Opening Week for a Moore Method Course: Getting Comfortable

I am teaching iteration number ten of my Modified Moore Method Euclidean Geometry course. This semester I am making an effort to refocus on the basics: managing and mentoring the students as much as I can.

At this point, my theorem sequence is very stable. (I am no longer surprised much by what happens in this course.) This allows me to work on the other aspects of the course. I feel like I have started to let some important things go in the last year or so, and now I want to sharpen up. What has been lacking? I don’t think I have kept on top of the students to keep them engaged as well as I might. And I don’t think I have done a good job selling the method of instruction, either.

So, I was much more deliberate about introducing myself to each of my sixteen students on the first day. I have been very explicit about my expectations and my willingness to help them meet those expectations (which are rather high). And I will be making a conscious effort to check in with as many students as possible each day.

The first week was a rousing success, I think. Each day we got at least one theorem. We have already set the expectation for what counts as an argument. (Well, surely, there is still some work to be done.) The class has made two conjectures. We took some time to discuss some basic points of what acceptable writing will look like. I even successfully navigated our first potential difficult situation and found something positive in it. All in all, I am feeling pretty good about this.

I think our next test comes when we have to finish conjecture 1.1. They haven’t addressed the second statement in that, yet.

And sometime next week I will have to steal ten minutes to talk with them about my Standards Based Assessment experiment for the term.


Running a Class Journal

For several years I have run an IBL Euclidean Geometry Course. You can find some of my thoughts about IBL courses in general, and about this course in particular, in other posts on this blog.

An important feature of the course is the class journal. I am writing an article about the rôle this plays in teaching students proof writing, so this post will serve as a first draft of my thoughts. I welcome comments and criticism, as that will help me write a better paper, and be a better educator. Also, my digital homey Bret Benesh asked for a blog post about exactly this subject.

As with all of my blogging, I intend to ramble on freely. Buckle up.


Most the students in my course are in a preservice teaching program that leads to certification for grades 5-12. Most students are getting their first college level introduction to what a mathematical argument is and how to write one coherently.

(This will be changing soon. We are instituting a new course that will explicitly teach proof writing and argument making. Though it won’t be a formal prerequisite, we will advise most students to take that course first. I intend to keep the introductory feel of my course for the time being. I am sure some students can benefit from extra layers of this type of course, and it is simpler to make a course harder than it is to make it more accommodating.)

I run the course as an instance of the Extreme Moore Method. So class time is dominated by student led presentations and discussions of their work. They spend a lot of time outside of class finding and constructing arguments. During class meetings, they defend their work, at the chalkboard, to their peers. This is all excellent training for how to work as a mathematician, but it doesn’t cover the skill set involved in carefully writing up results.

As the process of writing is an essential one, and a big part of what characterizes academic work, that needs to happen, too. This is where the journal comes in.

Basic Set Up

When a presentation is concluded, the student will get feedback from the class about the quality of the argument and its verbal exposition from the discussion that occurs. When the argument, or some portion of it, is accepted as correct and valuable to our class progress, the presenter is responsible for writing up the argument in the form of a short paper. This paper is due by the next class meeting.

This paper then is “submitted” to our class journal. It is refereed, and when accepted, published.


The Submission Process

In the past, students have used whatever word processing system they wish. Most students used Microsoft Word because they are familiar with it. As the course focuses on planar Euclidean geometry, there is not a great need for mathematical symbols, so Word is sufficient. In fact, I like that using Word encourages students to write with English words instead of mathematical symbols. Someone always figures out how to make a figure in GeoGebra, export it, and include it in a Word document, so I let that person be the class expert.

A student paper is expected to conform to the general format and style of a mathematical research paper.

At this point, the first submitted drafts usually come in on paper. In the past I have tried a class wiki, and submission of pdf by email.

The Referee Process

At the beginning, I am the sole referee for the journal. I mark up the papers much like I would when reading any other paper, and then make a short referee’s report. These are returned to the author. I try to return them by the next class period.

Of course, all papers are eventually accepted. This differs from standard journal practice, but I don’t see how to avoid this.

Some papers require several runs through the referee process. At some point, the changes required become very minor and the paper is deemed as “accepted with small changes” and the next version gets put in the queue for publication.

A few weeks into the semester, students who have proved themselves as competent authors are invited to become referees. Some care must be taken to train the students about how to do this, and some students have to be coached more than others about appropriate professionalism when acting as a referee. I try to monitor this work closely the first time through.

When I have a stable of student referees, the nature of my work changes. I act much less as a referee and more as an administrative assistant—shuffling papers and keeping things moving. Students then are engaged in the work of writing and evaluating writing.

The Publishing Process

Every two or three weeks I find enough papers have collected in the publishing queue that I can bundle them together to make an issue of a journal. (Four papers seems to be a minimum.) I have required papers to be turned in as .pdf files, so I can just bundle them together with the LaTeX pdfpages package. (I have designed a cover page that I can slap on top of each issue with a little graphic.)

I distribute the journal electronically: it is posted to the course web site. But as a treat, I print a copy of the issue for each author with a paper appearing. I hand these out at the beginning of a class meeting and say congratulations to the authors as I do so.

Students get a kick out of seeing their work in print, so this provides a little reward and motivation.

To Be Continued:

I promised to go to the local pool with my kids this afternoon, so I’ll just stop writing now. I look forward to your questions and comments. I do plan to write a little more about this issue, so look also for my next post: The Journal: What About Next Semester?

Help My Former Student: How To Teach Proof

I got a message recently from a former student, Tigh Bakker, who is now starting out as a practicing teacher. He sent me a thoughtful question, and seemed happy to have me share it here in hopes to generate a bunch of ideas.

Here is an excerpt of Tigh’s message to me:

I am currently directly in the middle of teaching proofs to my high school students, and all of my “high flyers” seem to grasp it, but a lot of my lower students are stumped and performing poorly. I know you don’t teach the lower level of Geometry that I do, but I was wondering if you have any tips/suggestions to helping students understand how a proof flows. Most of what we are doing now is simple triangle proofs using congruences such as ASA, SAS, SSS to then use the notion of CPCTC (corresponding parts of congruent triangles are congruent). Our book is called “the CME project” developed by Pearson and is rather open ended (I like to think of it as the high school version of how you teach). Knowing that you teach in an inquiry based fashion, how do I get my lower students to delve into the material and really work at learning some material without my guidance? Any tips at all would be great.

I sent him a few ideas yesterday. I don’t claim these are original to me.

This is indeed challenging. Getting students to change their mindset is hard. I imagine self conscious middle school and high school students would be tougher to convince.

I also struggle with the students at that margin. For some, there is safety in group work. But you don’t want a bunch of struggling students to latch on to each other and then drown together. As to the “flow of a proof”, here are some ideas. I have no idea of how they might work!

  1. Give them a proof written clearly, but cut up into individual sentences. Have them arrange the sentences in the correct order.
  2. Give them examples of poorly written proofs (not their own), and ask them to critique.
  3. Show them examples of various versions of a single proof. Range from good to bad, lots of style variation. Have them do some sort of ranking or vote on an order of best to worst. I hope that a subsequent class discussion on what makes a proof good or bad would help focus their attention.

Does anyone else have ideas for Tigh to try? Drop some in the comments!