Mental imagery in mathematics

Mental imagery in mathematicsAuthor(s): STANLEY M. JENCKS and DONALD M. PECKSource: The Arithmetic Teacher, Vol. 19, No. 8 (DECEMBER 1972), pp. 642-644Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41188129 .

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Mental imagery in mathematics

STANLEY M. JENCKS and DONALD M. PECK

An assistant professor of mathematics and of education at the University of Utah in Salt Lake City, Stanley Jencks devotes most of his professional time to the preparation of elementary teachers. He teaches courses in both mathematics and the teaching of arithmetic.

As assistant professor of education at the University of Utah in Salt Lake City, Donald Peck works extensively in the preparation of elementary teachers in the area of arithmetic methods. He also teaches classes in methods for secondary teachers of mathematics and conducts an experimental course for disadvantaged college students.

J't the elementary school level, teachers need to help each child form a basis for trusting his own thinking. For example, the authors recently asked numerous sixth- grade children for the sum of V2 + Уз. Many of the children, even those from "culturally advantaged" schools, were unable to find a correct answer or even to attack the problem with understanding. Furthermore, many of those producing correct answers were unable to relate the process to anything very meaningful. What is worse, the children had no place to go to construct a sensible answer for themselves. They were completely dependent on their teachers to remind them about common denominators before they could proceed. If a major objective of education is, in fact, "to enable a child to get along without his teacher," then there is still much work to be done.

A realistic framework from which to mount an attack on the problem is set forth in the following paragraphs. Work with fractions is used as the medium for illus- tration, but any process of arithmetic

642 The Arithmetic Teacher

would serve just as well, for all of the proc- esses have their roots in the world of objects and in common sense.

The learning of a new concept usually begins with a child's senses. He forms "pictures" or mental images of things perceived by those senses. These images provide the foundation for insight. When the child re- ceives firsthand experiences that help him build the necessary images, learning becomes deeper and more penetrating. On the other hand, when these initial experiences are omitted, learning becomes rote and superficial. To illustrate this point, try the following exercise with pencil in hand.

Suppose that each of the diagrams shown in figure 1 represents a dozen eggs. Shade in a part of each diagram as indicated. (It is really necessary to do this in order to get the impact of our point. ) Where shad- ing is already indicated, what part of the diagram is filled in?

Do you find yourself adding and sub- tracting fractions by counting parts of dozens? Many adults are astonished to find that these operations can be per-



ДД 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

! I 2. ! 2 3 3 6

1 1 1 ШШШЕш 1 1 1 1111111 1111 шШша 1 Filled in ,| Filled in

1 + 1 1-1 1+1 £- 1 34 32 63 64

Fig. 1

formed with simple objects and common sense. It may even come as a surprise that rules about common denominators and equivalent fractions are not necessary, even though they may be desirable at the appro- priate stage of development.

As the teacher provides initial expe- rience with egg cartons, a child who has never been exposed to formal rules for adding fractions finds himself developing confidence in his own thinking, since he can count his way to every answer. Through active participation, the child is finding out things for himself rather than applying rules given to him by someone else. Furthermore, he now has a way to test statements about addition or subtraction of fractions for himself. Later, when a rule is developed, the child has a way of checking to see if the rule really works.

Fundamental to the idea of mental imagery is the use of something - fre- quently concrete objects - from which the learner can find answers for himself. The teacher's role is to provide this foundation of concrete experiences and then to ask questions that require the learner to or- ganize his thinking in depth.

Even though children may quickly learn to add and subtract fractions from egg cartons with complete confidence, it is also highly important that many other objects be used - for example, yardsticks, clocks, pegboards, window panes, graph paper, and floor tiles. Each time one of these objects is used, addition and subtraction of fractions takes on greater generality and is seen to relate even more closely to common sense.

Thus, the problem of teaching addition of fractions reduces to one of finding suit- able objects or materials to portray the fractions involved. An example like % + % looks formidable to a child until he finds that both of these fractions can be easily associated with a clock - % hour is 10 minutes; % hour is 36 minutes. The total is 46 minutes, which corresponds to 4%o hour or, in simpler terms, 2%0 hour. Hence, % + % = 23/зо.

Sometimes children themselves find ap- propriate physical objects. For example, a boy had a piece of graph paper on his desk when he was given the exercise % + %. He marked off a rectangle 5 squares one way and 7 squares the other way. He

December 1972 643



easily found % of the rectangle because each column was y7 of it.

y is 20 squares

■ÉÉÉZZZ5

7 Similarly, each row was % of the rectangle.

~ is 14 squares

_ 5

7 Hence, 4/7 + % is 20 squares plus 14 squares, which totals 34 squares. There are 35 squares in the rectangle altogether; so % + % = 3Узг,

The graph paper this boy used as a model is particularly valuable because it can be used to do any problem involving addition or subtraction of fractions. There are other such universal models for each of the operations of arithmetic. When these models are coupled with significant questions, they provide mental pictures that lend substance to rules and algorithms.

Certainly it is important that children learn abstractions and that they learn symbolic ways to add fractions, such as the usual common-denominator rule. However, in introducing a new topic, if teachers will concentrate their initial efforts on providing firsthand experiences from which to reason, children will arrive at abstractions more quickly, more easily, and more surely. A mistake is made when symbolic rules pre- cede the mental imagery necessary to give an arithmetic process a common sense foundation.

644 The Arithmetic Teacher

In summary, building mental imagery provides an answer to a serious problem in the teaching of elementary school mathematics today. It provides children with a way to make sense out of ordinary computational procedures, it provides them with a basis for trusting their own reason- ing, and it brings them a giant step closer to being independent. Once a child can get his own common sense answers from physical objects, he is in a position to accept the algorithms suggested by his teacher, textbook, computer, or classmates, or perhaps to invent his own computational procedures. But without such a basis for making judgments of his own, his learning can be little more than rote and superficial.

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Mental imagery in mathematics