The Perils of Conditional Statements and the Notion of Logical Equivalence

1997 ◽  
Vol 90 (7) ◽  
pp. 544-548
Author(s):  
Jeff Gregg
Keyword(s):  
High School ◽  
Deductive Reasoning ◽  
Logical Reasoning ◽  
School Mathematics ◽  
Evaluation Standards ◽  
High School Geometry ◽  
School Geometry ◽  
Logical Equivalence ◽  
Conditional Statements ◽  
Tenth Grade

Increased attention should be given to the expression of deductive arguments in high school geometry, according to the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). In general, the standards for grades 9-12 emphasize helping students develop their logical-reasoning skills. One means by which high school geometry textbooks typically introduce the notion of deductive reasoning and the principles oflogic is by examining conditional statements and their converses, inverses, and contrapositives. This article describes an episode involving conditional statements and the notion oflogical equivalence that occurred in a tenth-grade college-preparatorygeometry class. The episode illustrates some of the confusion that can arise in connection with this topic for both students and teachers. The ensuing discussion

Prove It

1998 ◽  
Vol 91 (8) ◽  
pp. 726-728
Author(s):  
Amy A. Prince
Keyword(s):  
High School ◽  
School Mathematics ◽  
Evaluation Standards ◽  
High School Geometry ◽  
School Geometry

Ask anyone who has taken high school geometry, and he or she will have a notion of a proof— generally, a two-column proof of statements and reasons. The two-column proof has fallen out of favor in such reform documents as the NCTM's Curriculum and Evaluation Standards for School Mathematics, which seeks to emphasize “deductive arguments expressed orally or in sentence or paragraph form” (NCTM 1989, 126). The two-column proof is a somewhat rigid form, yet it demonstrates to the students that they may not just give statements or draw conclusions without sound mathematical reasons.

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.

Using The Geometer's Sketchpad to Visualize Maximum-Volume Problems

2000 ◽  
Vol 93 (3) ◽  
pp. 224-228 ◽  
Author(s):  
David C. Purdy
Keyword(s):  
High School ◽  
Graphing Calculator ◽  
School Mathematics ◽  
Teaching Mathematics ◽  
Geometer's Sketchpad ◽  
Maximum Volume ◽  
Evaluation Standards ◽  
High School Geometry ◽  
School Geometry ◽  
Traditional Courses

An underlying tenet of the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) and other movements toward reform in school mathematics is breaking down content barriers between traditional mathematical topics, with the goal of teaching mathematics as a logically interconnected body of thought. As teachers move toward integrating the various areas of mathematics into traditional courses, problems that were once reserved for higher courses, for example, precalculus and calculus, now surface earlier as interesting explorations that can be tackled with such tools as the graphing calculator. One such problem is the well-known maximum-volume-box problem. Although this problem and related optimization questions have been common in advanced algebra, precalculus, and calculus textbooks, they have only recently found their way into high school geometry textbooks, including Discovering Geometry: An Inductive Approach (Serra 1997).

Pixy Stix Segments and the Midpoint Connection

1993 ◽  
Vol 86 (8) ◽  
pp. 668-675
Author(s):  
Ruth McClintock
Keyword(s):  
High School ◽  
Problem Solving ◽  
Cooperative Learning ◽  
Communication Skills ◽  
School Mathematics ◽  
Peter Pan ◽  
Evaluation Standards ◽  
High School Geometry ◽  
Mathematical Connections ◽  
School Geometry

The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) offers a vision of mathematically empowered students embarking on exciting flights of discovery. This vision challenges teachers to look for ways to incorporate problem solving, cooperative learning, mathematical connections, reasoning, communication skills, and proofs into lesson plans. The Pixy Stix activities described in this article are not quite as magical as Peter Pan and Tinkerbell's prescription of sprinkling pixie dust over children who want to fly, but they do embody all the attributes mentioned above and may enable your high school geometry students to take off in some surprising directions.

Integrating Transformation Geometry into Traditional High School Geometry

1992 ◽  
Vol 85 (9) ◽  
pp. 716-719
Author(s):  
Steve Okolica ◽  
Georgette Macrina
Keyword(s):  
High School ◽  
Mathematics Curriculum ◽  
School Mathematics ◽  
High School Mathematics ◽  
Evaluation Standards ◽  
High School Geometry ◽  
School Geometry ◽  
Transformation Geometry ◽  
To Receive ◽  
School Mathematics Curriculum

The grades 9-12 section of NCTM's Curriculum and Evaluation Standards for School Mathematics defines transformation geometry as “the geometric counterpart of functions” (1989, 161). Further, the Standards document recognizes the importance of this topic to the high school mathematics curriculum by listing it among the “topics to receive increased attention” (p. 126). Also included on this list is the integration of geometry “across topics.”

Let’s Use Trigonometry

1975 ◽  
Vol 68 (2) ◽  
pp. 157-160
Author(s):  
John J. Rodgers
Keyword(s):  
High School ◽  
Subject Matter ◽  
School Mathematics ◽  
High School Mathematics ◽  
Mathematics Courses ◽  
High School Geometry ◽  
School Geometry ◽  
The Subject ◽  
Order Of Presentation

All too often in the teaching of high school mathematics courses, we overlook the inherent flexibility and interdependence of the subject matter. It is easy to fall into the trap of presenting algebra, trigonometry, geometry, and so on, as separate areas of study. It is because they were taught this way traditionally. With relatively minor changes in the order of presentation, we can demonstrate to the student the vital interconnectiveness of mathematics. For example, many courses in high school geometry include a unit on trigonometry. The student learns three trigonometric ratios, namely, the sine, the cosine, and the tangent. He also learns to use the trigonometric tables to solve for an unknown side of a right triangle. Generally this material comes quite late in the year.

The Forum: What Should Become of the High School Geometry Course?: The Present Year-Long Course in Euclidean Geometry Must Go

1972 ◽  
Vol 65 (2) ◽  
pp. 102-154
Author(s):  
Howard F. Fehr
Keyword(s):  
High School ◽  
Euclidean Geometry ◽  
School Mathematics ◽  
Study Group ◽  
High School Geometry ◽  
School Year ◽  
School Geometry ◽  
Long Course ◽  
Coordinate Geometry

It is assumed that the geometey course refers to one that is commonly taught in the tenth school year. It is traditional Euclidean synthetic geometry, 2- and 3-space, modified by an introduction of ruler and protractor axioms into the usual synthetic axioms. A unit of coordinate geometry of the plane is usually appended. It is a course that is reflected in textbooks prepared by the School Mathematics Study Group and in most commercial textbooks.

Experimental Programs: Coordinate Geometry with an Affine Approach

1964 ◽  
Vol 57 (6) ◽  
pp. 404-405
Author(s):  
Harry Sitomer
Keyword(s):  
High School ◽  
High School Teachers ◽  
School Mathematics ◽  
Study Group ◽  
School Teachers ◽  
High School Geometry ◽  
School Geometry ◽  
Coordinate Geometry

In the spring of 1961, the School Mathematics Study Group convened a group of college mathematicians and high school teachers of mathematics to consider plans for writing an alternate high school geometry course, in which coordinates would be introduced and used as early as feasible.

What Do Mathematics Teachers Think about the High School Geometry Controversy?

1975 ◽  
Vol 68 (6) ◽  
pp. 486-493
Author(s):  
George Gearhart
Keyword(s):  
High School ◽  
Secondary School ◽  
Mathematics Teachers ◽  
School Mathematics ◽  
Secondary School Mathematics ◽  
High School Geometry ◽  
School Geometry

A survey of the attitudes of secondary school mathematics teachers toward geometry.

A Quadrilateral Hierarchy to Facilitate Learning in Geometry

1993 ◽  
Vol 86 (1) ◽  
pp. 30-36
Author(s):  
Timothy V. Craine ◽  
Rheta N. Rubenstein
Keyword(s):  
High School ◽  
Multiple Representations ◽  
Evaluation Standards ◽  
Instructional Style ◽  
High School Geometry ◽  
School Geometry ◽  
Mathematics Educators ◽  
Conceptual Understandings

The NCTM's Curriculum and Evaluation Standards (1989) challenges mathematics educators to rethink both content and instructional style. Specifically, it calls for “a shift in emphasis from a curriculum dominated by memorization of isolated facts and procedures … to one that emphasizes conceptual understandings, multiple representations, and connections …”(p. 125). We would like to illustrate such an approach for the study of quadrilaterals in a year-long high school geometry course. This strand is part of a complete course described in more detail in Craine (1985).

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