Date: Apr 3, 1995 6:23 AM
Author: Tom McDougal
Subject: Re: Serra's _Discovering Geometry_
>Is this book referred to by Rhoad, Milauskas, ... written by teachers at
>a school called NewTrier(?) near Chicago.
This is the one.
> For example, in one of the sections there is presented the "Isosceles
>Triangle Theorem" without proof, with an indication that it will follow
>in the exercises. In the exercises, the following problem is presented:
>Given: AB = AC in Triangle ABC; Prove: <A = <B (I have used equals for
>Following is the "Proof"
>1. AB = AC 1. Given
>2. <A = <B 2. Isoscles Triangle Theorem
(This is odd. The authors never say "Isosceles Triangle THeorem" in any of
the example proofs. Instead they use an iconic shorthand for "If two sides
of a triangle are congruent then the base angles are congruent.")
I have the book in front of me and the isosceles triangle theorem is
thoroughly and rigorously proved immediately upon introduction in Section
3.6 (3.7 in the new edition).
I don't see the problem to which you refer, but in the previous section,
which introduces the definition of isosceles (et al.) triangles, there is
the following problem:
Given: AD and CD are legs of isosc. triangle ACD.
B is the midpoint of AC. (insert figure here)
Prove: <A = <C.
Except for the mention of "isosceles" (from which the student infers
AD=CD), this is virtually identical to problems in an earlier problem set
on drawing auxilliary lines. The student draws aux. line segment DB from
the vertex to midpoint B, proves the resultant triangles congruent by SSS,
and gets the angles by CPCTC. See also problem 5 in "Beyond CPCTC".
>Throughout, in the middle of a proof, suddenly a new "property",
>"theorem", or "postulate" is referenced without any foundation.
Where? What you may be objecting to is the way in which the authors use
sample problems to demonstrate how to use a new theorem or definition in a
proof. (E.g. final statement in sample problem 4 in section 3.6 (New
Edition: sample problem 3 in section 3.7).)
>Rarely, did I find a definition written with any degree of semblance of
The definitions are remarkably meaningful to the students I work with.
That's what's important, isn't it? how well it works for kids? The kids
end up knowing far more geometry and being far more capable of writing
mathematical arguments than the kids using any other book, rigorous and
deductive or exploratory and inductive. That's my experience with my small
sample size. Your mileage may vary -- but I note that in your message you
didn't say anything about how the book worked with kids.
>As far as the problems, I can not remember any problem contained therein
>that is not found in many other books.
You aren't serious! Look at the section on the three triangle congruence
theorems! Compare the variety of those problems to the problems in the
equivalent section of any other book! Notice how many of those problems
preview problems and ideas that will come later!
The merits of the Rhoad book didn't become apparent to me until I saw it in
use with kids. Similarly, I thought the Serra book was really neat until I
saw it in use with kids.
Tom McDougal University of Chicago Artificial Intelligence