AMB 176

MWF 10:15-11:15AM, T 11:00-12:00PM

dana.ernst@nau.edu

928.523.6852

danaernst.com

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

Mathematical reasoning in multi-step problems across different areas of mathematics. Focuses on problem solving and solution writing.

MAT 220 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.

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**.

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).

Perhaps this is sufficiently vague, but I believe that there are two essential elements to IBL. Students should as much as possible be responsible for:

- Guiding the acquisition of knowledge, and
- Validating the ideas presented. That is, students should not be looking to the instructor as the sole authority.

If you want to learn more about IBL, read my blog post titled What the Heck is IBL?

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!

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. This will be new to many of you and there may be some growing pains associated with it.

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.

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.

We will work through the problem-sequence as the semester progresses. The pace at which we cover the material depends largely on how quickly the class can digest the material in a meaningful way. In general, the difficulty of the problems will increase as the course progresses. Activities, assignments, and class discussions will be designed to introduce you to a variety of problem solving strategies.

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.

Student assessment 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.

We will not be using a 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?

We will make limited use of BbLearn this semester, which is Northern Arizona University’s default learning management system (LMS). Most course content (e.g., syllabus, course notes, homework, etc.) will be housed here on our course webpage that lives outside of BbLearn. I suggest you bookmark this page. The only thing I will use BbLearn for is to communicate grades.

Homework will be assigned each class meeting, and you are expected to complete (or try your best to complete) each assignment before walking into the next class period. All assignments should be carefully, clearly, and cleanly written. Among other things, this means your work should include proper grammar, punctuation and spelling. You will almost always write a draft of a given solution before you write down the final argument, so do yourself a favor and get in the habit of differentiating your scratch work from your submitted assignment.

Homework assignments will generally consist of solving problems from the problem sequence. Homework will be graded on a $\checkmark$-system. You are allowed (in fact, encouraged!) to modify your written solution in light of presentations made in class; however, **you are required to use the colored marker pens provided in class**. This will allow me to differentiate the work done in class versus the work you completed before class. The grade you receive on an assignment will be determined by the work you completed prior to class. I will provide more guidance with respect to this during the first couple weeks of the semester.

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.

You are allowed and encouraged to work together on homework. Yet, 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. When doing your homework, I encourage you to consult the Elements of Style for Proofs as a reference.

Your overall homework grade is worth 15% of your final grade.

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:

- 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.

Presentations will be graded using the rubric below.

Grade | Criteria |
---|---|

4 | Completely correct and clear solution/proof. Yay! |

3 | Solution/proof has minor technical flaws, some unclear language, or lacking some details. Essentially correct. |

2 | A partial explanation or solution is provided but a significant gap still exists. |

1 | Minimal progress has been made that includes relevant information & could lead to a solution/proof. |

0 | You were completely unprepared. |

However, 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 you 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! Yet, you should not come to the board to present unless you have spent time thinking about the problem and have something meaningful to contribute.

In general, I will curate the list of student presenters each class meeting. However, **students are welcome to volunteer to present**. This type of behavior will be rewarded. If more than one student volunteers, the student with the fewest number of presentations has priority. The problems chosen for presentations will come from the homework. Each student in the audience is expected to be engaged and respectful during another student’s presentation.

In order to receive a **passing grade** on the presentation portion of your grade, **you must present at least three times prior to the end of week 7 of the semester** (Friday, March 3) and **at least three times after week 7 of the semester**. Notice that I’ve described a minimum. To ensure a good grade, you should present as often as you can. Your overall performance during presentations, as well as your level of interaction/participation during class, is worth 15% of your overall grade.

We will have 7 quizzes this semester. The first one will be given during week 3 and then we will have a quiz every 2 weeks thereafter. Tentatively, the quizzes are scheduled for the following dates:

- Quiz 1: Wednesday, February 1
- Quiz 2: Wednesday, February 15
- Quiz 3:
~~Wednesday, March 1~~Friday, February 24 - Quiz 4:
~~Wednesday, March 22~~Friday, March 10 - Quiz 5:
~~Wednesday, April 5~~Wednesday, March 29 - Quiz 6:
~~Wednesday, April 19~~Wednesday, April 12 - Quiz 7:
~~Wednesday, May 3~~Wednesday, April 26

Each quiz will consist of a mixture of homework problems you encountered during the previous two weeks and new problems not previously attempted. Each problem will be graded using the following rubric:

Grade | Criteria |
---|---|

8 | This is correct and well-written mathematics! |

6-7 | This is a good piece of work, yet there are some mathematical errors or some writing errors that need addressing. |

4-5 | There is some good intuition here, but there is at least one serious flaw. |

1-3 | 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. Your overall quiz grade is worth 50% of your final grade and will be determined by your top 6 quiz scores.

We will also have a cumulative final exam that takes place on **Wednesday, May 10 at 7:30-9:30AM**. Your final exam score is worth 20% of your overall grade.

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. On the other hand, you may use each other, the course notes, your own intuition, and me.

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.

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

Category | Weight | Notes |
---|---|---|

Homework | 15% | See above for details |

Presentations & Participation | 15% | See above for details |

Quizzes | 50% | Best six out of seven |

Final Exam | 20% | Wednesday, May 10 at 7:30-9:30AM |

In general, you should expect the grades to adhere to the standard letter-grade cutoffs: A 100-90%, B 80-89%, C 70-79%, D 60-69%, F 0-59%.

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 Office of Student Life.

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.

Students are expected to treat each other with respect. Students are also expected to promote a healthy learning environment, as well as minimize distracting behaviors. In particular, you should be supportive of other students while they are making presentations. Moreover, 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.

Here are some important dates:

**Thursday, January 26:**Last day to drop/add (no W appears on transcript)**Monday, March 13-Friday, March 17:**Spring Break (no classes)**Friday, March 24:**Last day to withdraw from a course (W appears on transcript)**Wednesday, May 10:**Final Exam

There are many resources available to get help. First, I recommend that you work on homework in small groups as much as possible, and to come see me whenever necessary. 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*! Lastly, you can always email me.

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

Portions of "Class Presentations and Participation" are adapted from Carol Schumacher's *Chapter Zero Instructor Resource Manual*. The first paragraph of "An Inquiry-Based Approach" is borrowed from Robert Talbert. Lastly, I've borrowed a few phrases here and there from Bret Benesh.

Mathematics & Teaching

Northern Arizona University

Flagstaff, AZ

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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.