Partially Ordered Thoughts

I firmly believe that for effective teaching and learning in a class, you need two-way feedback. That means that not only do I give the students feedback, but they also get to give me feedback. Georgia Tech, like most places, have a standardized end-of-term course survey. However, I want to be able to fix things on the fly. To do this, I implement a quick midterm survey through our course management system. I provide a modest incentive to the students to take the anonymous survey (free credit for a reading assignment or three clicker questions), and this time I got a phenomenal response rate. (The survey really is anonymous since our course management system can record a score in the gradebook for taking the survey without attaching a name to the responses.) Read more »

While the course I’m teaching this semester is called Applied Combinatorics, I do try to incorporate some theory into the course. The handful of math majors in the course generally thrive on these days. However, the vast majority of the students are computer science and computer engineering majors. In the past, theory-intensive days were the days where their eyes glazed over and they followed nothing or else they nodded in their sleep and suggested that they understood. This semester, I’m trying to do a lot more active learning in my classes, and theory days are good candidates.

My first big foray into doing something other than using clickers and peer instruction or just some small-group problem solving in this course came today. There’s a famous result in graph theory that says there are triangle-free graphs with arbitrarily large chromatic number. This is a great way to help computer science majors understand that determining a graph’s chromatic number is not just about finding cliques or computing vertex degrees. There are two common constructions used to prove this result. Last time I taught the course, I lectured on one construction and told the students to read the other. Massive failure was the result. This time, I got creative and pulled out an active learning technique called jigsaw. I’d been hankering to find a place for it in one of my math classes, and I’m pleased to say it went well.

What is jigsaw? Here’s a summary from Arizona State. Let’s say you don’t want to read that, and I’ll give the quick outline:

1. Break up the material the students are to encounter based on the sizes of your groups. (In my case, we use groups of four.)
2. Give each “position” in the group a portion to learn. (I use playing cards, inspired by Barbara Millis.)
3. Give the students some time to work with the material outside class.
4. When they come to class, have students form “expert” groups. (Clubs in one corner, etc.)
5. Give students some time to clear up any issues they have and figure out the best way to present to group members.
6. Have students re-form their usual groups and explain the material to their peers.
7. Conduct some sort of debriefing for the whole class.

How did I implement this? I assigned two suits to each construction (one of the constructions is particularly hard for them to wrap their heads around, so having two heads helped). Of the two suits assigned to a construction, one was also responsible for establishing the graph’s chromatic number while the other tackled the lack of triangles. The handout I gave them in advance for reading took the proof from the book and broke it down into smaller steps with questions for them to think about as they worked through it while preparing and to guide the discussion in class.

Leading up to class, I was a little worried about how it was all going to turn out. What if half my class didn’t show up? What if the students resisted (even though we’ve been doing lots of group work)? What if they couldn’t figure it out and gave up in disgust? Fortunately, none of my fears came to pass. Out of 16 groups, only a handful reported only having two or fewer members present. (I had those groups come meet with me so that I could see about merging them for the day, but the suits didn’t work out, so we played it by ear.) The expert group meetings ran a little long, but it was good for them to work through things. Some of the expert groups got a lot from one or two very well-prepared group members. Others needed a little help from me. However, they were really engaged when I met with them. It was like teaching a small class of 16 inside my class of 64, and I did it with only my hands as visual aids.

When the students shifted back into their regular groups, things just kept on rolling. Some groups were really stuck, so I helped them out. However, I’d say that of my 16 groups, there were only three I spent any significant time with. One of those was a group that was shorthanded and wanted to work through both constructions, so that barely counts. Through the whole class, almost all the students were thoroughly engaged. They were really working hard to figure out what was going on and to get a grasp on things. Did the students leave class understanding every single detail? Of course not. However, I’m skeptical that any student other than my A students walked out understanding any details with straight lecture last time. This time, my students got to see some of how mathematicians struggle through things. The computer science majors got a feel for why computing a graph’s chromatic number is hard, something that it’s not clear their undergraduate CS classes provide.

Today I managed to save enough time to debrief the activity properly, too. We didn’t run through the constructions or anything like that. However, I made sure to help the students get the take-aways that I mentioned above. I made it super-clear to them that I wasn’t worried about them reproducing parts of the proofs or the constructions. Today was about the experience and helping my students develop the gut-level intuition you need in combinatorics. I hope I was successful. I left them with the result that there are graphs with arbitrarily large girth and chromatic number and told them that we have to get at those graphs through probability and not a construction. I was thrilled to look out at their faces and see a genuine appreciation of such a result.

On Wednesday, we move from graph theory to posets. This is the chapter of our book that needs the most work. (Tom and I need to take a step back and stop writing for our usual poset audience and write for our usual textbook audience.) However, I’ve got enough days that I can pace the material with some more experiences like today. Does anyone else have thoughts on topics in the math curriculum where jigsaw would be a good tool? I’m eager to try it with trig substitution next time I teach calculus but would be curious to hear about others’ successes or just ideas.

We’ve already covered the fact that I write long course policy documents. The Spring 2010 Applied Combinatorics (MATH 3012) Course Policies and Expectations are seven pages long. Last time I taught this course, they were four. However, last time I wasn’t doing as many different types of assessments and wasn’t using much active learning in class. This time, I had a lot of explaining to do up front. Inspired by Maria Anderson’s post over at Teaching College Math, I decided to get my students to actually read at least part of the course policies by crowdsourcing them. Read more »

This post started as a comment on Nick Hamblet’s blog post about changing calculus. When I realized it was incomplete and almost 700 words long, I decided that I should probably put my thoughts up here instead and link to Nick’s post.

I have to agree with Robert Talbert’s comment that most of this dialog is taking place at liberal arts colleges. This is great for liberal arts colleges and their students. Unfortunately, the behemoths that educate the masses really need to get on board and realize that the way we’ve been doing it for generations is not necessarily the best way any more. I have seen some signs at Georgia Tech that it’s possible things may start to change at the top-tier research universities in the future. The Harvard physicist Eric Mazur (leading proponent of peer instruction and clickers in science courses) is a great example of someone who asked these same questions about his physics classes a while back (and did something about it!) and is also widely-respected as a scientist. We need more people like him in every discipline to create change at the Georgia Techs and UVAs of the world. Another problem with using liberal arts colleges as the proving ground for new instructional methods is that faculty at Big State U almost always say “That’s great in your class of 25. I have 250 and just can’t imagine it working in my class.” Well, the specific things being used might not work, but there are things that can be effective in that context, too.

What are my thoughts on what calculus should be and how we should accomplish this? I’ll admit their not fully-formed, but they’re getting further along as time passes. I’ll do my best at outlining them, but I should point out that I’m writing this at 34,000 feet on four hours of not-so-great sleep.

Foremost, every course needs designated learning outcomes. These should be broad and determined by consensus from within the department, the professional community, and the “consumer departments” that require our courses for their majors. For Calculus I or II, I don’t think “Be able to integrate a function using partial fractions.” is a good learning outcome. Unfortunately, this is the most common type of learning outcome written now to satisfy accreditation bodies. I’ve seen Georgia Tech’s ABET outcomes for the lower-division math courses, and they’re sadly almost all like this. Instead, learning outcomes should lay out how we expect our students to think differently when they leave the course. What are the concepts they should know about? For calculus, I’m much more concerned about students understanding the derivative as a rate of change and how that connects to the integral and area. Instead of caring if students can graph functions, we should care about if they understand what the derivative tells us about the graph of a function. I bet if you asked most mathematics professors how they want their students to think differently after their calculus course, you’d get similar answers. Unfortunately, too many people don’t understand that our teaching and learning activities, feedback and assessment mechanisms, and learning outcomes all need to be aligned. It does no good to want your students to be able to do these things and then just teach them a bunch of algebra tricks and test them on meaningless manipulation of symbols.

Learning outcomes at every level (from remedial courses to graduate courses) need to include communication (reading, writing, and oral) in the discipline. We cannot teach our students everything, so they will have to learn by reading in the future. This is easier if we help them when they’re getting started. If we genuinely care about concepts, they need to communicate their understanding in a meaningful way, often in the context of an application. We need to teach our students to critically evaluate their work to determine if it makes sense. How many times have we seen students compute a volume using calculus techniques and give a negative number as an answer without batting an eye? In evaluating my precalculus students’ work at the end of this semester, I realized that some of them still don’t have BS detectors when reviewing others’ (or their own) work. Students need to be able to take a result that they get (by hand or from a computer) and determine if it makes sense in the context of the problem. (In case of my class, when answering a question that asks for the total amount of interest paid over the life of a loan, a percentage is usually not the right thing to give.)

Once you’ve got your learning outcomes laid out, let people teach the course how they want to (even graduate students). Those of us who think that students should not be glorified calculators should be allowed to focus on concepts, writing, and communicating. The old guard who like meaningless manipulation of symbols can go ahead and do that. They’ll retire soon enough, so it’s just not worth our time to change them (although it may be worth the effort to keep them away from the froshlings). Common exams need to be eliminated, as I’ve only seen them leave everyone unhappy. (The only exception being when a course “czar” writes the exam. Then he/she is happy and everyone else is unhappy.) With learning outcomes and good assessment schemes, you just hold every instructor accountable for providing evidence that students met the learning goals. This may be through questions from the final exam, mastery exams, projects, papers, whatever the instructor believes documents it (and is generally considered valid assessment in the professional community).

A potential pitfall here is the issue of sequence courses. At a liberal arts college, this is not so much of a problem, I imagine. It’s likely that if you took Calc I from Prof. X, then Prof. X will also be teaching Calc II the next semester. At Big State U, this is less likely (and almost unheard of at science/engineering school like Georgia Tech). A student who comes out of a concept-driven calculus course will struggle immensely in the sequel if his/her instructor wants fast-paced symbol manipulation. I don’t think there’s a great solution to this problem. However, Duke has come close. They’ve instituted two calculus sequences. One is traditional (no computers, symbol manipulation, etc.) and taught primarily by senior, tenured faculty. The second is reform (lots of calculators, computers, labs, but concept-heavy) and taught primarily by non-tenure-track (but full-time and under long-ish term renewable contracts) and graduate students. They help the advisors from the various majors understand which sequence is better for which types of students. I think the idea is that as the old guard retires, the traditional course will go by the wayside. (I should note here that I’m not a big proponent of the reform movement. I think they’ve done a lot of things wrong politically and caused an artificial schism that makes young mathematicians feel like they need to choose a “side”.)

Yes, this is all pie in the sky at the moment. We’re probably decades away from this, and it may not work at all. However, I think we’re failing our students if we don’t try. This is also the heart of my career conundrum. I’ve been educated solely at medium-large public research universities and know them well. Apparently there is some belief that I can make a career of working at one (at least that’s what I’m inferring from the fellowship I just got). However, I question whether I can do the research I enjoy and need to do in order to get tenure at such an institution while also trying to create change in how we teach mathematics, even if it’s just in my own courses. Where does the time come from for the professional development required to keep current on these matters? Is there a reward other than a pat on the back (at a research university) for really doing something about undergraduate education? One of the reasons I feel like I should take that career path is to influence the future generations of mathematicians. No one in the Georgia Tech School of Mathematics is seriously evaluating how we teach introductory courses other than graduate students. We were the beneficiaries of a course from our Center for the Enhancement of Teaching and Learning that made us reevaluate our views. We’ve all changed how we teach, although the degree varies, because we’re fortunate to teach in a unit that gives us a lot of latitude. Many mathematics graduate students, however, seem to just be exposed to the status quo when it comes to teaching. Others learn about different approaches but are discouraged from pursuing them because of the time it takes the first time around.

Again, apologies for any incoherence. This is pushing 1500 words, so I should probably edit it. However, that’s no fun on an airplane, so I’ll leave it as is.