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Re: Logic for 7th Graders [was Geometry/Alternatives/Gifted]
Posted:
Mar 7, 1998 2:12 PM


I just received your request for course descriptions, due to a break down in our online 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 thatperhaps with the exception of UICSMthe 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 collegelevel texts and into early graduate level. It is reasonable to conclude that the writers of such higherlevel 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 socalled "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 welltrained 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 WellFormed 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



