A. Goals of Math 214

A1) Teach students proof techniques so that they can read and write 
proofs.

A2) Cover topics that are needed in a plurality of upper level courses, 
e.g., mathematical induction and complex numbers (useful in linear 
algebra, group theory, complex analysis, and functional analysis).

A3) Provide students with a view of higher level mathematics so that they 
can decide early whether they want to major in mathematics, rather than 
waiting until they take Math 307 and Math 311.

B. Strategies to achieve these goals

The mathematical tools outlined in(B1) and (B2) below should be covered in 
every section of Math 214, while the topics in (B4) below are examples of 
optional topics that could provide a mathematical context in which the 
tools in (B1) and (B2) could be studied.  It is important for Math 214 
instructors to understand that the topics in (B1) and (B2) should be 
covered within a mathematical context.  Experience has shown (for example) 
that covering propositional calculus in the abstract (via truth tables) is 
not as effective as presenting the ideas of propositional calculus in a 
concrete context like number theory.  It is also important that Math 214 
continue to be a proof-oriented course, and not become merely a "useful 
mathematical facts course."

(B1) For Goal A1, Math 214 should include:

a) propositional calculus;

b) quantifiers;

c) proof techniques;

d) naive set theory;

e) mathematical induction.

(B2) For Goal A2, Math 214 should include:

a) functions -- injectivity, surjectivity, bijectivity;

b) functions -- domain, image/range, inverse images;

c) informal discussion of number systems -- integers, rational, real, and 
complex numbers;

d)introduction to the arithmetic of complex numbers, perhaps through a 
sequence of graded take-home projects, including at least a definition of 
complex numbers, and their geometric representation as points in the 
plane, addition, multiplication, and inverses, polar representation, and 
deMoivre's theorem.

e) equivalence relations and partitions; the notion of "well-defined" 
functions or operations.

(B3) Covering the topics in the above list may already meet Goal A3.  In 
addition, handing out information about the mathematics concentration 
(e.g., copies of the department's Advising Handbook) is a useful practice.

(B4) The following optional topics could provide the required mathematical 
context for Math 214; other topics may also be suitable mathematical 
contexts:

a) relations and orderings

b) a formal introduction to the natural, rational, and real number systems

c) introductory combinatorics

d) countability and uncountability