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DWCRTTPORS *Algebra, *Tnstr:Uction, Mathematical Concepts, *Mathematicq, Mathematics Education, *Secondary School Mathematics DOCUMENT RESUME ED 035 974 24 SE 007 912 AUTHOR nixon, Lyle J. TTmLE An Analysis of Certain Aspects of Approximation Theory as Related to Mathematical Instruction in. Algebra. Final Report. TmsTTTUmTON Kansas state Univ., Manhattan. MOONS AGENCY Office of Education (DHEW) , Washington, D.C. Bureau of Research. W17,ATT NO RP-8-F-03n PUP DAV' Aug 6() (1PANm OFG-6-9-008030-0035(097) NOTr 99D. "UPS DRICP ;?)RS price MP-s0.50 HC-4;3.05 DWCRTTPORS *Algebra, *Tnstr:uction, Mathematical Concepts, *Mathematicq, Mathematics Education, *Secondary School mathematics AsSTRACT This study is conc9rned with certain aspects of approximation theory which can he introduced into '.he mathematics curriculum at the secondary school level. The investigation examines existing literature in mathematics which relates to this subject in an effort to determine what is available in the way of mathematical concepts pertinent to this study. As a result of the literature review the material collected has been arranged in a structured mathematical form and existing mathematical theory has been extended to make the material useful to instructional problems in high school algebra. The results of the study are found in the expository material which comprises the ma-ior portion of this report. This material contains ideas of approximation theory which relate to elementary algebra. Also, included is a collection of references for this material. The report concludes that certain techniques of approximating a root of a polynomial by finding roots of a derived polynomial can be presented in a manner which is suitable for courses in elementary algebra. (DI') EDUCATION A WELFARE U.S. DEPARTMENT Of HEALTH, OFFICE OF EDUCATION REPRODUCED EXACTLY AS RECEIVEDFROM THE THIS DOCUMENT HAS BEEN POINTS OF VIEW OR OPINIONS PERSON OR ORGANIZATIONORIGINATING IT, REPRESENT OFFICIAL OFFICE 0EDUCATION STATED DO NOT NECESSARILY POSITION OR POLICY. FINAL REPORT Project No. 8-F-030 Grant No. 0EG-6-9-008030-0035(057) AN ANALYSIS OF CERTAIN ASPECTS OFAPPROXIMATION THEORY AS RELATED TO MATHEMATICAL INSTRUCTION IN ALGEBRA Lyle J. Dixon Kansas State University Manhattan, Kansas 66502 August 31, 1969 U. S. DEPARTMENT OF HEALTH, EDUCATION, AND WELFARE Office of Education Bureau of Research +IL Tv. nrm-75rms-Frrntrc.-4, FINAL REPORT Project No. 8-F-030 Grant No. OEG-6-9-008030-0035(057) AN ANALYSIS OF CERTAIN ASPECTS OF APPROXIMATION THEORY AS RELATED TO MATHEMATICAL INSTRUCTION IN ALGEBRA Lyle J. Dixon Kansas State University Manhattan, Kansas 66502 August 31, 1969 The research reported herein was performed pursuant to a grant withthe Office of Education, U.S. Department of Health, Education, andWelfare. Contractors undertaking such projects under Government sponsorship are encouraged to express freely their professional judgment in the conduct of the project. Points of view or opinions stated do not, therefore, necessarily represent official Office of Education position or policy. U.S. DEPARTMENT OF HEALTH, EDUCATION, AND WELFARE Officc of Education Bureau of Research ,miArrmvrr,u,,s TABLE OF CONTENTS Title Page 1 Table of Contents 2 Summary 3 I. Problem Under Consideration 5 II. Methodology for the Investigation 6 III. Findings and Results 8 IV. Conclusions and Recommendations 9 Appendix A . Expository Presentation of Approximation 11 Theory as Related to Secondary School Algebra. Appendix B - References 5? Appendix C Bibliography 59 2 SUMMARY mathematical principlethat certain solutions It is a well-known equation equation may beapproximated by considering an to an algebraic certain terms of the derived from the originalequation by "ignoring" introduced by this sortof procedure equation. Estimates of the errors At present,mathematical are given inadvanced courses inmathematics. include such principles. instruction at the secondarylevel does not have need This is true eventhough other areasof science instruction for the basic conceptsinvolved in thisprinciple. investigation is: Is there a The fundamentalquestion for this mathematical knowledgewhich covers thisapproxima- systematic body of in some con- tion technique and isit possible topresent this knowledge be established in venient form whereby a programof instruction can level?The basic courses inelementary algebra atthe secondary school task is to answerthis question. first necessary tomake a In order to answerthis question it was existing literature inmathematics rather extensiveexamination of the the way of mathematicalcon- to find out what wasalready available in of educationallitera- cepts pertinent to thestudy. A search was made these approxi- ture to determineif an attempt hadbeen made to relate the secondaryschool level. mation techniques toelementary algebra at of mathematical The search of theliterature revealed some sources literature on information but no reference wasfound in the educational concepts were classifiedby the subject of thisstudy. The mathematical put in a convenientstruc- areas and thosepertinent to the study were tured mathematical formfor future use. about the level ofmathemati- One significantobservation was made Almost all were at anadvanced cal concepts found inthe literature. to the study werefound in re- level and those whichmight be pertinent The trend in mathe- search journals in theperiod from 1900 to1930. included the areaof this spe- matical research inrecent years has not of the ideaspresented in the cific study. As a result of this, many developed and/orextended expository material inAppendix A had to be as originalresearch. concepts could beused in The criteria used todetermine if the to be included inthe elementary algebra andif these concepts were approximation expository material werethe following: Is it pertinent which is readableby the"average"high theory? Is it good mathematics in the teacher? Is it related tohigh school algebra school mathematics Is strictly algebraicand/or geometric concepts? sense it depends upon that a high in the expositorysection at a level the material presented what a high school school teacher canunderstand? Is it adaptable to student knows aboutmathematics? material in The results of thestudy are found inthe expository find the pertinentideas of approxi- Appendix A. In this material one can It is clear that the mation theory asrelated to elementaryalgebra. answered and in theaffirmative. basic question ofthis study can be be presented with somerigor Certain aspects ofapproximation theory can contribution to and/or in an intuitive mannerto make an additional at the secondaryschool level.This contribution mathematics education mathematical aspects. One, it broadensthe range of has two significant Secondly, it concepts which areavailable at a givenexperience level. which the appliedsciences provides the kind ofmathematical experience to be relevant tothe first courses can use andprovides it early enough in these sciencesat the secondaryschool level. additional insight into A by-product of thestudy has been the investigator duringthe course of mathematics obtainedby the principle way to measurethis, but it the investigation.There is no quantitative has been significant. should be made as theresult of The principlerecommendation which work to the last andultimate test the findings is thecontinuing of this and be used in an of its value. Some teaching unitsshould be prepared teachability of the conceptsout- experimental situationto determine the material. The expositorymaterial is essentially lined in the expository material at background for theunits and should notbe used as teaching for prospective the secondary schoollevel. It would be appropriate mathematics teachers. I. PROBLEM UNDER CONSIDERATION Certain equations which arisefrom physical problems proveto be difficult to solve because of thenumerical computations whichhave to 6olution of the original be completed. Sometimes an approximation for a equation can be obtained byignoring a given term of theequation and solving the "reduced" equation. For example: An equation such as \IT n3.1 (1) 7 iloo can betransformed into the equation d 6.2d d2 9.61 (2) 7 110o 1,210,000 = A solution to (2) isapproximated by solving theequation 6.2d 9.61 . (3) 7 if55 = Equation (3) is obtainedfrom (2) b-- "ignoring" ordeleting the term in reasonably d2. The solution of(3) is approximately 111 and this comes close to satisfying(2) and, hence, satisfying(1). One might be tempted toconclude that this is an"accident" of the equation involved but thisis not the case. Physicists use the proce- known to use a dure just described quitefrequently and chemists are similar procedure in thesolution of some of theirequations.Moreover, there appears to be a desire onthe part of teachers inthese sciences that some instruction takeplace in the mathematicscurriculum on precisely these procedures. The fundamental questionwhich should be answeredis: which covers Is there a systematicbody of mathematical knowledge such examples as justdescribed (and consequentlyothers) and is it form whereby a possible to present thisknowledge in some convenient program ofinstruction can be establishedin courses in elementary algebra at the secondarylevel? the first half of The last half of the questiondepends upon answers to readily lend them- the question and severalother factors which do not teachers, selves to research andanalysis. These other factors, such as However, curriculum, etc., are notnecessarily included inthis study. if the a significantportion of the fundamentalquestion can be answered specific objectives ofinvestigation are: 5 1(a) to determine if
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