Mathematics is an "artificial" deliberately constructed language, supported crucially by: (1) special alpha-numeric characters and usages; (2) extra-special non-alphanumeric symbols; (3) special written formats within a single line, such as superscripts and subscripts; (4) grouping along a line, including bracketing using round brackets, parentheses, and braces; and (5) the clever use of two or more lines at a time (as in fraction notation), and the set-theoretic and logical...
Topics: ERIC Archive, Geometric Concepts, Symbols (Mathematics), Artificial Languages, Mathematics...
In this paper, the author investigates the algebraic curve defined by the relation y[superscript 3] - 3y + 2x = 0. Treating this relation as a reduced cubic in the variable y, he uses a procedure first discovered by the mathematician Scipione del Ferro (Nahin, 1998, pp. 8-10) to obtain an expression for y in terms of x, namely y = (-x + [square root](x[superscript 2]-1))[superscript 1/3] - (x + [square root](x[squared]-1))[superscript 1/3]. By applying de Moivre's theorem to each term on the...
Topics: ERIC Archive, Geometric Concepts, Intervals, Mathematics Instruction, Mathematical Concepts,...
A search on the Internet for resources for teaching statistics yields multiple sites with data sets, projects, worksheets, applets, and software. Often these are made available without information on how they might benefit learning. This paper addresses potential benefits from resources that target trend and variability relationships in bivariate data. The paper is in five parts. The first is the introduction. In the second, trend and variability are defined. In the third, the author quotes...
Topics: ERIC Archive, Internet, Statistics, Mathematics Instruction, Web Sites, Worksheets, Computer...
The authors of this paper examine the use of formative assessment as a tool to assist teachers of mathematics to become more mindful developers of curricula. They focus on instructional design that is based on careful examination of student answers to questions. Empirical studies have shown the effectiveness of formative assessment for students, and recent theoretical work indicates that the positive feedback aspect of formative assessment stimulates self-regulation and transformation,...
Topics: ERIC Archive, Mathematics Teachers, Instructional Design, Feedback, Formative Evaluation,...
This article comes largely from observations made on-the-job while teaching mathematics in a government high school in the ACT. The issues canvassed will be familiar to those who have considered the arguments for and against ability grouping in mathematics education. It is speculative in nature, hinting at a synthesis of opposing views in the ability-grouping debate and ending with a proposal about how the practice of streaming might be aligned better with numeracy outcomes.
Topics: ERIC Archive, Numeracy, Ability Grouping, Mathematics Education, Mathematics Instruction,...
The author's secondary school mathematics students have often reported to her that quadratic relations are one of the most conceptually challenging aspects of the high school curriculum. From her own classroom experiences there seemed to be several aspects to the students' challenges. Many students, even in their early secondary education, have difficulty with basic multiplication table fact retrieval. Difficulty retrieving multiplication facts directly influences students' ability to engage...
Topics: ERIC Archive, Semantics, Secondary School Mathematics, Memory, Multiplication, Cognitive...
Sets of numbers where not only their sums are equal but the sums of other powers are also equal have been called multigrades. This article presents several mathematical equations that portray how multigrades are generated. By further extension of the process outlined in this article, students can generate higher-order multigrades. (Contains 1 figure.)
Topics: ERIC Archive, Mathematical Formulas, Mathematics Instruction, Numbers, Number Concepts, Equations...
Mathematics is like a language, although technically it is not a natural or informal human language, but a formal, that is, artificially constructed language. Importantly, educators use their natural everyday language to teach the formal language of mathematics. At times, however, instructors encounter problems when the technical words they use, as formal parts of mathematics, conflict with an everyday understanding or use of the same word, or related words. This article discusses this problem,...
Topics: ERIC Archive, Symbols (Mathematics), Artificial Languages, Vocabulary, Mathematics, Mathematics...
Inference, or decision making, is seen in curriculum documents as the final step in a statistical investigation. For a formal statistical enquiry this may be associated with sophisticated tests involving probability distributions. For young students without the mathematical background to perform such tests, it is still possible to draw informal inferences based on data of various sorts, for example by comparing two graphical representations. In doing so it is important to be able to state the...
Topics: ERIC Archive, Inferences, Probability, Prediction, Decision Making, Statistical Analysis,...
In the Western Gregorian Calendar, the date of Easter Sunday is defined as the Sunday following the ecclesiastical Full Moon that falls on or next after March 21. While the pattern of dates so defined usually repeats each 19 years, there is a 0.08 day difference between the cycles. More accurately, the system has a period of 70 499 183 lunations which is about 5 700 000 years: more details are at astro.nmsu.edu/~lhuber/leaphist.html. This website also provides one version of an algorithm,...
Topics: ERIC Archive, Computer Uses in Education, Mathematics Instruction, Educational Technology,...
For mathematics teachers who are continually looking for ways in which to engage their students in the learning process, the capabilities offered by technology answer the call. Whether the technology comprises computer based applications or graphics calculators, often boring aspects can be bypassed so that students can work on the "good bits" and build understanding. These tools, when used effectively, have been a great benefit to improving the cognitive development of many...
Topics: ERIC Archive, Adolescents, Teaching Methods, Mathematics Teachers, Mathematical Concepts, Graphing...
This article is about a very small subset of the positive integers. The positive integer N is said to be "perfect" if it is the sum of all its divisors, including 1, but less that N itself. For example, N = 6 is perfect, because the (relevant) divisors are 1, 2 and 3, and 6 = 1 + 2 + 3. On the other hand, N = 12 has divisors 1, 2, 3, 4 and 6, but since 1 + 2 + 3 + 4 + 6 = 16, rather than 12, 12 is not a perfect number. Ways and exercises for finding perfect numbers are presented....
Topics: ERIC Archive, Number Concepts, Arithmetic, Equations (Mathematics), Mathematics Instruction, Scott,...
Computers have been in schools and indeed some mathematics classrooms for more than 35 years. Some schools have chosen to centralise their computers in laboratories, while others have a mix of configurations and networks. Whatever the case, how extensive has been the classroom use of computers for teaching and learning in mathematics? What has their presence added to the classrooms and the learning experiences of students? What effect has there been on the pedagogy of teachers in this time? How...
Topics: ERIC Archive, Mathematics Instruction, Computer Software, Teacher Educators, Online Courses,...
This paper reports on a pilot study that examined the use of a science and technology curriculum based on robotics to increase the achievement scores of youth ages 9-11 in an after school program. The study examined and compared the pretest and posttest scores of youth in the robotics intervention with youth in a control group. The results revealed that youth in the robotics intervention had a significant increase in mean scores on the posttest and that the control group had no significant...
Topics: ERIC Archive, Pretests Posttests, School Activities, Intervention, Scores, Robotics, Informal...
This study examined the structure of two preservice teachers' understandings of educational software in mathematics using repertory grid techniques. Specifically, the study focused on how teacher educators can enable preservice teachers to discern the features of mathematics software, and develop pedagogical goals that reflect the affordances and constraints of available tools. Results showed a deepening of knowledge and a differentiation of knowledge following experiences of exploration,...
Topics: ERIC Archive, Teacher Educators, Preservice Teachers, Preservice Teacher Education, Computer...
This study examined activities used during elementary school math and reading instruction. Teachers reported their use of cooperative, competitive, and individual activities in math and reading, their subjective evaluations of teaching each subject, and their level of focus on promoting students' interests. Analyses indicated that teachers used more competitive activities in math than reading. Additionally, individual math activities increased across grades whereas individual reading activities...
Topics: ERIC Archive, Teaching Methods, Reading Instruction, Learning Activities, Class Activities,...
This paper is concerned with the use of spreadsheets within mathematical investigational tasks. Considering the learning of both children and pre-service teaching students, it examines how mathematical phenomena can be seen as a function of the pedagogical media through which they are encountered. In particular, it shows how pedagogical apparatus influence patterns of social interaction, and how this interaction shapes the mathematical ideas that are engaged with. Notions of conjecture, along...
Topics: ERIC Archive, Interpersonal Relationship, Interaction, Mathematics Instruction, Instructional...
Teacher quality has become a national policy concern in the US, especially in mathematics. This study provides insights into the conceptualisation of high quality mathematics teaching from the perspective of approximately 750 students in grades nine through twelve. Results from Rasch analysis yield information about the quality of the "Mathematics Quality Survey" constructed for this study and the hierarchy of items representing varying levels of quality as perceived by the students....
Topics: ERIC Archive, Mathematics Instruction, Teacher Effectiveness, Rating Scales, Student Attitudes,...
Two hundred and thirty-one students in six Grade 9 classes in two government secondary schools located near Chiang Mai, Thailand, attempted to solve the same 18 quadratic equations before and after participating in 11 lessons on quadratic equations. Data from the students' written responses to the equations, together with data in the form of transcripts of 36 interviews with 18 interviewees (a high performer, a medium performer, and a low performer from each of the six classes), were analysed....
Topics: ERIC Archive, Foreign Countries, Equations (Mathematics), Grade 9, Algebra, Mathematics...
In this article the development of two teachers as they make the transition from pre-service teachers to experienced teachers is examined. While these teachers participated in the same mathematics methods course and similar collaborative environments in their practicum experiences, their mathematics classroom instructional practices revealed stark differences by the time they were experienced teachers. In an effort to investigate these differences, the teachers' beliefs were explored in...
Topics: ERIC Archive, Teaching Methods, Methods Courses, Intellectual Development, Mathematics Education,...
An analysis of the 1990 National Assessment of Educational Progress (NAEP) found that only 46 percent of all high school seniors demonstrated success with a grasp of decimals, percentages, fractions and simple algebra. This article investigates error patterns that emerge as students attempt to answer questions involving the ability to apply fraction concepts and perform operations on fractions, providing a source to assist teachers in detecting and correcting common mistakes students make when...
Topics: ERIC Archive, Arithmetic, Word Problems (Mathematics), Numbers, National Competency Tests, Error...
It is widely acknowledged that what teachers believe influences their teaching, yet the focus of much professional learning remains on influencing the specific practices and tools that teachers employ in their classrooms. In this article, it is argued that a greater and more explicit focus on teachers' beliefs would be beneficial. To this end, an overview of aspects of the understandings of the nature of beliefs is presented followed by findings from a recent study that examined mathematics...
Topics: ERIC Archive, Mathematics Teachers, Mathematics Instruction, Teacher Attitudes, Foreign Countries,...
Problem solving and student-centred learning have received a great deal of attention in mathematics curricula for schools and in some universities. Much of this emphasis developed from the pioneering work of George Polya in heuristics, problem solving, and mathematics education. In this study, the author reviews Polya's work, and some of its later extensions, in the light of current research findings. It is argued that, despite the attention it has received, problem solving remains a difficult...
Topics: ERIC Archive, Mathematics Education, Problem Solving, Mathematics Instruction, Conceptual Tempo,...
Today's generation does not always have its clever ways of writing algebraic equations and expressions. This paper attempts to trace how this system has developed since the dawn of civilization. The author looks at a few snapshots taken at distinct times to illustrate this progress. (Contains 4 tables.)
Topics: ERIC Archive, Algebra, Teaching Methods, Educational History, Mathematics Instruction, Equations...
This article describes how a crystal ball known as "The Flash Mind Reader" is played. "The Flash Mind Reader" is a mathematics game in which the player is invited to select any-two digit number and then subtract the sum of these two digits from the original number. A chart is provided in which the (adjusted) number they obtained will have a symbol next to it. This crystal ball game is a wonderful example of how mathematics can be used in an absorbing way and provides an...
Topics: ERIC Archive, Educational Games, Mathematics Instruction, Arithmetic, Creative Teaching,...
The basic Pythagorean theorem for right-angled triangles is well-known in mathematical terms as a[squared]+b[squared]+c[squared] were "a," "b," and "c" are the lengths of the sides of the triangle with "c" as the hypotenuse. When "a," "b," and "c" are all integers and obey this equation, they are referred to as a Pythagorean triple. One property of right-angled triangles based on Pythagorean triples is that their areas are...
Topics: ERIC Archive, Geometric Concepts, Mathematical Formulas, Validity, Mathematical Logic, Mathematics...
In this note, we introduce and discuss convolutions of two series. The idea is simple and can be introduced to higher secondary school classes, and has the potential of providing a good background for the well known convolution of function.
Topics: ERIC Archive, Arithmetic, Mathematical Concepts, Validity, Mathematical Logic, Secondary School...
In this article, the author presents a paper folding activity that can be used for teaching fractions. This activity can be used to describe areas of folded polygons in terms of a standard unit of measure. A paper folding fractions worksheet and its corresponding solutions are also presented in this article. (Contains 2 figures.)
Topics: ERIC Archive, Measurement, Mathematics Instruction, Teaching Methods, Worksheets, Paper (Material),...
A solid understanding of equivalent fractions is considered a steppingstone towards a better understanding of operations with fractions. In this article, 55 rural Australian students' conceptions of equivalent fractions are presented. Data collected included students' responses to a short written test and follow-up interviews with three students from each year. This exploratory study found most participating Years 4, 6 and 8 students were familiar with geometric area models, particularly...
Topics: ERIC Archive, Geometric Concepts, Mathematics, Mathematics Instruction, Foreign Countries, Rural...
Several researchers have noted how children's whole number schemes can interfere with their efforts to learn fractions. An Australian study found that children who were successful with the solution of rational number tasks exhibited greater whole number knowledge and more flexible solution strategies. Behr and Post (1988) indicated that children needed to be competent in the four operations of whole numbers, along with an understanding of measurement, for them to understand rational numbers....
Topics: ERIC Archive, Numbers, Mathematics, Foreign Countries, Mathematics Skills, Mathematical Concepts,...
In this article, the author discusses the game of Hex, including its history, strategies and problems. Like all good games, the rules are very simple. Hex is played on a diamond shaped board made up of hexagons. It can be of any size, but an 11x11 board makes for a good game. Two opposite sides of the diamond are labelled "red," the other two sides "blue" (or, often, "black" and "white"). The hexagons at the vertices of the diamond belong to either side....
Topics: ERIC Archive, Games, Geometric Concepts, Puzzles, Mathematics Instruction, History, Scott, Paul
This paper examines the extent to which an intervention program on the use of practical work affected the following two variables: instructional techniques and mathematical ability. An attempt was made to correlate these two variables with age, educational attainment, and teaching experience to determine if there exists a relationship with and without the effect of the intervention. Differences and relationships between these two variables before and after the intervention were significant....
Topics: ERIC Archive, Teaching Experience, Teaching Methods, Intervention, Elementary School Mathematics,...
The mathematical performance of Chinese students, from mainland China, Hong Kong and Taiwan, is widely acclaimed in international comparisons of mathematics achievement. However, in the eyes of the Western educators, the environments established in Chinese schools are deemed relatively unfavourable for mathematics learning. This paper reports on a study that investigates the characteristics of effective mathematics teaching in five Shanghai schools. Findings reveal that those characteristics...
Topics: ERIC Archive, Foreign Countries, Mathematics Teachers, Mathematics Education, Mathematics...
This study implemented a multiplication program based on systematic practice, aimed at improving children's recall of basic multiplication facts. Four Year 5 classes were recruited to participate in the study. Two classes practised multiplication facts using pencil and paper worksheets and another two classes practised on computers. Eleven practice sessions (each of 15 minutes duration) were conducted over a four week period. Both groups increased their recall of basic multiplication facts and...
Topics: ERIC Archive, Mathematics Instruction, Multiplication, Mathematics Skills, Recall (Psychology),...
This study focuses on a group of practitioners from a school district that adopted reform-oriented curriculum materials but later rejected them, partially due to the inclusion of alternative algorithms in the materials. Metaphors implicit in a conversation among the group were analysed to illuminate their perspectives on instructional issues surrounding alternative algorithms. Several possible sources of resistance to folding alternative algorithms into instruction were found, including the...
Topics: ERIC Archive, Mathematics, Teaching Methods, Mathematics Education, Computation, Mathematics...
Mathematics games are widely employed in school classrooms for such reasons as a reward for early finishers or to enhance students' attitude towards mathematics. During a four week period, a total of 222 Grade 5 and 6 (9 to 12 years old) children from Melbourne, Australia, were taught multiplication and division of decimal numbers using calculator games or rich mathematical activities. Likert scale surveys of the children's attitudes towards games as a vehicle for learning mathematics revealed...
Topics: ERIC Archive, Foreign Countries, Grade 5, Student Attitudes, Negative Attitudes, Likert Scales,...
Math anxiety can begin as early as the fourth grade and peaks in middle school and high school. It can be caused by past classroom experiences, parental influences, and remembering poor past math performance. Math anxiety can cause students to avoid challenging math courses and may limit their career choices. It is important for teachers, parents and students to be aware of the effects of math anxiety so that if a student is affected, the student can receive the support necessary to lessen or...
Topics: ERIC Archive, Grade 4, Mathematics Anxiety, Mathematics Instruction, High School Students, Career...
This review of literature on progress monitoring was designed to examine the full array of curriculum-based measures (CBMs) in mathematics for students from preschool to secondary schools. We organized the article around two primary concerns: the approach used to develop the measures (curriculum sampling or robust indicators) and the type of research necessary to establish the viability of the tasks. Our review addressed the technical adequacy of the measures as indicators of performance and...
Topics: ERIC Archive, Teaching Methods, Predictive Validity, Curriculum Based Assessment, Academic...
The purpose of this article is two-fold: first, to explore the role played by linguistic and cultural factors in the mathematics classroom, particularly in relation to diverse learners; and, second, to provide insight into teaching, learning, and professional development that takes into account current mathematics education reform recommendations. In this article, the authors explore and discuss four selected cultural and linguistic factors to consider when teaching mathematics to diverse...
Topics: ERIC Archive, Teaching Methods, Mathematics Instruction, Educational Change, Cultural Influences,...
This paper examines the conflict in interest between teaching experiments and professional learning when they are combined in a research project. The study reported in this paper is the fourth year of a five year longitudinal study on the introduction of early algebraic concepts. The ten teacher participants are from five state primary schools in middle class areas in Brisbane, Queensland. The findings from this investigation suggest that potentially many conflicts exist between the interest of...
Topics: ERIC Archive, Algebra, Teaching Skills, Educational Research, Conflict of Interest, Longitudinal...
The notion of quality in undergraduate mathematics lectures is examined by using theoretical notions and research results from the literature and empirical data from a case study on lecturing on limits of functions. A systemic triangular model is found to catch critical quality aspects of a mathematics lecture, consisting of mathematical exposition, teacher immediacy, and general quality criteria for mathematics teaching. Mathematical exposition involves the dynamic interplay of mathematical...
Topics: ERIC Archive, Mathematics Instruction, Educational Quality, Lecture Method, Undergraduate Students,...
This study investigated teachers' attitudes towards proofs in the secondary school mathematics curriculum. The study was motivated by a desire to fill a gap existing in the literature in relation to teachers' attitudes towards proofs. Thirty-four secondary school mathematics teachers' responses to a Likert type questionnaire and interviews were summarised using the five themes of utility, positive attitudes, methods of proof, negative attitudes, and suitability of level of students to perform...
Topics: ERIC Archive, Secondary School Mathematics, Negative Attitudes, Foreign Countries, Mathematics...
This study compared number sense instruction in three first-grade traditional mathematics textbooks and one reform-based textbook ("Everyday Mathematics" [EM]). Textbooks were evaluated with regard to their adherence to principles of effective instruction (e.g., big ideas, conspicuous instruction). The results indicated that traditional textbooks included more opportunities for number relationship tasks than did EM; in contrast, EM emphasized more real-world connections than did...
Topics: ERIC Archive, Learning Problems, Textbooks, Comparative Analysis, Mathematics Instruction,...
This article describes the approach of a five-year initiative, funded by the National Science Foundation, to improve the teaching of mathematics and science in 10 rural school districts of Missouri. Traditional challenges of improving the professional practice of teachers are addressed through a regional partnership. External project evaluation results reveal specific teacher challenges, the change strategy of the Ozark Rural Systemic Initiative (ORSI), and what teachers value most. Continuous,...
Topics: ERIC Archive, Change Strategies, Rural Schools, Teacher Effectiveness, Educational Change,...
In this action research report 4 teachers and 1 teacher educator use the Japanese lesson study model of professional development for 15 months in rural Carlinville, Illinois. In March 2001, 4 teachers identified a goal to improve their students' understanding of two step word problems in 2nd grade elementary mathematics. Teachers completed three cycles of researching, planning, teaching, evaluating and reflecting. They were motivated, empowered, and found lesson study effective professional...
Topics: ERIC Archive, Action Research, Foreign Countries, Educational Change, Word Problems (Mathematics),...
Schools in 47 high-poverty school districts located mostly along the Atlantic Coast of North Carolina, South Carolina, and Virginia may have a head start on new requirements of the No Child Left Behind (NCLB) Act of 2001, thanks to a $6 million grant from the National Science Foundation. Begun in April 2000, the five-year Coastal Rural Systemic Initiative (CRSI) is striving to stimulate sustainable systemic improvements in science and mathematics education in school districts with a long...
Topics: ERIC Archive, Rural Schools, Poverty, Federal Legislation, Disadvantaged Youth, School Districts,...
Technology education teachers today have at their disposal the skills, opportunity, experience, ingenuity, expertise, equipment, and environment to greatly improve students' ability to learn and apply the knowledge they have gained in their academic programs. When a technology education teacher joins forces with an academic core teacher, the students reap the benefit of gaining empirical knowledge and skills not usually acquired within the confines of the traditional teacher-centered classroom....
Topics: ERIC Archive, Team Teaching, Technology Education, Core Curriculum, Aviation Education, Mathematics...
Classroom talk is regarded as essential in engaging and developing student understandings in the domain of mathematics. The processes of classroom talk may occur in quite different ways, ways that shape particular opportunities for learning mathematics. Little is known about how the talk produced in innovative approaches to education mediates the teaching/learning process and promotes student engagement in the practices of mathematics. Situated within a larger study that employed multiple forms...
Topics: ERIC Archive, Classroom Communication, Mathematics Instruction, Teaching Methods, Discussion...
There is evidence that spontaneous learning leads to relational understanding and high positive affect. To study spontaneous abstracting, a model was constructed by combining the RBC model of abstraction with Krutetskii's mental activities. Using video-stimulated interviews, the model was then used to analyze the behavior of two Year 8 students who had demonstrated spontaneous abstracting. The analysis highlighted the crucial role of synthetic and evaluative analysis, two processes that seem...
Topics: ERIC Archive, Cooperative Learning, Documentation, Mathematics Instruction, Secondary Education,...
This paper is structured in two sections. The first examines views of mathematical abstraction in two broad categories: empiricist and dialectical accounts. It documents the difficulties involved in and explores the potentialities of both accounts. Then it outlines a recent model which takes a dialectical materialist approach to abstraction in context. This model constitutes the basis of the second section where we describe an empirical study designed to investigate mathematical abstraction in...
Topics: ERIC Archive, Tutors, Abstract Reasoning, Student Development, Models, Mathematics Instruction,...