MATH
327 Foundations of Mathematics
Syllabus for Spring 2015
Mondays, Tuesdays, Thursdays and Fridays, 9:00-9:50am
Gildemeister Hall 326
Instructor: Dr. Eric Errthum Winona Email Username: eerrthum Office: 205 Gildemeister
Hall Office Hours: See homepage. Or by appointment on any
day.
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. Some additional examples may be taken from
Stewart’s Calculus and Pre-Calculus books.
Course
Website: http://course1.winona.edu/eerrthum/math327
Prerequisite: Passing grade in Linear Algebra (MATH242). Note: The real prerequisite
is something called “mathematical maturity”. This is a term that loosely means
one has been exposed to a variety of mathematics beyond the realm of
differential functions.
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, whether it 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”. The other side of mathematics 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 MATH448 and MATH453).
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) 150 points------- 23.44%
Reputation 70
points------- 10.94%
Peer Evaluations (drop 2
lowest) 70
points------- 10.94%
Contributed Written Work (10
highest) 120 points------- 18.75%
Miscellaneous Assignments 30
points-------- 4.68%
Exam on Text 50
points-------- 7.81%
Final Portfolio of Written
Work 50
points-------- 7.81%
Final 100
points------- 15.63%
--------
640
points total
All categories will be
scaled as needed.
Grades: A = 90% (576 pts), B = 80% (512
pts), C = 70% (448 pts), D = 60% (384 pts)
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, both in class and as hand-outs.
Periodically we will have quizzes over these definitions and theorems;
sometimes announced beforehand, sometimes as “pop” quizzes. Memorizing the
definitions and theorem statements is obviously the best strategy here, though
actually understanding them will also help in the other parts of the course.
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.
Homework: Homework for this class will
proceed in the following way:
1) Problems will be posted and/or discussed in class.
2) When a student feels confident they have arrived at a solution, they
sign up online to reserve that question. A problem reservation is held until
8am on the morning of the next class.
3) The student has until their reservation expires to email/upload a
solution. 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.
5) The instructor and the class will fill out a standard rubric for the
submission.
5) Feedback will be given to the original author and 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).
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.
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 class period and the process for that
problem will return to Step 3 above. (Note: if a revised and resubmitted
solution becomes accepted, it will only earn 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 will be updated in D2L containing the latest definitions, problems,
and solutions. Any solution using previous work should reference this document
appropriately (e.g. using the correct names and numbers of theorems,
propositions, etc.) You should be reading both the .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).
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’ve made a reservation, you have until 8am on the
morning of the next class day to upload to the solution to the D2L dropbox. If the 8am deadline passes without the
corresponding upload, the student who made the reservation will be penalized 5
reputation points. In other words, don’t sign up for a problem until you have a
solution at least sketched out. Students are limited
to having 1 problem reservation and 1 example reservation at any time.
Solutions: The solution should be typed
up in LaTeX using notation consistent with what’s
been used previously in class and in the class 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.
Work originating from an external source should also be explicitly documented.
Solutions/Examples may not use information, definitions, etc. that occur after the statement of the
problem. 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.) Before a
reservation expires, students should upload both the .tex file and
the compiled .pdf file containing the proposed solution. Make sure to name your files with the problem or
example number.
LaTeX: Resources for installing and getting started with LaTeX, as well as templates and the first assignment, can
be found here. Other LaTeX
resources can be found within the .tex version of the
class lexicon in D2L. Note: Graphics and images are a pain to do in LaTeX. Typically another program is used to create the
image and then that is imported into LaTeX. Likewise,
feel free to use any program you want to create an image file to be uploaded
with your solution. Make sure the file extension of the image is .jpg, .bmp,
.pdf, or .gif. If you’re having trouble with graphics, see the instructor for
help.
Co-authors: Working together is not
discouraged. In fact, you are encouraged to discuss problems, solutions, and
examples with classmates prior to submitting them. If you’re unsure of your
work, the best thing to do is show it to someone else to see if they can
understand what you’re doing. Additionally, groups of up to 3 students may
submit a joint written problem solution (but not example). To submit as a
group, list everyone in the group when making the online reservation. (Later
additions to the group will not be allowed.) If n people are in a group, this reservation will only count for 1/n towards the 1 reservation maximum.
(E.g. if you are part of a group of 3 reservation, this counts as 1/3 of a
reservation, so you have room for another group reservation, but not a solo
reservation.) In addition, each author must upload a
copy of the solution to the D2L dropbox. Any member
of the group not uploading a copy of the solution by the 8am deadline will be
removed from the group. If Accepted or Conditionally Accepted, this work may be
added to the student’s Final Portfolio and will count towards their Contributed
Written Work, however any reputation gained will be equally divided among the
authors.
Standard Rubric: All 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 between 0 – 3 points (half
points allowed) and the total will be reported back to the author. In addition,
after a solution is discussed in class, each student will score it according to
the rubric and report a total score to the instructor. 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
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
score of 9.5 – 11: Conditionally Accepted
score of 4.5 – 9: Revise and Resubmit
score of 4 or lower: Rejected
Solutions that are Accepted and/or Conditionally
Accepted will be entered into the class lexicon with the author’s name. A
student’s top ten solution scores will constitute the “Contributed Written
Work” portion of their grade. If a student contributes less than ten 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 submitting at least one example or problem solution
for each 1.5 class periods.
Examples: 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. (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, dropbox 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 to up an instructor-designated limit.
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.
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’ve already done for the course. 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. A detailed rubric for the
Final Portfolio can be found in the content section of D2L.
Reputation Points: Similar to some online
forums, positive participation results in an enhanced standing while negative
behavior results is a loss. At the beginning of the semester, all students will
start with 100 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 an example accepted |
+7 points |
|
Scoring perfect on a peer evaluation |
+5 points |
|
Top score on Vocab/Theorem quiz |
+5 points |
|
Answering a reading question in class |
+3 points |
|
Finding a typo in the lexicon |
+2 points |
|
Having a solution Rejected |
–5 points |
|
Having a reservation expire |
–5 points |
|
Plagiarizing |
–75 points |
|
Other |
(to be determined) |
Miscellaneous 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
unless otherwise indicated.
Final Exam: There will
be a final exam on Tuesday, May 5, 8:00
– 10:00am. The final exam may or may not include an oral exam to be
scheduled during finals week.
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.
Desire2Learn: The content section of D2L will include the class lexicon,
links to reservations, and LaTeX resources. Use the dropbox 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 the designated office hours don’t
won’t for you, feel free to suggest another time and/or email questions.
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
(subject
to change based on student in-class participation)
|
Week Beginning |
Monday |
Tuesday |
Thursday |
Friday |
|
1/12 |
Introductions |
LaTeX Introduction BRING YOUR LAPTOP |
Before Class: Read Chapter 1 (pgs
1 – 8) The Truth of It All |
Discussion on the Class
Rubric and Lexicon Basic Logic Section of
Lexicon
|
|
1/19 |
No Class |
Read Chapter 2 (pg 9 – 24) Student Examples and
Problem Solutions |
Student Examples and
Problem Solutions The Forward-Backward
Method |
Read Chapter 3 (pg 25 – 40) Problems: 3.3, 4, 5, 9, 10, 16, 19) Predicates and Quantifiers
Section of Lexicon On Definitions and
Mathematical Terminology |
|
1/26 |
Vocab/Theorem Quiz: Student Examples and
Problem Solutions Field Properties of the Reals |
Read Chapter 4 (pg 41 – 52) Quantifiers I: The
Construction Method |
Student Examples and
Problem Solutions Basic Set Theory Section |
Read Chapter 5 (pgs 53 – 68) Problems: 5.2, 3, 4, 6, 7, 10, 12, 14, 16 Quantifiers II: The Choose
Method |
|
2/2 |
Vocab Quiz: Student Examples and
Problem Solutions Basic Set Theory Section,
cont. |
Read Chapter 6 (pgs 69 – 80) Quantifiers III:
Specialization |
Student Examples and
Problem Solutions |
Read Chapter 7 (pgs 81 – 92) Quantifiers IV: Nested
Quantifiers |
|
2/9 |
Student Examples and
Problem Solutions Finish Section 3 Lectures Assignment: Peer Evaluations |
Read Chapter 8 (pgs 93 – 100 Nots of Nots
Lead to Knots |
Vocab Quiz: Assignment: Peer
Evaluations |
Read Chapter 9 (pgs 101 –
114) The Contradiction Method |
|
2/16 |
Student Examples and
Problem Solutions Assignment: Peer
Evaluations Lecture on Section 4.1 |
Read Chapter 10 (pgs 115 –
124) The Contrapositive Method |
Student Examples and
Problem Solutions Assignment: Peer
Evaluations Lecture on Section 4.1 |
Read Chapter 11 (pgs 125 –
132) The Uniqueness Methods |
|
2/23 |
Vocab Quiz: Sections ≤ 4.1, 3.2 Assignment: Peer Evaluations |
Student Examples and
Problem Solutions Q & A for Exam |
Exam on Chapters 1-11 |
No Class |
|
3/2 |
Student Examples and
Problem Solutions Assignment: Peer Evaluations |
Student Examples and
Problem Solutions Assignment: Who Can Eat
Both? |
Student Examples and
Problem Solutions Comments
on Exam |
Student Examples and
Problem Solutions Lecture:
Order Properties |
|
3/9 |
No Class SPRING BREAK |
|||
|
3/16 |
Student Examples and
Problem Solutions Assignment: Peer
Evaluations |
Student Examples and
Problem Solutions Lecture:
Congruences |
Domino
Activity |
Vocab
Quiz: Read Chapter 12 (pgs 133 – 144) Induction |
|
3/23 |
Induction, cont. Student Examples and
Problem Solutions |
Student Examples and
Problem Solutions |
Student Examples and
Problem Solutions |
Read Chapter 13 (pgs 145 – 154) The
Either/Or Methods |
|
3/30 |
Student Examples and
Problem Solutions Lecture: Equivalence
Relations |
Vocab
Quiz: Student Examples and
Problem Solutions Lecture: Equivalence
Relations, cont. |
Student Examples and
Problem Solutions |
No Class |
|
4/6 |
Construction of Standard
Sets Activity |
Student Examples and
Problem Solutions |
Student Examples and
Problem Solutions |
Student Examples and
Problem Solutions Lecture: Sequences |
|
4/13 |
Student Examples and
Problem Solutions Lecture: Sequences, cont. |
Student Examples and
Problem Solutions Lecture: Sequences, cont. |
Vocab Quiz: Sections ≤ 5.2, 6.2 Student Examples and
Problem Solutions |
Student Examples and
Problem Solutions Lecture: Algebraic
Structure |
|
4/20 |
Student Examples and
Problem Solutions |
Student Examples and
Problem Solutions Lecture: Algebraic
Structure, cont |
Student Examples and
Problem Solutions |
Vocab Quiz: Student Examples and
Problem Solutions Lecture: Basic Groups |
|
4/27 |
Class Cancelled |
Student Examples and
Problem Solutions |
Student Examples and
Problem Solutions |
Student Examples and
Problem Solutions |
Final Exam
Tuesday, May 5
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 of 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.
Campus Resources
Details about Campus
Resources
The Standard Disclaimer
applies.
© Eric
Errthum, May 2015, all rights reserved.