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Math Teacher Pub Date : 1970-10-01 DOI: 10.5951/MT.63.6.0471

J. E. Holmes

{"title":"Enrichment or Acceleration","authors":"J. E. Holmes","doi":"10.5951/MT.63.6.0471","DOIUrl":"https://doi.org/10.5951/MT.63.6.0471","url":null,"abstract":"THERE has been much difference of opinion among secondary mathematics teachers during the last several decades as to just how the curriculum should be changed. There appears to have been, however, general agreement that changes were necessary. This agreement has given rise to the so-called accelerated programs in which students are enabled to study the usual (but modernized) high school courses at an earlier stage. In many cases this acceleration allows the student to be exposed to a full year of calculus while still in high school. Many of the students who have followed this route are unable to qualify for Advanced Placement credit and are required to repeat the first year of calculus as college freshmen. This indi cates that we have allowed some students to enter an accelerated program who are not really ready. When this happens, have we really done our students any good? Or, have we succeeded only in making the first-year college mathematics course repetitive? We are confronted with the task of ade quately preparing those students who are average or above in ability but not quite ready for the accelerated program. After some years of teaching at both the high school and college level, the author has reached the opinion that we, as teachers, can enrich the secondary program to such an extent that when a student reaches college he is ready for calculus but, except for the truly outstanding student, has not studied it. A competent secondary school math ematics teacher will encounter no diffi culty in enriching a course if he tries to introduce, at a simplified or intuitive level, topics that he knows will be studied at a later time. During the second year of algebra, for example, it is possible to in troduce parabolic interpolation at an extremely low level. This could lead to better understanding of Newton's formula for approximating roots of a polynomial equation, which is customarily introduced about the twelfth year, and eventually to LaGrange's formula, which is usually not encountered until a course in differential equations is taken. Many secondary teachers introduce Newton's method as a mechanical process that involves only the ability to find the first derivative of a polynomial and to compute algebraically. This is done by using the fact that, by starting with a fairly good estimate of a zero r (call it Xi), you may obtain successively closer ap proximations x2, ?3, Xiy . . . by using the recursion formula f(xn) . 100 Xn+i = Xn --? (fi = 1, 2, 3, ... ) J %n)","PeriodicalId":144125,"journal":{"name":"Math Teacher","volume":"477 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121188630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

Discovery Teaching-From Socrates to Modernity. 发现教学——从苏格拉底到现代性。

Math Teacher Pub Date : 1970-10-01 DOI: 10.5951/AT.17.6.0503

Phillip S. Jones

引用次数: 2

Polygon Sequences--An Example of a Mathematical Exploration Starting with an Elementary Theorem. 多边形序列——一个从初等定理开始的数学探索的例子。

Math Teacher Pub Date : 1970-05-01 DOI: 10.5951/MT.63.5.0421

J. E. Mann

引用次数: 0

A Case for Flexibility in Classroom Instruction. 课堂教学的灵活性案例。

Math Teacher Pub Date : 1970-04-01 DOI: 10.5951/MT.63.4.0298

Elaine Kivy Genkins

{"title":"A Case for Flexibility in Classroom Instruction.","authors":"Elaine Kivy Genkins","doi":"10.5951/MT.63.4.0298","DOIUrl":"https://doi.org/10.5951/MT.63.4.0298","url":null,"abstract":"EVERY conscientious teacher of mathe matics attempts to create the basis for good teaching by attaining as deep and broad a knowledge of mathematics as he can. On the immediately practical level, he plans his lessons in as much detail as he feels necessary to guide him in the teaching of each class. A creative teacher anticipates student inquiry and provides opportunity for student conjecture and ex ploration. However, inevitably there are times when a student asks a question or poses a conjecture which has not been an ticipated and to which the answer is not immediately apparent. Over the years the writer has found that at these times it is beneficial in cases of genuine inquiry on the part of the students to depart from a planned lesson and follow up a students question or conjecture with class ?discus sion, thus capitalizing on the interest and motivation of the class. In the author's opinion, it is particularly in the working out of the answers to the questions to which neither students nor teacher know the answer, or the proof or disproof of the conjecture, that creative mathematics can be learned. These often prove to be the moments of maximum excitement in the classroom not only for the teacher, but for the students. No matter how small the \"new piece of mathe matics^ is and no matter how well known it may be to some, to others it is new, and finding the answer, the proof, or disproof brings excitement and satisfaction. The example that follows illustrates this point of view. The following problem had been as signed to the class :","PeriodicalId":144125,"journal":{"name":"Math Teacher","volume":"145 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1970-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116193771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The First International Congress on Mathematics Education. 第一届国际数学教育大会。

Math Teacher Pub Date : 1970-04-01 DOI: 10.5951/MT.63.4.0319

J. Hlavaty

引用次数: 0

A Construction of and Physical Model for Finite Euclidean and Projective Geometries. 有限欧几里德几何与射影几何的构造与物理模型。

Math Teacher Pub Date : 1970-04-01 DOI: 10.5951/MT.63.4.0301

W. A. Miller

引用次数: 1

Two Incorrect Solutions Explored Correctly. 正确探索了两个错误的解决方案。

Math Teacher Pub Date : 1970-03-01 DOI: 10.5951/MT.63.3.0257

Merle C. Allen

引用次数: 2

Definitions--Consensus or Confusion?. 定义——共识还是困惑?

Math Teacher Pub Date : 1970-03-01 DOI: 10.5951/MT.63.3.0223

David Rappaport

{"title":"Definitions--Consensus or Confusion?.","authors":"David Rappaport","doi":"10.5951/MT.63.3.0223","DOIUrl":"https://doi.org/10.5951/MT.63.3.0223","url":null,"abstract":"IN RECENT years, there has been considerable discussion and argument about the \"new\" or \"modern\" mathemat ics. A point on which everyone seems to agree, however, is that one of the charac teristics of the new math is greater preci sion in language, especially with defini tions. But, upon closer examination of the textbooks and manuals in current use, it becomes evident that there is indeed a great lack of precision and uniformity of definition. The many changes in the mathematics curriculum during the last decade were not designed by a single agency, but by a num ber of individuals and organizations who have produced new programs and text books at an increasingly rapid rate. These rapid changes have also produced some unfavorable effects. Many teachers were overwhelmed by the new math?they did not understand it and felt incompetent to teach it. To help them cope with the new programs, great investments have been made in in-service training. But imagine the frustration of one of these teachers when, after using one textbook series with the terms clearly defined, he reads another that defines the terms differently. Whose definitions should he accept? Logicians will say that it does not make any difference how a person defines his terms as long as he is consistent in their use. But elementary teachers and students are not logicians. Since these textbooks and manuals have been written by college teachers and high school teachers, it should be their concern and responsibility to bring about a uniformity of definitions. How many mathematics educators are aware of the variations in definitions? Following is a brief description of the extent to which writers disagree about some of the simple concepts.","PeriodicalId":144125,"journal":{"name":"Math Teacher","volume":"181 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1970-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126745102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

An Old Stumbling Stone Revisited. 重访一块古老的绊脚石。

Math Teacher Pub Date : 1970-03-01 DOI: 10.5951/MT.63.3.0259

R. R. Poole

引用次数: 0

The Geometric Continuum. 几何连续体。

Math Teacher Pub Date : 1900-01-01 DOI: 10.5951/at.17.5.0403

H. Fawcett

引用次数: 4