1 01:00:00,000 --> 01:00:04,416 [MUSIC] 2 01:00:15,541 --> 01:00:16,458 Hello, everybody. 3 01:00:16,750 --> 01:00:18,125 Welcome to another episode of 4 01:00:18,125 --> 01:00:18,958 Undercooled, a 5 01:00:18,958 --> 01:00:20,500 materials education podcast. 6 01:00:20,958 --> 01:00:22,500 As always, I'm your host, Tim Chambers. 7 01:00:22,666 --> 01:00:24,375 I'm here with my co-host, Steve Yalasov, 8 01:00:24,750 --> 01:00:26,041 and we are going to 9 01:00:26,041 --> 01:00:27,166 talk about math today, 10 01:00:27,541 --> 01:00:29,041 a topic that is very near and dear to 11 01:00:29,041 --> 01:00:30,000 both of our hearts, and 12 01:00:30,000 --> 01:00:31,208 hopefully to yours as well. 13 01:00:32,041 --> 01:00:33,791 Absolutely, especially 14 01:00:33,791 --> 01:00:35,541 in materials programs. 15 01:00:36,833 --> 01:00:39,500 I can't tell you how many of my 16 01:00:39,500 --> 01:00:41,500 colleagues here at Michigan and 17 01:00:41,500 --> 01:00:44,000 my colleagues all over the country are 18 01:00:44,000 --> 01:00:46,541 constantly complaining that our students 19 01:00:46,541 --> 01:00:50,000 are just not prepared with 20 01:00:50,000 --> 01:00:52,125 their math skills to handle 21 01:00:52,125 --> 01:00:53,458 the kinds of math we 22 01:00:53,458 --> 01:00:55,375 do in material science. 23 01:00:56,208 --> 01:00:57,208 This includes like 24 01:00:57,208 --> 01:01:01,833 thermo and kinetics and 25 01:01:02,541 --> 01:01:06,250 electronic materials, tensors for 26 01:01:06,250 --> 01:01:07,500 mechanical properties. 27 01:01:09,000 --> 01:01:11,500 We always seem to have a large number of 28 01:01:11,500 --> 01:01:12,583 our students who just 29 01:01:12,583 --> 01:01:13,625 don't know what we're 30 01:01:13,625 --> 01:01:15,583 talking about, and it's a problem. 31 01:01:16,291 --> 01:01:17,416 So I think this is a great 32 01:01:17,416 --> 01:01:18,791 thing to talk about today. 33 01:01:20,250 --> 01:01:22,541 Yeah, and we're going to talk about some 34 01:01:22,541 --> 01:01:23,791 of the curricular structuring, 35 01:01:24,041 --> 01:01:25,708 the sequence of courses, because we want 36 01:01:25,708 --> 01:01:27,416 to prepare our students for success. 37 01:01:27,875 --> 01:01:29,166 At the end of the day, that's really what 38 01:01:29,166 --> 01:01:30,416 it comes down to, is 39 01:01:30,416 --> 01:01:31,791 we want them to engage 40 01:01:31,791 --> 01:01:34,208 with these difficult high level topics in 41 01:01:34,208 --> 01:01:34,958 a way that's meaningful 42 01:01:34,958 --> 01:01:36,125 so they actually learn. 43 01:01:36,750 --> 01:01:38,750 And the mathematical foundation of that 44 01:01:38,750 --> 01:01:39,750 is a critical 45 01:01:39,750 --> 01:01:40,875 ingredient for their success. 46 01:01:41,750 --> 01:01:43,416 So how do we get them there? 47 01:01:43,416 --> 01:01:44,750 I guess that's our question today. 48 01:01:45,333 --> 01:01:46,958 Yeah, and even before that, I think we 49 01:01:46,958 --> 01:01:48,000 need to talk about some 50 01:01:48,000 --> 01:01:49,208 of the special problems 51 01:01:49,333 --> 01:01:52,916 that we're facing, because the students 52 01:01:52,916 --> 01:01:55,583 we're getting now all went through COVID. 53 01:01:57,166 --> 01:01:59,458 And in COVID days, we 54 01:01:59,458 --> 01:02:00,625 let everyone just pass. 55 01:02:01,250 --> 01:02:03,000 And so a lot of students just never even 56 01:02:03,000 --> 01:02:04,291 learned this material. 57 01:02:04,708 --> 01:02:07,083 What they learned were lots of bad habits 58 01:02:07,083 --> 01:02:08,291 that they can get away 59 01:02:08,291 --> 01:02:09,375 without doing anything 60 01:02:09,375 --> 01:02:11,041 and we'll excuse them. 61 01:02:11,958 --> 01:02:13,291 So this is all 62 01:02:13,291 --> 01:02:15,125 conspired to make it even worse. 63 01:02:16,916 --> 01:02:18,416 To be honest, we had 64 01:02:18,416 --> 01:02:19,916 this problem before COVID. 65 01:02:20,166 --> 01:02:20,500 Oh, sure. 66 01:02:20,791 --> 01:02:23,250 It got a little bit worse, but this is a 67 01:02:23,250 --> 01:02:24,583 never ending challenge 68 01:02:24,583 --> 01:02:26,208 of engineering education, 69 01:02:26,458 --> 01:02:28,833 is that it seems like in some ways the 70 01:02:28,833 --> 01:02:30,375 math departments aren't 71 01:02:30,375 --> 01:02:31,333 actually teaching the math 72 01:02:31,333 --> 01:02:33,208 that we need our students to know. 73 01:02:34,291 --> 01:02:35,708 Yeah, and it's even a little deeper than 74 01:02:35,708 --> 01:02:37,791 that, because we always 75 01:02:37,791 --> 01:02:40,625 get 30% of the students 76 01:02:40,625 --> 01:02:41,625 have no problem with 77 01:02:41,625 --> 01:02:43,000 the math we throw at them. 78 01:02:43,250 --> 01:02:44,666 And they went through those same courses. 79 01:02:45,125 --> 01:02:46,083 And so what gifts? 80 01:02:47,083 --> 01:02:50,291 And so I think the math department does 81 01:02:50,291 --> 01:02:52,500 teach the material, but 82 01:02:52,500 --> 01:02:53,916 they only teach it to a 83 01:02:53,916 --> 01:02:54,541 certain kind of 84 01:02:54,541 --> 01:02:56,375 student that can receive it. 85 01:02:56,833 --> 01:02:58,333 And I don't think they're being sensitive 86 01:02:58,333 --> 01:02:59,541 to the other students 87 01:02:59,541 --> 01:03:01,291 who really need to learn 88 01:03:01,291 --> 01:03:02,125 this material. 89 01:03:03,041 --> 01:03:06,333 And they're the ones that get Cs, C 90 01:03:06,333 --> 01:03:09,250 pluses, and they pass, but 91 01:03:09,250 --> 01:03:11,416 they really haven't learned 92 01:03:11,625 --> 01:03:13,458 the minimal amount to 93 01:03:13,458 --> 01:03:14,625 succeed in the future. 94 01:03:15,416 --> 01:03:17,625 And so part of this, 95 01:03:17,625 --> 01:03:18,750 I think, is pedagogy. 96 01:03:20,250 --> 01:03:21,500 Part of it is, you know, 97 01:03:22,083 --> 01:03:23,000 students are different. 98 01:03:24,541 --> 01:03:26,208 But none of that matters when they get it 99 01:03:26,208 --> 01:03:26,958 in our class, because 100 01:03:26,958 --> 01:03:27,750 we have to deal with 101 01:03:27,750 --> 01:03:28,083 it. 102 01:03:28,083 --> 01:03:28,750 That's right. 103 01:03:28,750 --> 01:03:30,500 So I think what we're going to be talking 104 01:03:30,500 --> 01:03:33,083 about today is how do 105 01:03:33,083 --> 01:03:34,458 we deal with it as a 106 01:03:34,500 --> 01:03:37,500 materials department without trying to 107 01:03:37,500 --> 01:03:39,875 cascade or, you know, 108 01:03:40,291 --> 01:03:42,291 complain about our math department? 109 01:03:43,125 --> 01:03:44,500 Well, if we're working on a solution, 110 01:03:44,500 --> 01:03:46,125 then let's start at the end. 111 01:03:46,500 --> 01:03:48,291 What is it that our students need? 112 01:03:48,583 --> 01:03:50,708 You know, as a materials department, what 113 01:03:50,708 --> 01:03:52,000 are the key 114 01:03:52,000 --> 01:03:53,958 components of understanding and 115 01:03:53,958 --> 01:03:56,416 being able to do math that our students 116 01:03:56,416 --> 01:03:57,541 have to be able to accomplish? 117 01:03:59,791 --> 01:04:00,541 That's a great question. 118 01:04:00,791 --> 01:04:02,791 And I really think we know the answer. 119 01:04:03,500 --> 01:04:05,500 And it's basically the sequence of 120 01:04:05,500 --> 01:04:08,000 calculus that has become 121 01:04:08,000 --> 01:04:10,833 Calc 1, Calc 2, Calc 3, none 122 01:04:10,833 --> 01:04:12,250 of which says anything about what they 123 01:04:12,250 --> 01:04:13,250 learn in their class. 124 01:04:13,958 --> 01:04:16,750 But you know, it's very interesting to 125 01:04:16,750 --> 01:04:18,000 me, at least at 126 01:04:18,000 --> 01:04:20,041 Michigan, and I kind of believe 127 01:04:20,041 --> 01:04:21,291 this has happened elsewhere. 128 01:04:22,791 --> 01:04:25,458 They turned four semesters of calculus, 129 01:04:25,833 --> 01:04:27,416 differential equations, and linear 130 01:04:27,416 --> 01:04:29,791 algebra into five semesters. 131 01:04:30,625 --> 01:04:34,333 For whatever reason, the students coming 132 01:04:34,333 --> 01:04:35,708 in now, the first 133 01:04:35,708 --> 01:04:37,583 calculus class they take 134 01:04:37,958 --> 01:04:41,916 is limited just to limits, series, review 135 01:04:41,916 --> 01:04:43,875 of trigonometry, and 136 01:04:43,875 --> 01:04:45,125 what a differential is. 137 01:04:45,875 --> 01:04:47,250 And then they wait for a whole other 138 01:04:47,250 --> 01:04:48,916 course to teach integration. 139 01:04:50,000 --> 01:04:52,833 And they do integration before they teach 140 01:04:52,833 --> 01:04:54,333 multivariate calculus. 141 01:04:55,000 --> 01:04:57,416 So they never even learn, you know, 142 01:04:57,583 --> 01:04:59,291 integration of volumes 143 01:04:59,291 --> 01:05:00,750 because that comes later. 144 01:05:01,791 --> 01:05:03,500 And then they take an ordinary 145 01:05:03,500 --> 01:05:04,500 differential equations 146 01:05:04,500 --> 01:05:06,791 course, and that's four courses, 147 01:05:07,333 --> 01:05:07,916 and they're done. 148 01:05:08,500 --> 01:05:10,750 But what happened to linear algebra? 149 01:05:11,250 --> 01:05:12,875 Yeah, what happened to statistics? 150 01:05:13,083 --> 01:05:14,708 What happened to complex variables? 151 01:05:15,208 --> 01:05:16,125 These are all critical 152 01:05:16,125 --> 01:05:17,500 topics that we also need. 153 01:05:18,000 --> 01:05:18,458 They are. 154 01:05:19,000 --> 01:05:22,541 So our students come to us, and they 155 01:05:22,541 --> 01:05:24,041 might have some inkling 156 01:05:24,041 --> 01:05:26,208 of those things, but you 157 01:05:26,208 --> 01:05:28,708 know, I would have rather they turn this 158 01:05:28,708 --> 01:05:29,791 into a five course 159 01:05:29,791 --> 01:05:32,083 sequence where they combined 160 01:05:32,416 --> 01:05:34,875 differentiation and integration into one 161 01:05:34,875 --> 01:05:36,708 course, taught a course 162 01:05:36,708 --> 01:05:38,416 on multivariate calculus, 163 01:05:39,166 --> 01:05:42,583 taught another course on ordinary 164 01:05:42,583 --> 01:05:44,750 differential equations, and a fourth 165 01:05:44,750 --> 01:05:45,958 course on linear algebra, 166 01:05:46,500 --> 01:05:49,750 and a fifth course on engineering math, 167 01:05:50,041 --> 01:05:52,708 where they cover orthogonal 168 01:05:52,708 --> 01:05:54,375 series, they cover complex 169 01:05:54,750 --> 01:05:58,041 numbers, they cover statistics, and even 170 01:05:58,041 --> 01:05:59,166 discrete math for 171 01:05:59,166 --> 01:06:01,333 computational applications. 172 01:06:02,625 --> 01:06:04,416 So that would be ideal. 173 01:06:05,208 --> 01:06:07,500 But instead of, again, complaining about 174 01:06:07,500 --> 01:06:10,333 what would be ideal, why 175 01:06:10,333 --> 01:06:12,416 don't we talk about what 176 01:06:12,458 --> 01:06:18,750 are the things that students often have 177 01:06:18,750 --> 01:06:20,166 trouble with when they get 178 01:06:20,166 --> 01:06:22,416 to our higher level courses? 179 01:06:23,208 --> 01:06:24,458 And so I'll go with my 180 01:06:24,458 --> 01:06:26,250 first one, my favorite one. 181 01:06:27,166 --> 01:06:28,041 Our students don't even 182 01:06:28,041 --> 01:06:29,375 know how to do trigonometry. 183 01:06:30,791 --> 01:06:32,166 They have an idea of 184 01:06:32,166 --> 01:06:33,833 what sines and cosines are. 185 01:06:34,583 --> 01:06:35,375 None of them know what 186 01:06:35,375 --> 01:06:37,416 hyperbolic sines and cosines are. 187 01:06:38,291 --> 01:06:41,000 And more importantly, they don't 188 01:06:41,000 --> 01:06:41,708 understand the 189 01:06:41,708 --> 01:06:45,375 relationship of series expansions and 190 01:06:45,375 --> 01:06:48,083 exponentials to sines and cosines. 191 01:06:48,625 --> 01:06:50,708 So when they get to where we're trying to 192 01:06:50,708 --> 01:06:53,625 explain structure factor and diffraction, 193 01:06:54,291 --> 01:06:57,125 you know, we put e to the ik dot x, and 194 01:06:57,125 --> 01:06:58,333 they like look at us 195 01:06:58,333 --> 01:06:59,958 like we're from outer space. 196 01:07:00,166 --> 01:07:02,291 I just had that experience two weeks ago, 197 01:07:02,291 --> 01:07:04,250 I was doing a review of x ray diffraction 198 01:07:04,333 --> 01:07:07,583 for the lab class and e to the ik dot, 199 01:07:08,083 --> 01:07:09,500 you know, ik vector dot x 200 01:07:09,500 --> 01:07:11,500 vector and we were gone. 201 01:07:12,041 --> 01:07:13,500 And I, you know, I backed up, we worked 202 01:07:13,500 --> 01:07:14,458 through it, but I had to 203 01:07:14,458 --> 01:07:15,875 do a lot of unpacking there 204 01:07:15,875 --> 01:07:19,416 that I was surprised had to be taught 205 01:07:19,416 --> 01:07:21,083 that far into the sequence. 206 01:07:21,500 --> 01:07:21,791 That's right. 207 01:07:22,041 --> 01:07:24,041 And of course, this is where series 208 01:07:24,041 --> 01:07:26,083 expansions really help. 209 01:07:26,875 --> 01:07:28,583 Because the only way to prove that any of 210 01:07:28,583 --> 01:07:29,666 this stuff is correct 211 01:07:29,666 --> 01:07:31,375 is by doing this series 212 01:07:32,083 --> 01:07:33,500 expansions and, you know, taking science 213 01:07:33,500 --> 01:07:35,208 and cosines and, and 214 01:07:35,208 --> 01:07:36,666 doing their series expansions 215 01:07:36,666 --> 01:07:39,208 and multiplying one of them by I, and you 216 01:07:39,208 --> 01:07:40,500 add it up and woohoo, 217 01:07:40,500 --> 01:07:41,541 it's the same as a series 218 01:07:41,541 --> 01:07:44,250 expansion of e to the ik dot x. 219 01:07:44,916 --> 01:07:46,958 And so that gives students, you know, 220 01:07:46,958 --> 01:07:48,083 it's not that that proof 221 01:07:48,083 --> 01:07:49,666 is such an important thing 222 01:07:49,958 --> 01:07:52,875 for the practical use, but it gives 223 01:07:52,875 --> 01:07:54,541 students a lot of 224 01:07:54,541 --> 01:07:57,500 confidence that they understand why 225 01:07:57,500 --> 01:07:58,833 that equation works. 226 01:07:59,916 --> 01:08:01,833 And so they were supposed to learn that a 227 01:08:01,833 --> 01:08:04,125 long time ago, and they didn't. 228 01:08:04,583 --> 01:08:06,333 And probably because they learned a 229 01:08:06,333 --> 01:08:08,000 little bits of it across 230 01:08:08,000 --> 01:08:09,958 many courses, and nobody 231 01:08:09,958 --> 01:08:11,875 ever coupled it together. 232 01:08:12,333 --> 01:08:14,083 That's the exact experience I've been 233 01:08:14,083 --> 01:08:15,416 having, especially in the 234 01:08:15,416 --> 01:08:16,791 last year or two for students 235 01:08:16,791 --> 01:08:18,666 who did get a lot of 236 01:08:18,666 --> 01:08:20,083 this core math during COVID. 237 01:08:20,916 --> 01:08:22,958 A lot of the puzzle pieces are there, but 238 01:08:22,958 --> 01:08:24,250 the puzzle has never been built. 239 01:08:24,833 --> 01:08:25,000 Right. 240 01:08:25,625 --> 01:08:26,916 So that was my first one. 241 01:08:27,125 --> 01:08:28,291 What's, what's your next one? 242 01:08:29,250 --> 01:08:32,000 Well, I have some personal feelings about 243 01:08:32,000 --> 01:08:33,958 this because I, once upon 244 01:08:33,958 --> 01:08:35,000 a time as an undergraduate 245 01:08:35,166 --> 01:08:36,791 student at the University of Michigan, 246 01:08:37,083 --> 01:08:39,208 had an amazing linear 247 01:08:39,208 --> 01:08:41,000 algebra for scientists and 248 01:08:41,000 --> 01:08:41,666 engineers course. 249 01:08:42,083 --> 01:08:44,916 And it was a very practical, essentially 250 01:08:44,916 --> 01:08:46,458 semester and how to set up 251 01:08:46,458 --> 01:08:47,791 and solve eigenvalue problems 252 01:08:47,958 --> 01:08:49,291 was really the punchline of it. 253 01:08:49,541 --> 01:08:51,458 And that completely transformed the way I 254 01:08:51,458 --> 01:08:52,541 thought about math, 255 01:08:52,541 --> 01:08:53,041 the way I thought about 256 01:08:53,041 --> 01:08:54,208 how math is used to 257 01:08:54,208 --> 01:08:55,541 solve scientific problems. 258 01:08:56,375 --> 01:08:57,375 Really my understanding of quantum 259 01:08:57,375 --> 01:08:59,041 mechanics, a lot of it 260 01:08:59,041 --> 01:09:00,000 stemmed from that course. 261 01:09:00,708 --> 01:09:02,791 And so when I'm looking at what our 262 01:09:02,791 --> 01:09:04,666 students here in MSC need 263 01:09:04,666 --> 01:09:09,750 to know, I am often finding 264 01:09:09,833 --> 01:09:13,625 ways in which students don't think about 265 01:09:13,625 --> 01:09:15,041 functions, about series, 266 01:09:15,041 --> 01:09:16,791 about vector spaces in the way 267 01:09:16,791 --> 01:09:17,958 that I do. 268 01:09:18,500 --> 01:09:19,750 And that can be a challenge in 269 01:09:19,750 --> 01:09:21,458 communicating with them when I want to 270 01:09:21,458 --> 01:09:22,500 teach something mathematical 271 01:09:22,750 --> 01:09:24,500 effectively, but we're just using a 272 01:09:24,500 --> 01:09:25,708 completely different framework. 273 01:09:26,291 --> 01:09:27,291 And I feel like if they had that 274 01:09:27,291 --> 01:09:28,166 understanding of vector 275 01:09:28,166 --> 01:09:30,750 spaces, it would enable so many 276 01:09:31,375 --> 01:09:33,500 more discussions and so much more 277 01:09:33,500 --> 01:09:35,208 understanding of other topics. 278 01:09:35,208 --> 01:09:36,208 Like for example, discrete 279 01:09:36,208 --> 01:09:37,500 math, as you brought up earlier. 280 01:09:39,458 --> 01:09:41,083 It's funny you say that, you know, I, I 281 01:09:41,083 --> 01:09:42,791 was a math major, so I 282 01:09:42,791 --> 01:09:44,458 had a lot of this in a 283 01:09:44,541 --> 01:09:45,625 different way than I think 284 01:09:45,625 --> 01:09:47,583 a lot of engineers get it. 285 01:09:47,833 --> 01:09:49,375 But I probably got it more like a 286 01:09:49,375 --> 01:09:50,958 physicist got it because 287 01:09:50,958 --> 01:09:52,791 physicists need this too. 288 01:09:53,708 --> 01:09:56,666 And while I've got to say even three 289 01:09:56,666 --> 01:09:58,833 space, you know, just XYZ 290 01:09:58,833 --> 01:10:01,291 vectors, I wish our students 291 01:10:01,291 --> 01:10:02,208 knew more about that. 292 01:10:02,416 --> 01:10:04,791 I teach our introductory course and I'm 293 01:10:04,791 --> 01:10:06,375 blown away by students 294 01:10:06,375 --> 01:10:07,375 who don't know how to take 295 01:10:07,375 --> 01:10:09,750 a dot product in vector notation. 296 01:10:10,166 --> 01:10:11,041 Like what could be easier? 297 01:10:11,833 --> 01:10:13,291 And they just don't understand that. 298 01:10:13,625 --> 01:10:15,166 And they don't understand its connection 299 01:10:15,166 --> 01:10:16,416 again to trigonometry. 300 01:10:18,500 --> 01:10:21,916 That's at least they have some inkling of 301 01:10:21,916 --> 01:10:23,416 what a vector space is. 302 01:10:24,208 --> 01:10:26,208 Now, of course, the big thing I wish our 303 01:10:26,208 --> 01:10:27,625 students would learn, and 304 01:10:27,625 --> 01:10:29,458 no one teaches this in the 305 01:10:29,958 --> 01:10:31,541 first few courses of math. 306 01:10:31,541 --> 01:10:32,625 I never learned it in the 307 01:10:32,625 --> 01:10:33,708 first few courses of math. 308 01:10:33,708 --> 01:10:35,541 I learned it in applied math. 309 01:10:35,541 --> 01:10:37,458 Well, I actually learned a real analysis. 310 01:10:38,250 --> 01:10:41,791 But the simple idea, and it really is a 311 01:10:41,791 --> 01:10:44,000 simple idea, that the 312 01:10:44,000 --> 01:10:46,541 space of all functions is a 313 01:10:46,583 --> 01:10:47,333 vector space. 314 01:10:48,500 --> 01:10:50,958 That blew my mind when I learned that. 315 01:10:51,250 --> 01:10:53,875 And once you realize that, then you can 316 01:10:53,875 --> 01:10:57,333 define a basis set where the basis 317 01:10:57,333 --> 01:10:58,750 elements are mutually 318 01:10:58,791 --> 01:11:00,916 orthogonal, meaning their dot products 319 01:11:00,916 --> 01:11:03,375 are zero, and you 320 01:11:03,375 --> 01:11:05,291 normalize it so they're unity. 321 01:11:05,500 --> 01:11:06,458 That's all you have to do. 322 01:11:06,958 --> 01:11:10,666 But then to know that by a linear 323 01:11:10,666 --> 01:11:13,041 combination of these, you 324 01:11:13,041 --> 01:11:15,916 can map out any function. 325 01:11:16,583 --> 01:11:18,083 Well, not any function. 326 01:11:18,791 --> 01:11:19,875 You can't do real functions. 327 01:11:20,458 --> 01:11:22,875 But for any countable space, and of 328 01:11:22,875 --> 01:11:23,666 course, that's another 329 01:11:23,666 --> 01:11:26,041 thing, how hard would it be 330 01:11:26,041 --> 01:11:28,375 to just explain the different sizes of 331 01:11:28,375 --> 01:11:30,291 infinity to all of our students? 332 01:11:30,541 --> 01:11:31,291 I love having that 333 01:11:31,291 --> 01:11:32,458 conversation with students. 334 01:11:32,708 --> 01:11:34,250 They feel so enlightened, so 335 01:11:34,250 --> 01:11:35,500 philosophical, like I'm 336 01:11:35,500 --> 01:11:36,541 thinking about infinity. 337 01:11:36,916 --> 01:11:38,250 But it's interesting as well. 338 01:11:38,791 --> 01:11:39,000 Yeah. 339 01:11:39,166 --> 01:11:42,416 And it goes back to the 1850s, where they 340 01:11:42,416 --> 01:11:43,500 figured all this stuff. 341 01:11:44,250 --> 01:11:48,000 You know, Cantor wrote his book, which 342 01:11:48,000 --> 01:11:49,583 was called The Laws of 343 01:11:49,583 --> 01:11:51,500 Thought, because we're 344 01:11:51,500 --> 01:11:52,791 stuck here as humans. 345 01:11:53,375 --> 01:11:54,791 This is another one of my pet peeves. 346 01:11:55,208 --> 01:11:57,000 Psychologists get all flipped out when 347 01:11:57,000 --> 01:11:58,583 they say things like, 348 01:11:59,000 --> 01:12:00,583 "Don't anthropomorphize." 349 01:12:01,166 --> 01:12:03,000 And of course, what does that word mean? 350 01:12:03,458 --> 01:12:05,916 It means don't ascribe human 351 01:12:05,916 --> 01:12:09,375 characteristics to inanimate objects or 352 01:12:09,375 --> 01:12:10,333 animals, because they're 353 01:12:10,333 --> 01:12:10,916 not humans. 354 01:12:11,958 --> 01:12:12,500 I'm sorry. 355 01:12:12,750 --> 01:12:14,291 When you take physics, you learn about 356 01:12:14,291 --> 01:12:16,333 how the electron feels the proton. 357 01:12:16,666 --> 01:12:16,875 Right. 358 01:12:17,166 --> 01:12:19,083 And how the system wants to reach its 359 01:12:19,083 --> 01:12:20,291 lowest free energy state. 360 01:12:20,875 --> 01:12:22,958 And yeah, it's just how people think. 361 01:12:23,375 --> 01:12:24,166 Well, we're human. 362 01:12:24,833 --> 01:12:26,625 And it's how logic is. 363 01:12:26,875 --> 01:12:27,958 Logic is all about 364 01:12:27,958 --> 01:12:29,208 how we think as humans. 365 01:12:29,625 --> 01:12:30,125 And we're kind of 366 01:12:30,125 --> 01:12:31,708 stuck, because we're humans. 367 01:12:31,708 --> 01:12:32,458 We're not dogs. 368 01:12:33,291 --> 01:12:34,708 We're not rocks. 369 01:12:35,416 --> 01:12:37,083 And so there's only one way for us to 370 01:12:37,083 --> 01:12:37,958 think, and that's the 371 01:12:37,958 --> 01:12:39,666 anthropomorphize everything. 372 01:12:40,708 --> 01:12:42,458 But you should be 373 01:12:42,458 --> 01:12:43,958 aware of what you're doing. 374 01:12:44,500 --> 01:12:46,750 That's very important, because we know 375 01:12:46,750 --> 01:12:47,500 that just because we 376 01:12:47,500 --> 01:12:48,708 think that way doesn't mean 377 01:12:48,708 --> 01:12:49,541 that other things are 378 01:12:49,541 --> 01:12:50,500 going to think that way. 379 01:12:51,583 --> 01:12:53,791 But that said, The Laws of Thought was 380 01:12:53,791 --> 01:12:55,041 the first book on logic. 381 01:12:55,875 --> 01:12:59,250 And Cantor introduced his ordinal numbers 382 01:12:59,250 --> 01:13:00,000 and his cardinal 383 01:13:00,000 --> 01:13:01,333 numbers and all that, the 384 01:13:01,333 --> 01:13:02,541 basis for number theory. 385 01:13:03,166 --> 01:13:05,541 But it's such an important thing, because 386 01:13:05,541 --> 01:13:07,166 ordinary differential 387 01:13:07,166 --> 01:13:08,708 equations are easy to 388 01:13:08,750 --> 01:13:11,166 solve, because they, the 389 01:13:11,166 --> 01:13:12,833 solutions are countable. 390 01:13:13,750 --> 01:13:15,958 Whereas partial differential equations 391 01:13:15,958 --> 01:13:17,500 are really difficult to 392 01:13:17,500 --> 01:13:19,583 solve, because the solution 393 01:13:19,750 --> 01:13:21,708 space is uncountable. 394 01:13:23,166 --> 01:13:25,250 And it goes right back to what you said, 395 01:13:26,000 --> 01:13:27,500 the eigenvalue problem. 396 01:13:28,166 --> 01:13:29,958 What are those? 397 01:13:29,958 --> 01:13:31,000 They're really just the 398 01:13:31,000 --> 01:13:32,458 weights on the basis functions. 399 01:13:33,416 --> 01:13:36,083 And the way that you solve a partial 400 01:13:36,083 --> 01:13:38,041 differential equation is 401 01:13:38,041 --> 01:13:39,583 you map out the characteristic 402 01:13:39,750 --> 01:13:40,333 lines. 403 01:13:40,791 --> 01:13:42,375 And then every point of the 404 01:13:42,375 --> 01:13:44,291 characteristic line contains 405 01:13:44,291 --> 01:13:46,583 a new set of solutions that 406 01:13:46,583 --> 01:13:49,208 lie on a cone called a mange cone. 407 01:13:49,875 --> 01:13:51,875 And of course, there's a uncountable 408 01:13:51,875 --> 01:13:53,083 number of those points, 409 01:13:53,083 --> 01:13:55,000 and the cones sweep out. 410 01:13:55,416 --> 01:13:57,000 So that's how you do it. 411 01:13:57,000 --> 01:13:57,875 And that's how people 412 01:13:57,875 --> 01:13:59,666 solve these problems. 413 01:14:00,291 --> 01:14:02,875 But it all goes back to very fundamental 414 01:14:02,875 --> 01:14:03,875 things, and you need 415 01:14:03,875 --> 01:14:05,083 linear algebra for that. 416 01:14:05,875 --> 01:14:08,833 So how can we ignore linear algebra? 417 01:14:10,125 --> 01:14:12,041 We deal with tensors. 418 01:14:13,125 --> 01:14:13,958 What are tensors? 419 01:14:13,958 --> 01:14:14,791 They're matrices. 420 01:14:15,041 --> 01:14:16,458 You've got to diagonalize stuff. 421 01:14:16,458 --> 01:14:16,958 You've got to look at 422 01:14:16,958 --> 01:14:19,041 all the off-axis elements. 423 01:14:19,041 --> 01:14:19,250 Right. 424 01:14:19,250 --> 01:14:19,833 Think of it as a 425 01:14:19,833 --> 01:14:21,500 transformation acting on a space. 426 01:14:22,166 --> 01:14:22,583 Precisely. 427 01:14:23,250 --> 01:14:24,958 So a little bit of 428 01:14:24,958 --> 01:14:26,875 understanding, not with proofs. 429 01:14:26,916 --> 01:14:27,416 You don't have to sit 430 01:14:27,416 --> 01:14:28,375 there and prove it all. 431 01:14:28,875 --> 01:14:32,666 Just explain that functions represent a 432 01:14:32,666 --> 01:14:34,416 vector space, that you have 433 01:14:34,416 --> 01:14:36,583 these basis sets of functions 434 01:14:36,583 --> 01:14:39,500 that you use linear sums of. 435 01:14:40,208 --> 01:14:42,416 And you tell them, we can prove this if 436 01:14:42,416 --> 01:14:44,125 you really want, but you 437 01:14:44,125 --> 01:14:46,291 can approximate any real 438 01:14:46,291 --> 01:14:49,541 function arbitrarily closely if you take 439 01:14:49,541 --> 01:14:51,208 enough terms in your series. 440 01:14:52,291 --> 01:14:53,583 So brings it back to you. 441 01:14:53,583 --> 01:14:54,958 You've got to understand what series are. 442 01:14:55,583 --> 01:14:58,250 You've got to understand vector space is 443 01:14:58,250 --> 01:14:59,208 more than just three 444 01:14:59,208 --> 01:15:00,791 space and how functions 445 01:15:00,791 --> 01:15:02,000 can be a vector space. 446 01:15:02,583 --> 01:15:04,250 And then you need the mechanisms of 447 01:15:04,250 --> 01:15:05,666 linear algebra to actually 448 01:15:05,666 --> 01:15:07,583 be able to do computations 449 01:15:07,583 --> 01:15:08,916 in all of these things. 450 01:15:10,166 --> 01:15:12,916 So with that little bit of 451 01:15:12,916 --> 01:15:16,291 information, it can go so long. 452 01:15:16,291 --> 01:15:17,416 So I totally agree with you. 453 01:15:17,958 --> 01:15:18,500 Second thing. 454 01:15:19,208 --> 01:15:20,541 And my next thing is 455 01:15:20,541 --> 01:15:21,583 kind of related to that. 456 01:15:23,000 --> 01:15:25,750 You tell students that the solutions to 457 01:15:25,750 --> 01:15:26,458 these are Bessel 458 01:15:26,458 --> 01:15:28,166 functions and you write them down 459 01:15:28,791 --> 01:15:29,416 and they freak 460 01:15:29,416 --> 01:15:31,000 because they're really ugly. 461 01:15:32,083 --> 01:15:34,250 And unless you understand that a Bessel 462 01:15:34,250 --> 01:15:36,250 function is just a basis 463 01:15:36,250 --> 01:15:38,708 set that makes life easier. 464 01:15:39,000 --> 01:15:39,375 It's a convenient 465 01:15:39,375 --> 01:15:40,666 collection of functions. 466 01:15:41,000 --> 01:15:41,958 That work in circular 467 01:15:41,958 --> 01:15:43,916 boundary condition areas. 468 01:15:44,541 --> 01:15:47,291 And the same is true for any orthogonal 469 01:15:47,291 --> 01:15:49,083 series, any orthogonal 470 01:15:49,083 --> 01:15:51,250 series, Fourier series, Legendre 471 01:15:51,291 --> 01:15:54,416 polynomials, you know, Bessel functions, 472 01:15:54,666 --> 01:15:56,416 Hermit functions, they're all same. 473 01:15:57,125 --> 01:16:00,000 They're all exactly the same from a 474 01:16:00,000 --> 01:16:02,750 simplistic view of 475 01:16:02,750 --> 01:16:04,625 vector spaces and bases. 476 01:16:05,666 --> 01:16:07,000 Why don't we teach it like that? 477 01:16:07,541 --> 01:16:08,333 I wish we did. 478 01:16:08,916 --> 01:16:10,916 Well, if that's our finish line, then the 479 01:16:10,916 --> 01:16:11,833 question becomes, how 480 01:16:11,833 --> 01:16:12,875 do we get students here? 481 01:16:13,208 --> 01:16:15,083 What can we do from a 482 01:16:15,083 --> 01:16:16,125 curricular point of view? 483 01:16:16,208 --> 01:16:17,583 What can we do in the classroom to 484 01:16:17,583 --> 01:16:19,458 achieve that goal so that 485 01:16:19,458 --> 01:16:21,375 we can help our students 486 01:16:21,375 --> 01:16:23,625 think in this more sophisticated, but 487 01:16:23,625 --> 01:16:24,750 also simpler and more 488 01:16:24,750 --> 01:16:27,416 effective way about the mathematical 489 01:16:27,541 --> 01:16:28,458 problems they're dealing 490 01:16:28,458 --> 01:16:30,166 with in their MSE courses? 491 01:16:31,250 --> 01:16:32,250 So I think there's two approaches. 492 01:16:33,416 --> 01:16:35,583 One is the simple, 493 01:16:35,583 --> 01:16:37,041 but really stupid idea. 494 01:16:37,458 --> 01:16:39,416 Just make them take more math courses. 495 01:16:40,041 --> 01:16:42,583 Again, why academics think the solution 496 01:16:42,583 --> 01:16:43,666 to any problem is... 497 01:16:43,875 --> 01:16:44,500 Is classes and lectures. 498 01:16:46,416 --> 01:16:48,583 And, you know, yeah, that might work, but 499 01:16:48,583 --> 01:16:49,458 it would take forever. 500 01:16:49,916 --> 01:16:51,333 And we don't really have room in the 501 01:16:51,333 --> 01:16:52,500 curriculum to do that. 502 01:16:53,208 --> 01:16:56,625 So we need to be more thoughtful and more 503 01:16:56,625 --> 01:16:58,833 innovative, I think, than that. 504 01:16:59,125 --> 01:17:03,375 And so we've talked a lot about this, but 505 01:17:03,375 --> 01:17:04,000 Tim, why don't you 506 01:17:04,000 --> 01:17:06,333 talk about what you think 507 01:17:06,333 --> 01:17:07,666 might be a solution to this? 508 01:17:08,000 --> 01:17:11,500 Well, first I should unpack the genius 509 01:17:11,500 --> 01:17:12,833 idea of the more courses, 510 01:17:13,250 --> 01:17:15,166 because even though throwing 511 01:17:15,250 --> 01:17:18,625 classes at a problem is exactly, as you 512 01:17:18,625 --> 01:17:20,000 said, Steve, the academic way. 513 01:17:22,375 --> 01:17:25,500 In our case, the way this manifested a 514 01:17:25,500 --> 01:17:26,583 couple years ago at Michigan 515 01:17:26,583 --> 01:17:28,625 was having this conversation 516 01:17:28,750 --> 01:17:31,208 that you, dear audience, just heard, what 517 01:17:31,208 --> 01:17:32,125 are our students missing? 518 01:17:32,375 --> 01:17:35,000 And then finding the courses where those 519 01:17:35,000 --> 01:17:36,833 topics are taught around campus. 520 01:17:37,125 --> 01:17:38,625 Many of them math, but some of them in 521 01:17:38,625 --> 01:17:40,791 stats, some even in science departments. 522 01:17:42,125 --> 01:17:45,416 And we used one of our elective course 523 01:17:45,416 --> 01:17:47,375 slots to require a 524 01:17:47,375 --> 01:17:48,458 fifth math class out of this 525 01:17:48,458 --> 01:17:50,250 list so that students would have a little 526 01:17:50,250 --> 01:17:51,625 bit of agency and have an 527 01:17:51,625 --> 01:17:53,083 opportunity to specialize 528 01:17:53,625 --> 01:17:55,666 more in an area that they found valuable 529 01:17:55,666 --> 01:17:56,791 to themselves, be it 530 01:17:56,791 --> 01:17:58,041 statistics, linear algebra, 531 01:17:58,041 --> 01:17:58,375 or whatever. 532 01:17:59,500 --> 01:18:02,000 And this idea was great on paper. 533 01:18:03,000 --> 01:18:05,125 The factor that we neglected to account 534 01:18:05,125 --> 01:18:05,958 for is that our students 535 01:18:05,958 --> 01:18:08,208 are smart and they understand 536 01:18:08,250 --> 01:18:10,666 how the school works very well. 537 01:18:11,958 --> 01:18:12,791 And so they all took their 538 01:18:12,791 --> 01:18:14,333 extra math class senior year. 539 01:18:15,000 --> 01:18:16,333 Long after the point at which it would 540 01:18:16,333 --> 01:18:17,375 have actually done them 541 01:18:17,375 --> 01:18:18,750 some good, I'm very happy 542 01:18:18,750 --> 01:18:20,583 for their future employers that they're 543 01:18:20,583 --> 01:18:22,291 getting a more complete 544 01:18:22,291 --> 01:18:24,041 product, but it did not do 545 01:18:24,041 --> 01:18:26,166 any good for solving the preparation 546 01:18:26,166 --> 01:18:27,666 issue of students being 547 01:18:27,666 --> 01:18:29,916 ready to succeed in advanced 548 01:18:29,958 --> 01:18:30,916 MSC coursework. 549 01:18:31,458 --> 01:18:34,083 So we're going to chalk that up as a 550 01:18:34,083 --> 01:18:35,333 noble experiment that 551 01:18:35,333 --> 01:18:36,375 didn't quite pan out. 552 01:18:37,833 --> 01:18:38,208 That's right. 553 01:18:39,833 --> 01:18:43,666 In addition to this, I think the student 554 01:18:43,666 --> 01:18:44,625 in the median of the 555 01:18:44,625 --> 01:18:46,791 distribution needs more 556 01:18:47,000 --> 01:18:51,000 math to be able to do our courses in 557 01:18:51,000 --> 01:18:53,208 mechanics, our courses in electronic 558 01:18:53,208 --> 01:18:55,500 materials, in thermo, 559 01:18:55,916 --> 01:18:58,500 in our, you know, partial differential 560 01:18:58,500 --> 01:19:00,416 equations course, which is transport. 561 01:19:02,041 --> 01:19:04,833 We also need to think about the students 562 01:19:04,833 --> 01:19:07,916 who are even more poorly prepared. 563 01:19:08,541 --> 01:19:10,833 How do we help meet them where they are 564 01:19:10,833 --> 01:19:12,500 and bring them forward 565 01:19:12,500 --> 01:19:15,375 so that they can succeed? 566 01:19:16,416 --> 01:19:18,416 And there's a lot of pedagogical tricks 567 01:19:18,416 --> 01:19:19,625 to do that, like 568 01:19:19,625 --> 01:19:21,875 team-based teaching, but you 569 01:19:21,875 --> 01:19:24,125 need to do it at the right time and you 570 01:19:24,125 --> 01:19:25,791 need to make sure that 571 01:19:25,791 --> 01:19:28,083 you're not trying to overcome 572 01:19:28,083 --> 01:19:29,458 a massive gap. 573 01:19:30,208 --> 01:19:32,666 So instead of waiting till their senior 574 01:19:32,666 --> 01:19:34,625 year, this is something 575 01:19:34,625 --> 01:19:35,875 we need to do as soon as 576 01:19:35,916 --> 01:19:37,750 they declare as 577 01:19:37,750 --> 01:19:39,208 students in our department. 578 01:19:39,958 --> 01:19:41,333 So we catch them early. 579 01:19:41,916 --> 01:19:44,500 We can certainly add more complex math 580 01:19:44,500 --> 01:19:47,333 topics when we get to 581 01:19:47,333 --> 01:19:49,583 thermo, when we get to kinetics, 582 01:19:49,875 --> 01:19:50,750 but we really need to 583 01:19:50,750 --> 01:19:51,875 bring them up to speed. 584 01:19:52,083 --> 01:19:55,708 So I think it speaks to the point that 585 01:19:55,708 --> 01:19:57,541 when we design a system 586 01:19:57,541 --> 01:20:00,458 to help our students all 587 01:20:00,625 --> 01:20:04,708 get to a certain level of competency for 588 01:20:04,708 --> 01:20:06,791 skills, ready to move 589 01:20:06,791 --> 01:20:07,875 on to the next step, 590 01:20:08,208 --> 01:20:09,333 we need to do that early. 591 01:20:10,125 --> 01:20:11,958 So I think we're going to get rid of our 592 01:20:11,958 --> 01:20:13,750 fifth math course and 593 01:20:13,750 --> 01:20:15,875 instead require a sophomore 594 01:20:15,958 --> 01:20:17,625 level math course. 595 01:20:19,375 --> 01:20:21,791 Before we get to that, I had an idea for 596 01:20:21,791 --> 01:20:23,416 something that could be 597 01:20:23,416 --> 01:20:24,708 its own episode, but briefly 598 01:20:24,875 --> 01:20:26,291 here as you're talking about 599 01:20:26,291 --> 01:20:29,000 interventions and what can we do with the 600 01:20:29,000 --> 01:20:31,250 students pedagogically 601 01:20:31,250 --> 01:20:35,583 socially to help those who need some 602 01:20:35,583 --> 01:20:37,333 extra support in the math area. 603 01:20:37,833 --> 01:20:40,000 I know you've been working on some 604 01:20:40,000 --> 01:20:41,500 tutoring programs lately 605 01:20:41,500 --> 01:20:43,750 and being very intentional 606 01:20:43,750 --> 01:20:44,666 about how those peer 607 01:20:44,666 --> 01:20:46,125 tutoring groups are set up. 608 01:20:46,125 --> 01:20:47,916 Is that something that you could talk 609 01:20:47,916 --> 01:20:48,666 about for a couple of 610 01:20:48,666 --> 01:20:49,416 minutes and how that might 611 01:20:49,416 --> 01:20:50,625 help in the math area? 612 01:20:51,583 --> 01:20:51,833 Sure. 613 01:20:52,083 --> 01:20:55,083 This is something that's come to my 614 01:20:55,083 --> 01:20:57,041 attention as the program advisor. 615 01:20:58,458 --> 01:21:01,666 Many of our, right now we tell our 616 01:21:01,666 --> 01:21:03,833 students, "Get in a study group. 617 01:21:03,833 --> 01:21:05,000 It's really valuable." 618 01:21:06,000 --> 01:21:07,291 But we don't do anything 619 01:21:07,291 --> 01:21:08,833 to create the study groups. 620 01:21:09,208 --> 01:21:10,833 We leave that up to the students. 621 01:21:11,666 --> 01:21:12,791 And what happens is 622 01:21:12,791 --> 01:21:14,750 something that's not very inclusive. 623 01:21:15,750 --> 01:21:17,166 So a lot of our students, like our 624 01:21:17,166 --> 01:21:19,333 transfer students, students with 625 01:21:19,333 --> 01:21:21,208 disabilities, underrepresented 626 01:21:21,250 --> 01:21:22,458 minority students, 627 01:21:23,291 --> 01:21:26,333 they're not in the club. 628 01:21:26,666 --> 01:21:28,666 They're not in this group of students 629 01:21:28,666 --> 01:21:31,083 that naturally wants to 630 01:21:31,083 --> 01:21:33,166 get together to work out 631 01:21:33,166 --> 01:21:34,458 problems and help each other. 632 01:21:34,750 --> 01:21:35,833 So they get excluded. 633 01:21:36,875 --> 01:21:38,125 And I found that out when I was talking 634 01:21:38,125 --> 01:21:39,083 to a student and I 635 01:21:39,083 --> 01:21:40,208 said, "Why don't you join 636 01:21:40,208 --> 01:21:41,000 a study group?" 637 01:21:41,416 --> 01:21:42,541 And I was told no one will 638 01:21:42,541 --> 01:21:43,791 let me in their study group. 639 01:21:44,750 --> 01:21:45,916 And that just broke my heart. 640 01:21:46,958 --> 01:21:49,000 And so then I started looking around at 641 01:21:49,000 --> 01:21:50,208 what other people did. 642 01:21:50,208 --> 01:21:55,208 And I found out in our literature science 643 01:21:55,208 --> 01:21:56,375 and arts college, 644 01:21:56,375 --> 01:21:58,958 this is where physics and 645 01:21:58,958 --> 01:22:01,500 math and chemistry, biology are, they 646 01:22:01,500 --> 01:22:02,875 have a science learning center. 647 01:22:03,333 --> 01:22:04,416 And they've created 648 01:22:04,416 --> 01:22:08,708 facilitated study groups. 649 01:22:09,291 --> 01:22:11,791 You just go to a website and you choose a 650 01:22:11,791 --> 01:22:13,083 time you can make it. 651 01:22:13,458 --> 01:22:15,083 You have no idea who's going to be in 652 01:22:15,083 --> 01:22:16,583 that group, but it's all 653 01:22:16,583 --> 01:22:18,166 groups focused on a particular 654 01:22:18,166 --> 01:22:20,416 course and sometimes even a particular 655 01:22:20,416 --> 01:22:21,583 section of that course. 656 01:22:22,083 --> 01:22:23,000 And you just sign up. 657 01:22:23,708 --> 01:22:26,541 And then, you know, on Wednesdays from 658 01:22:26,541 --> 01:22:27,791 eight to nine o'clock at 659 01:22:27,791 --> 01:22:29,250 night, you get on a Zoom 660 01:22:29,250 --> 01:22:29,708 meeting. 661 01:22:30,250 --> 01:22:33,000 But what they do, which is awesome, they 662 01:22:33,000 --> 01:22:35,083 hire a student who got a 663 01:22:35,083 --> 01:22:36,750 B plus or better in the 664 01:22:36,750 --> 01:22:37,041 course. 665 01:22:38,375 --> 01:22:41,166 Eighteen bucks an hour to be on every 666 01:22:41,166 --> 01:22:42,875 single Zoom call to 667 01:22:42,875 --> 01:22:45,416 facilitate the conversation. 668 01:22:46,125 --> 01:22:49,875 It's not quite tutoring, but these people 669 01:22:49,875 --> 01:22:51,291 help all the students 670 01:22:51,291 --> 01:22:52,750 do their homework, help 671 01:22:52,750 --> 01:22:53,791 them understand things 672 01:22:53,791 --> 01:22:54,750 they don't understand. 673 01:22:55,500 --> 01:22:57,583 But what I love about it is how inclusive 674 01:22:57,583 --> 01:22:59,333 it is because anyone 675 01:22:59,333 --> 01:23:00,416 who wants to join can 676 01:23:00,458 --> 01:23:00,833 join. 677 01:23:00,833 --> 01:23:02,541 There's no barrier to joining. 678 01:23:03,041 --> 01:23:04,916 There's no social capital involved about 679 01:23:04,916 --> 01:23:05,916 who you know and who 680 01:23:05,916 --> 01:23:07,500 you don't and what you 681 01:23:07,500 --> 01:23:08,250 look like and whether 682 01:23:08,250 --> 01:23:09,250 you're the cool kid. 683 01:23:10,000 --> 01:23:11,500 All it is is a common need 684 01:23:11,500 --> 01:23:13,458 and schedules that line up. 685 01:23:14,166 --> 01:23:15,791 And then what they do, which I think is 686 01:23:15,791 --> 01:23:17,750 also genius, you worry 687 01:23:17,750 --> 01:23:20,125 that, well, you're the people 688 01:23:20,125 --> 01:23:22,000 who still want to form their own study 689 01:23:22,000 --> 01:23:23,541 groups will just ignore this. 690 01:23:24,333 --> 01:23:26,083 They might, but once they hear that 691 01:23:26,083 --> 01:23:27,250 there's a facilitator 692 01:23:27,250 --> 01:23:28,833 there who's going to help them, 693 01:23:29,416 --> 01:23:30,750 I think it's going to draw 694 01:23:30,750 --> 01:23:32,750 everybody into this process. 695 01:23:33,750 --> 01:23:36,458 So anyone can really start this. 696 01:23:36,458 --> 01:23:37,791 You don't need fancy software. 697 01:23:38,000 --> 01:23:41,250 You can use signup genius for free and 698 01:23:41,250 --> 01:23:44,000 cap the number of people for a group. 699 01:23:45,125 --> 01:23:46,875 And so we're going to actually try to 700 01:23:46,875 --> 01:23:48,666 pilot it this term with 701 01:23:48,666 --> 01:23:50,416 our kinetics course and 702 01:23:50,416 --> 01:23:51,291 see how it goes. 703 01:23:51,666 --> 01:23:54,875 But it's a way to help students find 704 01:23:54,875 --> 01:23:56,083 study groups, at least 705 01:23:56,083 --> 01:23:57,458 for the upper level courses. 706 01:23:58,250 --> 01:23:59,333 And hopefully the college 707 01:23:59,333 --> 01:24:00,708 will see this experiment. 708 01:24:00,708 --> 01:24:02,750 It's funny, I just saw our associate dean 709 01:24:02,750 --> 01:24:03,875 told him about this 710 01:24:03,875 --> 01:24:05,000 and he loves this idea. 711 01:24:05,666 --> 01:24:07,083 So they're going to be watching what we 712 01:24:07,083 --> 01:24:08,000 do and we'd probably do 713 01:24:08,000 --> 01:24:10,250 it through our academic, 714 01:24:10,250 --> 01:24:12,083 I forgot the name, we have something like 715 01:24:12,083 --> 01:24:13,291 a science learning center. 716 01:24:13,916 --> 01:24:15,666 So they're actually very interested in 717 01:24:15,666 --> 01:24:16,750 maybe rolling this out 718 01:24:16,750 --> 01:24:19,208 for our lower level courses 719 01:24:19,208 --> 01:24:20,166 for the whole college. 720 01:24:21,000 --> 01:24:22,541 So I think it's going 721 01:24:22,541 --> 01:24:23,583 to be a good experiment. 722 01:24:24,250 --> 01:24:26,500 And obviously this math course is a 723 01:24:26,500 --> 01:24:28,166 perfect place to 724 01:24:28,166 --> 01:24:29,708 institute something like this. 725 01:24:30,041 --> 01:24:31,583 Yeah, I think that could translate really 726 01:24:31,583 --> 01:24:33,083 well to the sophomore 727 01:24:33,083 --> 01:24:35,000 level, especially imagine 728 01:24:35,000 --> 01:24:37,083 a student has just declared their major, 729 01:24:37,083 --> 01:24:38,000 I'm brand new to MSE. 730 01:24:38,583 --> 01:24:39,291 I don't know anyone. 731 01:24:39,708 --> 01:24:40,625 I don't know anything. 732 01:24:41,083 --> 01:24:42,041 Where do I even start? 733 01:24:42,416 --> 01:24:44,666 And to create this guided experience of 734 01:24:44,666 --> 01:24:45,666 just tell us when 735 01:24:45,666 --> 01:24:46,791 you're free and we'll handle 736 01:24:46,791 --> 01:24:47,291 the rest. 737 01:24:47,833 --> 01:24:50,291 I think that can be a really great method 738 01:24:50,291 --> 01:24:51,375 to reduce that barrier 739 01:24:51,375 --> 01:24:52,583 to entry and get them 740 01:24:52,625 --> 01:24:55,541 meeting their peers in an effective and 741 01:24:55,541 --> 01:24:57,416 helpful social environment right away. 742 01:24:58,041 --> 01:24:58,333 That's right. 743 01:24:58,541 --> 01:25:01,375 And of course, this speaks to something I 744 01:25:01,375 --> 01:25:02,541 know you believe and I 745 01:25:02,541 --> 01:25:04,958 believe the idea that 746 01:25:05,333 --> 01:25:08,000 the best people to learn from are other 747 01:25:08,000 --> 01:25:09,125 students who just learned 748 01:25:09,125 --> 01:25:10,541 it or learned it within a 749 01:25:10,541 --> 01:25:12,875 year because nothing is obvious to them. 750 01:25:13,291 --> 01:25:14,333 Whereas we're like some of 751 01:25:14,333 --> 01:25:15,416 the worst people to teach. 752 01:25:15,625 --> 01:25:15,958 Right. 753 01:25:15,958 --> 01:25:17,041 Because to us, it's like, what do you 754 01:25:17,041 --> 01:25:18,125 mean I don't understand that? 755 01:25:18,125 --> 01:25:18,541 It's obvious. 756 01:25:19,083 --> 01:25:19,166 Yeah. 757 01:25:19,166 --> 01:25:20,541 I've known that for 30 years. 758 01:25:21,125 --> 01:25:22,208 Everyone know that at this point. 759 01:25:22,625 --> 01:25:22,833 Yeah. 760 01:25:23,208 --> 01:25:23,416 Right. 761 01:25:23,833 --> 01:25:27,750 And, um, but it also is really valuable 762 01:25:27,750 --> 01:25:28,875 even to the better 763 01:25:28,875 --> 01:25:29,958 students who are going to be 764 01:25:29,958 --> 01:25:32,291 helping the students who aren't as 765 01:25:32,291 --> 01:25:34,458 advanced because they'll get to teach it. 766 01:25:35,041 --> 01:25:38,208 And we all know we learn much better when 767 01:25:38,208 --> 01:25:39,583 we teach the material. 768 01:25:40,125 --> 01:25:41,333 That's how you really learn something. 769 01:25:41,958 --> 01:25:45,125 So it's just a win-win and, uh, can be 770 01:25:45,125 --> 01:25:46,250 done in parallel with a 771 01:25:46,250 --> 01:25:48,000 regular course and should 772 01:25:48,083 --> 01:25:51,333 ultimately save the student time because 773 01:25:51,333 --> 01:25:52,500 it's a very efficient 774 01:25:52,500 --> 01:25:54,250 process to figure out 775 01:25:54,250 --> 01:25:55,208 what you're doing wrong. 776 01:25:57,541 --> 01:26:01,541 So I think we've come upon this idea that 777 01:26:01,541 --> 01:26:02,666 need for this course. 778 01:26:02,958 --> 01:26:05,291 And, and I guess, uh, we should come 779 01:26:05,291 --> 01:26:07,833 clean and say, uh, uh, 780 01:26:07,833 --> 01:26:09,333 Tim here has actually been 781 01:26:09,333 --> 01:26:12,333 charged with, uh, teaching a course just 782 01:26:12,333 --> 01:26:13,833 like this next fall. 783 01:26:14,166 --> 01:26:15,000 For better or worse, 784 01:26:15,000 --> 01:26:15,708 it's going to happen. 785 01:26:16,041 --> 01:26:16,250 Yes. 786 01:26:17,000 --> 01:26:18,458 And I've been thinking about this a lot. 787 01:26:19,333 --> 01:26:21,208 So, um, Tim, why don't you talk about 788 01:26:21,208 --> 01:26:22,500 your process of how 789 01:26:22,500 --> 01:26:24,458 you're approaching this and 790 01:26:24,458 --> 01:26:27,416 how you would, what the constraints and 791 01:26:27,416 --> 01:26:29,250 what the opportunities might be. 792 01:26:29,500 --> 01:26:29,625 Yeah. 793 01:26:30,666 --> 01:26:33,291 My approach to a curriculum design 794 01:26:33,291 --> 01:26:36,041 problem is that word design. 795 01:26:36,583 --> 01:26:38,000 This is an engineering problem. 796 01:26:38,625 --> 01:26:40,416 So you have to use an engineering toolkit 797 01:26:40,416 --> 01:26:42,125 to solve an engineering problem. 798 01:26:42,791 --> 01:26:44,500 Step one, define the problem. 799 01:26:44,500 --> 01:26:45,583 Step two. 800 01:26:46,291 --> 01:26:47,666 And then everything goes from there. 801 01:26:48,125 --> 01:26:49,000 Who are the stakeholders 802 01:26:49,000 --> 01:26:50,583 who's involved in this? 803 01:26:50,583 --> 01:26:51,750 What are the different needs of the 804 01:26:51,750 --> 01:26:53,166 different populations involved? 805 01:26:53,166 --> 01:26:54,625 What does success look like? 806 01:26:54,625 --> 01:26:55,375 What's a minimum 807 01:26:55,375 --> 01:26:56,916 viable product look like? 808 01:26:57,291 --> 01:27:00,000 So in my mind, this is completely an 809 01:27:00,000 --> 01:27:01,125 engineering problem of 810 01:27:01,125 --> 01:27:05,500 creating this solution to a need 811 01:27:05,500 --> 01:27:06,875 that's been identified. 812 01:27:08,041 --> 01:27:10,125 And so a lot of this at this point is 813 01:27:10,125 --> 01:27:11,833 having the conversations 814 01:27:11,833 --> 01:27:13,041 with all the people involved, 815 01:27:13,375 --> 01:27:15,541 talking to faculty and saying, what do 816 01:27:15,541 --> 01:27:16,583 you wish your students 817 01:27:16,583 --> 01:27:18,083 were better prepared for 818 01:27:18,083 --> 01:27:19,291 when they get to your course? 819 01:27:20,125 --> 01:27:22,000 And some of it is talking to students and 820 01:27:22,000 --> 01:27:23,000 saying, what do you 821 01:27:23,000 --> 01:27:23,750 feel like you're struggling 822 01:27:23,750 --> 01:27:25,750 with this semester that you wish you had 823 01:27:25,750 --> 01:27:27,250 additional help or 824 01:27:27,250 --> 01:27:28,583 additional instruction in? 825 01:27:29,000 --> 01:27:32,875 Some of it is even consulting friends 826 01:27:32,875 --> 01:27:33,958 that I have an industry 827 01:27:33,958 --> 01:27:35,625 and saying, what's the 828 01:27:35,625 --> 01:27:37,666 math that you wish you actually knew when 829 01:27:37,666 --> 01:27:39,250 you graduated that you never got? 830 01:27:40,083 --> 01:27:40,875 And so we start to 831 01:27:40,875 --> 01:27:42,875 define these sets of topics. 832 01:27:43,791 --> 01:27:45,500 And I'm hoping that as these 833 01:27:45,500 --> 01:27:46,833 conversations continue, that 834 01:27:46,833 --> 01:27:48,291 it'll turn into a Venn diagram 835 01:27:48,500 --> 01:27:50,083 and there will be some sweet spot, some 836 01:27:50,083 --> 01:27:52,166 overlap of what different people want. 837 01:27:52,708 --> 01:27:53,458 And then that can really 838 01:27:53,458 --> 01:27:54,958 start to define course topics. 839 01:27:57,000 --> 01:28:01,666 Beyond that, there is also just the 840 01:28:01,666 --> 01:28:04,916 personal aspect of what do 841 01:28:04,916 --> 01:28:07,541 I, the instructor believe 842 01:28:07,583 --> 01:28:07,875 in? 843 01:28:07,875 --> 01:28:09,000 What can I teach effectively? 844 01:28:09,583 --> 01:28:10,500 What can I teach in a way that I think is 845 01:28:10,500 --> 01:28:11,500 meaningful and useful? 846 01:28:12,708 --> 01:28:16,583 And then trying it, throw the pasta at 847 01:28:16,583 --> 01:28:17,916 the wall, see what sticks. 848 01:28:17,916 --> 01:28:20,250 That's always the last step in a new 849 01:28:20,250 --> 01:28:21,541 course is just giving it 850 01:28:21,541 --> 01:28:22,791 your best and then finding 851 01:28:22,791 --> 01:28:24,916 out what you were right about and what 852 01:28:24,916 --> 01:28:25,833 you were wrong about. 853 01:28:25,833 --> 01:28:27,333 And then you iterate because it's an 854 01:28:27,333 --> 01:28:28,416 engineering problem and that 855 01:28:28,416 --> 01:28:29,791 requires iteration as well. 856 01:28:31,750 --> 01:28:33,041 The other thing I think that's a real 857 01:28:33,041 --> 01:28:36,500 opportunity is that you're 858 01:28:36,500 --> 01:28:40,416 very aware and understand a 859 01:28:40,416 --> 01:28:41,875 lot of literature of 860 01:28:41,875 --> 01:28:43,541 evidence-based teaching. 861 01:28:44,291 --> 01:28:46,291 So I presume you're going to be putting 862 01:28:46,291 --> 01:28:49,708 in a massive amount of 863 01:28:49,708 --> 01:28:51,500 active learning, lots 864 01:28:51,583 --> 01:28:56,541 of feedback loops so that students get 865 01:28:56,541 --> 01:28:58,000 their reps in and 866 01:28:58,000 --> 01:29:00,333 actually do the work themselves, 867 01:29:01,541 --> 01:29:04,458 without fear of failing, so that they can 868 01:29:04,458 --> 01:29:05,750 actually learn the material. 869 01:29:06,166 --> 01:29:06,750 Yeah, absolutely. 870 01:29:07,083 --> 01:29:09,041 Something that I know a lot of students 871 01:29:09,041 --> 01:29:10,833 don't appreciate and I 872 01:29:10,833 --> 01:29:12,333 find also quite a few faculty 873 01:29:12,333 --> 01:29:15,500 don't appreciate is that you need to have 874 01:29:15,500 --> 01:29:17,833 the conceptual understanding of the math, 875 01:29:18,875 --> 01:29:21,458 but it's also just a skill. 876 01:29:22,000 --> 01:29:24,000 It's a practice that you develop through 877 01:29:24,000 --> 01:29:26,708 repetition, but that the 878 01:29:26,708 --> 01:29:27,750 goal for our students should 879 01:29:27,791 --> 01:29:31,041 be to achieve a level of automaticity 880 01:29:31,041 --> 01:29:32,750 with this skill, not to 881 01:29:32,750 --> 01:29:33,791 have to sit down and think 882 01:29:33,791 --> 01:29:35,333 for three hours about how to do this 883 01:29:35,333 --> 01:29:37,250 integral, but to sit down 884 01:29:37,250 --> 01:29:38,458 and have the integral solve 885 01:29:38,458 --> 01:29:40,166 itself while their pen just walks across 886 01:29:40,166 --> 01:29:41,041 the paper because 887 01:29:41,041 --> 01:29:42,958 they've done it before and to 888 01:29:42,958 --> 01:29:44,833 say, "Oh, I guess I didn't have to think 889 01:29:44,833 --> 01:29:45,958 so hard about that after 890 01:29:45,958 --> 01:29:47,416 all," because that frees 891 01:29:47,416 --> 01:29:49,583 up their cognitive resources for the 892 01:29:49,583 --> 01:29:50,416 stuff that we really 893 01:29:50,416 --> 01:29:51,791 care about, which is what's 894 01:29:51,791 --> 01:29:53,250 the physical meaning of this math? 895 01:29:53,666 --> 01:29:56,125 How do you use this to describe or 896 01:29:56,125 --> 01:29:56,833 predict or do 897 01:29:56,833 --> 01:29:58,000 something in the real world? 898 01:29:58,750 --> 01:30:00,041 If they're spending all of their 899 01:30:00,041 --> 01:30:02,625 brainpower saying, "Where 900 01:30:02,625 --> 01:30:03,916 do I put the x squared?" 901 01:30:04,916 --> 01:30:07,000 they don't have the bandwidth left over 902 01:30:07,000 --> 01:30:08,666 to think about these 903 01:30:08,666 --> 01:30:11,041 more scientific topics that 904 01:30:11,041 --> 01:30:14,083 we hope to get them into and how that 905 01:30:14,083 --> 01:30:15,416 transforms or how that 906 01:30:15,416 --> 01:30:17,416 translates into the pedagogical 907 01:30:17,416 --> 01:30:18,333 structure of the course. 908 01:30:19,625 --> 01:30:22,083 Steve's still having some ideas here, but 909 01:30:22,083 --> 01:30:23,083 my initial vision is 910 01:30:23,083 --> 01:30:23,833 something that's a lot 911 01:30:23,833 --> 01:30:25,875 more like a studio course where the 912 01:30:25,875 --> 01:30:27,666 students come in and it's 913 01:30:27,666 --> 01:30:29,375 more like these guided practice 914 01:30:29,541 --> 01:30:29,916 sessions. 915 01:30:30,166 --> 01:30:31,125 Here are some problems we're going to 916 01:30:31,125 --> 01:30:31,958 work through together, 917 01:30:32,583 --> 01:30:33,833 get in those reps, get in 918 01:30:33,833 --> 01:30:36,166 that practice, but also get real-time 919 01:30:36,166 --> 01:30:37,875 feedback exactly as you said, Steve. 920 01:30:38,166 --> 01:30:39,166 There has to be that, 921 01:30:39,541 --> 01:30:40,625 "I attempted something. 922 01:30:40,625 --> 01:30:41,500 Where did I succeed? 923 01:30:41,500 --> 01:30:44,875 Where did I fail?" and have someone there 924 01:30:44,875 --> 01:30:47,875 who can keep pushing the learners forward 925 01:30:48,291 --> 01:30:50,500 and slowly climb that tall, tall mountain 926 01:30:50,500 --> 01:30:52,333 of someday looking down and saying, "You 927 01:30:52,333 --> 01:30:52,666 know what? 928 01:30:52,666 --> 01:30:53,291 I'm actually pretty 929 01:30:53,291 --> 01:30:54,500 okay at math after all." 930 01:30:55,125 --> 01:30:55,541 That's the dream. 931 01:30:56,166 --> 01:30:56,916 That's fantastic. 932 01:30:57,875 --> 01:31:00,625 Obviously, we're going to be able to talk 933 01:31:00,625 --> 01:31:01,625 about this in about 934 01:31:01,625 --> 01:31:04,416 maybe 10, 12 months from 935 01:31:04,416 --> 01:31:06,375 now and see how it actually went, 936 01:31:06,583 --> 01:31:08,041 probably 10 months from now. 937 01:31:09,625 --> 01:31:10,958 I'm very hopeful that 938 01:31:10,958 --> 01:31:13,333 this will be a good thing. 939 01:31:14,833 --> 01:31:16,708 I know that I just read a book. 940 01:31:16,750 --> 01:31:18,125 I think Tim's probably going to take a 941 01:31:18,125 --> 01:31:18,958 look at it too, 942 01:31:20,458 --> 01:31:23,125 "Greeting for Growth," which was 943 01:31:23,125 --> 01:31:23,208 written- 944 01:31:23,208 --> 01:31:24,291 I just started reading it today. 945 01:31:24,833 --> 01:31:26,083 Written by two math 946 01:31:26,083 --> 01:31:28,000 professors at Grand Valley State. 947 01:31:30,000 --> 01:31:35,250 I think there's a lot of really good 948 01:31:35,250 --> 01:31:37,166 material in that book 949 01:31:37,166 --> 01:31:40,375 that meshes very nicely with 950 01:31:40,416 --> 01:31:42,208 what Tim and I have just been talking 951 01:31:42,208 --> 01:31:45,958 about for the pedagogy to 952 01:31:45,958 --> 01:31:48,208 choose for teaching this 953 01:31:48,208 --> 01:31:48,583 class. 954 01:31:49,500 --> 01:31:54,166 I think I'm extremely excited about what 955 01:31:54,166 --> 01:31:56,250 Tim is about to undertake. 956 01:31:57,500 --> 01:31:58,291 Of course, I'll help 957 01:31:58,291 --> 01:31:59,625 you any way I can, Tim. 958 01:31:59,750 --> 01:32:00,666 I appreciate that. 959 01:32:00,666 --> 01:32:01,916 I'm only mildly terrified. 960 01:32:07,083 --> 01:32:07,958 It's not going to be 961 01:32:07,958 --> 01:32:09,208 just a bunch of lectures. 962 01:32:09,208 --> 01:32:10,250 It could be on YouTube. 963 01:32:10,958 --> 01:32:12,541 It's got to be a whole process. 964 01:32:12,541 --> 01:32:13,750 That's already out there, and it's been 965 01:32:13,750 --> 01:32:14,583 done better by 966 01:32:14,583 --> 01:32:15,625 someone who knows it better. 967 01:32:16,208 --> 01:32:16,958 It doesn't work. 968 01:32:18,625 --> 01:32:22,041 Our students try to learn with video, and 969 01:32:22,041 --> 01:32:22,958 it just doesn't work. 970 01:32:24,291 --> 01:32:26,875 I think it's because they really need 971 01:32:26,875 --> 01:32:30,041 these social engagement with each other. 972 01:32:31,083 --> 01:32:33,458 I think that's the secret sauce. 973 01:32:35,416 --> 01:32:37,625 Yeah, I'm sure most of our students don't 974 01:32:37,625 --> 01:32:38,750 think of math as a 975 01:32:38,750 --> 01:32:40,375 social activity, but it 976 01:32:40,375 --> 01:32:41,500 can be, and it should be. 977 01:32:42,583 --> 01:32:44,250 That's something that I hope we can help 978 01:32:44,250 --> 01:32:45,166 them discover through 979 01:32:45,166 --> 01:32:48,291 this new attempt at teaching 980 01:32:48,291 --> 01:32:50,416 the math that our students need at the 981 01:32:50,416 --> 01:32:51,333 time and place where they 982 01:32:51,333 --> 01:32:53,083 need it in our curriculum. 983 01:32:54,333 --> 01:32:55,250 All right. 984 01:32:55,541 --> 01:32:58,000 With that, I think we'll say goodbye. 985 01:32:58,000 --> 01:32:58,750 Was there anything you 986 01:32:58,750 --> 01:32:59,916 want to say at the end? 987 01:32:59,916 --> 01:33:00,833 No, I think that's a 988 01:33:00,833 --> 01:33:02,000 great place to wrap up. 989 01:33:02,666 --> 01:33:03,375 To the audience out 990 01:33:03,375 --> 01:33:04,500 there, thanks for coming. 991 01:33:04,500 --> 01:33:05,500 We'll see you next time.