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.

Integrated Course Design for Euclidean Geometry, Part II

I continue the process of redesigning Math 3600 Euclidean Geometry using Fink’s Integrated Course Design Guide.

Step Three: Feedback and Assessment

This is where I am ready to do some tweaking. I have tried to use a “Standards-Based Assessment” mindset in this course lately, but it still doesn’t meet my needs. Also, I have been part of an on-going conversation about Linda Nilson’s new book on Specifications Grading. (Hop over to Google+ and look for Robert Talbert or Bret Benesh.)

Fink makes a distinction between “educative assessment” and “audit-ive assessment.” At first glance it feels like the distinction between formative and summative assessment. But Fink means more than just this.

In his terms, educative assessment has four crucial features:

  • It is forward-looking, in that it mimics the ways in which students will be required to use the knowledge or skills from the course after the course is over. His focus is on life after school, but I think it might reasonably apply to subsequent courses.
  • It has clear criteria and standards. By criteria, Fink means those features of quality work that you will seek in your students work. By standards, he means figuring out what counts as good enough and what counts as awesome.
  • There is opportunity for self-assessment and reflection.
  • It provides for high-quality feedback. His guide for “high-quality” involves an acronym: FIDeLity
    • F — frequent
    • I — immediate
    • D — discriminating
    • L — loving

To check myself against all of this, I started by making a list of all of the bits of feedback and assessment that happens in the course. Students have always done these things:

  • Presentations of their arguments in class
  • Writing for the class (scientific) journal
  • Refereeing for the class journal
  • Writing a Final Exam

Of those only the Final Exam counts as a summative assessment. For an active student, the other three things happen pretty frequently and give a lot of formative assessment and self-assessment opportunities, but only about the quality of the mathematics done. Early on, it became clear that students needed a “early warning system,” so I started using a

  • Midterm Exam

It is important to note that my midterm is purely a formative assessment and feedback mechanism. I do not record anything from this in my gradebook, but I comment on them as much as I can.

About a year-and-a-half ago, I realized I needed to put in something which more seriously focused on self-assessment (and self-regulated learning). So now I also have a sequence of

  • Three self-assessment interviews. I ask the students to write a short reflection and then meet with me to discuss it for 15 – 20 minutes.
The big problems

This leaves me with some issues. First, my only summative assessment is a single (week-long, take-home) final exam. Also, the process of self-assessment does not play into my final assessment at all. The whole thing gets dropped after the third one (usually in week 12). Finally, I have no summative assessment which covers the learning goal for oral communication.

My Adjustments
  1. Next term, I will ask students to complete a portfolio of their work. This should involve almost no extra work, they should simply bundle together the things they have already done together with their solutions to final exam questions. I think of this a bit like a tenure portfolio document.
  2. To bring the level up on meta-cognition and self-regulated learning, I will ask students to write three additional essays. These will be written sometime in the last half of class, I think. I don’t know exactly how I will handle the details, yet.
    • An essay on “productive failure” of some sort. Maybe My best mistake or something similar.
    • A “how it happened” essay on “my best paper.” I want some sort of introspection on process as it pertains to success.
    • An essay on “what it means to do mathematics.”

Each of these will be less than two pages, I think. Unlike the rest of our work, they should be personal narrative storytelling.

Step Four: Teaching and Learning Activities

This part is a longer section of Fink’s guide. He tries to push the reader to set up some sort of active learning course. Consider getting the students to do the subject; think about the cycle of work in class and between classes, etc. I don’t need this push.

The class is run as a (Modified) Moore Method course. Students will spend time between classes finding and constructing arguments for a carefully designed sequence of mathematical conjectures. During our meetings, students will present their work and critique the arguments of others. After a successful presentation (deemed by the students to be a theorem), the presenter will write a paper for the class journal and submit it. It will go through a peer-review process and eventually get published. The whole environment is structured to be a miniature version of the wider mathematics community. I serve as the benevolent dictator and direct traffic. Everything is meant to mimic the professional work of a mathematician as much as is feasible.

Step Five: Integration

Now Fink encourages us to check that things fit together.

Are the situational factors accounted for? I think so…

Do the learning goals and the feedback & assessment line up? Mostly. I still have a problem with the lack of summative assessment for oral communication. I don’t think I have anything extraneous that don’t serve one of my goals.

Do the learning goals align with the teaching and learning activities? Yes. Both are aligned with how a professional mathematician works.

Does the feedback and assessment loop help students deal with the teaching and learning activities? Well, the feedback part does. Most of the true assessment comes pretty late. But feedback happens at the point of contact.

Just to check, I made a little table of my learning goals vs. my assessment methods (which are just my learning activities by other names). I noted when formative assessment and feedback is happening and when summative assessment is happening. I won’t share it here, but I was happy with the result. The only missing bit is that summative assessment of oral communication.

With that, I have finished the initial design phase. Next up, the intermediate phase where things get “assembled.” I think I might get through that quickly.

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


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.

Next Projects: “professional” course design

I have done a lot of thinking about my teaching this term, but I have been so swamped by the circumstances of my linear algebra assessment structure that I haven’t had any time to write.

In particular, I have been thinking about both assessment (from reading and discussing Linda Nilson’s Specifications Grading), and about general structure of college education (from reading and discussing Teaching Naked by José Bowen).

After a decade of teaching while actually thinking about what I am doing, I am starting to feel like a professional. So my next task is to take a more professional approach to designing courses for next term.

I will be using this paper by L. Dee Fink as a map for my work:

A Self-Directed Guide to Designing Courses for Significant Learning

I invite you to play along. I will start by rethinking the design of a course that I have run with great success, Math 3600 Euclidean Geometry, so that I can learn the process. Then I will design MATH 3630 Differential Geometry, which needs some attention. I am assigned both of these for next term, so I can count this as work on schedule.

If you are not convinced this is a good idea, consider this from the opening of the guide:

When we teach, we engage in two closely related, but distinct activities. First, we design the course by gathering information and making a number of decisions about the way the course will be taught. Second, we engage in teacher-student interactions as we implement the course we have designed.

However, of these two activities, our ability to design courses well is usually the limiting factor. Most of us have had little or no training in how to design courses. In addition during the last two decades, research on college teaching and learning have led to some new ideas about course design that have, in essence, “raised the bar” in terms of what is possible.

So if you care to join me as I take a more purposeful, professional approach to course design, stay tuned. First up: an outline of what to do, in order.

(As I look ahead, I think it is a 12 step program…)

Better Standards for Linear Algebra: Organize by the Big Questions

I want to write a big post with lots of pictures summarizing my experience at the last Iowa Section of the MAA meeting. But that will have to wait as I have lots of real work to do. One piece of that work I want to share makes up this post.

In response to a bunch of whining on Google+, many internet friends who have tried standards based grading gave me advice on how to handle my linear algebra class better. Basically, by trying to be complete, I was making an enormous, unwieldy list of tiny little standards to track. This was clearly a disaster in the making, so I begged for help.

The main piece of advice I got was this: group the standards. Josh Bowman told me this quite clearly, and remarked that he had a standard for every day or two of a calculus course. Then either Bret Benesh or Kate Owens reminded me of Kate’s organizational scheme: use the big questions from your course to decide your standards.

With this pair of wonderful ideas, I ran back to my office to rework things. This took a lot of thought to sort out. But I am so grateful to my internet friends for sharing their expertise with me. I am slowly becoming a better teacher in part because they are willing to help me.

Anyway, I worked hard on this, but I don’t feel like it is totally mine because I needed so much help to get rolling. So, I am sharing it with you, my faithful readers. (uh, hi… Doug? I am pretty sure there is a nonzero chance someone named Doug will read this.) This is version 0.12, or something like it. It could use some criticism, which I welcome. Still, I feel like this is a public beta product. Maybe an official release could happen the next time I am assigned linear algebra.

Without further ado, here is the current status of my Linear Algebra Content Standards, Organized Thematically by The Big Questions of a first matrix algebra course. In this course, I like to use Gil Strang’s Introduction to Linear Algebra (4th Ed), so items are keyed to the chapters in that text where the relevant material is developed. I have done this for my student’s sake, but I bet it can’t hurt that someone tells you that this is text worth reading.

Standards for Linear Algebra Content Learning

These are organized by the “big questions” that we address throughout the course. At first, the questions are pretty straightforward and focus on solving systems of equations. Eventually, the questions become more internal to linear algebra, and address things that come up in our study of systems, and are definitely at a “second level.”

Foundational Goals

Question One: What are the basic objects of linear algebra?

  • Vector Algebra (Chapter 1)

    add vectors, plot vectors, compute scalar multiplication of number and vector, compute linear combinations, geometric interpretations of these operations

  • Matrix Algebra (Chapter 1 and 2)

    add matrices, take transpose, multiply matrix times vector (two ways), multiply two matrices (three ways), identify troubles with matrix multiplication: commutativity, inverses

  • The Dot Product (Chapter 1)

    compute the dot product of two vectors, compute angles between vectors, compute length of a vector, normalize a vector, use connection between dot product and linear equations to work with normal vectors

Question Two: How can we solve a square system of linear equations?

  • Gauss-Jordan Elimination (Chapter 2)

    Use Gauss-Jordan and back-solving to solve a system, find LU decomposition, identify when Gauss-Jordan breaks, identify when matrix does not have an LU decomposition and discuss workaround, compute determinant of a square matrix, compute the inverse of a square matrix

Question: How can we solve a general system of linear equations?

How can we tell if there is a solution? What shape will the solution set have? When will the solution be unique? Is there a computationally effective way to find the solution set?

  • Solving Systems of Equations (Chapter 3)

    Solve a general (rectangular) system of linear equations using the reduced row-echelon form, special solutions, a particular solution. Give the general solution to a system of linear equations. Compute the rank of a matrix. Use pivots and free variables to reason about the solution set to a system of equations

  • The Four Subspaces (Chapter 3)

    Compute the nullspace, column space, row space, and left nullspace of a matrix. describe these subspaces by giving bases

Question: What are the good ways to understand subspaces?

  • Implicit and Explicit Descriptions (Chapter 3)

    determine when a set of vectors is linearly dependent or linearly independent, determine the span of a set of vectors. determine if a collection of vectors is a basis for a subspace Find a basis for a subspace described using equations, find equations to describe a subspace described using a basis use the row space algorithm and the column space algorithm to find a basis

Question: Can we find approximation solutions to systems that do not have an actual solution?

  • Approximate Solutions and Least Squares (Chapter 4)

    Find the “best” available approximate solution to an unsolvable system of equations, draw pictures explaining how orthogonal projection is relevant, use approximate solutions to fit curves to data

Question: Is there a good way to test if a square matrix is invertible?

  • Determinants and the Invertible Matrix Theorem (Chapter 5)

Question: How can we understand the geometry of square matrices as transformations?

  • Eigenvalues, Eigenvectors, and the Spectral Theorem (Chapter 6)

Advanced Goals

Question: Are there any good geometric interpretations of a system of linear equations?

  • The Three Viewpoints (Chapters 1 and 2)

    The row picture, the column picture, and the transformational picture. pass back and forth cleanly pass between the representations, and describe what a solution means in each case.

Question: How do we understand matrices as transformations?

  • Four Subspaces and the Fundamental Theorem of Linear Algebra (Chapters 3 and 4)

    Use the four subspaces to describe the action of a matrix as a transformation (function) Draw reasonably accurate schematic of the transformational picture using information about the four subspaces, make conclusions about the nature of a matrix using the four subspaces

Question: Is there a way to choose a geometrically good basis for a subspace?

  • Orthonormal Bases and the QR Decomposition (Chapter 4)

    Use Gram-Schmidt to compute an orthonormal basis for a subspace, decide if a matrix is orthogonal or not, compute the QR decomposition of a matrix

Question: Is there a good geometric way to understand the behavior of a general matrix as a function?

  • Singular Value Decomposition (Chapter 6)

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.

Starbird’s IBL

During MathFest a month ago, I was fortunate to have dinner with Dana Ernst, Matthew Leingang, Stan Yoshinobu, and Mike Starbird.

Photo courtesy of Stan Yoshinobu. Pictured (right to left): Matthew, me, Dana, and Mike. Mike is playing the game Wuzzit Trouble for the first time on my iPad.

Mike is pretty famous in the mathematics community: he is a well-known topologist, and for a while now has been one of the more public faces of inquiry-based teaching in college mathematics. Mike is a wonderful storyteller, an entertaining speaker, a good listener, and so positive and friendly with everyone, it is easy to see how he would have a good impact on students. I was pleased that events conspired so that I got to spend some time talking with him.

We walked a couple of blocks from the conference site to the place Matthew had picked out for dinner, and so I got ten minutes of beautiful Portland evening to talk with Mike one-on-one. During our short conversation he mentioned something about how he and Ed Burger (now President of Southwestern University) went about writing their book The Heart of Mathematics. The key point was that in writing the book, he had to give up some of the IBL approach, but he could keep other parts which he found essential. He said something like, “IBL really has two parts, and though we had to give up on students doing all of the development of the mathematics for themselves, we could keep the other part.”

At this point, my ears picked up. Stan had asked me last summer to formulate my own definition of what it is to do IBL. Regular readers (‘sup Vince and Paul), might recall I wrote about that a while back. Dana took a step forward to catch up and listen, too. He had overheard, and the two of us were in the middle of planning for the workshop in Wales to happen a few weeks later. Here was a big chance: we would get an attempt at what it means to teach with an IBL bent from one of the masters. Since Mike’s definition is a little different from what I had, I filed it away to share with you someday. Today is that day.

Mike Starbird’s two part definition of what it means to run an IBL class

  1. The students are responsible for developing and presenting the mathematics.
  2. The mathematics is presented with what might be called a “plausible false history,” in that questions are presented in a natural order that gives them some meaning.

That first part I expected. The second one is close to addressing the issue of intellectual need that I read about in some of Guershon Harel’s work. Why should students care about what you are teaching them? Well, if you can connect the ideas to questions that they can imagine asking and build a sequence of tasks so that this motivation stays with them, they just might care.

Certainly Mike’s definition challenges me to reexamine the way I have structured my courses. I don’t have to follow the history of a particular question, but have I invented a reasonable alternate one?