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Topic: Mathematical Concepts

NLM copy, advertisement for Fuller's computing telegraph affixed to front pastedown; directions for using the scale affixed to back pastedown

Topics: Computers, Mathematical Concepts

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Oct 26, 2019
10/19

by
Cadwell, J. H. (James Henry)

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xii, 180 pages 24 cm

Topics: Mathematical Concepts, Mathematical recreations, Unterhaltungsmathematik, Matematica

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Oct 29, 2020
10/20

by
Cadwell, J. H. (James Henry)

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xii, 180 pages 24 cm

Topics: Mathematical recreations, Mathematical Concepts, Unterhaltungsmathematik, Matematica

[10] leaves (last leaf blank) ; 20 cm

Topics: Astronomical Phenomena, Moon, Solar System, Mathematical Concepts

Calculus is a mathematical concept that is fundamental to how we understand the world around us. Whether it is in the world of technology, finance, astronomy, sociology, medicine, calculus in one form or another can be found. This brief article describes the origins of calculus in Greece, further developments by Newton and Leibniz, and the fundamentals of integration and differentiation.

Topics: ERIC Archive, Calculus, Mathematical Concepts, Harding, Simon|Scott, Paul

This article traces the history of the number [Pi] from 3000 BC (the construction of the Egyptian pyramids) to 2005 (the calculation of the first 200 million digits of Pi).

Topics: ERIC Archive, Mathematical Concepts, Mathematics, History, Computation, Measurement, Scott, Paul

The multivariate analog of Hays omega-squared for estimating the strength of the relationship in the multivariate analysis of variance has been proposed in this paper. The multivariate omega-squared is obtained through the use of Wilks' lambda test criterion. Application of multivariate omega-squared to a numerical example has been provided so as to help understand the mechanics of the formulas necessary for the computation of multivariate omega-squared. (Author/DB)

Topics: ERIC Archive, Analysis of Variance, Mathematical Concepts, Statistical Analysis, Sachdeva, Darshan

We present a mathematical analysis that distinguishes two quantitative perspectives on ratios and proportional relationships: Multiple Batches and Variable Parts. We argue that (a) existing research on proportional relationships has addressed Multiple Batches but has largely overlooked Variable Parts, (b) Multiple Batches makes the co-variation aspect of proportional relationships more explicit, while Variable Parts makes the fixed multiplicative relationship between two quantities more...

Topics: ERIC Archive, ERIC, Beckmann, Sybilla Izsák, Andrew, Mathematical Concepts, Mathematics, Algebra

Multiplication and division have in general been much more difficult to perform than addition and subtraction. Perhaps, if we could find some device for reducing multiplication and division to addition and subtraction, computational loads could be lightened. One such device is that of logarithms of course. This note outlines another such device with a brief outline of its history; that being the use of trigonometric tables to effect multiplication and division. The technique is called...

Topics: ERIC Archive, Trigonometry, Mathematical Concepts, Arithmetic, Multiplication, Mathematics,...

The principal objects of the investigation reported were, first, to study qualitative probability relations on Boolean algebras, and secondly, to describe applications in the theories of probability logic, information, automata, and probabilistic measurement. The main contribution of this work is stated in 10 definitions and 20 theorems. The basic concern in this technical report was to show that probability, entropy, and information measures can be studied successfully in the spirt of...

Topics: ERIC Archive, Logic, Mathematical Applications, Mathematical Concepts, Mathematics, Probability,...

This article presents a problem set which includes a selection of probability problems. Probability theory started essentially as an empirical science and developed on the mathematical side later. The problems featured in this article demonstrate diversity of ideas and different concepts of probability, in particular, they refer to Laplace and Bernoulli models as well as to geometric probability.

Topics: ERIC Archive, Problem Sets, Probability, Mathematics Instruction, Mathematical Concepts,...

A "convex" polygon is one with no re-entrant angles. Alternatively one can use the standard convexity definition, asserting that for any two points of the convex polygon, the line segment joining them is contained completely within the polygon. In this article, the author provides a solution to a problem involving convex lattice polygons.

Topics: ERIC Archive, Plane Geometry, Geometric Concepts, Mathematical Concepts, Equations (Mathematics),...

In "Just Perfect: Part 1," the author defined a perfect number N to be one for which the sum of the divisors d (1 less than or equal to d less than N) is N. He gave the first few perfect numbers, starting with those known by the early Greeks. In this article, the author provides an extended list of perfect numbers, with some comments about their discovery. He also briefly discusses Euclid's proof and Mersenne primes. [For Part 1, see EJ769967.]

Topics: ERIC Archive, Mathematical Concepts, Numbers, Validity, Mathematical Logic, Number Concepts, Scott,...

Archimedes, the famous Greek mathematician, lived from 287 BCE until approximately 212 BCE. He thought that the figure of two semi-circles on a straight line enclosed by a larger semi-circle resembled a shoemaker's knife. Archimedes called this figure an "arbelos" since arbelos is the Greek word for a shoemaker's knife. The author describes the properties of the arbelos and how the arbelos continues to fascinate mathematicians.

Topics: ERIC Archive, Mathematics Instruction, Mathematical Concepts, Graphing Calculators, Teaching...

Using the concept of exploded and compressed numbers the author constructs the supercone which is able to turn upon the border of three dimensional space and breaks through it. The introduction of super-cone gives a possibility for students to see the properties of traditional cone while the super-cone is not a traditional cone. Moreover we show that an unbounded super-cone is a proper subset of an unbounded super-paraboloid such that they have the same infinitely large highness.

Topics: ERIC Archive, Mathematical Concepts, Geometric Concepts, Equations (Mathematics), Measurement,...

The major results of this dissertation are theorems to the effect that certain classes of relational structures are not axiomatizable by universal sentences. Some of the particular classes considered are theories of measurement in the sense of the Scott-Suppes definition while others are theories of measurement according to a natural generalization of the above definition. Part of the significance of the results is that they are closely related to problems of proving representation theorems in...

Topics: ERIC Archive, Mathematical Applications, Mathematical Concepts, Mathematical Models, Measurement,...

This paper presents a new theoretical viewpoint blended from the perspectives that mathematical meaning is extracted (from objects falling under a particular concept) and that mathematical meaning is given (to objects that an individual interacts with). It is elaborated that neither uni-directional framing (whether involving extracting meaning or giving meaning) provides a comprehensive account of the complex emergence of evolving forms of meaning. It is argued for a framing that construes...

Topics: ERIC Archive, ERIC, Scheiner, Thorsten, Comprehension, Mathematical Concepts, Teaching Methods,...

Before primitive man had grasped the concept of number, the written word or even speech, he was able to count. This was important for keeping track of food supplies, sending messages, trading between villages and even keeping track of how many animals were in their herd. Counting was done in various ways, but in all cases, the underlying principle was one-to-one correspondence. This correspondence between the objects being counted and their counting aid, enabled primitive man to make the first...

Topics: ERIC Archive, Mathematical Concepts, Computation, Numbers, Manipulative Materials, Numeracy, Box,...

In this article, John Gough describes the "haiku" and its link to mathematics. A haiku is a short Japanese form of poetry, of three lines, with five syllables in the first, seven in the second, and five in the last. Although brief, a haiku is like a meditation on, or observation of, an experience, conveyed directly through objective images or sensory feelings with no personal judgment or analysis. A haiku is like a captivating photo of something in nature. Gough uses this article to...

Topics: ERIC Archive, Mathematics Instruction, Poetry, Teaching Methods, Interdisciplinary Approach,...

The problem considered in this paper demonstrates that quite profound and inherently fascinating mathematics is accessible to students who have a sound number sense and deep conceptual understanding of very basic mathematics. This is one of many reasons why we should teach mathematics in ways that promote these attributes in students.

Topics: ERIC Archive, Numbers, Mathematical Concepts, Mathematics Skills, Mathematics Instruction,...

Vygotsky posed a variety of meaningful ideas for education in his short life. This paper focuses on everyday concepts and mathematical concepts or scientific concepts from his theory, reorganizing these ideas according to a new idea of sublated concepts. Using a series of interviews from a third grade fraction class in Japan, the paper discusses how everyday and mathematical concepts arise out of discussions among children and a teacher, and how they develop into sublated concepts. [For...

Topics: ERIC Archive, Foreign Countries, Grade 3, Scientific Concepts, Mathematical Concepts, Interviews,...

The aim of the study (Johnsen Hoines 2002) reported here was to shed light on the process through which student teachers express and interpret their understandings about mathematical notions and thereby gain insight into them. This paper focuses on how students cooperate and move between different ways of understandings. It emphases in particular on how differences in the ways of explaining a mathematical content make different conditions for the dialogue and learning processes that develop....

Topics: ERIC Archive, Learning Processes, Student Teachers, Mathematical Concepts, Teaching Methods,...

Probability is more increasingly recognized as a priority in elementary school. In this article, the author argues that if students have a strong understanding of experimental probability, notions of theoretical probability will be easier for the student to understand. In particular, students need extensive experience with experiments using objects and simulators, allowing student understanding to be grounded in the concrete, and then progressively built into an abstract principle.

Topics: ERIC Archive, Probability, Elementary Education, Mathematics Education, Elementary School...

This is part of a student text which was written with the aim of reflecting the thinking of The Cambridge Conference on School Mathematics (CCSM) regarding the goals and objectives for mathematics. The instructional materials were developed for teaching geometry in the secondary schools. This document is chapter six and titled Motions and Transformations. Presented is the concept of rigid motion in the plane. Various kinds of rigid motions are considered, certain mathematical ideas about rigid...

Topics: ERIC Archive, Geometric Concepts, Geometry, Instructional Materials, Mathematical Concepts,...

The number Pi (approximately 3.14159) is defined to be the ratio C/d of the circumference (C) to the diameter (d) of any given circle. In particular, Pi measures the circumference of a circle of diameter d = 1. Historically, the Greek mathematician Archimedes found good approximations for Pi by inscribing and circumscribing many-sided polygons about this circle, and calculating their perimeters. Since Pi stands for an infinite decimal, for practical purposes it is useful to find fractions which...

Topics: ERIC Archive, Arithmetic, Numbers, Mathematics Instruction, Equations (Mathematics), Mathematical...

Math can sometimes seem like a strange language from foreign land--one communicated in symbols, numbers, and geometric figures. When teachers talk about mathematical concepts, even familiar, garden variety words such as "parallel," "power," "even," "odd," "multiply," "difference," "product," "positive," and "negative" take on brand-new meanings. What's the best way for teachers to help students master this...

Topics: ERIC Archive, Geometric Concepts, Vocabulary Development, Teaching Methods, Mathematics...

Understanding the equivalence in relation to fractions is centrally important. Students need many experiences of flexibly combining simple fractions in ways that reinforce equivalences among them. This article presents several ideas that are contributed by a range of people. They are designed to help students to develop understandings of equivalence among fractions and of fractions as division. (Contains 7 figures.)

Topics: ERIC Archive, Mathematics Instruction, Mathematical Concepts, Arithmetic, Elementary School...

This is a reprint of the historical capsules dealing with algebra from the 31st Yearbook of NCTM,"Historical Topics for the Mathematics Classroom." Included are such themes as the change from a geometric to an algebraic solution of problems, the development of algebraic symbolism, the algebraic contributions of different countries, the origin and development of topics in algebra, and the search for generality and abstract structures. (Author/JG)

Topics: ERIC Archive, Algebra, Enrichment, Instructional Materials, Mathematical Concepts, Mathematics,...

This article presents situations involving perimeter, area, volume and mass, and the misconceptions often encountered with these measurements. The author suggests possible interventions that teachers can use to correct these misconceptions and help students to better understand these properties.

Topics: ERIC Archive, Misconceptions, Mathematics Instruction, Mathematical Concepts, Algebra, Measurement...

The beta coefficient of an intermediate variable in a causal direction remains relatively constant as other system variables are introduced and controlled in stepped regression, whereas that in the acausal direction changes noticeably. Normalized random numbers (200x5) were generated and substituted in interdependent equations to produce five scores for each of 200 pseudo-people. Stepped-regression analysis was then applied on all possible three-variable paths. Beta differentials on a given...

Topics: ERIC Archive, Experiments, Logic, Mathematical Concepts, Mathematical Models, Mathematics,...

A synthesis of reasons for the production of this monograph is presented with a focus on contemporary research in the context of the Ninth Congress of the European Society for Research in Mathematics Education. Within the domain of mathematics and language, three lines of concern are addressed: (1) classroom discourse, (2) language diversity, and (3) conceptualization through language. Each line of concern is respectively illustrated by pioneering results from Ruthven and Hofmann, Barwell, and...

Topics: ERIC Archive, Mathematics Education, Language Usage, Classroom Communication, Teaching Methods,...

Tennis is a sport in which the mathematics involves an unusual scoring system together with other applications pertinent to the draw for different types of tournaments and the relative ratios of points won and lost. The name of the sport is thought to have originated from the French word "tenez", which translates roughly as "to receive (the ball)". However, there is an alternative possibility connected with the Egyptian city, Tennis, which was noted in the Middle Ages for...

Topics: ERIC Archive, Mathematics Activities, Racquet Sports, Scoring, Mathematics Instruction,...

This study examined the effect of language on acquiring the concept of place value. The sample for the study consisted of 211 Arabic-speaking children aged between 6.6 years and 7.8 years. Children were interviewed individually and asked to represent written two-digit numbers using base-10 blocks. A new approach for testing the linguistic relativity hypothesis, explicit grouping language approach, was used. Following a screening task, participants were randomly divided into four groups: (1)...

Topics: ERIC Archive, Elementary Education, Language, Mathematical Concepts, Mathematics Instruction, Place...

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Topics: Algorithms, Algorithms -- History, Computer algorithms, Mathematical Concepts -- history, Computing...

The stabilities of standardized (ß) and structure (rs) coefficients in canonical (CA) and discriminant analyses (DA) were studied. Four different situations were studied--two pertaining to CA and two to DA. The situations were meant to represent "somewhat typical" and yet varying research conditions that often would not be thought to be notably objectionable among informed users of CA and DA. Data were sampled from a real population. For each of three situations, 100 random samplings...

Topics: ERIC Archive, Discriminant Analysis, Multivariate Analysis, Reliability, Mathematical Concepts,...

This paper will examine the use of software environments and dynamic visualization in mathematics education. This examination will be based on theoretical papers and research reports from substantial literature on visualization. How the term "visualization" is used in mathematics education field is also discussed. The thesis of this essay is that visualization and visual reasoning play vital roles in mathematical thinking. Therefore, the software environments could be integrated in...

Topics: ERIC Archive, Visualization, Computer Software, Mathematics Education, Definitions, Mathematical...

The paper explores and clarifies the similarities and differences that exist between proof by contradiction and proof by contraposition. The paper also focuses on the concept of contradiction, and a general model for this method of proof is offered. The introduction of mathematical proof in the classroom remains a formidable challenge to students given that, at this stage of their schooling, they are used to manipulating symbols through sequential steps. There is a consensus that learners do...

Topics: ERIC Archive, Validity, Mathematical Logic, Mathematics Instruction, Mathematical Concepts, Models,...

Presented is an elementary approach to areas, columns and other mathematical concepts usually treated in calculus. The approach is based on the idea of average and this concept is utilized throughout the report. In the beginning the average (arithmetic mean) of a set of numbers is considered and two properties of the average which often simplify the arithmetic is noted. Averages are further used to solve a number of important practical problems - to find the work done in stretching a spring,...

Topics: ERIC Archive, Calculus, Elementary School Mathematics, Geometry, Instructional Materials,...

This paper describes a model of student attainment of mathematical concepts and its development. In this model three types of activities (developmental, connecting, and abstract) are considered as an overlay on the three ways of representing mathematical concepts (physical/visual, oral, and symbolic). Each activity type involves some means of representing mathematical concepts, and the activities are sequentially presented as the learner attempts to master a mathematical idea. Initial exposure...

Topics: ERIC Archive, Concept Formation, Elementary Secondary Education, Mathematical Concepts, Mathematics...

These days, multiplying two numbers together is a breeze. One just enters the two numbers into one's calculator, press a button, and there is the answer! It never used to be this easy. Generations of students struggled with tables of logarithms, and thought it was a miracle when the slide rule first appeared. In this article, the author discusses how to multiply by adding and offers ways on how to make a simple slide rule.

Topics: ERIC Archive, Arithmetic, Graphs, Calculus, Mathematics Instruction, Multiplication,...

In this paper, I describe how undergraduates can develop their understanding of the concept of proof by viewing the act of proving as a procedure. Such undergraduates first understand proof as an algorithm, or a step-by-step mechanical prescription for proving certain types of statements. The students can then condense this algorithm into a processor, a shorter list of global, qualitative steps. By reflecting on the process, successful students can view proof as an argument, something that...

Topics: ERIC Archive, Mathematics Instruction, Mathematical Logic, Validity, Undergraduate Students,...

Mathematics teachers are frequently looking for real-life applications and meaningful integration of mathematics and other content areas. Many genuinely seek to reach out to students and help them make connections between the often abstract topics taught in school. In this article the author presents ideas on integrating literature and mathematics to illustrate how easily mathematics can be found in books that initially do not come across as containing mathematics content. Among the examples...

Topics: ERIC Archive, Mathematics Instruction, Teaching Methods, Literature, Interdisciplinary Approach,...

In this paper seventh-grade pupils' ways of handling aspects of probability have been investigated. The aspects in question were embedded in a dice game, based on the total of two dice. Four different set-ups of dice were included in the situation in which they were up to explore optimal strategies for winning the game. How children understand concepts is regarded from the perspective of how the pupils' understanding varies with their interpretation of the situation, in which the concepts are...

Topics: ERIC Archive, Probability, Middle School Students, Mathematics Activities, Mathematical Concepts,...

The objectives of the Los Angeles Model Mathematics Project (LAMMP) are stated by the administration as improvement of mathematical skills and understanding of mathematical concepts; improvement of the pupils' self-image; identification of specific assets and limitations relating to the learning process; development and use of special instructional materials and programs and assessment of their values; selection and use of appropriate commercially developed equipment, instructional materials,...

Topics: ERIC Archive, Evaluation, Evaluation Methods, Instructional Materials, Mathematical Concepts,...

This paper discusses the existence of theoretical distributions subsumed within mathematical reality and it highlights several meaningful implications for psychological researchers to consider in the application of statistical procedures. It begins with a brief overview of the misinterpretation of statistical testing, followed by a discussion of mathematical reality. In addition, the paper presents ideas for and against theories of mathematical reality developed by noted scholars, and it...

Topics: ERIC Archive, Mathematical Concepts, Psychological Studies, Researchers, Statistical Analysis,...

The bilingual dictionary has been adapted from a dictionary originally developed by Teresa Kow and Euphine Cheung in New York to assist Chinese students in their understanding of mathematics vocabulary and concepts in English. A total of 204 terms and definitions are provided in English with Chinese translations directly below the entries. Each entry has diagram/number representations. Students may use this publication as a dictionary, workbook, and supplement to their English language...

Topics: ERIC Archive, Chinese, Dictionaries, Elementary Secondary Education, Mathematical Concepts,...

These materials were written with the aim of reflecting the thinking of The Cambridge Conference on School Mathematics (CCSM) regarding the goals and objectives for school mathematics. Presented are plans for teaching 15 inequality lessons for above average first grade students. The discovery approach is utilized by the teacher in order to involve students in the classroom discussions. Ten wooden sticks of varying lengths and thicknesses were available for these inequality lessons and they were...

Topics: ERIC Archive, Arithmetic, Curriculum Development, Elementary School Mathematics, Grade 1,...

In this article, the author responds to the paper "Exploring pre-service teachers' understanding of statistical variation: Implications for teaching and research" by Sashi Sharma (see EJ779107). In that paper, Sharma described a study "designed to investigate pre-service teachers' acknowledgment of variation in sampling and distribution environments." This author raises issues about one of the two questions presented in Sharma's survey of pre-service teacher education...

Topics: ERIC Archive, Preservice Teacher Education, Preservice Teachers, Statistical Analysis, Probability,...

Mathematics instruction in the United States has historically focused on procedures, facts, and algorithms. Because of that focus, mathematics instruction, in essence, becomes lessons in arithmetic and efficiency. While there are promising changes occurring in mathematics instruction, we still need to help both teachers and students develop a more conceptual understanding of mathematics. This issue of "SEDL Insights" explains how educators can shift from helping students memorize...

Topics: ERIC Archive, ERIC, Molina, Concepción, Mathematics Instruction, Teaching Methods, Mathematical...