Course Policies

**Devote time to the course.** The Boulder Faculty Assembly has moved that

An undergraduate student should expect to spend approximately 3 hours per week outside of class for each credit hour earned.

That adds up to **9 hours per week, outside of class**, for Math 2001.

Not all of the work you need to do is collected, or even assigned. One of the goals of this course is to *learn to teach yourself mathematics*. Reading the textbook means more than casting your eyes across the page: pick apart the definitions, theorems, and proofs; *organize and reorganize them within your own mind*; *create examples* to develop intuition. Doing homework means more than handing in problems when they are assigned: *find—and create—exercises* that push the limits of your understanding, and learn to *evaluate for yourself* whether your answers are correct.

**Ask questions.** If you are studying actively, you will have questions. Use this principle to measure whether you are actively engaged.

The textbook for the course is Book of proof by Richard Hammack. It is available for free online, and bound copies can be purchased inexpensively in the usual places.

We will use this book primarily a *reference* for tools and techniques. A great deal of content that is not in the text will be introduced in class.

Instructor: Jonathan Wise

Office: Math 204

Office hours: calendar

Phone: 303 492 3018

e-mail: jonathan.wise@colorado.edu

I maintain a calendar showing times I will be available in my office. I am also happy to make an appointment if these times are not convenient for you.

Beginning in the second week of class, all homework will be submitted online, via D2L. Homework should be typed unless specified otherwise, preferably using LaTeX.

Each week, you are allowed to make **independent submissions** for
comments, but you will not receive a grade. You may later include such
a submission in a portfolio (see below). These should be submitted via
D2L.

Suggested independent submissions: a problem you have worked
independently, revision of a writing assignments, a video explaining a
concept or giving a solution to a problem, a question about a
mathematical object you have constructed. Remember to **cite any resources you use** in creating your submission, even if if the resource was just the source of a practice problem.

I encourage you to consult outside sources, use the internet, and collaborate with your peers. However, there are important rules to ensure that you use these opportunities in an academically honest way.

- Groups may submit a single assignment with multiple names on it. The effect of group work on course-wide scores will be in inverse proportion to the number of group members. Groups can make independent submissions as well, but group work in a portfolio will be considered in inverse proportion to the number of group members.
- You should not do consecutive assignments in groups unless you are specifically assigned to do so.
- Anything with your name on it must be your work and accurately reflect your understanding. You are responsible for every word in a joint submission. Division of labor, in which one group member does one half of an assignment and the other member does another half, is not permissible and will be treated as plagiarism.
- Plagiarism will be dealt with harshly. Deliberate plagiarism will be reported to the Honor Code Office.

To avoid plagiarism, you should always **cite all resources you consult**,
whether they are textbooks, tutors, websites, classmates, or any other
form of assistance. Using others' words verbatim, without attribution,
is absolutely forbidden, but so is using others' words with small
modifications. The ideal way to use a source is to study it, understand
it, put it away, use your own words to express your newfound
understanding, and then cite the source as an inspiration for your work.

The final grade is the sum of a **skills** score and a **communication** score. Each is computed initially as an average, but you may improve them by submitting **portfolios**.
The skills score is computed based on in-class quizzes and exams.
Each question is scored between 0 and 2, and the skills score is the
average of all of these scores. The communication score is based
primarily on at-home writing assignments, although some in-class
activities may contribute as well. Communication scores are also
assigned between 0 and 2 and are averaged.

With each assignment, I will also assign numerical values for A, B,
and C. At the end of the semester, I will compare your averages to the
averages of the typical A, B, and C scores and assign your score
accordingly. It is important to be aware that **scores are assigned based on standards for the end of the course**.

At any time during the semester, you may submit a **portfolio** documenting your progress towards the goals of the class. You can use a portfolio to **add new scores** to your quiz or communication average, to **replace** quiz or communication scores, or even to **remove** quiz or communication scores.

Except in special circumstances, a portfolio should be a **single PDF file** or a **physical packet of papers**
containing all of the relevant documentation. The first page of the
portfolio should be a statement explaining what the contents of the
portfolio are, which course goals they demonstrate achievement of, and
how they demonstrate that achievement. Be as detailed and explicit as
possible on the cover page. If you believe the portfolio should replace
scores you earned previously, you should explain which scores those are
and why the materials you are enclosing are a suitable replacement.
The remaining pages of the portfolio should be **previously assessed**
materials (either previously graded materials, or independent
submissions with comments), including any comments I made on those
materials.

It is incumbent on you to make it easy for me to understand your portfolio. If it is not clear what you are requesting, or if it is difficult for me to understand or find your documentation, I will return the portfolio to you with a request for clarification.

There is also a **limit on the number of changes** you can request
in one portfolio. A portfolio submission cannot change more than the
equivalent of one whole quiz or one whole writing assignment.

I will not always follow your suggestions, of course, but I will follow suggestions that are sufficiently well-argued and supported. Below are some examples of ways you could use a portfolio:

*Example 1:* Argue that you received a low score on an early
assignment because you did not understand a concept that you later
understood better. Include both the early assignment and the later
assignment, make clear what the concept is, and how both assignments
test it. If I am persuaded, I will remove the earlier assignment from
the calculation of your grade.

*Example 2:* After receiving a low score on a paper, you
independently submit a series of revisions. You may request that I
replace your score on the earlier paper with a score for a revision.
You should submit both the original paper and the revisions, including
all comments I have made on them. Your cover page should address the
goals of the course and how your revision better achieves those goals
than your original version.

*Example 3:* You miss a quiz, but find or create a collection
of problems that examine the same concepts. You submit those problems
as an independent submission and receive my comments on them. You may
then request that the quiz you missed be replaced with a grade for the
independent submission. You should submit the original quiz and your
substitute problems, along with my comments, in your portfolio.

*Example 4:* You find an interesting problem in the textbook
(or some other source). You submit a solution to this problem
independently and receive comments on it. You may then submit your
solution in a portfolio and ask for it to be included in your
communication score. Make sure to document which course goals your
independent work addresses.

If you miss a quiz, you will receive a score of zero on it. However, you can replace the quiz by making an independent submission and then submitting a portfolio. I will usually offer small time extensions on late assignments, provided that it does not interfere with my grading. However, very late assignments should go through the independent submission / portfolio process.

To achieve an A, you should be able to do all of the following:

- Reformulate mathematical statements so that they are suitable for proof or disproof. Reformulate statements that are not explicitly about mathematical objects in mathematical terms. Prove or disprove the statements you come up with.
- Write proofs that involve multiple intermediate steps and multiple proof techniques.
- Construct mathematical objects, verify their properties, and use them effectively in proofs.
- Construct examples and counterexamples without prompting to achieve understanding and clarity.
- Identify gaps and errors in mathematical reasoning. Discern the essential assumptions of a mathematical argument.

To achieve a B, you should be able to do all of the following:

- Reformulate mathematical statements so that they are suitable for proof or disproof.
- Correctly select and implement the different proof techniques (direct, indirect, contradiction, induction).
- Use axioms, concepts (sets, functions, numbers, etc.), and definitions correctly and effectively in proofs.
- Use clear and consistent notation.
- Construct examples and counterexamples to determine the truth or falsity of a mathematical statement.

To achieve a C, you should be able to do all of the following:

- Reformulate mathematical statements as suggested.
- Write coherent mathematical paragraphs.
- Write proofs using any of the proof techniques (direct proof, indirect proof (proof by contraposition), proof by contradiction, or induction) when the technique is suggested.
- Prove statements about mathematical objects by applying the definitions.
- Determine, with justification, whether a mathematical object satisfies a particular definition.
- Construct examples and counterexamples as prompted.

To achieve a D, you should be able to:

- Make direct and indirect (contrapositive) logical inferences.
- Manipulate the mathematical objects we study correctly in explicit situations.
- Write clear, unambiguous, mathematical sentences.

One goal of this course is to develop skills working with various mathematical objects. When assessing your skills with these objects, I use the following principles to obtain a letter grade:

- A: Create instances of the objects and use them to solve problems and prove statements not already phrased in terms of the objects.
- B: Understand relationships between the topic and other topics; reason about them and draw conclusions.
- C: Understand relationships between the objects and work with them in less explicit examples.
- D: Perform operations with the objects in explicit examples.

Here are the actual objects we will be working with. This list is *subject to change* depending on time constraints and interest of the class.

- Sets
- Perform set operations (union, intersection, product, powerset).
- Discern relationships between objects and sets (membership, containment, subset, superset).
- Compute the cardinalities of sets and understand the relationships between set operations and cardinalities.
- Parse set builder notation to understand the members of a set, and use set builder notation to construct sets.
- Functions
- Determine whether function is well-defined.
- Determine whether functions are injective, surjective, bijective.
- Deduce things about the cardinalities of sets from functions.
- Relations
- Verify properties of a relation (reflexive, irreflexive, symmetric, antisymmetric, transitive, equivalence relation, partial order, total order, function, injective, surjective, bijective).
- Equivalence classes, relation to surjections.
- Translate between equivalence relations, partitions, and surjections.
- Logic
- Translate correctly between symbolic sentences and English.
- Find logical sentences equivalent to or opposite to a given one.
- Counting
- Use inclusion-exclusion to compute the sizes of sets.
- Translate counting problems to questions about sets.
- Rephrase counting problems as sampling with or without replacement to solve them.
- Number theory
- Know and correctly apply the definitions of divisibility, prime, composite.
- Make calculations using modular addition, subtraction, and multiplication.