Department of Teaching and Learning, Steinhardt School of Culture, Education, & Human Development
Department of Mathematics, Courant Institute of Mathematical Sciences
New York University
MTHED-UE-1049 / MTHED-GE-2050: Mathematical Proof and Proving (MPP)
SYLLABUS – Spring, 2015
Monday, 11:50-1:50, East Bldg. 403
Professor: Inbar Aricha-Metzer
Office: 623 East Bldg., 239 Greene St.
Email: [email protected]
Office Hours: By appointment
Welcome to the class!!
Course Description
The course introduces significant aspects and perspectives of mathematical proof and proving, e.g.: The need
to prove (including functions and goals of proof and proving); Various types of mathematical proofs and their
logical foundation (including direct deductive proof, proof by exhaustion, indirect proof by contradiction or
by contrapositive, and disproof by counterexample); Ways of communicating and presenting proofs
coherently and flawlessly; Alternative approaches to proving a given statement. Special attention will be
given to the meaning of mathematical statements – universal/existential, and the roles of examples in
determining the validity of mathematical statements. This is a problem-based course. Most lessons are
structured around activities that engage students in doing proofs that hopefully are meaningful to them and
based on mathematical topics with which they are familiar (e.g., elementary number theory, school algebra,
geometry).
Objectives
Students should be able to:
Interpret the meaning of mathematical statements (universal/ existential) and identify examples that
satisfy or contradict them;
Determine the validity of mathematical statements using examples and rules of inference;
Solve problems that require reasoning and proof;
Present arguments/proofs in an accurate and coherent way;
Construct formal proofs of mathematical statements by selecting and employing mathematically
valid ways of proving;
Prove mathematical statements in multiple ways;
Detect mathematical flaws and evaluate the validity of arguments/proofs.
Recommended Books
Chartrand, G., Polimeni, A. D., & Zhang, P. (2008). Mathematical proofs: A transition to advanced
mathematics. NY: Pearson, Addison Wesley.
Fendel, D., & Resek, D. (1990). Foundations of higher mathematics: Exploration and proof. NY: AddisonWesley Publishing Company.
2
Additional Resources
Jacobs, H. R. (1974). Geometry. NY: W. H. Freeman and Company. Chapter 1: The nature of deductive
reasoning (pp. 8-70).
Movshovitz-Hadar, N., & Webb, J. (1998). One Equals Zero and other mathematical surprises, paradoxes,
fallacies, and mind bogglers. Key Curriculum Press.
Nelson, R. B. (1993). Proofs without words. MAA.
Course Requirements
1. Full Participation (20%)
The quality of this class depends on your attendance, preparation for scheduled discussions and
activities, and engagement in and contributions to class discussion. I will take attendance and keep notes
on your participation in class. Good contributions to classroom discussion do more than simply push the
conversation along. Positive, substantive contributions make your thinking available to the class, raise
new questions, and challenge the group and yourself. I really encourage you to speak up, question what
you do not understand, and make proposals in class. You cannot go wrong by engaging.
For each class you will receive 0-2 points for active participation. For your final grade, I will count the 10
classes for which you received the highest scores.
2. Weekly Assignments (36%)
Every week you will receive an assignment; the assignment will be posted in NYU Classes by midnight of
Tuesday, to be submitted manually in the beginning of the following lesson.
Your assignments will include tasks requiring application of proof methods or principles that were
illustrated and discussed in the previous classes, as well as problems and questions that prepare you for
the following class.
Weekly assignment will not be graded solely on correctness, accuracy, and coherence, but also on the
seriousness of the attempts you made to tackle with the problems and the extent to which you shared
your thinking and reasoning.
For each weekly assignment you will receive 0-4 points. For your final grade I will count the 9
assignments with the highest scores that you received.
I recommend that you submit a printout of a Word document, and reduce any handwritten responses to
a minimum. Otherwise, make sure your handwritten is clear and easy to read. If you cannot attend class,
make sure to submit the homework via email to me ([email protected]) before the beginning of the lesson.
The subject of the email should be: MPP-HW1_LastName FirstName. If you are scanning parts that are
handwritten, please write these parts in a black ink pen, so it is clear and easy to read! Your full name
should appear on the top of each page; You should number the pages. The file name should be: MPPHW1_LastName FirstName.
3. Midterm Exam (15%)
We will have a midterm exam that will require the review and application of proof methods and
principles that have been addressed so far in the course. The final date will be determine by the end of
the 4th week of the course.
4. Final Exam (29%)
At the end of the course you will take a final exam that will cover all of the material from the course.
Syllabus for MPP course
October 2010 Spring, 2015
3
Attendance Policy
Full participation is required.
Unexcused absences are not allowed in this course. Each unexcused absence will result in a 2%
reduction of your final grade, in addition to missing points for participation in the non-attended
class. Unexcused tardiness (15 minutes or more) or early departures count as unexcused absences.
Absences will be excused at the discretion of the instructor. Please contact the instructor as soon as
you anticipate missing class. For an excused absence, I will waive the 2% reduction of your final
grade, however, you will not receive any points for participation on that date.
Three or more absences will result in no credit for the course.
Grading Policy
In order to get credit for the course, you must attend at least 12 (of the 14) classes, and submit on time
all weekly assignments. Late submissions may not be accepted. You also need to take both midterm exam
and the final exam.
The final grade will be determined by the sum of points accumulated for each component (participation,
weekly assignments, midterm exam, and final exam), according to Steinhardt School of Education Grading
Scale:
A
AB+
B
BC+
93-100
90-92
87-89
83-86
80-82
77-79
C
CD+
D
F
73-76
70-72
65-69
60-64
Below 60
Course Overview*
Week
Topic
Introduction to the course
Week 1
1.26.2015 Mathematical statements, rules of inference, and proving
Week 2
2.2.2015
Mathematical statements, rules of inference, and proving
Establishing the need to prove
Week 3
2.9.2015
Mathematical statements, rules of inference, and proving
The role of examples and counterexamples in making conjectures and verifying or
disproving them
Week 4
Direct Proofs- Number Theory
2.23.2015
Syllabus for MPP course
October 2010 Spring, 2015
4
Week
Topic
Week 5
3.2.2015
Direct Proofs- Number Theory
Week 6
3.9.2015
Direct Proofs- Geometry
Week 7
Midterm Exam (Tentative)
3.23.2015
Week 8
Direct Proofs- Geometry
3.30.2015
Week 9
4.6.2015
Direct Proofs- Geometry
Week 10 Indirect Proof – Part 1
4.13.2015 Proof by contrapositive
Week 11 Indirect Proof – Part 1
4.20.2015 Proof by contrapositive
Week 12 Indirect Proof – Part 2
4.27.2015 Proof by contradiction
Week 13
5.4.2015
Indirect Proof – Part 2
Proof by contradiction
Week 14
Reflection and Review
5.11.2015
* Modifications may be made as the course progresses, to address students’ strengths and
weaknesses
Note that throughout the course there will be opportunities to Evaluate Proofs. We will discuss "faulty
proofs" and learn to detect flaws in what may seem as a convincing proof.
Syllabus for MPP course
October 2010 Spring, 2015
5
Accommodation for NYU Students with Disabilities
Any student attending NYU who needs an accommodation due to a chronic, psychological, visual, mobility,
and/or learning disability, or is Deaf or Hard of Hearing should register with the Moses Center for Students
with Disabilities at 212 998-5980, 240 Greene Street. See www.nyu.edu/csd.
Academic Integrity
The relationship between students and faculty is the keystone of the educational experience in The
Steinhardt School of Culture, Education, and Human Development at New York University. This relationship
takes an honor code for granted. Mutual trust, respect and responsibility are foundational requirements.
Thus, how you learn is as important as what you learn. A university education aims not only to produce high
quality scholars, but to also cultivate honorable citizens.
Academic integrity is the guiding principle for all that you do; from taking exams, making oral presentations
to writing term papers. It requires that you recognize and acknowledge information derived from others, and
take credit only for ideas and work that are yours.
You violate the principle of academic integrity when you:
Cheat on an exam;
Submit the same work for two or more different courses without prior permission from your professors;
Receive help on a take-home examination that calls for independent work;
Plagiarize.
Plagiarism, one of the gravest forms of academic dishonesty in university life, whether intended or not, is
academic fraud. In a community of scholars, whose members are teaching, learning and discovering
knowledge, plagiarism cannot be tolerated. Plagiarism is failure to properly assign authorship to a paper, a
document, an oral presentation, a musical score and/or other materials, which are not your original work.
You plagiarize when, without proper attribution, you do any of the following:
Copy verbatim from a book, an article or other media;
Download documents from the Internet;
Purchase documents;
Report from other's oral work;
Paraphrase or restate someone else's facts, analysis and/or conclusions;
Copy directly from a classmate or allow a classmate to copy from you.
For more on academic integrity see http://steinhardt.nyu.edu/policies/academic_integrity.
Syllabus for MPP course
October 2010 Spring, 2015