The objective of this module is to give you a high-level overview of the various mathematical thinking processes used in all advanced math courses. The goal is not for you to become proficient in using these thinking processes but rather, to give you a general understanding of what they are and how they work.

The objective of this module is to enable you to work with visual images, which really involves the following two related thinking processes: 1. Creating visual images, in which you learn to associate a metal or visual image with a particular mathematical object. This allows you to use that visual image when solving problems that involve those objects. 2. Converting visual images to written form. To convey a solution to a problem that you have solved using a visual image, you must learn to convert that mental image back to a written form that can then be communicated to someone else.

The objective of this module is for you to learn the following two opposite thinking processes: (1) Generalization, in which you create, from an original mathematical concept (for example, a formula, an equation, a problem, a definition, and so on), a new and more encompassing concept that includes not only the original one, but also something new and different. The advantage of doing so is that any result you obtain for the more general concept applies not only to the original concept but also to other similar concepts you might encounter in the future. (2) Anti-Generalization. When the problem you are working on is too general, you might not be able to obtain the type of solution you wish (or any solution at all, for that matter). With anti-generalization (also called specialization or simplification), you learn to look at more restricted versions of the problem that are simpler and thus allow you to obtain appropriate solutions.

The objective of this module is for you to learn the time-saving technique of unification, in which you combine two different, but related, mathematical concepts into a single unified concept. The advantage of doing so is that any result you obtain about the unified concept also applies to the original concepts you started with. To create a unification, you will also learn how to identify similarities and differences so you can create a unified concept that includes the similarities of the two original concepts while eliminating the differences.

In advanced math courses, you are generally given mathematical definitions that often capture desirable properties of a collection of objects you are working with. In this module, you will learn how to create such definitions. You will also learn that for a definition to be valid, you must be sure that all objects in the collection satisfy the properties in your definition while all other objects do not satisfy these properties.

The objective of this module is for you to learn about an axiomatic system, which is a collection of objects together with one or more operations on those objects, in which the mathematical properties the operations satisfy are specified explicitly. To create an axiomatic system, you will first learn about abstraction, which is the process of thinking in terms of general objects instead of specific items (for example, when working with adding two numbers, say x + y, with abstraction you learn to think of x and y as general objects rather than numbers). You also learn to use “general” operators when working with objects (for example, when x and y are objects, the “+” symbol in x + y becomes a general operator that combines object x with object y in some unspecified way). However, general operators satisfy no properties at all and so you will learn that an axiomatic system includes a list of properties that the general operators are assumed to satisfy (for example, it might be assumed that, for objects x and y, x + y = y + x).

The objective of this module is to bring together your knowledge of many of the mathematical thinking processes you learned in this course using a complete example. These thinking processes—together with mathematic proofs (for which there is a separate course)—are needed to understand the material in all advanced math courses.