The semester just ended, and I’m now in full retrospect mode. This semester I was fortunate to have only one prep — two sections of Linear Algebra. Linear algebra, for me, is the cornerstone of a modern mathematics education precisely because its concepts and its mechanics lie at the heart of so much real-world stuff — from web search algorithms to scheduling problems to computer graphics and many other areas. And yet, in a typical one-semester course on linear algebra you only get to touch on a handful of applications, and those tend to be sort of domesticated. A few years ago, I decided I wanted students to explore more than just the stock examples in the textbook, and I wanted them to do so in an authentic way that reflects real-world mathematical practice.

About that time, Derek Bruff published this blog post about his use of Application Projects, and I gleefully appropriated his ideas and materials for my own class. I’ve modified the process and the workflow over the last few years, but the whole experience is largely the same as when I first started, and I’ve started using Application Projects not only in linear algebra but in any 200- or 300-level course I teach that has an applied slant to it, including discrete math and my cryptography course coming up in January 2014.

The overall goal of the Application Project is simple: **Find a real-world problem that can be solved using linear algebra, and then solve it.** The only stipulations are (1) the problem has to be significantly more in-depth than a simple homework problem, (2) the problem has to be small enough that it can be completed in 6–8 weeks, and (3) the problem has to involve real-world situations or — very preferably — real data. Around week 6 of the semester, I introduced the Application Project and had students form into teams of 2 or 3 to start researching potential problems. Students were encouraged to look for problems in their textbook, from MAA publications, from professors in their major field (we had a lot of students from other disciplines taking the class for a math minor), or from library research. They were given three days to form their teams and two weeks to choose a problem. The main information document I gave to the students is here, and there’s a list of suggested problem areas in the back of that document — not problems themselves, because I didn’t want teams just picking problems “off the rack” with no personal investment, but general areas for exploration.

At the end of the two weeks, teams submitted a 3–5 page project proposal in which they formally stated their team members, stated the problem or problems they intended to solve, gave some background on the problem, brainstormed ways they intended to approach their problem and resources they might use, and set goals for a status report they were required to give a little later in the course. The purpose of the proposal was to make sure students had clarified in their own minds what they were going to do and what their goals were for the next three weeks. It was also intended for me to catch any problems with projects early in the process.

The biggest problem was (and has always been) teams declaring a problem that was either too general, too simple, or already taken by another team. Usually if there was an issue in one of these areas, there was an issue on all of them. For example, cryptography was one of the major areas students could explore. It turned out that six different teams ended up proposing the same problem, namely to focus on the Hill cipher and show how it could be used to encrypt and decrypt data. The reason six different teams collided on this problem was that the “problem” was way too simple. Encryption and decryption with the Hill cipher is exceedingly well-understood, and if this were covered in our textbook, it would be a basic homework problem if the book treated it. I had to contact all six of those teams and have them take another few days to retool their project topics. For example, could they come up with a novel cryptanalytic technique for the Hill cipher? Or, what if the Hill cipher were applied to another language besides English with a different vowel/consonant structure — how would the cryptanalysis change? Or, would they like instead to explore this cipher based on Hilbert matrices? In the end, four of the six teams ended up going with a different problem altogether. One team stuck with it and did an analysis of the Hill cipher when applied to Italian (one of the team members is fluent in Italian)— not extremely difficult but novel enough to be interesting and raise some interesting questions that can be solved with linear algebra.

About three weeks into the project (and three weeks from the end of the semester) teams submitted a progress report intended to catch me up to speed on their work and show that they’d met the goals they set for themselves in the proposal. Some teams met with me personally to do this — a few were in my office almost every day! So I had a sense all along the timeline of how teams were doing, and the built-in accountability throughout the process helped keep teams on track.

I’m a big believer that if you are going to do interesting work, the results of it need to be displayed in public. So the heart of the application project is a poster session that took place on the next-to-last day in class. I showed students the blog post from Derek along with some photos from past poster presentations to give them an idea of what a poster is and what a poster session is like, and I gave them this web site for ideas and guidelines on how to make good posters. The poster session was scheduled for the usual class hour, but in an atrium area that joins two wings of GVSU’s science building — it’s a very public place with a high level of foot traffic. Students set up their posters around the periphery of the atrium so that anybody and everybody coming through would see them and have the chance to visit.

The session was broken into three 15-minute shifts. During each shift, one team member was responsible for being at the table to host any visitors while the other member(s) circulated through all the other posters and asked questions of the people “on duty” there. Students were asked to develop a 3-minute “elevator pitch” for their project — something that would communicate the gist of their project in a quick way to an average person. One of the challenges of the presentation is that students would have to adjust their pitch to their audience on the fly — bring it down a notch if a visitor didn’t understand the math, kick it up a notch if it’s a faculty member visiting. I’d call out the shift changes every 15 minutes and team members would rotate through either being the presenter at their project or the person being presented to at someone else’s project.

We used this rubric for grading the poster presentations. The students who were not on duty at their poster were responsible for grading the presentations they visited, using the rubric. I spent about 4–5 minutes at each station asking questions or listening in to people’s presentations and grading them on the fly, and doubled the points from the student rubric. With this level of assessment, every student in each team had to be knowledgeable enough about the project to be able to present it well and answer questions intelligently.

For fun and a little extra credit, I also had a balloting station open where visitors could vote for their favorite project in three categories: Most Attractive Poster, Most Interesting Topic, and Most Sophisticated Use of Mathematics. The top winner in each category got +2 points (out of 80) on the project. This year I had a first — a single project that won, actually dominated is a better word for it, in all three categories. Yeah, it’s a little crass to be so loose with extra credit, but it added to the fun and many students went all out for it.

As far as the grading went, it was easy. The whole project was worth 80 points and between 15% and 25% of their semester grade — students got to choose how much they wanted it to count within that range. Six of those points came from just meeting key deadlines. Twenty points were for the proposal and the progress report; 48 came from the poster session grades; and 4 points came from an executive summary that students had to write and submit at the end. It was a lot of small, low-stakes assessment that provided support and accountability.

On the final exam, I asked students what was the most interesting thing they did or saw in the entire course, and the vast majority said the application project. Among the reasons application projects really stick with students are:

- Students have a great deal of personal buy-in with application projects. They are working on problems
*they chose*. - Students develop a surprising amount of depth with an application project. They pick one basic linear algebra topic and go deep with it, which is helpful because we have to cover so much in the course, sometimes it can lead to shallow understanding.
- Students encounter problems that are authentically real-world. The biggest example of this was from two of the teams using linear systems to model traffic flow through city streets. This is a simple application in the book, and the teams took the idea and tried to apply it to an actual city street grid. But the problem they encountered wasn’t the math — it was getting the data! They spent hours on the phone with state and local transportation departments trying to get data for how much traffic flows along a street at a certain time of the day. One team even traveled into Grand Rapids to meet face-to-face with a traffic engineer. Another team just got completely stonewalled and eventually had to go out in the freezing cold with a clicker and a stopwatch and measure the traffic flow on their own. In the book, these problems present the traffic numbers as if the data are just obviously out there and available — it glosses over the hardest problem about using math in the real world, which is the problem of incomplete information. These students had to deal with it head-on, and I think that’s an outstanding lesson to learn when you’re in the second year of math major.
- Students end up teaching themselves stuff and actually becoming experts on it. Several teams chose problems related to Markov chains, and during the poster sessions we had great Q&A about steady-state vectors, stochastic matrices, and so on. Amount of class time spent going over Markov chains? Zero.

Application Projects are hard, fun, and an awesome learning experience for students whose shelf life far exceeds the date of the final exam for the course. As far as I am concerned, they’re a permanent feature of any application-friendly class at the sophomore level or above I teach from now on.