Project Laboratory in Mathematics A Taste of Research

no one could figure out what a math lab was as in a laboratory have Erlenmeyer flasks and things where are the flasks where's the equipment in a in mathematics my contention is that this lab follows the scientific method students take many courses most of them have a set syllabus it's organized it goes from a beginning through definitions and theorems to an end they're asked to solve problems which are well posed and have a have an expected answer this course is completely different because here the questions are not well posed they have to decide what part of this question they're going to focus on they have to decide where the exciting mathematics is going to be it's actually very hard to get beyond the smooth outer shell of a problem we rarely give ourselves a specific target and aim only at that target it's very easy just to you know write down the statement of the problem and some equations that you know which of elevant and then just sit and stare at them for a while and think what next we look at a general area and try to find the weak points find the points that we can make progress in that's a skill to turning something over in your head until you find the point of entry and I think that's one of the things that the class is designed to teach here we go through the scientific process we look at some mathematical situation we frame a hypothesis hypothesis might be called a conjecture in mathematics and then we test the hypothesis we test it by doing numerical experiments and so on but the ultimate verification of a hypothesis is a proof so this course is unique in that it doesn't have any syllabus per se there's no body of material that I hope students will come away with I hope they come away with an experience I want them to have some sense of the frustration and the challenge and the excitement of doing mathematical research I know a lot of people are terrified of doing it you know it's pretty much original research for most of these problems like you came up with this cool thing and you get to tell this whole group of interested people ostensibly interested people sort of what you did the involvement of the class is 27 so everything in the class is threes and we each get three groups of three undergrads each can work on three projects over the course of the term they will open the box do some experiments maybe do some computer simulations think about the question maybe they'll do some literature search after a few weeks they have enough material they start to write a paper and then I ask them to come into my office here with their advisor and tell me what they've discovered briefly a 10-minute presentation of what they've discovered and I have their first draft in front of me I've read it so it gives me an opportunity and their advisor an opportunity to comment on the mathematics make suggestions for directions they might go next and comment on the writing comment on the presentation it's a small opportunity to make mid-course correction those debriefings I think are very valuable for a number of reasons being able to give feedback on the writing in person is valuable because when you give in-person feedback then you can emphasize what's most important then they give a presentation to the entire class lecture of the sort that you would see in a mathematical conference each of these projects is a three week process for them in which they're supposed to learn about a new problem that's probably completely unfamiliar to them which involves learning some new math and then try and say something about the problem that has create something new three weeks is a very short time to go about that process most mathematicians take at least a semester and they know something about the problem beforehand but sometimes many years you're working on not only your and loop skills but your presentation you know communication skills how do you connect with your audience how do you explain a concept perhaps that some of our peers who are not familiar with what makes a good problem for this class the best thing about a project is if it can go off in different directions that's the way mathematical research works take a prime number like five or seven and look at its reciprocal 1 over 3 1 over 5 if you expand it out as a decimal you're likely to notice some patterns a fraction has a repeating decimal expansion whenever 3 is 0.33 3 repeating the decimal expansion of 111th is point zero nine zero nine zero nine and sometimes you'll get longer repeating strings of numbers how many digits are there before the decimal expansion repeats itself can you make any conjectures about that string of repeating numbers if I give you a random prime maybe a prime with 20 digits in it a big prime some random prime what's the likelihood that their decimal expansion of its reciprocal will have period exactly P minus 1 and so the students take this open-ended question in whatever direction seems like it's likely to be fruitful to them this is much closer to the experience of doing research in mathematics than anything they see in a standard class in this course in particular if the students come from many different backgrounds with many different levels of preparation some teams will be very sophisticated no a lot of higher mathematics and apply those tools to understand the math that they see that the regularities that they see in these projects some teams will take the same project but they won't have as much background they may have a lot of computational skills and they'll do an extensive computer simulation and learn from that some teams won't be strong in either dimension and they'll find their own set of things to study in these projects it's quite remarkable to look at what different student teams do to the same project and see the diversity of results that come out it's very encouraging somehow that students of many different levels and many different interests will find interesting and exciting things in these projects taking projects in your own direction is something that we definitely learned how to do I feel like this infant in many directions can go but we have to know like what's a good ending point for each of them how to kind of like just be confident in like whatever direction we're going is another thing that we learned so each semester this class is staffed with a professor a postdoc and a graduate student this combination seems to work very well their role in this is to help students overcome obstacles that they might have helped them understand what mathematical research is about help them overcome the frustrations that go along with this and reassure them that the instructors had the same experience himself that there can be weeks that go by when no visible progress happens it's important not to point the students too much in one direction because the whole point of the course is for students to discover mathematics on their own in meeting with these groups of students I think of it as like meeting with a graduate student I sit and think as clearly as I can about the problem that they're working on I'm honest about it I don't pretend to be dumber than I am I do what I can to help them so being sincere about helping them to make progress as part of the game there's a very subtle thing that the people leading the course have to do they have to be both encouraging and avoid blind alleys but also let the students find their own way I would love to see this style of mathematics subject flourish and grow across the country because I think that it fills a need so often our students see mathematics is something which is cut and dried and it seems like it was all done long ago and this is the one course where our students mathematics is a living thing that they can contribute to and they really understand how mathematics evolves this is a great class and it's so much fun watching the students learn and improve and play with the math and present it to each other it's a great class it's my favorite course to teach you get to engage with students really closely you get to know them well and see them change and grow in the course of the semester I hope that they take away something of the flavor of doing math research once you feel you've made a worthwhile insight you become more dedication it's like this is what I do now this course is unlike any other course that these students have ever seen because it's so the the goals are so experiential and so one student said this course is kind of strange but it comes together and it works you