Integrated Course Design for Euclidean Geometry, III

Today I finished grades for all of my Fall 2014 courses, so it is time to get back to planning next term. As my last “pre-holiday” bit of work, I’ll finish the course design template for Euclidean Geometry following Fink. I previously did the initial design phase which consisted of “building strong primary components” in Steps 1-5.

Intermediate Phase: Assemble the Components into a Coherent Whole

Step 6: Create a Thematic Structure for the Course

Fink encourages us to find 4-7 segments of the course, each focusing on key concepts or topics. I think I have five, or maybe four. It depends on how you want to count them. The first one on this list might be two things.

  1. Polygons, and the axiomatic method, conjectures, and definitions. This is mostly about using triangles to study other polygons. But there is a little mini-unit on arguments with parallel lines mixed in here.
  2. Circles
  3. Straightedge and Compass Constructions (as an efficiency game!)
  4. Area and the regular pentagon

Segment 1 is really long, it can take half a semester. Then segments 2 and 3 are shorter. We almost never finish segment 4 completely, but the fastest class I had finished 4 and started another (bonus) segment!

Step 7: Instructional Strategy

I have no desire to make changes here. For ten iterations of this course I have used a flavor of Inquiry-Based Learning called a Modified Moore Method. Really, I have done something called The Extreme Moore Method (EMM), which I have written about before. Students will spend their time out of class finding and composing arguments for conjectures. We will spend class time presenting and critiquing this work. Then students that will write papers for the class journal. Oh, just go read the other post.

Step 8: The Overall Scheme of Learning Activities

Here Fink wants the instructor to think about the variety of activities the students should do, and when. Also, how does the sequence of tasks for before, during, and after class meetings mesh together? This is all pretty well decided with my EMM.

Final Design Phase

I won’t be able to finish all of this tonight in detail, but I can generally get through the rest of the guide.

Step 9: How are you going to grade?

Well, I have been thinking about this a lot lately. For years. Next term, I will try something called “Specifications Grading” following the work of Linda Nilson. I have been part of conversation about this for a few weeks now. Go find the work Robert Talbert has been doing in this direction for a list of places to start.

I will wrap up work on this and make a new post soon.

Step 10: What Could Go Wrong?

I have run this class often enough that I only have one worry: The new grading system is (a) not a solution to my problem, or (b) actively messes with the other parts of class which were working fine. I guess (a) isn’t too big a deal. I’ll just keep looking in that case. But I worry about (b). So far, the assigning of grades has been kinda vague, and this pushes students to keep working. (It is just like the real mathematics community.) And I want to make things more reliable and, I hope, more “fair.” But I don’t want students to start micro-managing publication counts instead of trying to solve more problems. For now, I need an experiment.

Step 11: Make a Plan for Communicating with Students

Fink writes this as if the big deal is to just write a syllabus. Well, yes, and no. I see these tasks ahead of me.

  1. Rewrite my syllabus
  2. Rework my first day handout (which is like a mini textbook)
  3. Update the course web page (which is like a digital syllabus and record of class).
  4. Flesh out the detailed specifications for the grading system. In particular, this means I will have to update

* The class journal style guide
* the instructions for referees
* my grading policy document
* specifications for the “non-mathematical” writing assignments (reflections and essays)

Step 12: Make a Plan to Assess your Teaching

This is a challenge. How many of us do this with such foresight? Well, my plan is this: I will use a simplified SALG instrument at final exam time to assess student satisfaction and understanding of the new grading system.

Let’s go fly a kite!

Today my Euclidean Geometry class discussed the construction of a kite with a compass and straightedge. They had four different constructions, all of them different from the one I had in mind. And these constructions led us to interesting discussions and new questions. It was a truly glorious day. I love teaching this class so much.

Anyway, it made me think of this song, so I played it in class. I had forgotten the two lines of dialogue right before the song starts, but they are appropriate too. I hope I can get my students to this state of wonder and excitement. This particular course makes me feel this way a lot.

A reflection on “Assessment Interviews, Phase 2”

I have spent a large portion of today in one-on-one conversation with the students in my Euclidean Geometry course. To prepare the students for these meetings, I asked them to complete a one page reflection paper, with this prompt. If you don’t want to click through, I basically ask the students to read through the “standards for assessment,” which is just a fancy name for my student learning goals, and do a self-assessment. Then I want them to make a plan of action for improvement during the next three weeks.

The striking part is the strength of the negative correlation between student self-assessment and my assessments.

Students who I recognize as having developed strong skills come it with focused critiques and tight plans for how to improve.

Students who I recognize as having not yet demonstrated many of our foundational skills show up with some confidence that they are doing everything just fine, and weak plans for self-improvement.

(This relationship is not perfect. Some students were spot on, of course.)

I have enough experience that I expected this, but to watch in unfold all day was really something.

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.

Meditations on Feedback

I am taking part in a “Talking Teaching and Learning” group, and my homework this week was to think about the last few details of my new assessment structure for Euclidean Geometry. In particular, how will I handle the “regular, daily feedback” part of the process?


So, if I am to provide regular feedback to my students at “assessment opportunities” they take, how shall I do it? I want this to be meaningful and effective. And it would be nice if it didn’t consume my working time.

I think I will try a format beloved by politicians: I will ask and answer my own questions. If any of you wishes to play investigative journalist and ask other questions that I should be forced to answer, go hit the comment box. I would like to play.

What counts as an assessment opportunity?

Any student presentation, meaningful engagement in class discussion, a discussion with me outside of class where I learn something. Those things count as opportunities for me to assess student performance that don’t necessarily have written feedback attached to them. In each case there is plenty of verbal feedback from classmates–but I don’t always participate. In fact, I prefer to leave it to the students.

Why are written papers not on this list?

Students will get feedback in the form of a referee report on each paper. I am not as concerned about providing more structured feedback here because I feel it is adequately covered.

Why do you prefer to leave the process of verbal feedback to the students?

One of the skills I am trying to encourage is the ability to evaluate arguments critically and thoughtfully.

If there is a reason to leave the verbal feedback to the students, might written feedback from the instructor corrupt this process?

Oh, yes. That is my main worry.

How can you avoid this trouble?

um. uh. [blink. blink.]

I hope this writing will spur me to some ideas about that…

What are the goals for this written feedback?

I want to focus student attention on some aspects of what they did. Ideally, quality feedback should help speed up a student’s process of improvement by directing his or her attention to something concrete.

What kind of constraints are you going to impose upon yourself?

I am a constructivist at heart. The student must come to grips with the material and how to do it. Each one should do this on his or her own terms. One idea would be to give feedback by asking questions.

I am just not sure what kinds of questions I would ask that are detached from the process of running a class meeting. We handle lots of things in class, and I almost always do it by asking questions. Maybe I will just reiterate some of the unanswered questions. That doesn’t feel like a very good answer.

Another idea is to use a sandwich approach: mention something positive, make a suggestion for improvement, reiterate the positive outcomes. And be relentlessly optimistic.

Now I’ve run out of questions. So.

I think whatever I do will have to play to my strengths. I am at my best when I split my time as a cheerleader, mentor, and coach. Students are capable of amazing things, and sometimes they just need for me to believe in them and expect it out of them. Sometimes they just need a little bit of commiserating about how frustrating it is to do mathematics. Sometimes they need a concrete suggestion of what to do when they are stuck and at their wit’s end.

That was unsatisfying.

Here we are. 500+ words in, and no answers I feel wonderful about.

Never mind, time for some unbridled confidence.

When I got into IBL teaching, I recognized that a major asset I had was hubris. I just believed that I could do this. Usually it works.

What? So I will have to help each student in as individual a way as possible, thinking on
my feet and being careful about everyone’s feelings? Why should I worry? I can do that.
I’ll think about this some more, and just try to roll with it.

I don’t feel like I finished my homework.

Standards Based Assessment for a Moore Method Course

Motivation

I have been working on developing a reasonable assessment model for my IBL Euclidean Geometry Course for a while now. I have several reasons for this:
1. It would be more fair, and better for my students, if I found a way to communicate with them about their progress. At the very least, I need to open the line of communication, so students feel they can have a conversation with me about how things are going.
2. So far, I have been going with a “you will have to trust me” approach. I have gotten away with it. But someone who wants to raise hell will make my undocumented life difficult.
3. This class is conducted as a lightly modified Moore Method course. Standard assessment with homework, quizzes and exams just doesn’t feel right.
4. The accountability movement is coming. Sooner or later, I will have to deal with a top-down mandate to deal with how I assess my students, and how I assess my teaching. I choose to start, on my own, with the parts I can control before that pressure gets here. First up: how I assess students.

The Main Idea

I will try to use a Standards Based Assessment scheme. I will attempt to focus on this mainly as a feedback mechanism. Grades will only happen to the minimal extent that is required.

What didn’t work well enough, and why.

I tried to implement a simple SBG/SBAR scheme in each of the last two semesters. Neither worked because I had not found a method of dealing with the administrative details. At first, I asked too much of myself. Then, I asked even more of myself, but on deadlines. Ugh.

What is working

I am happy with my set of standards (read that as learning goals). I am very proud that they are weighted toward process goals: what one does and how one behaves as a mathematician. This is intentional—I want students to become acculturated to doing mathematics, and to acquire some of a mathematicians habits of mind.

A New Attempt

For next semester, I have devised a two-prong approach to administering a standards based assessment mechanism.

The First Prong: Face to face meetings

In order to make for better communication about expectations, I will meet with each student individually every three weeks. This will involve splitting the class. I will meet with half one week, half the next, and then take a week break.

Before each meeting, I expect the student to write a one page reflection about their progress in the course. To tighten this process up, I have written specific prompts to which the students must respond. This must be done before the meeting. It can either be sent to me electronically, or it can be brought to the meeting on paper, but it has to be done before the conversation. Really, the paper is not important. But the time for reflection is crucial. The meeting could too easily be wasted without it.

Second Prong: Professional Feedback at each Assessment Opportunity

Each time a student participates in some sort of assessment opportunity (a presentation at the board, turning in a written paper), I will provide feedback. I have a little electronic system built (with the help of my friend Stephen Hughes) using a Google Docs form/spreadsheet/script combo. I have a web form into which I will type comments. When I click the “enter” button, my comments are saved in a spreadsheet, emailed to me, and emailed to the students.

It is too much to manage class and write out feedback at the same time, so I will be doing this during the hour after my class meeting. I normally take time to convert my notes into a blog post for the students anyway. Now I will just add a little bit to the “post meeting decompression” that I do.

What is left to do?

I need to think some more about how I will provide feedback. I want this to be a narrative process, but what are my aims? What constraints should I observe?

That should be my next post. 🙂

Where is all of my stuff?

Well, I keep a blog for the students, and it has a page all about assessment. Go have a look. Not all of the links are live, yet, but they will be at the appropriate time of the semester.

In the end, what about grades?

Here, I have no substantive changes, but Ed Parker has pushed me a bit…

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.

Context

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.

Details

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?