bluuebunny/crossref_metadata_2025 huggingface dataset:
International Mathematical Education: A course in probability theory for secondary schools
Authors: Råde Lennart
Published: November 2020

Editor's note.—Professor Lennart RÅde is a member of the Scandinavian Committee for the Modernizing of School Mathematics and is responsible for most of the work in probability and statistics produced by this committee. He is a University Lecturer at the Chalmers Technical High School, Göteborg, Sweden. This paper reflects the type of high school course that is now given in the Swedish Gymnas or senior high school. The Scandidinavian countries—through the works of Cramer, Fisher, and others—have made major and important contributions to probability and statistical theory. It is of value to study the approach given in this paper, with current books in the U.S.A. intended for secondary school study.—Howard F. Fehr.
Problem solving in mathematics, II
Authors: Botts Truman
Published: November 2020

Every positive rational number r is a sum of distinct unit fractions
A digital problem for 1965
Authors: Niebaum Jerome
Published: November 2020

Some mathematics just for fun
Points and Viewpoints: Makers of a magazine
Authors: Brune Irvin H.
Published: November 2020

A column of unofficial comment
Implications of an Operation Uniquely Defined with Respect to an Equivalence Relation: The “Freeing” of Vectors
Authors: Brown Stephen I.
Published: November 2020

The method of “postulating” what we wont has many advantages; they are the same as the advantages of theft over honest toil.—Bertrand Russell.
The Problem of the Three Prisoners
Authors: Brown David H.
Published: November 2020

Our school recently received a copy of Oswald Jacoby's little book Mathematics for Pleasure.1 It is a charming little book which will give many hours of pleasure to students and teachers alike, but I respectfully take issue with the distinguished authors on their solution to the problem of the Three Prisoners. For the convenience of readers who may not have a copy of the book at hand, I quote the problem and the given solution in full.
A Proposal for the High School Mathematics Curriculum: Principles of the Modern Mathematics Curricula
Authors: Kline Morris
Published: November 2020

Before I present the major features of a new curriculum for high school mathematics, I should like to state what I understand to be the principles of the reform movement generally known as modern mathematics. My purpose in reviewing these principles is to make more apparent the differences between these curricula and what I shall propose. Though the many modern mathematics curricula differ slightly from one another, I believe it is fair to treat them collectively insofar as the underlying principles are concerned.
A Computer for Every Classroom!
Authors: Guttersen Granville
Published: November 2020

A FREE computer just for the asking? What teacher wouldn't jump at this offer? Yet, it's really true—and if the metals shop in your school can make a couple of handles, you can have one for every mathematics classroom at a cost of just saying “Please”! I believe they could be used in arithmetic classes as well, but we have found that they make excellent teaching aids in our general mathematics classes—particularly since many of the boys may become service-station attendants in the future.
A Pattern for Research in the Mathematics Classroom
Authors: Johnson Donovan A.
Published: November 2020

We have reached a crucial time in mathematics education. Questions are being asked and criticisms made of school mathematics programs. Some of these are difficult to answer. Although several important studies have been made, we have not established criteria; we have not devised measuring instruments; and we have not completed research that would give us valid information about the effectiveness of a mathematics program. However. lack of evidence doesn't mean that a given program or given procedure is not effective. It merely means that we withhold judgment until we have adequate information.
Historically Speaking—: the Remarkable Mental Arithmetic of the Egyptian Scribes, II
Authors: Gillings R. J.
Published: November 2020

Many of the additions shown in the Recto list1 are obvious, and others may be very simply read off from the tables, as indicated in the right-hand column, where groupings are suggested. Many alternative groupings are also possible. However, some are not so obvious, and these we show here.
Steers, Pigs, and Chickens: A Study in Semantics
Authors: Heineman E. Richard
Published: November 2020

One of the objectives of the new PS (post-Sputnik) mathematics is the insistence upon precise phraseology. In dealing with a given situation it is necessary that all conditions and requirements be listed explicitly—not just implied. These restrictions, which may be called “the rules of the game,” must be stated, understood, and agreed upon before action begins.
Networks
Authors: Tammadge Alan
Published: November 2020

An Airline has routes which serve four cities, A, B, C, and D. They are shown in Figure 1(a). The director wishes to present his pattern of routes diagrammatically and chooses to show his routes as straight lines as in Figure 1(b).
The Rule of 72
Authors: Brown Richard G.
Published: November 2020

The “Rule of 72” provides a shortcut for determining the approximate number of years one must wait to double an investment earning interest compounded annually. The rule states that if the interest rate is r percent, then the investment can be doubled in 72/r years. The rule was suggested in class by a student whose father used it in business. I do not know how well known the rule is in business or banking, but it does not seem to be known generally by teachers of mathematics. I havefound students like the rule and find its justification interesting.
A Nontrivial Automorphism of the Field of Real Numbers
Authors: Lawler Robert F.
Published: November 2020

When one discusses the basic principles of a mathematical field, one must be aware that the meaning of some of these principles is hidden by the fact that the examples giYen with addition and multiplication are too obvious. For a student to say that (3 + 4) + 5 = 3 + (4 + 5) because of the associative principle does not necessarily mean that he understands the nature of this principle.
Comprehensive Examinations
Authors: Seymour F. Eugene
Published: November 2020

The present impetus given to the consideration of comprehensive examinations in our schools had its origin in an address by Superintendent Maxwell, of New York City, delivered before the Association of College and Preparatory Schools of the Middle States and Maryland at their 25th annual convention held at Columbia University, New York City, December, 1911. Many liberal interpretations of this particular phase of the address have been made, but I shall define the question and attempt to discuss it, conforming as closely as possible to what seem to be the intentions of the author.
Some Experiences with a Class of Reviewers in Geometry
Authors: Walsh Sara C.
Published: November 2020

There was offered, at the Buffalo Central High School last January, a review geometry class in an attempt to aid, in a particular way, those who had failed in the regents’ examination one or more times. On looking up the records of the applicants to the course, it was discovered that some had taken it two or more times and in one extreme case, the boy had registered ten terms. None of these pupils were mentally deficient, as was shown by exceptional work in other lines. Such remarks as these reached me at the first regular session of the class: “I’ll never get it.” “What’s the good of it anyhow?” “I hate the stuff.” “I’d like to get together all the geometry books in the school and watch them burn.”
Euclidean and Other Geometries
Authors: Meserve Bruce E.
Published: November 2020

Geometry has been my favorite subject for years. My topic, Euclidean and other geometries, has been selected to enable me to consider Euclidean geometry as one of several geometries. Such an approach at the secondary school level makes it possible to illustrate the need for the postulates used for Euclidean geometry. This approach also helps the teacher present any given geometry as a particular mathematical system. Especially when properties of several geometries are considered with reference to various models for the geometries, students recognize that postulates are assumptions rather than “self-evident truths.”
Mathematics for the Below-Average Achiever in High School
Authors: Elder Florence
Published: November 2020

When President Johnson signed the $1.3 billion aid-to-education bill on April 11, 1965, he expressed hope that the bill would “bridge the gap between helplessness and hope for more than 5 million educationally deprived children.”
Mathematics Leagues: Stimulating Interest through Competition
Authors: Mattson Robert J.
Published: November 2020

The post-Sputnik syndrome has put great stress on the value of the study of mathematics and the sciences. This stress has been evident in the development of the new mathematical programs with the hope that students will gain a deeper insight into mathematics, which would promote greater understanding and interest in mathematics, and which would further result in the development of more mathematicians, engineers, physicists, and scientists.
Classroom Inquiry into the Conic Sections
Authors: Coxford Arthur
Published: November 2020

The study of the conic sections in a high school algebra or analytic geometry class offers a unique opportunity for students to engage in productive mathematical inquiry. When accurate graphs of the conic sections are available, simple measurements lead to conjectures that can be proved by the students, using only elementary algebraic skills. However, a typical treatment of the conic sections defines each in terms of the properties, and the possibilities for student discovery are thus reduced.
Fermat's Factoring
Authors: Ransom William R.
Published: November 2020

Fermat's method of factoring is an ideal topic for a mathematics club lecture; the discovery of factors is intriguing, and the mathematical basis is easily understood.
International Mathematical Education: An Example of the Axiomatic Method in Instruction: The Mathematization of a Political Structure
Authors: Steiner Hans-Georg
Published: November 2020

In the following, we wish to show by example how the axiomatic method, content-wise as well as pedagogically, can be brought to the fore in instruction.
Experimental Programs: Innovations and Research in the Teaching of Mathematics to the Terminal Student
Authors: Weiss Sol
Published: November 2020

Just two years ago, about fifty people interested in improving the mathematics education of low achievers met in Washington, D.C. The meeting was sponsored jointly by the U.S. Office of Education and the National Council of Teachers of Mathematics.
Effect of Teaching Concepts of Logic on Verbalization of Discovered Mathematical Generalizations
Authors: Retzer Kenneth A. , Henderson Kenneth B.
Published: November 2020

A method of teaching that is lauded by many mathematics educators is the method of guided discovery. Among the advocators of this method, there are some who believe that asking the student to verbalize what he has discovered as soon as he has made the discovery is ineffective because he does not have the linguistic capacity.1 Moreover, some argue that if the premature verbalization is incorrect, the basic insight may actually be mutilated.2 Others doubt this and question the adequacy of the evidence advanced by the former group to support their contention.3
What is a Trapezoid?
Authors: Dye David L.
Published: November 2020

Until relatively recently, almost all students or former students of geometry would have been in complete agreement when answering the question in the title. They would have said that a trapezoid is a quadrilateral in which two and only two sides are parallel, or words to that effect. However, a present-day geometry student might answer the question by stating that a trapezoid is a quadrilateral with one or more pairs of sides parallel or with at least one pair of sides parallel.
What has Happened to Probability in the High School?
Authors: Mosteller Frederick
Published: November 2020

The teaching of probability and statistics both in school and secondary school mathematics has come a very long way since the Commission on Mathematics first produced the Gray Book.1 Many commercial publishers have attractive texts, and SMSG and others have produced sections on probability and statistics for various grade levels.
Experimental Programs: An Experiment in High School Calculus
Authors: Hubley Marlin F.
Published: November 2020

In 1965 the School Mathematics Study Group (SMSG) released a calculus text. The writing team for this text considered the study of calculus to be in transition from college to the high school. The text was the team's thinking on what the course should be during the transition.
Factor Lattices
Authors: Kenney Margaret J.
Published: November 2020

Factor lattices can be an interesting and instructive diversion for the junior or senior high school student. Divisors, multiples, and the prime factorization of a positive integer are joined in a meaningful presentation. The student is motivated to relate these three topics and is also challenged to contribute his own ideas.
Modular Arithmetic Divisibility Criteria
Authors: Singer Richard
Published: November 2020

There are well-known tests to determine when a positive integer k written in decimal notation is divisible by such numbers as 2, 3, 4, 5, 9, 10, or 11.1 These tests substitute for k some testing expression that can be checked more easily than k itself. For example, the testing expression for divisibility by 2 is the last digit of k, while the testing expression for 9 is the sum of the digits of k. In this paper we shall show how concise proofs of these tests can be obtained using the remainder function from the integers to the integers modulo n, obtain testing expressions for divisibility by other numbers such as 7 and 13, and examine how the integers modulo n can be used to determine divisiility in the absence of testing expressions.
Historically Speaking—: The Early Beginnings of Set Theory
Authors: Johnson Phillip E.
Published: November 2020

Georg Canttor created and largely developed the theory of sets in approximately the year. 1874-1897. In contrast to such developments as the calculus and non-Euclidean geometry, The creation of set theory was, according to all indications, Cantor's alone. AIso, set theory was not preceded by a long evoIutionary period such as is usually the case with big mathematical breakthroughs. The present article will concern itself primarily with the very earliest set-theoretic works of Cantor, namely, his first two papers in this area.1
The NCTM and the Two-Year Colleges
Authors: Meserve Bruce E.
Published: November 2020

The National Council of Teachers of Mathematics is the world's largest professional organization concerned with the teaching of mathematics. The two-year colleges are rapidly becoming the colleges where the largest number of college students are taught mathematics.
On Proofs of the Irrationality of √2
Authors: Harris V. C.
Published: November 2020

In This paper there are given thirteen proofs that √2 is irrational. Indications are given as to whether the method employed extends to proving the irrationality of other square roots or of roots of higher order. In addition, a reference is provided to an incorrect proof recently published and its criticism.
Gauss, Computer-Assisted
Authors: Hughes Helen S.
Published: November 2020

The computer is almost certainly the most valuable aid we have acquired in the past few decades for teaching concepts of maihematics, but it will not make its maximum contribution until we have discovered the best techniques for using it.
SoIving Absolute-Value Problems
Authors: Atherton Ruth C.
Published: November 2020

College professors inform high school teachers that secondary school graduates are deficient in working inequalities and absolute-value problems. Most students who have had two years of algebra remember having been exposed to a few rules dealing with these, but their memory is foggy. In attempting to solve problems they can sometimes get half of the answers, but they have trouble when the numbers in absolute-value symbols are negative. Since this topic appears early in calculus, it is likely to be one of the first deficiencies to show up in college.
A Simpler “7” Divisibility Rule
Authors: Long Calvin T.
Published: November 2020

The choice of title for this article is prompted by the title of an interesting article, by Professor Matthews concerning a so-called scratch method for determining divisibility by the primes 7, 11, and 13.1 If the reader has not read that earlier article, he would probably find it interesting and helpful to do so, since some of the simple divisibility properties of integers discussed there will be used here without further explanation. The purpose of the present discussion is to present a modifi-cation of Professor Matthews's test which seems easier to apply and to show how the test can be extended to primes other than 7, 11, and 13.
A New Look At Heuristic Teaching
Authors: Higgins Jon L.
Published: November 2020

Not Much has been said or written lately about heuristic teaching in mathematics. Perhaps this is because few people can agree on just what heuristic teaching encompasses. Or perhaps we have assumed that heuristic teaching is synonymous with discovery teaching. Things are further complicated by the fact that the term heuristic has a specialized meaning and lengthy history in the field of mathematics. To a mathematician the word heuristic has an infinitely richer meaning than simply discovery.
Evaluating the Cosine Function
Authors: Warren John C.
Published: November 2020

The cosine function is a mapping of real numbers into real numbers. The function is called cos, and the mapping is usually described with reference to a geometric figure that involves a number circle and a number line. (See fig. 1.)
A Digital Bracelet For 1971
Authors: Trigg Charles W.
Published: November 2020

A Bracelet is one period of a simply periodic series considered as a closed sequence with terms equally spaced around a circle. Thus distances between terms may be measured in degrees or in steps. A bracelet may be regenerated by starting at any arbitrary point to apply the generating law. A bracelet may be cut at any arbitrary point for straight-line representation without loss of any properties.
Fractions In S-Adic Form
Authors: Griffin Harriet
Published: November 2020

We Are used to writing integers in base ten and know that in so doing both the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and their relative positions are significant. Thus we recall that 326 = 3(102) + 2(10) + 6, which is called the 10-adic expansion of the integer. We say 326 is its 10-adic, or decimal, form. We have also learned to convert integers written in the base ten to a different base.
Persuasive Arguments: .9999… = 1
Authors: Hall Lucien T.
Published: November 2020

WHEN teaching high school students, prospective teachers, and in-service classes, I have used the usual method for changing the repeating-decimal format of a rational number to an a/b format where a and b are integers and b ≠ 0.
Picture of the Rational Numbers: Dense but Not Complete
Authors: Allison Joe Frank
Published: November 2020

The rational numbers have the propty that for any two distinct rational numbers, say a and b, with a < b, there ists another rational number c, distinct om both a and b and such that a << b. For example, c = (a + b)/2 is such a number.
Student Teachers Ask for Help
Authors: Lipsey Sally Irene
Published: November 2020

In methods courses in the teaching of secondary mathematics at Brooklyn College, we asked the student teachers every week to write out the pedagogical questions that most concerned them. Twenty-five student teachers submitted 170 questions over a period of one school term. Most questions were inspired hy discipline problems, planning for particular lessons, difficulties in relationships between the student teachers and their cooperating teachers, testing, and the low achiever. What the students wrote served as excellent motivation for class discussion and provided data on the stresses and strains of student teaching. The questions were a productive source of material for the instructor to consider in planning the methods course and gave the students a feeling of being involved in that planning.
New Publications
Authors: Dodes Irving Allen , Peak Philip , Price Jack , Schaumberger Norman
Published: November 2020

No reviewer (especially an unpaid reviewer for the Mathematics Teacher) can afford the time to read hooks thoroughly and completely. However, this book was definitely an exception. It is written so cleady and beautifully that it is very difficult to put down. The topics are: groups (including permutation groups, congruence and cosets, and quotient groups), mappings, polynomials, Galois theory, and radicals. The exercises are interspersed with, alas, no answers.-DODES.
Permutation Patterns
Authors: Ranucci Ernest R.
Published: November 2020

The field of permutations deals with the order of things. Order can be interpreted spatially. Thus, the patterns associated with permutations can be related to the geometric world of points, lines, planes, and space.
New Programs
Authors: Hight Donald W. , Kansky Robert J. , Richards Don K.
Published: November 2020

The Profrssional-Year Program. a component of the Training Teacher Trainers Project (or TTT), was initiated during the 1969-70 school year al Indiana University in cooperation with the Monroe County Community School Corporation. Presently the project involves 92 undergraduate students who are completing methods courses and student teaching simultaneously in three schools in Monroe County.
Collapsible Models of the Regular Octahedron
Authors: Trigg Charles W.
Published: November 2020

THE faces of a right regular prism consist. of two regular n-gona (the bases) connected by n rectangles (the lateral faces). Thus there are two rectangles and one n-gon at each vertex. When the rectangles are squares and n = 4, the prism is a cube.
Constructing the Square Root
Authors: Jordan James H.
Published: November 2020

Given a line segment and a unit distance |OI|, the problem is to construct a line segment such that |OP|2 = |OM|. The simplest construction and proof is desirable. The following is one possibility.
Mathematical competencies and skills essential for enlightened citizens
Authors: Edwards E. L. , Nichols Eugene D. , Sharpe Glyn H.
Published: November 2020

Following World War II, the crucial importance of mathematics in the technological and other aspects of life in the complex modern society has been unquestionably recognized. With it the question How much mathematics is a “must” for every citizen? reappeared with new urgency.
Computer Programming for the Seventh Grade
Authors: Allison Ronald
Published: November 2020

During the past few years, high schools have made a commitment to the computer-science age by making available a computer and terminals for student use. The equipment is usually assigned to the mathematics department, and the department is responsible for developing a program for student use of the computer.
Mathematics And Its Relations To Other Disciplines
Authors: Wilder R. L.
Published: November 2020

Not long after agreeing to discuss this topic, I became aware that I had committed myself to a very controversial subject, I found, too, that I could not possibly read, or even locate, the major part of what has been written on it. Educators, mathematicians, engineers, natural scientists, administrators (including college presidents), and just about every class of professionals who use any mathematics at all seem to have expressed their views, Moreover, nonprofessionals, nonusers of mathematics, and especially parent groups have been quite vocal and, as might be expected, the most dogmatic in their pronouncements.
Pythagorean Triples: A New, Easy-to-Derive Formula with Some Geometric Applications
Authors: Rothbart Andrea , Paulsell Bruce
Published: November 2020

Elementary algebra is used to prove a formula that generates all Pythagorean triples.
Why Do We Teach Mathematics?
Authors: Hendrickson Dean
Published: November 2020

This answer to the question raised in the title of the article should be of interest to all teachers of mathematics and to students in grades seven through junior college.
The Limits of Parabolas
Authors: Sconyers James M.
Published: November 2020

What are the limiting cases when the distance between the focus and the directrix of a parabola is varied?
Pythagorean Numbers
Authors: Cohen Israel
Published: November 2020

This article shows a simple method for finding the sets of integers that are the measures of sides of right triangles. It provides a method for the geometry teacher or students to generate right triangles.
First Proofs
Authors: Ehrlich Amos
Published: November 2020

Suggestions for a student's first encounter with a deductive system and his first experiences in constructing proofs.
Clubs: Mu alpha theta in two-year colleges: A good bet
Authors: Boldt Chris
Published: November 2020

Large numbers of two-year college mathematics faculties are familiar with some of the more obvious advantages of having a Mu Alpha Theta (MAO) chapter on campus, such as—
The Mathematics of Buying a Car: A Basic Skills Unit
Authors: Doebling Mary Jo
Published: November 2020

Since few things interest young people as much as automobiles, a unit on buying one creates great enthusiasm. Such a unit has been a repeated success in a consumer mathematics course for eleventh- and twelth-grade students who have taken at least one previous high school mathematics course. Because of the wide variety of abilities in a class, materials should be highly adaptable.
A Primer of Domestic Mathematics
Authors: Rosenthal Dorothy B.
Published: November 2020

A lighthearted view of some timely problems
Probability and Statistics for Grades 9-11
Authors: Richbart Lynn A.
Published: November 2020

What has happened with probability and statistics in recent years? Have these topics been integrated into the regular naathematics program or do they still remain as extras to be taken possibly as a senior elective? Fey (1979, p. 492) compared course enrollments for 1972-73 and 1976-77. He stated that “… the modest growth in probability and statistics enrollments suggests that, despite strong support for inclusion in the curriculum, these topics have not become a major option to either algebra, geometry, or general mathematics.”
An Ancient Problem
Authors: Flusser Peter
Published: November 2020

Some problems have easier solutions than we think.
Mathematics Term Paper!
Authors: Lipman Myra R.
Published: November 2020

“Whoever heard of doing a term paper for a mathematics course?”
Getting Started in a Junior High School: A Case Study
Authors: Robbins Bill , Taylor Ross
Published: November 2020

Ramsey Junior High School has been making instructional use of computers for over a dozen years. Although our efforts at getting started seem like ancient history to us, we believe that our experience over the years may benefit those of you who are getting started today.
Computer Applications
Authors: Inman Don , Clyde Donald
Published: November 2020

The teaching of problem solving has been one of our most difficult tasks. Problem solving is more than just a single skill. The ability to solve problems quickly and efficiently calls on several elusive skills that must be combined in an organized manner. The computer is a tool that strengthens the student's skills in problem solving. In fact, it might be said to force the student to use good problem-solving techniques.
Soundoff: High School Geometry Should Be a Laboratory Course
Authors: Woodward Ernest
Published: November 2020

Present day instruction in geometry is ineffective. Results of the fourth mathematics assessment of the National Assessment of Educational Progress (NAEP) (Brown et al. 1988) indicate that fewer than half the eleventh-grade students who had taken geometry could apply the Pythagorean theorem in a routine problem and that fewer than a third of these students could find the perimeter of a rhombus drawn on grid paper. Eleventh-grade students who had taken geometry performed only slightly better on spatialvisualization tasks than eleventh-grade students who had not taken geometry.
Exploring Fractals a Problem—solving Adventure Using Mathematics and Logo
Authors: Kern Jane F. , Mauk Cherry C.
Published: November 2020

The concept of fractal is relatively new yet is quickly being recognized as a realistic way to model nature. Fractals can be used to generate stimulating problems that can be solved using no more than high school algebra and geometry.
Get Your Asymptote Intercepts Here
Authors: Kmiecik Joan
Published: November 2020

The determination of asymptotes-horizontal, vertical, or oblique-is important in analyzing and sketching the graphs of functions. Although the graph of a function cannot intersect its vertical asymptotes, it is important to note that it might intersect its horizontal or oblique asymptotes. The purpose of this article is to present a simple way to determine the points of intersection if they do exist.
Popular Measures of Central Tendency
Authors: Mogull Robert G.
Published: November 2020

Introductory textbooks in applied statistics typically present a discussion of descriptive statistics as well as of inferential techniques. The presentations include measures of central tendency, such as the arithmetic mean, the median, and the mode, with definitions and examples given for each location parameter.
Using Problems to Implement the NCTM's “Professional Teaching Standards'
Authors: Showalter Millard E.
Published: November 2020

As set forth in the Professional Standards for Teaching Mathematics (NCTM 1991), a primary goal for teaching and learning mathematics is the development of mathematical power for all students. To accomplish this goal, the teaching standards document recommends that teachers select interesting and intellectually stimulating mathematical tasks, present opportunities for students to deepen their understanding of mathematics and its applications, promote the investigation of mathematical ideas, use technology to pursue these investigations, find connections to previous and developing knowledge, and employ cooperativelearning experiences (NCTM 1991, 1).
Deconstructing Constructivism
Authors: Gadanidis George
Published: November 2020

Mathematics education suffers from a condition that resembles schizophrenia. One of its personalities is exhibited in the day-to-day realities of classroom learning; another is evident in journal articles, in-service presentations, and other such forums where educators present alternative realities of learning. For the purposes of this article, these personalities will be labeled, respectively, as the practice and theory of mathematics education. This article focuses on the latest form of the theoretical personality of mathematics education, constructivism, by asking what is constructivist learning theory and what does it imply for the practice of learning mathematics?
Exploratory Geometry—Let the Students Write the Text!
Authors: Stallings-Roberts Virginia
Published: November 2020

The NCTM's Curriculum and Evaluation Standards calls for a decreased attention to Euclidean geometry as a complete axiomatic system (NCTM 1989). To clarify this charge, the following was stated in the recommendations for mathematics in grades 9-12 (p. 159):
Connecting Research to Teaching: Research on Graphing Calculators
Authors: Dunham Penelope H. , Dick Thomas P.
Published: November 2020

The National Council of Teachers of Mathematics has long advocated the use of calculators at all levels of mathematics instruction, and graphing calculators are no exception.
Reflections on Miniature Golf
Authors: Powell Nancy Nore , Anderson Mark , Winterroth Stanley
Published: November 2020

To make geometry meaningful, fun, and applicable to the lives of our students, we adapted the game of miniature golf as a way to teach transformatiOnal geometry. Geometry textbooks often feature twodimensional pictures of miniature-golf holes to illustrate transformational geometry. Geometry students in this project actually brought the golf holes out of the textbook by constructing a life-sized miniature golf course.
Helping Students Write Paragraph Proofs in Geometry
Authors: Brandell Joseph L.
Published: November 2020

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) suggests that students should learn to express deductive arguments orally and in sentence form. Writing proofs in paragraph form—hereafter known as paragraph proof—accomplishes this objective.
Implementing the “Assessment Standards for School Mathematics”: Valuing What We See
Authors: Clarke Doug , Wilson Linda
Published: November 2020

As teachers explore alternative forms of assessment for the classroom, interest increases in all aspects of observational assessment-what to look for, how to look for it, how to document it, and how to use it. This article offers some guidance in each of these areas and some hints from the experiences of teachers who have experimented with observational assessment.
Sherlock Holmes, Master Problem Solver
Authors: Ballew Hunter
Published: November 2020

In the Curriculum and Eualuation Standards for School Mathematics (1989), the National Council of Teachers of Mathematics emphasizes problem solving at every grade level.
Connecting Research to Teaching: Classroom Research and Classroom Practice: Blurring the Boundaries
Authors: Tinto Patricia P. , Shelly Barbara A. , Zarach Nancy J.
Published: November 2020

Traditionally, academic research on teaching has been carried out by university faculty independent of both teachers and students. Although the audience of journal readers may gain from such research, the subjects of the studies—the teachers and students—often do not. Proponents of research by teachers (Cochran-Smith and Lytle 1993; Duckworth 1986; Houser 1990; Lerman 1993) argue that for substantial progress in improving classrooms to occur, teachers themselves must become researchers and take charge of the research process in their classrooms.
Making Connections: Journal Writing and the Professional Teaching Standards
Authors: Stewart Carolyn , Chance Lucindia
Published: November 2020

Can the national focus on the NCTM's Standards be combined with the focus on writing to learn to streng then mathematics instruction? The NCTM's Professional Standards for Teaching Mathematics (1991) supports the use of writing as an instructional technique. The possibilities suggested for journal writing in the mathematics classroom challenged these authors to explore the available literature. Several studies appeared promising (Linn 1989; Miller and England 1989; Powell and Lopez 1989; Richards 1990; Rose 1989; Skiba 1990; Vukovich 1985; White and Dunn 1989). Where as most of these projects indicated that journal writing increased mathematical ability and decreased anxiety, too few studies actually delineated the content of the journal entries to any great extent. Moreover, many of the investigations did not consider the context within which students and teachers operate. The relevance and interaction of teacher, student, and subject matter were often overlooked. This situation further challenged the authors to extend and broaden the investigation of journal writing into a more global study-one of practical inquiry, not only exploring the cognitive and affective influences of journal writing but seeking situational insight into, and understanding of, the mathematics classroom.
How Rough Is Your State?
Authors: Bosch William W. , Devine Karen M. , Petherick John C. , Wellman Eldon C.
Published: November 2020

Do you think the land mass of your state is rough? Is West Virginia rough? Colorado? Is Colorado rougher than West Virginia? When these questions were posed to a university class in mathematical modeling for preservice teachers, a lively discussion ensued. Various definitions for roughness or rugged-ness were advanced and debated. Most of these definitions or measures involved some aspect of altitude change, such as maximum altitude differences within the state; frequency of altitude change; or rate of changes in altitude expressed as steepness, or slope.
The Return of Matt Dillon
Authors: Paul Clyde
Published: November 2020

Recently CBS resurrected Matt Dillon from Gun-smoke fame in a two-hour television special appropriately named "The Return of Matt Dil-lon.” Yielding to the attraction of nos-talgic memories of Miss Kitty, Chester, and Dodge City, many of us watched the show. Although an appealing opportunity exists, I will leave the job of television critic to others.
Connecting Research to Teaching: Making the Transition: Tensions in Becoming a (Better) Mathematics Teacher
Authors: Jones Doug
Published: November 2020

Becoming a better mathematics teacher, particularly during a time of reform, is not an easy task. A variety of information must be learned—a wide range of mathematical content, pedagogical theory and techniques, behaviors of a teacher—and even more must be balanced—administrative and instructional tasks, teaching for procedural and conceptual understanding, and making time for school and family. Although it is impossible to make a definitive statement regarding what is necessary and sufficient for teachers to learn or how to strike the right balance, it is important to view development as an ongoing process throughout a teacher’s
Sharing Teaching Ideas: A Simple Way to Teach Logarithms
Authors: Hammack Richard , Lyons David
Published: November 2020

We have taught logarithms in high school and college algebra and in introductory calculus courses. Before we began using the method of teaching logs described in this article, we found that many students had difficulties mastering the concept, more so than with other functions. Other teachers reported similar experiences.
Changes in Teaching Practices: The Effects of the Curriculum and Evaluation Standards
Authors: Garet Michael S. , Mills Virginia L.
Published: November 2020

Since the publication of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989), the NCTM, state affiliates, mathematics teachers, textbook authors, the mathematics education research community, and many others have engaged in sustained efforts to change practice in the teaching of mathematics. The Standards document has attracted much attention as a national model of reform distinguished by the major leadership role played by the teaching profession. However, the proposed changes are a substantial departure from conventional practice, and considerable evidence suggests that reforms of the type advocated by the curriculum and evaluation standards are difficult to translate into action and hard to sustain (Cohen, McLaughlin, and Talbert 1993; Cuban 1992). Furthermore, the reforms advocated are intended to apply across a wide variety of contexts, including schools with many resources and schools with few resources, urban schools and rmal schools, and large schools and small schools.
Implementing the Assessment Standards for School Mathematics: Enhancing Mathematics Learning with Open-Ended Questions
Authors: Hancock C. Lynn
Published: November 2020

This composite vignette about Louise Mason is based on a series of classroom observations of five secondary mathematics teachers working to bring some new methods, including the use of open-ended questions, into their assessment practice (Hancock 1994). Although the teachers recognized that opening up the possibilities for responses to questions “really makes the students think,” they encountered difficulties when bringing such questions into their daily practice. As expressed by Ms. Mason, a common concern among the teachers was that using open-ended questions increased the amount of time required for assessment. My consideration of the teachers' dilemma led to some thoughts about the relationship between learning and new forms of assessment that may be helpful to teachers dealing with similar difficulties.
Is This a Mathematics Class?
Authors: Alper Lynne , Fendel Dan , Fraser Sherry , Resek Diane
Published: November 2020

Let us tiptoe down the hall and peek into a “classroom of the future” that is here today.
A Technology-Intensive Approach to Algebra
Authors: Heid M. Katheleen , Zbiek Rose Mary
Published: November 2020

Computer-lntensive Algebra (CIA) focuses on the use of technology to help develop a rich understanding of fundamental algebraic concepts in real-world settings. CIA students learn about functions, variables, systems, equivalent expressions, and equivalent equations in the context of developing, critiquing, and exploring mathematical models of real-life situations. Students use computing tools in their mathematical explorations for easy access to numerical, graphical, and symbolic representations of mathematical ideas. CIA is a fundamentally reformulated beginning-algebra curriculum whose concepts and techniques differ essentially from those of traditional algebra courses. CIA focuses on mathematical modeling instead of classic word problems; it centers on algebraic concepts, such as function and variable, instead of on pencil-and paper algebraic routines; it replaces symbolic manipulation with symbolic reasoning; and it encourages using multiple representations and strategies in algebraic problem solving instead of mastering and using one representation or strategy at a time.
Sharing Teaching Ideas: Bringing Pythagoras to Life
Authors: Ericksen Donna , Stasiuk John , Frank Martha
Published: November 2020

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) states that “[o]ne of the most important properties in geometry, the Pythagorean theorem, is introduced in the middle grades” (p. 113). Although the Standards document assigns much prominence to the Pythagorean theorem, our experience teaching at the university level has revealed that students know the theorem by name and can recite a2 + b2 = c2 but that they often cannot handle even simple computations using the formula. Students' experience with the Pythagorean theorem in high school needs to be broadened by their continually using the standard formula as well as applying the formula to geometric figures and special right triangles-in particular, the 30°-60°-90° and the 45°-45°-90° right triangles. The following game was developed to afford high school students more opportunity for practicing the formula in an engaging way. This game was created by the second author of the article, a high school classroom teacher, while he was a student in a class taught by another of the authors.
Connecting Research to Teaching: Mathematics Performance Assessment: A New Game for Students
Authors: Shannon Ann , Zawojewski Judith S.
Published: November 2020

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) encourages a much broader view of mathematics education than that which is associated with traditional curriculum.
The Incredible Three-by-Five Card!
Authors: Lufkin Dan
Published: November 2020

I have used one 3″ × 5″ card so much that it measures 2″ × 4″! Much mathematics is embedded in this well-known card. First, I show my students a 3″ × 5″ card. Then I show them a card of another commonly used size, a 5″ × 8″ card. Finally, I mention the standard legal pad, with dimension of 8 1/2″ × 14″. These shapes lead to a discussion of pleasing-looking rectangles.
Using Clock Arithmetic to Send Secret Messages
Authors: Gorini Catherine A.
Published: November 2020

Computers and electronic communications have influenced all aspects of our lives, from supermarket checkout to understanding the structure of DNA and the origins of the universe. These applications require the transmission, storage, and retrieval of large quantities of data. Two major concerns when working with these data are the transmission accuracy and security of the data, which are the subjects of coding theory and cryptology, respectively. Both use mathematical techniques extensively. This article describes a classroom activity in which students explore a real-life application of algebra by sending secret messages to each other. By so doing, students see the necessity of the laws of exponents for making calculator or computer calculations feasible. It is an excellent way for students to review the laws of exponents and to learn new ways to apply them.
Class Activities with Student-Generated Data
Authors: Somers Kay , Dilendik John , Smolansky Bettie
Published: November 2020

College-level instructors in mathematics, the sciences, and the social sciences—or in any discipline involving the analysis of quantitative information—are well aware of the devastating effects of “symbol shock” and “mathematics anxiety” on otherwise successful undergraduates. The simplest algorithms seem hopelessly baffling to some students, and the presentation of formulas is met with emotions that range from resistance to outright panic. One effective way of helping students overcome this anxiety is to involve them in concrete exercises in which they collect and organize data and draw inferences from the data. These exercises address the Curriculum and Evaluation Standards for School Mathematics of the National Council of Teachers of Mathematics (1989), which calls for students to be able to collect, organize, and describe data and to be able to draw inferences from real-world data. The exercises also involve an active learning approach as advocated in the Standards document.
A Visual Approach to Solving Mixture Problems
Authors: Bennett Albert B. , Maier Eugene
Published: November 2020

In the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989), the 9–12 standards call for a shift from a curriculum dominated by memorization of isolated facts and procedures to one that emphasizes conceptual understandings, multiple representations and connections, mathematical modeling, and mathematical problem solving. One approach that affords opportunities for achieving these objectives is the use of diagrams and drawings. The familiar saying “A picture is worth a thousand words” could well be modified for mathematics to “A picture is worth a thousand numbers.” As an example of visual approaches in algebra, this article uses diagrams to solve mixture problems.
Sharing Teaching Ideas: Understanding the Composites
Authors: Hurwitz Marsha
Published: November 2020

In his article “An Attractive View of Composite Functions,” Hansen (1993) uses an infinite composition of cos (x) to help students gain an intuitive idea of a limit. Fundamental to the students' understanding of the convergence of the composition is the awareness that an output value becomes an input value for the subsequent function evaluation. This insight eludes some students as they compose functions.
Technology and Reasoning in Algebra and Geometry
Authors: Hirschhorn Daniel B. , Thompson Denisse R.
Published: November 2020

If one topic is likely to be stressed by algebra and geometry teachers, it is reasoning. In algebra classes, students are constantly being asked to show their work and justify their simplifications, often without formal connection to proof concepts or the proof process. In geometry classes, students are expected to learn how to write simple proofs. However, evidence shows that students are not learning these reasoning skills. In the 1985–86 National Assessment of Educational Progress, Silver and Carpenter (1989, 18) found that “many eleventhgrade students are confused about the fundamental distinctions among mathematical demonstrations, assumptions, and proofs.” Most students thought a theorem was a demonstration or an assumption. Senk (1985) found that only about 30 percent of students mastered proof wTiting in geometry, despite being enrolled in a year-long course emphasizing proof. Thompson (1992) found that roughly 60 percent of precalculus students were successful at trigonometric-identity proofs, more than 30 percent could complete number-theory proofs dealing with divisibility, and less than 20 percent could handle indirect arguments or proof by mathematical induction.