Illustrating Mathematical Connections: A Geometric Proof of Euler's Theorem

1996 ◽  
Vol 89 (1) ◽  
pp. 62-65
Author(s):  
Erin K. Frye ◽  
Peter L. Glidden
Keyword(s):  
Mathematical Reasoning ◽  
School Mathematics ◽  
Geometric Proof ◽  
Evaluation Standards ◽  
Mathematical Connections ◽  
Euler’S Theorem ◽  
Problem Solvers

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) calls for teachers to emphasize mathematical connections, promote mathematical reasoning, and help students become better problem solvers. If teachers are to achieve these goals, they need compelling examples, problems, and theorems that address all these elements.

One Point of View: Using Manipulatives Successfully

1990 ◽  
Vol 38 (2) ◽  
pp. 6-7
Author(s):  
Jeane M. Joyner
Keyword(s):  
Mathematical Problem ◽  
Point Of View ◽  
School Mathematics ◽  
Major Difficulty ◽  
Evaluation Standards ◽  
Mathematics Standards ◽  
Mathematical Connections ◽  
Learning To Reason ◽  
Problem Solvers

A curriculum with goals for students of valuing mathematics, being confident in their abilities, making mathematical connections, becoming mathematical problem solvers, and learning to reason and communicate mathematically is a call for classrooms in which students are actively involved in learning. It is a call for teachers to establish environments that encourage the use of manipulatives to assist students in attaining these goals proposed by the NCTM's Curriculum and Evaluation Standards for School Mathematics (Standards) (1989). A major difficulty, however, is how to manage the materials efficiently.

Beyond the Golden Ratio: A Calculator-Based Investigation

2001 ◽  
Vol 94 (2) ◽  
pp. 138-144
Author(s):  
Peter L. Glidden
Keyword(s):  
Problem Solving ◽  
Mathematical Reasoning ◽  
Golden Ratio ◽  
School Mathematics ◽  
Opportunities To Learn ◽  
Mathematical Communication ◽  
Evaluation Standards ◽  
Mathematical Connections ◽  
Principles And Standards ◽  
Classroom Use

NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) calls for increased emphasis on problem solving, mathematical reasoning, mathematical communication, and mathematical connections. This call is reaffirmed in Principles and Standards for School Mathematics (NCTM 2000). A preferred way of achieving these goals is by having students perform mathematical investigations in which they explore mathematics, search for patterns, and use technology when appropriate. In short, students should be given opportunities to learn mathematics by doing mathematics. Of course, if students are to learn mathematics through investigations, teachers must have a ready supply of such investigations available for classroom use.

Photographs and Committees: Activities That Help Students Discover Permutations and Combinations

2000 ◽  
Vol 93 (2) ◽  
pp. 93-96
Author(s):  
Jennifer Earles Szydlik
Keyword(s):  
Mathematics Curriculum ◽  
Mathematical Reasoning ◽  
School Mathematics ◽  
Evaluation Standards ◽  
Mathematical Ideas ◽  
Mathematical Connections ◽  
Problem Situations ◽  
Guiding Principles ◽  
Support Students

The vision of Mathematics Curriculum promoted by the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) is based on two guiding principles: “First, activities should grow out of problem situations; and second, learning occurs through active as well as passive involvement with mathematics” (1989, 9). In particular, curriculum should be designed to support students in constructing their own mathematical ideas and connections. Students should solve problems, communicate ideas both orally and in writing, engage in mathematical reasoning, and search for mathematical connections.

Implementing the Standards: Uses of Technology in Prealgebra and Beginning Algebra

1990 ◽  
Vol 83 (3) ◽  
pp. 194-198
Author(s):  
M. Kathleen Heid
Keyword(s):  
Problem Solving ◽  
School Mathematics ◽  
Evaluation Standards ◽  
Mathematics Classrooms ◽  
Beginning Algebra ◽  
Grade Levels ◽  
Mathematical Connections

The NCTM's Curriculum and Evaluation Standards for School Mathematics (Stan dards) (1989) designates four standards that apply to all students at all grade levels: mathematics as problem solving, mathematics as communication, mathematics as reasoning, and mathematical connections. These and NCTM's other standards are embedded in a vision of technologically rich school mathematics classrooms in which students and teachers have constant access to appropriate computing devices and in which students use computers and calculators as tools for the investigation and exploration of problems.

Problem Solving: Is More Than Solving Problems

1998 ◽  
Vol 4 (1) ◽  
pp. 20-25
Author(s):  
Michael G. Mikusa
Keyword(s):  
Problem Solving ◽  
Correct Answer ◽  
Mathematical Problem ◽  
School Mathematics ◽  
Evaluation Standards ◽  
Students First ◽  
Mathematics Class ◽  
General Goals ◽  
Problem Solvers

The curriculum and evaluation Standards for School Mathematics (NCTM 1989) states that one of its five general goals is for all students to become mathematical problem solvers. It recommends that “to develop such abilities, students need to work on problems that may take hours, days, and even weeks to solve” (p. 6). Clearly the authors have not taught my students! When my students first encountered a mathematical problem, they believed that it could be solved simply because it was given to them in our mathematics class. They also “knew” that the technique or process for finding the solution to many problems was to apply a skill or procedure that had been recently taught in class. The goal for most of my students was simply to get an answer. If they ended up with the correct answer, great; if not, they knew that it was “my job” to show them the “proper” way to go about solving the problem.

Mathematical Connections: Free Rides For Kids

1990 ◽  
Vol 38 (3) ◽  
pp. 10-14
Author(s):  
Harry Bohan
Keyword(s):  
School Mathematics ◽  
Evaluation Standards ◽  
Free Ride ◽  
Mathematical Connections ◽  
First Time

If one of your goals is to teach students to think mathematically for themselves (Curriculum and Evaluation Standards for School Mathematics, 1989; An Agenda for Action 1980), you may want to take them for some “free rides.” During a fourthgrade unit on fractions I incidentally used the term “free ride” for the first time in the introduction to a lesson on multiplication of mixed numbers. The class was told they were about to be taken on what could be called a free ride. A free ride was described as a situation in which they were studying a topic that was new, but one that could be attained by using what they know.

Implementing The Standards: Students' Microcomputer-Aided Exploration in Geometry

1990 ◽  
Vol 83 (8) ◽  
pp. 628-635
Author(s):  
Daniel Chazan
Keyword(s):  
High School ◽  
Problem Solving ◽  
Mathematical Structure ◽  
School Mathematics ◽  
Evaluation Standards ◽  
Mathematics Standards ◽  
Mathematical Connections ◽  
Teachers And Students ◽  
K 12

Four important themes presented in the K–12 Curriculum and Evaluation Standards for School Mathematics (Standards) (NCTM 1989) are mathematics as problem solving, mathematics as communication, mathematics as reasoning, and mathematical connections. The high school component also stresses mathematical structure. Furthermore, the Standards calls for new roles for teachers and students and suggests that microcomputer technology can help support teachers and students in taking on these new roles.

Ideas for Developing Students’ Reasoning: A Hungarian Perspective

1998 ◽  
Vol 91 (8) ◽  
pp. 677-681
Author(s):  
Anita Szombathelyi ◽  
Tibor Szarvas
Keyword(s):  
Twentieth Century ◽  
Mathematics Education ◽  
Teaching Assistants ◽  
Mathematical Reasoning ◽  
School Mathematics ◽  
Evaluation Standards ◽  
American University ◽  
Graduate Teaching ◽  
Reasoning Skills ◽  
End Results

As the end of the twentieth century approaches, we start to realize again the significance of proof in mathematics education. The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) cautions against the tendency to completely abandon proofs and focus only on the end results and formulas. In this article, we reemphasize the importance of proofs in teaching by sharing some of our experiences as students and teachers in Hungary, in addition to our experiences as graduate teaching assistants at an American university. We offer examples and ideas that might help educators develop students' mathematical reasoning skills.

Connecting Research to Teaching: Mathematics as Reasoning

1995 ◽  
Vol 88 (5) ◽  
pp. 412-417
Author(s):  
Peter Galbraith
Keyword(s):  
High School ◽  
Mathematical Reasoning ◽  
School Mathematics ◽  
Teaching Mathematics ◽  
Evaluation Standards ◽  
High School Geometry ◽  
School Geometry ◽  
Mathematical Quality

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) defines a role for reasoning in school mathematics that is far different from the norm of recent practice. Until recently, the study of mathematical reasoning was largely confined to high school geometry. Further, as Schoenfeld (1988) pointed out, the approach used in geometry was often so rigid that it conveyed the impression that the style of the response—for example, the two-column-proof format—was more important than its mathematical quality. The Standards document notes that reasoning is to have a role in all of mathematics from the earliest grades on up and that the form of justification need not follow a pre scribed format. Indeed, students are encouraged to explain their reasoning in their own words. Teachers are asked to present opportunities for students to refine their own thoughts and language by sharing ideas with their peers and the teacher.

A Popcorn Project for All Students

1997 ◽  
Vol 90 (3) ◽  
pp. 194-200
Author(s):  
Lydotta M. Taylor ◽  
Joann L. King
Keyword(s):  
Problem Solving ◽  
Cooperative Learning ◽  
Real World ◽  
Data Gathering ◽  
School Mathematics ◽  
Combine Data ◽  
Evaluation Standards ◽  
Mathematical Ideas ◽  
Mathematical Connections ◽  
Reasoning Problem

The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) encourages teachers to include activities that help students “construct and draw inferences from charts, tables, and graphs that summarize data from real-world situations” (p. 167) and “express mathematical ideas orally and in writing” (p. 140). The following activities combine data gathering and analysis with cooperative learning, mathematical connections, reasoning, problem solving, and communication.

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