WEBVTT 00:00:00.000 --> 00:00:01.782 align:middle line:90% [MUSIC PLAYING] 00:00:01.782 --> 00:00:03.990 align:middle line:84% HAYNES MILLER: We all think we know what numbers are. 00:00:03.990 --> 00:00:05.490 align:middle line:90% We use them every day. 00:00:05.490 --> 00:00:08.940 align:middle line:84% But very quickly, you get into some real mathematical 00:00:08.940 --> 00:00:12.540 align:middle line:84% questions that aren't buried at the end of some course. 00:00:12.540 --> 00:00:14.952 align:middle line:90% They're right on the surface. 00:00:14.952 --> 00:00:15.660 align:middle line:90% PAIGE BRIGHT: Hi. 00:00:15.660 --> 00:00:18.360 align:middle line:84% My name is Paige Bright, and I'm a junior studying mathematics 00:00:18.360 --> 00:00:21.480 align:middle line:84% at MIT, and I'll be your guest host for today's episode 00:00:21.480 --> 00:00:23.310 align:middle line:90% of Chalk Radio. 00:00:23.310 --> 00:00:26.790 align:middle line:84% For this episode, I sat down with Professor Haynes Miller. 00:00:26.790 --> 00:00:28.830 align:middle line:84% HAYNES MILLER: I'm a professor, now emeritus, 00:00:28.830 --> 00:00:30.540 align:middle line:90% of mathematics at MIT. 00:00:30.540 --> 00:00:33.930 align:middle line:84% And I got here in 1986, a long time ago. 00:00:33.930 --> 00:00:37.080 align:middle line:84% I've taught a lot of subjects to undergraduates, especially 00:00:37.080 --> 00:00:38.962 align:middle line:90% 18.03, Differential Equations. 00:00:38.962 --> 00:00:40.920 align:middle line:84% And I hope we have a chance to talk about that. 00:00:40.920 --> 00:00:42.228 align:middle line:90% That's who I am. 00:00:42.228 --> 00:00:43.770 align:middle line:84% PAIGE BRIGHT: I first met with Haynes 00:00:43.770 --> 00:00:46.260 align:middle line:84% when he was my instructor for a special First-Year Seminar 00:00:46.260 --> 00:00:46.992 align:middle line:90% at MIT. 00:00:46.992 --> 00:00:49.200 align:middle line:84% I'll let Haynes tell you more about these seminars he 00:00:49.200 --> 00:00:50.850 align:middle line:90% facilitated. 00:00:50.850 --> 00:00:54.090 align:middle line:84% HAYNES MILLER: MIT has a wonderful series 00:00:54.090 --> 00:00:57.930 align:middle line:84% of classes called First-Year Advising Seminars. 00:00:57.930 --> 00:01:00.810 align:middle line:84% These are small seminars, maybe eight students. 00:01:00.810 --> 00:01:03.120 align:middle line:84% The seminar that you and I were in, 00:01:03.120 --> 00:01:05.580 align:middle line:84% Paige, was unusual because this was 00:01:05.580 --> 00:01:09.060 align:middle line:84% the first Fall of the pandemic, so it was entirely remote. 00:01:09.060 --> 00:01:10.050 align:middle line:90% Very interesting. 00:01:10.050 --> 00:01:12.360 align:middle line:84% We had many time zones represented. 00:01:12.360 --> 00:01:16.200 align:middle line:84% One student was calling in from Vietnam, another from Kuwait, 00:01:16.200 --> 00:01:19.500 align:middle line:84% another from Nepal, and places in the US. 00:01:19.500 --> 00:01:21.960 align:middle line:84% And it was the most successful seminar 00:01:21.960 --> 00:01:26.250 align:middle line:84% that I've run because it was a chance for students to interact 00:01:26.250 --> 00:01:28.920 align:middle line:84% on a personal level with their peers, 00:01:28.920 --> 00:01:31.762 align:middle line:84% even though they were halfway across the world. 00:01:31.762 --> 00:01:33.720 align:middle line:84% PAIGE BRIGHT: One of the most important aspects 00:01:33.720 --> 00:01:35.262 align:middle line:84% of these First-Year Advising Seminars 00:01:35.262 --> 00:01:37.830 align:middle line:84% is its focus on communication and mathematics. 00:01:37.830 --> 00:01:40.050 align:middle line:84% The seminars start by asking a simple question 00:01:40.050 --> 00:01:42.300 align:middle line:84% that hooks in mathematicians of any background 00:01:42.300 --> 00:01:45.690 align:middle line:84% like, how do you define symmetry, or what is a number? 00:01:45.690 --> 00:01:48.450 align:middle line:84% And in the end, it covers complex mathematical topics 00:01:48.450 --> 00:01:51.570 align:middle line:84% like moduli spaces and surreal numbers. 00:01:51.570 --> 00:01:53.730 align:middle line:84% Part of what makes Haynes's classes so compelling 00:01:53.730 --> 00:01:57.240 align:middle line:84% is his focus on the ways we communicate as mathematicians. 00:01:57.240 --> 00:02:00.150 align:middle line:84% HAYNES MILLER: In a way, I envy my engineering colleagues 00:02:00.150 --> 00:02:02.040 align:middle line:90% because they produce artifacts. 00:02:02.040 --> 00:02:04.110 align:middle line:84% They produce actual physical things. 00:02:04.110 --> 00:02:07.350 align:middle line:84% And my lab science colleagues do the same thing. 00:02:07.350 --> 00:02:10.199 align:middle line:84% They interact in complicated ways with the world. 00:02:10.199 --> 00:02:13.050 align:middle line:84% Mathematicians, all we have is communication. 00:02:13.050 --> 00:02:14.820 align:middle line:84% All we have is the writing that we do. 00:02:14.820 --> 00:02:17.910 align:middle line:84% We publish and we try to explain our ideas, 00:02:17.910 --> 00:02:20.730 align:middle line:84% and there isn't any physical object that we produce. 00:02:20.730 --> 00:02:24.668 align:middle line:84% So communicating is the whole game in mathematics in a way. 00:02:24.668 --> 00:02:26.460 align:middle line:84% PAIGE BRIGHT: Given that communicating math 00:02:26.460 --> 00:02:28.920 align:middle line:84% is so integral in Haynes's teaching and research, 00:02:28.920 --> 00:02:30.660 align:middle line:84% I was curious how he thinks about it. 00:02:30.660 --> 00:02:32.880 align:middle line:90% What's his approach? 00:02:32.880 --> 00:02:35.340 align:middle line:84% HAYNES MILLER: Whenever you're writing or communicating, 00:02:35.340 --> 00:02:37.980 align:middle line:84% the main thing is to keep your audience in mind. 00:02:37.980 --> 00:02:39.120 align:middle line:90% Who are you talking to? 00:02:39.120 --> 00:02:40.260 align:middle line:90% Where are they? 00:02:40.260 --> 00:02:42.810 align:middle line:84% What is their background interest? 00:02:42.810 --> 00:02:44.100 align:middle line:90% Why are they there? 00:02:44.100 --> 00:02:45.960 align:middle line:84% What is their background knowledge? 00:02:45.960 --> 00:02:49.800 align:middle line:84% And when you're teaching, that's paramount importance. 00:02:49.800 --> 00:02:52.830 align:middle line:84% So when I was teaching Differential Equations, 18.03, 00:02:52.830 --> 00:02:56.250 align:middle line:84% I always had in mind that I was teaching mainly engineers. 00:02:56.250 --> 00:02:59.070 align:middle line:84% And there were some science majors and some math majors 00:02:59.070 --> 00:03:01.140 align:middle line:84% in my audience, but for them, this 00:03:01.140 --> 00:03:03.270 align:middle line:84% was often as close to an engineering course 00:03:03.270 --> 00:03:05.160 align:middle line:84% as they were going to see here at MIT. 00:03:05.160 --> 00:03:08.630 align:middle line:84% So I tried very hard to use engineering language. 00:03:08.630 --> 00:03:10.380 align:middle line:84% In fact, one of the things that I would do 00:03:10.380 --> 00:03:13.890 align:middle line:84% was to invite an engineer into the classroom one day 00:03:13.890 --> 00:03:16.500 align:middle line:84% during the semester, and we would talk the talk 00:03:16.500 --> 00:03:17.250 align:middle line:90% with each other. 00:03:17.250 --> 00:03:20.280 align:middle line:84% We'd use the language that we'd been developing in the class. 00:03:20.280 --> 00:03:24.030 align:middle line:84% And it became clear to the students that this was real. 00:03:24.030 --> 00:03:27.390 align:middle line:84% This was the language that engineers used. 00:03:27.390 --> 00:03:29.490 align:middle line:84% Of course, the engineers always knew much more 00:03:29.490 --> 00:03:31.680 align:middle line:84% about the kind of systems and signals language 00:03:31.680 --> 00:03:33.690 align:middle line:84% than I was talking about, so I formed 00:03:33.690 --> 00:03:36.270 align:middle line:84% a kind of translating device between the students who 00:03:36.270 --> 00:03:39.383 align:middle line:84% knew little and the engineers who knew a great deal. 00:03:39.383 --> 00:03:41.050 align:middle line:84% PAIGE BRIGHT: So let's talk for a minute 00:03:41.050 --> 00:03:43.150 align:middle line:84% about this class that Haynes is referring to, 00:03:43.150 --> 00:03:45.700 align:middle line:90% Differential Equations or 18.03. 00:03:45.700 --> 00:03:47.230 align:middle line:84% The study of differential equations 00:03:47.230 --> 00:03:48.730 align:middle line:84% is the study of how small changes 00:03:48.730 --> 00:03:52.750 align:middle line:84% in a function or derivatives impact the function itself. 00:03:52.750 --> 00:03:55.630 align:middle line:84% For instance, how does heating a metallic rod at one point 00:03:55.630 --> 00:03:58.060 align:middle line:84% affect the temperature of the rod everywhere? 00:03:58.060 --> 00:04:01.330 align:middle line:84% In a class like 18.03, one hopes to systematically understand 00:04:01.330 --> 00:04:02.347 align:middle line:90% these types of problems. 00:04:02.347 --> 00:04:04.180 align:middle line:84% But Haynes has a bit of a different approach 00:04:04.180 --> 00:04:06.310 align:middle line:84% to teaching differential equations, one 00:04:06.310 --> 00:04:08.853 align:middle line:84% that changes how students think about functions as a whole. 00:04:08.853 --> 00:04:11.270 align:middle line:84% HAYNES MILLER: Students come into a Differential Equations 00:04:11.270 --> 00:04:13.690 align:middle line:84% course, and they've seen calculus, 00:04:13.690 --> 00:04:16.029 align:middle line:84% and they're getting used to the idea of a function. 00:04:16.029 --> 00:04:18.190 align:middle line:84% But still for many of them, a function, 00:04:18.190 --> 00:04:19.959 align:middle line:90% it's a complicated thing. 00:04:19.959 --> 00:04:21.610 align:middle line:90% It has many parts. 00:04:21.610 --> 00:04:25.750 align:middle line:84% There's many input values, and each one has an output value. 00:04:25.750 --> 00:04:27.790 align:middle line:90% It's a complicated thing. 00:04:27.790 --> 00:04:30.280 align:middle line:84% And what happens in the course of a Differential Equations 00:04:30.280 --> 00:04:33.160 align:middle line:84% course is you come to think of a function as a single thing. 00:04:33.160 --> 00:04:35.110 align:middle line:90% It's a composite thing. 00:04:35.110 --> 00:04:39.700 align:middle line:84% It has smaller parts, but you think of it as a single object. 00:04:39.700 --> 00:04:43.360 align:middle line:84% That process of collecting a group of things 00:04:43.360 --> 00:04:46.150 align:middle line:84% and coalescing them into a single object, that's 00:04:46.150 --> 00:04:49.660 align:middle line:84% a fundamental mathematical operation that you can see 00:04:49.660 --> 00:04:52.390 align:middle line:84% in action already very clearly in a differential 00:04:52.390 --> 00:04:53.230 align:middle line:90% equations course. 00:04:53.230 --> 00:04:55.750 align:middle line:84% And it's one of the main skills that I 00:04:55.750 --> 00:04:59.440 align:middle line:84% hope students come away with from taking a course like this. 00:04:59.440 --> 00:05:01.870 align:middle line:84% They think of a function as a single object 00:05:01.870 --> 00:05:03.010 align:middle line:90% that you can manipulate. 00:05:03.010 --> 00:05:05.200 align:middle line:84% You can compare two of them because one 00:05:05.200 --> 00:05:08.230 align:middle line:84% is the derivative of another or they enter into a differential 00:05:08.230 --> 00:05:09.040 align:middle line:90% equation together. 00:05:09.040 --> 00:05:12.257 align:middle line:90% 00:05:12.257 --> 00:05:14.590 align:middle line:84% PAIGE BRIGHT: Before we get too deep into the math here, 00:05:14.590 --> 00:05:17.500 align:middle line:84% I want to take a minute to focus on Haynes's innovative approach 00:05:17.500 --> 00:05:18.580 align:middle line:90% to teaching. 00:05:18.580 --> 00:05:21.610 align:middle line:84% I asked him about how he makes math not just comprehensible, 00:05:21.610 --> 00:05:24.490 align:middle line:90% but also engaging for students. 00:05:24.490 --> 00:05:26.200 align:middle line:84% HAYNES MILLER: We human beings are-- 00:05:26.200 --> 00:05:28.030 align:middle line:84% most of us are equipped-- we're all 00:05:28.030 --> 00:05:29.395 align:middle line:90% equipped with a visual cortex. 00:05:29.395 --> 00:05:32.380 align:middle line:84% We're all equipped with an extremely highly 00:05:32.380 --> 00:05:36.520 align:middle line:84% developed, elaborate, and sophisticated visual system. 00:05:36.520 --> 00:05:40.660 align:middle line:84% Even if we're blind, that system is still at work in our brain. 00:05:40.660 --> 00:05:42.790 align:middle line:84% It's still available in our brain. 00:05:42.790 --> 00:05:44.410 align:middle line:90% And what does it do? 00:05:44.410 --> 00:05:47.290 align:middle line:84% An infant looks around in the world 00:05:47.290 --> 00:05:52.270 align:middle line:84% and sees a million pixels, sees a million spots of color. 00:05:52.270 --> 00:05:56.680 align:middle line:84% But one of the first operations that an infant performs 00:05:56.680 --> 00:05:59.770 align:middle line:84% in growing up is learning the notion of an object, 00:05:59.770 --> 00:06:02.230 align:middle line:84% putting together a collection of pixels 00:06:02.230 --> 00:06:06.190 align:middle line:84% and seeing that they constitute an object that 00:06:06.190 --> 00:06:07.780 align:middle line:90% has its own identity. 00:06:07.780 --> 00:06:11.830 align:middle line:84% So our visual cortex is really well suited 00:06:11.830 --> 00:06:15.130 align:middle line:84% to doing exactly this operation of assembling things 00:06:15.130 --> 00:06:17.500 align:middle line:84% and seeing them as a single unit. 00:06:17.500 --> 00:06:21.130 align:middle line:84% It's a special feature that we have in our brain. 00:06:21.130 --> 00:06:25.000 align:middle line:84% So one of the things that I did to help students make 00:06:25.000 --> 00:06:27.250 align:middle line:84% this step in the Differential Equations course 00:06:27.250 --> 00:06:30.910 align:middle line:84% was to create a series of computer manipulatives, 00:06:30.910 --> 00:06:32.260 align:middle line:90% I call them. 00:06:32.260 --> 00:06:32.830 align:middle line:90% Programs. 00:06:32.830 --> 00:06:35.830 align:middle line:84% You can find them on the web at mathlets.org. 00:06:35.830 --> 00:06:39.970 align:middle line:84% And they display functions as units, 00:06:39.970 --> 00:06:42.610 align:middle line:84% and they show how the functions will change 00:06:42.610 --> 00:06:44.020 align:middle line:90% when you change parameters. 00:06:44.020 --> 00:06:46.510 align:middle line:84% They show you the relationship between functions 00:06:46.510 --> 00:06:48.970 align:middle line:84% that participate in a differential equation. 00:06:48.970 --> 00:06:54.250 align:middle line:84% And I think that for many students, seeing the graph, 00:06:54.250 --> 00:06:56.890 align:middle line:84% visually seeing the graph of a function 00:06:56.890 --> 00:07:02.587 align:middle line:84% is a huge asset in making them see this as a single object. 00:07:02.587 --> 00:07:04.170 align:middle line:84% PAIGE BRIGHT: In keeping with his work 00:07:04.170 --> 00:07:06.330 align:middle line:84% helping students visualize differential equations, 00:07:06.330 --> 00:07:07.997 align:middle line:84% Haynes noticed that there was a big part 00:07:07.997 --> 00:07:09.750 align:middle line:84% of the academic experience that was 00:07:09.750 --> 00:07:14.440 align:middle line:84% missing a mathematical element beyond visualization: lab work. 00:07:14.440 --> 00:07:17.610 align:middle line:84% HAYNES MILLER: MIT has a general institute requirement. 00:07:17.610 --> 00:07:21.300 align:middle line:84% All undergraduates are required to take a course that has 00:07:21.300 --> 00:07:23.940 align:middle line:90% a designation as a lab course. 00:07:23.940 --> 00:07:26.430 align:middle line:84% Historically, mathematics didn't have a lab course. 00:07:26.430 --> 00:07:28.770 align:middle line:84% The wisdom was that, how could you have 00:07:28.770 --> 00:07:30.090 align:middle line:90% a laboratory in mathematics? 00:07:30.090 --> 00:07:31.470 align:middle line:90% There was no artifact. 00:07:31.470 --> 00:07:36.360 align:middle line:84% But I felt that mathematics is a science. 00:07:36.360 --> 00:07:38.400 align:middle line:90% We employ the scientific method. 00:07:38.400 --> 00:07:41.850 align:middle line:84% We collect data, perhaps by doing examples, perhaps 00:07:41.850 --> 00:07:42.810 align:middle line:90% by computer. 00:07:42.810 --> 00:07:44.520 align:middle line:90% We form hypotheses. 00:07:44.520 --> 00:07:46.290 align:middle line:90% Maybe we call them conjectures. 00:07:46.290 --> 00:07:48.660 align:middle line:84% And we try to get evidence for them. 00:07:48.660 --> 00:07:51.190 align:middle line:84% Our evidence typically is called a proof. 00:07:51.190 --> 00:07:54.060 align:middle line:84% There's not a bench with lab equipment on it 00:07:54.060 --> 00:07:55.620 align:middle line:84% that people are working on together, 00:07:55.620 --> 00:07:58.350 align:middle line:84% but there is the blackboard or the whiteboard 00:07:58.350 --> 00:08:02.370 align:middle line:84% or the tablet these days of Zoom where 00:08:02.370 --> 00:08:05.337 align:middle line:90% people are working in parallel. 00:08:05.337 --> 00:08:07.170 align:middle line:84% PAIGE BRIGHT: So together with his colleague 00:08:07.170 --> 00:08:09.270 align:middle line:84% in the math department, Professor Michael Artin, 00:08:09.270 --> 00:08:12.180 align:middle line:84% Haynes pulled together a series of small, open-ended research 00:08:12.180 --> 00:08:14.730 align:middle line:84% projects that undergraduate students could work on 00:08:14.730 --> 00:08:18.300 align:middle line:84% to fulfill their lab requirement in a math-focused environment. 00:08:18.300 --> 00:08:20.790 align:middle line:84% This led to the creation of a class entitled the Project 00:08:20.790 --> 00:08:23.070 align:middle line:90% Laboratory in Mathematics. 00:08:23.070 --> 00:08:26.490 align:middle line:84% HAYNES MILLER: It's a very high communication kind of course, 00:08:26.490 --> 00:08:30.540 align:middle line:84% but it also gives a pretty honest introduction 00:08:30.540 --> 00:08:33.450 align:middle line:84% to the research process in mathematics. 00:08:33.450 --> 00:08:35.580 align:middle line:84% It's something that's been useful for students 00:08:35.580 --> 00:08:38.370 align:middle line:84% who come to it never having engaged in any kind of research 00:08:38.370 --> 00:08:39.210 align:middle line:90% before. 00:08:39.210 --> 00:08:41.190 align:middle line:84% It opens their eyes to the notion 00:08:41.190 --> 00:08:44.820 align:middle line:84% that not everything is a textbook exercise. 00:08:44.820 --> 00:08:48.360 align:middle line:84% And for people who have had research experience, 00:08:48.360 --> 00:08:53.040 align:middle line:84% there also it's a great training in teamwork and in writing, 00:08:53.040 --> 00:08:54.612 align:middle line:90% in communicating. 00:08:54.612 --> 00:08:56.820 align:middle line:84% PAIGE BRIGHT: One part of my conversation with Haynes 00:08:56.820 --> 00:08:59.100 align:middle line:84% that I really loved was his explanation of how 00:08:59.100 --> 00:09:02.340 align:middle line:84% teaching these courses led to his own understanding of what 00:09:02.340 --> 00:09:05.010 align:middle line:84% goes into effective math communication. 00:09:05.010 --> 00:09:06.690 align:middle line:84% HAYNES MILLER: After I got involved 00:09:06.690 --> 00:09:08.580 align:middle line:84% with these communication-intensive 00:09:08.580 --> 00:09:11.880 align:middle line:84% subjects, I became much more aware 00:09:11.880 --> 00:09:15.540 align:middle line:84% of the sort of stylistic issues in mathematics writing. 00:09:15.540 --> 00:09:18.420 align:middle line:84% And I have to say, I became a much more severe editor 00:09:18.420 --> 00:09:24.300 align:middle line:84% and reviewer, and worked quite hard to help the author use 00:09:24.300 --> 00:09:28.470 align:middle line:84% language correctly and be more careful about their audience, 00:09:28.470 --> 00:09:31.830 align:middle line:84% because it's very common in mathematics 00:09:31.830 --> 00:09:34.290 align:middle line:84% to begin a paragraph with a statement that comes out 00:09:34.290 --> 00:09:38.040 align:middle line:84% of the blue, because there's an image that we're 00:09:38.040 --> 00:09:40.980 align:middle line:84% struck by lightning, and we have this inspiration. 00:09:40.980 --> 00:09:43.230 align:middle line:90% That's of course not true. 00:09:43.230 --> 00:09:45.090 align:middle line:84% There's a big background to that. 00:09:45.090 --> 00:09:48.600 align:middle line:84% There's a lot of preparation that goes into that. 00:09:48.600 --> 00:09:50.490 align:middle line:84% And writing should be the same way. 00:09:50.490 --> 00:09:53.640 align:middle line:84% You build up to a point, and then you connect. 00:09:53.640 --> 00:09:55.058 align:middle line:90% You make the next step. 00:09:55.058 --> 00:09:56.850 align:middle line:84% PAIGE BRIGHT: Haynes has been collaborating 00:09:56.850 --> 00:09:59.912 align:middle line:84% with Susan Ruff, a lecturer in Writing Across the Curriculum, 00:09:59.912 --> 00:10:01.620 align:middle line:84% who has been working with the Mathematics 00:10:01.620 --> 00:10:05.130 align:middle line:84% Department and the Project Lab for many years now. 00:10:05.130 --> 00:10:08.910 align:middle line:84% HAYNES MILLER: Susan Ruff is very clear on the importance 00:10:08.910 --> 00:10:11.640 align:middle line:84% of what she calls "connectivity," which 00:10:11.640 --> 00:10:15.180 align:middle line:84% is to say you don't introduce a lot of new concepts 00:10:15.180 --> 00:10:16.350 align:middle line:90% simultaneously. 00:10:16.350 --> 00:10:18.810 align:middle line:84% You begin where the last sentence ended 00:10:18.810 --> 00:10:20.910 align:middle line:90% and you take the next step. 00:10:20.910 --> 00:10:24.240 align:middle line:84% And when you think of mathematics writing 00:10:24.240 --> 00:10:27.780 align:middle line:84% in that way, you really improve the comprehensibility 00:10:27.780 --> 00:10:28.950 align:middle line:90% of the writing. 00:10:28.950 --> 00:10:31.050 align:middle line:84% So I think that my work is an editor 00:10:31.050 --> 00:10:33.540 align:middle line:84% and as a reviewer of math papers has really 00:10:33.540 --> 00:10:35.790 align:middle line:84% been influenced by the communication-intensive 00:10:35.790 --> 00:10:38.767 align:middle line:84% teaching that I've been involved with here. 00:10:38.767 --> 00:10:40.350 align:middle line:84% PAIGE BRIGHT: Haynes's clear intention 00:10:40.350 --> 00:10:43.920 align:middle line:84% to constantly be improving his teaching is truly remarkable. 00:10:43.920 --> 00:10:46.360 align:middle line:84% One way he's helped both himself and other instructors 00:10:46.360 --> 00:10:49.650 align:middle line:84% develop their teaching skills is through a program he created 00:10:49.650 --> 00:10:51.840 align:middle line:84% and hosts called the Online Seminar 00:10:51.840 --> 00:10:55.650 align:middle line:84% on Undergraduate Mathematics Education, or OLSUME. 00:10:55.650 --> 00:10:57.660 align:middle line:84% HAYNES MILLER: So about six years ago or so, 00:10:57.660 --> 00:11:00.960 align:middle line:84% I was sitting in a coffee shop with a former postdoc, Grace 00:11:00.960 --> 00:11:03.300 align:middle line:84% Lyo, from the Math Department here. 00:11:03.300 --> 00:11:07.710 align:middle line:84% And she was making the point that you research universities 00:11:07.710 --> 00:11:10.290 align:middle line:84% like Stanford where she was employed at the time and MIT 00:11:10.290 --> 00:11:13.590 align:middle line:84% where she used to be employed, the professors there think hard 00:11:13.590 --> 00:11:18.360 align:middle line:84% about teaching, but they don't talk to each other very much 00:11:18.360 --> 00:11:20.970 align:middle line:84% about their thoughts about teaching. 00:11:20.970 --> 00:11:24.300 align:middle line:84% We really don't talk too much with each other normally 00:11:24.300 --> 00:11:27.300 align:middle line:84% within a given faculty about teaching. 00:11:27.300 --> 00:11:30.360 align:middle line:84% So we thought, well, there are these few campuses. 00:11:30.360 --> 00:11:32.220 align:middle line:84% Maybe we should have a meeting together. 00:11:32.220 --> 00:11:37.020 align:middle line:84% But it occurred to me that we have Zoom, which existed back 00:11:37.020 --> 00:11:41.040 align:middle line:84% before COVID, believe it or not, and we could do this remotely 00:11:41.040 --> 00:11:43.710 align:middle line:90% and start up a monthly seminar-- 00:11:43.710 --> 00:11:45.720 align:middle line:84% every two weeks, actually, seminar-- 00:11:45.720 --> 00:11:49.590 align:middle line:84% asking faculty to talk about the teaching initiatives 00:11:49.590 --> 00:11:52.350 align:middle line:84% that they'd engaged in in their own campus. 00:11:52.350 --> 00:11:55.800 align:middle line:84% Sometimes it was a specific, different way of grading. 00:11:55.800 --> 00:12:00.510 align:middle line:84% Sometimes it was a different way of organizing basic courses. 00:12:00.510 --> 00:12:03.450 align:middle line:84% Sometimes it was interacting with engineers 00:12:03.450 --> 00:12:07.140 align:middle line:84% to optimize the basic courses that the Math Department was 00:12:07.140 --> 00:12:08.100 align:middle line:90% offering. 00:12:08.100 --> 00:12:10.050 align:middle line:84% And we've been running it every two weeks 00:12:10.050 --> 00:12:12.600 align:middle line:84% during the academic year ever since. 00:12:12.600 --> 00:12:15.480 align:middle line:84% It's become quite a useful resource. 00:12:15.480 --> 00:12:18.930 align:middle line:84% I think my original thought that there 00:12:18.930 --> 00:12:24.150 align:middle line:84% was a lot of activity going on around the country, 00:12:24.150 --> 00:12:27.510 align:middle line:84% in different departments, with a lot of imagination 00:12:27.510 --> 00:12:30.210 align:middle line:84% and creativity and passion, it was true. 00:12:30.210 --> 00:12:34.920 align:middle line:84% And we have it on record now, archived and available 00:12:34.920 --> 00:12:36.990 align:middle line:90% at the OLSUME website. 00:12:36.990 --> 00:12:38.942 align:middle line:90% 00:12:38.942 --> 00:12:41.400 align:middle line:84% PAIGE BRIGHT: On top of teaching the Differential Equations 00:12:41.400 --> 00:12:43.140 align:middle line:84% course and running the Project Lab, 00:12:43.140 --> 00:12:45.060 align:middle line:84% Haynes also runs the First-Year Seminars 00:12:45.060 --> 00:12:46.590 align:middle line:90% that we mentioned earlier. 00:12:46.590 --> 00:12:49.747 align:middle line:84% The title of the seminar I was in was titled What Is a Number? 00:12:49.747 --> 00:12:52.080 align:middle line:84% And while you might be surprised that a math major would 00:12:52.080 --> 00:12:54.120 align:middle line:84% take a class titled What Is a Number?, 00:12:54.120 --> 00:12:56.262 align:middle line:84% it's actually a rather fascinating question. 00:12:56.262 --> 00:12:58.470 align:middle line:84% HAYNES MILLER: We all think we know what numbers are. 00:12:58.470 --> 00:12:59.970 align:middle line:90% We use them every day. 00:12:59.970 --> 00:13:03.420 align:middle line:84% But very quickly, you get into some real mathematical 00:13:03.420 --> 00:13:07.020 align:middle line:84% questions that aren't buried at the end of some course. 00:13:07.020 --> 00:13:08.400 align:middle line:90% They're right on the surface. 00:13:08.400 --> 00:13:11.760 align:middle line:84% And you realize, for example, that you can't really 00:13:11.760 --> 00:13:14.040 align:middle line:84% answer that question, what is a number? 00:13:14.040 --> 00:13:16.080 align:middle line:90% It doesn't make sense in itself. 00:13:16.080 --> 00:13:19.620 align:middle line:84% You can answer a question like, what is a number system? 00:13:19.620 --> 00:13:21.330 align:middle line:84% Because the whole point about numbers 00:13:21.330 --> 00:13:23.040 align:middle line:84% is how they relate to each other. 00:13:23.040 --> 00:13:26.850 align:middle line:84% So we get back to this issue about things forming complexes, 00:13:26.850 --> 00:13:29.670 align:middle line:84% things forming combinations before they really 00:13:29.670 --> 00:13:30.840 align:middle line:90% have meaning. 00:13:30.840 --> 00:13:33.270 align:middle line:84% That was one of the most interesting 00:13:33.270 --> 00:13:35.520 align:middle line:84% and fun things about the seminar that we were in, 00:13:35.520 --> 00:13:38.970 align:middle line:84% is thinking carefully about the concept of number 00:13:38.970 --> 00:13:43.500 align:middle line:84% and the many different varieties of numbers that there are. 00:13:43.500 --> 00:13:45.660 align:middle line:84% But every time, every one of them, 00:13:45.660 --> 00:13:48.180 align:middle line:84% you see that it doesn't make sense until you 00:13:48.180 --> 00:13:50.610 align:middle line:90% think of them in a system. 00:13:50.610 --> 00:13:53.880 align:middle line:84% Other mathematical concepts that came out very clearly 00:13:53.880 --> 00:13:57.780 align:middle line:84% was the notion that you get at something by characterizing 00:13:57.780 --> 00:13:59.430 align:middle line:90% its properties. 00:13:59.430 --> 00:14:03.000 align:middle line:84% So I guess this is the axiomatic system. 00:14:03.000 --> 00:14:05.040 align:middle line:84% You gain much better insight into what 00:14:05.040 --> 00:14:08.910 align:middle line:84% an integer is by thinking about it axiomatically. 00:14:08.910 --> 00:14:11.160 align:middle line:84% What are the properties that you expect 00:14:11.160 --> 00:14:13.380 align:middle line:90% that number system to have? 00:14:13.380 --> 00:14:15.930 align:middle line:84% You can construct them in different ways, 00:14:15.930 --> 00:14:17.730 align:middle line:84% but every single construction of them 00:14:17.730 --> 00:14:23.280 align:middle line:84% is artificial and in a sense irrelevant, because in the end 00:14:23.280 --> 00:14:25.080 align:middle line:84% you're just using their properties. 00:14:25.080 --> 00:14:28.380 align:middle line:84% So this is a very deep mathematical perspective 00:14:28.380 --> 00:14:30.840 align:middle line:84% actually that came out quite clearly right 00:14:30.840 --> 00:14:33.030 align:middle line:84% away, without a great deal of background, 00:14:33.030 --> 00:14:36.607 align:middle line:84% and it was accessible to freshmen pretty easily. 00:14:36.607 --> 00:14:38.690 align:middle line:84% PAIGE BRIGHT: Haynes is truly dedicated to helping 00:14:38.690 --> 00:14:42.050 align:middle line:84% people understand complex and intimidating subjects. 00:14:42.050 --> 00:14:44.030 align:middle line:84% He also cares about crafting his materials 00:14:44.030 --> 00:14:46.297 align:middle line:84% in a way that gets students thinking differently. 00:14:46.297 --> 00:14:48.380 align:middle line:84% And there's one seminar where that certainly comes 00:14:48.380 --> 00:14:50.870 align:middle line:90% to fruition, the Kan Seminar. 00:14:50.870 --> 00:14:53.390 align:middle line:84% HAYNES MILLER: So Dan Kan was a faculty member here. 00:14:53.390 --> 00:14:54.770 align:middle line:90% He was here when I got here. 00:14:54.770 --> 00:14:56.360 align:middle line:90% He's now deceased. 00:14:56.360 --> 00:15:01.610 align:middle line:84% He ran a graduate course, but he declined to lecture. 00:15:01.610 --> 00:15:03.680 align:middle line:84% Instead, he had students lecture. 00:15:03.680 --> 00:15:05.690 align:middle line:84% And this was a revolutionary concept 00:15:05.690 --> 00:15:10.100 align:middle line:84% I think when it began back in about 1965. 00:15:10.100 --> 00:15:13.730 align:middle line:84% The Kan Seminar has been in action every fall 00:15:13.730 --> 00:15:17.570 align:middle line:84% ever since then, run by him first and then by other people, 00:15:17.570 --> 00:15:19.220 align:middle line:84% and then for a number of years by me. 00:15:19.220 --> 00:15:21.530 align:middle line:84% Now it's passed on to other people. 00:15:21.530 --> 00:15:25.820 align:middle line:84% It's a literature seminar on the classic papers 00:15:25.820 --> 00:15:28.280 align:middle line:84% in this subject, algebraic topology. 00:15:28.280 --> 00:15:32.780 align:middle line:84% And it gives students-- it gives the graduate students-- and now 00:15:32.780 --> 00:15:35.600 align:middle line:84% increasingly undergraduates from MIT 00:15:35.600 --> 00:15:37.850 align:middle line:90% and from Harvard will take it. 00:15:37.850 --> 00:15:40.310 align:middle line:84% It gives them a chance to appreciate 00:15:40.310 --> 00:15:44.630 align:middle line:84% how the subject developed and to really learn 00:15:44.630 --> 00:15:48.530 align:middle line:84% the basic examples that are still 00:15:48.530 --> 00:15:51.320 align:middle line:90% motivating the subject today. 00:15:51.320 --> 00:15:56.300 align:middle line:84% It gives students a chance to talk to a faculty member 00:15:56.300 --> 00:15:56.870 align:middle line:90% regularly. 00:15:56.870 --> 00:15:58.880 align:middle line:84% I talk with each student every week 00:15:58.880 --> 00:16:02.420 align:middle line:84% about the paper that they're going to present, and learn 00:16:02.420 --> 00:16:04.310 align:middle line:90% to read a paper in two ways. 00:16:04.310 --> 00:16:07.253 align:middle line:84% They learn to read a paper in great detail 00:16:07.253 --> 00:16:09.170 align:middle line:84% because they're going to give a talk about it. 00:16:09.170 --> 00:16:10.295 align:middle line:90% They have to understand it. 00:16:10.295 --> 00:16:12.262 align:middle line:84% And they will pick some part of the paper 00:16:12.262 --> 00:16:13.970 align:middle line:84% that they think is especially interesting 00:16:13.970 --> 00:16:17.450 align:middle line:84% and they want to tell their peers about. 00:16:17.450 --> 00:16:20.960 align:middle line:84% But also, they're expected to read the papers that everybody 00:16:20.960 --> 00:16:21.920 align:middle line:90% else is talking about. 00:16:21.920 --> 00:16:25.490 align:middle line:84% So they also learn to read a paper very quickly 00:16:25.490 --> 00:16:30.057 align:middle line:84% and get something useful from it very efficiently. 00:16:30.057 --> 00:16:31.890 align:middle line:84% PAIGE BRIGHT: I was curious about how he had 00:16:31.890 --> 00:16:33.720 align:middle line:90% students teach the lectures. 00:16:33.720 --> 00:16:35.640 align:middle line:84% Haynes told me a bit about his own beginnings 00:16:35.640 --> 00:16:37.218 align:middle line:84% as an instructor, and also how he 00:16:37.218 --> 00:16:38.760 align:middle line:84% gets students to present in ways that 00:16:38.760 --> 00:16:42.180 align:middle line:84% are both engaging and useful to others. 00:16:42.180 --> 00:16:44.670 align:middle line:84% HAYNES MILLER: I remember when I first got here, 00:16:44.670 --> 00:16:48.600 align:middle line:84% I was videotaped, as many beginning teachers are here, 00:16:48.600 --> 00:16:51.660 align:middle line:84% and I watched the video of myself teaching. 00:16:51.660 --> 00:16:56.820 align:middle line:84% And I thought, why is this guy trying to act so cool? 00:16:56.820 --> 00:17:01.410 align:middle line:84% And why doesn't he just say things like they are? 00:17:01.410 --> 00:17:07.290 align:middle line:84% And so I learned to be more direct from that experience. 00:17:07.290 --> 00:17:08.797 align:middle line:84% Students have a tendency to assume 00:17:08.797 --> 00:17:10.380 align:middle line:84% that everybody in their audience knows 00:17:10.380 --> 00:17:15.089 align:middle line:84% everything they're going to say, and so they'll rush through it. 00:17:15.089 --> 00:17:18.932 align:middle line:84% They'll not do an adequate job in explaining 00:17:18.932 --> 00:17:21.390 align:middle line:84% what they have to say, because they think there's no point. 00:17:21.390 --> 00:17:23.670 align:middle line:84% This is just a formality of some kind. 00:17:23.670 --> 00:17:27.780 align:middle line:84% So really persuading them that they're saying something 00:17:27.780 --> 00:17:32.820 align:middle line:84% that the audience doesn't know, and it's up to them [LAUGHS] 00:17:32.820 --> 00:17:34.740 align:middle line:90% to explain it. 00:17:34.740 --> 00:17:37.142 align:middle line:90% That's a big deal. 00:17:37.142 --> 00:17:38.850 align:middle line:84% PAIGE BRIGHT: To close our episode today, 00:17:38.850 --> 00:17:40.267 align:middle line:84% I wanted to share some reflections 00:17:40.267 --> 00:17:41.842 align:middle line:90% from Haynes about OCW. 00:17:41.842 --> 00:17:43.800 align:middle line:84% Having personally worked on some course content 00:17:43.800 --> 00:17:46.860 align:middle line:84% for OpenCourseWare, I can say that it's an amazing experience 00:17:46.860 --> 00:17:48.900 align:middle line:84% to get to make materials that help educators 00:17:48.900 --> 00:17:52.740 align:middle line:84% and students alike learn from the amazing resources at MIT, 00:17:52.740 --> 00:17:55.620 align:middle line:84% regardless of where they are or what academic avenues are 00:17:55.620 --> 00:17:58.050 align:middle line:90% available to them. 00:17:58.050 --> 00:18:00.900 align:middle line:84% HAYNES MILLER: The MIT faculty are so incredibly 00:18:00.900 --> 00:18:04.800 align:middle line:84% lucky to have OpenCourseWare as a conduit, 00:18:04.800 --> 00:18:06.960 align:middle line:84% as a way of expressing yourselves, 00:18:06.960 --> 00:18:09.540 align:middle line:84% as a way of publishing this work that we do, 00:18:09.540 --> 00:18:12.300 align:middle line:84% that we spend a great part of our professional life working 00:18:12.300 --> 00:18:12.870 align:middle line:90% on. 00:18:12.870 --> 00:18:16.200 align:middle line:84% Other universities don't have that luxury, typically. 00:18:16.200 --> 00:18:20.850 align:middle line:84% They don't have a way to memorialize and contribute 00:18:20.850 --> 00:18:23.745 align:middle line:84% their teaching work to the world at large. 00:18:23.745 --> 00:18:26.040 align:middle line:84% It's such a great resource that we have here. 00:18:26.040 --> 00:18:29.130 align:middle line:84% One of the great things about OCW is you run the course, 00:18:29.130 --> 00:18:32.760 align:middle line:84% and then you tell OpenCourseWare to go run with it, 00:18:32.760 --> 00:18:34.710 align:middle line:90% and something amazing happens. 00:18:34.710 --> 00:18:37.705 align:middle line:90% 00:18:37.705 --> 00:18:40.080 align:middle line:84% PAIGE BRIGHT: Professor Miller shares additional thoughts 00:18:40.080 --> 00:18:41.520 align:middle line:84% on the courses talked about today, 00:18:41.520 --> 00:18:44.340 align:middle line:84% such as the Kan Seminar and the Project Lab in Mathematics, 00:18:44.340 --> 00:18:45.690 align:middle line:90% on OCW. 00:18:45.690 --> 00:18:50.167 align:middle line:84% You can view these resources at ocw.mit.edu. 00:18:50.167 --> 00:18:51.750 align:middle line:84% You can also learn more about his work 00:18:51.750 --> 00:18:54.360 align:middle line:84% with the Online Seminar on Undergraduate Mathematics 00:18:54.360 --> 00:18:59.550 align:middle line:84% Education, or OLSUME at olsume.org. 00:18:59.550 --> 00:19:01.800 align:middle line:84% Thank you so much for listening to this guest podcast. 00:19:01.800 --> 00:19:07.350 align:middle line:84% Until next time, I'm Paige Bright from MIT OpenCourseWare. 00:19:07.350 --> 00:19:10.260 align:middle line:84% MIT Chalk Radio's producers include Sarah Hanson, 00:19:10.260 --> 00:19:12.630 align:middle line:90% Brett Paci, and Dave Lishansky. 00:19:12.630 --> 00:19:15.840 align:middle line:84% Show notes for this episode were written by Peter Chipman. 00:19:15.840 --> 00:19:18.240 align:middle line:84% We're funded by MIT Open Learning and supporters 00:19:18.240 --> 00:19:20.000 align:middle line:90% like you. 00:19:20.000 --> 00:19:25.000 align:middle line:90%