MATH
327 Foundations of Mathematics
Syllabus for Fall 2020
Mondays, Tuesdays, Thursdays and Fridays, 9:00 - 9:50am
Gildemeister Hall 325
Instructor: Dr. Eric Errthum Winona Email Username: eerrthum {at}
winona {dot} edu
Office Hours: TBD, but feel free at
any time to email me and/or request a zoom meeting
Text: “How to Read and Do Proofs:
An Introduction to Mathematical Thought Processes” by Daniel Solow. – The
point of the text is to provide a mental framework for understanding proofs
while providing detailed examples.
Course
Website: http://course1.winona.edu/eerrthum/math327
Prerequisite: Some sort of Discrete Math course (MATH247 or CS275, etc.), and
something called “mathematical maturity”. This is a term that loosely means one
has been exposed to mathematics beyond the realm of Calculus.
About This Course: Mathematics has two sides to
it. The first side, sometimes called Computational or Applied Math, is one that
you’re probably very familiar with. Most of the classes you’ve probably taken
so far have mainly focused on how to get an answer that is a number, function,
or an equation. This is the side of math that usually appeals to Engineers,
Physicists, and other physical scientists who are looking for answers in the
“real world”. This course is on the other side of mathematics which is
sometimes called Theoretical or Pure Math. In this area of mathematics, the focus
is on using deductive logic to construct and communicate sound, rigorous
arguments while striving for abstraction and the recognition of underlying
structure. This area of mathematics is beneficial to those who wish to study
Law, Philosophy, or, of course, higher Mathematics. This course, specifically,
will focus on the written mathematical proof with the goal of training students
how to make mathematical arguments. This course can serve as an introduction to
the other Pure Math courses in the WSU curriculum (specifically MATH447 and
MATH452).
Expectations: Students who complete this
course with a passing grade are expected to be able to demonstrate the
following skills: (i) Mastery of prerequisite material, (ii) Use technology
(e.g. LaTeX) to correctly format mathematical writing, (iii) Demonstrate
knowledge (i.e. memorization) of pertinent definitions and theorems, (iv)
Reason deductively in a variety of proof forms, (v) Communicate mathematics
correctly, clearly, and concisely.
Grading: Vocab/Theorem Quizzes (drop
lowest) 120 points------- 13%
Contributed Written Work (20
highest) 240 points------- 27%
Reputation 200
points------- 22%
Peer Review Work 100
points------- 11%
Notetaking 40
points--------- 4%
Other Assignments 40
points--------- 4%
Exam on Text 90
points------- 10%
Final Portfolio of Written
Work 70
points--------- 8%
All categories to be scaled as needed TOTAL:
900 points
Grades: A = 90% (810 points), B = 80%
(720 points), C = 70% (630 points), D = 60% (540 points).
Vocab/Theorem Quizzes: A large part of writing and
communicating mathematically is understanding the words, phrases, and theorem
statements. Definitions and theorems will be introduced throughout the semester
– in class, as hand-outs, or in the
Lexicon. Periodically we will have quizzes over these definitions and theorems;
sometimes announced beforehand, sometimes as “pop” quizzes. Memorizing the
definitions and theorem statements word-for-word is obviously the best strategy
here, though actually understanding them will also help in the other parts of
the course. Your lowest quiz score will be dropped at the end of the semester.
The student(s) with the highest mark on a quiz will be awarded reputation
points.
Readings: Students are expected to
have completed the readings in the text and the corresponding problems before
the class discussion on that chapter. Students will be called on in a somewhat
cyclic manner to answer reading questions. A student who gives a thoughtful,
prepared response (as opposed to an “on the fly” response) will be awarded
reputation points. Students who wish may pass on a question. You do not need to
typeset your answers, nor will they be handed in. You are advised to keep them
in an organized manner to facilitate class discussion.
|
|
Don't just read it; fight it! Ask your own questions, look for your own
examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? ---
Paul R. Halmos |
Homework: Homework for this class will
proceed in the following way:
1) Problems will be posted in the Lexicon and/or discussed in class.
2) When a student feels confident that they have arrived at a solution,
they sign up online to reserve that question. (Links to reservations can be
found on D2L.) A problem reservation is held until 8am on the morning of the
next class.
3) The student has until their reservation expires to email the
instructor the Overleaf editable link to their work. Students who let a
reservation expire will receive a small penalty in reputation points.
4) The submitted solution will be displayed (anonymously) in class and
the proof’s correctness will be discussed by the whole class.
5) The instructor (and sometimes the class) will fill out a standard
rubric for the submission.
5) Feedback will be given to the original author and occasionally the
other students will be graded on their application of the rubric.
6) Depending on the correctness of the submission, the work will be
designated into one of the following statuses:
a)
Accepted. The piece will be inserted into the class lexicon as is (or
with very minor corrections) along with the contributor’s name. (Earning them
20 reputation points.)
b)
Conditionally Accepted. The piece will be accepted once the original
author makes suggested corrections. Once the work is deemed acceptable by the
instructor, it will be inserted into the class lexicon along with the
contributor’s name. (Earning them 15 reputation points.)
c)
Revise and Resubmit. The student will have the choice of
(i) dropping the problem (for no penalty) and the process for that
problem will return to Step 2 above, or
(ii) obtaining an extension on their reservation until 8am on the
second next class day and the process for that problem will return to Step 3
above. (Note: if a revised and resubmitted solution becomes accepted, it will
earn at most 15 reputation points.)
d)
Rejected. The problem re-enters the problem pool. The student will
receive a small penalty in reputation points.
Class Lexicon: A tex document can be found
in Overleaf containing the latest definitions, problems, and student solutions.
Any solution using previous work in the Lexicon should reference it
appropriately (e.g. using the correct names and numbers of theorems,
propositions, etc.) You should be reading both the compiled pdf version of the
lexicon (to find new examples/problems to work on, to study
definitions/theorems, etc.) and the tex version (to learn how to reproduce the
notation and formatting).
LaTeX/Overleaf: Overleaf.com is a platform that allows users
to create LaTeX documents in a way similar to Google Docs. Once you have
created a free account, you can create LaTeX documents and share links to
readable and/or editable versions of them.
i)
Link to Lexicon: https://www.overleaf.com/read/nzcthyjnmvhg
ii) Link to Homework Template: https://www.overleaf.com/read/qzzghwgvnrmy
iii) General Overleaf
documentation/help: https://www.overleaf.com/learn/latex/Mathematical_expressions
Problem Reservations: In the content section of
D2L are links to a Doodle reservation page for each collection of problems.
When you have a solution worked out, follow the link, enter your name, and
check the appropriate box and click submit. I will receive an email
notification that you have done so, but your name will remain anonymous on the
sign-up sheet. Once you have made a reservation, you have until 8am on the
morning of the next class day to email me your Overleaf link. If the 8am
deadline passes without the corresponding email, the student who made the
reservation will be penalized 5 reputation points. In other words, do not sign
up for a problem until you have a solution at least sketched out. At any given
time, students are limited to having
2 or fewer problem reservations and at most 1 example reservation at any time.
Solutions: The solution should be typed
up in LaTeX on Overleaf using notation consistent with what has been used
previously in class and in the Lexicon. Problem set-ups should be rephrased
into mathematical statements. (For example, if a problem says, “Problem 3.2:
Prove or disprove that the square of a real number is positive,” then your
solution should start by stating “Proposition 3.2: If x is a real number, then
x2 is nonnegative.”) References to previous definitions,
propositions, problems, etc. and should be explicitly documented. Solutions/Examples
may not use information, definitions, etc. that occur after the statement of the problem in the Lexicon.
Solutions/Examples may use previous unsolved Problems, but you must state
explicitly that your argument also requires the solution to the previous
problem. (If, for instance, you know a friend has/will submit a solution to
that problem, your submission may operate under that assumption.) Some
mathematical writing tips can be found in the document
“MathematicalWriting.pdf” located in the content section of D2L. Before a
reservation expires, students should email me the Overleaf editable link to
their work. Make sure to indicate the
problem or example number.
Using and Citing Outside
Work: Work originating from an external
source should be explicitly cited. As a general rule, you are expected
to wrestle with the problems mostly on your own, consulting with the instructor
if you get stuck and/or looking through the Lexicon for inspiration, until you
arrive at a logical argument that accomplishes your goal. At times you will be
tempted to search the internet for a solution; often students will claim they
were just looking for a “hint on how to get started.” Doing so will short
circuit the learning process and rob you of the educational benefits. It is
through struggle and difficulty that we make new connections and learn. So, I strongly
advise against searching online for your problem (or one similar) before,
during, or after writing up a solution. Understandably, we all give in
to temptation at times and make mistakes. If you do find a solution online, you
should cite that in your submission. (If you are unsure how to do a proper
citation, email me or look at other citations in the Lexicon, if they exist.)
If un-cited submitted work appears to be inauthentic to the student, they may
be penalized with a rejection of their work and a penalty to their reputation.
Cited work that appears to have been expanded and/or has added value by the
student is allowed. (Note: just changing variable or function names does not
count as added value.)
Checkpoint Goals: To keep you on track to
producing 20 written solutions, the following checkpoints are required.
Approximately they require 1 proof a week for the first part of the course and
2 proofs a week during the second part. Recall, at any given time, students are limited to having 2 or
fewer problem reservations.
Plan accordingly.
|
Checkpoint Date |
Minimum Total Number of |
|
September 20 |
3 |
|
October 11 |
6 |
|
November 1 |
12 |
|
November 22 |
18 |
|
December 2 |
20 |
Standard Rubric: Most written problem solutions will be shown in
class and discussed. Authors of solutions may remain anonymous during the
discussion or they may defend their work. Ultimately the solution will be
graded using the standard rubric that can be found in D2L. Each row will
receive a score from the instructor (half points allowed) and the total (out of
12) will be reported back to the author. A written solution’s status will
mostly be determined by the instructor’s score (these are guidelines, not
definite cut-offs):
score
of 11.5 or higher: Accepted (+20 rep)
score of 9.5 – 11: Conditionally Accepted (+15 rep)
score of 4.5 – 9: Revise and Resubmit
score of 4 or lower: Rejected (-5 rep)
Solutions that are Accepted and/or Conditionally
Accepted will be entered into the class lexicon with the student’s name
revealed. A student’s top twenty solution scores will constitute the
“Contributed Written Work” portion of their grade. If a student contributes
less than twenty solutions, zeroes will be used in place of the missing scores.
To ensure that you are creating a solid Final Portfolio, each student should be
meeting the Checkpoint dates above.
Classroom Etiquette: When
critiquing submitted work in class, please provide polite and constructive
criticism. Refrain from saying unhelpful things such as “that’s totally wrong,”
or “that’s really dumb.” For very erroneous submissions, you should try to find
the very first incorrect statement, beyond which there is often no need to
discuss. Also, refrain from saying, “The right way to prove this is to do…” and
then giving an alternate proof. We want to judge the merits of the solution
given. Helpful hints are allowed.
Peer Review
Work: Peer Review Work comes in two parts.
1) Occasionally all the
students will be required to score a problem submission according to the rubric
and report a total score to the instructor prior to class discussion. Students’
responses will be compared to the instructor’s and graded accordingly:
equal:
4/4 and +5 reputation points
within ±1 point: 4/4
within ±2 points: 3/4
within ±3 points: 2/4
within ±4 points: 1/4
Off by 4.5 points or more: 0/4
2) In the second part of the
course, students will be assigned as reviewers of another student’s work. Like
the process for real academic journals, this process will be anonymous on both
ends with the instructor/editor as the go-between. A peer reviewer should not
only score the submission on the rubric, but also provide helpful/constructive
feedback on the piece. The reviewer will be graded out of 4 on their response.
If the instructor deems the response to be appropriate, it will be passed along
to the contributor and the reviewer will receive reputation points.
Examples/Corollaries: Most definitions
will also come with the chance for students to create examples. Similar to
Problems, appropriate rewording should be done by the student so that it can be
directly copied into the lexicon, replacing the directions of the example with self-standing
prose. (For example, if the lexicon originally states, “Example 2.3: Give
an example of a prime factorization containing 5 different primes,” then the example
submission should start “Example 2.3: Consider the integer 2310. … .”)
Submitting an example will follow the same procedure as a problem solution
(reservation, assignment folder submission, in-class discussion) but will not
be graded according to the rubric. Instead the class (with possible instructor
veto) will decide whether or not to accept the example into the lexicon (with
revise/resubmit as a middle option). Type-setting issues will not prohibit an
example from being accepted, however a lack of explanation and/or justification
may. If accepted, the author will be awarded 7 reputation points and this
example may be added to their Final Portfolio. There is no penalty if the
example is rejected. In some instances, the definition will remain open for
others to submit additional (significantly different) examples up to an
instructor-designated limit. Likewise, occasionally propositions will have
immediate applications to specific cases. These are called corollaries and
proofs of corollaries must either use the statement or proof structure of the
proposition immediately preceding them. (In this way, a corollary is typically
very easy to write and/or figure out, hence the lower designation.) Otherwise,
corollaries will be graded and treated like examples.
Final Portfolio of Written Work: All written work (problem
solutions and examples) that are accepted should be compiled into one document
that will be handed in at the Final Exam. In other words, no new work is
expected beyond organizing what you have already done for the course. Make
sure you are saving final versions of all your contributed work. This
portfolio will be graded on diversity of proof types, difficulty of problems
solved, and quality of examples. An ideal Final Portfolio would contain at least
one problem solution and example from each section, at least one solution in
each proof type, a handful of solutions that involve longer strings of
reasoning, and a handful of nontrivial examples. The portfolio will also
include a reflection section. Reflection questions will be posted after the
Book Exam. The Final Portfolio will be due during the usual final exam time: Tuesday, December 8, at 8:00am.
Reputation Points: Like some online forums and
real-life professional communities, positive participation results in an
enhanced standing while negative behavior results in a loss. At the beginning
of the semester, all students will start with 25 reputation points. At any
point (including the end of the semester) your grade for reputation will be
determined as a percentage of the top quartile, and then scaled to the
appropriate course points. Reputation Points can be earned or loss in the
following ways:
|
Action |
Result |
|
Having a solution
Accepted |
+20 points |
|
Having a solution Conditionally
Accepted |
+15 points |
|
Having a Revised and
Resubmitted Solution Accepted |
+15 points |
|
Meeting a Checkpoint
Goal |
+10 points |
|
Pioneer (First accepted
solution in a section) |
+8 points |
|
Having an
example/corollary accepted |
+7 points |
|
Award Winning (Accepted
solution to a long-standing/difficult problem) |
+7 points |
|
Finisher (Completes one
of the last three problems in a section) |
+5 points |
|
Scoring perfect on a
peer evaluation or successfully reviewing peer work |
+5 points |
|
Top score on
Vocab/Theorem quiz |
+5 points |
|
Answering a reading
question in class or in the online forum |
+3 points |
|
Finding a typo in the
lexicon |
+2 points |
|
Having a solution
Rejected |
–5 points |
|
Having a reservation
expire |
–5 points |
|
Rejected due to not citing
an outside source/inspiration |
–20 points |
|
Rejected due to
plagiarizing |
–75 points |
|
Other |
(to be determined) |
Notetaking: On
days when mathematical or proof-writing content is being delivered in a
face-to-face lecture, two students will be assigned to take notes and post them
to D2L. This allows other students to focus on the material without being
afraid they will forget to write something down. It will also assist those who
are unable to make it to class. The schedule of notetaker pairs can be found on
D2L.
Other Assignments: There will also be a
collection of disconnected assignments emerging from in-class activities,
specific tasks, etc. These responses should be typed in LaTeX/Overleaf unless
otherwise indicated.
Late/Missed Work: Students who miss class because of an unexcused absence will not be
able to retake Vocab/Theorem quizzes.
Students are allowed to miss a maximum of 5 solution evaluations because of
unexcused absences. They will receive a score of 0 for any additional missed
evaluations. Excused absences will be generally granted to students who
communicate with the instructor at least 2 hours before class time. In other
words, email me as soon as you think you might not make it to class.
Desire2Learn: The content section of D2L will include the class lexicon,
links to reservations, and LaTeX resources. Use the assignment folder to submit
written solutions and examples. Approximate grades will be
maintained as well, though at any time you may email the instructor for your
exact grade.
Resources: Your main resource for this course
is the instructor. The course text should provide some general guidelines for
writing proofs, but the instructor is the best resource for the problems in the
lexicon. If you are struggling with a problem, you should email and/or set up a
zoom meeting with the instructor.
Academic Dishonesty: Any type of academic
dishonesty (cheating, copying, plagiarism, using a solutions manual to do
homework, etc.) will result in failure and will be reported to school
authorities. If elements of submitted work were derived from another source,
you must cite that source. The instructor reserves the right to Google all
submitted work. The use of online mathematics forums is prohibited. If you are
having trouble with an assignment, please see the instructor for hints.
Note: This syllabus is subject to change if deemed
necessary by the instructor.
Tentative Schedule of Events – Math 327
(as of November 16, 2020, subject to change based on student in-class participation)
Due Before Class Notetaking Sessions
|
Week Beginning |
Monday |
Tuesday |
Thursday |
Friday |
|
8/24 |
Introductions |
LaTeX Introduction BRING YOUR LAPTOP |
Before Class: Read Chapter 1 (pgs 1 – 8) Chap 3: The Truth of It All |
Discussion on the Class
Rubric and Lexicon Basic Logic Section of Lexicon Predicates and Quantifiers Section of Lexicon
|
|
8/31 |
LaTeX DuplicateThis Due Read Chapter 2 (pg 9 – 24) Chap 2: The Forward-Backward Method |
Chapter 2, cont |
Read Chapter 3 (pg 25 – 40) Problems: 3.3, 4, 5, 9, 10, 16, 19) Chap 3: On Definitions and Mathematical Terminology Definition Assignment #1 |
Student Work Section 2.1 of Lexicon: Field Properties of the Reals |
|
9/7 |
No Class |
Student Work Chapter 3, cont. |
CLASS ZOOM Read Chapter 4 (pg 41 – 52) Quantifiers I: The Construction Method |
CLASS ZOOM Read Chapter 5 (pgs 53 – 68) Problems: 5.2, 3, 4, 6, 7, 10, 12, 14, 16 Quantifiers II: The Choose Method |
|
9/14 |
Vocab Quiz Student Work Definition Assignment #1 Due |
Student Work |
Read Chapter 6 (pgs 69 – 80) Quantifiers III: Specialization |
Student Work
|
|
9/21 |
Student Work Section 3 of Lexicon: Basic Set Theory |
Read Chapter 7 (pgs 81 – 92) Quantifiers IV: Nested Quantifiers |
Student Work |
VOCAB QUIZ Read Chapter 8 (pgs 93 – 100 Nots of Nots Lead to Knots |
|
9/28 |
|
Read Chapter 9 (pgs 101 – 114) The Contradiction Method |
Read Chapter 10 (pgs 115 – 124) The Contrapositive Method |
|
|
10/5 |
|
No Class |
|
Read Chapter 11 (pgs 125 – 132) The Uniqueness Methods |
|
10/12 |
|
Read Chapter 13 (pgs 145 –
154) The Either/Or Methods |
Q & A for Exam |
Exam on Chapters 1-11, 13 (End of Part 1
of the Course) |
|
10/19 |
|
|
Read Chapter 12 (pgs 133 –
144) Induction |
Induction,
cont. |
|
10/26 |
|
4.2 |
|
5.2 |
|
11/2 |
Vocab Quiz 6.1 |
|
6.2 |
|
|
11/9 |
Vocab Quiz 7.1 |
|
Construction of Reals |
|
|
11/16 |
Vocab Quiz 8.1 |
8.2 |
|
|
|
11/23 |
Vocab Quiz 3.3 |
|
No Class |
|
|
11/30 |
ZOOM ONLY Construction of the
Integers and Rationals |
ZOOM ONLY |
ZOOM ONLY Last Day for Submitted
Work |
ZOOM ONLY |
Final Exam
Tuesday,
December 8
8:00am –
10:00am
|
Commitment
to Inclusive Excellence WSU recognizes that our individual differences can deepen our understanding of one another and the world around us, rather than divide us. In this class, people of all ethnicities, genders and gender identities, religions, ages, sexual orientations, disabilities, socioeconomic backgrounds, regions, and nationalities are strongly encouraged to share their rich array of perspectives and experiences. If you feel your differences may in some way isolate you from WSU’s community or if you have a need for any specific accommodations, please speak with the instructor early in the semester about your concerns and what we can do together to help you become an active and engaged member of our class and community. If you or a friend has been a victim of sexual assault, dating violence, domestic violence, or stalking, you can talk to a trained, confidential advocate by calling 507.457.5610. |
The Standard Disclaimer
applies. © Eric Errthum, November 2020, all rights reserved.