Mathematical Thinking for Advanced Mathematics

Case Western Reserve University via Coursera

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Coursera
Pricing

Paid Course
Languages

English
Certificate

Certificate Available
Effort

8 weeks, 3 hours a week
Sessions

Self-Paced
Level

Intermediate
Subtitles

English

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Overview

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The objective of this course is to teach the mathematical thinking processes that are used in all college-level math-theory courses (such as discrete math, abstract algebra, and real analysis, for example). The goal is to reduce the time and frustration involved in learning such courses and to provide mathematical skills that are needed for graduate work and research in areas such as engineering, statistics, computer science, physics, the pure sciences, operations research, and economics, for example. This goal is accomplished by describing the following fundamental thinking processes that are used in advanced mathematical reasoning, each of which is illustrated with carefully explained examples: -Working with visual images. -Generalization (creating, from an original mathematical concept, a new and more encompassing concept that not only contains the original concept but something new and different as well) -Unification (combining two or more mathematical concepts into a single concept that includes the original ones). -Creating and working with mathematical definitions. -Creating and working with axiomatic systems (used in all abstract math courses).

Syllabus

Some Thinking Processes in Applied Mathematics

The objective of this module is to give you an introduction to the course and then to show you what is meant by a “mathematical thinking process” using examples that you have already seen in previous courses. In particular, a mathematical thinking process is a way of approaching a problem that you can use over and over again in different settings. The examples given here should be familiar to you and the ones presented in the remaining modules are used repeatedly in all advanced math courses.

Overview of the Thinking Processes for Advanced Mathematics

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.

Working with Visual Images

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.

Generalization and Anti-Generalization

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.

Unification

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.

Creating Definitions

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.

Axiomatic Systems

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 Thinking Process in Action

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.