what do children see in mathematics textbook pictures?

What Do Children See in Mathematics Textbook Pictures?Author(s): Patricia F. CampbellSource: The Arithmetic Teacher, Vol. 28, No. 5 (January 1981), pp. 12-16Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41162219 .

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What Do Children See in Mathematics

Textbook Pictures? By Patricia F. Campbell

In communicating with words it is obvious that if the receiver does not know the language he will not be able to decode the message. It is obvious that reading is a skill which must be learned. It is not obvious - indeed it is sometimes com- pletely unnoticed - that a parallel condition ex- ists in the case of pictures. Children must learn how to "read" pictures.

-(Levie and Dickie 1973, p. 865)

Pictures are an important tool used in the teaching of primary mathematics. Textbooks and other teaching materials use pictures as one way of communicating mathematical concepts to young children. However, pictures must be interpreted, and children interpret pictures differently than adults. Generally, children obtain less information from pictures than adults. Psy- chological research indicates that children initially key in on single items depicted in a picture without much concern as to whether those items are central to the theme of the picture (Travers 1969). At the next level, children's interpretations of pictures tend to be a listing of many of the objects or characters shown in the pictures. Only gradually do young children develop the ability to describe a picture in terms of motion and to see relationships between the characters (Amen 1941, Carpenter 1964, Miller 1938, Travers and Alvarado 1970). Thus the ability to interpret pictures is not natu- ral; it must be learned.

One type of picture commonly used in primary mathematics textbooks is the dynamic picture or sequence of pictures. These pictures portray motion by either postural cues (i.e., depicting arms or legs in the state of nonequi-

Patricia Campbell is a visiting assistant professor of mathematics and education at Purdue Univer- sity. She teaches preservice and inservice mathematics education courses and supervises student teachers.

librium) or conventional cues (i.e., depicting clouds of dust or wavy lines about the figures). These pictures are meant to serve as a reference either to assist the students in solving a related problem or to define a setting in which the problem may be interpreted. How- ever, it is necessary for the children to relate the characters depicted in the pictures and to see the action portrayed

before they can associate the pictures with the addition or subtraction of whole numbers. Yet these are precisely the picture interpretation skills which psychological studies indicate that first-grade children may be developing, but have not yet obtained. Motion is one of the more difficult representations for young children to understand (Friedman and Stevenson 1975, Schnall 1968), but perception of motion is essential to understanding the mathematical relationships in dynamic primary textbook pictures.

The purpose of this article is to identify some common stages of interpretations of mathematical pictures that are exhibited by first-grade children and to offer some suggestions for teachers to use when working with students in these stages. The stages that are identified were observed in interviews with 192 first-grade students ran- domly selected from five elementary schools, three in Lafayette, Indiana, and two in Tallahassee, Florida. (See table 1) The interviews were conducted during late February and March in two consecutive years. Table 1 Distribution of Students by Site, Sex and Socio-economic Status

Sex

Site Males Females SES Tallahassee

School 1 28 27 Lower through middle class

School 2 22 19 Middle class

Lafayette School 1 13 20 Middle class School 2 19 13 Middle class School 3 20 11 Middle class

The Interview The interview consisted of questions about five addition and five subtraction illustrations. The interviewer

12 Arithmetic Teacher

Fig. 1



was allowed to repeat questions, but not to rephrase them. Initially, an at- tempt was made to put the child at ease by conversing with the child about school, pets, or brothers and sisters. Then the interviewer told the child that they would be looking at some pictures and that the interviewer would be ask- ing some questions about the pictures. The interviewer then placed either a single picture or a sequence of three pictures (fig. 1) before the child.

The children were told that the pictures^) told a story. At this point, one of two procedures was followed. Some of the children were asked to tell a story to go with the picture(s). The other children had a pile of small toys corresponding to the characters in the picture(s) placed before them, below the pictures. They were asked to use the toys to show the story being told by the picture(s). The same procedure was repeated for each of the ten pictures or sequences in the interview.

Every child who was asked to show the story with toys also, at the same time, told a story to go with the place- ment of the toys. This verbal story-tell- ing with the toys occurred without any prompting from the interviewer. The stories told by the children with the toys available did not differ from the stories told by the children who saw the pictures without accompanying toys.

Each interview was recorded. The children's stories were evaluated with respect to the following criteria:

(a) Did the story indicate perception of two disjoint sets?

(b) Did the story indicate perception of a total set formed by the relationship of the two disjoint sets?

(c) Did the response indicate perception of motion or of an operation between the disjoint sets?

Categories of Interpretation of Mathematics Pictures The stories told by the children may be described as being in one of four stages or categories. At the first stage, the stories were general in nature, elaborating on one character or item in the picture^) without concern as to the theme of the illustration or the mathematical concept being portrayed. At the next level, the stories revealed perception of disjoint sets, but neither perception of a mathematical relationship between the sets nor perception of motion was indicated. Stories in the third category described disjoint sets and indicated recognition of motion, but did not relate the sets. The fourth level of stories described a total set formed by the relationship of two disjoint sets with an action either joining or separating the sets.

Stage one: General story Some of the children told general stories which indicated recognition of a character or item in the illustration, but their stories were not pertinent to the mathematical theme depicted in the picture or sequence. For example, when asked to tell a story for the pic-

ture in figure 2, a child responded, "Some guy's trapped on an island and he wants some food, so he's trying to catch some crabs to eat. He can't get off the island 'cause there's some sharks swimming around the island." Or, shown the illustration in figure 3, another child's story was, "Worms can walk. Worms can swim. Worms can go under dirt. You can pick up worms."

Children who told general stories seemed to focus on the character depicted in the illustration and indicated no perception of the mathematical concept being portrayed. In fact, their stories did not reflect the setting or theme of the pictures. Rather, upon recogniz- ing a character in the illustration, they made up a story about that character without reference to the picture. The length and complexity of their stories seemed bounded only by their imagi- nation and their knowledge of facts about the type of character being portrayed.

Stage two: Set recognition The stories at the second level revealed perception of groups of characters in the setting portrayed in the illustration, but no recognition of either motion or a mathematical relationship between the sets. For example, upon seeing the sequence in figure 4, a child responded, "Two ducks. Two ducks. Four ducks. Six ducks." Or for the picture in figure 5, one child responded, "There's three over there and four others over there and they all want to play together. But they can't get together 'cause - I don't know why."

January 1981 13

Fig. 2 Fig. 3



For these children, the desire to count or group the characters was dominant. Some of the stories were merely listings of how many characters were portrayed in each set, while other stories also had further elaboration related to a theme. But the emphasis was on enumerating the members of the sets. Because these children did not perceive any motion between the depicted sets, they did not relate the sets. Hence, no mathematical relationship was perceived by these children.

Stage three: Set and motion recognition At the third level, the stories indicated perception of disjoint sets of characters with one group portraying motion, but no recognition of a relationship between the sets. For example, for the picture in figure 6, one child's story was, "Two seals were coming along. Two seals were playing. These two wanted to play with them, but they just keep coming and they don't get to play." For the picture in figure 7, another child responded, "Once there was four birds and two went away."

The listing of characters by groups is still dominating at this stage, but recognition of motion is evident. How- ever, the children do not interpret motion as indicating a relationship between the sets of characters. Rather, motion is an additional feature to be noted when describing a group.

Stage four: Perception of a mathematical relationship

Only at the fourth level do students perceive a mathematical relationship between the groups of characters. Mo- tion is recognized as causing the sepa- ration of a total set into disjoint subsets or the joining of two groups of characters into a total set. For example, shown the picture in figure 6, a child responded, "There were these two seals playing games and then two more came. They played with them." Or, for the sequence in figure 8, "One day there was, hmmm, five bears. And three of 'em had to go home and so two of them were left. And so - they - so they played all night."

Each child's stories during the ten-

14 Arithmetic Teacher

Fig. 4



item interview tended to fall into one or two categories. Those children who told general stories (stage one) tended to do so on each item. Other children also consistently told similar stories at either stage two, three or four. How- ever, there were a number of children who seemed to be in a state of transi- tion between stages two and three or between stages three and four. On some items they would perceive motion, on others they would not (stage two or three). Other children would see a mathematical relationship between the sets and identify a total set with some illustrations, but on other items they would not perceive a mathematical relationship (stage three or four).

Implications for teaching Pictures are an important tool used in the teaching of primary mathematics. They are used not only as a communication medium in primary textbooks, but also as a means of aiding young children in making the transfer from concrete manipulative settings to ab- stract, symbolic representations. How- ever, it is first necessary for each child to correctly interpret the pictures. Ask- ing the child to tell the story told by the picture is a simple but effective means of determining young children's perception of the depicted mathematical relationship in textbook pictures. It has been suggested (Poage and Poage 1977) that if a teacher finds that more than 20 percent of the class is misinter- preting a picture, then that picture should be replaced with an alternative form of instruction.

Little is known concerning which characteristics of mathematical pictures influence children's interpretations and hence which forms of pictorial representations best convey the intended concepts to young children. Two recent research studies with first-grade children, however, have identified some artistic variables which do effect children's interpretations (Campbell 1978, 1979).

Sequences of three pictures (as in figure 4 or in figure 8) prompt stories at a higher level than do single pictures. In fact, viewing and interpreting sequences of three pictures has a learning effect which aids a student in later in-

Januaryl981 15

Fig. 5

Fig. 6

Fig. 7



terpreting single pictures. If a child is having difficulty perceiving or interpreting the mathematical relationship in a single picture, use a related sequence or ask the child to imagine what may have occurred before and af- ter the depicted event.

The way motion is portrayed also significantly effects picture interpretation. Postural cues (as in figure 5) are more readily interpreted, but first- grade children may "see motion" via

16

conventional artistic cues (as in figure 3). It is suggested that teachers discuss with their students what the wavy lines or clouds of dust mean before assum- ing that the children perceive motion in the illustration. Another procedure may be for students to enact the illustration with objects or with each other.

The type of figure used has no sig- nificant effect on children's per- ceptions. Students seem to interpret pictures of personified animals or ob-

jects with an ease equivalent to their interpretation of realistic pictures of children. Use of both types of illustrations within primary mathematics textbooks may provide variety without ad- versely affecting understanding.

Conclusion Pictures are an important form of rep- resentation which young children ex- perience as they learn mathematical concepts. But young children have dif- fering interpretations of pictures and may not perceive the mathematical relationships which are depicted. The transfer from concrete objects to pictures and from pictures to numerals must be taught; it cannot be assumed. Illustrations may assist the children in understanding the concepts of addition and subtraction, but only if the children understand the pictures.

References Amen, E. W. Individual Differences in Apper-

ceptive Reaction: A Study of the Response of Preschool Children to Pictures." Genetic Psy- chology Monographs 23 (1941):3 19-85.

Campbell, P. F. (a) "Textbook Pictures and First-Grade Children's Perception of Mathe- matical Relationships." Journal for Research in Mathematics Education 9 (November 1978):368-73.

(b) "Artistic Motion Cues, Number of Pictures, and First-Grade Children's Inter- pretation of Mathematics Textbook Pictures." Journal for Research in Mathematics Education 10 (March 1979): 148-53.

Carpenter, H. M. "Study Skills: Developing Pic- ture Reading Skills." The Instructor 74 (1964):37-38.

Friedman, S. L., and M. B. Stevenson. "Devel- opmental Changes in the Understanding of Implied Motion in Two-Dimensional Pic- tures." Child Development 46 (1975):773-78.

Levie, W. H., and K. E. Dickie. "The Analysis and Application of Media." In Second Hand- book of Research on Teaching edited by R. M. W. Travers. Chicago: Rand McNally, 1973.

Miller, W. A. "What Children See in Pictures." Elementary School Journal 39 (1938):280-88.

Poage, M., and E. G. Poage. "Is One Picture Worth One thousand Words?" Arithmetic Teacher 24 (May 1977):408-14.

Schnall, M. "Age Differences in the Integration of Progressively Changing Visual Patterns." Human Development 11 (1968):287-95.

Travers, R. M. W. A Study of the Advantages and Disadvantages of Using Simplified Visual Pre- sentations in Instructional Materials. Final Report. USOE Grant No. OEG- 1-7-070 144- 5235, June, 1969. ERIC Document Reproduc- tion Service No. ED 031 951.

Travers, R. M. W., and V. Alvarado. "The De- sign of Pictures for Teaching Children in the Elementary School." A V Communication Re- view 18 (1970):47-64. W

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what do children see in mathematics textbook pictures?

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