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Re: Logic for 7th Graders [was Geometry/Alternatives/Gifted]
Posted:
Mar 7, 1998 2:12 PM
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I just received your request for course descriptions, due to a break down
in our on-line system. I will send one per post as I have time.
Book 1 Introductory Logic [usually taught only to precocious 7th graders]
The material was written with the following aims in mind: to develop an
accurate and efficient symbolic language suitable for expressing
mathematical statements and names; and to develop essential notions of a
proof theory. Here's what the writers said at the time:
"The Elements of Mathematics materials contain a far more
complete treatment of logic than can be found in current school
mathematics materials. The idea of a more explicit treatment of formal
logic did not, of course, originate with EM. Earlier pioneering projects
dealt with formal logic to a degree that seemed unusual at the time.
From today's viewpoint it appears that--perhaps with the exception of
UICSM--the logic was approached quite gingerly. It is hard to say
whether these projects were responding to, or setting, a trend. In any
event, even a cursory check of school mathematics texts confirms that
most of them have at least one chapter on logic and set theory. Indeed,
this practice extends up through college-level texts and into early
graduate level. It is reasonable to conclude that the writers of such
higher-level texts regard this attention to logic and set theory as
worthwhile, and assume that their readers do not have the necessary
competence in these areas, or at least need a review."
"Textbook writers have a special attitude toward language, for
they know that they will not be there to mediate their expositions when
they reach the reader. They welcome any device that makes communication
easier. No modern writer would think of giving up the precision and
space economy of mathematical symbolism, and it is clear that many wish
to use the precision and accuracy of logical symbolism as well. Most
texts are written with the expectation that a teacher will play a role in
the use of the exposition, with the greater that role, the lower the
academic level of the materials. In view of the instructional procedures
and objectives of the EM program, it is not surprising that we want to
experiment with the use of a strong, formal language for logic and set
theory."
"In view of the recurrent, repetitive treatment of logic from at
least the seventh grade up to graduate school, it is strange that a
course in logic does not have some place in the mathematics curriculum.
At the college level it is still more commonly found in philosophy
departments than in mathematics departments. Certainly, most present
treatments of logic must be considered unsatisfactory, both as
presentations of logic itself and as tools for mathematics. As one
mathematician states, the student 'must read the usual canonical chapter
on logic and sets.' At the school mathematics level it must be read
again and again, for although the student matures, the successive
canonical chapters do not show a like development. They deal with the
same dreary truth tables, inadequate semantic discussion of proof, with
syllogisms and an occasional bit of predicate language. The set theory
treats the same dreary Venn diagrams."
"There is nothing inherently dreary about truth tables or Venn
diagrams the first time through. They become dreary with repetition and
when the student discovers that, generally, the subject may be left in
the first chapter, since the rest of the book makes no essential use of
it or is even inconsistent with it. IN THE EM PROGRAM WE WANT TO TEACH
LOGIC TO STUDENTS WITH THE EXPECTATION THAT THEY WILL LEARN IT AND USE IT
IMMEDIATELY AND CONTINUALLY." [caps mine]
"A second role of logic is to present some sort of proof theory.
For years we have taught students about proof in a way that is analogous
to the so-called "direct method" for teaching languages, i.e., by
exposing them to lots of proofs and requiring them to construct
replicas. In the geometry class there is often some analysis of what a
proof is, but the analysis and description of proof is invariably
inconsistent with the actual proofs that appear in the text. If students
are to go beyond replicating models, they must abstract for themselves some
concept of proof from the examples seen, from inadequate descriptions,
and always subject to the authority of the teacher. This is hard on the
student, since teachers do not all agree on what a satisfactory proof
form is. Of course, many students do eventually attain a workable
concept of mathematical proof, but we believe the length of time spent to
reach this goal is far too long, and the number of students succeeding
far too small."
"To be consistent with our aim to develop the students' resources
to check the correctness of their own work, we must provide them with
some criteria for judging their proofs other than approval by a teacher.
The plan is to break the large sequence of inferences down into very
small, immediate reference rules, leaning very heavily on the students'
command of language for acceptance of the reasonableness of the basic
rules. Since at this stage we are necessarily concerned with language
and form, the mathematical content of Book 1 is low compared with that of
later books. Also, we do not want to complicate the problem of learning
what a proof is with the much harder problem of learning how to find
proofs. Thus, most of the proofs that the students see and write are
quite long, but, as the students gain experience, immediate inferences
are grouped in larger chunks, are abbreviated, or are left tacit. It is
the aim that, eventually, the overt manifestations of the logical
machinery will whither away from the students' written proofs, leaving
the stage to the mathematical content. The students learn several
colloquial styles of proof, and eventually they write proofs that look
like those of any well-trained mathematics student able and inclined to
use logical symbolism."
[At this point, the writers give three examples of student
proofs, which I will skip over in the interest of time and space!]
Summary of Contents
The Formal Language: Introduction, Negation, Conjunction and
Disjunction, Sentences and Well-Formed Formulas, Truth Tables,
Implication, Tautologies, Equivalence of Formulas, Some Useful
Equivalences, the Substitution Principle, Instantiation of Formulas by
Formulas, the Tautology Principle.
Introduction to Demonstrations: Modus Ponens, a First Look at
Demonstrations, Conjunctive Inference and Conjunctive Simplification,
Contrapositive Inference and Modus Tollens, Syllogistic Inference and
Inference by Cases, Subroutines Involving the Biconditional, the General
Substitution Principle, Are All the Connectives Necessary?, the Deduction
Theorem, Using the Deduction Theorem, Object Language and Metalanguage,
the Principle of Indirect Inference.
The Propositional Calculus: A Review, the Propositional Calculus
and Open Sentences, the Limitations of the Propositional Calculus.
Ron Ward/Western Washington U/Bellingham, WA 98225
ronaward@henson.cc.wwu.edu
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