abstractmath.org

help with abstract math

Produced by Charles Wells.   Home    Website TOC     Website Index

Last edited 6/24/2008 2:22:00 PM

 

TABLE OF CONTENTS

This is a list of all the major sections on this site.  Each section (in most cases) has its own table of contents.  The index gives a more detailed access to the topics covered.

 

Home page

     Background and attitude

 

Diagnostic Examples (incomplete)

 

Index

 

The languages of math

    References

    Mathematical English

        Assertions

        Names

        Glossary:  A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

    The symbolic language of math

Symbolic expressions  1

Grammar of the symbolic language  2

Variables

Substitution

Symbols

            Alphabets 

            Delimiters

            Other symbols  (only a few symbols included so far)

         More about the symbolic language

    More about math languages 

Context

Embedding. 1

Parenthetic assertions. 2

Context-sensitive interpretation. 3

Incomplete notation. 9

Suppression of parameters. 9

Synecdoche. 9

Overloaded notation. 10

Redundancy. 10

    Variations in meaning

         Conventions

         Defaults

         Scope

 

Proofs

    The nature of proofs

    The role of proofs in math

    Presentation of proofs

    Mathematical reasoning

         Universally true assertions

         And, or, not

         Conditional assertions

         Existence assertions

         More about quantifiers

              Order of quantifiers

              Negating quantifiers

              Reading and writing quantifiers

    Forms of proof

   

Understanding math

    Definitions

    Mathematical objects

    Mathematical structures

    Isomorphism and identity

    Representations and models

    Images and metaphors

    Conceptual and Computational   (STUB)

    Other aspects of understanding math

         Abstraction

         The axiomatic method

         The ratchet effect

         Pattern recognition

         Cognitive dissonance

 

Doing math

    Useful behaviors

       Self-monitoring

       Zooming and chunking

       More useful behaviors

       Useful behaviors for proofs

    Dysfunctional attitudes and behaviors

 

Numbers

 Natural numbers

 Integers

 Rational numbers

 Real numbers

 Algebra of the real numbers

 Decimal representation of real numbers

 Irrational numbers

 Density of the reals

 Complex numbers

 

Sets

    Sets: Notation

                 Russell’s paradox

            Some specific sets

            Sets: Specification

            Sets: Rules of Inference

            Sets: Metaphors and images

            Operations on sets (outline only)

            

 

Functions

Examples of functions  1

Specification and definition of function. 3

Objects associated with a function. 3

The name of a function

Domain and codomain

The input to a function  not written

The graph of a function

The rule for evaluation

            The value of a function at an input

             Quotient and image  not written

Properties of functions

Composition of functions

Inverses of functions. 4

Special kinds of functions. 4

Images and metaphors for functions. 4

 

Relations

Relations: Basics 

Relations: Examples 

Operations on relations 

Properties of relations  

Equivalence Relations

Relations and functions 

 

Number Theory (small fragment only)

Limits

Continuous Functions

Derivatives  (some nice illustrative graphs but nothing else so far)

Integrals

Abstract Algebra

Groups (fragment only)

Rings

 

Astounding Math Stories