Proofs and Concepts

About the Textbook

Proofs and Concepts: the fundamentals of abstract mathematics

Description:  The text provides an introduction to upper-division mathematics, covering propositional logic, intuitive set theory, first-order logic, and proofs from an elementary starting point. The text then uses the tools developed earlier to examine topics touching on divisibility, abstract functions, cardinality, equivalence relations, and mathematical induction.

Authors:

• Dave Witte Morris - University of Lethbridge
• Joy Morris - University of Lethbridge

Formats:  The text is made freely available online by the authors. The authors provide a printable PDF document, typeset for 8.5 in. by 11 in. paper.

The utility of the text would be improved greatly by re-typesetting against other formats: specifically, e-reader and presentation formats. The extra formats could be typeset if the source code were made available. 

Supplemental resources:  No supplemental resources are provided. I have compiled solutions to a portion of the exercises, which I will make available on MERLOT and which are now available on my website: Morris Solutions. 

Cost savings:  Similar texts retail anywhere from $40 to $120 and contain one quarter/semester worth of material. Like this text, these other textbooks provide no supplementary material, no online homework systems, and no solutions manuals; the biggest difference is that this text provides an electronic copy whereas the similar texts do not.

The book that was previously used in this class is Mathematical Proof: A Transition to Advanced Mathematics, by Chartrand, Polimeni, and Zhang, which currently retails for $135 on Amazon. Since I teach about 105 students annually, this would be a potential annual savings for students of $14,175.

Accessibility and diversity statement:

The text is offered in electronic format, capture-able by standard screen-reader software.

The text references hypothetical individuals by using given names taken from a variety of cultures and languages, and it does so always in a positive context.PDF

License

The text is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Generic license. This means you can copy and redistribute the material in any medium or format and remix, transform, and build upon the material. If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original. You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. You may not use the material for commercial purposes.  

About the Course

MATH 300: Sets and Logic

Description:

Investigation of the fundamental tools used in writing mathematical proofs, including sentential and predicate calculus, topics from naive set theory, Cartesian products, partitions, equivalence relations, functions, countability, recursion, the binomial theorem and mathematical induction. This course relies heavily on problem-solving and writing complete, logically consistent arguments to illustrate the correct use of the logical tools and methods discussed.

Prerequisites:

C- or better for MATH 202 Calculus II, or departmental approval.

GE credit:

• Required for Mathematics majors and certain Computer Science majors. Does not provide GE credit.
  
• 5 Quarter Units.

Learning outcomes:

After completing this course, students will:

• Be able to read a mathematical proof and decide if it is correct or incorrect.
  
• Be able to write correct mathematical proofs.
• Understand the basics of naive Set Theory and be comfortable manipulating sets.
• Understand the notion of cardinality and be able to determine the cardinality of various sets.
• Understand the notion of equivalence relations and know the connection between equivalence relations, partitions, and functions.

Curricular changes:

No curricular changes needed to be made in order to adopt this OER textbook. 

Teaching and learning impacts:

Collaborate more with other faculty: No 
Use wider range of teaching materials: No 
Student learning improved: Unsure
Student retention improved: Possibly
Any unexpected results: No

Student retention possibly improved, in the sense that students who would have otherwise ignored the textbook requirement for the course (who would have simply done without a textbook, photocopying pages from a classmate when needed) now own a full copy of the textbook that they can keep and use as a reference for their future classes.

Sample assignment and syllabus:

Assignments
The text provides meaningful and appropriate exercises. Comprehension-focused exercises are interspersed throughout the prose of the text, and a number of mastery-focused exercises are provided at the end of each chapter.I recommend doing the comprehension-focused exercises during class and assigning a selection of the end-of-chapter exercises for graded homework.

Solutions to Assignments
These are the solutions to the assignments made in class.

Exams
These are the exams that I used for the M300 course.

Syllabus
This is the syllabus I used for the Winter 2016 term of Math 300.

Textbook Adoption

OER Adoption Process:

The primary motivation for adoption is the potential cost savings for students. The material covered in the typical introduction-to-proofs textbook is hardly new or novel in any way - it is all standard material. There is no justification for charging students up to $120 for one-quarter's worth of material that is abstract in nature, general in application, and essentially amounts to an "open standard" among mathematicians.

Aside from cost, this textbook offers several advantages over other similar texts. 

• The prose is clear and enjoyable. 
• The presentation is elementary and self-contained. 
• The exercises are meaningful, challenging while still accessible, and are strongly aligned to the flow of development of the included topics.

As this textbook includes all of the material routinely covered in an introduction-to-proofs course, no outside material was required.

Student access:

Printed, comb-bound copies of the text were made available to students for a nominal fee from the CSU Bakersfield print shop. The PDF document was delivered to the students via Blackboard.

Student feedback or participation: 

I do not have any specific comments from students about the textbook. However, the text book acquisition was phenomenally quick. You often lose a week of instruction while students complain about waiting for their textbooks in the mail. Students tended to read the textbook on their phone, so I'm currently working on typesetting it for viewing on a screen instead of in print.

Daniel Brice, Ph.D.  

I am a Mathematics lecturer at CSU Bakersfield. I teach the Calculus sequence, Sets and Logic, and occasionally Mathematics Education courses.

I do research in Lie algebras, linear and multilinear algebra, and zero product determined algebras.

I prefer teaching methods that empower students. I believe that education can be used to promote equity and that teaching methods play a large role in facilitating this goal. Towards this end, I use teaching methods that encourage a spirit of independence, exploration, and critical thinking. I believe that all students should come to understand that, rather than consisting of facts passed down from long ago, knowledge is something that they can and should actively take a part in creating. Teaching methods should result in situations where students’ actions and thoughts are relevant. I find that the methods that support a sense of agency in the student, promoting empowerment for all students, are those based on inquiry and problem solving.