**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
students.
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.
**

**References**

**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.

Return to Kelli's Homepage