 Okay, in any case, as I say, we've come a long road these three weeks, and I'm really pleased to introduce our final speaker for the cross-program events. And so, because it's a special occasion, I wanted to invite Tom Garrity. So, Tom is, hopefully, most of you who have seen him around these weeks have been talking to him. So, Tom is a longtime member of the steering committee, the undergraduate steering committee. He's lectured in the undergraduate summer program three years ago, four years ago, something like that. A longtime professor at Williams College works in number theory and, you know, just one of the great additions to the PCMI family. And I'm really pleased to introduce Tom Garrity, who will speak on mathematical maturity. Thank you. Most of us learned, at the feet of our parents, the following truth. Describe the world. Everything is described by functions. The sound of my voice on your eardrum function. The light that's kind of hitting your eyeballs right now function. The entries you put in your random matrices function. It's all function. Different classes in mathematics, different areas in mathematics study, different kinds of function. High school math studies, second degree one variable polynomials, calculus studies, smooth one variable functions, and it goes on and on. Functions describe the world. I start most of my courses that way. And the question is, why? For the classes, it's because I really believe it, and of course the purpose of academia is a search for truth. And I believe that's true. Even though if you push me, I'm not quite sure how to define all those words. In fact, I don't know how to define any of those words. But I believe it. And it conveys to students that mathematics is really important. Not just important to get a job, but if you want to get a good job, you should know how the world works. But at a fundamental level, I also say this is kind of funny, but why did I do it here, when almost everyone in this room, except for possibly the photographer on the side, is a true believer? Two reasons. One, it's kind of been entering into the notion of mathematical maturity, a word that many college and university professors use, but not all, and which I've learned in the last couple of weeks, is not commonly used by educators in K through 12. And we'll be talking about the nature of mathematical maturity. And you can interpret mathematical maturity almost as how sophisticated you think about functions, though that's not how we're going to be doing it. The real reason I did it is I am uncharacteristically nervous right now. I'm going to give a fair number of talks, and I don't really get nervous. I don't really think about it that much. But because most talks that I give, it's the mathematics that's driving it. You know, if it's a research talk, I might be a little nervous that someone who stands up if it's about my own reason says that's trivial, but it's a different kind of nervousness. I'm going to be talking today about mathematics, about the teaching of mathematics, and that makes me uncomfortable. So just to let you know, I am definitely more nervous than normal. Now, I don't know about the rest of you. I did not go into mathematics to teach. When I went to graduate school, I was wanting to learn how to do research in mathematics. Now, to be clear, I was not a naive 22-year-old. I understood full well that the day job of a mathematician in the United States involved teaching. And I felt that if you ever forced me to teach, I would try to bring the best of my abilities to it. I would try to recognize the people in that room were human beings who deserve my care and attention. I just had no interest in doing that, I mean, honest. I'm going to step down here a bit because the light's in my eye. I just had no interest in that. It's also the case, and I didn't have to teach for quite a while, probably until my mid-20s, when they eventually dragged me into a room to teach. It was also by my mid-20s I certainly had the belief that I certainly have very strongly now is that I kind of bought the pop psychologist who think that all of us think we're better than we are. You know, the study is that we all think we're better drivers than we all. We all think we're better at this than we really are. So I've, since my mid-20s, probably early 20s, unless I have strong evidence to the contrary, I believe I am average. It's sort of what average means. If I think of the things that are most important to me in my personal life, let's say, how am I as a spouse or as a parent? I hope, as I say as a spouse, that I'm wonderfully warm and encouraging, providing a framework of support for our entire lives. It's conceivable, unfortunately, that I'm actually a spouse that's constantly passive-aggressive, putting my wife down every step of the way. I hope not. Probably I'm average. It's the way it works. I assume that if I had to teach, I would be average. I saw many of my fellow graduate students teach quite a bit while I did not. I, they all thought they were great teachers. I thought they were all average. You know, some were pretty good, some not. And I think average is fine. I mean, it's not bad. So it was one of the great surprises of my life when I eventually had to teach. I was going to take it seriously. There were human beings in that room. But I was shocked at how enjoyable it was, of how, and which is one difficulty in this room, is by trying to look in people's eyes to see who was understanding, who was not understanding. You know, trying to figure out how to touch that person way in the back that I can't see right now, but I'm just pretending. I mean, it was just exhilarating. I mean, it was just, when I teach, it's all I'm thinking about. The number of times the crap of, oh, there's high school students, I apologize. The number of times when bad things happen in personal life intrude on my teaching, I think it's like four or five times in 35 years. And even then, I think I covered it. Teaching is exhilarating. And I learned a tremendous amount about human beings. Before I started teaching, I walked into that first class and I made a quick guess of who I thought was going to do well and who was going to do poorly. Of course, I thought the people who were like me in school were going to do well. So the people who were sitting in the back row looking bored and disaffected, ah, they were the really talented ones. They were going to do well. The people who were sitting in the front row, the people who I thought of as shameless toadies, of course, they're not really creative, though they might get good grades, but they just suck up. And I was deeply, I was surprised at how it was all over the map. I can't tell at all by looking at people how good, how well they're doing tests, how well they write, and it opened up an entire perspective on humanity for me. But still, I don't really think of myself as a teacher. I identify myself as a mathematician. So it was somewhat surprising to my department back in 2004, 2005, when I was put in charge of the Williams College Project for Effective Teaching. I was suddenly the teaching guy at a school that prides itself on good teaching. And what I had to do was help mentor all new faculty in all disciplines. People in my department thought that was hilarious. Yeah, they said, you tell me, you just can't walk in and make a bunch of jokes. Not everyone does that. But I found out it was great, but you don't even imagine Williams College is a serious school. You know, it's people who care about their teaching, they care about their research. You know, I was talking to a bunch of people probably roughly in their late 20s, maybe early 30s, they were young, they were ambitious, they were full of ideas. I learned all kinds of things. It was really exciting. I don't think I helped them that much, but they really helped me learn about all kinds of things. But one thing I did learn is that they have absolutely no notion of mathematical maturity. I would sometimes say, well, the historical analog and mathematical, don't worry, I'll talk more about what mathematical maturity is. It's in some crude sense an ability to pick up things quickly, but it's deeper than that. And I said, well, there must be the analog of historical maturity. I'm thinking that you can imagine an historian who studies France in the 1730s and surely they could teach China in 1000 more quickly and better than me. So I mentioned this to the young faculty and they just said, we do not have such a concept. And I talked, I'm looking at an historian and saying, well, you must have the analog. I'm not saying you have that word and they went, it does not exist. I'm going, no, let me explain it more. We understand what you're saying. It does not exist. I talked to a chemist, an alum, who's at Harvey Mudd, actually. And I tried to explain it to him and he went, really? No, we don't have it. Maybe we have it if you're looking at certain kinds of the shapes of molecules, you had a certain kind of geometric maturity. We're just getting close to math maturity. The only group that I found that had it, the close to it, and this took me years of asking faculty was a Japanese professor who talked about language readiness. And I think in K through 12 you all talk about readiness and that might be close and I think it's because in a foreign language, I've certainly tried to learn foreign languages but I've never thought in another language. But there's a point where people tell me where suddenly you go from translating everything to where you understand the conversations. And I'm guessing there's a point where there's a sudden change. Is that true? And they talk about that, they talk about that as language readiness. So let's talk about what mathematical maturity is. There's no well-defined meaning to it. For a lot of college and university professors, a lot of the maturity comes down to do you know what is a proof? Are you capable of cleanly and easily writing down proofs? And you recognize that for many students, that first serious proof course is really hard. At Williams, that's traditionally either abstract algebra or real analysis. Whoever's teaching that recognizes you have people are going to hit a wall and they're going to be frustrated. There will be tears in your office. They'll just boom, boom, boom. Now maybe a lot of people here did not find that kind of class hard. I remember that well. It was I was a first year student at University of Texas, Austin. Me and 45,000 of my closest friends. There was a young mathematician at the time who was ambitious and a very good teacher, Bruce Polka, who many of the mathematicians know because in recent years he's been handing out money from the National Science Foundation. He was an inspiration to us. He decided to start an honors program in mathematics. This is honors program in University of Texas. Doesn't mean that much. It means people who are okay at math and we're all from Texas. But really, so it's calculus. It was kind of before the days of the AP exams, though I think most of us had taken some calculus in high school. But it was really a real analysis course. We use Spivak's calculus book. How many of them are mathematicians with no Spivak's calculus book? It's an absolutely beautiful. It's still one of my favorite books of all time. So I walked in there and I was gonna do well. And the first day he assigned 18 problems. It was a Monday. They were due that Friday. I was not gonna blow it off like I did high school. Because I would lived in mortal fear of having to return to the small town in Texas that I grew up in, and it was a real fear. And I says, I'm not gonna wait till the last minute. So that Monday night, I walked across the street from my dorm, from people from Texas, it was Jester. Into the brand new pericostinal library, still had a new library smell. I was ready to start working the problems. It was 18 problems. How long could it take me? I'm good at math. I'm in an honors program. Three hours later, I had at best worked three problems. And I dragged myself out. I felt horrible. I felt really bad. Somehow, because I did not want to return to my hometown. Well, the visit my parents was okay, but I mean, somehow I got through that first assignment. I don't know how. I did not do that well on it. Now, Paul Co was a great professor. He understood that working in isolation is not good. He was encouraging us constantly to talk to each other. Talk to each other. And so a group, a couple of us, met for the second assignment. There were three of us. One whose name I do not remember. The other, far more mathematically mature than me at the time. Some of you might know Michael Lacey. He's a pretty big time mathematician now at Georgia Tech. Salem prize winner. He's still more mathematically mature than me. And we were doing the problems, but saying we were doing the problems is not really true. He was doing the problems, and I was copying them down. It wasn't a violation of any kind of honor code, but I knew where the information was flowing. It was almost at the level as Michael would say, we do problem 13, but can I go get you a soda? I mean, it was at that level. So I got through the second problem set. But I felt bad. I felt that I was just, I identified my, I so much want to do math actually. It's all I could do. I was a failure at everything else. And here I'm a failure. And the second week of college, third week came along, a group of us met to do the homework problem. Michael did not bother to show up. I understood full well why. And then it happened. It was a Thursday night. My roommate had a lot of David Bowie records. Music was hard to get at the time. You had to use actual physical records. And the song Rebel, Rebel was in my mind. Rebel, Rebel. And I was looking at, I was struggling. I'd been thinking about this course all the time. I was dry. I just didn't know what was going on. I was in the undergraduate library, which is this big library where all the people study. And suddenly it happened. I saw what was going on. I could work the problems. Suddenly problems. I didn't even understand what they were talking about went, hmm, that's interesting. I think we just let epsilon equal to that. Let delta be epsilon over. Ba, ba, ba, ba, ba, ba, ba. It was the greatest, one of the greatest nights in my life. I mean, I was just, I was mathematically mature. And thank goodness the first test was four days later. And I did well on it. That was wonderful. And that's an example of mathematical maturity. After that moment, when I made mistakes in mathematics, they were sort of honest mistakes. I just got some fact mixed up. It wasn't because I couldn't express it. I just thought I didn't understand what was going on. I could see the moves. Do other people have had experience? I'm just going to teach. But there's other types of mathematical maturity. And the point is, now frequently among college professors, when we talk about mathematical maturity, we frequently mean that sort, right? I'm talking to someone from Harvey Mudd. Michael, is that true? Absolutely. There's other types. One big type of mathematical maturity is being comfortable with high school algebra, being able to do algebraic manipulations. So this is now another of how I became high school algebraically mature. There was no moment of epiphany. There was no moment when I was so frustrated and I know what's going on. And suddenly, I saw. Absolutely not. It was seventh grade. We had an ambitious seventh grade teacher who, in November or December of that year, decided to teach us high school algebra. I had no clue what was going on. It was probably one equation and one unknown. And we probably just had to isolate the x. I don't remember. All I remember there was equations, and he kept saying the equal sign. It's a balance beam. It's a balance beam. And every time it kept changing, you're trying to figure out, but the equation keeps changing. But it's a balance beam. It's a balance beam. And I did not know what's going on. And I remember actually going to talk to him, which was frustrating for me because I did not like teachers, and I still feel uncomfortable around you all. And I asked him, I don't know what's happening. And he talked to me. And then we left it. We went back to the stuff we normally did, the standard seventh grade math back in the early 1970s, which was set theory. No, I'm serious. Talk to older people, right? It was set theory. I understood it. It was good. But then we came back to it in May of that year, and it was perfectly obvious to me. I didn't remember why. It's not like I thought about it. I was seventh grader. I didn't think about school that much. And so somehow in that ensuing time, I became algebraically or high school algebraically mature. And that happens to us all. Also, you struggle with something. You go away, and you come back, go, I don't know what the problem was. No moment of epiphany, no great experience of light bulbs going off. And it happens throughout our careers. I don't think about math now the same way as I did, let's say, 20 years ago. And 20 years ago, I was a tenured professor. I certainly don't think of it the same way as I did 30 years ago when I was a postdoc and already had a PhD and was already proving real results. Certainly 40 years ago when I was starting to college, I've explained it. My knowledge about that is amazing compared to it. Not just knowledge, it's not just facts. It's just an awareness of how it all kind of fits together. So I suspect if I'm teaching beginning calculus, the words I say now are probably the same as I would have said 30 years ago. But the underlying nuance behind it is there's a richness to it. So let's try to sketch out the stages of our lives. We're all trying to become better people, better understanding of the world. And so to understand the world, we have to understand functions, which means we really have to understand math. Now none of what I'm gonna be talking about should be viewed as like curriculum. If you, I mean, then you have to worry about real details. This is just a sketch. So let's go through. I don't know much about primary school. I have no clue how to teach third graders. I do not know, even begin to what to do. But I would think that by the end of, by the time they're 12, basic arithmetic infractions, graphs and charts, Fermi type problems, believe the recognition that is starting their patterns and the patterns have reasonings. I talked to a lot of people apparently mentoring new faculty. Many of them say they've never made sense out of math. I can remember learning multiplication tables and being enchanted by them in primary school. There's so many patterns. Factor trees, oh God, I love factor trees. Sitting in a boring church, looking up and seeing the hymnal number. How fast can I factor it? I still do that now that I think about it with airplane tickets when we're landing. I'm thinking, I'm probably gonna die. I've had a good life. Huh, I think seven goes into it. I assume there's other people here who do that also. But the record is there's patterns as opposed to mere technique. Secondary school, of course, high school algebra, trig, more controversial Euclidean geometry, basic counting. But again, the idea that these hang together, they're not just random things. One person I know is teaching a course at a college, community college, and took off some points from a student. The student came in and said, I'm wondering why you said X was equal to 37. He said, well, that's the right answer. I thought you told us last week X was 17. It's a true story. And the person was, oh my God, this person's level of mathematical maturity is the wrong place. One thing that's probably mildly controversial is Euclidean geometry. That's been really eliminated from the secondary schools, at least in the US, starting in the 60s. The probably people my age were still doing it. Did you do a rigorous, did you do in Slovenia, rigorous, rigorous, a little bit? Now they don't do it that much. I think that's actually hurt our humanities colleagues. Our humanities colleagues make fun of axiomatic methods and they never experienced its power. They don't really know what they're even arguing against. They were good intellectual reasons in 1930 to argue against it. Now they don't even know what they're arguing against and it creates problems in the humanities. But that's it, just moving into a crank view. Moving into the world that I know, college, certainly in the first two years, calculus and linear, especially linear algebra. And it's pretty much what I'm saying. It's the second, no one fits in all this. As I said, I was taking analysis my first year and not my third year. But that's the real thing in the last two years. A full understanding of what rigor means. In graduate school, you make huge leaps. I certainly tell students that if you're adhering to people learning multiplication, here's where you are with high school algebra, here's where you are when you get through an undergraduate math major, here's where you're in your first or second year. Graduate school, the scale is mathematical maturity. Here is when you're finishing your PhD. Up there is when you're gonna finish your postdoc and then you keep climbing higher and higher. You're not proving better results, but you're getting higher and higher. And so the last two years is when you really starts working on your PhD thesis. As I mentioned earlier, your first two years, you're learning an amazing ability to learn mathematics quickly. I'm putting this on very explicitly because I know for K through 12, many of you have not gone to graduate school. And it's awe-inspiring at the end of those first two years. You can just, whoop, whoop, whoop. You know, you can make connections very fast. You can pick up a text and you go, whoom, right through it. Suddenly, you have to do original mathematics. And you crash because you're used to learning amazing amounts of mathematics really fast and efficiently. And you're feeling powerful. You remember what you were when you were 18 and now you're 23. And you can just, whoom, and then you come across a real problem and you go, you try something, it doesn't work. You try something, it doesn't work. You do that for a year, two years. And if you're lucky, you get someplace. But it's a real, also helps with mathematical maturity. So, then you become a poet. First years after a PhD is more vague, it depends on what kind of job you get. If you're, certainly you should be starting to move away from your thesis area or at least make it richer. And that's when you start talking about having your own research program. And that's what you probably, certainly at Williams, if we start looking at when we hire people and get them tenure, do they have a vision of where they want their research to go? Dude, are they just solving little problems? That's not as good as, no, no, I'm really interested in the broad spectrum of what's going on. Mid-career though, actually, I'm not really living it, probably. Retirement, of course, and then nursing home. And I don't know about those yet. But with nursing home, here's how I think probably most of us want to die. I want to be 103 in a hospital surrounded by loved ones. Surrounded by loved ones. I want my spouse, my children, grandchildren. Maybe even a great-grandchildren, I'm 103. And then a great-great-grandchild. Natasha, a baby in arms. And I want to be lying there feeling the love. And then, at that moment, having my last mathematical insight. And then the machine's going, ah, I mean, that's the way to live and die. And that's the aim for, at the nursing home. But how can we use this? I mean, to help us, to help our careers, to help our teaching. Well, teaching is pretty clear. One thing is to recognize, I'm not gonna need it anymore. To recognize is that when you teach a course, you have to figure out what level of mathematical maturity your students are at. It's very common, especially at colleges and universities, for everyone to complain because they're not teaching the students they have, they're teaching the students they want. And that's bad. You have to recognize where they are and consciously think about it. If I'm teaching beginning real analysis, I'm thinking about it that these are people who are probably pretty good at math in terms of algorithmically, but they're gonna make a leap. If I was teaching in high school, I think I'd probably be feeling the same things. So that's one way to use it. It's more than that. Mathematicians, regularly in academic politics, get rolled over by administrations. It's a fact. It's because we have a tendency to listen to authority. We have a tendency to like rules and say, it's a rule, the dean said it, it must be real. And so part of that meant that the math world was far behind other sciences in terms of lobbying the government. And the math world hired its first lobbyists in the mid 1980s, long after every other discipline. The guy they hired was Ken Hoffman, who had been a longtime chair of the MIT Math Department and a very serious person in several complex variables. He was speaking once when I was in grad school. I think it was 1985. And he was talking about his experiences. This is gonna be connected to mathematical treatment. And he was talking about his experiences in Washington. And one thing he said is that everyone in Washington thought that mathematics was important. He did not have to convince anybody that math was important. What he had to convince them is that math is still going on. They all thought it was over. Now, when he said that, I was a young, arrogant graduate student. But I felt the entire auditorium getting very smug. I think, ignorant Philistines. Not to say that mathematics is still going on. And they turned to us and said, and whose fault is it? And he said, it's ours. He said, people are in math classes every day throughout the country. How are they gonna know math is going on if we don't tell them? And we don't tell them. I took that to heart. And it's also tied into mathematical maturity is we wanna get through to students that math is an ongoing process that is never ending in its richness. And part of that is say, there are still people who are very serious and very interesting who are struggling with new mathematics. I use that at the college level to try to drag in research whenever possible in the following way. It's not like I'm gonna give a 30 minute lecture on my research, absolutely not. But let's say I was teaching first semester calculus. We were doing the second derivative test. In the 80s, early 90s, I was extremely concerned with curvature conditions. I was, technically, I was concerned about ample vector bundles. But it actually came down to algebraic and just positive, I was concerned about curvature. So if I was teaching the second derivative, I'd say, oh, no, we're doing the second derivative test. It goes like this, one kind of second derivative, other kind of second. By the way, when you go to higher variables becomes much trickier. And I say, it's actually tied into curvature, which is not a single thing. It's very, very, very tricky. My research is really concerned about it. And then I back go back to the subject. Let's say in the late 80s, I was actually working in computer algebra. I was concerned about factoring polynomials. I think I had the time, the fastest algorithm that factor multivariable polynomials over the complex numbers. I was proud of that work. Whenever I came across any kind of factoring problem it shows up every place. I'd say, by the way, factoring is a touchstone problem in mathematics. You always wanna take a given area math and break it up into its primitive parts and then ask how you can put it back together again. This permeates mathematics. And it does. Much of high school algebra is doing the quadratic equation, right? Is that true? Factoring. High or low, you're factoring all the time. Mention it quickly. If I would say for those of you in K through 12, I would mention PCMI. I would say, oh yeah, yeah, people are still doing math all the time. I spent the summer research institute. It was great. I was kind of wacky. But, you know, and do that. So create the image of it. And that's part of the mathematical maturity. Any questions? There's other things you can do. And this is now to help departments. Especially now this is trying to mentor young faculty. It's good if a department has, for example, a lunch crowd.