by Tien Y. Chih
Montana State University, Billings
Since the COVID19 pandemic hit during the Spring of 2020, I’ve been nothing short of impressed and amazed at my colleagues’ resourcefulness and creativity in shifting their courses to an online modality. So when I was asked to teach an online Modern Geometry course this past Summer, I was eager to roll out an inquirybased version of this course. But when planning this course I realized that I would face unique challenges that would make this difficult.
MSUBillings is a comprehensive regional state university which serves central and eastern Montana and northern Wyoming, parts of the nation that are often very rural and distant from the physical campus. For this reason, the school has had a strong focus on its online course offerings even prior to the pandemic. In particular, Modern Geometry serves as the last course in a Math Teaching Minor, which certifies current teachers in the state to teach Mathematics in addition to their other certifications. Because this minor is intended for current working professionals from across the state, it is necessary for the courses in it to be online.
But this poses its own challenges. I know from previous experience that internet access in rural Montana can be spotty and unreliable, so I was unsure that a Zoom or WebEx based course would allow for equal opportunity amongst students to share and communicate with each other. Moreover, as working professionals, they would all be saddled with myriad other responsibilities, particular this year. Since our school does not designate a specified meeting time for online classes, I did not think it would be feasible to be able to designate a meeting time without excluding some students. I concluded that a synchronous model for Inquiry Based Learning (IBL) would not support the pillars of Equity and Collaboration.
I concluded that I would have to develop an asynchronous model for inquiry learning. I was highly skeptical about how this would go. IBL experts who generously lent me their time gave me advice in general on inquiry learning, but were likewise skeptical of the asynchronous approach. I didn’t see that I had any choice in the matter.
The Course Design.
I decided to use Charles Coppins Euclidean and NonEuclidean Geometries as the primary resource for the class. Given that we had an 8week schedule, I thought that the layout of the course would match well with that of the text. We also scheduled a single synchronous meeting on the first day of the summer session to discuss the flow of the course. I encouraged them to think of themselves as a research team, uncovering the structure of neutral geometries together, and that the discussion board posts would serve as a replacement for the in person conversation and discussion they would have had, both in and out of the class.
The layout of the course was as follows:
 Week One: This week served as an introductory/ice breaker week. Students were asked to post a thread containing a short Mathematical Biography: their experiences with Mathematics up to this point, and what they hoped to get out of this course. They were asked to comment on each other’s biographies. They were also given some videos to watch, such as the Extra Credits History of NonEuclidean Geometry, post a thread about their thoughts and reactions to these videos, and comment on each other’s threads.
 Week Two through Six: For each week, students worked through a section of Coppin. Each student would post a thread dedicated to one of the problems in the discussion board. The initial post could contain a proposed solution, but could also contain an idea for a solution, a proposed strategy, an example, or even some initial thoughts. The point was that I wanted them to make an honest start towards thinking of a solution. The students then commented on each other’s posts and progressed together until they arrived at an agreed upon solution. The goal for each week was to arrive at a complete proof or solution for each problem.
 Week Seven: I saved four problems from Coppin for this week, where they were asked to define and conjecture a theorem about Planes and Quadrilaterals. They spent this week proposing and discussing definitions, conjecturing theorems, and helping each other prove their theorems.
 Week Eight: This would be a wrap up week, where they would finish writing up any proofs from the previous week, and work on typesetting a final document of their findings in LaTeX. I originally intended that the students submit their own personal documents, but the class made an argument early on that since they were collaborating as a research team, it only made sense that they collaborate on the writing. I agreed, and rolled with it, so they worked together on an Overleaf document that they collaborated on through the semester and during this week.
My role in all of this, especially in the beginning, was to facilitate discussion and conversation. Each week I would grade each student’s participation based on a rubric, and give qualitative feedback for each student, including things they posted that were good, and suggestions for the coming weeks. The final grade was based on the final document, discussion posts, and Desmos Geometry interactives I asked them to create.
The Good.
Right away, I noticed that there were advantages to running a class in this format as opposed to synchronously or face to face. In face to face classes, conversations can be dominated by those who speak more quickly or more openly. Even in a class where everyone was willing to share, time constraints only allowed for so much conversation to be had. On the boards, I found that when everyone could take the time to craft their posts and responses, that there was far greater and deeper participation. During the ice breaker phase, I found students writing quite illuminating posts and responses to each other. On the other hand, I had the time to write detailed individual responses to each student, which other students could read and respond to. It was kind of a best of both worlds of one on one conversation and group discussion.
Hello,
My name is XXX, I live in XXX and teach 7^{th}, 8^{th} and Algebra 1 at XXX high school. I have three grown children and have lived in XXX 29 years – raised in XXX. I am returning to college to get an endorsement in mathematics. I taught 58^{th} grade at a colony school for the last 13 years and was asked if I would come into town to teach just math. I wasn’t sure I was up to the task, but working with XXX and creating a fantastic math program has been a lot of fun. My goal with mathematics is to reach tohe struggling students and encourage them to trust me and we will understand math together. Most of my students are terrified of math (As am I – just a bit) and hopefully when they leave our school they will feel mor secure in their math skills. I feel like I will learn to ask better questions and I like the fact that we are working together as a term to reach our final solutions. I think it will free us up to explore and take risks. Thank you for taking the time to read this and I look forward to working with everyone.
I don’t know if I’m more blown away with how geometry relates to all math or that I didn’t realize this years ago! These videos weren’t very long, but they started to open my mind about how geometry relates to algebra and calculus in ways that I’ve never thought of before. The math concepts in the videos were not new, but the way the information was presented is a different way of thinking.
Hi,
I know exactly where you are coming from…. Math does not come easy to me either and I truly feel it is to our advantage – we understand the “not understanding” and I do believe that we have great empathy! I am like you – for some reason I really like working my way through a problem – I never would have guessed that one day I would be a math teacher.
XXX,
I too was blown away with how all other math subjects seem to have stemmed from Geometry! I really didn’t know the history of math at all and never thought to explore it any. Like you said, the videos really opened my eyes to the connections between all the disciplines of math that I didn’t actually see before!
Likewise, there were unexpected benefits of the discussion board for collaborative proof writing. The asynchronous format allowed for wide flexibility in responses. Students could write long detailed posts, or short quick responses. They could also include pictures, videos, or links to interactives such as Desmos in their responses to help illustrate their ideas. Having a written record of the collective conversation also made it easier to reach back and pull from an earlier idea, or make minor edits to someone else’s argument.
Here are my initial thoughts on Problem 5, which reads:
“Problem 5. Prove or disprove the following: Each point belongs to a line.
I think you would agree that any geometry that contains two points not belonging to any line is not interesting. Therefore, we need the next axiom, Axiom 4. If A and B are different points, then there exists at least one line that contains both A and B.”
I felt this statement could be proven.
It makes sense to me that any point either has to be on line L (since A3 says at least one line exists) or

 There exists at least one line L. Axiom 3.
 That line L contains at least two points. Axiom 1.
 Any point outside L makes a line with points in line L. Axiom 4
 Therefore, each point belongs to a line.
outside of line L, in which case A4 says that point has to make a line with one of the points on line L. Thoughts?
Problem 18: Suppose ABC and ACD.
Q1: Is it possible that two of A, B, C, and D are the same?
Initial Thoughts: By A5, A, B, and C are different points and A, C, and D are different points. It could be possible that B and D could be the same point. Suppose A, B, C, and D are all points on a circle, where B and D are the same point. ABC results from going clockwise around the circle and ACD (which equals ACB) results from going counter clockwise. This would prove that it is possible for B and D to be the same point.
Edited to add: I’m not sure this would satisfy A4.
Q2: Do they all belong to the same line or is it possible that A, B, and C belong to one line and A, C, and D belong to a second line?
My initial thoughts: if a line exists, do all of the points need to be notated. For example, if there are points W, X, Y, Z on a line, would I need to list all of them? Or could I say WYZ – X would still be on the line, just not listed?
So for this problem, if I had a line ABCD… would it be appropriate to say within ABCD, there is ABC and ACD? If that’s the case, then it is possible that A, B, C, and D belong to one line.
XXX: I like your circle for matching points for question 1. I was thinking of an analog clock where 12 am and 12 pm are the same point but different times.
This is what I imagined for question 2…
XXX:
Initially, I thought the same as XXX. Then after rereading AB(b), I think there exists a point between A and B (labeled “D” in AB).
I think AB could apply to intervals (it doesn’t specifically mention lines). Based on Abb, I think an interval must contain at least three points.
[reply]
XXX:
I approached this in terms of AB (b) as well. If two distinct points exists it makes sense that there have to be points in between them. Part of Definition 3 says that interval AB is “the set of all points between A and B inclusive.” That, combined with A8(b), which says there has to be at least one point between any two distinct points, makes me think that any interval has to contain at least 3 points.
[reply]
Here is another run at it from the top, considering A9’s role in forcing location of points. I’m still not sure that I am properly applying A9 though.

 A, B, and C are distinct, noncollinear points (Given)
 Lines AB, AC, and BC exist (A4)
 Points D and E exist such that AEC and BDC (AB)
 Lines BE and AD exist (A4) and intersect at X (applying A9 to triangle BEC)
 Line XC exists (A4) and crosses AB at point F (applying A9 to triangle ABE)
 Interval XA is on the Aside of BC (Defn.4)
 Interval XB is on the Bside of AB (Defn.4)
 Interval XC is on the Cside of AB (Defn.4)
 The intersection of XA, CB, and XC = [X], so X lies on the Aside of BC, the Bside of AC, and the Cside of AB.
https://www.desmos.com/geometry/ohgnzbybvs
I found that the students were active and engaged, treated each other with respect, and worked diligently towards the goal of solving each problem. At the start of the course, I was very active on the boards, responding to each post right away, asking questions, prodding or challenging them. This was mostly to model the type of interaction I wanted to see in the course. As the class continued, I was able to step back as they took on these roles themselves, prodding and nudging in them occasionally. It was great to see them come into their own as mathematicians!
At the end of the course, they turned in a beautifully written document, including several proven conjectures, that they wrote together in Overleaf, coordinated by a student driven effort on the discussion board. Their feedback, both on the boards and in evaluations, were overwhelmingly positive, I could not be happier or more proud.
The Ugly.
I had vastly underestimated the number of students who would be in this class. I ended up with 16 students, and in some weeks there would only be a handful of problems. The result was that there would be sometimes redundant or disjointed conversations, which could be confusing to peruse. Even within a thread, discussions would grow quite long, and after a few layers of replies, my learning management system no longer kept track of who was replying to whom, making conversation difficult to track at times. In the future, I plan on assigning small groups, either per section or through the semester, so that conversations can be more streamlined and tractable.
I had not planned on having the final document be written collaboratively. I don’t regret acquiescing to the students, as their argument was cogent and I believe it gave them a greater sense of ownership of the course. I do wish I had built a collaborative writing component into the course from the ground up, following the ideas of Wikitextbooks by Brian Katz and Elizabeth Thoren. As a result, there was quite a bit of inequitable work distribution by the end of the course. With the classes permission, we resolved this by having some students write bookend introductions, summaries and conclusions in the document, which did improve the final document, but it was an adhoc solution, and could have gone badly.
The Future.
I found the Discussion Board format to be an extremely effective way of delivering an inquiry course, and in fact the only way I could have reasonably delivered one this Summer. I’ve incorporated the boards in other classes, incorporating the “Investigate!” sections of Oscar Levin’s Discrete Mathematics text into my Discrete course, and in lieu of face to face office hours due to the pandemic. This coming Spring, I plan on teaching a Linear Algebra using Drew Lewis and Steven Clontz’s Team Based Linear Algebra, adapted for discussion board conversation.
Some advice I would offer is:
 Be active and present on the boards, especially early on. The students told me that they are used to boards being a place of minimal, loweffort posts. Model the type of interaction you want to see.
 Aside from IceBreakers, make sure each thread is dedicated to accomplishing a particular task. This ensures that the conversations go for as long as long or in depth as necessary for the task to be completed.
 Give frequent and qualitative feedback. There will be some students who make few or low effort posts. You want to correct this early, while not penalizing them in a way they cannot recover from. Give concrete suggestions, and highlight posts of theirs that you consider high quality.
 Include an IceBreaker period. If you and the students will have little or no facetoface or out of class conversations, then the boards will be their only chance to know each other. I learned a lot about my students reading their biographies and responses to the History of NonEuclidean Geometry videos, and they also found commonalities and bonded over these conversations as well. This is especially important for establishing a team rapport for the course. (One idea, from Sarah Bockting, is to adopt Matt Salomone’s Sandwiches and Ontology of Definitions activity as an icebreaker)
 For taskbased discussions, break the class into groups. Smaller groups allow for more intimate focused interactions, as well as keeping the threads and discussions from becoming unreadable.
 If you have a collaborative writing component, build it into the course from the ground up, and have a system in place which ensures an equitable distribution of tasks and responsibilities.
In an uncertain and precarious time, most of us have already delivered courses in modalities we may never have considered. Many of these modalities may work very well for most students, but not for all. Depending on the needs and limitations of your students, know that it is possible to design a rich, engaging and meaningful inquiry experience for students in an asynchronous format.