Preservice teachers clarify mathematical percepts through field experiencesAuthor(s): MALCOLM D. SWAN and ORVILLE E. JONESSource: The Arithmetic Teacher, Vol. 16, No. 8 (DECEMBER 1969), pp. 643-645Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41187566 .

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Forum on teacher preparation Francis J. Mueller

Preservice teachers clarify mathematical percepts through field experiences MALCOLM D. SWAN and ORVILLE E. JONES Lorado Taft Field Campus of Northern Illinois University Oregon, Illinois

Malcolm Swan and Ovviile Jones both have considerable interest and experience in the development of teaching techniques utilizing outdoor and community resources.

Among the educational objectives for teaching social science and mathematics is that of developing the pupil's ability to understand the world in which he lives. In- cluded in this ability is the development of percepts - commonly referred to as mental images - relating to distance, weight, height, volume, temperature, etc. If this ob- jective is to be achieved, the teacher should provide pupils with experiences that will help them develop such understandings. Some teachers believe it almost impossible to do so without direct experience, while other teachers assume that pupils develop sufficiently adequate images through solv- ing mathematical problems found in text- books and other written source materials.

It is the writers' opinion that a pupil's understanding of measurement percepts has meaning only to the extent that he has had direct or firsthand experiences in applying intervals of measure in problem-solving situations.

Although writers and curriculum spe-

cialists generally express concern over the matter, they give little attention to ways of improving a child's mental imagery in measurement.1 A few authors maintain that improvement will occur if teachers provide pupils with a greater number and variety of direct or firsthand experiences in estimat- ing, followed by accurate measurement. These authors encourage teachers to use every opportunity to have pupils estimate and measure (and remeasure using instru- ments of successive degrees of accuracy) size, height, weight, volume, temperature, etc., both inside and outside the classroom.2

i Glen O. Blough, Julius Schwartz, and Albert J. Huggett, Elementary School Science and How to Teach It (New York: Dryden Press, 1959), p. 158; and Vene RockCastle, Manual for Measurement Science (Union, New Jersey: OHAUS Scale Corp., 1965).

2 Peter Spencer and Marguerite Brydegaard, Build- ing Mathematical Concepts in the Elementary School (New York: Holt, Rinehart & Winston, 1952), p. 29; and Malcolm D. Swan, "How Long is 100 Feet?" Science and Children, vol. 4, December 1966.

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It is unrealistic, however, to expect teachers to provide such direct or firsthand experiences for pupils if the teachers have not previously had such experiences. If the development of such percepts is a valid objective for social science and mathe- matics curricula, then teachers should de- velop their ability to perceive "guessti- mated" units of measure prior to assuming responsibilities as classroom teachers.

Few teacher education programs provide opportunities for firsthand or direct ex- perience to develop these sensory abilities. At Northern Illinois University, however, through its unique program of outdoor teacher education at Lorado Taft Field Campus, we seek to develop some of these competencies in preservice elementary teachers. Elementary education majors at this institution have two resident experi- ences at the field campus: three days as juniors and five days as seniors. During the junior experience, students go through a series of firsthand outdoor activities de- signed to apply measurement percepts in problem-solving situations.

For an initial experience at the outdoor education center, junior students receive a mimeographed card on which they are to record their estimation of several distances, heights, weights, areas, and temperatures. Results are tabulated and incorporated into frequency polygons and graphs. These serve as the basis for discussions and ex- periences designed to improve the student's perception of related intervals and quanti- ties. Most disturbing is the students' in- ability to apply metric units. Although they are asked to estimate intervals and quanti- ties using both English and metric units, only a few can estimate at all in meters, kilograms, and centigrade. It is as if they are totally ignorant of the relative relation- ships of the units in the two systems. For example, many students indicate that there are more meters than feet in a given dis- tance interval.

Becoming aware of their individual per- formances as revealed by the tabulated re- sults, students are amazed at how unreli-

able are their "guesstimations." Usually, they want to repeat the outdoor experi- ences so as to improve their scores. Im- proving one's score, they are informed, is not the purpose of the activity. Thus, they see the desirability of planning for first- hand experiences to enhance development of these abilities.

Another learning experience for juniors is referred to as the "100-feet exercise." Each student is given a white flag to place along a string line at the spot he visually estimates to be 100 linear feet from some arbitrarily established point on the line. Each student then checks his estimate by stepping off the distance from the starting point (using the length of his shoe, stride, or Boy Scout pace, etc.) and marks the spot with a black flag. A third measure- ment is then taken by using a 100-foot steel tape, after which each student reviews his estimates by the first three methods. Ordinarily the amount of error decreases with each estimate, but there is always a perceptible range and difference, even when the steel tape is used. By repeating this exercise and using different linear in- tervals, distance percepts improve signifi- cantly.

Juniors also estimate the size of various plots of land to improve acreage and area percepts. In one activity, two lines are staked out in an open field to form a 90- degree right angle. Each student uses these lines for two sides and a corner and lays out a rectangular plot containing what he estimates by sight to be an acre, or some other designated area. Each student com- pares his own with other students' plots, noting differences in shape and perimeter. The student subsequently "steps off" the plot using a side of his original plot as one dimension. Finally, a steel tape is used to lay out accurately the designated areas.

Another experience relates to estima- tions of weight. Each student is asked to estimate, by sight, the weight of a stone and a block of wood in units of pounds and kilograms. He is then asked again to esti- mate the weight by lifting each object.

644 The Arithmetic Teacher /December 1969

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Finally, the student makes an accurate measurement by weighing the stone and wood on a spring scale. Here, too, answers generally indicate little understanding of the mathematical relationship between pounds and kilograms.

Estimating the current temperature in an open field is always an interesting experi- ence, especially as students are asked to make a judgment in terms of centigrade and Fahrenheit degrees. This initial esti- mate of temperature is then followed by several observations from cheap and inac- curate thermometers. Some students find it difficult to understand and accept the fact that different thermometers produce differ- ent data. Finally, the students take a more valid reading from the weather station thermograph.

Our students are also asked to estimate, by sight, the number of gallons of water per minute flowing from an open spring. They then check their estimates by a more ac- curate measurement procedure, using a gal- lon can and a stopwatch.

Students seem to do a better job of estimating by sight the height of a tower or a tree than they do with other intervals and quantities. This may be due to some of their experiences in art classes where they are taught how to approximate the height of an object using a pencil or some other object of known length. However, they are given additional instruction in how to use the shadow-ratio method for deter- mining height, and the 1 1 -to- 1 ratio or similar triangles method. Their findings are then compared with the more valid data obtained by the instructor who used more reliable mathematical procedures.

Another exercise found to be particu- larly valuable in developing and clarifying certain measurement percepts is that of simple map construction. Students are taught how to use magnetic compasses or

simple transits for laying out shapes found in nature, for determining differences in elevation, and for showing relative loca- tions of campus buildings, trees, etc., on a scaled map.

Through these experiences in the out- doors, preservice teachers generally im- prove their percepts of scale and direction, and begin to recognize the relationships be- tween various units of measure.

These, we have found, are the essential steps in each experience: (1) recording the estimate of the interval or quantity under consideration; (2) preliminary measure- ment using inaccurate, and sometimes crude, instruments; (3) remeasurement us- ing available precise instruments; (4) analytical comparisons of the estimates of all students.

During the five-day senior student teach- ing experience at the outdoor education center, the preservice teacher lives and learns with fifth- or sixth-grade pupils. He uses what he learns as a junior and in- cludes the measurement percept experi- ences in curriculum planning. Thus he takes his fifth- or sixth-grade pupils out- doors to have firsthand experiences with the 100-feet exercise, measuring heights of objects, guesstimating weights, volumes, temperatures, etc. In this experience the preservice teacher senses the motivating influence of firsthand experiences on pupil behavior and attitude. Later, when he as- sumes responsibility as a classroom teacher, he will recall some of the satisfying ex- periences he had as a learner and teacher in the outdoor education center, and will utilize some of these activities with his own pupils as he seeks to teach measurement percepts. Moreover, it is expected that his creative imagination as a teacher will sug- gest additional ideas for using "hands-on" experiences for developing and understand- ing other phases of mathematics.

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