Course Syllabus

Welcome to MA111: Introduction to Mathematical Reasoning

Specific information about this course and its requirements can be found below. For more general information about taking Saylor Academy courses, including information about Community and Academic Codes of Conduct, please read the Saylor Student Handbook.

 

Course Description

This course is an examination of the language and practice of set theory, and the theory and practice of mathematical proof, with the purpose of guiding you from “doing mathematics” at an elementary (i.e. problem-solving) level to “doing mathematics” at an advanced level.

 

Course Introduction

The main purpose of this course is to bridge the gap between introductory mathematics courses in algebra, linear algebra, and calculus on one hand and advanced courses like mathematical analysis and abstract algebra, on the other hand, which typically require students to provide proofs of propositions and theorems. Another purpose is to pose interesting problems that require you to learn how to manipulate the fundamental objects of mathematics: sets, functions, sequences, and relations. The topics discussed in this course are the following: mathematical puzzles, propositional logic, predicate logic, elementary set theory, elementary number theory, and principles of counting. The most important aspect of this course is that you will learn what it means to prove a mathematical proposition. We accomplish this by putting you in an environment with mathematical objects whose structure is rich enough to have interesting propositions. The environments we use are propositions and predicates, finite sets and relations, integers, fractions and rational numbers, and infinite sets. Each topic in this course is standard except the first one, puzzles. There are several reasons for including puzzles. First and foremost, a challenging puzzle can be a microcosm of mathematical development. A great puzzle is like a laboratory for proving propositions. The puzzler initially feels the tension that comes from not knowing how to start just as the mathematician feels when first investigating a topic or trying to solve a problem. The mathematician "plays” with the topic or problem, developing conjectures which he/she then tests in some special cases. Similarly, the puzzler "plays” with the puzzle. Sometimes the conjectures turn out to be provable, but often they do not, and the mathematician goes back to playing. At some stage, the puzzler (mathematician) develops sufficient sense of the structure and only then can he begin to build the solution (prove the theorem). This multi-step process is perfectly mirrored in solving the KenKen problems this course presents. Some aspects of the solutions motivate ideas you will encounter later in the course. For example, modular congruence is a standard topic in number theory, and it is also useful in solving some KenKen problems. Another reason for including puzzles is to foster creativity.

This course is comprised of the following units:

  • Unit 1: Logic
  • Unit 2: Sets
  • Unit 3: Introduction to Number Theory
  • Unit 4: Rational Numbers
  • Unit 5: Mathematical Induction
  • Unit 6: Relations and Functions
  • Unit 7: Sets, Part II
  • Unit 8: Combinatorics

 

Course Learning Outcomes

Upon successful completion of this course, you will be able to:

  • read and dissect proofs of elementary propositions related to discrete mathematical objects such as integers, finite sets, graphs and relations, and functions;
  • translate verbal statements into symbolic ones by using the elements of mathematical logic;
  • determine when a proposed mathematical argument is logically correct;
  • determine when a compound sentence is a tautology, a contradiction, or a contingency;
  • translate riddles and other brainteasers into the language of predicates and propositions;
  • solve problems related to place value, divisors, and remainders;
  • use modular arithmetic to solve various equations, including quadratic equations in Z6, Z7, Z11 and Diophantine equations;
  • Prove and use the salient characteristics of the rational, irrational, and real number systems to verify properties of various number systems;
  • use mathematical induction to construct proofs of propositions about sets of positive integers;
  • classify relations as being reflexive, symmetric, antisymmetric, transitive, a partial ordering, a total ordering, or an equivalence relation;
  • determine if a relation is a function, and if so, whether or not it is a bijection;
  • manipulate finite and infinite sets by using functions and set operations;
  • determine if a set is finite, countable, or uncountable;
  • use the properties of countable and uncountable sets in various situations;
  • recognize some standard countable and uncountable sets; and
  • determine and effectively use an appropriate counting tool to find the number of objects in a finite set. 

Throughout this course, you'll also see related learning outcomes identified in each unit. You can use the learning outcomes to help organize your learning and gauge your progress.

 

Course Materials

The primary learning materials for this course are readings, lectures, video tutorials, and other resources.

All course materials are free to access, and can be found through the links provided in each unit and subunit of the course. Pay close attention to the notes that accompany these course materials, as they will instruct you as to what specifically to read or watch at a given point in the course, and help you to understand how these individual materials fit into the course as a whole. You can also access a list all of the materials used in this course by clicking on Resources in the course's "Activities" menu.

 

Evaluation and Minimum Passing Score

Only the final exam is considered when awarding you a grade for this course. In order to pass this course, you will need to earn a 70% or higher on the final exam. Your score on the exam will be tabulated as soon as you complete it. If you do not pass the exam on your first attempt, you may take it again as many times as needed, following a 7-day waiting period between each attempt. Once you have successfully passed the final exam you will be awarded a free Saylor Certificate of Completion.

There are also 5 unit assessments and other types of quizzes in this course. These are intended to help you to gauge how well you are learning and do not factor into your final course grade. You may retake all of these as many times as needed to feel that you have an understanding of the concepts and material covered. You can locate a full list of these sorts of assessments by clicking on Quizzes in the course's "Activities" menu.

 

Tips for Success

MA111: Introduction to Mathematical Reasoning is a self-paced course in which you the learner determines when you will start and when you will complete the course. There is no instructor or predetermined schedule to follow. While learning styles can vary considerably and any particular student will take more or less time to learn or read, we estimate that the "average" student will take 112.25 hours to complete this course. We recommend that you work through the course at a pace that is comfortable for you and allows you to make regular (daily, or at least weekly) progress. It's a good idea to also schedule your study time in advance and try as best as you can to stick to that schedule.

Learning new material can be challenging, so below we've compiled a few suggested study strategies to help you succeed:

  • Take notes on the various terms, practices, and theories as you read. This can help you differentiate and contextualize concepts and later provide you with a refresher as you study.
  • As you progress through the materials, take time to test yourself on what you have retained and how well you understand the concepts. The process of reflection is important for creating a memory of the materials you learn; it will increase the probability that you ultimately retain the information.
  • Although you may work through this course completely independently, you may find it helpful to connect with other Saylor Academy students through the discussion forums. You may access the discussion forums at https://discourse.saylor.org.

 

Technical Requirements

This course is delivered fully online. You will be required to have access to a computer or web-capable mobile device and have consistent access to the internet to either view or download the necessary course resources and to attempt any auto-graded course assessments and the final exam.

  • To access the full course including assessments and the final exam, you will need to be logged into your Saylor Academy account and enrolled in the course. If you do not already have an account, you may create one, free of charge, here. Although you can access some course resources without being logged into your account, it's advised that you log in to maximize your course experience. For example, some of the accessibility and progress tracking features are only available when you are logged in.

For additional technical guidance check out Saylor Academy's tech-FAQ and the Moodle LMS tutorial.

 

Fees

There is no cost to access and enroll in this course. All required course resources linked throughout the course, including textbooks, videos, webpages, activities, etc are accessible for no charge. This course also contains a free final exam and course completion certificate.



Last modified: Friday, August 10, 2018, 2:37 PM