Mathematics, multiple embodiment, and elementary teachers

Mathematics, multiple embodiment, and elementary teachersAuthor(s): Robert E. ReysSource: The Arithmetic Teacher, Vol. 19, No. 6 (OCTOBER 1972), pp. 489-493Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41188080 .

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emplary of the spirit of this new focus. This month's article, "Mathematics, Multiple Embodiment, and Elementary Teachers," is another attempt to put some of these ideas into practice.

Your reactions to, comments on, suggestions for, and questions about this renewed focus on the elementary teacher are invited - also, articles!

Robert E. Reys

Mathematics, multiple embodiment, and elementary teachers

Introduction

Many recent elementary school mathematics programs place heavy emphasis on physical materials and activity-oriented learning. Teachers are encouraged to use concrete materials to help children develop fundamental mathematical ideas. It is generally accepted that pupils must be involved in the learning process - often phys- ically, and always mentally.

One of the basic assumptions underly- ing the use of physical materials is that pupils learn best through active involve- ment with concrete experiences. One cannot say, at least with any degree of cer- tainty, that using physical materials with active learning experiences will be effective in helping all children master all types of mathematical objectives. Too much depends on the complex interaction among and between pupils, teachers, and materials.

Multiple embodiment and related research

Nevertheless, it is important, when physical materials are used in instruction, that they provide the proper embodiment of the mathematical concept being developed. That is, the material should provide a concrete representation, or embodiment, of a mathematical principle. When different, yet appropriate, concrete materials are used to develop the same mathematical idea, a "multiple embodiment" is provided. This approach to instruction demands that teachers use a variety of perceptually different materials in developing a mathematical concept.

Consider a multiple-embodiment approach to teaching the concept of area in a third- or fourth-grade class. Some direc- tions for possible activities follow.

1 . Draw the outline of a leaf on a sheet of paper. See how many beans or peas are needed to cover the region. Have pupils compare their results to see who has the biggest leaf.

2. Place your hand on a piece of heavy cardboard or construction paper and trace. Cut along the trace lines and weigh the cutout to determine who has the largest hand. Would the largest also be the biggest?

3. Using your mathematics books, determine the number of books needed to cover the top of a desk.

4. Cut out several congruent geometric shapes (such as triangles, both scalene and isosceles, and concave and convex quadri- laterals) from cardboard or construction paper. Tessellate congruent pieces and see how many of each shape are needed to cover a sheet of paper. Can you always cover the paper with these shapes?

5. Use a geoboard to construct figures and then calculate the number of units included in each region.

6. Provide some drawings of rectangles, triangles, and so on. Have the pupils use a transparent grid to determine the number of units included in each figure.

Although these activities are similar, they look different. In fact, they are perceptually different, especially to third- and fourth-grade children. Nevertheless, these

October 1972 489



perceptually different activities are struc- turally the same. That is, each individual activity develops the concept of area as a covering scheme, and therefore together they provide a multiple-embodiment approach to teaching the concept of area. Proponents of this instructional strategy claim that it enhances the likelihood of the formation of meaningful concepts.

Multiple embodiment requires that mathematical concepts be developed in perceptually different situations; however, it does not require a mathematics laboratory or the actual manipulation of materials. The amount of "hands-on" experience required depends on several factors, including the developmental level of the learner. Al- though actual work with various embodiments is necessary for young children, the mere exposure of older pupils to different embodiments (perhaps by teacher demon- stration) may be sufficient to provide the desired concept formation. Few mathematics teachers challenge the rationale for concrete embodiments, but many feel that in most situations one good embodiment is sufficient.

What does research say about multiple embodiment and concept formation, or multiple embodiment versus a single-embodiment approach to concept formation? In short, the research is inconclusive. Sev- eral recent reviews of research (Beougher 1967; Kieren 1969; and Suydam and Weaver 1970) have identified investiga- tions related to multiple embodiment. A critical review of these published studies reveals a wide range in the quality of research and therefore raises doubts about the credibility of certain findings. Never- theless, it is clear that the research does not consistently support or rejute a multiple- embodiment approach to teaching mathematics. In fact, the one common thread among these studies is that learning mathematics depends more on the teacher than on the embodiment used.

In addition to research specifically related to mathematics education, several more general findings also support a mul-

490 The Arithmetic Teacher

tiple-embodiment approach to instruction: 1. Pupils learn differently. Increasing

the number of embodiments increases the likelihood of correctly matching an instructional approach with a child's learning preference.

2. Pupils enjoy new and different activities. A change in the physical setting is usually accompanied by renewed interest and enthusiasm.

3. Pupils often overgeneralize and therefore incorrectly transfer ideas from one situation to another. Instruction based on a variety of experiences encourages the abstraction of essential ideas that are common to several activities. Consequently, the learner is less likely to incorrectly jump to a conclusion in a situation where only one of several necessary ideas is presented. Although none of these statements deal exclusively with mathematics learning, they uphold the spirit of a multiple-embodiment approach to teaching mathematics.

Multiple embodiment in the classroom

The rationale for multiple embodiment is generally accepted; yet it has not re- ceived widespread use by teachers. The biggest stumbling block is not the lack of research evidence, but rather the lack of ideas and ways to provide multiple embodiment for concepts. Dienes (1960) has ably presented the case for multiple embodiment and illustrated it with many ex- cellent examples. Most teacher's editions of elementary textbooks suggest several strategies for developing each mathematical concept that is to be studied. Often such strategies suggest ideas for multiple embodiment, but it is the teacher who must gather the necessary physical materials and then prepare the appropriate lessons or activities. This is, of course, a very demand- ing task and is without doubt the strongest deterrent to a more widespread use of the multiple-embodiment approach to teaching mathematics.

Let's take another topic - a set of pri-



тагу lessons designed to develop the concept of addition through various number families - and see how it might be presented using a multiple-embodiment approach. In the following activities the "seven family" is explored in four embodiments:

1. Balance beam. (See figure 1.) The teacher hangs one weight on hook seven. The child is given two weights and asked to hang them on the other side so that the beam will balance. The child's responses may be recorded, and then the task is repeated to see if weights can be hung on other hooks to balance the beam.

2. Bowling game. (See figure 2.) The teacher places seven pins in a given arrangement. The child is given a ball and allowed to roll the ball to knock down the pins. After each roll, the number of pins standing and number of pins knocked down are recorded. Then the teacher places the pins back in the original arrangement, and the task is repeated.

3. Minnebars, Cuisenaire rods, or strips of construction paper cut in different lengths. (See figure 3.) The teacher places a seven-unit rod before the child and asks the child to make a train of the same length using only two rods. The child's response may be recorded, and then the child is asked to continue constructing two-car trains of the same length until all possible trains have been made.

4. Two transparent plastic measuring cups. (See figure 4.) One cup is filled with seven units of sand. The child is asked to pour some sand from one cup to another. The amount in each cup is then recorded. The task is repeated until all integral com- binations have been made. (It may be necessary for the teacher to decide on a unit and calibrate the cups himself. The sand could also be weighed. Another possibility is to use sugar cubes instead of sand.)

Besides these embodiments, counters and the number line could also be used. The different embodiments could help the child develop the seven family in a context that is not associated with any one embodiment, but incorporates the essential ideas that are common to all of them.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

October 1972 491



This set of embodiments was specifically designed for developing addition within the seven family. However, concepts cannot be developed or learned in isolation. Conse- quently, these activities also provide developmental experiences in other mathematical topics such as number, subtraction, addition within other families, open sen- tences, missing addends, and measurement.

Practical questions and classroom implications Research evidence related to multiple

embodiment of mathematical concepts is not conclusive. More research needs to be done. Many issues related to embodiments and levels of concept formation need to be considered. Here are some specific questions, related to quality, number, sequence, and time allocation for embodiments, that need answering.

For a given mathematical concept: 1. Which embodiments are most effec-

tive in fostering concept formation? Are some easier, or more difficult, for pupils to understand? Are some appropriate for slow learners, but not necessary for fast learners?

2. To how many embodiments should each pupil be exposed? Is one embodiment sufficient? Two? What is the optimal number of embodiments? Do too many embodiments confuse a slow learner?

3. Does a given set of embodiments have a fixed order that maximizes learning? (That is, does it make any difference if we use the balance beam before or after the bowling game?)

4. Does each embodiment require the same amount of time to develop? How much time should be devoted to each embodiment? Should different embodiments be developed together (same day or week), or should they be spread over a period of time (several weeks or months) using a spiral approach?

These issues are complex. It seems un- likely that research will provide simple answers such as "three embodiments are always needed" to develop a mathematical

492 The Arithmetic Teacher

concept. It seems more likely that answers to the previous questions will need to be qualified in terms of teachers, mathematical content, and embodiments used, as well as by the pupil's ability, background, and achievement level. Nevertheless, answers to questions such as these have real sig- nificance for the elementary teacher.

What are some of the classroom implications? Since current research is inconclusive, the only clear implication is that more experimentation is needed. If the idea of multiple embodiment "turns you on," then accept the challenge by getting involved in some classroom trials. Here are some suggestions to get you started.

1. Identify the next big mathematical concept to be taught.

2. Describe at least three or four appropriate, yet perceptually different, embodiments for the concept.

3. Develop a teaching strategy for each embodiment.

4. Share your ideas with colleagues. They may have some suggestions for other embodiments, or may be willing to use these embodiments (perhaps presenting them in a different sequence) in their classes.

5. Determine an order for these embodiments and present them to your pupils.

6. Record problems associated with each embodiment along with suggestions for im- provement.

7. Revise embodiments and presenta- tions.

8. See if you have answers to some of the practical questions that were stated earlier.

9. Identify another mathematical topic and repeat steps 2-8.

10. Share your results with colleagues and the Arithmetic Teacher.

It is not enough to just think about different ways to embody a mathematical concept; you need to try them with your pupils. Such experimentation by elementary teachers is desperately needed. See what effect



multiple embodiment has on your pupils as they learn mathematics.

References

Beougher, Elton E. The Review of the Literature and Research Related to the Use of Manipu- lative Aids in the Teaching of Mathematics. Pontiac, Mich.: Special publication of the Di- vision of Instruction, Oakland Schools, 1967.

Dienes, Zoltan P. Building Up Mathematics. London: Hutchinson Education, 1960.

Kieren, Thomas E. "Review of Research on Ac- tivity Learning." Review of Educational Re- search 39 (October 1969):509-22.

Reys, Robert E. "Considerations for Teachers Using Manipulative Materials." Arithmetic Teacher 18 (December 1971) :551-58.

Suydam, Marilyn N. and J. Fred Weaver. Inter- pretive Study of Research and Development in Elementary School Mathematics. University Park, Pa.: Center for Cooperative Research with Schools, 1970.

Books and materials [Continued from p. 417.]

Revolution in the British Primary Schools. Sir Alec Clegg. Washington, D.C.: National As- sociation of Elementary School Principals, 1971. Pp. 48.

Right Angles: Paper-Folding Geometry. Jo Phil- lips. New York: Thomas Y. Crowell Co., 1972. Pp. 33, $3.75.

Rubber Bands, Baseballs and Doughnuts. Robert Froman. New York: Thomas Y. Crowell Co., 1972. Pp. 33, $3.75.

Science Book List for Children. 3d ed. Edited by Hilary J. Deâson. Washington, D.C.: Amer- ican Association for the Advancement of Sci- ence, 1972. Pp. xiii + 253, $8.95.

Set and Think. Lee Jenkins and Peggy McLean. San Leandro, Calif.: Educational Science Con- sultants, 1972. Pp. 8 + 15 cards, $1.95.

Teaching Elementary School Mathematics. Rob- ert Underhill. Columbus, Ohio: Charles E. Merrill Publishing Co., 1972. Pp. xviii + 490, $10.50.

Teaching Modern Mathematics in the Elemen- tary School. Howard F. Fehr and Jo McKeeby Phillips. Reading, Mass.: Addison-Wesley Pub- lishing Co., 1972. Pp. xviii + 513, $9.50.

Tomorrow's Math: Unsolved Problems for the Amateur. C. Stanley Ogilvy. New York: Ox- ford University Press, 1972. Pp. 198, $7.50.

Understanding the Young Child and His Cur- riculum. Belen Collantes Mills. New York: Macmillan Co., 1972. Pp. xvii + 489, $5.25.

Arithmetic Is Important. 11 min., 16mm, color, 1971. CCM Films, 34 MacQuestern Parkway South, Mount Vernon, N.Y. 10550.

Cosmic Zoom. 8 min., 16mm, color, 1968. Mc- Graw-Hill Book Co., Film Preview Library, Hightstown, N.J. 08520. Purchase, $115.00.

Environmental Math. Series of six filmstrips. Color. BFA Educational Media, 2211 Michi- gan Ave., Santa Monica, Calif. 90404. Pur- chase, $63.00.

First Things: Mathematics. Series of six filmstrips and records or cassettes. 45 frames av., color, 1971. Guidance Associates, 41 Wash- ington Ave., Pleasantville, N.Y. 10570. Pur- chase, $18.00 each for record, $20.00 each for cassette.

Functional Arithmetic in the Elementary School - Kindergarten through Grade 8 (with supplementary notes). 45 frames, color. Bureau of Educational Research, Teaching Aids Labora- tory, 13 Page Hall, The Ohio State University, Columbus, Ohio 43210. Purchase, $3.00.

Material for the Teaching of Arithmetic (with supplementary notes). 45 frames, color. Bu- reau of Educational Research, Teaching Aids Laboratory, 13 Page Hall, The Ohio State University, Columbus 10, Ohio 43210. Pur- chase, $3.50.

Measuring Length. Series of six filmstrips. 35 frames av., color, 1970. McGraw-Hill Films, 330 West 42d St., New York, N.Y. 10036.

Measuring Things. Series of six filmstrips, cap- tioned or with three 12-inch records. 49 frames av., color. Coronet Films & Filmstrips, 65 E. South Water St., Chicago, 111. 60601.

Primary Math. Series of thirty filmloops. 8mm, color. BFA Educational Media, 2211 Michi- gan Ave., Santa Monica, Calif. 90404.

Science Processes, Sets 1-4. Series of 24 filmstrips. 37 frames av., color, 1971. McGraw- Hill Films, 330 West 42d St., New York, N.Y. 10036.

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Mathematics, multiple embodiment, and elementary teachers