abstractmath.org

help with abstract math

Produced by Charles Wells.   Home    Website Index   Blog

Last edited 8/21/2009 10:25:00 AM

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.

 

Introduction

     Background and attitude

             Diagnostic Examples

 

Index

 

The languages of math

 

    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

                 Some symbols used in math

            Alphabets 

            Delimiters

            Other symbols

 

    More about math languages 

Context 1

Embedding. 6

Displayed symbolic expressions  6

Parenthetic assertions  6

Preconditions and postconditions. 9

Incomplete notation. 16

Suppression of parameters  16

Synecdoche  16

Overloaded notation. 17

Redundancy. 17

             Parameters

     Intent of assertions

     Variations in meaning

         Conventions

         Defaults

         Scope

     Speaking math

       

 

   Proofs

    The nature of proofs

    The role of proofs in math

    Presentation of proofs

    Mathematical reasoning

Logical Constructions. 1

Methods of Deduction. 2

Translation problem.. 2

Mathematical Logic. 2

Universally true assertions

And, or, not

Conditional assertions

Forms of proof

Existence statements

More about quantifiers

              Order of quantifiers

              Negating quantifiers

              Reading and writing quantifiers

   

   

Understanding math

    Definitions

    Mathematical objects

    Mathematical structures

    Isomorphism and identity

    Representations and models

    Images and metaphors

    Conceptual and Computational  

    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 value of a function

            The image of a function

            The quotient of a function

Properties of functions

Composition of functions

Inverses of functions. 4

Special kinds of functions. 4

Images and metaphors for functions. 4

     Function Spaces

 

Relations

Relations: Basics 

Relations: Examples 

Operations on relations 

Properties of relations  

Equivalence Relations

Relations and functions 

 

Number Theory (small fragment only)

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

Groups

        Astounding Math Stories

The unreliability of numerical evidence

Finding factors, or not

Sizes of infinity