Children’s Abilities With the Four BasicArithmetic Operations in Grades K-2

Ron GoodThe Florida State UniversityTallahassee, Florida 32306

A position paper entitled "The Traditional Sequencing of Mathemat-ics for Young Children Should be Changed" appeared in the January1977 issue of School Science and Mathematics, In that paper I proposedthat the four basic operations of addition, subtraction, multiplication,and division should be totally interrelated, in a concrete way, from thevery beginning of a child’s instruction in mathematics. The "content" ofthe four operations would be mostly non-symbolic experiences withmanipulative materials, with references to multi-digit numbers being de-emphasized. Thus, the increasing difficulty of such a mathematics pro-gram for young children would be with the number system itself ratherthan the operations. This is consistent with what teachers recognize asthe most difficult area for children to conceptualize, namely place value.The proposal was also related to Jean Piagefs work in which he stressesthe importance of experience with the physical environment and howknowledge develops as an interrelated whole.

During the Fall of 1975, a pilot research project was begun to deter-mine the feasibility of interrelating the four basic operations with youngchildren. The design, results, and conclusions of that study are reportedhere along with implications for curriculum and instruction.

RELATED LITERATURE

There is apparently no research directly related to the effectiveness ofinterrelating the four basic arithmetic operations from the beginning of achild’s introduction to mathematics. A study by Coxford (1966) suggest-ed that there is greater understanding by young children when the rela-tionship between addition and subtraction is emphasized, but no attemptwas made to include multiplication and division as well. This would, ofcourse, require a relatively long-term study�a phenomenon seldomfound in educational research where a treatment is required for an ex-perimental sample and the effects are compared to a control group.A conclusion by Anderson (1949) also supports the general proposal of

interrelating the, four basic arithmetic operations, in a concrete way,from the beginning of a child’s introduction to mathematics. He con-cluded that understanding and the ability to transfer are facilitated wheninterrelationships and generalizations are stressed in mathematics in-struction.There seem to be no other studies which have attempted to look at the

importance of stressing interrelationships among the basic operations for

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young children. The 1975 NCTM yearbook, "Mathematics Learning inEarly Childhood" seems to support the conclusion that little, if any, rele-vant research exists in the area.The other basic assumption, that young children should have broad,

concrete experiences with their physical environment as the basis formathematics understanding, has been researched more widely. Althoughthe issue is far from settled, it appears that certain advantages accrue tochildren who are taught mathematics with manipulative aids involved. Agreater variety of concrete objects was found by Zweng (1964) to helpchildren better conceptualize solutions to problems. It was also found byEkman (1967) that the use of manipulative materials prior to the use ofsymbols increases both understanding and the ability to transfer. A morerecent study by Steffe and Johnson (1971) also confirmed that mathe-matical problem-solving is facilitated by the use of manipulative mate-rials.

All of the conclusions from these studies support the contention byPiaget (1953) that children develop mathematical concepts from theirown experience with their physical environment.The paucity of research directed at the question of whether or not

mathematics instruction should be directed toward interrelating the fourbasic arithmetic operations for young children, combined with Piaget’stheory of cognitive development, which suggests that these operationsare totally interrelated and have the same origin of development, suggestmuch more work must be done in this area.

THE STUDY

During the Fall and Winter of the 1975-76 academic year, a pilot studywas conducted at Florida State University’s Developmental ResearchSchool (DRS) in grades K, 1, and 2 to determine the ability of young chil-dren to understand the operations of addition, subtraction, multiplica-tion, and division from the point of view of sets of concrete objects. For-ty-five children from grades K-2 were individually interviewed usingseven tasks.Task 1: One-To-One Correspondence. Eight red and nine white poker

chips were used to determine the child’s ability to establish one-to-onecorrespondence between the two sets.Task Two: Number Conservation. Red and white poker chips were

again used to determine each child’s ability to conserve the concept ofnumber when a spatial rearrangement of the chips occurred.

Task Three: Counting. The poker chips were used to test each child’sability to count up to ten.

Task Four: Adding. For this task and the remaining three tasks, boththe poker chips and a specially constructed number board (See Figure 1)

Children *s Abilities v^ith Basic Math Operations 95

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FIGURE 1. Number board used in the interviews,depicting sets of poker chips.

depicting sets of chips from one to nine were used to assess each child’sunderstanding of the process of combining two sets of objects. The fol-lowing questions are examples of those asked of each child:

1. Do you see a group on the board just like my group (of poker chips)? How can youtell?

2. Do you see a group on the board just like your group of chips (child has eight chips)?How do you know?

3. Which group on the board do you need to add to yours to make mine? How do youknow?

4. If you add this group of chips (three) to this group (four) which group on the boardwill be just the same? How can you tell?

Task Five: Subtracting.

1. How many chips do I have to remove to make this group (of chips) just like thisgroup (on board)?

2. How many chips are in this group (six)? If I take these two away how many will beleft?

3. If I start out with this group of chips (seven) and I end up with this group (four onboard), how many chips did I lose?

Task Six: Multiplying.

1. If I give you two groups like this (set of 3 chips), how many will you have?2. If I give you three groups like this (3 chips), how many will you have?3. If I give you four groups like (2 chips), how many will you have?

Task Seven: Dividing

1. Can you make two equal groups out of this group (six)?2. Can you make two equal groups out of this group (seven)?3. Can you make three equal groups out of this group (six)?4. Can you make three equal groups out of this group (eight)?

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For each task, questions were asked using both the poker chips and thenumber board, depicting sets of poker chips, until the interviewer wassatisfied that each child had indicated his/her best ability to respond.Each interview lasted about 20-30 minutes.

THE RESULTS

All 45 students were able to count to 10 and establish one-to-one corre-spondence using the poker chips. Table 1 shows the data for Task Two;Number Conservation.

TABLE lCONSERVATION OF NUMBER TASK

Conserver Transitional NonconserverGrade K 64 5Level 1 84 3

2______________H____________1______________0______28 9 8

By second grade, all but one of the students were capable of conservingthe one-to-one correspondence between the two sets of objects, eventhough spatial rearrangements occurred.

Table 2 shows the relative numbers of students in grades K, 1, and 2who were successful on Tasks 4-7 (adding, subtracting, multiplying, anddividing).

TABLE 2STUDENTS’ SUCCESS ON TASKS 4-7 BY GRADE LEVEL

Conserver Transitional NonconserverK 1 2 K 1 2 K 1 2

Task 4(Adding) 6814 4 4 1 5 2 0Task 5(Subtracting) 6814 3 4 1 5 2 0Task 6(Multiplying) 1 6 14 1 2 1 0 1 0Task?(Dividing) 2 4 14 0 2 1 0 1 0

At first glance, it would appear that little relationship exists between theability to conserve the concept of number and the ability to add or sub-tract sets of objects. However, most of the children who were categorizedas transitional or nonconserver also tended to rely exclusively on count-ing in responding to questions in Tasks 4 and 5. They seemed to be un-able to see a pattern in the sets of outlines of poker chips on the numberboard even though the outlines were circled with pieces of string andregularly ordered from one to nine. The apparent failure to relate ordinaland cardinal properties of the sets caused many of these children to countor match perceptually the sets of poker chips with the set outlines on the

number board. Although a correct answer was eventually attained, eachquestion was treated individually, and counting was again required.

97 Children’s Abilities with Basic Math Operations

Table 2 shows that Tasks 6 and 7 were more difficult for the childrenin both kindergarten and first grade. Only 2 of 15 children in kindergar-ten were successful on both tasks and 7 of 15 first grade children succeed-ed. All of the second grade children were capable of responding correctlyto Tasks 4-7, and so a nearly perfect correlation existed between numberconservation ability and performing set operations with objects.

INTERPRETATIONS, CONCLUSIONS, AND RECOMMENDATIONS

It appears that counting can be used as a successful procedure in deal-ing with simple addition and subtraction problems, when concrete ob-jects are present. Children who are unable to conserve number seem torely heavily on counting as a means to cope with such problems, and theygive little indication of relating ordinal to cardinal properties of number.Each addition or subtraction problem is done independently of others,regardless of their similarity. The ability to conserve number seems to

make a sharp reduction in the kind of "mindless" counting employedearlier for addition and subtraction problems.Even for simple multiplication and division problems using concrete

objects, children were generally unable to use simple counting as a meansto successfully solve the problems. The sharp drop in success on Tasks 6and 7 for children in grades K and I can be explained by their reliance onperception and/or counting as means for solving number-related prob-lems. By second grade, where nearly all of the children were conserversof number (14 of 15), the processes of multiplying and dividing could bedealt with more successfully.

Making simple classes and series, conserving number, and using rever-sibility all begin to appear at about 6 to 7 years of age and it is not untilthis point that children can get beyond counting as a means to solve"operations on numbers" problems. By second grade, it appears thatmost children have a concrete, intuitive concept of the operations ofaddition, subtraction, multiplication, and division. If multiplication anddivision are dealt with in the context of manipulative materials, it ap-pears that children who are able to conserve number can understandthese operations much as they conceptualize addition and subtraction.

If the previous results and interpretations are accurate, then it wouldseem reasonable to question certain common practices in helping youngchildren learn mathematics. First, for children who are developmentallyunable to relate classification and seriation, conserve number, and usereversibility, it should not be expected that they use logical operations onnumbers, except at the simplest, concrete level. Much of the kindergartenand first grade mathematics curriculum should consist of concrete activi-ties related to classification, seriation, conservation, and reversibility.

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Second, when logical operations are introduced to children who are de-velopmentally ready to understand their meaning and usefulness, all 4operations should be interrelated. Concrete activities as well as paper andpencil activities, games, etc. should stress the interrelatedness of addi-tion, subtraction, multiplication and division. Third, since place value isone of the most difficult concepts for children to understand, multi-digitnumbers should be avoided until children show that they understand the4 basic operations. This means that the order of difficulty of the mathe-matics curriculum would be with the number system itself rather thanwith the operations of addition, subtraction, multiplication, and divi-sion. Just how fast the concept of place value can progress will have to becarefully researched, but it is clear that most children in first and secondgrade should not be working with multi-digit numbers until much later.Even by fourth or fifth grade, few children give indications that theyhave finally conceptualized place value in any operational sense.

In second grade, rather than asking such questions as 25 - 14 =? or56 - 28 = ? the children should be searching for all possible concretemeanings of the concept of, say, "sixness." What do the operations ofaddition, subtraction, multiplication, and division have to say about"sixness?" How many different groups can be made using six pennies?How many different patterns can be made using six pegs in a pegboard?What will balance six pennies? How many different shapes can be madeusing six triangles? How big is a piece of paper if you fold it six times? Ifdice are rolled six times, what are the results? The possibilities are endlessand such an approach allows for the wide range of developmental abili-ties found in any classroom. Using such an approach in helping childrenlearn mathematics should result in a greater, concrete understanding ofour number system and in more positive attitudes toward a subject whichneeds improvements in that area.

REFERENCES

ANDERSON, G. "Quantitative Thinking as Developed Under Connectionist and FieldTheories of Learning" in Learning Theory in School Situations. University of MinnesotaStudies in Education, No. 2, pp. 40-73. Minneapolis: University of Minnesota Press,1949.

COXFORD, A. "The Effects of Two Instructional Approaches on the Learning of Additionand Subtraction Concepts in Grade One." (University of Michigan, 1965) DissertationAbstracts 26 (May 1966): 6543-44.

EKMAN, L. "A Comparison of the Effectiveness of Different Approaches to the Teachingof Addition and Subtraction Algorithms in the Third Grade." 2 vols. (University of Min-nesota, 1966) Dissertation Abstracts 27A (February 1967): 2275-76.

PIAGET, J. "How Children Form Mathematical Concepts." Scientific American 189(1953): 74-79.

STEFFE, L. and D. JOHNSON. "Problem-Solving Performances of First-Grade Children."Journal for Research in Mathematics Education 2 (January 1971): 50-64.

ZWENG, M. "Division Problems and the Concept of Rate." Arithmetic Teacher 11 (Decem-ber 1964): 547-56.