Mathematics in elementary science

Mathematics in elementary scienceAuthor(s): SAM S. BLANCSource: The Arithmetic Teacher, Vol. 14, No. 8 (DECEMBER 1967), pp. 636-640, 670Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41185682 .

Accessed: 20/06/2014 15:15

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Arithmetic Teacher.

http://www.jstor.org

This content downloaded from 195.34.79.174 on Fri, 20 Jun 2014 15:15:36 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=nctm

http://www.jstor.org/stable/41185682?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Mathematics in elementary science

SAM S. BLANC

San Diego State College, San Diego, California

Dr. Blanc is professor of elementary education at San Diego State College.

JVA athematics and science seem to be terms that are joined together in our com- mon language almost as firmly as day and night, salt and pepper, and love and mar- riage! Everywhere, that is, but in our typical elementary school program. Here, for some reason, a dichotomy of emphasis occurs. As every experienced teacher knows, mathematics is commonly taught quite separately from science. Children may not be led to discover the relationships that exist between these two major investigative activities of the human mind.

Certainly, teachers read in educational journals and hear in professional and in-service meetings that mathematics is the foun- dation on which all sciences are built, that mathematics is the language of science, that science is the logical application of mathematics, and so on. But what is being done in the elementary school to help children discover some of these quantitative relationships between mathematics and science? Some nationally funded curriculum studies in elementary science are developing materials that can be used to teach children how to observe, measure, quantify, record, and interpret results of scientific investigations.1 But how can the average elementary school teacher organize classroom activities that will give the children

1 The Science Curriculum Improvement Study (SCIS), University of California, Berkeley, California; the Ele- mentary-School Science Project (ESSP), University of Illinois, Urbana, Illinois; the Commission on Science Education (AAAS), Washington, D.C.; and the Ele- mentary Science Study (ESS), Watertown, Massa- chusetts.

opportunities for mathematical interpretations?

Quantitative investigations can be designed for elementary school children so that they learn how to treat mathematical data as well as how to apply principles of science. At this level, counting is probably the most obvious means of obtaining quantitative results. Rulers, stopwatches, balances, and volumetric measures are ideal instruments for measuring at the elementary level. The role of the teacher should be that of designing activities or adapting existing activities in the text- book, so that the children are given the opportunity to use these measuring instruments in collecting and organizing quantitative data.

Children working with quantitative measurement soon learn that accuracy, organization, and planning are very im- portant. If the same activity is done several times with widely varying results, the

636 The Arithmetic Teacher



child soon realizes that some error has crept in or that the control situation has not been well planned. Possibly, some children may find repeated measurements of the same thing boring, but this also is a technique for mathematical understanding that must be developed and ac- cepted. From beginning to end, mathematical interpretations of data are significant factors in building careful observation, organization, and analysis in developing a true understanding of science and mathematics.

The foregoing is not to be interpreted as a plea for the integration of mathematics into science or science into mathematics. Each is a distinct discipline and needs to be emphasized as such in the curricular pattern. But when it is possible to observe and record certain findings in the study of science in such a way that mathematical interpretation is possible, then the teacher interested in developing this type of understanding in children must take ad- vantage of the opportunity. These activities must be planned as a part of the lesson. They are not haphazard happen- ings. Children can learn the skills involved in interpreting results, analyzing them mathematically, and forming inferences and predictions based upon the mathematical treatment of the data.

Many laboratory activities have been designed by teachers and science educators interested in this problem. For example, the SCIS has developed several simple quantitative experiments suitable for the primary level.2 Children are led to discover such interactions as the relationship of the quantity of sugar dissolved in water to the amount and weight of the solution, the relationship of the amount of electric current to the attractive force of an electro- magnet, or the relationship of the amount of water absorbed to the type and number of beans soaked in the water.

2 Herbert D. Thier, "Quantitative Approaches to Ele- mentary Science," The Instructor, January 1966, pp. 66-92.

A lesson for middle-grade students

The following is a description of a lesson used successfully by the author with middle-grade children to develop scientific observation and mathematical rea- soning. The equipment is very simple, and the children are involved in the process as the lesson progresses. Setting Up a Bridge for Motivation

Two simple pendulums, of different lengths and having differently shaped bobs, are set in motion for the children to observe. Within a few seconds, the two pendulums are no longer swinging in uni- son. The children are asked to observe this phenomenon several times and to "guess" (hypothesize) why the two pendulums are swinging at different rates of speed.

Developing Logical Hypotheses After some discussion, the children

usually present a number of hypotheses, which are listed on the board without comment. Some of these would logically be as follows:

1. The bobs are of different sizes. 2. The strings are of different lengths. 3. The bobs have different weights. 4. The bobs have different shapes. The teacher goes over the list with the

class to remove those hypotheses that are so obviously unrelated that they can have little bearing on the problem. The children are then asked to suggest how they can find which one or ones of the remain- ing "guesses" are actually the cause of the pendulums' swinging at different rates of speed. Almost universally, the children exclaim, "Let's try different things to see what happens!" And this attitude, of course, is basic to any good teaching situation.

Equipment Necessary As a means of conserving time and help-

ing the children make more or less accurate observations, simple equipment is assem- bled beforehand and is brought into the

December 1967 637



classroom. As can be seen in the accom- panying picture, the pendulum lengths and the pendulum bobs are readily interchange- able. The only additional piece of equipment needed is a watch with a sweep sec- ond hand or, preferably, a stopwatch.

The Teaching Procedure After some discussion, a set number of

swings is decided upon as a constant in the investigation (20 swings of the pendulum for each observation has been found to be excellent). The children are brought into the actual investigation by letting var- ious members of the class serve as time- keepers and "pendulum counters." Each child in the class prepares a table on which to record the data, as in Table 1. The results for a typical series of observations are included as an illustration.

Table 1

Observations on rates of pendulum swing

Length of Size of bob Average time pendulum S M L for 20 swings 6 inches 16 16 16 16 seconds 9 inches 19 19 19 19 seconds 12 inches 22 22 22 22 seconds

Mathematical Interpretation The children are asked to study the

results obtained so far and to make some tentative inferences. It is a very slow child who will not see at this point that the length of the pendulum is the significant factor in determining the rate of swing.

Now the children are asked if they would like to draw a "picture" of the information they have recorded. This usually brings forth an enthusiastic response, and half-inch squared graph paper is dis- tributed. The children may need to be shown how to make a simple graph, although some children at this grade level may already have some experiences in graphing two variables.

Some discussion may be necessary at this point to help the class determine how the points are to be plotted on the graph. Once they understand, the three points are quickly plotted, and the children draw a line connecting the three points and ex- tending to the limits of the graph. The sample drawing in Figure 1 shows a typical graph derived from the data in the table.

-12 - ¿L

I ■ /

! 5 y¿- * /

2 - _ ■

1 _

0 ' 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Average Time for 20 Swings (in seconds)

Fig. 1. - Relationship of length to period of pendulum.




The Evaluation Now, the teacher needs to check wheth-

er the children really understand what they have observed, recorded, and graphed. Let- ting the children draw inferences and make predictions from their graphs is probably as good a way to check this as any other technique. Ask the class how many seconds a 15-inch pendulum would take to complete 20 swings, and how long a 10- inch pendulum would take. Ask them how long a pendulum would have to be in order to take 13 seconds or 20 seconds to complete 20 swings. If they can interpret the graph to answer questions such as these satisfactorily, the children are developing an understanding of the power of mathematics.

A lesson for junior high school students

Most of the quantitative activities developed by investigators deal with direct relationships of one factor to another. For the upper grades and for junior high school children it should be possible to design experiments that call for more sophisticated mathematical interpretations. A lesson that has been used to help children discover the meaning of an inverse relationship in an energy field is described below. Setting Up a Bridge for Motivation A large U magnet or other strong mag-

net is displayed for the class; and a few simple concepts relating to magnetism and magnetic force, already familiar to most of the children, are reviewed. By empirical observation the class determines that the magnetic field will exert an attraction on an iron object only within a limited distance from the pole. Let us say this is estimated to be about one-half inch. The children are allowed to experiment with other mag- nets for a few minutes before the problem is posed.

Developing the Problem After some directed discussion, the class

will readily agree that the magnetic force seems to become weaker as the distance

from the pole is increased. The question is then raised, "How can we measure the change in magnetic force, within the half- inch limit previously determined, as the distance from the pole is increased?" The class usually has a number of suggestions for designing activities that will enable them to obtain data to solve this problem. De- pending on the time available, the children should be allowed to test some of their ideas before bringing out the equipment as follows.

Equipment Necessary The children can readily work in groups

of four to six in this activity. Each group will need a ring stand, a ring, a clamp, a bar or U magnet, a box of paper clips, and about fifteen glass microscope slides.

Teaching Procedure After some experimentation and discus-

sion, the children will see that the glass slides have a standard thickness (approx- imately 1/25 of an inch) and can be used to increase the distance from the magnetic pole at a constant rate. The number of paper clips supported by the magnetic field passing through the glass slides will then give a measure of the magnetic force as the distance is increased to the pre- determined limit of about one-half inch. The results for a typical series of observations are included as an illustration. (See Table 2.)

Table 2

Number of ' Distance from Number of paper slides

' magnetic pole clips supported

1 1/25 inch 10 2 2/25 inch 8 3 3/25 inch 7 4 4/25 inch 6 5 5/25 inch 5 6 6/25 inch 4 7 7/25 inch 4 8 8/25 inch 3 9 9/25 inch 3

10 10/25 inch 2 11 11/25 inch 2 12 12/25 inch 1 13 13/25 inch 1 14 14/25 inch 0

Mathematical Interpretation As the investigation progresses the chil-

December 1967 639



12 i

•11

: '

1 8- -'

^ 7 ^ .1 6 X

° 5 ^- S ' 1 X

2 ^VJ '

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Distance from Magnetic Pole (in number of slides)

Fig. 2. - Relationship of distance to strength of magnetic field.

dren will usually jump to the conclusion that there is a straight-line relationship between distance from a magnetic pole and force of attraction. But as the distance increases beyond about one third of an inch, the data begin to vary from the pattern. In answer to the questions and looks of dismay, the teacher can only suggest that the activity be repeated and that the results be checked with those obtained by other groups.

When the class is finally convinced that the observations are correct for this activity, the teacher can suggest that to really understand what is happening, a graphical representation would be helpful. The children then construct a graph similar to the one shown in Figure 2.

The Evaluation

When the graphs are completed and compared by the several groups, the teacher needs to check whether the children really understand what happened to the magnetic field as the distance from the pole was increased, bit by bit. The children are asked to interpret the graph and make a statement as to what is observed in the graph line. In almost all cases the

children will come to the conclusion that the magnetic field gets weaker as the distance increases, and more importantly, that this change in strength changes at a different rate as the distance is increased. At this point, the idea of an inverse-square relationship can be introduced; and the children, having done the activity, should be able actually to understand what is meant by this rather sophisticated concept.

Although many science and mathematics curriculum groups are working to develop ideas of quantitative interactions in these areas, the main effort in developing understanding of mathematical relationships with science must come from the minds of creative teachers throughout our schools. Good teaching, teaching for the process of inquiry, teaching that allows children to discover and apply relationships is surely taking place. What is needed is some means by which other teachers can become aware of what is taking place, what techniques are proving successful, and where some of these creative ideas are being used. If these ideas were to become commonly available, many teachers would certainly be stimulated to develop their own

[Continued on p. 670]




Another interesting investigation is an extension of Virginia's idea. Systems for determining the number of permutations of a given combination are sometimes hard to find. Also, being sure all combinations have been listed as numbers grow larger taxes clear thinking.

Mathematically keen students from the sixth through the eighth grade will find several challenging problems in this exer- cise. Working on these problems will en- courage children to make reasonable guesses, but to keep open minds and to check their guesses systematically.

Mathematics in elementary science [Continued from p. 640]

techniques and philosophies, whereby they could instill in their students an understanding of the relationship of mathematics and science.

There are many principles of science that can be taught so that they have a quantitative interpretation and that can be used to help children discover some of the mathematical relationships. Why not

write out some of your techniques that have proved successful for showing the mathematics of science and send them to the editor of The Arithmetic Teacher? Teachers are always interested in more creative ways to teach children; and by drawing on the experience of many teachers, more effective lines of communication become possible.




Documents

Mathematics in elementary science