Syllabus

Euclidean Geometry
Math 3600, Spring 2017

Course Description:

This course is devoted to classical, planar geometry, including basic properties of triangles, circles, quadrilaterals and other polygons, compass and straightedge constructions, and the notion of area.

Students who complete this course will learn to work as a mathematician does. To meet this challenge, we will run class as an Inquiry Based Learning (IBL) environment.

Successful students will be proficient in working with axiomatic arguments, and will improve their abilities to construct, read, write, present, and critique mathematical proofs. Students will gain insight into the nature and role of definitions and the process of mathematical work. Students will also learn something about how to ask and answer their own well-formed mathematical questions.

Basic Information:

• Instructor: Theron J Hitchman
• Office: 327 Wright Hall
• Office Hours: This semester I am using the service YouCanBookMe.com as an experiment. I expect to be generally available M-F 3-4:30pm. To schedule an appointment, use the service at https://theronhitchman.youcanbook.me/. If you take a chance and just drop by I might be free, or I might have a meeting. If these hours won’t work for some reason, send me an email.
• contact info:
  email: theron.hitchman@uni.edu
  office phone: 319-273-2646

• Course Web Page: http://theronhitchman.github.io/euclidean-geometry

  Check this regularly. Lots of important things are at this page.

Materials:

• Text: Euclid’s Elements, Heath translation, Green Lion Press, 2002, ISBN 1888009-19-5.
• Software: GeoGebra is a free, open-source geometry program. www.geogebra.org.
• Drawing Tools: You may wish to use a compass and straightedge set. If you do, I recommend a “modern compass” that may be fixed at a given radius.

Assessment:

Grades in this course are determined by the instructor’s evaluation of student progress towards stated course goals. Evidence will be collected throughout the term in a variety of assessments, with emphasis on the first two items:

• class participation, focusing on finding and presenting arguments, and publishing in the class journal
• work as a referee for the class journal
• two reflective self-assessments
• a week-long, take-home, final examination

More details and discussion follow below.

If you feel uncertain about your progress in the course at any time, please contact me.

Dates to Observe:

• Week of January 16-20: First self-assessment conferences
• Wednesday, March 1: Midterm formative assessment, in class
• Week of March 6 - 10: Second self-assessment conferences
• Week of March 13 - 17: No Classes: Spring Break
• Friday, April 28: Final class meeting. Final Examination distributed
• Thursday, May 4: Final Exam and Curriculum Vita due at 5pm

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Accommodations: If you have a disability and require accommodations, please contact the instructor early in the semester so that your learning needs may be appropriately met. You will need to provide documentation of your disability to the Student Disability Services (SDS) office, located on the top floor of the Student Health Center, (319) 273-2677.

Academic Learning Center Syllabus Statement I encourage you to utilize the Academic Learning Center’s free assistance with writing, math, science, college reading, and learning strategies. UNI’s Academic Learning Center, located in 007/008 ITTC, also provides advising services and is the University’s testing center for many standardized tests, including the PLT, GRE, and Praxis Core. Visit the website at http://www.uni.edu/unialc/ or call (319) 273-6023 for more information or to set up an appointment.

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MORE DETAILS

Inquiry-Based Learning

I conduct each of my classes in some sort of Inquiry Based Learning format. Essentially, this means that I strive for an active, student-centered environment. Lectures will be infrequent (if they happen at all), and instead students will engage in activities that have them participate as working mathematicians by conducting research at an appropriate level. It is important to note two fundamental differences between how Inquiry Based classes operate and traditional lecture classes do:

1. Students will be responsible for producing mathematical understanding. They must grapple with the material and construct their own arguments for why things are as they appear to be.
2. Students will be the arbiters of correctness in the classroom. The instructor’s role is to guide discussion and set the parameters for student inquiry. But the students are the ones who say whether a particular argument is correct or not—students must take ownership of the mathematics being developed.

If you have not experienced an inquiry based learning environment before, this can be a bit unsettling. Have no fear! It is my job as the instructor to put you in a position to have success, and to help guide you back toward success when you miss the mark. Pretty soon, you will be much more comfortable.

Why are we doing this?

It is my goal to help you increase your power and ability as a mathematician. Since I want you to help you improve at doing mathematics, I have structured our class as an environment where you must do mathematics. This gives you opportunity to practice and grow, and it gives me an opportunity to see what you are capable of doing and where I can help you the most.

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Technology: GeoGebra

GeoGebra is a free*, open-source, interactive and dynamic mathematics package which has good support for playing with Euclidean geometry concepts.

You can use the software in many ways:

• download your own copy from the web site
• use a Google Chrome Web App
• use the Java-based “web start” version
• one of the Android or iOS apps

Often, I use the chrome web app, which will connect with a UNI Google Drive to save your work. I encourage you to play around with the software. It is very useful.

* There is a bit of funny business here. GeoGebra is not totally free to everyone for everything. But it will be free for you as long as you do not try to use it to build another product to sell.

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There are some games worth your attention, too.

Dragonbox Elements is not free, but it is not too expensive. It is marketed as a way to learn the process of making axiomatic arguments in the context of classical geometry. I played it and had fun, and I can see the potential.

Euclid the Game is a free online game based on GeoGebra. It has a large number of construction challenges which are typical objects of study in classical geometry built in. You can learn something by completing these, too.

Euclidea is a mobile game (Android or iOS) built around very challenging construction problems. Some of these are hard! But I find this very enjoyable for passing time during dull meetings or a few idle moments. There is a free version that you have to succeed in to unlock further levels, or you can pay a small amount to unlock all of them.

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About Assessment

The structure and aims of Math 3600:01 are so radically different from a traditional “computation based” mathematics course that I find the general structure of assigning grades from an accumulation of points on homework, quizzes, and exams to be completely unworkable. If I could get away with it, I would not assign grades at all, and focus all of my energy on helping you develop. However, the registrar demands that I assign grades.

I can usually sort out a reasonable “grade” for each student by the end of the semester by some sort of mysterious process of watching and listening in class. There are several difficulties with this, which are probably apparent to you:

1. It requires that you trust me and my professionalism.
2. It mixes up formative assessment and summative assessment pretty badly.
3. It has no formal feedback mechanism.
4. Students do not necessarily identify the same things as important that I do.

I can not see any way around problem (1). Even the traditional “points accumulation” process requires this. I choose not to deal with item (2). In fact, I am deliberately leaving it mixed up so that I can help you all as much as possible. But items (3) and (4) are about clear communication of expectations and of perceived progress. That I can do something about. I choose to use a communication scheme inspired by things called “Standards Based Assessment,” and “Specifications Grading.”

So, I will set out some description of those things I deem as important for the students in this class to improve upon. This makes clear the kinds of things I really hope you are working on this term. These are somewhat nebulous, as we are working on the kinds of qualities and habits you need to be a competent mathematical professional. We will refer to the items in this list as the standards.

Also, I will describe a scheme for documentation and communication of your demonstrated proficiency on the standards, and a little bit about how I will try to use this information to assign a grade at the end of the term.

Of course, this scheme is as likely to have failures as another, but I hope that it won’t distort incentives and cloud communication in the way that traditional grades do. It is my belief that if you tried to “game the system” to improve your grade, you would be doing exactly the kinds of things I wanted you to do, and improving your abilities as a mathematician. That is, after all, my only goal.

At any point in the semester, if you would like more detailed discussion of my view of your mathematical skills and how you can work to improve them (i.e. your grade), please come talk with me.

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The Standards

Every one of these points is the conclusion to a sentence which begins: Students will demonstrate that they can…

Mathematical Investigative Process:

Foundational Proficiencies

• explore examples
• make appropriate choice & use of technology in investigation
• look for and use structure in an uncertain environment
• persevere by trying multiple approaches

Advanced Proficiencies

• extend ideas to find or create new mathematics
• devise a relevant new conjecture
• modify hypotheses or conclusions to make work more tractable

The Axiomatic Method:

Foundational Proficiencies

• use definitions to justify assertions (in both directions)
• use literature (Euclid or class work) appropriately to justify assertions
• prove or disprove statements by making correct logical arguments (direct, indirect, by cases)

Advanced Proficiencies

• make a clear definition to fit a new concept
• prove a difficult theorem requiring an intricate argument or a deep, original idea

Planar Geometry Content:

Foundational Proficiencies

• write arguments involving the properties of
  • congruence for polygons
  • parallel lines
  • compass and straightedge constructions
  • circles
  • Euclid’s conception of area

Oral Communication:

Foundational Proficiencies

• present ideas clearly with precision about mathematics
• handle questions respectfully and directly
• ask questions respectfully and directly

Advanced Proficiencies

• Make effective use of presentation technology (chalk, computer, other)
• engage in meaningful critique

Written Communication:

Foundational Proficiencies:

• write in clear English prose
• use proper mathematical style
• use language with precision and care
• effectively construct and use figures
• produce professional quality documents
• read new mathematics for understanding

Advanced Proficiencies:

• write concisely, balancing clarity and brevity
• critique written mathematics critically in a peer review setting

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Students who wish to pass the course should demonstrate mastery of the foundational proficiencies. To earn a higher grade, a student must demonstrate mastery of some portion of the advanced proficiencies.

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The Assessment Process

Assessment comes in a variety of forms, but there are only two for us to be concerned with: formative assessment where I help you evaluate your learning and pick a path to improve, and summative assessment where I decide at the end of the course to what extent you have demonstrated mastery of the course goals and use that information to assign grades.

From your point of view, formative assessment will play into the kind of feedback you receive from me and your classmates. (From mine, formative assessment is how I gather such information and use it to guide the development of class. This will likely be invisible to you.)

Formative Assessment: Feedback on Your Work

I will be keeping track of what goes on during class to the best of my ability. After each presentation, the class will have a conversation about your work: this is your primary source of feedback. Sometimes, I may send an email with some extra comments about your progress. I think of this mostly as a formative assessment. It is a way for me to gauge your progress and help you focus on improving your work.

Through the journal process, you will receive referee reports on your submitted papers. These are the equivalent type of feedback on your written work.

How Summative Assessment Works

I will use the following things to assess your work this term:

• Published papers derived from your work and class presentations.
• Reports you write while serving the class journal as a referee.
• Reflective self-assessments you write at conference times.

At the end of the semester, your published papers will have appeared in the class journal. But I want a clear way to summarize your work, so you will prepare a short curriculum vita (CV) to turn in with your final exam. I will use the your CV and your final exam to set your course grade.

Each individual item you produce contributes to my understanding of your learning. I will make available detailed and clear specifications for what constitutes acceptable work. Most of our class time will be spent doing and discussing mathematics, so the way to meet the specifications will become apparent as we go.

The Curriculum Vita Assignment

Preparing a CV should be a short (15 minute) exercise at the end of the term. I will make available a latex template and an example as we get close to the end of the term so that you see what is expected of you. But the main thing is to make lists of which papers you have published and which papers you have refereed during the term. So keep track of those as you go!

Published Papers

The normal mode of work for this class leads to publication of your work in the class journal. Specifications for this work are as follows:

• The mathematics presented must be complete, correct, and clearly presented.
• The article should follow conventions for mathematical writing as detailed in the Style Guide for Authors on the Journal’s web page.

Referee Work

As part of the journal process, you will serve as a referee for the work of other students. The specifications for referee reports are listed in the journal web page.

Reflections & Conferences

To facilitate the communication process, we will schedule regular conferences to discuss your progress. You will meet with me for a ten or twenty minute appointment at three points during the term. The first meeting is the longest, and each of you will meet with me during week 2 or 3. I will meet with each of you again during weeks 8 and 9.

To give these meetings some substance, I require that you write a one page reflection and either bring it to our meeting or email it to me beforehand. The point of these reflections is for you to engage in an honest self-assessment. This is an important part of learning and growing!

• Reflection Prompt #1 (weeks 2 and 3)
• Reflection Prompt #2 (week 8 and 9)

Of course, you are welcome to come by and talk about mathematics, or your progress in the course, at any time I am available.

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What is Typical?

I have taught this course a great many times. The clearest thing I can say about assigning grades is this: I know approximately what the work looks like at each grade.

What does an A look like?

A student who earns an A has typically published about ten papers, at least one of which contains something really amazing: something showing a beautiful insight, real perseverance, or extreme cleverness. (By the way, extreme cleverness usually lies underneath a huge pile of work.) This student has also been the referee for several papers by peers.

Such a student usually has shown some intellectual leadership, and has demonstrated some of the “advanced” levels of proficiency in more than one standard. This student’s final exam paper has solid work on all of the tasks assigned.

What does a B look like?

A student who earns a B has typically published five or more papers, and has demonstrated competence in all of the “foundational” standards, and at least one of the “advanced” standards. This student has also refereed a few papers by peers. This student’s final exam paper has solid work on most of the tasks assigned.

What does a C look like?

A student who earns a C typically has published more than one paper, and has demonstrated proficiency with all of the “foundational” standards, though those performances may be uneven enough that at the close of the semester I am not certain all those standards have been mastered. This student may or may not have been a referee. This student’s final exam paper successfully navigates only one or two of the tasks assigned, and shows a lack of understanding on the others.

What does a D look like?

A student who earns a D has typically published only one paper, usually very late in the term. I lack confidence in the fact that such a student has a good grasp of the “foundational” standards. This student’s final exam paper perhaps has one task successfully completed, but the others show a serious lack of depth.

What does an F look like?

A student who earns an F has usually taken an extremely passive approach to the course. This student may have a publication of dubious quality, or none at all. This student’s attendance is spotty, and participation on days present is almost non-existent. This student’s final exam paper demonstrates an inability to complete the most straightforward tasks.