Grabbing Attention with Student Inquiry

I have been pretty lazy the last two days, but I have been thinking about the upcoming semester “as a background process.” Right now, my main concern is that I will be teaching at least one section of Math in Decision Making, our liberal arts/quantitative reasoning course. The classes tend to be large (50-60 students), many of the students are either mathmatically bruised or have small motivation for the class.

Usually, I combat this by picking topics to study that I find interesting to let my enthusiasm shine through. Also, I try to sprinkle in a few high energy activity days. That energy comes from me, though. It doesn’t always infect the students. This term, I will try to get a better hook into the students by supplementing our intro days by asking them to ask questions about the subject the activity introduces. I don’t have a better plan, yet. It can be hard to get students to generate meaningful questions when confronted with new material. So, I will spend some time this week thinking about how to support them through this activity.

Tomorrow, Spring 2016 begins in earnest. I have a dangerously large to-do list forming for the week, but I refuse to worry about it until 9am.

Back to Writing

I haven’t written much in the last year. This is not particularly a problem—this blog is mostly an outlet for thoughts in progress.

But the lack of posting here is a symptom of my distance from deep thinking about my work in the last year. I have plenty of excuses, even some good ones. But I did find it useful to write here at one point. I found it even more useful when people decided to read and comment. I am still amazed that some of you have done this. Thank you.

Anyway. I am going to try regular reflective writing again.

Here I go.

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An Approach to Specifications Grading: Guest Post by John Ross

I have been involved in a lot of discussions about assessment strategies lately. There is a bit of a swell of young faculty who are rethinking their assessment strategies carefully. For some, this is a first serious step to rethinking their jobs as educators, and for others it is further step into the details of how to be effective.

Today we have a guest post by John Ross of Southwestern University. I met John at the Legacy of R.L. Moore meeting this summer, so I already know he is interested in effective teaching methods. This past weekend he mentioned lightly on twitter that he is using a new assessment setup. I wanted to hear the details, so I invited him to write about it. I am very pleased that he accepted my challenge.

My Version of Specs-Based Grading

by John Ross, Southwestern University
This semester I am running my calculus class using a specifications-based grading system. The decision to do this was made after discovering Robert Talbert’s blog and reading the many informative things he had to say about specs grading. If you’re unfamiliar with this style of grading, I’d recommend starting there (

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Change of assessment: SBG & Specs down

In the last few years, I have experimented with different types of assessment strategies. In particular, I used something that would be recognizable as Standards Based Grading, and something that would be recognizable as Specifications Grading.

But this term, for at least one course, I am abandoning both. My best course is Euclidean Geometry. Standards based Grading didn’t work because the learning goals are big, process-oriented things. This made it seem like Specs Grading would be a decent fit, especially because a typical assignment (“find a proof of this theorem, present it to the class, and write a paper about it”) is a complicated, professional activity. In the end, the quality might vary, but you either did it well enough, or you didn’t. But the explicit statements like “do seven of these to earn an A” broke my spring class. Students rushed to their desired grade and then stopped. So, I won’t do either of them in the canonical way for this course again.

I have learned a lot from trying SBG and Specs. I think that a knowledgeable person can still see evidence of each in my work. I now have a clear statement of what the (previously nebulous) learning goals are (SBG). And I have much better language to describe what acceptable work looks like (Specs).

But I won’t use explicit promises about grade conversions anymore. Instead, I have described what typical achievement looks like for each grade in looser terms.

If you want to peek, go here.

Eugenia Cheng and Knowledge, Belief, and Understanding

Recently I finished How to Bake π by Eugenia Cheng. It is nice to pick up a bit of mathematics IMG_20150816_173003129_HDRpopularization every now and again and look for new ways of explaining the essential nature of our subject with the uninitiated. Since Cheng is a category theorist, and I don’t know any category theory, this gave me some hope for a new perspective or two. These hopes were reinforced by the fact that she has chosen to use making dessert as the analogy to describe ideas!

The book is enjoyable, and Cheng does an admirable job of trying to explain what mathematics is and how one does it. Of course, she then moves on to focus on what the essential nature of category theory is (her answer: “the mathematics of mathematics”). I found a few nuggets of new analogies and viewpoints to share with my students who are struggling to write their first proofs and find their first conjectures.

The part I most enjoyed was in the last few pages. Cheng describes three different viewpoints on truth:

  1. Knowledge
  2. Belief
  3. Understanding

I really like that these are clearly differentiated. Most importantly, she describes the process of mathematical communication really well, and with a useful diagram. (I suppose that a useful diagram is to be expected from a category theorist.) Here is the diagram.


I think what is truly great here is that the picture is a narrow ravine (she points out that it is difficult and dangerous to jump across). And her accompanying description of mathematical communication is as a process by which you (1) somehow pack your understanding into a proof, and then (2) share that rigorous proof. This plays up the role of a axiomatic proof as the kind of thing which can be done without (or maybe just with a lot less) ambiguity, while recognizing that it might obscure the understanding a little bit. But it also highlights that there is a job for the reader, which is really an inverse to the job of the writer. The reader must (2′) confirm receipt of and agreement with the logical argument, but then (1′) unpack it to construct their own understanding of the ideas.

This is a model I can share with my new proof-writers to help them do their jobs better.

An IBL Special Interest Group of the MAA

Over the last five to ten years, the number of people interested in using Inquiry Based Learning techniques in college mathematics classrooms has grown rapidly. Naturally, a lot of those people are members of the Mathematical Association of America.

The MAA has a structure, called a SIGMAA, for helping subsets of members interested in some topic to organize a little bit. Running a SIGMAA makes it easier to hold events at the big meetings (MathFest and the Joint Mathematics Meetings). Also, the MAA helps with some logistical support for community needs. There are lots of different organizations and projects that support the IBL community, and we see the SIGMAA as a projection of the IBL community upon the MAA. (Linear algebra metaphors are the best, right?)

So, a few of us are starting an IBL SIGMAA. The charter is almost done and we are close to empaneling a first slate of officers, so what we really need now is a list of charter members. This list of members has to be submitted with the application for a SIGMAA as a proof that people are interested. I don’t doubt that we have plenty of interest, but the list still needs to be made.

Interested? Remember that the “Member Plus” membership category includes three SIGMAA memberships. We would be honored if you would use one to join us.

Sign Up to Be a Charter Member of IBL SIGMAA

If you are interested in being “more involved,” know that we are required to appoint the first set of officers as part of the application, but later open elections will be held for all positions.

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.