Learning Goals for Linear Algebra: Content and Process

As part of my reworking of linear algebra, I have been reconsidering the course goals. I went back and read some of what I have written before about the mathematics to be discussed and the working habits on which we will focus. There is also this post from early in last semester. I think I only have small tweaks to make on these counts.

Well, I am a long way from having a fully developed set of Student Learning Objectives and a Student Outcomes Assessment set-up. (The SLO/SOA regime is the language used to sound official at UNI.) I have got a reasonable set-up for the department level technology SLO/SOA. At some point, I should get such an explicit set-up for my course. Maybe next summer I will work on that? Uh, sure, next summer.

Changes to Content Coverage

Last semester, the class got just within shouting distance of the three act content plan I wrote in the post linked above. Our discussion of determinants was superficial, and we just didn’t talk about the finite dimensional spectral theorem. Given that we basically wasted the first two weeks, I think those things can be addressed easily.

I would like to add just the smallest thing. Given all of the things listed here, and the focus I give to treating a matrix as a function, it would be great to add a discussion of the singular value decomposition.

Oh! Also, I didn’t talk about the cross product in Euclidean 3-space at all. I am comfortable with this.

Changes to the Meta-mathematical, process goals

I still see these as primary working habits that students must gain proficiency with to succeed:

  • make and explore examples
  • gain experience with abstraction
  • engage in more careful use of language

I will have to address these goals by writing my course so that students are forced to grapple with them, and also by being explicit with my students that this is expected of them.

And I still have these goals to deal with:

  • learn to use technology in an appropriate way
  • learn to read a textbook for understanding

I have a plan for the technology part. I have to work out something reasonable to help students with reading for understanding. Again, I have to be explicit about expectations and give focused instruction. (Somewhere in the back of my mind I have an evil plan to implement some sort of analog of a grad school foreign language reading exam.)

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An Assessment Idea: Git

I just had a short conversation with Dave Grant. Dave and I started at UNI at the same time, we have kids of similar ages, so I saw him tonight when he came to pick up my son for a sleepover. While the kids were playing we chatted about “stuff,” and the topic of assessment came up.

At this point, I am talking about assessment of a program, or its pieces, rather than assessment of students. Dave mentioned that he had described a process for several of the Languages and Literature courses involving keeping drafts and comments on those drafts as evidence of student work and improvement.

It immediately made me think of Git. Basically, the problem of documenting all of these things is one about version control. Computer scientists have solved this problem (and more).

So, is it crazy to imagine a world in which college students are asked to use a version control system to document their work?

This doesn’t solve the problem of how to take that data and turn it into some useful measurement of “success.” Still, I think it might be useful.

Tell me why I am wrong.

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…

Monthly Checkup? Committee Work

It has been about a month since I last took time to write here. Of course a lot is going on. I am going to blather on in a rambling way now. This will probably be a couple of posts–as I likely should have written one a week.

The Committee Assignment:

I have been participating in a University subcommittee with an insanely bureaucratic name: The LACC1CCC. That stands for “Liberal Arts Core Category 1C Coordinating Committee.” At this point, we are trying to sort out some kind of large scale SLO and SOA (Student Learning Objectives and Student Outcomes Assessment) for those courses we offer that fulfill the Liberal Arts Core requirement labelled 1C: Quantitative Literacy. I find this really challenging. On one hand it is important to have a clear idea of what you are trying to do and how you will decide if you are getting it done effectively. But it also feels like a giant nightmare: a lot of talking that will just disappear and not be used by anyone. For now, I have decided to be altruistic and believe there is a chance this will work well.

At UNI, we have five courses listed that meet the requirement:

  1. Math 1100: Math in Decision Making
  2. Math 1420: Calculus 1
  3. Stat 1772: Introduction to Statistical Methods
  4. Math 1201: Mathematical Reasoning for Teaching 1
  5. Computer Science 1025: Computer Modeling and Simulation

Those are very different courses. Somehow, we are to find a description of the commonalities in what we are trying to achieve with them and then decide on a measurement scheme for our own effectiveness at meeting those goals. I have taught the first three on the list (3, 1 and 2 times, respectively), but I don’t know much about the other two.

Right now, my biggest hangup is that my goals for Math in Decision Making are all about attitude and affect for the subject. I feel challenged by this process to improve my approach to “quantitative literacy.”

My course is split into three units: (1) counting and the idea of the infinite, (2) something about probability and/or statistics, and (3) classifying surfaces.

Almost all of the things that sound like strict “quantitative literacy” are in unit (2), which is clearly the worst part of the course. I am on my third time teaching this, I am using my third different approach to that unit, and it still doesn’t click. Students don’t like it. I don’t like it. Generally they don’t learn what I am aiming at and no one has any fun.

I worry that this is largely a problem of my attitude. Units (1) and (3) feel like this: “woo-hoo. let’s do some crazy math!” Unit (2) makes me feel like Ben Stein’s character from Ferris Bueller’s Day Off. So, this is something I am noodling over. Anyone got a good idea for a unit that teaches some basic prob & stat, but feels like interesting math is going on? Anyone? Anyone?

 

Oh, and if you have experience with designing SLO/SOA, I’d like to hear about that, too.

 

 

Examples for Differential Geometry

So, I must be some sort of ancient fuddy-duddy. It is true that I went to college before the internet was really a pervasive thing like it is today, but there is not excuse for not executing some simple searches of likely places.

I was goofing off and looking through my Google+ stream, and I came across this nice link in a post by Alexander Kruel:

http://en.wikipedia.org/wiki/List_of_curves

Of course, on it there is a link to their list of surfaces:

http://en.wikipedia.org/wiki/List_of_surfaces

Now I just have to sift through and pick some examples to show off to the students. I have a few to add of my own that don’t appear on those lists exactly. Hey, that gives me an idea!

Students could improve some wikipedia pages as a way of sharing their work. I have never done that. Has anyone tried that? Is it plausible as an assessment?

Objectives for Differential Geometry

Regular readers (all three of you) are aware that

  1. I am scheduled to teach differential geometry this coming term.
  2. I am very excited about this.
A minimal surface

a minimal surface

Less well known is that I am also terrified of this experience. I taught the course in the Spring of 2010 and the experience was not all warm and fuzzy. The biggest problem was the disconnect between what my students knew coming in, and what I thought they would know. Suffice it to say that I have a much better understanding of what it means to have a student who passed linear algebra and multivariable calculus at UNI.

Let’s take it as given that this iteration of the course will also contain surprises, but I hope they will be much smaller ones. And I am at a place now as an instructor where I am much more alert for this kind of trouble, so that can only help. (Oh, please, please let that be a true statement.)

So, now I have spent a fair amount of time in the last two weeks worrying about how this course will all work out when I should have been properly enjoying time with my family. I started by trying to write a set of learning goals for the students.

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Student Outcomes Assessment, Standards Based Assessment

Originally Posted 06-05-2012

The Problem of Outcomes Assessment

Student Outcomes Assessment (SOA) is a major pain in the ass, and I dislike it.

OK. Now that I have done that, I feel better and I can get on with my work. Today I am trying to figure out how to appropriately handle SOA for my Euclidean Geometry course. I recognize the importance of this type of reflective practice, and I implicitly do it, of course.

The trouble is finding a way to assess and document if UNI students are learning what the department wants, without negatively impacting the learning process. Also, whatever I use has to be workable if someone else gets a chance to run the course, and has to pass muster with the UNI administration and the accrediting bodies it kow-tows to, um, uses for cover, no…uh, asks for certification. (Yeah, that’s the ticket!)

Also, the process really has to be something manageable. I do a lot of work as it is, and so this can’t be too big a structure on top of everything else. I would like to find a way to either integrate this into my standard practice, or extract it from what I am already doing. The distinction there is really minor. I will certainly have to do something new. I just don’t want it to be huge.

As long as we are talking about assessment

I have been thinking for a while that I want to try a Standards Based Assessment experiment in this course. For one thing, my Euclidean Geometry is a course that is running so well that I am comfortable running experiments in it. And for another, it really needs some kind of paper trail to avoid problems.

Why does it need tightening up?

I run my Euclidean Geometry course as a Modified Moore Method experience. This is a pretty old-fashioned version of Inquiry Based Learning. If you know about the traditional Moore Method, then my major adaptations are
* I use Euclid as a text book. (In particular, I use the Green Lion Press edition as it is Heath’s translation without the commentary and is a beautifully constructed book.) My task seqence is “around and nearby” Euclid.
* I don’t usually cold-call students.

Now, such a class is all assessement, all the time. The line between formative and summative assessments is completely obliterated. Students work hard and give presentations on their ideas. Each presentation and conversation, the student bares his or her soul upon the chalkboard, and I simultaneously (1) update my opinion of their capabilities and performance, (2) communicate to the student what they can do to improve and where I think their work as fallen down, (3) mentally adjust the rest of the semester to help address what the student and the class need to work on.

That is a lot to juggle. So much so, that I can take only cursory notes. But at the end of the semester deciding letter grades for the registrar and provost is easy: I just know.

The thing is, I am getting away with it. I know I am getting away with it. The students trust me enough (so far) that they haven’t complained. But if a student complained, I have nothing to say except, “This is my professional opinion.” I need something more concrete to use for cover um, … document my opinions.

That need to document student progress really doesn’t just come from above. I want it to make sure I try to be fair. I mean, I try, but I need to make a larger effort to be objective.

Past SOA Approach

Shortly after starting at UNI, I got asked to do a very simple SOA for geometry. The Euclidean Geometry course fits in our department’s SOA plan as a (the?) designated place that we instruct our teaching majors on what a proof is and how to construct one. I was asked to make an appropriate problem for my final exam that I could reuse and a rubric for scoring it so that we could track progress.

What Am I Trying to Achieve?

The department’s Student Learning Objectives statement includes the following relevant language. (I’m taking this from the SLO for the liberal arts major, but it applies about as well to the pre-service teaching track, I think.)

Analytical reasoning and problem solving skills specification:

Students will state problems and definitions carefully, modify problems when necessary to make them tractable, articulate assumptions, reason logically to conclusions, and interpret results intelligently. Students will approach problem solving with a willingness to try multiple approaches, persist in the face of difficulties, assess the correctness of solutions, explore examples, pose questions, and devise and test conjectures.

Proof and Argument specification:

Students will be able to compose and explain proofs in clear mathematical style, both orally and in writing, and to critically evaluate mathematical arguments made by others. Students will be able to use a variety of techniques of proof, including direct proof, proof by contradiction, and mathematical induction.

Now, as a class we address all of those except for mathematical induction.

My Magical Exam Question

Here is the question from my final exam, with set-up.

Definition: A quadrilateral is said to be circumscriptable when there exists a circle inside the quadrilateral which is tangent to each of the four sides.

Task: Settle the following conjecture as completely as you can.

Conjecture: Let ABCD be a quadrilateral. Then ABCD is circumscriptable if, and only if, AB and CD taken together are congruent to AC and BD taken together.

I don’t know why I felt the need to write that out, except that I am proud of this one. It hits all of those specifications as best I can manage.

My stupid “rubric”

I split the assessment into two phases: The first is “mathematics” like recognizing the bidirectional statement, noticing that one direction is false, construction of a counter-example for this direction, noticing that the false direction can be repaired with the addition of a hypothesis (hint: we talk about convexity alot in this course), and basic correctness of the arguments for each direction. The second phase is more about “writing and communication” like clarity and precision. Each part was judged on a 0–4 scale.

Looking back, I seriously don’t know what any of that is supposed to mean.

Why I Think That My Magical Exam Question is Inadequate

The more I think about repairing the rubric (or tossing it out and writing a new one), the more I think that the whole enterprise is doomed to failure.

This assessment happens once. It happens during a week long take-home final exam with five of six questions. Not every student turns in an attempt at that question.

How do I meaninfully separate the mathematics from the communication standards? How do I assess those students who spent six hours and tried the kind of strategies they should be learning, but didn’t close the circuit on this particular problem? How do I assess the partial progress that most students make?

How do I take data from the last five years and use it to either defend what I am doing, or inform me about the kinds of changes I need to make? As far as I can tell, the little columns of numbers don’t mean anything to me, and I made them.

The Big Reveal

So, the rest of today (and probably tomorrow) I will spend thinking about how to design and implement a Standards Based Assessment regime for this course. I have to meet the following guielines:
* minimaly intrusive
* fits my current practice and set of goals
* better reporting of expecations and progress to students
* ease of extraction for SOA purposes

So, anyone got any advice?