MATH
327 Foundations of Mathematics
Syllabus for Spring 2024
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, see the office hours calendar link in D2L
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.
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 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------- 25%
Reputation 200
points------- 21%
Peer Review Work 100
points------- 11%
Group Projects and Other
Assignments 80
points--------- 8%
Exam on Text 90
points--------- 9%
Final Portfolio of Written
Work 70
points--------- 7%
Final Exam 50
points--------- 5%
All categories to be scaled as needed TOTAL:
950 points
Grades: A = 90% (855 points), B = 80%
(760 points), C = 70% (665 points), D = 60% (570 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: See D2L.
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 reservation page for each collection of problems. When you
have a solution worked out, follow the link, click “Sign Up” for the desired
item, click “Save & Continue” at the bottom, enter your name
and click “Sign Up Now”. 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 at most 2 problem reservations and/or
at most 2 example/corollary reservations at any one 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.
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.
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.
(This includes using Large Language AI Models.) 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 |
February 5 (Beginning of Week 5) |
3 |
February 26 (Beginning of
Week 8) |
6 |
March 25 (Beginning of Week 11) |
12 |
April 15 (Beginning of
Week 14) |
18 |
April 30 (Beginning of Finals Week) |
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. Either way, being present for the
discussion will help you learn what the mistakes are and how to fix them.
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: Regularly
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
Other times 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, similar
to the role of the instructor during in-class discussions. 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.
Final Portfolio of Written Work: All written work (problem
solutions and examples/corollaries) that are accepted should be compiled into
one document that will be handed in at the Final Exam. In other words, no new
mathematical work is expected beyond organizing what you have already done for
the course. However, you will also be expected to reflect on two problems that
required multiple submissions and describe your learning process. Make
sure you are saving all intermediate versions of all your contributed work.
The final versions can be easily copied out of the lexicon, but the initial
drafts may be hard to replicate if lost. Make sure to work from a copy of
an initial draft rather than writing over your draft. The problems
portion of the 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 reflection
portion of the portfolio will be graded separately. Reflection questions will
be posted after the Book Exam. The Final Portfolio will be due during the usual
final exam time: Tuesday, April 30, at
8:00am.
Lean Solutions: The
future of mathematics will in some way incorporate programming languages called
proof assistants. One popular example of this is Lean. To
familiarize yourself with the language and the use of a proof assistant, you
can earn reputation by completing nontrivial levels of “Natural Number
Game” or “Set Theory
Game”. Reservations will be done like
problems/examples/corollaries and will be available as we cover related
sections in class. Solutions should be emailed to the instructor. Working
solutions will be published in the Lexicon, but we will not go over solutions
in class.
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 |
Having a Group Project Submission
Accepted |
+12 / +6 / +4 / +3 points |
Meeting a Checkpoint
Goal |
+10 points |
Having an
example/corollary accepted |
+7 points |
Solve a level in Natural
Number Game and/or Set Theory Game |
+6 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) |
Group Projects: Sometimes
the class will work on a problem together, hopefully arriving at an outline for
the proof. You will then be tasked with creating the formal write
up. You may work in groups of 1, 2, 3 or 4. Submissions will be given a
grade and the best solution submission splits 12 reputation points equally
among the group members. Group projects do NOT count toward your 20 submissions
but can be included in your final portfolio.
Other Assignments: There will also be a
collection of disconnected assignments emerging from in-class activities,
specific tasks, etc. Unless otherwise indicated, these responses should be
typed in LaTeX/Overleaf and submitted as pdfs to D2L. Note: The more often
students submit written work to be evaluated in class, the less there will be
of these types of assignments.
Final Exam: The final portfolio is due at the time of the final
exam. There will also be a short final exam that asks you to critique given
proofs and write one final proof. The final exam is Tuesday, April 30, at 8:00am.
Late/Missed Work: Students who miss class because of an unexcused absence will not be
able to retake Vocab/Theorem quizzes.
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 April 15, 2024, subject to change based on
student in-class participation and submissions)
Due Before Class Solow Text Lexicon Lectures
Week Beginning |
Monday |
Tuesday |
Thursday |
Friday |
1/8 |
Introductions |
LaTeX Introduction BRING YOUR LAPTOP |
Before Class: Read Chapter 1 (pgs
1 – 8) The Truth of It All |
SNOW DAY |
1/15 |
No Class |
Discussion on the Class
Rubric and Lexicon Section 1.1: Basic Logic
|
Read Chapter 2 (pg 9 – 24) The Forward-Backward Method |
Read Chapter 3 (pg 25 – 40) Problems: 3.3, 4, 5, 9, 10, 16, 19) Chap 3: On Definitions and Mathematical Terminology |
1/22 |
Student Submissions Section 1.2: Predicates and Quantifiers |
Read Chapter 4 (pg 41 – 52) Quantifiers I: The Construction Method |
Student Submissions Section 2.1:
Field Properties |
Read Chapter 5 (pgs 53 – 68) Problems: 5.2, 3, 4, 6, 7, 10, 12, 14, 16 Quantifiers II: The Choose Method |
1/29 |
VOCAB QUIZ Student Submissions Definitions Discussion |
Read Chapter 6 (pgs 69 – 80) Quantifiers III: Specialization |
Student Submissions |
Read Chapter 7 (pgs 81 – 92) Quantifiers IV: Nested Quantifiers |
2/5 |
Checkpoint Student Submissions |
Read Chapter 8 (pgs 93 – 100 Nots of Nots Lead to Knots |
Student Submissions |
Read Chapter 9 (pgs 101 –
114) The Contradiction Method |
2/12 |
VOCAB QUIZ Student Submissions |
No Class |
Read Chapter 10 (pgs 115 –
124) The Contrapositive Method |
Student Submissions 4.1 Basic Number
Theory |
2/19 |
Read Chapter 11 (pgs 125 –
132) The Uniqueness Methods Intro to Lean |
Work on |
Group Project #1: Prove It! |
Definitions #2 Assignment
Due Read Chapter 13 (pgs 145 – 154) The Either/Or Methods |
2/26 |
Checkpoint Student Submissions Section 2.2 |
VOCAB QUIZ Q & A for Exam |
Exam on Chapters 1-11, 13 (End of Part 1 of the Course) |
Student Submissions |
3/4 |
SPRING BREAK |
|||
3/11 |
No In-Person
Class Group Project #1 Work Period |
Student Submissions Section 4.2 |
Read Chapter 12 (pgs 133 – 144) Induction |
VOCAB QUIZ Section 1.3 |
3/18 |
Group Project
Due Student Submissions 1.3, cont. |
Section 5.1 |
Student Submissions |
VOCAB QUIZ Section 3.2 |
3/25 |
Checkpoint Student Submissions |
3.2, cont. Section 6.1 |
VOCAB QUIZ Student Submissions |
No In-Person Class Section 5.2 Video Lecture on D2L |
4/1 |
Student Submissions |
Section 6.2 |
Student Submissions |
VOCAB QUIZ Section 8.1 |
4/8 |
Student Submissions |
Section 7.1 |
VOCAB QUIZ Student Submissions |
No Class |
4/15 |
Checkpoint Student Submissions Section 8.2 |
Student Submissions |
No Class |
VOCAB QUIZ Student Submissions |
4/22 |
Axiomatic Construction of Standard Number Systems, cont. |
Last Day for Submitted
Work |
VOCAB QUIZ – ALL SECTIONS |
No In-Person
Class Checkpoint |
Final Exam and
Final Portfolio Deadline
Tuesday, April
30
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,
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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
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violence, or stalking, you can talk to a trained, confidential advocate by
calling 507.457.5610.
Reach out
to the OASIS
Advocacy Center for more information about your
rights and resources.
Campus
Resources
Student
Support Resources
Two good places to help you find resources of all kinds on
campus are TRIO Student Support and the Equity & Inclusive Excellence
Office. Both offices are dedicated to helping students of all racial, ethnic,
economic, national, sexual, and gender identities.
They can facilitate tutoring and point you to a wide range of
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Services office can document it for your professors and facilitate
accommodations.
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please contact Access Services as soon as possible.
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·
WSU Confidential Advocate: 507.457.2956
·
Advocacy Center of Winona Crisis Line: 507.452.4453
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WSU Health, Counseling, and Wellness Services: 507.457.5160
Academic
Student Resources
To find out about web registration, placement tests, program
requirements, and support tools to help students succeed, visit the Advising
Services office and website for answers to all your questions!
·
On the Rochester campus, the UCR Learning Center provides help
with both the development and the writing of papers.
·
On the Winona campus, for help with understanding the concepts
of a particular class or understanding the requirements of an assignment,
Tutoring Services offers three types of tutoring: drop-in appointments, 1-on-1
tutoring, and group sessions. You can visit them in Krueger Library 220 or
schedule a session online
·
For help specifically with writing and the development of
papers, the English department has a Writing Center that is staffed by trained
graduate students pursuing their Master’s degree in
English. The Writing Center is located in Minné Hall 348. You can make an appointment on the sign-up
sheet on the door or call 457.5505.
The Standard Disclaimer
applies. © Eric Errthum, April 2024, all rights reserved.