Using Quality Control Activities to Develop Scientificand Mathematical Literacy

L. Diane MillerMiddle Tennessee State UniversityMurfreesboro, Tennessee

Charles E.MitchellTarleton State UniversityStephenville, Texas

The daily activities of business and industry provide many fascinating opportunities to study basicconcepts of mathematics and science. These activities often lend themselves to an activity-based,manipulative-enhanced learning environment. This article focuses upon a quality control activity usedin the manufacturing of camera film. It suggests how this activity can be adapted to science andmathematics classrooms, and discusses some basic concepts of scientific methodology, probability, andstatistics.

The activities of business and industry provideclassroom teachers with many settings which can beadapted to the classroom to convey the relevance andvalue of the curricula in mathematics and science.Furthermore, these settings often lend themselves tobeing adapted to an activity-based, manipulative-en-hanced learning environment. One of the most usefuland interesting settings involves the quality controlactivities of business and industry. These settings,which can vary greatly from business to business.provide opportunities for students to become involvedin the use ofscientific methodology, and the manner inwhich new knowledge is acquired, and a wide varietyofdata collection activities involving concepts ofprob-ability and statistics. It is the purpose of this article tosuggest how one of these activities, the manufacturingof film for cameras, can be adapted to the classroom.

This article describes a general outline for a weekto three weeks of class work. The material is appropri-ate for Grade 5 through high school level classes, andit offers the opportunity for a variety ofindependent orspecial group efforts of varying levels of difficulty.The direction the actual lessons take will vary fromclassroom to classroom as the individual interests ofthe students and the decisions of each class impactupon the scope and sequence. Teachers should encour-age this individual expression of interests and allowinstruction to move away from a teacher-centered,textbook-driven environment in which the teacher isthe authority figure. The teacher assumes the role of acatalyst for the student investigations which lead to thesolutions the students seek. In this environment thestudents have the opportunity to construct their ownunderstandings of the concepts and principles whichare objectives of the lessons. These activities alsoprovide ample opportunities to evaluate students innontraditional activities such as independent projects,

small group collaborative efforts, oral presentations,and written reports and exercises.

The Quality Control Context

In this lesson the students are to be givenmanagingcontrol of a factory which produces camera film usedin taking photographs. Although factory supervisorsclosely monitor each stage of the manufacturing pro-cess, at times things go wrong and substandard film isproduced. Forthis lesson it is stated that the productionprocess produces batches of 500 rolls of film at a timeand from pastexperience approximately 5% ofthe rollsoffilm are defective. The class should be informed thattesting each roll is not feasible, not only because thetesting process exposes good film. but testing each rollof film is time consuming and unnecessary. If no onein the class suggests the possibility of randomly sam-pling some of the rolls of film in each batch, theinstructor should raise a question concerning the effi-cacy of this approach. If it is feasible to use a smallsample to test the quality of the batch, both time andmoney would be saved. There are additional sources ofinformation regarding sample sizes, such as Hays(1973), for teachers who are interested.

Most students quickly perceive that testing 499 or490 rolls of film is as good as testing 500 rolls, so theproblem becomes one of determining just how smallthe sample can be and still accurately predict theoverall quality of a batch of film. A method can bediscovered and developed by the class to whateverdegree ofprecision that the class decides is appropriateand this activity lends itself to small group investiga-tions. For each group identified (four to six are recom-mended). 500 congruentobjects (straws, popsicle sticks,metal washers) are needed with an additional 50 ob-jects marked in some way with a pen or magic marker

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Using Quality Control Activities

to indicate the defective film. The initial set of 500objects representnondefectfve film. Foreach setof500objects, a specified number of nondefective objectswill be replaced in the bag by the correspondingnumber of defective objects. The defective objectsmust be marked in some way so that handling them ina bag prior to drawing them out will not reveal theirstatus.

The objective ofthe first class activity is to collectdata in order to make a preliminary decision as to howsmall the sample size can be and still accurately predictthe percent of defective objects in each bag. This is asubjective decision and the student must balance thetime and expense involved in investigating largersamples with the threat to the factory’s reputation iftoomany rolls of defective film are produced, and theexpense of replacing defective film returned by thecustomers. The instructor should encourage the stu-dents to be open to the idea that the use ofsmall samplescould be both highly effective and reduce costs dra-matically.

To begin, each group is given abag in which 5% ofthe objects are defective. Group leaders, orthe instruc-tor, should insure that 25 unmarked objects are re-placed by25 marked objects in each bag. The value5%is used because of the factory’s history which wasstated previously. Different groups will be assigned toinvestigate different sample sizes. One group willwork with sample sizes of five, recording the numberof defective objects in each sample, returning thesample to the bag, thoroughly mixing the objects, andrepeating the process. The remaining groups will workwith larger sample sizes in increments of five follow-ing the same process in each case. Each group shouldtest its sample size 5 to 10 times. After completing theanalysis of each sample size, additional sample sizesshould be assigned to the groups until a sample size of75 has been investigated. The class should conveneagain, and the findings for each sample size should begraphed. Table 1 contains a sample of the data whichmight be produced.

The students should observe the initial trend thatsuggests that the larger the sample size. the moreaccurately thepercent ofdefective objects is predicted,yet there is a point ofdiminishing returns. The data willsuggest that the points of the graph will level out andbegin to fluctuate. This first activity provided the classwith a rough idea of the optimal sample size and thepoint at which the trend levels out is the starting pointfor the second activity.

Suppose that the initial data suggest that the trendbegins to level out at sample sizes of 30. The smallgroups will now focus their attention on sample sizesbetween 25 and 35, orperhaps 20 to 40. Each group willbe assigned a sample size in this range (25,26.27. etc.),and repeat earlier procedures until data for all samplesizes in the rangehave been collected. The class shouldthen chart the findings and make its preliminary deci-sion on the sample size to be used by the factory.Suppose that the class agrees on a sample size of 26objects.

It should be noted that students will raise manyquestions as the investigation proceeds. Students mayask whether a number different than 26 would havebeen indicated had they started with a different percentofdefective objects. The teachermay wishto assign thequestion to an individual or small group for a laterpresentation. The teacher might also wish to facilitatethe investigation by suggesting that students use bagsin which 50% of the objects are defective, instead of5%.

In the third phase ofthese activities, the class willexamine the reliability of their choice of sample size.The class must decide the maximum percentage ofdefective items in a batch that it will permit beforeinsisting that the entire batch be rejected. Assume thatthe class decides that whenever the percentage ofdefective objects in the batch is 8% or more, or twodefective objects for the sample size of 26. that theentire batch should be rejected. Although 8% is anarbitrary choice, it serves as a starting point. At thispoint the concept of reliability should be introduced.

Table 1. Initial Determination of Sample Size.

Sample Size102030405060

Number of Defective0-0-1-1-2-2-

Objects322334

Per Sample Range of Percent0% - 30%0% -10%3% - 7%3% - 8%4% - 6%3% - 7%

Volume 95(2). February 1995

Using Quality Control Activties

When a sample is used to make a determination as tothe quality of the batch, two kinds of errors can bemade, and reliability is related to how often an error ismade. A batch meeting quality standards can be mis-takenly rejected or a batch which is below standardscan be approved for marketing. The discussion shouldfocus on the idea that taking steps to limitone ofthetwoerrors increases the likelihood ofmaking the otherkindoferror. Ifone mistake is more important to avoid thanthe other, then the class must respond accordingly.Most students will understand that protecting thefactory’s reputation is of crucial importance, so themistake of approving an inferior batch is the morepotentially damaging mistake. Thus, when the deci-sion is made to approve a batch offilm. how often willthe percent of defective rolls of film be 8% or more ifa sample size of 26 is used to predict the quality of theentire batch?

Some quick examples should suggest to the classthat as the percentage of defective rolls climbs (from10% to 20% to 30%. etc.). the better the sample size of26 will operate in determining that the percent ofdefective rolls of film is 8% or more. Thus. the classshould focus its attention on batches in which 8% oftheobjects arc defective. The frequency with which poorquality batches arc mistakenly marketed with thispercent will be used to compute the reliability figurefor the sample size of 26. If mistakes arc made 5% ofthe time, then the sample size is 95% reliable basedupon this definition of reliability.

Group leaders should insure that their bag of ob-jects contains 40 defective objects. Each group shouldtest the reliability of the sample size of 26 at least 10times, and then the class should compile the results. Itis interesting to note that frequently when the results ofone group markedly differed from the othergroups thatthe contents of the bag it used did not conform tospecifications. Aberrant results arc often an indicatorthat mistakes have been made in the investigatingprocess.

Forpurposes ofdiscussion, assume that 15% ofthetime inferiorbatches were not rejected. Thus. a reliabil-ity figure of85% is associated with a sample size of26.Experience suggests that students will believe that85% is an acceptable figure. It is often difficult forstudents to understand that the level of security sug-gested by the 85% figure varies from situation tosituation. With one’s life at stake the figure of 85% isa poor one to face. The class may then decide toinvestigate the reliability of slightly larger samplesizes.

In the fourth and final stage ofthese activities, theteacher simulates the operation ofthe factory machin-ery and produces batches of objects of varying levelsof quality. The class, again in groups, tests the qualityof the batches based upon the decision rules set downpreviously. The investigations should suggest that thecloser the percent of defective objects is to 8%, thegreater the numberofmistakes that aremade.The classshould also be exposed to the idea that the closer anunacceptable batch is to 8%, the threat that the batchposes to the factory’s reputation is diminished. Bagswith higherlevels ofdefective objects pose the greatestthreat.

Ifpossible, completion ofthe investigations shouldbe followed by the visit from a quality control experttothe classroom, or a class field trip to a factory. In a realsense the investigations of the class only touch uponreal life quality control activities and much is to begained from the added exposure to these activities.

Conclusion

The daily activities of business and industry oftenprovide fascinating settings for the study ofscience andmathematics. The myriad of interdisciplinary activi-ties provides opportunities for students to see howmathematics and science unite to solve many problemsfaced by people on a daily basis in theireveryday lives.These activities often promote small group activitiesand lend themselves to nontraditional methods ofevalu-ation such as oral presentations and written reports.Visits from members of society to the schools or fieldtrips to organizations within the community often leadto the development of working relationships whichserve all involved. An increased awareness of the useof mathematics and science in real life should have aprofound impact both upon what is taught in theclassrooms, and how it is taught.

References

Hays.W.L. (1973). Statistics for the socialsciences (2nd ed.). New York: Holt. Rinehartand Winston, Inc.

Note: L. Diane Miller’s address is P.O. Box 34.Department of Mathematics, Middle Tennessee State Uni-versity, Murfreesboro, TN. Charles E. MitchelFs address isDepartment of Mathematics & Physics, Tarleton State Uni-versity,BoxT-519.TarletonStation,Stephenvine,TX76402.

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