Date: Apr 2, 1995 7:04 PM
Author: Tom McDougal
Subject: Serra's _Discovering Geometry_

In article <> 
Art Mabbott, writes:
>I cannot more strongly
>recommend Michael Serra's Discovering Geometry - An Inductive Approach.
>It is an incredible text. In my opinion, it is the best thing since
>sliced bread.

Since other posters seem to agree with this view, I want to provide a
different opinion.

Working as a volunteer at an after-school tutoring program for inner-city
kids, I have had a very negative experience with the Serra book. I have
three complaints with it.

My first complaint is with the problems. The selection is very small and
the problems get hard very quickly. There is also very little variety,
and little connection between problem sets.

My second complaint is with the lack of examples. Students are expected
to go out and use new relationships and new facts with almost no problem-
solving examples to guide them.

Related to this, the book does not pay attention to the difficulty
often have in (visually) recognizing certain patterns. For example,
it does not help students learn to identify overlapping triangles.

Third, it is very difficult to go back and look stuff up. The kids often
forget the various theorems/concepts. When we flip back several
pages, all we find are uncompleted conjectures.

Finally, when it comes to proof, the loosy-goosey approach does not
seem to be effective. The book asks students to make arguments supporting
various conclusions (just like any proof-based book) but provides no
help to the students in learning *how* to make such arguments.
Imagine what a student would do if she were home sick for an extended

Now, many of the complaints listed above are true of other, proof-based
books as well. But they are not true of _Geometry for Enjoyment &
Challenge_, by Rhoad, Whipple, & Milauskas, published by McDougal,

(Truth in advertising: I used to work for McDougal, Littell, but only
since then, in my tutoring experience, have I come to appreciate the
merits of that book. Furthermore, although my father started McDougal,
Littell, the company is now owned by Houghton Mifflin. So I have no
current connection with the company or this book.)

This book was written by three teachers, two of whom have won the
Presidential award for teaching. Their teaching skill and their
understanding of students is evident in the book.

The problem sets in the Rhoad book are large & diverse and build slowly
in difficulty. Each problem has its own diagram, so kids don't get
confused about what is given. The problems build on similar problems in
earlier lessons.

The Rhoad book provides lots of sample problems showing how each new
idea can fit into a proof or be used to solve a problem.

The Rhoad book helps students learn to recognize visual patterns. When
the three main triangle congruence theorems are introduced, the book
devotes considerable space -- in the sample problems and in the problem
set -- to showing diagrams and asking students merely to identify which
theorem (if any) applies. It devotes an entire lesson to overlapping
triangles. It shows students the "N", "Z", and "F" patterns associated
with parallel lines cut by a transversal and also shows students how
alternate interior angles can occur in more complicated figures, esp.
parallelograms with diagonals drawn in.

The bottom line is, it all seems to work. The kids I have worked with
who use the McDougal, Littell book perform head and shoulders above all
the others in terms of their understanding of geometry concepts and
their ability to write mathematical arguments.

I agree with the goal that students should discover geometry relations
for themselves. One can pursue this goal no matter what text one uses.
In fact, I conjecture that the success people have had with the Serra
book is due more to a change in their teaching than to the contents of
the book. However, as a source of problems, as a source of examples, and
as a reference for the student to use while working at home, the Serra
book is a disaster.

Tom McDougal University of Chicago Artificial Intelligence