Integrated Course Design for Euclidean Geometry (following Fink)

I recently learned of the work of L. Dee Fink on course design for college instruction. I am still amazed that (a) this hasn’t existed for a lot longer, and (b) no one had mentioned it to me before in its ten years of existence. Anyway, I have grabbed a copy of A Self-Directed Guide to Designing Courses for Significant Learning, and I am trying to use it to “redesign” my course Math 3600 Euclidean Geometry. I put that in quotes because I don’t think the course really needs a redesign. In fact, this one course is easily the best work of my career and runs pretty well with almost no friction. But that doesn’t mean I can’t maybe tweak a little thing. And it definitely makes a reasonable subject for learning the process set out in Fink.

picture of fink's 2003 paper

I took a picture of a printed copy of
the pdf because this blog is a first-rate web-based operation. Also, this visual
is somehow important to me.

Currently, my only concern with the course is in my assessment structure, and I have been thinking about standards based grading and specifications grading lately. We’ll see how that plays out when they pop up.

This is going to be a long post. I am going to work until I can’t, and then pick up again with a fresh post in the morning.

General Structure

The Guide is structured as a bundle of worksheets and reflection prompts for the instructor to complete. There are twelve steps bundled into three phases: an initial design phase where you “build strong primary components”; an intermediate phase where you “assemble the components into a coherent whole”; and a final phase where you “finish important remaining tasks.”

A nice feature of this is the sequencing. Fink guides us to work with a backwards design philosophy where we identify outcomes and a scheme for assessment before thinking about regular coursework. That seems like a good idea, but

It is too late for me, my son.

I have so much of this course settled that I know I cannot reasonably claim to be thinking about outcomes and assessment first. So be it.

Initial Design Phase: Build Strong Primary Components

Step One: Identify Situational Factors

Students

Math 3600 Euclidean Geometry typically has between 15 and 22 students. Students can take the course anytime after completing a first semester of calculus, though many take it much later, so a class is usually mixed from sophomores to seniors.

Physical Facilities

We meet face-to-face three times a week for 50 minutes. The classroom always has a large chalkboard, but only some semesters do we have a full set of presentation equipment. (There is always a projector, an ELMO, and a set of VGA hookups, but not every room has a computer in it.) Most often the classroom has long, narrow tables and movable chairs. Some classrooms have a pile of chairs with attached desks. Next term, I have a room without a computer and with the “attached desks.”

Nature of the Subject

The course is dedicated to the study of classical planar geometry: triangles, polygons, circles, constructions, and area. Really, the idea is to teach How to work like a mathematician. Students need to get involved in the processes of finding, making, and critiquing axiomatic arguments. So, the subject is HARD CORE MATHEMATICS on REALLY BASIC OBJECTS.

Characteristics of the Learners

The primary audience consists of preservice secondary teachers, but usually there are students in a standard mathematics major, too. I have encountered a wide variety of attitudes towards (1) geometry, and (2) argument-based mathematics. Most are a bit apprehensive about “proofs.” A fair number did not enjoy high school geometry.

Characteristics of the Teacher

Well, I am pretty awesome. I also have an unreasonably high level of confidence in my abilities in the classroom. And I am pretty humble. Don’t underestimate how humble I am. I am really good at that.

I am a convert to Inquiry-Based Learning. I believe that students learn best by doing. I really like coaching, mentoring, and playing cheerleader for my students. Students tell me that I am a little scary, but I can’t tell if that is because I have inherited an unfriendly face or because I expect students to do challenging things. I don’t really consider myself an expert in classical geometry, but I am a mathematician and I can certainly handle this material. I do enjoy geometry, and I enjoy watching students grow.

Step Two: Identify Learning Goals

I have written about this many times before, so I will be brief. The goals in this course are mostly about process. I have tried a standards-based assessment scheme (sorta), where the listed standards are bundled into groups like:

  • Mathematical Investigative Process
  • The Axiomatic Method
  • Planar Geometry Content
  • Oral Communication
  • Written Communication

For this list, I have tried to pick out the behaviors a professional mathematician engages in when going about the job of making mathematics.

Now, Fink has a spiffy diagram of “A Taxonomy of Significant Learning” at this point. In this model, there are six sorts of learning (broadly stated) that one can do, and learning counts as significant if it involves more than one of them. I like to think that my course hits four out of the six:

  • Application
  • Foundational Knowledge
  • Learning how to learn
  • Caring

But I don’t think I hit these last two, at least not directly:

  • Human dimension
  • Integration


Okay. Now it is midnight. I am two steps in, with ten more to go. I’ll attempt to finish the initial design phase tomorrow morning.

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