Course Info

Title: MAT 232: Introduction to Mathematical Reasoning
Semester: Fall 2020
Credits: 3
Section: 1
Time: 10:20-11:10AM MWF
Location: SAS 201A

Instructor Info

  Dana C. Ernst, PhD
  AMB 176
  12:30-1:30PM on MTh, 11:30AM-12:30PM on WF
  dana.ernst@nau.edu
  928.523.6852
  danaernst.com

Prerequisites

MAT 136 with a grade greater than or equal to C.

Course Description

MAT 232 is an introductory course in mathematical reasoning in multi-step problems across different areas of mathematics. The goal is to use elementary mathematical tools to solve more complex problems in already familiar areas of study such as precalculus, basic number theory, geometry, and discrete mathematics, instead of teaching new mathematical tools that are used in straightforward one-step exercises. The focus is on problem solving and solution writing.

Learning Outcomes

Upon successful completion of the course, you will be able to:

  • Solve multi-step, complex problems in elementary areas of mathematics using common problem solving strategies;
  • Judge what constitutes a solid mathematical argument;
  • Write readable and concise solutions using correct English with some mathematical notation.

Assessment of learning outcomes will be based on regular class attendance, participation during class meetings, consistent progress on assigned homework problems, in-class quizzes, and a comprehensive final examination. Homework may include newly assigned problems, as well as formal write-ups of previously explored problems.

Course Content

The content of the course includes, but is not limited to:

  • Problem solving strategies such as: use of figures and diagrams, use of variables, considering simpler cases, recognizing patterns, conjectures, counterexamples, breaking up into sub-problems, working backwards, case analysis, considering an extreme case, contradiction, induction, pigeon hole principle, symmetry, algorithms, coding, persistence;
  • Writing solutions such as: communicating a solution, planning, organization, lemmas, naming, figures, concise vs. detailed, proofreading;
  • Mathematical thinking such as: generalization, converse, hidden connections, new problem construction, open ended problems, ill-defined problems.

What is This Course Really About?

The world is changing faster and faster. An education must prepare a student to ask and explore questions in contexts that do not yet exist. That is, we need individuals capable of tackling problems they have never encountered and to ask questions no one has yet thought of.

The focus of this course is on reasoning and communication through problem solving and written mathematical arguments in order to provide students with more experience and training early in their university studies. The goal is for the students to work on interesting yet challenging multi-step problems that require almost zero background knowledge. The hope is that students will develop (or at least move in the direction of) the habits of mind of a mathematician. The problem solving of the type in this course is a fundamental component of mathematics that receives little focused attention elsewhere in our program. There will be an explicit focus on students asking questions and developing conjectures.

In addition to helping students develop procedural fluency and conceptual understanding, we must prepare them to ask and explore new questions after they leave our classrooms—a skill that we call mathematical inquiry.

An ounce of practice is worth more than tons of preaching.

An Inquiry-Based Approach

This is not a lecture-oriented class or one in which mimicking prefabricated examples will lead you to success. You will be expected to work actively to construct your own understanding of the topics at hand with the readily available help of me and your classmates. Many of the concepts you learn and problems you work on will be new to you and ask you to stretch your thinking. You will experience frustration and failure before you experience understanding. This is part of the normal learning process. If you are doing things well, you should be confused at different points in the semester. The material is too rich for a human being to completely understand it immediately. Your viability as a professional in the modern workforce depends on your ability to embrace this learning process and make it work for you.

In order to promote a more active participation in your learning, we will incorporate ideas from an educational philosophy called inquiry-based learning (IBL). Loosely speaking, IBL is a student-centered method of teaching mathematics that engages students in sense-making activities. Students are given tasks requiring them to solve problems, conjecture, experiment, explore, create, and communicate. Rather than showing facts or a clear, smooth path to a solution, the instructor guides and mentors students via well-crafted problems through an adventure in mathematical discovery. Effective IBL courses encourage deep engagement in rich mathematical activities and provide opportunities to collaborate with peers (either through class presentations or group-oriented work). If you want to learn more about IBL, read my blog post titled What the Heck is IBL?.

Don’t fear failure. Not failure, but low aim, is the crime. In great attempts it is glorious even to fail.

Much of the course will be devoted to students presenting their proposed solutions/proofs on the board and a significant portion of your grade will be determined by how much mathematics you produce. I use the word “produce” because I believe that the best way to learn mathematics is by doing mathematics. Someone cannot master a musical instrument or a martial art by simply watching, and in a similar fashion, you cannot master mathematics by simply watching; you must do mathematics!

In any act of creation, there must be room for experimentation, and thus allowance for mistakes, even failure. A key goal of our community is that we support each other—sharpening each other’s thinking but also bolstering each other’s confidence—so that we can make failure a productive experience. Mistakes are inevitable, and they should not be an obstacle to further progress. It’s normal to struggle and be confused as you work through new material. Accepting that means you can keep working even while feeling stuck, until you overcome and reach even greater accomplishments.

You will become clever through your mistakes.

Furthermore, it is important to understand that solving genuine problems is difficult and takes time. You shouldn’t expect to complete each problem in 10 minutes or less. Sometimes, you might have to stare at the problem for an hour before even understanding how to get started. In fact, solving difficult problems can be a lot like the clip from the Big Bang Theory located here.

In this course, everyone will be required to

  • read and interact with course notes and textbook on your own;
  • write up quality solutions/proofs to assigned problems;
  • present solutions/proofs on the board to the rest of the class;
  • participate in discussions centered around a student’s presented solution/proof;
  • call upon your own prodigious mental faculties to respond in flexible, thoughtful, and creative ways to problems that may seem unfamiliar at first glance.

As the semester progresses, it should become clear to you what the expectations are.

Tell me and I forget, teach me and I may remember, involve me and I learn.

Course Structure

Class meetings will consist of discussion of problems, student-led presentations, and group work focused on problems selected by the instructor. A typical class session may include:

  • Informal student presentations of progress on previously assigned homework problems;
  • Summary of major steps and techniques of the solution of a finished problem;
  • Exploration of alternative approaches, possible generalizations, consequences, special cases, converse;
  • Discussion of relationships to previously assigned or solved problems;
  • Assignment of new problems;
  • Explanation of unfamiliar mathematical concepts as needed.

The impediment to action advances action. What stands in the way becomes the way.

Rights of the Learner

As a student in this class, you have the right:

  1. to be confused,
  2. to make a mistake and to revise your thinking,
  3. to speak, listen, and be heard, and
  4. to enjoy doing mathematics.

You may encounter many defeats, but you must not be defeated.

Commitment to the Learning Community

In our classroom, diversity and individual differences are respected, appreciated, and recognized as a source of strength. Students in this class are encouraged and expected to speak up and participate during class and to carefully and respectfully listen to each other. Every member of this class must show respect for every other member of this class. Any attitudes or actions that are destructive to the sense of community that we strive to create are not welcome and will not be tolerated. In summary: Be good to each other. I would appreciate private responses to the following question: Are there aspects of your identity that you would like me to attend to when forming groups, and if so, how?

Students are also expected to minimize distracting behaviors. In particular, every attempt should be made to arrive to class on time. If you must arrive late or leave early, please do not disrupt class. Please turn off the ringer on your cell phone. I do not have a strict policy on the use of laptops, tablets, and cell phones. You are expected to be paying attention and engaging in class discussions. If your cell phone, etc. is interfering with your ability (or that of another student) to do this, then put it away, or I will ask you to put it away.

Problem Collection

We will not be using a formal textbook this semester, but rather a problem sequence designed for this course. The problem collection will be available on the course webpage. We will not be covering every detail of the notes and the only way to achieve a sufficient understanding of the material is to be digesting the reading in a meaningful way. You should be seeking clarification about the content of the notes whenever necessary by asking questions in class or posting questions to the course forum. Here’s one of my favorite quotes about reading mathematics.

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?

Rules of the Game

Reviewing material from previous courses and looking up definitions and theorems/facts you may have forgotten is fair game. However, when it comes to completing assignments for this course, you should not look to resources outside the context of this course for help. That is, you should not be consulting the web, other texts, other faculty, or students outside of our course in an attempt to find solutions to the problems you are assigned. On the other hand, you may use each other, the course notes, me, and your own intuition. If you feel you need additional resources, please come talk to me and we will come up with an appropriate plan of action. Please read NAU’s Academic Integrity Policy.

Quizzes and Final Exam

We will have 6 quizzes this semester. Tentatively, the quizzes are scheduled for the following Fridays:

  • Quiz 1: August 28 (week 3)
  • Quiz 2: September 11 (week 5)
  • Quiz 3: September 25 (week 7)
  • Quiz 4: October 9 (week 9)
  • Quiz 5: October 23 (week 11)
  • Quiz 6: November 6 (week 13)

Each quiz will consist of a mixture of homework problems you encountered during the previous two weeks and new problems not previously attempted. During each quiz, you will have the opportunity to collaborate with your peers for a few minutes. The amount of time you have to collaborate may diminish over the course of the semester. Each problem will be graded using the following rubric:

Grade Criteria
4 This is correct and well-written mathematics!
3 This is a good piece of work, yet there are some mathematical errors or some writing errors that need addressing.
2 There is some good intuition here, but there is at least one serious flaw.
1 I don't understand this, but I see that you have worked on it.
0 I believe that you have not worked on this problem enough or you didn't submit any work.

Make-up quizzes will only be given under extreme circumstances, as judged by me. In general, it will be best to communicate conflicts ahead of time. Each Quiz is worth 10% of your overall grade (Quizzes constitute 60% of your overall grade). We will also have a cumulative final exam that takes place on Friday, November 20 at 10:00AM-12:00PM. Your Final Exam grade is worth 20% of your overall grade.

Homework

You are allowed and encouraged to work together on homework. However, each student is expected to turn in their own work. In general, late homework will not be accepted. However, you are allowed to turn in up to three late homework assignments with no questions asked. Unless you have made arrangements in advance with me, homework turned in after class will be considered late. Your overall homework grade will be worth 10% of your final grade.

On each homework assignment, please write (i) your name, (ii) name of course, and (iii) Homework number. You can find the list of assignments on the homework page. I reserve the right to modify the homework assignments as I see necessary.

Homework will be assigned almost every class meeting, and students are expected to complete (or try their best to complete) each assignment before walking into the next class period. Homework will generally consist of completing exercises and proving theorems from the textbook. In addition to completing the assigned problems, I also want you to assess your own work. Next to each problem, I want you to write down a score between 0 and 4 that represents your perception of the validity and quality of your proposed solution/proof. Consider using the following rubric.

Grade Criteria
4 This is correct and well-written mathematics!
3 This is a good piece of work, yet there are some mathematical errors or some writing errors that need addressing.
2 There is some good intuition here, but there is at least one serious flaw.
1 I don't understand this, but I see that you have worked on it; come see me!
0 I believe that you have not worked on this problem enough or you didn't submit any work.

Not completing the self-assessment step may impact the score on your homework. On the day that a homework assignment is due, the majority of the class period will be devoted to students presenting some subset (maybe all) of the proofs/solutions that are due that day.

The following framework assumes that we will be in a remote setting. If we happen to return to face-to-face classes, we will adjust in a natural way. With a few exceptions, homework assignment will consist of two parts.

Part 1: Prior to the start of class, you will need to capture your handwritten work digitally and then upload a PDF to BbLearn. There are many free smartphone apps for doing this. I use TurboScan on my iPhone. Submitting your work prior to class allows me to see what you accomplished outside of class. Part 1 will be assessed using the following rubric.

Grade Criteria
4 Sufficient effort was put into nearly all the problems.
2 Some problems were omitted and/or sufficient effort was not exhibited.
1 Many problems omitted and/or minimal effort exhibited.
0 Assignment was not turned in.

Part 2: During class, we will discuss most of the problems that are due that day. While we are discussing them, you should either annotate your work and/or take notes on separate paper. It’s expected that most of the work you did prior to class will need to be refined. It is your responsibility to process this in some way. Annotating your work or taking notes will increase the chances that you are processing the work in a meaningful way. If you choose to annotate your work, please use a different color than what you originally used to complete your assignment. After class, you will need to capture your annotations/notes digitally and then upload a PDF to BbLearn. Part 2 will be assessed using the following rubric.

Grade Criteria
2 Student annotated their work in a meaningful way and/or took sufficient notes.
1 Student annotated their work and/or took notes but effort and/or attention to detail was insufficient.
0 Student did not submit this portion of the assignment.

I write one page of masterpiece to ninety-one pages of shit.

x

Please understand that the purpose of the homework assignments is to teach you to solve problems. It is not expected that you started the class with this skill; hence, some low grades are to be expected. However, I expect that everyone will improve dramatically. Your Homework grade is worth 10% of your overall grade.

Presentations and Participation

General Comments

The problems chosen for presentations will come from the Daily Homework assignments. Though the atmosphere in this class should be informal and friendly, what we do in the class is serious business. In particular, the presentations made by students are to be taken seriously since they spearhead the work of the class. Here are some of my expectations for the presenter:

  • The purpose of class presentations is not to prove to me that the presenter has done the problem. It is to make the ideas of the solution clear to the other students.
  • Presenters should explain their reasoning as they go along, not simply write everything down and then turn to explain.
  • Fellow students are allowed to ask questions at any point and it is the responsibility of the person making the presentation to answer those questions to the best of their ability.
  • Since the presentation is directed at the students, the presenter should frequently make eye-contact with the students in order to address questions when they arise and also be able to see how well the other students are following the presentation.

Alone we can do so little; together we can do so much.

Assessing Presentations

Presentations will be assessed using the following criteria.

Specification Criteria
U Unsatisfactory. Minimal progress was made that included relevant information or the student was unprepared.
I In progress. The student made an honest attempt at the problem but recognized a flaw that prevented them from being able to complete the problem during the presentation. Alternatively, the student reported on their current progress on a problem and attempted to convey where or why they are currently "stuck".
M Meets Expectations. The student demonstrated an understanding of the problem and presented the key ideas. Perhaps some details were omitted or interesting mistakes were made. The presentation led to fruitful class discussion.
E Exceeds Expectations. The presentation was flawless and the student demonstrated keen insight into the problem. The presentation led to fruitful class discussion.

You should aim to avoid unsatisfactory (U) presentations. An in progress (I) presentation should not be viewed as a bad thing as each of us will occasionally get stuck. However, you should strive for the majority of your presentations to meet (M) or exceed (E) expectations. Most presentations will meet expectations (M) while presentations that exceed expectations (E) will be rare. You should not let the rubric deter you from presenting if you have an idea about a solution/proof that you’d like to present but are worried that your solution/proof is incomplete or you are not confident your solution/proof is correct. You will be rewarded for being courageous and sharing your creative ideas! In my view, an interestingly wrong solution or proof makes for the best presentation since it generates the best discussion. This is really what we are after. On the other hand, you should not come to the board to present unless you have spent time thinking about the problem and have something meaningful to contribute.

I alone cannot change the world, but I can cast a stone across the waters to create many ripples.

Selecting Presenters

In general, I will curate the list of student presenters each class meeting. A presenter is a student that either volunteered (V) or was chosen (C) by me. Volunteering is encouraged, but being chosen without volunteering isn’t bad. If more than one student volunteers for a specific problem, the student with the fewest number of presentations has priority. I reserve the right to decline your offer to present. This may happen if you are volunteering too often (and hence removing another student’s opportunity to present) or if I know in advance that another student’s presentation will lead to a fruitful discussion.

If you are chosen to present but would prefer not to present that particular problem, you can either negotiate presenting a different problem or take a pass (P). You may elect to pass at most two times during the semester, after which a presentation will be deemed unsatisfactory (U). By default, if you have an unexcused absence on a day when you have been chosen to present, then your presentation will be recorded as a pass (P) unless you have already exhausted your two passes, in which case the presentation will be recorded as unsatisfactory (U).

Specification Criteria
V Volunteered. Student volunteered during class or in advance to present.
C Chosen. Student was selected by the instructor and agreed to present.
P Pass. Student was selected to present, but asked to take a pass. Allowed at most two.

In summary, for each student presentation, I will record one of V, C, or P. In the case of V or C, I will also record one of U, I, M, or E based on the rubric given above.

Participation

You are expected to respectfully participate and contribute to class discussions. This includes asking relevant and meaningful questions to both the instructor and your peers. Moreover, you are expected to be engaged and respectful during another student’s presentation. Posting questions and/or responses in our Discord server counts as class participation, but posting in Discord is not required. Your class participation will be assessed as follows.

Specification Criteria
U Unsatisfactory. Student was often disengaged or disrespectful. Alternatively, the student regularly missed class.
M Meets Expectations. Student was consistently respectful, engaged, and contributed to meaningful class discussions. In addition, the student regularly attends class.
E Exceeds Expectations. Student's presence in the classroom truly enhances the learning environment.

Determining Presentation and Participation Grades

Your Presentation and Participation grade is determined by your frequency and ability to foster productive class discussions through presentations and audience participation. The greatest determining factor in your Presentation and Participation grade is your willingness to present often. You should aim to present at least once prior to each quiz. The table below provides a summary of how your Presentation and Participation grade will be determined.

Grade Range Criteria
90-100% Student receives M or E for participation. Student averages at least 3 presentations every four weeks. Student often volunteers to present and some of these problems are challenging. Most presentations receive M or E.
80-89% Student receives M or E for participation. Student averages at least 2 presentations every four weeks. Student occasionally volunteers to present. Most presentations receive M.
70-79% Student receives M for participation. Student averages less than 2 presentations every four weeks. Student rarely volunteers to present and actively avoids presenting challenging problems. Some presentations receive U.
60-69% Student receives U for participation. Student rarely presents and actively avoids presenting challenging problems. Some presentations receive U.
Below 60% Student receives U for participation. Student rarely or never presents and has completely disengaged from the class community.

I anticipate that most students will fall in the 80-89% range. Your Presentation and Participation grade is worth 10% of your overall grade.

I must not fear.
Fear is the mind-killer.
Fear is the little-death that brings total obliteration.
I will face my fear.
I will permit it to pass over me and through me.
And when it has gone past I will turn the inner eye to see its path.
Where the fear has gone there will be nothing.
Only I will remain.

Attendance

Regular attendance is expected and is vital to success in this course, but you will not explicitly be graded on attendance. Yet, repeated absences may impact your participation grade (see above). Students can find more information about NAU’s attendance policy on the Academic Policies page.

Extra Credit

The only thing I will award extra credit for is finding typos on course materials (e.g., course notes, quizzes, syllabus, webpage). This includes broken links on the webpage. However, it does not include the placement of commas and such. If you find a typo, I will add one percentage point to your next quiz. You can earn at most two percentage points per quiz and at most five percentage points over the course of the semester. They’re is a typo right here.

Basis for Evaluation

In summary, your final grade will be determined by your scores in the following categories.

Category Weight Notes
Homework 10% See above for details.
Presentations & Participation 10% See above for details.
Quizzes 60% Mixture of previously encountered problems and new problems.
Final Exam 20% Friday, November 20 at 10:00AM-12:00PM

Department and University Policies

You are responsible for knowing and following the Department of Mathematics and Statistics Policies (PDF) and other University policies listed here (PDF). More policies can be found in other university documents, especially the NAU Student Handbook (see appendices) and the website of the Student Outreach and Support (SOS).

As per Department Policy, cell phones, MP3 players and portable electronic communication devices, including but not limited to smart phones, cameras and recording devices, must be turned off and inaccessible during in-class tests. Any violation of this policy will be treated as academic dishonesty.

Important Dates

Here are some important dates:

  • Friday, August 21: Last day to Drop courses without a “W”
  • Monday, September 7: Labor Day (no classes)
  • Monday, October 19: Last day to drop individual courses without a petition
  • Wednesday, November 11: Veteran’s Day (no classes)
  • Wednesday, November 11: Last Day to withdraw from all classes in session
  • Thursday, November 19: Reading Day (no classes)
  • Friday, November 20: Final Exam (10:00AM-12:00PM)

Getting Help

There are many resources available to get help. First, you are allowed and encouraged to work together on homework. However, each student is expected to turn in their own work. You are strongly encouraged to ask questions in our Discord discussion group, as I (and hopefully other members of the class) will post comments there for all to benefit from. You are also encouraged to stop by during my office hours and you can always email me. I am always happy to help you. If my office hours don’t work for you, then we can probably find another time to meet. It is your responsibility to be aware of how well you understand the material. Don’t wait until it is too late if you need help. Ask questions!

Changes to the Syllabus

Any changes to this syllabus made during the term will be properly communicated to the class.

If you want to sharpen a sword, you have to remove a little metal.


Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
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Land Acknowledgement

  Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.