AMB 176

MoTu 2-3:30PM, Tu 10-11:30AM, Fr 10:15-11:15AM

dana.ernst@nau.edu

928.523.6852

danaernst.com

(MAT 137 and (MAT 226 or MAT 232) with grades of C or better in each); and (Foundation English Requirement or HON 190)

The course trains students on methods and techniques of mathematical communication, focusing on proofs but also covering expository writing and problem-solving explanations.

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

- Write a readable and mathematically rigorous proof.
- Express in writing, knowledge of the terminology, concepts, basic properties and methodology of symbolic logic, set theory, relations and functions, mathematical induction, cardinality, and number systems.
- Propose useful definitions and make correct deductions from definitions.
- Identify correct proof structures and criticize incorrect proof structures.

The primary objective of this course is to develop skills necessary for effective proof writing. Students will improve their ability to read and write mathematics. Successful completion of MAT 320 provides students with the background necessary for upper division mathematics courses.

Aside from the obvious goal of wanting you to learn how to write rigorous mathematical proofs, one of my principle ambitions is to make you independent of me. Nothing else that I teach you will be half so valuable or powerful as the ability to reach conclusions by reasoning logically from first principles and being able to justify those conclusions in clear, persuasive language (either oral or written). Furthermore, I want you to experience the unmistakable feeling that comes when one really understands something thoroughly. Much “classroom knowledge” is fairly superficial, and students often find it hard to judge their own level of understanding. For many of us, the only way we know whether we are “getting it” comes from the grade we make on an exam. I want you to become less reliant on such externals. When you can distinguish between really knowing something and merely knowing about something, you will be on your way to becoming an independent learner. Lastly, it is my sincere hope that all of us (myself included) will improve our oral and written communications skills.

This course will likely be different than any other math class that you have taken before for two main reasons. First, you are used to being asked to do things like: “solve for $x$,” “take the derivative of this function,” “integrate this function,” etc. Accomplishing tasks like these usually amounts to mimicking examples that you have seen in class or in your textbook. The steps you take to “solve” problems like these are always justified by mathematical facts (theorems), but rarely are you paying explicit attention to when you are actually using these facts. Furthermore, justifying (i.e., proving) the mathematical facts you use may have been omitted by the instructor. And, even if the instructor did prove a given theorem, you may not have taken the time or have been able to digest the content of the proof.

Unlike previous courses, this course is all about “proof.” Mathematicians are in the business of proving theorems and this is exactly our endeavor. For the first time, you will be exposed to what “doing” mathematics is really all about. This will most likely be a shock to your system. Considering the number of math courses that you have taken before you arrived here, one would think that you have some idea what mathematics is all about. You must be prepared to modify your paradigm. The second reason why this course will be different for you is that the method by which the class will run and the expectations I have of you will be different. In a typical course, math or otherwise, you sit and listen to a lecture. (Hopefully) These lectures are polished and well-delivered. You may have often been lured into believing that the instructor has opened up your head and is pouring knowledge into it. I absolutely love lecturing and I do believe there is value in it, but I also believe that in reality most students do *not* learn by simply listening. You must be active in the learning you are doing.

All of the secondary skills you will develop in this course are highly valued by society. Whether you become a teacher, a lawyer, an engineer, or an artist, what differentiates you from your competition is your ability to think critically at a high level, collaborate professionally, and communicate effectively.

The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.

Aside from the obvious goal of wanting you to learn how to write rigorous mathematical proofs, one of my principle ambitions is to make you independent of me. Nothing else that I teach you will be half so valuable or powerful as the ability to reach conclusions by reasoning logically from first principles and being able to justify those conclusions in clear, persuasive language (either oral or written). Furthermore, I want you to experience the unmistakable feeling that comes when one really understands something thoroughly. Much “classroom knowledge” is fairly superficial, and students often find it hard to judge their own level of understanding. For many of us, the only way we know whether we are “getting it” comes from the grade we make on an exam. I want you to become less reliant on such externals. When you can distinguish between really knowing something and merely knowing about something, you will be on your way to becoming an independent learner. Lastly, it is my sincere hope that all of us (myself included) will improve our oral and written communications skills.

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

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 Inquiry-Based Learning Resource page.

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.

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.

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

- to be confused,
- to make a mistake and to revise your thinking,
- to speak, listen, and be heard, and
- to enjoy doing mathematics.

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

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.

Our textbook for the semester is *An Introduction to Proof via Inquiry-Based Learning*, which is a free and open-source textbook that was written by me (Dana C. Ernst). The textbook is designed to be used with an inquiry-based learning (IBL) approach to an introduction to proof course. The textbook is avaiable in multiple formats:

- PDF of textbook
- PreTeXt/HTML version of textbook
- Low cost print version for sale via AMS/MAA Press
- PDF of textbook that matches the current print version by MAA Press
- Errata
- Source code available via GitHub

You can also find a link to the textbook one the Course Materials page.

I will not be covering every detail of the textbook 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 whenever necessary by asking questions. 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?

Reviewing material from previous courses and looking up definitions and theorems you may have forgotten is fair game. Since mathematical reasoning, problem solving, and critical thinking skills are part of the learning outcomes of this course, all assignments should be prepared by the student. Developing strong competencies in this area will prepare you to be a lifelong learner and give you an edge in a competitive workplace. When it comes to completing assignments for this course, unless explicitly told otherwise, you should *not* look to resources outside the context of this course for help. That is, you should not be consulting the web (e.g., Chegg and Course Hero), generative artificial intelligence tools (e.g., ChatGPT), mathematics assistive technologies (e.g., Wolfram Alpha and Photomath), 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 textbook, me, and your own intuition. You are allowed and encouraged to work together on homework. Yet, each student is expected to turn in their own work.

In this course, we may use generative AI tools (such as ChatGPT) or AI mathematics assistive technologies (such as Wolfram Alpha) to examine the ways in which these kinds of tools may inform our exploration of mathematics content. You will be informed as to when and how these tools will be used, along with guidance for attribution if/as needed. Any use of generative AI tools outside of these parameters constitutes plagiarism and a violation of the University’s Academic Integrity Policy. Please read NAU’s Academic Integrity Policy.

The ultimate goal is for each individual student to learn and to be successful. So, if you feel you need additional resources or support, please come talk to me and we will come up with an appropriate plan of action.

The following are examples (not an exhaustive list) of behaviors that could constitute cheating and/or plagiarism. You should *not* be doing these things.

- Copying solutions or portions of solutions from another person
- Submitting solutions (in part or whole) by multiple students that identically match, especially in peculiar details
- Having another person complete your homework problems for you
- Using any applications or websites (e.g., Course Hero, Chegg, ChatGPT, WolframAlpha, PhotoMath) to complete problems or portions of problems (even if only used on one step that you are stuck on)
- Anything that takes solutions or portions of solutions and attempts to pass them off as your own ideas and work

The following are examples (not an exhaustive list) of behaviors that do not constitute cheating and/or plagiarism. You should be doing these things.

- Having a conversation with a classmate about a homework problem to compare methods and discuss strategy
- Collaborating with a classmate on a homework problem (not copying)
- Asking questions about a homework problem on our course forum
- Responding to questions on our course forum in the form of feedback or guidance
- Asking the instructor for assistance or a hint

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 **three late homework assignments**. 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” appendix in the textbook. Your overall homework grade will be worth 20% of your final grade.

On each homework assignment, please write (i) your name, (ii) name of course, and (iii) Daily/Weekly 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. Not completing the self-assessment step may impact the score on your homework. 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 worked on the problem, but made minimal progress. |

0 | I did not attempt this problem. |

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. You are allowed (in fact, encouraged!) to modify your written work in light of presentations made in class; however, **you are required to use a different color than what you used to complete your homework**. In the past, I provided colored pens in class, but due to the pandemic, please provide your own colored pen. 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. Homework 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. |

In addition to the Daily Homework, we will also have Weekly Homework assignments. For most of these assignments, you will be required to submit 2-3 formally written proofs. Some or all of these problems will come directly from the Daily Homework assigned the previous week. You are required to type your submission using LaTeX. Each problem on the Weekly Homework assignments is subject to 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. |

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 be prepared to address questions when they arise.

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

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.

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. The most common pair will likely be CM (i.e., student was chosen to present and presentation met expectations).

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 Q&A Discussion board counts as class participation, but posting in Q&A Discussion board 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. |

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 twice prior to each midterm exam. 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 prior to each of the exams. 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 prior to each of the exams. Student occasionally volunteers to present. Most presentations receive M. |

70-79% | Student receives M for participation. Student averages less than 2 presentations prior to each of the exams. 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 20% 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.

There will be two midterm exams and a cumulative final exam. Each exam will consist of an in-class portion and possibly a take-home portion. Exam 1 and Exam 2 are *tentatively* scheduled for **Monday, October 9** (week 7) and **Wednesday, November 15** (week 12), respectively. Each of Exam 1 and Exam 2 will be worth 20% of your overall grade. The final exam will be on **Wednesday, December 13** at **7:30-9:30AM** and is worth 20% of your overall grade. Each of the exams are likely to also include a take-home portion, which you will have a few days to complete. Make-up exams will only be given under extreme circumstances, as judged by me. In general, it will be best to communicate conflicts ahead of time.

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

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.

The only thing I will award extra credit for is finding typos on course materials (e.g., textbook, exams, 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 exam. You can earn at most two percentage points per exam. Extra credit points are only available to the first person to notify me of the error. They’re is a typo right here.

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

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

Homework | 20% | See above for requirements |

Presentations & Participation | 20% | See above for requirements |

Exam 1 | 20% | In-class portion on October 9, possible take-home portion |

Exam 2 | 20% | In-class portion on November 15, possible take-home portion |

Final Exam | 20% | Wednesday, December 13, 7:30-9:30AM |

It is not the critic who counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood; who strives valiantly; who errs, who comes short again and again, because there is no effort without error and shortcoming; but who does actually strive to do the deeds; who knows great enthusiasms, the great devotions; who spends himself in a worthy cause; who at the best knows in the end the triumph of high achievement, and who at the worst, if he fails, at least fails while daring greatly, so that his place shall never be with those cold and timid souls who neither know victory nor defeat.

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).

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.

Here are some important dates:

**September 4:**Labor Day (no classes)**November 10**Veteran’s Day (no classes)**November 23-24:**Thanksgiving (no classes)**December 13:**Final Exam (7:30-9:30AM)

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 Q&A Discussion board, 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*!

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.

Portions of "Presentations and Participation" are adapted from Carol Schumacher's *Chapter Zero Instructor Resource Manual*. The first and fourth paragraphs of "An Inquiry-Based Approach" are borrowed from Robert Talbert and Joshua Bowman, respectively. The "Rights of the Learner" were adapted from a similar list written by Crystal Kalinec-Craig. The first paragraph of "Commitment to the Learning Community" is a modified version of statement that Spencer Bagley has in his syllabi. Lastly, I've borrowed a few phrases here and there from Bret Benesh.

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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 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.