review. The Saxon approach is to automatize arithmetic
in order to begin more abstract ideas. It is designed to teach students
to be problem solvers after concept mastery. The recommended audience is
7th grade advanced, 8th grade average, and 9th grade remedial mathematics
According to the authors, repetitious drill with skills is necessary in order to build automaticity. When automaticity is achieved, the student's mind is available for higher levels of thinking. The Saxon developers interpret this to mean that students learn by doing, but not on the day a topic is introduced. In one section, they state "we have memorized formulas, but formulas are easy to forget. It is poor practice to memorize a formula that can be developed quickly." At this point, the lesson reveals the development of the formula that students have been using in previous exercises.
In the Algebra 1/2 textbook, polygons are singularly addressed in a lesson titled "Polygons ~ Congruence". It begins with a discussion of convex and concave polygons. Polygon classification (based on number of sides, measures of angles, and number of sides parallel) is illustrated and the lesson is completed with an exposition on transformational geometry. Students express their understanding of new materials by answering between 4 and 10 practice problems relating to the lesson. Then they answer a variety of questions reviewing any previous topics in the book. The polygon practice problems begin to surface in the problem sets several sections later and sporadically from that point on. This indicates that in order to use this series, the teacher must proceed through the book in the sequence designated by the writers.
The teacher's edition does not provide additional resources or strategies for teaching. The teacher's role is to briefly dispense information and then help students individually as they encounter areas of difficulty. Problem Solving is viewed as the use of concepts in new situations after mastery has been obtained; therefore, learning basic facts by rote memorization is a prerequisite. Application is left up to the student.
The Mathematics: Exploring Your World materials addresses "analyzing polygons" throughout a unit of study. The instructional approach of this series is to pursue mathematical literacy by addressing the needs of the middle grades student including investigation focused around their energy level and diverse learning styles. Students are introduced to new topics by guided explorations with an emphasis on questioning strategies in order to share discoveries. At the end of an activity, students answer between ten and twenty exercises that build transition between concrete and abstract understanding.
In the Analyzing Polygons unit, the teacher's edition begins with a variety of instructional strategies to teach the objective of 'determining the sum of the interior angles of a polygon.' Among the different approaches are: tearing triangle corners, computer investigation, and protractor usage. Other topics introduced in this unit of study are congruence and polygon classification. Every lesson includes recommendations for: promotion of group work, teaching to encompass a variety of learning styles, involvement of students in active learning, enhancement of written and oral communication, integration of mathematics with other disciplines, relating new knowledge to old, promoting higher level thinking, and use of technology.
Collaborative learning hints are provided to encourage successful group work. These include recommendations of group size and procedures as well as questions to encourage reflection for groups to sum up what they have accomplished.
Teachers can actively involve students in learning by following the "Alternate Teaching Modality" suggestions. This kinesthetic approach to learning is motivating and meaningful because students construct their knowledge. Students are also asked to share their results by describing any patterns they find. This written communication, along with the indicated oral reflection and kinesthetic involvement, help insure that teaching encompasses a variety of learning styles.
The "Anytime Math" hints integrate mathematics with other disciplines. For example, polygons can be found in artwork. Also, new topics are related to previous mathematical learning to enhance understanding. This is shown in the use of triangles to determine the sum of the degrees in other polygons.
The teacher's edition also provides oral questioning strategies to promote higher level thinking. For example, "Why should you draw all of the triangles in a polygon" (Analysis), or "How is the number of triangles formed related to the sum of the angle measures of the polygon" (Ordering).
The optional use of technology allows flexibility in choosing the best tool for building understanding. Click here to see a sample computer lesson for analyzing polygons that I developed from the guidance provided in this unit.
The traditional mathematics texts, Algebra 1/2: An
Incremental Development (from the Saxon series) and Mathematics: Exploring
Your World (from the Silver Burdett Ginn series) have several fundamental
differences summarized in the following table:
If I were on the committee choosing the textbook for my county, I would definitely choose the guided-generation model developed by the Silver Burdett Ginn series over the basics-first curricula presented by the Saxon series. First of all, even though the Saxon materials provide constant reoccurrence of topics, it does not provide multiple teaching strategies. I do agree that students do not master topics upon first introduction of a topic. However, this focus on review takes away flexibility for teachers to progress through topics in the manner that best suites their purposes.
Furthermore, teachers can easily include mixed review opportunities, but what is difficult is the pedagogy of introducing new topics. In the constructivist perspective, people learn by being actively involved in the construction of one's own knowledge. The teacher's responsibility, thus, is to arrange a learning environment with situations and contexts in which the learner constructs appropriate knowledge rather than passively receiving information. I believe that these learning experiences should be the focus of instructional preparation.
Also, I believe that a basic premise upon which the Saxon series is established is faulty. Studies, such as the "Inert Knowledge Project", have shown that students do not naturally apply appropriate knowledge without prompting. Thus, student application after mastery is not likely unless they have the appropriate experiences.
Learning tends to be more generalizable when the learner is actively involved rather than passively receiving knowledge form a teacher. This indicates that instructional decisions should be determined by the teacher's knowledge of the learning process based on the "best fit" for meeting student's needs. "Students are not empty vessels into which information is poured, but rather, they are products of experiences on which knowledge and skills are built." Teachers must flexibly structure learning situations for optimum learning.
Cognition and Technology Group at Vanderbilt (1992) "Anchored Instruction and Situated Cognition." website: http://alcor. concordia.ca/~tbolton/edcomp/mod9art1.html; Printed: Monday, February 1, 1999.
Carroll, William M. Illinois Mathematics Teacher. The Van Hiele Model of Geometry: Research and Implications for Classroom Instruction. Chicago, Illinois.: 1993. p. 138- 148
Curcio, F. R. Mathematics Teaching in the Middle School. "Dispelling Myths about Reform in School Mathematics."
Hake S. and J. Saxon. Algebra 1/2: An Incremental Development
(2nd Edition). Saxon. Norman, Oklahoma.: 1990.
Hake, S. and J. Saxon. Math 65: An Incremental Development. Saxon Publishers, Inc.: 1987. p. 348-353.
Jackiw, Nicholas. The Geometer's Sketchpad, Version 2.1. Berkeley, CA: Key Curriculum Press.: 1994.
Mathematics: Exploring Your World. Silver Burdett Ginn. Morristown, New Jersey; Needham, Massachussetts; Atlanta, Georgia; Irving, Texas; San Jose, California.: 1995.
National Council of Teachers of Mathematics. Principles and Standards for School Mathematics: Discussion Draft. Reston, Virginia.: National Council of Teachers of Mathematics, 1998.
Shaughnessy, J. M. and W. F. Burger. Illinois Mathematics
Teacher. Spadework Prior to Deduction in Geometry. Chicago, Illinois.: September,
1995. p. 419-429.
Appendix 1: Some Characteristic Indicators of the van Hiele Levels
Appendix 2: A guided investigation of polygons including a Geometer's Sketchpad unit.
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