Introduction to Mathematical Reasoning
Title: Introduction to Mathematical Reasoning
Course Number: MAT 220
Semester: Spring 2015
Time: MWF at 11:30-12:20PM
Location: AMB 162
Instructor: Dr. Dana C. Ernst
Office: AMB 176
Office Phone: 928.523.6852
Office Hours: 10:00-11:15AM on Mondays, Wednesdays, Fridays (or by appointment)
Course Information and Policies
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.
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.
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.
Learning Management System
We will make limited use of Bb Learn 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. In addition, we will utilize a Google Group to facilitate out of class discussion. I will send the class an invite to our Google Group and briefly discuss its use. The only thing I will use BbLearn for is to communicate grades.
Course Structure and Approach
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 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 what Paul Halmos has to say 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 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. At the beginning, students will work in small groups most of the time, but the course will gradually become more individualized as you gain confidence and experience.
Student Learning Expectations/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 Student Learning Outcomes
Student assessment will be based on regular class attendance, participation during class meetings, consistent progress on assigned problems, 2 midterm examinations, and a comprehensive final examination. Homework may include newly assigned problems, as well as formal write-ups of previously explored problems. In addition, some assignments may require students to write simple computer programs.
There will be two midterm exams and a cumulative final exam. Each exam will be worth 20% of your overall grade and may consist of both an in-class portion and a take-home portion. The in-class portions of the midterm exams are tentatively scheduled for
Monday, February 23 Wednesday, February 25 and Monday, April 13, and the in-class portion of the final exam will be on Wednesday, May 6 at 10:00–12:00PM. 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.
You are allowed and encouraged to work together on homework. Yet, each student is expected to turn in his or her own work. In general, late homework will not be accepted. However, you are allowed to turn in up to 5 homework assignments late 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 25% of your final grade.
Generally, homework will fall into 3 categories.
Daily Homework: 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.
The Daily Homework will generally consist of solving problems from the problem sequence. Daily 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. I will provide more guidance with respect to this during the first couple weeks of the semester.
Weekly Homework: In addition to the Daily Homework, you will also be required to submit 2-3 formally written solutions/proofs each week. Many of the problems chosen for the Weekly Homework will be a designated subset of the Daily Homework, but there may also be a few new problems thrown into the mix. You are required to type your submission using $\LaTeX$. I will walk you through how to do this.
The Weekly Homework assignments described above are subject to the following rubric:
|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.|
Please understand that the purpose of the written 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. Improvement over the course of the semester will be taken into consideration when assigning grades.
(Adopted from Chapter Zero Instructor Resource Manual) 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 very seriously since they spearhead the work of the class. Here are some of my expectations:
- In order to make the presentations go smoothly, the presenter needs to have written out the solution in detail and gone over the major ideas and transitions, so that he or she can make clear the path of the proof to others.
- 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 his or her 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.
|4||Completely correct and clear proof or solution. Yay!|
|3||Proof has minor technical flaws, some unclear language, or lacking some details. Essentially correct.|
|2||A partial explanation or proof is provided but a significant gap still exists to reach a full solution or proof.|
|1||Minimal progress has been made that includes relevant information & could lead to a proof or solution.|
|0||You were completely unprepared.|
However, you should not let the rubric deter you from presenting if you have an idea about a solution that you’d like to present, but you are worried that your solution is incomplete or you are not confident your solution 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.
I will always ask for volunteers to present proofs, but when no volunteers come forward, I will call on someone to present their proof. If more than one student volunteers, the student with the fewest number of presentations has priority. The problems chosen for presentation will come from the Daily Homework assignments. Each student in the audience is expected to be engaged during another student’s presentation.
In order to receive a passing grade on the presentation portion of your grade, you must present at least 3 times prior to each of the two midterm exams and at least twice after the second midterm exam for a total of at least 8 times during semester.
Your overall performance during presentations, as well as your level of interaction/participation during class, will be worth 15% of your overall grade.
Basis for Evaluation
Your final grade will be determined by your scores in the following categories.
- Homework: 25%
- Midterm Exams: 40% (each exam is worth 20%)
- Presentations/Participation: 15%
- Final Exam: 20%
Determination of Course Grade
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%.
Rules of the Game
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. Of course, institutional excuses will be honored.
Additional Comments and More Propaganda
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. Likely 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. I’m sure that each of you has said, “Hmmm, I understood this concept when the professor was going over it, but now that I am alone, I am lost.”
In order to promote a more active participation in your learning, we will incorporate ideas from an educational philosophy called inquiry-based learning (IBL). If you want to learn more about IBL, go here. Much of the course will be devoted to students presenting their ideas 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 problems is difficult and takes time. You shouldn’t expect to complete a single problem in 10 minutes. Sometimes, you might have to stare at the statement for an hour before even understanding how to get started. In fact, solving problems can be a lot like the clip from the Big Bang Theory located here.
Aside from the obvious goal of wanting you to learn how to solve problems, one of my principal 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.
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.
You are expected to treat each other with respect. You 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.
Department and University Policies
You are responsible for knowing and following the Department of Mathematics and Statistics Policies and other University policies listed here. 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.
Here are some important dates:
- Monday, January 19: Martin Luther King Day (no classes)
- Thursday, January 22: Last day to drop/add (no W appears on transcript)
Monday, February 23Wednesday, February 25: Exam 1
- Friday, March 13: Last day to withdraw from a course (W appears on transcript)
- Monday, March 16-Friday, March 20: Spring Break!
- Monday, April 13: Exam 2
- Wednesday, May 6: Final Exam
(Adopted from pages 202-203 of The Moore Method: A Pathway to Learner-Centered Instruction by C.A Coppin, W.T. Mahavier, E.L. May, and G.E. Parker) There are two ways to approach this class. The first is to jump right in and start wrestling with the material. The second is to say, “I’ll wait and see how this works and then see if I like it and put some problems on the board later in the semester after I catch on.” The second approach isn’t such a good idea. If you try every night to do the problems, then either you will get a problem (Shazaam!) and be able to put it on the board with pride or you will struggle with the problem, learn a lot in your struggle, and then watch someone else put it on the board. When this person puts it up you will be able to ask questions that help you and the others understand it, as you say to yourself, “Ahhh, now I see where I went wrong and now I can do this one and a few more for the next class.” If you do not try problems each night, then you will watch the student put the problem on the board, but perhaps will not quite catch all the details and then when you study for the exams or try the next problems you will have only a loose idea of how to tackle such problems. And then the anxiety will build and build and build. So, take a guess what I recommend that you do.