Mathematics Experiments: An Alternative Teaching Strategy

Mathematics Experiments: An Alternative Teaching StrategyAuthor(s): Charlotte L. WheatleySource: The Arithmetic Teacher, Vol. 27, No. 2 (October 1979), pp. 18-21Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41189455 .

Accessed: 10/06/2014 02:27

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 188.72.127.85 on Tue, 10 Jun 2014 02:27:45 AMAll use subject to JSTOR Terms and Conditions

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

http://www.jstor.org/stable/41189455?origin=JSTOR-pdf

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


n~2 - ̂ Mathematics Experiments: An Alternative Teaching Strategy By Charlotte L. Wheatley

For whatever the reason, teachers often rely heavily on a mathematics textbook, having students complete page after page most of which are filled with computational exercises. Al- though students need to develop computational proficiency, there are other important objectives of the mathematics curriculum that are not as read-

dealt with in the textbook approach. Furthermore, the work of Rosenshine (1971) has shown that instructional variety is one of the few factors that makes a difference in pupil achievement.

The purpose of this article is to pre- sent a teaching strategy for school mathematics with a rationale, examples, and procedures for implemen- tation. A case is built for mathematical experiments as one instructional approach teachers should use in teaching mathematics.

What Is a Math Experiment? To some, a math lab is a room where teachers send pupils who need help or where manipulative materials are kept.

An assistant professor in the Department of Edu- cation at Purdue University, Charlotte Wheatley teaches mathematics methods courses for prospec- tive and inservice elementary teachers. She is a former elementary school mathematics teacher and supervisor of student teachers.

18

To others, it is a set of manipulative materials for small-group use. I am sure the term math lab evokes still other ideas to some persons. To avoid confusion, I shall use the term math experiment to name a particular instructional strategy for mathematics instruction in the elementary and junior high school.

A math experiment is a small-group activity in which a problem is posed that can best be solved by collecting data, forming and testing hypotheses, and verifying the results. When a problem is posed, children must decide how to approach the problem, what data to collect, and how to interpret the results. Finally, they must verify the so- lution. Such classroom activities as games or the use of individualized learning kits or some specific manipu- latives are not necessarily math experiments. Some specific examples of math experiments are provided near the end of this article.

Why Use Math Experiments? As the work of Brownell established more than twenty years ago, children need to see meaning in the mathematics they learn. He stated in the first issue of the Arithmetic Teacher (1954, P. 5), To be intelligent in quantitative situations children must see sense in the arithmetic they learn. Hence, instruction must be meaningful and must be organized around the ideas and relations in- herent in arithmetic as mathematics. But they

must also have experiences in using the arithmetic they learn in ways that are significant to them at the time of learning, and this require- ment makes it necessary to build arithmetic into the structure of living itself.

Since they are set in a problem situa- tion, math experiments help children "see sense in the arithmetic they learn." For this and other reasons, math experiments tend to be highly motivating to students.

Probably the most important reason for using the math experiment strategy lies in the learning of problem-solving techniques that results. Improvement of children's problem-solving ability is near the top of everyone's list of goals for mathematics instruction. In its recent "Position Paper on Basic Mathe- matical Skills," the National Council of Supervisors of Mathematics stated (1977, p. 19, 21). "Learning to solve problems is the principal reason for studying mathematics."

The importance of this topic today can be judged by the fact that the No- vember 1977 issue of the Arithmetic Teacher was devoted to problem solving. The current Mathematics Guide- lines (Negley 1977) emphasizes the place of problem solving in the curriculum.

In spite of the importance of problem-solving skills, we have not been very successful in attaining this goal of mathematics education. The word problems in texts serve primarily as a vehicle for practicing arithmetic skills. Math experiments, on the other hand,

Arithmetic Teacher



are specifically designed to promote problem solving. In a math experiment, students are encouraged to try alternatives, explore the problem, and choose an approach that can then be tested.

According to Piaget (Piaget and In- helder 1969) children have a need to interact with their peers to stimulate higher-level cognitive functioning. The small-group interactions of math experiments provide a setting for an es- sential aspect of children's develop- ment. Piaget is not the only scholar to advocate pupil interactions in the learning process. He is joined by such educators as John Dewey, Jerome Bru- ner, and Edith Biggs in advocating small-group learning activities.

Experimental evidence for the supe- riority of small-group learning is also provided in a study by O'Brien and Shapiro (1977). They compared the re- tention of number patterns discovered from a grid. In one group of subjects, children worked independently of one another; in another group they worked in groups of three. The subjects working in small groups remembered twice as many number patterns as those working independently.

There is also recent evidence that inquiry training can produce long-term effects in the study of mathematics. Scott (1977) found that students ex- periencing an inquiry strategy five years earlier were more analytic and better in mathematics than pupils not receiving inquiry training.

How Can Math Experiments Be Used?

Mathematics experiments can be im- plemented in the classroom in a variety of ways. Four are described here.

1 . Integrated with content

Math experiments can be used with the whole class performing an experiment appropriate to the topic being taught. For example, for a measurement unit with a class of thirty students, a teacher might form six groups of five students and have six duplicate sets of materials available for them to use.

Alternatively, the teacher could have six different experiments, with one set of materials for each experiment. In

October 1979

this way in any one period each of the five groups would be doing a different experiment. On subsequent days, students might rotate through the six experiments so that ultimately all students would have a chance to do each experiment.

Having the entire class use a math experiment on a given day is probably the most effective way to use math experiments in the classroom. In so doing, the math experiment strategy is used to teach the topic under consid- eration; it is not a diversion, but a part of the basic presentation. Students find the instructional variety very appealing and enjoy the hands-on approach. The experience is particularly meaningful for students when the math experiment is directly related to the topic being studied.

2. As a classroom learning center A particular math experiment can be set up as a learning center in the classroom. As time is available, students may elect to visit the learning center for math experiment activities. Teach- ers may also schedule students to the learning center at certain times during the school day.

3. With special groups of students Certain students can work on math experiments while the remainder of the class is engaged in other activities. Math experiments are particularly use- ful with the talented students. Too often very capable students are ne- glected, and the math experiment approach is ideally suited to their abili- ties and interests. Bright students are generally curious and certainly have the ability to perform at the level of thought involved in a math experiment. They also will often find exten- sions and continue exploration long after a particular experiment is completed - the most ideal of situations. A good math experiment should stimulate exploration beyond the specific problem given.

A teacher could have bright students work on math experiments certain days of the week. For example, they might be able to complete the week's regular classwork in three days and then spend two days on math experiments.

Although the math experiment ap-

proach is particularly effective with the talented, all students can profit from doing math experiments. And every student should have the opportunity to profit from them.

4. In a special room

A specially equipped room in the school can be designated for math experiment activities, and a teacher could schedule the class into this room on Certain days. The advantages of this approach are that the materials are available to more people and individual teachers are not burdened with pre- paring all materials. But, in practice, teachers find it difficult to schedule use of the room in advance and often will not use math experiments for this reason. One possible alternative would be to have materials that are available commercially collected on a cart, which would then be circulated in the department. A teacher could schedule use of the cart as needed.

Mathematics experiments are more natural for some topics than others. A review of The Mathematics Laboratory: Readings from the ̂Arithmetic Teacher" revealed that the following topics predominate: measurement, probability, statistics, estimation, patterns, and applications. Mathematics experiments do not seem particularly appropriate for learning facts, rules, al- gorithms (computational skills), and concepts. Sometimes it is possible to devise a mathematics experiment for one of the latter set of topics, but usually another approach is more effective. I have attempted to construct mathematics experiments for these topics, but usually felt strained to do so. On the other hand, mathematics experiments are easily and naturally written for the first set of topics listed. Teach- ers, however, must not expect one approach to apply to every topic and situ- ation.

Realistically speaking, mathematics experiments should not be used every day. Children need to learn facts, develop computational proficiency, and learn a broad range of concepts. Many of these topics can best be taught by using other teaching strategies.

Ideas for the use of math experiments in the classroom can be found in Kidd, Myers, and Cilley (1970) and

19



Reys and Post (1973). Specific sugges- tions for math experiments appear in Cathcart (1977), Gawronski et al. (1975), and Kulm (1976). Many other sources are available.

Examples of Math Experiments Three examples of math experiments for different grade levels are included here. The materials used in the experiments are inexpensive and readily available to any teacher.

Example 1

Topic: Statistics Problem: What type of pet is most pop-

ular in our class? Level: Primary Materials: A list of popular pets, with a

space for the pupils to record the types of pets they have; graph paper; colored pencils

Procedure: Divide the class into three groups. Then direct the children in each group to do the following:

1. Predict what type of pet they think is most popular.

2. Indicate what types of pets they have by making a chart like the one shown in figure 1.

3. Tally the number of pets of each kind.

4. Report the results to the class.

When the results have all been re- ported, have the children compare the findings of the individual groups. (The composite results of the groups could be compiled by the teacher.) Have the children develop a baT graph of the results of the groups' investigations. Pro- vide further follow-up activities for the children - other uses of graphs, repeat- ing the experiment with a larger sample, and so on.

Example 2

Topic: Number patterns Level: Intermediate Problem: Which sums are most likely

to occur in adding two numbers? Materials: A telephone directory, a

data-collecting sheet Procedure: Have the students doing the

20

Fig. 1 What Type of Pet Do You Have?

Dog

Cat

Monkey

Gerbil

Guinea Pig

Fish

Bird

Turtle

Frog

Fig. 2 Data-Collecting Sheet

Last two digits Sum of last two digits

1 2

_3

_4 ^

48 49

~~50

Arithmetic Teacher



Fig. 3 a. 13 b. 16 e. 18 d. 20 e. 25

Circle Circumference Diameter Circumference of circle of circle Diameter

1

2

3

4

5

6

7

8

9

0

experiment go through the following steps: 1. From a telephone directory, copy

the last two digits of fifty consecutive telephone numbers.

2. Find the sum of each pair of digits.

3. Record the sum of each pair of digits on the data-collecting sheet as shown in figure 2.

4. Arrange the results so that one can see the number of times each sum occurs.

5. Consider these questions: Are all the sums just as likely to occur? Do you see any patterns? Why would there be a pattern?

Example 3

Topic: Measurement Level: Upper elementary or junior high

school Problem: What is the relationship be-

tween the circumference and the diameter óf a circle?

Materials: Circles or circular shapes of varied sizes, string, metersticks, re- cording sheets

October 1979

Procedure: Have the students do the following: 1. Measure the circumference and

diameter of each circle given. 2. Record the measurements on a

chart like the one in figure 3. 3. For each circle, divide the cir-

cumference by the diameter. 4. Record the results in the spaces

on the chart.

When the charts are completed, pose questions like the following for the students:

Is the ratio of the circumference to the diameter (c/d) about the same for every circle? The distance around each circle is about how many times as long as the distance across each circle? Which of the following would be the closest to the distance across a circle if the distance around the circle is 25? a. 5 b. 8 C.6 d. 7 e. 9 Which of these would be closest to the distance around a circle if the distance across the circle is 6?

The results that the students get trom dividing the circumference of a circle by the diameter will approximate the number pi, which is the ratio of the circumference of a circle to its diameter.

Summary In this article, reasons have been given for incorporating the math experiment teaching strategy into the school curriculum. When we examine a single instructional strategy, we must not lose sight of the fact that many other effective strategies exist. Mathematics experiments are not being touted as THE strategy; they are but one among many that should be used in the classroom. By studying mathematics through experiments, pupils can learn to solve problems and to understand the utility of mathematics. I hope you will give mathematics experiments a try in your class.

RofcrGncGS

Brownell, W. A. "The Revolution in Arith- metic." Arithmetic Teacher 1 (1954): 1-5.

Cathcart, G. The Mathematics Laboratory: Read- ings from the "Arithmetic Teacher." Reston, Va.: National Council of Teachers of Mathe- matics, 1977.

Gawronski, J., V. Hansen, A. Hendnckson, R. Jackson, and D. Johnson. Laboratory Mathe- matics, 1-7. Glenview, 111.: Scott, Foresman & Co., 1975.

Kidd, K., S. Myers, and D. Cilley. The Labora- tory Approach to Mathematics. Chicago: Sci- ence Research Associates, 1970.

Kulm, G. Laboratory Activities for Teachers of Secondary Mathematics. Boston: Prindle, Weber, & Schmidt, 1976.

National Council of Supervisors of Mathe- matics. "Position Paper on Basic Skills." Arithmetic Teacher^ (October 1977): 19, 21.

Negley, H. H. Mathematics Guidelines. In- dianapolis: Indiana Department of Public In- struction, 1977.

O'Brien, T., and B. Shapiro. "Number Patterns: Discovery Versus Reception Learning." Jour- naif or Research in Mathematics Education 8 (1977): 83-87.

Piaget, J., and B. Inhelder. The Psychology of the Child. New York: Basic Books, 1969.

Reys, R., and T. Post. The Mathematics Labora- tory: Theory to Practice. Boston: Prindle, Weber, <& Schmidt, 1973.

Rosenshine, B. Teaching Behaviors and Student Achievement. London: National Foundation for Educational Research, 1971.

Scott, N. "Inquiry Strategy, Cognitive Style, and Mathematics Achievement." Journal for Re- search in Mathematics Education 8 (1977): 132-43. D

21



Documents

Mathematics Experiments: An Alternative Teaching Strategy