constant-summethod conform with our traditional sense of arrangementof objects, its addition facts have been rearranged fromTable 4 into Table 5. The first nine steps cf Table S aretaught to a child chiefly through his basic training in count-ing, concepts and unthrstanding of numbers. The remain-ing six steps arc taught with the help of the first nine steps,and by use cf reasoning or derivation. In this method, achild must complete the addition facts of the first ninesteps before he can use reasoning to derive each fact of theremaining six steps. In a way therefore we may regard thefirst nine steps of Table 5 as the fundamental steps of addi-tion and the remaining vix steps of Table f as the additionalsteps of addition, in the constant-sam method. The readermay have observed that the flash system of training cannotbe used in the constant-sum method, since the sum is con-stant for all the facts of the same step. An elaboration ofthis point is found in a later section of this paper.
FIGURE 2. The summary card of Milton Bradley Flash Cards (By permission)
S I
3 2 1
+1+
2+3 T
X-W 4 3 14 +
1
+2
+3
2+
4
5 4 3 1
+I
+2
+3 +4
+5I T
TiT
T6 5 4 3
+1
+2
+3
+4
7777 7 6 5 4
+I
+2
+3
+4
WT
-FT
TA
BLE
4. TH
E
BA
SIC
AD
DIT
ION
BLO
CK
arranged
for
TH
E
CO
NS
TA
NT
-SU
M
ME
TH
OD
%/777777,A P 7 A
ir.
OF ?A
Aargi
FA ri7. 7
TA
BLE
4A.
TH
E
DIR
EC
TIO
N
NE
T
for
TH
E
CO
NS
TA
NT
-SU
M
ME
TH
OD
7 6
10 +2
9+
38
+4
+5T
i
Ti
Ti
if
9 8 7 6I I +3
+4
+5
+6
WIF
TE
-17 9 8 7 6
+4
+5
+6
+73.T
i S i Ts9 8 7 6
+5
+6
+7
+8
+7
+6
4
IT IT
5 4
+7+
8 UM
5 4
+8
+9'IT
I 5+
9
3+
8 II 3+
9T
Y
2+
9 II
9 7
15 +7
+8
8+
9
16 16 169
16 +8
+9
rr 1179
17+9
18T
AB
LE
5.
TH
E
AD
DIT
ION
TA
BLE
by TH
E
CO
NS
TA
NT
-SU
M
ME
TH
OD
6
TABLE 6. THE BASIC ADDITION BLOCKarranged for THE CONSISTENT-LOGIC METHOD
THE CONSISTENT LOGIC METHODThe method presented in this paper will be referred to
as the consistent-logic method. The reason for the namefollows shortly. The order in which the basic addition factswill be introduced to a child in the consistent-logic methodis shown in Table 6. Again, it may be observed that thebasic addition block has not been altered; only a newdirection, shown by the direction net of Table 6A, hasbeen superimposed on it. In this new direction, a childtravels the basic addition block as follows:
He travels the first row first from left to right, as hedid in the old method. He then travels the diagonals,starting with 2 -I- 2, 3 -I- 2, 4 + 2, respectively. Next,he travels the column starting with 9 ± 2, the row start-ing with 5 -I- 2, the column starting with 8 + 3, andfinally, the small triangle starting with 6 -I- 3.
Arranged to conform with our traditional sense of arrange-ment of objects, the addition facts of the consistent-logicmethod are shown from Table 6 into Table 7. It may beobserved that we did not include the addition facts of thelower triangle of Table 6 in Table 7. We shall see laterwhy we did not do so.
The first two steps of Table 7 are fundamental steps,and we may refer to the remaining six steps as the addi-tional steps of addition, defined by the same criteria ofdefinition we employed for the fundamental and additionalsteps of addition in the constant-sum method. As in theconstant-sum method, the fundamental steps of the con-
TABLE 6A. THE DIRECTION NETfor THE CONSISTENT-LOGIC METHOD
sistent-logic method are taught to a child through his basictraining in counting, concepts, and understanding of num-bers, and the additional steps, with the help of the funda-mental steps, and hy use of reasoning or derivation.
The use of reasoning for derivation of addition facts isnot a new concept in the consistent-logic method. Again,for the benefit of those who may not be familiar with :henew math as applied to elementary grades, it may bepointed out that children arc now trained in the use ofreasoning, whenever applicable, to acquire basic additionfacts. For example, a child is now trained to learn thefact 9 -I- 4 = 13, as follows.t
He first learns the fact 10 -I- 3 = 13, through histasic training in counting by tens and ones (10 and 1are 11, 10 and 2 are 12, . . . are counting by tens andones). Then he is trained to reason, "take 1 from 4 andput it (1) to 9, we have 10 and 3 are 13" (Figure 3).
4
miTake lfro
r-10-and 3 are
13.--I
and put it (11t we have1
FIGURE 3. An example of use of reasoningfor derivation of addition facts
t Sec page 245 of "Elementary School Mathematics," cited pre-viously.
I
2
7
2+2-4i
3+3T
3+1
4-
33
+2
4+2
6
59
+21 I
65
+2-r7
8+3
1 I
86
+3T
4+3"75
+3If9
+3
4+4-ff5
+4
4+1
T5
+51W6
+53.a_33_..-7
5 I 6+1 +1T 7
7+6
1
8+614
8+7
1
9+716
8+8W
+II I +?T T9
+9IT
9+817
6+410
+512
IT6
+2if8
+4TE7
+3I0
9+413
9+514
9+61-6
7+2-ES8
+5IS7
+41 I
8+2
,75'
TABLE 7. THE ADDITION TABLEby THE CONSISTENT-LOGIC METHOD
He is demonstrated the reasoning with some visualaids or objects, and he is exposed to the use of reasoningonly after he has received and acquired a reasonable train-ing in concepts and understanding of numbers. Althoughthe use of reasoning is not a new concept in the consistent-logic method. we shall sec later that reasoning is systematicand consistent in this method. For this reason, we havereferred to this method as the consistent-logic method.And as to the use of flash system of training, we sec thatit can still be used in the consistent-logic method.
Reasoning of the kind discussed here with the help ofFigure 3 cannot be very easily explained without a per-sonal demonstration. For readers other than tcachcrs, itmay not have been very clear. Since, however, this reason-ing is an important part for the discussion to follow, itwould be appropriate for a reader to pause here for amoment to review the reasoning, and to grasp it fairly well.
SUMMARY OF METHODSOne could conceive of a time when the addition facts
were not organized in order. Man's first attempt was toorganize them in some order. and he did so, as shown inTable 2. Later, different methods were developed forteaching basic addition, depending upon how the additionfacts were introduced to a child. In one of the old methods,the addition facts were introduced to a child in the orderas shown in Table 3. The order in which they as., intro-duced in the constant-sum method and in the consistent-logic method arc shown in Table 5 and Table 7, respec-tively. The method currently in use in the United Stateshas been called the constant-sum method. and the methoddeveloped by the author is called consistent-logic method.
ADOPTION OF THECONSISTENT-LOGIC METHOD
As stated in the Abstract, the consistent-logic methodcan be used in the arithmetic books for first and secondgrades. The adoption of the method will not require anychange in the arithmetic books for upper grades, unless itis desired to upgrade the curriculum and the standard ofinstruction. The adoption of the method also will not re-quire any change in the preparatory training of a childfrom kindergarten to that gage of the first grade when hefirst begins his formal training in basic addition facts, norwill it require any change in the training for application ofbasic addition facts, after he has acquired them. All mate-rials, methods, or tools used in the preparatory trainingand in the training for application will remain unaffected,as well as the ways they arc used. The change will be madeduring the actual period of training in basic addition facts,and anly in the order of presenting them according to theconsistent-logic method.
In view of the preceding discussion, which is pictoriallyillustrated in Figure 4, we may conclude that the r )nsistent-logic method will not require new or special training fortcachcrs, nor will it require complete rewriting of first andsecond grade arithmetic books. It will be sufficient fortcachcrs to have an understanding of the method, andauthors may revise their books, to write only those portions
PERIOD OF PREPARATORYBASIC TRAINING
ACTUAL PERIOD OFTRAINING IN THEBASIC ADDITION FACTS
PERIOD OF TRAINING INTHE APPLICATION OFBASIC ADDITION FACTS
t t tNO CHANGE CHANGE NO CHANGE
FIGURE 4. Changes relative to periods of trainingwhen the consistent-logic method is used
8
which were devoted to the training in basic addition facts,according to the constant-sum method.
A second paper is being prepared to systematize theconsistent-logic method further and to outline broad guide-lines for revision of existing books and for wfiting of newbooks. The paper will also list the different war in whichthe consistent-logic method can be presented in teachingmaterials and will describe the flexibility with which it canbe used in various situations.
COMPARISON AND ANALYSISOF METHODS
Why Comparison And Analysis?
The consistent-logic method was developed through ex-perimentation with children for more than one year. Afterit was developed, the method was tested on the basis ofindividual instruction. Tests showed that, by the consistent-logic method, a child learned the basic addition facts moreeasily and quickly. Comparison and analysis are presentedto explain the results of tests and to give some insight intothe various methods. Intent into the various methods maypoint out some of the reasons that could have contributedto the improvement of the consistent-logic method. Otherreasons for analyses are found in the Conclusion. It maybe more interesting to a reader to have his own compari-son and analysis before proceeding to those that follow.
Fundamental Facts Reduced In NumberIn the old method, all 81 basic addition facts were fun-
damentals to a child, since he learned each of them byfinger-counting or by rote memorization. In the constant-sum method, there are 9 fundamental steps, with 45 basicaddition facts. In the consistent-logic method, there aretwo fundamental steps, with 17 basic addition facts. Theconsistent-logic method has, therefore, the least number offundamentals.t
The fact that the consistent-logic method has the leastnumber of fundamentals remains true also after we excludethe inverse addition facts from Table 5 of the constant-summethod much as we have excluded them from Table 7 ofthe consistent-logic method. (An inverse addition fact hereis defined to be one in which two numerals to be addedare reversed, for example, 1 + 7 = 8 is the inverse addi-tion fact of 7 + 1 = 8.) However, we must not overlookthe fact that the inclusion of the inverse addition facts inTable 5 was a necessary part of the constant-sum method,whereas their exclusion from Table 7 is not an exclusionof a necessary part of the consistent-logic method.
Fundamental Facts EasierTo Learn And Remember
Tests showed that the fundamental facts of the con-sistent-logic method were easier to learn and rememberthan those of the constant-sum method. One could intui-
3 2 4 3+2 +2 +3 +3
4+1 +1
5 7
FIGURE 5. First example of use of sons logic and fundamental shipsfor derivation of addition facts by theeandstwit4aaie method
tively conclude so by inspecting them. However, some ex-planation is provided to support test results.
The facts 1 + I, 2 + 1, 3 I- 1, . . are present in bothsets of fundamentals. They could not alter the ease withwhich a child learns the fundamental facts in either method.Fortunately, therefore, we need compare the remainingfundamental facts. We shall ch, so with the help of exam-ples. For example, we take the fact 6 + 6 = 12 fromstep 2 of the consistent-logic method, and see how a childlearns this fact. We see that a child could learn this factdirectly after his training in counting,* as a natural secondstep. The point being brought out is that there is no addi-tional fact in between a child's training in counting and hislearning of the fact 6 + 6 = 12. But such is not the casefor all the fundamental facts of the constant-sum method.For example, a child mit know the fact 7 + 2 = 9 beforehe could learn the fact 7 + 3 = 10 of step 9 of the con-stant-sum method.
We also see that the facts of step 2 of the consistent-logic method are symmetrical. They are symmetrical in thesense that the two numerals to be added are same about theplus (+) sign of the problem. This symmetry couldhave helped a child learn these facts more easily, and re-member them longer (or permanently) once they arelearned. The fundamental facts of the constant-sum methddare not all symmetrical.
More Effective GroupingIn the old method, each fact of the basic addition block
was new to a child, because there were no fundamentalsteps that could be advantageously used to derive the rest.This serious drawback of the old method, of having nofundamental steps that could be advantageously used toderive tile rest, was overcome in both the constant-summethod and in the consistent-logic method. This was doneby special grouping of basic addition facts of Table 2according to certain rules or concepts. (This concept ofspecial grouping has been known as the concept of setsand subsets in the new math.)
Grouping in the constant-sum method is mechanical;that is, addition facts having the same sum are mechan-ically grouped together. For example, addition facts havingthe sum 13 are grouped in step 12 of Taole 5, and additionfacts having the sum 16 are grouped in step 15.
Grouping in the consistent-logic method is logical; thatis, addition facts whose sums are derived by the use of
t Fundamental facts were defined earlier. * in the new math a child is trained to count upward to 20 by twos.
9
same logic and same fundamental steps are grouped to-gether. For example, all the addition facts of step 3 ofTable 7 use the logic of I more t and the fundamental step2. This is explained with the help of Figure 5. For the firstfact, the logic is"2 and 2 arc 4 and 1 more are 5." Forthe second fact, the logic is"3 and 3 are 6 and I moreare 7." All the addition facts of step 4 use the logic of 1more and I less with the fundamental step 2. This is ex-plained in Figure 6. For the first fact, the logic is"take1 from 4 and put it to 2, we have 3 and 3 are 6." For thesecond fact, the logic is"take 1 from 5 and put it to 3,we have 4 and 4 arc 8."
4 3 5 4+2 +3 +3 +4
6 8
FIGURE 6. Second exempla of use of selle logic and fundamental shopsfor derivation of addition facts by the consi:ient-logic method
Logic Systematic And ConsistentWe have seen that grouping has reduced the number of
fundamentals both in the constant-sum method and in theconsistent-logic method from those present in the oldmethod. Grouping has also made it possible to use logic(reasoning) in the derivation of additional steps of additionin these two methods. Logic is systematic and comistent inthe consistent-logic method, but not so in the constant-summethod. This is shown below.
In the constant-sum method, logic changes from oneaddition fact to another in the same step. For example, thefirst fact of step 12 of Table 5 uses the logic, "take 1 from4 and put it to 9, we have 10 and 3 are 13," as shown inFigure 7. But the second fact of the same step uses the
[
9 10 8 10+4 +3. +5 +3
13 13
FIGURE 7. An example of use of different logic and fundamental stepsfor derivation of addition facts by the constant-sum method
logic, "take 2 from 5 and put it to 8," etc., which is adifferent logic. In the second case, we are taking away 2instead of 1. The third addition fact uses another logic yet.But in the consistent-logic method, logic does not changefrom one addition fact to another of the same step, as wesaw in the description under "More Effective Grouping."
Because logic changes from one addition fact to anotherin the same step of the constant-sum method, each additionfact of the constant-sum method seems new to a learning
child, a drawback which we wanted to eliminate L JM theold method. So, we can say that the constant-sum methodhas not used grouping very effectively, while, on the otherhand, we see that the consistent-logic method has.
Continuing our analysis, we can also say that this draw-back of the old method, still existing in the constant-summethod, has been eliminated from the consistent-logicmethod by logical grouping of its basic addition facts. For-tunately enough, grouping of the consistent-logic methodstill does or is capable of doing what grouping of theconstant-sum method does. In addition, it uses logic con-sistently for all the addition facts of the same step. Becauselogic does not change from one addition fact to another inthe same step of the consistent-logic method, we nay ex-pect a child to remember which logic to use in the subse-quent fact once he 74.5 given the logic in the first fact, espe-cially after some practice.
The Special Class-Room TechniqueEarlier in the discussion, we stated that the flash method
of training could not be employed in the constant-summethod. This is elaborated here. If we take a set of flashcards containing the addition facts of one of the groups ofthe constant-sum method, we see that the sum remains thesame for all the facts of the group. Since the sum remainsthe same, a child will know the answers to all the problemsonce he knows the answer to one problem. Therefore, inthe constant-sum method, a child could not be drilled inthe addition facts of a group which was being taught at thetime. This may explain why there are no flash cards, in themarket, using the grouping of the constant-sum method.
In line with the use of flash cards, some class-roomteaching cards have been conceived. These are 9 x 7 inchcards for the teacher as shown in Figure 8, and a set of5 x3 inch cards for the students as shown in Figure 9.These class-room teaching cards may be used to drillstudents in basic addition by the consistent-logic method.Drilling may begin after an understanding of logic has beenestablished, and, simultaneously may be conducted asfollows.
The teacher will take a set of 9x 7 inch cards containingthe addition problems of a step of the consistent-logicmethod. Each student will be given a set of 5 x3 inch cardsconsisting of the sums of the given addition problems, plusan extra card not containing an answer. The teacher willface the class and show one problem to the class. The classwill search for the card that has the correct answer andwill show it to the teacher. The teacher can easily checkall the answers at the same time when the students willhold up their answer cards.
No Loss Of New Math ConceptsOne of the important advantages of the new math, as
used in the constant-sum methal, is that it trains a childin tie practice of different arithmetic operations with thesame three numerals of an addition fact. These differentarithmetic operations with the same three numerals of an
t An example of the simple logic of 1 more is"5 and 1 moreare 6."
FIGURE 10. Different arithmetic operationswith same three numerals of an addition fact
addition fact are shown in Figure 10 fcr the fact 9 4- 3 =12. Practice in performing these operations has resulted inimproving a child's proficiency in application of a basicaddition fact, and at the same time, a child learns subtrac-tions and equations involving numbers up to 20. In theconsistent-logic method also, a child can perform these op-erations. If we, further, considered other beneficial aspectsof the constant-sum method or other new math concepts,we would see that the consistent-logic method has lost noneof them. However, tests showed that the same results couldbe expected in these Iperations with less work on the childif he were trained by the consistent-logic method. Theresults of tests are explained as follows.
In the consistent-logic method, a basic addition fact suchas 9 + 3 = 12 is taught with emphasis on it as funda-mental information. The remaining seven operations aretaught as examples of how to aps lundamental informa-tion in practical problems. It was found that if a childacquired high proficiency in a basic addition fact in lesstime, he could learn the remaining seven derivatives of itvery easily with a slight adjustment in his thinking. Sincewe found that, by the consistent-logic method, a child ac-quired high proficiency in less time, we could, at our option,take some advantage of the consistent-logic method inreducing the work-load which otherwise would be placedupon the child if he practiced basic addition facts and theirderivatives with equal emphasis on both. For this reason,Table 7 dots not contain the addition facts of the lowertriangle of Table 6, namely, the inverse addition facts, nordoes Table 6 include them in the direction assignment.
SUMMARY OFCOMPARISON AND ANALYSIS
In the old method, all 81 basic addition facts were fun-damentals. In the constant-sum method, this number wasreduced to 45, and in the consistent-logic method, thisnumber was further reduced to 17. In the consistent-logicmethod, not only are the fundamentals minimum in num-ber, but they are also easier to learn and remember.
In *he old method, each addition fact of the basic addi-tion olock was new to a child. To overcome this drawback,both the constant-sum method and the consistent-logicmethod group basic addition facts, with the latter methodgrouping them more effectively. Both the constant-summethod and the consistent-logic method use reasoning inthe derivation of addition facts, with the latter methodusing it systematically and consistently.
The flash method of training can be employed in the oldmethod and in the consistent-logic method, but it cannotbe employed in the constant-sum method. Lastly, no newmath concepts or beneficial aspects of the constant-summethod have been lost in the consistent-logic method.
CONCLUSIONThe consistent-logic method is not a new method. It has
some new features or concepts. In all other respects, it canbe used similarly as the existing methods. Stated differently,the added features of the consistent-logic method can beincorporated in the existing methc.!s. Some readers maysee enough strength in the analyses for the conclusion thatthe consistent-logic method should help a child learn basicaddition more easily and quickly; others may like to seethe analyses verified uy formal tests on the basis of groupinstruction.t
t As the authors tests were neither formal nor based on groupinstruction.
12
I
DISCUSSION
BY PROFESSOR SCHINDLER: t
1 want to commend Mr. Aziz for his presentation ofaddition facts. Especially, I want to commend Table 7 as away of showing the goal to be achieved in teaching audi-tion and subtraction. Some teachers may not clearly see thegoal to be achieved. As for pupils, the comment of a thirdgrader is significant. When a teacher showed pupils a tableof the facts, one pupil remake& "If there are only thatmany, we can learn them. I thought that there were thou-sands of them." Mr. Aziz's table make the task look likeone that can be accomplished, and that may be a greatfountain of motivation.
Mr. Aziz does not lunv a total program. It seems to methat teachers and supervisors will want to see a total pro-gram into which his plan is integrated. They will want toknow what the learning activities with children should bebefore they are ready for step I.
In some respects there is very little that is absolutelynew in a field which has been worked as much as ele-mtntary school arithmetic. For example, generalization(understanding and logic) ha-.e been emphasized as ap-proaches to mastery of facts. Grossnickle and Bruecknerhave encouraged teachers to help pupils develop groupingof facts which resemble Aziz's steps in Table 7. Theirgroups were: the zero facts, the ones (his step 1), thedoubles and near doubles, and the couple of others. As faras I know, a concise arrangement such as Mr. Aziz has inTable 7 was not used before. The method has merits, andit will be a contribution if a total program is developedsatisfactorily.
THE AUTHOR'S CLOSUREProfessor Schindler has viewed Table 7 as a goal to be
achieved. In the second paper on the method, Table 7 hasbeen further divided into two separate tables (or goals).They are Table 8 and Table 9 as shown. The developmentand usefulness of these tables have been fully described inthat paper.
Table 8 contains only the fundamental steps, and Table9, the additional steps of addition. Steps of Table 8, in thesecond development, may be more significantly classifiedas fundamental steps with the following criteria.
1. A ch"d naturally goes to these steps after histraining in counting.
2. These steps are easier than the others.
I w. av s s s
STEP IPREPARATORYBASK; TRAINING
3. And once learned. a child does not easily forget them,they become a part of his permanent knowledge.
As for Table 9 each new step of additional facts a childlearns is based mainly on the four fundamental steps andonly e.,...,. the four basic concepts contained in them. Timefour concepts are:
I. The concept of one more, that is, going up by one.2. The concept of one less, that is, coming down by one.3. The concept of symmetry (physically), or doubles
(mathematically).4. The concept of uniqueness (physically), or tens and
ones (mathematically).
Professor Schindler's discussion was made on the firstmanuscript.* The discussion of this paper about the adop-tion of the method was made after his comment regardingthe integration of tHs method. As to the development of atotal program, it is hoped that the interested teachers andauthors will develop such programs or use the method forpreparation of their teaching materials. While broad guide-lines have been outlined in the second paper for revisionof existing books and for writing of new books, some sug-gestions in advance about the integration of the method interms of goals are presented here.
Figure 1 I shows the steps in the process of a child'slearning of addition and subtractior,. Step I will prepare achild for Table 8, the first goal to bu achieved; and Step II,for Table 9, the second goal to be achieved. In Step I, thechild will be trained in counting, in understanding, and, inthe four concepts of numbers previously described. In StepII, he will be trained in the fundamental facts and in theirapplication limited to the derivatives of the facts only. StepIII will trait: a child in the additional facts as applicationof the four fundamental steps and the four basic concepts.The accomplishment of goal 1 and goal 2 automaticallyaccomplishes the final goal in basic addition, namely, Table7, and prepares the child for Step IV, that is, for longaddition and subtraction.
Two addition facts of zero and one inverse addition factare shown :n Table 9 without any step number, becausetheir presentation is not totally dependent on the consist-ent-logic method; S they may be best presented by an authorin a sequence most suitableand effective to his planof presentation.
STEP IIIGOAL 2: TABLE 9
STEP IIGOAL 1: TABLE 8
SUBTRACTION-1
STEP IVADDITION &
FiGURE 11. The steps in the process of learning addition and subtraction
t a member of the National Council of Teachers of Mathematics $ All the discussions were made on the first 'manuscript.5 The names, the constant-sum method anu tl.e consistent-logicmethod, were not used in the first manuscript.
STEP I
I
i-i2
+i3
+4.
41.
5
16
++7
18
+-4-
I 9
+6
STEP 2 -I.1
-I2 -13 I4 15 -716 1 487
STEP 32
+z-4
3+-3-6
4+4-8
5
4-5.10
6+-6.12
7
+7-14
8
+a16
9+.218
STEP 41
10
+1II0 10
+-a +3.12 I 13
10 10 10
16
10 10
18
10
14 15 17 19
TABLE 8. THE FUNDAMENTAL STEPS OF ADDITION
STEP 53
+2
I+3+.7
5
+119
6+5
7+6
I
+8I15
9+8
5 I I 13 17
STEP 64
+25
+3i3
6+4lo +5
7
12
8+6 +7
9
F6--g 14
STEP 79
+2TT
9+3
12
9+4
9+514
9+5-
1513
STEP 8 +56
+2 +27
+2 +82,Iv77 -8 9
STEP 932
4-a2
52-
I
STEP 10 4-18
I I
84.11
12+
85
2+8io13
STEP II6_q9
7+310
7+4II
0+00
5+05
TABLE 9. THE ADDITIONAL STEPS OF ADDITION
14
BY Dr. llieNtrl'T:Mr. Aziz has applied mathematical logic to the teaching
of mathematics in a way so simple, it appeared to me, thatit is surprising that it was not done before. He has observedthat of the one hundred ba.iic addition facts of decimalarithmetic, a few are particularly easy to learn, and theothers can be derived using logical devices understandableto the student.
Eventually the student must commit all the addition factsto memory so that recall is immediate. To make this taskeasy for a child, new math introduced grouping of basicaddition facts and use logic in their derivation. Mr. Aziz,too, groups basic addition facts and uses logic, but in amanner in which logic becomes more systematic and con-sistent. As a result, I believe, he has made the task of achild simpler and easier. To this end, he also encouragesthe use of flash cards for drill. In his method, the additionfacts are introduced in an order in which sums vary sothat subsets of the flash cards can be used before all thefacts have been taught.
This work I hope would be considered a start in an im-portant field. What has been done here, I believe, could beextended immediately to addition facts in other numberbases and multi, lication. Similar techniques involv ing de-tailed analysis materials to be taught may well revolu-tionize the teaching of elementary mathematics. I wouldurge teachers to understand not only how a child mightfind Mr. Aziz's method easier and quicker, but also whyhe should find it so.
THE AUTHOR'S CLOSURE
I may point out some distinction between committing afact to memory and deriving it with high speed. To recallthe fact 5 + 5 = 10 is memorization. To recall the result25 + 15 = 40 is derivation, and to recall the fact 9 + 8= 17 in the new math is also derivation. If the derivationof the fact is so rapid that it seems as if the fact werememorized, there is little distinction in the end betweenrecall by memorization and by rapid derivation. What Dr.McNutt refers to as recall by committing to memory maytruly be referred to as recall by rapid derivation for somefacts in the new math concepts.
BY PROFESSOR BERG:
I have read Mr. Aziz's paper on teaching addition tochildren. He suggests an order of presentation of additionfacts which should give two benefits. Firstly, it should bepossible for the child to learn addition more easily andquickly. Secondly, the child's sense of logic and organiza-tion, or thought patterns, should be greatly improved orrapidly developed by this method.
It is easier to leant facts if they are presented in anorganized manner. The organization is most helpful if it isdirectly related to the learning task at hand. Mr. Aziz'spaper organizes addition facts in such a way. Rather thanming the ordering of the numbers as a principle of organi-zation, Mr. Aziz bases his organization on logic. The facts
are divided into two classesthe basic addition facts, andadditional addition facts. The basic addition facts aresimple, and few in number. The additional addition factsare then grouped into steps (as in the basic facts) and thechild progresses one step at a time. Within each step, allfacts can be derived from the fundamental facts, by thesame logic. The child thus is given a method he can usewith confidence for all the facts to be learned within thesame step. The author points out that such methods ofreasoning out the answers are taught in the present method,but the logic is not used in the consistent manner.
This grouping by logic has another advantage. Drilling,by use of flash cards, is a generally accepted manner ofincreasing speed and proficiency. lf, as is presently done,the facts arc grouped on the basis of the sum, then obvi-ously the flash method is inoperable. In Mr. Aziz's methodit is the logic which is constant and not the sum, thusenabling the use of flash cards.
hope Mr. Aziz's work would be widely read, as Ibelieve, it would lead to the significant development ofteaching of arithmetic in the lower grades.
BY PROFESSOR ELL1NG:
read the manuscript of Mr. Aziz' paper with consider-able interest. I can recognize the great amount of time,thought and elTxt he has devoted to this work. I believe,this warrants sincere congratulation.
I find the author's approach to the development of hismethod creative and ingenious. He studied the past and thepresent methods with a view to discover a pattern or a cluethat could lead to an improvement in the present method.His interpretation of grouping of addition facts in termsof direction, I believe, gave him the clue. He then discov-ered a direction for grouping in which the use of logic wassystematic and consistent.
I have never been directly involved with the teaching ofa child, and hence have little knowledge of what the ele-mentary thinking processes are that a child can bring tobear on a subject. I learned addition by the old method,but as best I can remember we did not use finger countingand were still able to master long addition after the firstfew grades. I have some question about whether imposinga need for exercising a deductive process in the masteringof basic addition is truly an advantage. I have no reasonfor believing this other than my intuition regarding a child'slearning capacity. Suitable tests may well prove the author'spoint. I think that many people, such as myself, who arenot intimately familiar with a child's learning process, maywell be skeptical until tests on the basis of class-room in-struction are conducted and that results have been shownto be conclusive. My suggestion, therefore, would be toencourage the author to find some means by which thesetests can be conducted, even if the initial tests are of lim-ited scope.
Overall, I find the author's ideas very interesting andsomething that is worth pursuing, for if it were a contribu-tion, it would benefit our children for all generations.
I-
0 .9ki
tD
I.-
4
W
a' 0I II M BE SE VIE
GRADES
FIGURE 12. Expected progress of laming of addition factsby different methods
THE AUTHOR'S CLOSURE
Comments of Professor Elling were made on the firstmanuscript in which, I believe, it was not as clear as in thefinal paper that logic was being employed in the methodcurrently in use (the constant-sum method). The consist-ent-logic method does not introduce logic, it uses logicsystematically and consistently that was introduced by theconstant-sum method. We could assume that if logic helpeda child in the constant-sum method, using it systematicallyand consistently should increase its effectivenrss. There-fore, Professor Elling's question about the need for logiccould rightfully belong to the constant-sum method.
The constant-sum method has been used for many years.It has been found to be an efficient and effective method,and an improvement to the old method to the extent thatit has practically replaced the old method. Some reasonswere discussed frr the adoption of the constant-sum methodfrom an analytical point of view. Other reasons may bediscussed from a practical point of view. Let us considerthat a child is mastering the fact 9 + 4 = 13. After theconstant-sum method was introduced, he could still masterthe fact by the old method; but in case he forgot the fact,he could fall back on the derivation if he knew it. More-over, by the constant-sum method, not only a child knowsthat 9 4 = 13, but also he knows why. Another pointmay be brought out. In adopting the constant-sum method,we did not dump the old method, we use the direction ofthe old method for drill by flash cards. These are the prac-tical reasons for which the constant-sum method was pre-ferred to the old method.
By the old method in this paper was implied a methodwhich did not use logic and derive addition facts. Fingercounting was an example. That the finger counting was anexample was not clear in the first manuscript. To respondto Professor Elling's comment on the mastery of additionfacts, we take the old method he has referred to, andassume that the child's progress in mastering the additionfacts by this method is indicated by the curve labelled "old
method- in Figure 12. The ordinate of Figure 12 repre-sents the percent average adult $peed per fact with whicha child can answer the addition problems [cf., "LearningAbout Numbers' previously cited] or recall the additionfacts (cf., Dr. McNutt). Expected progress of a child bythe constant-sum method and the consistent-logic methodarc also shown in Figure 12. The practical reasons forwhich the constant-sum method was preferred to the oldmethod has been listed previously. In terms of recall - speed,the constant-sum method may not have an appreciable im-provement. or perhaps any, but the consistent-logic methodshould result in a marked improvement, as shown by theuppermost curve of Figure 12.
Coming to the need for testing the method on the basisof class-room instruction I agree with Professor Elling.Teachers, principals, supervisors and superintendents forelementary schools also have strongly suggested the needfor testing the method on the basis of class-room instruc-tion under controlled conditions. I believe tilt, methodshould be tested for two reasons.1. To determine a quantitative measure of improvement
of the consistent-logic method, or in other words, to de-termine the progressive values of AB of Figure 12.To demonstrate to practical educators, in a short timeand without much analytical effort, that the consistent-logic method is an improvement to the constant-summethod, as concluded.
ACKNOWLEDGMENTI gratefully acknowledge the cooperation given to me during
the work by many parents. children, and elementary schools ofPrince Georges County, Maryland. without whose help this workwould not be possible. I am indebted to Professors Schindler,Berg and Elling. and Dr. McNutt for their discussions. I amalso indebted to Dr. Robert J. Shockley. Assistant Superintendentfor elementary schools of Prince Georges Couniy, and Dr. RobertB. Ashlock.t Associate Professor of Education. University ofMaryland. for their reading of the manuscript and suggestionswhich resulted in many improvements in the paper. The paperwas edited by Robert E. Clark. MI viiior of the Naval ResearchLaboratory. I appreciate his help and the constructive criticismwith which he discussed the paper during his work. Appreciationis also expressed to my wife.* with whom I discussed the thoLghtsfor the first examination of their validity and who was the firstreader of the manuscript. Lastly. almost the entire credit is dueto my boy Fuad (a first grader in 1967-68), who gave me logic,the idea, the need and the inspiration for such a development.
ABOUT THE AUTHORThe author holds a Master of Science degree in Mechanical
Engineering from Michigan State University. He did furthergraduate work at MIT. Columbia and Johns Hopkins, and isnow working for the doctoral degree at the University of Mary-land with an Edison Memorial Scholarship from the Naval Re-search Laboratory under -the work-study plan. He previouslyworked for the General Electric Company, was an AssociateProfessor of Mathematics at the Hudson Valley Community Col-ege, and an Instructor of Mathematics at the Chicago campussf the Northwestern University. He is a member of the National
Council of the Teachers of Mathematics. (The present work wasundertaken and conducted by him outside his duties at theNaval Research Laboratory.)
t a member of the National Council of Teachers c Mathematics11 former elementary school teacher
![Merhi [15] Sleiman Aziz Youssef M. - Harvard John A ...people.seas.harvard.edu/~aziz/familytree/All-in-one-tree-Merhi-Aziz... · Charbel Aziz 1968 - Born: 1968 in Lebanon Email: 2004](https://static.documents.pub/doc/80x56/5ec4b1630cd59f6a275d4c94/merhi-15-sleiman-aziz-youssef-m-harvard-john-a-azizfamilytreeall-in-one-tree-merhi-aziz.jpg)
