1 00:00:00,040 --> 00:00:02,460 The following content is provided under a Creative 2 00:00:02,460 --> 00:00:03,970 Commons license. 3 00:00:03,970 --> 00:00:06,910 Your support will help MIT OpenCourseWare continue to 4 00:00:06,910 --> 00:00:10,660 offer high quality educational resources for free. 5 00:00:10,660 --> 00:00:13,460 To make a donation or view additional materials from 6 00:00:13,460 --> 00:00:17,390 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,390 --> 00:00:18,640 ocw.mit.edu. 8 00:00:21,560 --> 00:00:22,060 PROFESSOR: OK. 9 00:00:22,060 --> 00:00:23,780 So misconceptions. 10 00:00:26,530 --> 00:00:30,550 So misconceptions, alternative conceptions, understanding the 11 00:00:30,550 --> 00:00:34,740 student view of the world, why is that important? 12 00:00:34,740 --> 00:00:40,460 Well one of the best ways to introduce the importance of 13 00:00:40,460 --> 00:00:42,860 the idea is a story. 14 00:00:53,170 --> 00:00:56,660 Now I'm not 100% sure the story is true, but it's so 15 00:00:56,660 --> 00:00:58,960 good that it ought to be true even if it isn't. 16 00:00:58,960 --> 00:01:00,530 I think it is actually true. 17 00:01:00,530 --> 00:01:08,320 So this was two, let's say physics, TAs, physics grad 18 00:01:08,320 --> 00:01:11,550 students, who were in a philosophy class. 19 00:01:11,550 --> 00:01:15,370 And the teacher in the philosophy class said well-- 20 00:01:15,370 --> 00:01:18,140 they were talking about things being relative, and morals 21 00:01:18,140 --> 00:01:23,650 being relatives, and relativity in philosophy-- 22 00:01:23,650 --> 00:01:28,180 as an example of relativity, and things being relative 23 00:01:28,180 --> 00:01:33,210 depending on the situation, if you drop a pencil on the 24 00:01:33,210 --> 00:01:35,430 Earth, it will fall to the ground. 25 00:01:35,430 --> 00:01:37,800 But if you drop the pencil on the moon, it 26 00:01:37,800 --> 00:01:38,720 won't fall to the ground. 27 00:01:38,720 --> 00:01:40,270 It will just float. 28 00:01:40,270 --> 00:01:43,650 And so most of the people in the class just nodded, 29 00:01:43,650 --> 00:01:46,320 basically thinking, oh that's a good example. 30 00:01:46,320 --> 00:01:50,520 But the two physicists in the class were a bit shocked and 31 00:01:50,520 --> 00:01:53,040 said well wait a minute, wait a minute. 32 00:01:53,040 --> 00:01:55,720 That can't be true. 33 00:01:55,720 --> 00:01:57,840 Surely, something is wrong here. 34 00:01:57,840 --> 00:01:58,640 So they raised their hand. 35 00:01:58,640 --> 00:02:00,810 They said, well wait a minute. 36 00:02:00,810 --> 00:02:05,570 If you drop a pencil on the moon, surely it's going to 37 00:02:05,570 --> 00:02:08,620 fall to the ground, to the ground of the moon. 38 00:02:08,620 --> 00:02:11,730 And the other people in the class said, no, 39 00:02:11,730 --> 00:02:13,150 no, that's not right. 40 00:02:13,150 --> 00:02:13,880 That's not right. 41 00:02:13,880 --> 00:02:18,440 So they thought, OK, well we have some way of convincing 42 00:02:18,440 --> 00:02:20,700 them that the pencil's not going to float. 43 00:02:20,700 --> 00:02:23,590 And they said, OK well you all saw the pictures of the 44 00:02:23,590 --> 00:02:25,190 astronauts walking on the moon, right? 45 00:02:25,190 --> 00:02:27,100 And they said, oh yeah. 46 00:02:27,100 --> 00:02:32,860 Well, when the astronauts walk, did they just float 47 00:02:32,860 --> 00:02:34,580 away, or did they stay on the moon? 48 00:02:34,580 --> 00:02:36,670 Oh, they stayed on the moon. 49 00:02:36,670 --> 00:02:38,980 Well, how'd that happen? 50 00:02:38,980 --> 00:02:40,780 They thought, oh we have them now. 51 00:02:40,780 --> 00:02:41,790 And what was the response? 52 00:02:41,790 --> 00:02:43,870 The response was, oh, well that was because the 53 00:02:43,870 --> 00:02:45,470 astronauts were wearing heavy boots. 54 00:02:48,340 --> 00:02:52,930 So here was a chance to do a bit of educational research. 55 00:02:52,930 --> 00:02:54,200 So they thought, oh my god. 56 00:02:54,200 --> 00:02:56,790 Maybe is this just this class? 57 00:02:56,790 --> 00:02:58,390 Let's actually do a survey. 58 00:02:58,390 --> 00:03:00,590 So they took out the university phone directory and 59 00:03:00,590 --> 00:03:02,370 just randomly dialed people. 60 00:03:02,370 --> 00:03:08,090 And they called them up and said, OK if you drop a pencil 61 00:03:08,090 --> 00:03:10,020 on the moon, will it float away or 62 00:03:10,020 --> 00:03:11,260 will fall to the ground? 63 00:03:11,260 --> 00:03:13,990 And most of the people said, it's going to float away. 64 00:03:13,990 --> 00:03:17,080 So then they said, oh yeah, but what about the astronauts? 65 00:03:17,080 --> 00:03:17,860 And they gave them the whole thing about 66 00:03:17,860 --> 00:03:19,050 the astronauts again. 67 00:03:19,050 --> 00:03:22,290 And about half of the people said, oh yeah, that's because 68 00:03:22,290 --> 00:03:25,270 the astronauts were wearing heavy boots. 69 00:03:25,270 --> 00:03:27,870 So again, they were shocked and flabbergasted. 70 00:03:27,870 --> 00:03:31,790 And I think that reaction is a very healthy reaction. 71 00:03:31,790 --> 00:03:36,110 But what it shows is that the students-- 72 00:03:36,110 --> 00:03:38,900 so in this case they weren't officially their students-- 73 00:03:38,900 --> 00:03:39,950 but students-- 74 00:03:39,950 --> 00:03:41,470 they could easily be your students-- 75 00:03:41,470 --> 00:03:45,240 can have a very different view of the world, of the physical 76 00:03:45,240 --> 00:03:47,090 world, than you have. 77 00:03:47,090 --> 00:03:52,300 And if that's true, that means that as you're teaching them, 78 00:03:52,300 --> 00:03:55,530 what you're teaching is actually being interpreted in 79 00:03:55,530 --> 00:03:58,740 this completely different way. 80 00:03:58,740 --> 00:04:00,190 I think Goethe said that-- 81 00:04:00,190 --> 00:04:02,630 he said that mathematicians, or maybe it was Frenchmen. 82 00:04:02,630 --> 00:04:04,410 It was either Frenchmen or mathematicians. 83 00:04:04,410 --> 00:04:06,260 Maybe it was mathematicians. 84 00:04:06,260 --> 00:04:08,380 They have their own language, and everything you tell them, 85 00:04:08,380 --> 00:04:10,540 they first translate into their own language, and only 86 00:04:10,540 --> 00:04:12,240 then do they understand it. 87 00:04:12,240 --> 00:04:13,570 So that's the same thing. 88 00:04:13,570 --> 00:04:16,390 The students are translating the things you're saying into 89 00:04:16,390 --> 00:04:17,600 their own language. 90 00:04:17,600 --> 00:04:19,880 But to understand what they're translating into, you have to 91 00:04:19,880 --> 00:04:21,800 know what their language is. 92 00:04:21,800 --> 00:04:24,640 So in other words, you have to understand not just the 93 00:04:24,640 --> 00:04:28,970 misconceptions, but in general the conceptions that students 94 00:04:28,970 --> 00:04:31,420 have, so that you can actually plan your teaching. 95 00:04:31,420 --> 00:04:33,948 And here is a diagram to illustrate that. 96 00:04:56,070 --> 00:05:07,040 So this is a model of teaching to see where all 97 00:05:07,040 --> 00:05:08,600 the pieces fit in. 98 00:05:08,600 --> 00:05:12,630 So here, this is the system, which is how students think. 99 00:05:16,550 --> 00:05:22,570 This is the output, which is what students can do after 100 00:05:22,570 --> 00:05:24,490 they go through our teaching. 101 00:05:24,490 --> 00:05:30,850 And this is the input, which is what we do, what kind of 102 00:05:30,850 --> 00:05:34,200 classroom structure we set up, what assignments we do. 103 00:05:34,200 --> 00:05:37,030 And by this, I mean how students think individually 104 00:05:37,030 --> 00:05:39,570 and collectively, how students work together, as well. 105 00:05:39,570 --> 00:05:43,030 So generally speaking the way-- 106 00:05:43,030 --> 00:05:45,040 so this is the system. 107 00:05:45,040 --> 00:05:48,450 And call this the result. 108 00:05:48,450 --> 00:05:52,470 And this is the teaching. 109 00:05:52,470 --> 00:05:54,490 Our problem is trying to figure out the teaching. 110 00:06:04,270 --> 00:06:05,540 And how do you figure out the teaching? 111 00:06:05,540 --> 00:06:09,140 Well you start with figuring out what result you want. 112 00:06:09,140 --> 00:06:10,850 What do you want students to be able to do? 113 00:06:10,850 --> 00:06:13,200 What is your goal for the class? 114 00:06:13,200 --> 00:06:16,600 And then you have to run it back through the system. 115 00:06:16,600 --> 00:06:22,310 Right so you figure out the result, and you run it through 116 00:06:22,310 --> 00:06:24,190 the inverse of s. 117 00:06:24,190 --> 00:06:25,970 And then there's your teaching. 118 00:06:25,970 --> 00:06:28,170 That's the teaching equation let's call it. 119 00:06:36,100 --> 00:06:39,420 Now those of you from engineering fields know that 120 00:06:39,420 --> 00:06:42,480 basically the only thing you can do anything 121 00:06:42,480 --> 00:06:44,560 with is linear systems. 122 00:06:44,560 --> 00:06:47,390 And unfortunately anything that's even slightly non 123 00:06:47,390 --> 00:06:49,450 linear basically is a bag of tricks, and there's nothing 124 00:06:49,450 --> 00:06:49,780 you can do. 125 00:06:49,780 --> 00:06:51,340 And you're just hosed. 126 00:06:51,340 --> 00:06:54,580 Well unfortunately this system is incredibly non-linear. 127 00:06:54,580 --> 00:06:58,070 So it's probably kind of hard to work out s inverse, even if 128 00:06:58,070 --> 00:07:00,180 you can work out r. 129 00:07:00,180 --> 00:07:03,060 But it's hard, but it doesn't mean you can't start. 130 00:07:03,060 --> 00:07:06,220 And a necessary condition for even doing anything in this 131 00:07:06,220 --> 00:07:09,980 direction is you have to know what is in s, 132 00:07:09,980 --> 00:07:11,340 how do students think. 133 00:07:11,340 --> 00:07:14,440 Another necessary condition is you need to work out r. 134 00:07:14,440 --> 00:07:16,180 And that's actually something-- 135 00:07:16,180 --> 00:07:17,670 what is the result you want? 136 00:07:17,670 --> 00:07:19,270 So that we're going to talk about in the 137 00:07:19,270 --> 00:07:20,710 session on course design. 138 00:07:20,710 --> 00:07:23,640 That's something that's often forgotten in teaching, and 139 00:07:23,640 --> 00:07:25,090 it's fundamentally important. 140 00:07:25,090 --> 00:07:28,060 That if you don't plan your goals, if you don't plan where 141 00:07:28,060 --> 00:07:31,750 you want to go, your chance of getting there is pretty low. 142 00:07:31,750 --> 00:07:33,780 So assume you've done that. 143 00:07:33,780 --> 00:07:36,100 The next thing you want to do is you want to understand how 144 00:07:36,100 --> 00:07:39,000 students think, so that you can plan your teaching to 145 00:07:39,000 --> 00:07:41,450 reach them, in other words, so you have some chance of 146 00:07:41,450 --> 00:07:42,980 working out s inverse. 147 00:07:42,980 --> 00:07:44,970 Now you can't actually write down some equation for s 148 00:07:44,970 --> 00:07:47,360 inverse, but you can invert it mentally in your head. 149 00:07:47,360 --> 00:07:50,310 And a necessary condition for doing that is to understand s. 150 00:07:50,310 --> 00:07:52,700 So you want to understand where students are coming 151 00:07:52,700 --> 00:07:57,710 from, so that you can work out t. 152 00:07:57,710 --> 00:08:02,700 So that's the signals and systems model of teaching. 153 00:08:02,700 --> 00:08:08,820 Now, an example again to show the importance of doing that 154 00:08:08,820 --> 00:08:11,110 and what happens if you don't actually understand where the 155 00:08:11,110 --> 00:08:14,430 student's are is yet another example from freshman physics. 156 00:08:14,430 --> 00:08:16,710 I'll leave this here. 157 00:08:16,710 --> 00:08:20,210 So this is one of the examples from the readings. 158 00:08:20,210 --> 00:08:22,050 So not everyone may have read that reading. 159 00:08:22,050 --> 00:08:24,450 It's the reading by Reif. 160 00:08:24,450 --> 00:08:25,700 And it's this problem. 161 00:08:33,870 --> 00:08:37,890 So this is a pendulum at five points in its arc. 162 00:08:37,890 --> 00:08:39,549 So these are the endpoints. 163 00:08:39,549 --> 00:08:43,929 And the question people were asked was to draw the 164 00:08:43,929 --> 00:08:47,340 acceleration vector at these five locations. 165 00:08:47,340 --> 00:08:49,350 So what's the acceleration right here, here, 166 00:08:49,350 --> 00:08:52,490 here, here, and here? 167 00:08:52,490 --> 00:08:56,490 So I'm not so interested in what the actual answer is as 168 00:08:56,490 --> 00:08:59,700 so much as how people did on this. 169 00:08:59,700 --> 00:09:01,850 So this is actually really shocking. 170 00:09:05,280 --> 00:09:09,660 So of students in intro physics who've had 171 00:09:09,660 --> 00:09:10,380 acceleration-- 172 00:09:10,380 --> 00:09:11,630 so they'd done acceleration-- 173 00:09:19,780 --> 00:09:22,760 zero out of 124 students could do that problem. 174 00:09:32,180 --> 00:09:34,800 So that's 0%. 175 00:09:34,800 --> 00:09:41,580 So of grad TAs, it was 15%. 176 00:09:41,580 --> 00:09:46,130 So these are grad TAs in physics who TA that course. 177 00:09:46,130 --> 00:09:47,550 And PhD students-- 178 00:09:51,520 --> 00:09:54,320 so was PhD students on their qualifying exam. 179 00:09:54,320 --> 00:09:55,570 It went up to 22%. 180 00:09:58,830 --> 00:10:04,320 OK so now this is a fundamentally important idea, 181 00:10:04,320 --> 00:10:06,680 acceleration. 182 00:10:06,680 --> 00:10:09,710 But you'd expect PhD students on their qualifying 183 00:10:09,710 --> 00:10:11,930 exam to be at 100%. 184 00:10:11,930 --> 00:10:15,530 And you'd like the TAs in the course to be at 100%. 185 00:10:15,530 --> 00:10:17,860 Because there's no chance of them teaching the students an 186 00:10:17,860 --> 00:10:20,020 idea that they themselves don't understand. 187 00:10:20,020 --> 00:10:22,600 As Polya said, there's no method of teaching yet 188 00:10:22,600 --> 00:10:26,700 invented that will allow the students to understand what 189 00:10:26,700 --> 00:10:28,450 their teacher does not. 190 00:10:28,450 --> 00:10:29,220 And that's true. 191 00:10:29,220 --> 00:10:33,000 There's no amount of sort of trickery with style and things 192 00:10:33,000 --> 00:10:35,810 you can do to convey an understanding 193 00:10:35,810 --> 00:10:37,450 that you don't have. 194 00:10:37,450 --> 00:10:41,710 So this is a very shocking number to me. 195 00:10:41,710 --> 00:10:42,440 And-- 196 00:10:42,440 --> 00:10:42,740 AUDIENCE: Who said that? 197 00:10:42,740 --> 00:10:43,300 PROFESSOR: Pardon? 198 00:10:43,300 --> 00:10:44,260 AUDIENCE: Who said that? 199 00:10:44,260 --> 00:10:45,510 PROFESSOR: George Polya. 200 00:10:50,190 --> 00:10:54,300 So Polya was a great mathematics teacher from 201 00:10:54,300 --> 00:10:58,020 Hungary and then eventually at Stanford. 202 00:10:58,020 --> 00:10:59,170 And he's written-- 203 00:10:59,170 --> 00:11:02,870 I'll put references to his books on the website. 204 00:11:02,870 --> 00:11:04,700 But they are some of the best books on math 205 00:11:04,700 --> 00:11:06,470 teaching ever written. 206 00:11:06,470 --> 00:11:08,700 And so he said that in one of his books. 207 00:11:08,700 --> 00:11:12,110 And it's I think very true. 208 00:11:12,110 --> 00:11:17,530 So what this example shows is that, a, the teachers didn't 209 00:11:17,530 --> 00:11:18,250 understand it. 210 00:11:18,250 --> 00:11:21,590 But that really fundamental ideas can be completely 211 00:11:21,590 --> 00:11:24,230 skipped over for years and years and years. 212 00:11:28,190 --> 00:11:29,120 Yes, [INAUDIBLE]. 213 00:11:29,120 --> 00:11:31,454 AUDIENCE: It's a special example of some vague, 214 00:11:31,454 --> 00:11:32,897 conceptually difficult problem. 215 00:11:32,897 --> 00:11:35,783 I'm just wondering-- so I read the article. 216 00:11:35,783 --> 00:11:38,030 Did they focus on how to figure the answer out? 217 00:11:38,030 --> 00:11:40,980 Because a lot of physics majors don't. 218 00:11:40,980 --> 00:11:42,500 PROFESSOR: OK, so the question is, how 219 00:11:42,500 --> 00:11:44,700 difficult is this problem? 220 00:11:44,700 --> 00:11:48,760 It's an interesting problem, because it's difficult if you 221 00:11:48,760 --> 00:11:51,450 try to do it by just blasting out the equations. 222 00:11:51,450 --> 00:11:54,930 So they actually asked a bunch of faculty 223 00:11:54,930 --> 00:11:56,830 members the same question. 224 00:11:56,830 --> 00:11:59,280 And one of them just got completely tangled in knots, 225 00:11:59,280 --> 00:12:01,830 physics faculty, trying to write out the differential 226 00:12:01,830 --> 00:12:03,390 equation and figure out what the 227 00:12:03,390 --> 00:12:05,690 acceleration is by brute force. 228 00:12:05,690 --> 00:12:07,440 So yeah, it is hard if you do it that way. 229 00:12:07,440 --> 00:12:09,770 But if you actually understand-- 230 00:12:09,770 --> 00:12:10,930 I'll show you-- 231 00:12:10,930 --> 00:12:14,410 what the acceleration means, then it's no problem. 232 00:12:14,410 --> 00:12:17,670 So for example, here, the thing isn't moving. 233 00:12:17,670 --> 00:12:19,080 So it has v equals zero. 234 00:12:19,080 --> 00:12:22,440 But, what velocity is it going to have just a bit later. 235 00:12:22,440 --> 00:12:24,190 Well it's going to be going that way. 236 00:12:24,190 --> 00:12:25,840 So that means acceleration is that way. 237 00:12:25,840 --> 00:12:28,640 AUDIENCE: But were they given enough time to do this, to do 238 00:12:28,640 --> 00:12:29,690 this thinking problem? 239 00:12:29,690 --> 00:12:32,360 PROFESSOR: Yeah, so they were given quite a while. 240 00:12:32,360 --> 00:12:34,380 I think it was-- 241 00:12:34,380 --> 00:12:36,640 when the study on the faculty members, I think it was a big 242 00:12:36,640 --> 00:12:37,715 long think aloud protocol. 243 00:12:37,715 --> 00:12:39,350 So they were given like half an hour. 244 00:12:39,350 --> 00:12:42,190 In fact, the time is what hurt them. 245 00:12:42,190 --> 00:12:45,210 Because they all thought, oh I have so much time. 246 00:12:45,210 --> 00:12:48,080 I'm going to just blast it out with equations. 247 00:12:48,080 --> 00:12:49,780 And if you were only given one minute, you might have 248 00:12:49,780 --> 00:12:50,850 actually had a chance. 249 00:12:50,850 --> 00:12:52,270 Because you realized, oh I have to think about this 250 00:12:52,270 --> 00:12:54,260 intuitively. 251 00:12:54,260 --> 00:12:57,060 So here, same thing by symmetry. 252 00:12:57,060 --> 00:12:58,240 And then this one is hard. 253 00:12:58,240 --> 00:12:58,940 And this one is hard. 254 00:12:58,940 --> 00:13:01,280 But this one right at the center is much easier. 255 00:13:01,280 --> 00:13:02,930 So this one catches a lot of people. 256 00:13:02,930 --> 00:13:08,240 But here it's moving in a circle, and it's at its 257 00:13:08,240 --> 00:13:09,800 maximum speed. 258 00:13:09,800 --> 00:13:11,880 So here it's moving at its maximum speed. 259 00:13:11,880 --> 00:13:15,510 So it's not going to be going faster that way or that way. 260 00:13:15,510 --> 00:13:17,950 So its speed is actually at a max, which means there's no 261 00:13:17,950 --> 00:13:18,930 change in speed. 262 00:13:18,930 --> 00:13:19,910 So you're not going to get any 263 00:13:19,910 --> 00:13:21,430 acceleration in that direction. 264 00:13:21,430 --> 00:13:23,100 But because it's moving in a circle, 265 00:13:23,100 --> 00:13:25,090 somebody's yanking it upwards. 266 00:13:25,090 --> 00:13:27,330 So it has to be accelerating upwards to move in a circle, 267 00:13:27,330 --> 00:13:28,880 because it was going this way. 268 00:13:28,880 --> 00:13:30,800 And then later, it's going that way. 269 00:13:30,800 --> 00:13:32,780 So it has to be accelerating upwards. 270 00:13:32,780 --> 00:13:40,050 So there's an acceleration vector upwards, like that. 271 00:13:42,990 --> 00:13:44,110 So this is a bit exaggerated. 272 00:13:44,110 --> 00:13:46,460 It's generally not that big. 273 00:13:46,460 --> 00:13:48,700 And then the question is, with these guys, well it's just 274 00:13:48,700 --> 00:13:49,600 interpolation. 275 00:13:49,600 --> 00:13:51,400 It's somewhere between that and that. 276 00:13:51,400 --> 00:13:53,310 So it's going to be something like that and something like 277 00:13:53,310 --> 00:13:56,400 that, depending on where exactly they are. 278 00:13:56,400 --> 00:14:01,460 But this is the one that tripped up a lot of people. 279 00:14:01,460 --> 00:14:03,730 And in fact, a lot of people got tripped up because they 280 00:14:03,730 --> 00:14:09,110 memorize the following thing, which is an object in simple 281 00:14:09,110 --> 00:14:11,430 harmonic motion has no acceleration at the 282 00:14:11,430 --> 00:14:12,980 equilibrium point. 283 00:14:12,980 --> 00:14:14,780 And that's true, except this isn't a 284 00:14:14,780 --> 00:14:15,750 simple harmonic motion. 285 00:14:15,750 --> 00:14:16,310 It's almost. 286 00:14:16,310 --> 00:14:17,020 But it's a pendulum. 287 00:14:17,020 --> 00:14:18,400 It's not a pure spring. 288 00:14:18,400 --> 00:14:19,680 So it has circular motion. 289 00:14:19,680 --> 00:14:21,890 So if you reason about it from the motion, it's 290 00:14:21,890 --> 00:14:22,990 actually very quick. 291 00:14:22,990 --> 00:14:25,020 But if you try to blast it out with equations, you're 292 00:14:25,020 --> 00:14:26,270 basically hosed. 293 00:14:29,790 --> 00:14:35,560 So I wanted to check also, how deep are these misconceptions? 294 00:14:35,560 --> 00:14:39,370 So I actually did a survey of students in Cambridge in the 295 00:14:39,370 --> 00:14:40,940 physics major. 296 00:14:40,940 --> 00:14:44,850 And so one of the questions I asked them was the following-- 297 00:14:44,850 --> 00:14:48,670 so I made a survey of nine questions. 298 00:14:48,670 --> 00:14:51,865 And I wanted to see how they reasoned intuitively on these 299 00:14:51,865 --> 00:14:52,560 nine questions. 300 00:14:52,560 --> 00:14:54,690 And to force intuitive reasoning, I only gave a few 301 00:14:54,690 --> 00:14:56,900 minutes to get around the problem. 302 00:14:56,900 --> 00:14:59,960 So they knew that basically writing out a bunch of 303 00:14:59,960 --> 00:15:01,810 equations wasn't going to help, because you had no time 304 00:15:01,810 --> 00:15:03,143 to actually write them all out. 305 00:15:17,000 --> 00:15:19,230 So the question was, which is going to go 306 00:15:19,230 --> 00:15:21,010 faster down the plane? 307 00:15:21,010 --> 00:15:24,840 So these are two identical planes, inclined planes with 308 00:15:24,840 --> 00:15:25,950 angle alpha. 309 00:15:25,950 --> 00:15:29,290 And there is either going to be a disk like a coin rolling 310 00:15:29,290 --> 00:15:32,560 down the plane, or a ring, like somebody's wedding ring 311 00:15:32,560 --> 00:15:33,760 rolling down the plane. 312 00:15:33,760 --> 00:15:36,240 And the question is, which goes faster, or which has a 313 00:15:36,240 --> 00:15:37,850 bigger acceleration. 314 00:15:37,850 --> 00:15:43,960 So this we'll call the hoop, and this is the disk. 315 00:15:43,960 --> 00:15:46,626 And they're going down the same angle. 316 00:15:46,626 --> 00:15:48,640 AUDIENCE: The mass is the same. 317 00:15:48,640 --> 00:15:50,190 PROFESSOR: I didn't say that. 318 00:15:50,190 --> 00:15:51,690 The question is, is the mass the same? 319 00:15:51,690 --> 00:15:53,390 So I'm going to give you that choice. 320 00:15:53,390 --> 00:15:56,690 So the question is, which goes faster in 321 00:15:56,690 --> 00:15:58,310 acceleration or velocity? 322 00:15:58,310 --> 00:16:00,821 So the choices were the hoop-- 323 00:16:16,230 --> 00:16:19,170 So to answer the question about mass, one of the choices 324 00:16:19,170 --> 00:16:20,420 is that it depends on mass. 325 00:16:24,950 --> 00:16:33,630 Depends upon radius or depends on alpha. 326 00:16:33,630 --> 00:16:36,820 So it could depend on the mass, the radius, or on alpha. 327 00:16:36,820 --> 00:16:38,080 It could be the same. 328 00:16:38,080 --> 00:16:39,100 The disk could be faster. 329 00:16:39,100 --> 00:16:41,030 Or the hoop could be faster. 330 00:16:41,030 --> 00:16:43,015 OK, so just for fun-- 331 00:16:43,015 --> 00:16:45,090 and I realize this is not a physics class-- 332 00:16:45,090 --> 00:16:47,450 but it's fun to actually try to guess the answer to this, 333 00:16:47,450 --> 00:16:49,030 even if you're not a physicist. 334 00:16:49,030 --> 00:16:55,190 So choose A, B, C, D, E, or F in cooperation 335 00:16:55,190 --> 00:16:57,880 with one or two neighbors. 336 00:16:57,880 --> 00:17:01,460 And then I'll talk about my analysis of this problem. 337 00:17:08,200 --> 00:17:11,780 OK, take another 30 seconds and collect a vote. 338 00:17:11,780 --> 00:17:15,849 Again, it doesn't really matter whether you get it 339 00:17:15,849 --> 00:17:16,569 right or not. 340 00:17:16,569 --> 00:17:19,540 I just want you to think about it, so you realize it is a 341 00:17:19,540 --> 00:17:20,790 subtle question. 342 00:17:24,490 --> 00:17:30,655 OK, so who votes for the hoop being faster? 343 00:17:33,330 --> 00:17:35,670 Who votes for the disk being faster? 344 00:17:39,500 --> 00:17:40,780 Who votes that they're the same? 345 00:17:43,867 --> 00:17:47,280 That it depends on the mass, you'd like to know the mass? 346 00:17:47,280 --> 00:17:49,180 Fair enough. 347 00:17:49,180 --> 00:17:50,430 That you'd like to know the radius? 348 00:17:53,870 --> 00:17:54,790 Maybe 12. 349 00:17:54,790 --> 00:17:57,880 That you'd like to know the inclined plane angle? 350 00:17:57,880 --> 00:17:59,722 Nobody like that one, OK. 351 00:18:06,500 --> 00:18:11,370 And now who's sure of their answer. 352 00:18:11,370 --> 00:18:15,260 OK, so that's a much lower number. 353 00:18:15,260 --> 00:18:19,995 So now I can show you what numbers came up when I asked 354 00:18:19,995 --> 00:18:22,610 the students this. 355 00:18:22,610 --> 00:18:26,000 So this was 11%. 356 00:18:29,560 --> 00:18:31,980 35%. 357 00:18:31,980 --> 00:18:33,850 13%. 358 00:18:33,850 --> 00:18:38,310 This one is 11% as well. 359 00:18:38,310 --> 00:18:39,775 Radius was 26%. 360 00:18:42,440 --> 00:18:44,300 And 3%. 361 00:18:44,300 --> 00:18:49,320 So kind of similar, kind of similar to your percentages. 362 00:18:49,320 --> 00:18:53,080 And again, only about a third of the people were sure of 363 00:18:53,080 --> 00:18:54,720 their answer. 364 00:18:54,720 --> 00:18:57,400 OK, so why is this interesting? 365 00:18:57,400 --> 00:19:00,560 Well first of all, all the students had the technical 366 00:19:00,560 --> 00:19:03,030 knowledge to do the problem. 367 00:19:03,030 --> 00:19:05,670 These were all physics majors in various years, one through 368 00:19:05,670 --> 00:19:08,310 four, so freshman through senior. 369 00:19:08,310 --> 00:19:11,840 But the freshman had seen the material to actually do this 370 00:19:11,840 --> 00:19:15,090 and calculate it if they need to. 371 00:19:15,090 --> 00:19:18,250 So now this problem illustrates well first of all 372 00:19:18,250 --> 00:19:22,120 that even an intuitive question like this can 373 00:19:22,120 --> 00:19:23,780 actually trip people up, even if they've done all the 374 00:19:23,780 --> 00:19:25,410 calculations. 375 00:19:25,410 --> 00:19:27,990 So the question is, how are they reasoning? 376 00:19:27,990 --> 00:19:31,510 Now in Cambridge I had the advantage of also doing 377 00:19:31,510 --> 00:19:34,620 tutorials, because that's how half the teaching goes. 378 00:19:34,620 --> 00:19:38,460 So in tutorials I actually I set this problem for my 379 00:19:38,460 --> 00:19:40,750 students, and I got a chance to talk to them about it. 380 00:19:40,750 --> 00:19:43,580 So I got to figure out, how are they reasoning about it? 381 00:19:43,580 --> 00:19:46,410 So I asked them, well what do you think is going to happen? 382 00:19:46,410 --> 00:19:48,680 And then please explain why, more than you can get just 383 00:19:48,680 --> 00:19:49,990 from multiple choice. 384 00:19:49,990 --> 00:19:52,960 And the answers were very revealing. 385 00:19:52,960 --> 00:19:57,460 What they said was, many of them said, well, just as 386 00:19:57,460 --> 00:20:01,740 heavier objects fall faster, so should heavier 387 00:20:01,740 --> 00:20:03,860 objects roll faster. 388 00:20:03,860 --> 00:20:08,890 So that was basically to justify they wanted to know 389 00:20:08,890 --> 00:20:11,590 either the radius or the mass, because that would affect the 390 00:20:11,590 --> 00:20:13,580 mass, and that would also affect the mass. 391 00:20:13,580 --> 00:20:15,900 So, let me put that up again, because that's a really 392 00:20:15,900 --> 00:20:17,150 interesting statement. 393 00:20:35,700 --> 00:20:40,830 So actually what I call this, this is like the American 394 00:20:40,830 --> 00:20:43,040 theory of the British accent. 395 00:20:43,040 --> 00:20:45,160 Some of you know my explanation of that. 396 00:20:45,160 --> 00:20:49,020 So the American theory of the British accent is that if you 397 00:20:49,020 --> 00:20:51,630 go to an English person and you step on their feet in the 398 00:20:51,630 --> 00:20:55,340 middle of the nigh, they'll actually cry out in an 399 00:20:55,340 --> 00:20:57,130 American accent. 400 00:20:57,130 --> 00:21:00,080 Which is to say that the American, sort of folk theory 401 00:21:00,080 --> 00:21:02,810 of the British accent is that the British accent is a fake 402 00:21:02,810 --> 00:21:06,110 that people just put on, and in moments of stress they 403 00:21:06,110 --> 00:21:08,220 forget to do the put on act. 404 00:21:08,220 --> 00:21:09,920 OK, so now why do I bring that up? 405 00:21:09,920 --> 00:21:12,450 Well because it's very similar. 406 00:21:12,450 --> 00:21:14,780 So if you ask the students-- 407 00:21:14,780 --> 00:21:16,310 suppose you didn't tell them this question 408 00:21:16,310 --> 00:21:17,210 about inclined planes. 409 00:21:17,210 --> 00:21:19,910 You just say, do heavier objects fall faster? 410 00:21:19,910 --> 00:21:22,090 You don't know about air resistance or anything. 411 00:21:22,090 --> 00:21:24,690 They say, oh no definitely not. 412 00:21:24,690 --> 00:21:27,440 Galileo showed that. 413 00:21:27,440 --> 00:21:31,340 But what this shows is that in a new situation, in other 414 00:21:31,340 --> 00:21:34,840 words where they don't have their automatic reactions, all 415 00:21:34,840 --> 00:21:38,360 of a sudden the folk theory comes out again. 416 00:21:38,360 --> 00:21:40,960 They really do think heavier objects fall faster. 417 00:21:40,960 --> 00:21:43,160 Right, so how can you reconcile that with if you ask 418 00:21:43,160 --> 00:21:44,520 them, they'll say no they don't? 419 00:21:44,520 --> 00:21:48,590 Well that's because they had so much experience with every 420 00:21:48,590 --> 00:21:50,940 time they said that, someone said, whack. 421 00:21:50,940 --> 00:21:52,230 Heavier objects fall at the same speed. 422 00:21:52,230 --> 00:21:54,090 Galileo showed that, Leaning Tower of Pisa. 423 00:21:54,090 --> 00:21:57,790 So it becomes this linguistic string which hasn't actually 424 00:21:57,790 --> 00:22:02,080 been incorporated into their way of thinking about how the 425 00:22:02,080 --> 00:22:03,470 world works. 426 00:22:03,470 --> 00:22:06,420 They think force is related to velocity, heavier 427 00:22:06,420 --> 00:22:07,830 objects fall faster. 428 00:22:07,830 --> 00:22:10,430 And to get that to come out, you have to, for example, 429 00:22:10,430 --> 00:22:13,020 equivalent to stepping on them in the middle of a night, 430 00:22:13,020 --> 00:22:14,580 surprising them. 431 00:22:14,580 --> 00:22:17,970 So you surprise them with a completely new context, which 432 00:22:17,970 --> 00:22:19,610 is the rolling. 433 00:22:19,610 --> 00:22:21,600 Rolling, they haven't thought much about rolling. 434 00:22:21,600 --> 00:22:24,400 So all of a sudden, they're now thinking about rolling. 435 00:22:24,400 --> 00:22:27,840 And they're not thinking about all the policing rules that 436 00:22:27,840 --> 00:22:30,300 they were told about what happens to heavier objects. 437 00:22:30,300 --> 00:22:32,760 And so now their true way of looking at the world comes 438 00:22:32,760 --> 00:22:36,760 out, which is that heavier objects fall faster. 439 00:22:36,760 --> 00:22:40,180 So again, if you don't realize that that's what they're 440 00:22:40,180 --> 00:22:44,640 doing, your teaching is going to be completely pointless. 441 00:22:44,640 --> 00:22:48,600 And what's going to happen is the following, which is that 442 00:22:48,600 --> 00:22:51,900 here are two models. 443 00:22:51,900 --> 00:22:54,470 The students will develop two models of the 444 00:22:54,470 --> 00:22:55,640 world in their mind. 445 00:22:55,640 --> 00:22:58,190 One let's call it the school model. 446 00:23:01,020 --> 00:23:03,320 And another is their intuitive model. 447 00:23:06,750 --> 00:23:10,370 And let's say they start out a bit different from each other. 448 00:23:10,370 --> 00:23:13,660 And now you do a whole bunch more teaching, and you don't 449 00:23:13,660 --> 00:23:16,060 take account of the fact that this belongs in their 450 00:23:16,060 --> 00:23:17,400 intuitive model. 451 00:23:17,400 --> 00:23:20,200 And you teach a bunch of stuff about rolling down the plane. 452 00:23:20,200 --> 00:23:22,060 You do it with a bunch of equations. 453 00:23:22,060 --> 00:23:26,110 By the way, I should tell you what the right answer is. 454 00:23:26,110 --> 00:23:29,280 The right answer is this one. 455 00:23:29,280 --> 00:23:32,140 It actually doesn't matter how heavy they are, because it's a 456 00:23:32,140 --> 00:23:34,110 gravitation problem. 457 00:23:34,110 --> 00:23:36,680 But suppose you just calculated it out. 458 00:23:36,680 --> 00:23:38,050 Well what have you done? 459 00:23:38,050 --> 00:23:41,760 You've overlaid on top of this, you've put in-- so this 460 00:23:41,760 --> 00:23:44,380 is their intuitive model. 461 00:23:44,380 --> 00:23:46,060 You've done a whole bunch of calculations 462 00:23:46,060 --> 00:23:47,400 which contradict it. 463 00:23:47,400 --> 00:23:50,800 And the two models diverge farther and 464 00:23:50,800 --> 00:23:52,030 farther from each other. 465 00:23:52,030 --> 00:23:55,330 So now next time they have to solve a problem, they have to 466 00:23:55,330 --> 00:23:57,230 decide whether I'm going to use the school model or the 467 00:23:57,230 --> 00:23:58,540 intuitive model. 468 00:23:58,540 --> 00:24:00,380 And they learn, OK, well the intuitive model 469 00:24:00,380 --> 00:24:00,870 doesn't work very well. 470 00:24:00,870 --> 00:24:01,830 Let me use the school model. 471 00:24:01,830 --> 00:24:04,860 So the school model basically keeps diverging. 472 00:24:04,860 --> 00:24:07,610 And the intuitive model and the school model never meet 473 00:24:07,610 --> 00:24:10,930 again, which means that they can't use their intuition for 474 00:24:10,930 --> 00:24:12,720 solving any of the problems you give them, or that they're 475 00:24:12,720 --> 00:24:13,790 going to solve later. 476 00:24:13,790 --> 00:24:17,760 So that is an educational tragedy. 477 00:24:17,760 --> 00:24:30,900 So the divergence, and in my view, that is the fundamental 478 00:24:30,900 --> 00:24:33,470 reason why most teaching produces no 479 00:24:33,470 --> 00:24:35,240 results a year later. 480 00:24:35,240 --> 00:24:38,650 Because what you've done is you've developed the school 481 00:24:38,650 --> 00:24:41,270 model, the symbolic crunching, whatever it may be. 482 00:24:41,270 --> 00:24:43,635 But you haven't actually yanked the intuitive model and 483 00:24:43,635 --> 00:24:44,860 the school model together. 484 00:24:44,860 --> 00:24:47,690 You need to actually work on bringing them together. 485 00:24:47,690 --> 00:24:51,410 So now the natural question is, how do you do that? 486 00:24:51,410 --> 00:24:53,370 So let me give you another example of a similar 487 00:24:53,370 --> 00:24:56,500 misconception and a way of yanking the two together. 488 00:24:56,500 --> 00:24:57,040 Yes, question. 489 00:24:57,040 --> 00:24:58,378 AUDIENCE: Would you mind explaining why 490 00:24:58,378 --> 00:24:59,720 the disk rolls faster? 491 00:24:59,720 --> 00:25:02,430 PROFESSOR: Oh, yeah, why is it the disk? 492 00:25:02,430 --> 00:25:05,390 Yeah, sure. 493 00:25:05,390 --> 00:25:08,390 So first of all, why doesn't it depend on 494 00:25:08,390 --> 00:25:10,220 let's say the mass? 495 00:25:10,220 --> 00:25:13,690 So that's one thing to clear up right away. 496 00:25:13,690 --> 00:25:17,470 And one way to reason about that, maybe the best way in a 497 00:25:17,470 --> 00:25:19,770 physics class is to do it by dimensions. 498 00:25:19,770 --> 00:25:22,970 But another way is the following thought experiment. 499 00:25:22,970 --> 00:25:26,020 Which is, suppose I have two disks-- 500 00:25:26,020 --> 00:25:27,520 let's see, I might be poor today. 501 00:25:27,520 --> 00:25:29,870 I only have one disk in my pocket. 502 00:25:29,870 --> 00:25:32,370 Well here's two disks. 503 00:25:32,370 --> 00:25:33,930 The chalk is also a disk. 504 00:25:33,930 --> 00:25:36,940 So now, suppose I have one piece of chalk, and I want to 505 00:25:36,940 --> 00:25:38,690 roll it down the incline plane. 506 00:25:38,690 --> 00:25:41,570 And now I ask about a second piece of chalk. 507 00:25:41,570 --> 00:25:42,900 Let's say they're identical pieces of chalk. 508 00:25:42,900 --> 00:25:44,790 They're both going to go at the same speed. 509 00:25:44,790 --> 00:25:48,550 OK, so now suppose I stick a bit of glue on this chalk and 510 00:25:48,550 --> 00:25:50,480 that chalk, and I stick them together. 511 00:25:50,480 --> 00:25:53,110 They're going to go, again, at the same speed, as if the glue 512 00:25:53,110 --> 00:25:53,690 weren't there. 513 00:25:53,690 --> 00:25:55,170 Because the glue isn't being stressed. 514 00:25:55,170 --> 00:25:56,430 They're just moving identically. 515 00:25:56,430 --> 00:25:58,820 So what I've shown by that experiment is that a heavy 516 00:25:58,820 --> 00:26:00,850 piece of chalk moves just the same as a 517 00:26:00,850 --> 00:26:01,970 light piece of chalk. 518 00:26:01,970 --> 00:26:05,240 So it's quite plausible from that reasoning that mass 519 00:26:05,240 --> 00:26:07,110 doesn't matter. 520 00:26:07,110 --> 00:26:08,530 Why doesn't radius matter? 521 00:26:08,530 --> 00:26:11,240 Well you could do a couple of scaling arguments to show that 522 00:26:11,240 --> 00:26:14,520 the extra torque you get from being bigger is exactly 523 00:26:14,520 --> 00:26:19,570 canceled out by the extra mass because you're bigger. 524 00:26:19,570 --> 00:26:22,970 So the radius doesn't matter. 525 00:26:22,970 --> 00:26:24,080 Mass and radius don't matter. 526 00:26:24,080 --> 00:26:26,710 So then it's just a question of shape, which is a 527 00:26:26,710 --> 00:26:28,200 dimensionalist thing. 528 00:26:28,200 --> 00:26:32,300 And what it is intuitively is how concentrated is the mass 529 00:26:32,300 --> 00:26:34,700 near the center versus near the edge. 530 00:26:34,700 --> 00:26:40,010 So suppose I actually made a extreme version of the hoop. 531 00:26:40,010 --> 00:26:42,450 And actually I did this for the students in Cambridge. 532 00:26:42,450 --> 00:26:43,640 We manufactured-- 533 00:26:43,640 --> 00:26:45,820 and you could actually imagine what it is. 534 00:26:45,820 --> 00:26:47,850 So imagine this were hollow inside. 535 00:26:47,850 --> 00:26:49,960 That would be your ring or your hoop. 536 00:26:49,960 --> 00:26:52,270 Well can you make anything that would move even slower? 537 00:26:52,270 --> 00:26:54,320 Because I'm planning the hoop moves slower than the disk. 538 00:26:54,320 --> 00:26:59,170 Yes, what you do is you put giant ears on the thing. 539 00:26:59,170 --> 00:27:02,710 So it rolls like those dumbbell things you do for 540 00:27:02,710 --> 00:27:03,580 weightlifting. 541 00:27:03,580 --> 00:27:07,840 So it rolls on this radius, but it has these giant ears. 542 00:27:07,840 --> 00:27:10,450 Well it's just going to move so slowly, because it's just 543 00:27:10,450 --> 00:27:12,850 being yanked by this really weak force. 544 00:27:12,850 --> 00:27:15,320 And it has to accelerate this giant mass. 545 00:27:15,320 --> 00:27:15,830 No chance. 546 00:27:15,830 --> 00:27:16,690 So it moves really slow. 547 00:27:16,690 --> 00:27:19,600 In fact, that thing it can take 20 seconds to go down a 548 00:27:19,600 --> 00:27:21,750 meter long inclined plane. 549 00:27:21,750 --> 00:27:23,350 So that's an extreme version. 550 00:27:26,500 --> 00:27:29,490 So it rolls on this, and this is these big ears. 551 00:27:29,490 --> 00:27:30,580 So that's really slow. 552 00:27:30,580 --> 00:27:31,560 That's medium slow. 553 00:27:31,560 --> 00:27:32,910 And that's faster. 554 00:27:32,910 --> 00:27:34,870 And if you put all the mass right at the center, it would 555 00:27:34,870 --> 00:27:37,290 be even faster, because the rolling wouldn't suck up any 556 00:27:37,290 --> 00:27:39,790 of the energy. 557 00:27:39,790 --> 00:27:41,040 So that's the intuitive explanation. 558 00:27:45,470 --> 00:27:49,400 OK so now an example of a question showing a kind of 559 00:27:49,400 --> 00:27:53,000 similar misconception, a related misconception. 560 00:27:53,000 --> 00:27:55,620 But I'm going to show it to you to illustrate what you can 561 00:27:55,620 --> 00:27:58,030 do about this. 562 00:27:58,030 --> 00:27:58,910 So what can you do about it? 563 00:27:58,910 --> 00:28:00,780 So the question is the following. 564 00:28:12,620 --> 00:28:16,820 So that's a steel ball, which I have in my hand, and I drop 565 00:28:16,820 --> 00:28:19,520 on a steel table, say from a meter up high. 566 00:28:19,520 --> 00:28:20,850 Forget about air resistance. 567 00:28:20,850 --> 00:28:22,780 It bounces off the steel table. 568 00:28:22,780 --> 00:28:23,930 And the question is I'm interested in 569 00:28:23,930 --> 00:28:25,670 the forces on it. 570 00:28:25,670 --> 00:28:26,540 So what are the forces. 571 00:28:26,540 --> 00:28:29,100 So here's my table. 572 00:28:29,100 --> 00:28:49,620 So while it's falling, I'm interested in the forces on it 573 00:28:49,620 --> 00:28:52,190 at three different points in its flight. 574 00:28:52,190 --> 00:28:53,550 First while it's falling. 575 00:28:53,550 --> 00:28:56,110 I've let it go, and it hasn't yet hit. 576 00:28:56,110 --> 00:28:59,910 At the instant that it's stationary during its bounce. 577 00:28:59,910 --> 00:29:02,780 And while it's rising. 578 00:29:02,780 --> 00:29:06,070 So when you ask the students this-- and I've done this 579 00:29:06,070 --> 00:29:07,450 several times-- 580 00:29:07,450 --> 00:29:10,650 mostly they know it's just gravity while 581 00:29:10,650 --> 00:29:16,590 it's falling, mg. 582 00:29:16,590 --> 00:29:19,830 So if you actually use this question, don't ask-- 583 00:29:19,830 --> 00:29:22,200 this is chronologically next, but don't ask that next. 584 00:29:22,200 --> 00:29:25,090 Ask this one, when it's rising. 585 00:29:25,090 --> 00:29:26,860 And what you'll find-- 586 00:29:26,860 --> 00:29:27,980 what do you think many students 587 00:29:27,980 --> 00:29:30,530 say, not all but many? 588 00:29:30,530 --> 00:29:32,230 What are the forces on it while it's rising? 589 00:29:36,130 --> 00:29:37,276 Yeah, [? Adrian. ?] 590 00:29:37,276 --> 00:29:39,070 AUDIENCE: So of them will say the force goes up. 591 00:29:39,070 --> 00:29:41,990 PROFESSOR: Right, some will say the force goes up. 592 00:29:41,990 --> 00:29:44,078 They'll say mg up. 593 00:29:46,810 --> 00:29:49,110 I see people laughing, and that's a good reaction, 594 00:29:49,110 --> 00:29:51,090 because that shows you're learning something about how 595 00:29:51,090 --> 00:29:51,680 students think. 596 00:29:51,680 --> 00:29:52,620 That's kind of amazing. 597 00:29:52,620 --> 00:29:55,410 Why would they possibly say that? 598 00:29:55,410 --> 00:29:59,640 Well let's talk about this one, and you'll see that it's 599 00:29:59,640 --> 00:30:02,620 actually a similar mode of reasoning to this. 600 00:30:02,620 --> 00:30:05,170 So what you'll find is even students 601 00:30:05,170 --> 00:30:06,410 who get this correct-- 602 00:30:06,410 --> 00:30:08,400 so this isn't right. 603 00:30:08,400 --> 00:30:10,440 It's mg down. 604 00:30:10,440 --> 00:30:12,940 But there's a deep seated misconception that if 605 00:30:12,940 --> 00:30:14,690 something's moving upwards, it has to have a 606 00:30:14,690 --> 00:30:16,400 force pushing it upwards. 607 00:30:16,400 --> 00:30:18,070 And then the force somehow gets used 608 00:30:18,070 --> 00:30:20,480 up, and then it stops. 609 00:30:20,480 --> 00:30:22,040 OK, well what about here? 610 00:30:22,040 --> 00:30:25,540 So now they'll agree that there's weight pushing down. 611 00:30:25,540 --> 00:30:26,790 So that's the mg. 612 00:30:29,790 --> 00:30:31,700 And then the question is, what's the other force? 613 00:30:31,700 --> 00:30:33,190 And they call it by different names. 614 00:30:33,190 --> 00:30:35,900 In America, it's called the normal force, which is a 615 00:30:35,900 --> 00:30:37,370 terrible name. 616 00:30:37,370 --> 00:30:40,070 And in England it's called the reaction force, which is a 617 00:30:40,070 --> 00:30:41,960 terrible name for another reason. 618 00:30:41,960 --> 00:30:44,780 Which is that it implies that it's the 619 00:30:44,780 --> 00:30:47,570 result, somehow a reaction. 620 00:30:47,570 --> 00:30:49,920 So it's a terrible name. 621 00:30:49,920 --> 00:30:52,880 But let's ask the students, how big is this force? 622 00:30:52,880 --> 00:30:53,860 OK, so it's upwards. 623 00:30:53,860 --> 00:30:54,950 Everyone agrees that it's upwards. 624 00:30:54,950 --> 00:31:01,530 But how big, r or n, depending on which country you're in? 625 00:31:01,530 --> 00:31:07,140 So take a minute or two and for yourself figure out how 626 00:31:07,140 --> 00:31:12,270 big the reaction force or the normal force is, in comparison 627 00:31:12,270 --> 00:31:14,200 with the weight. 628 00:31:14,200 --> 00:31:18,330 So this is a small steel ball dropped from say a meter and 629 00:31:18,330 --> 00:31:19,650 bouncing off a steel table. 630 00:31:22,580 --> 00:31:25,980 So two questions, first figure out what you think. 631 00:31:25,980 --> 00:31:27,690 And second, figure out what you think students 632 00:31:27,690 --> 00:31:29,020 are going to say. 633 00:31:29,020 --> 00:31:32,460 OK so you're doing both, trying to get 634 00:31:32,460 --> 00:31:33,710 into the student mind. 635 00:31:38,940 --> 00:31:41,220 Let's do a quick vote. 636 00:31:41,220 --> 00:31:43,310 So what's the ratio between that normal 637 00:31:43,310 --> 00:31:44,560 force and the weight? 638 00:31:47,370 --> 00:31:48,900 So who votes for equal? 639 00:31:48,900 --> 00:31:51,350 First we'll do what you think, and then we'll do what 640 00:31:51,350 --> 00:31:52,020 students think. 641 00:31:52,020 --> 00:31:53,270 So who thinks they're equal? 642 00:31:57,600 --> 00:31:58,850 Two to one? 643 00:32:02,430 --> 00:32:06,740 Greater than 10, but not quite 1,000? 644 00:32:06,740 --> 00:32:10,840 Greater than 1,000 to 1? 645 00:32:10,840 --> 00:32:13,320 OK now what do you think students say? 646 00:32:13,320 --> 00:32:15,560 How many say one to one? 647 00:32:19,610 --> 00:32:21,080 How many say two to one? 648 00:32:23,620 --> 00:32:24,870 How many say greater than 10? 649 00:32:27,870 --> 00:32:30,840 Greater than 1,000? 650 00:32:30,840 --> 00:32:34,790 And how many are not sure what the students are going to say? 651 00:32:34,790 --> 00:32:37,600 So actually when you ask the students it depends on the 652 00:32:37,600 --> 00:32:38,230 population. 653 00:32:38,230 --> 00:32:45,870 But generally, what I find is it's 90% for one to one, And 654 00:32:45,870 --> 00:32:49,320 about 10% for two to one. 655 00:32:49,320 --> 00:32:52,130 And no one says anything else. 656 00:32:52,130 --> 00:32:54,790 OK so now why would students say one to one? 657 00:32:57,310 --> 00:33:00,640 What's the reasoning going on in their mind? 658 00:33:00,640 --> 00:33:02,420 Has no velocity. 659 00:33:02,420 --> 00:33:05,160 So that is exactly the same reasoning as 660 00:33:05,160 --> 00:33:07,340 this, with mg upwards. 661 00:33:07,340 --> 00:33:09,470 But what's interesting is even the students who know it's mg 662 00:33:09,470 --> 00:33:13,830 downwards still say, well the velocity is zero. 663 00:33:13,830 --> 00:33:16,130 So v equals zero. 664 00:33:16,130 --> 00:33:17,790 Therefore, it's in equilibrium. 665 00:33:17,790 --> 00:33:19,890 Therefore, there's no forces acting on it. 666 00:33:19,890 --> 00:33:22,520 Therefore, the reaction force, or the normal force, and the 667 00:33:22,520 --> 00:33:25,800 weight have to cancel. 668 00:33:25,800 --> 00:33:30,030 N equals mg because it's stopped. 669 00:33:30,030 --> 00:33:34,920 So that what I call that is I call that the f equals mv 670 00:33:34,920 --> 00:33:37,550 theory of physics. 671 00:33:37,550 --> 00:33:39,270 Now maybe like in the linguistic books, I'll put a 672 00:33:39,270 --> 00:33:42,210 star by that saying don't say that. 673 00:33:42,210 --> 00:33:44,870 That's not how you translate that sentence into real life. 674 00:33:44,870 --> 00:33:46,500 But why f equals mv? 675 00:33:46,500 --> 00:33:48,330 Why is it so deep seeded? 676 00:33:48,330 --> 00:33:51,760 Well we have a gazillion hours of experience with it. 677 00:33:51,760 --> 00:33:55,930 Suppose I move a object, and I push it. 678 00:33:55,930 --> 00:33:58,180 Stop pushing it, and it stops moving. 679 00:33:58,180 --> 00:34:00,800 If I have a light object, I just push it a 680 00:34:00,800 --> 00:34:01,880 little less and it moves. 681 00:34:01,880 --> 00:34:03,640 And I stop pushing, and it stops moving. 682 00:34:03,640 --> 00:34:08,130 So f is related to m, and f is related to v. This seems to 683 00:34:08,130 --> 00:34:10,110 explain most of our world. 684 00:34:10,110 --> 00:34:11,360 It's hard to get around that. 685 00:34:11,360 --> 00:34:14,310 So the question is, how do you actually try to force the 686 00:34:14,310 --> 00:34:17,460 students to see that that's actually completely junk what 687 00:34:17,460 --> 00:34:21,150 they said, and that it can't be equal? 688 00:34:21,150 --> 00:34:26,500 And the way I want to do it is I want to do it in a way that 689 00:34:26,500 --> 00:34:29,440 recombines their intuitive model and their school model. 690 00:34:29,440 --> 00:34:31,250 I want to meld them together. 691 00:34:31,250 --> 00:34:33,350 So the way I do that is the following, 692 00:34:33,350 --> 00:34:35,070 which is I get a rock. 693 00:34:35,070 --> 00:34:36,850 So let's call this a rock. 694 00:34:36,850 --> 00:34:38,659 And I ask one of the students, could I 695 00:34:38,659 --> 00:34:40,239 please borrow your hand? 696 00:34:40,239 --> 00:34:41,659 And so somebody gives me their hand. 697 00:34:41,659 --> 00:34:44,310 And I say, OK, I'm going to put the rock in your hand. 698 00:34:44,310 --> 00:34:47,560 Now, what's the weight of the rock? 699 00:34:47,560 --> 00:34:47,825 mg. 700 00:34:47,825 --> 00:34:48,360 OK, great. 701 00:34:48,360 --> 00:34:50,580 So what's the force of the rock on your hand? mg. 702 00:34:50,580 --> 00:34:52,110 Great. 703 00:34:52,110 --> 00:34:56,300 Now I hold the rock about the same place I'm going to drop 704 00:34:56,300 --> 00:34:58,770 the steel ball, and I say, OK on the count of three, I'm 705 00:34:58,770 --> 00:35:00,630 going to drop the rock on your hand. 706 00:35:00,630 --> 00:35:03,080 One, two, and then I'm ready. 707 00:35:03,080 --> 00:35:04,120 What are they going to do? 708 00:35:04,120 --> 00:35:05,130 Like that. 709 00:35:05,130 --> 00:35:07,180 And soon as they try to do that, I grab their hand and 710 00:35:07,180 --> 00:35:08,080 say, wait a minute. 711 00:35:08,080 --> 00:35:10,280 Let's reason about this with physics. 712 00:35:10,280 --> 00:35:12,330 Why are you moving your hand? 713 00:35:12,330 --> 00:35:17,570 You just told me that when the thing bounces, it's still mg. 714 00:35:17,570 --> 00:35:20,010 The reaction force and the normal force is still mg. 715 00:35:20,010 --> 00:35:22,600 And you just showed me that mg on your hand 716 00:35:22,600 --> 00:35:23,590 didn't hurt your hand. 717 00:35:23,590 --> 00:35:25,620 You could hold the rock on your hand no problem. 718 00:35:25,620 --> 00:35:27,250 So why are you moving your hand? 719 00:35:27,250 --> 00:35:29,680 So then we'll have a discussion, and many of them 720 00:35:29,680 --> 00:35:31,530 will then change to two to one. 721 00:35:31,530 --> 00:35:34,420 They'll say, oh, well it's actually two to one. 722 00:35:34,420 --> 00:35:36,840 And I say OK, two to one, well I have an answer for that. 723 00:35:36,840 --> 00:35:38,740 Two rocks. 724 00:35:38,740 --> 00:35:41,830 So I have two rocks, and I put them on their hand. 725 00:35:41,830 --> 00:35:46,410 So now compared to the first rock, mg, 2mg. 726 00:35:46,410 --> 00:35:47,820 So now I take the first rock, and I'm going to 727 00:35:47,820 --> 00:35:49,090 drop it on your hand. 728 00:35:49,090 --> 00:35:50,530 And one, two, three. 729 00:35:50,530 --> 00:35:51,700 And they try to move. 730 00:35:51,700 --> 00:35:52,740 And I say wait, wait. 731 00:35:52,740 --> 00:35:54,370 You just told me 2mg was fine. 732 00:35:54,370 --> 00:35:56,180 This felt fine to you, right? 733 00:35:56,180 --> 00:35:59,780 And they say, I know it's going to hurt like hell. 734 00:35:59,780 --> 00:36:04,230 So now what they're doing is they now have some kind of 735 00:36:04,230 --> 00:36:06,850 contradiction that's evident between the school model and 736 00:36:06,850 --> 00:36:07,890 the intuitive model. 737 00:36:07,890 --> 00:36:13,010 And now is your opportunity to introduce and explain, yeah 738 00:36:13,010 --> 00:36:14,040 you're right. 739 00:36:14,040 --> 00:36:17,550 Your intuitive model is actually correct in this case. 740 00:36:17,550 --> 00:36:20,240 And your school model reasoning is incorrect. 741 00:36:20,240 --> 00:36:21,810 So actually what we're going to do is try to move the 742 00:36:21,810 --> 00:36:24,440 school model towards a piece of your intuitive model that 743 00:36:24,440 --> 00:36:25,290 is correct. 744 00:36:25,290 --> 00:36:27,690 We're trying to get away from the f equals mv part of the 745 00:36:27,690 --> 00:36:31,690 intuitive model and emphasize the part that does know. 746 00:36:31,690 --> 00:36:33,990 And the part that says, I don't want to have that rock 747 00:36:33,990 --> 00:36:35,730 in my hand is correct. 748 00:36:35,730 --> 00:36:39,600 So then they're very ready to do a school model calculation 749 00:36:39,600 --> 00:36:41,500 of how big the force is. 750 00:36:41,500 --> 00:36:44,850 And it turns out to be something like 100,000 or 751 00:36:44,850 --> 00:36:47,200 10,000 times the weight. 752 00:36:47,200 --> 00:36:51,090 So it's huge because the impact time is so short. 753 00:36:51,090 --> 00:36:54,070 And then to make that plausible to them, we can 754 00:36:54,070 --> 00:36:56,770 actually calculate the impact time or estimate it. 755 00:36:56,770 --> 00:36:59,530 But then you can actually say, look you already know this. 756 00:36:59,530 --> 00:37:01,260 You intuitively understand that. 757 00:37:01,260 --> 00:37:03,742 Because if you-- 758 00:37:03,742 --> 00:37:05,570 again I'll stand on the table. 759 00:37:08,990 --> 00:37:13,080 Suppose you're going to jump off an object like this, what 760 00:37:13,080 --> 00:37:15,050 do you do when you land? 761 00:37:15,050 --> 00:37:16,940 What does everyone do? 762 00:37:16,940 --> 00:37:17,680 Bend your knees. 763 00:37:17,680 --> 00:37:19,610 Right, so you do that. 764 00:37:19,610 --> 00:37:20,710 And what does that do? 765 00:37:20,710 --> 00:37:22,995 That increases the contact time. 766 00:37:22,995 --> 00:37:25,030 And why are you increasing the contact time? 767 00:37:25,030 --> 00:37:28,430 Because that decreases the acceleration. 768 00:37:28,430 --> 00:37:32,000 The acceleration is a change in velocity divided by the 769 00:37:32,000 --> 00:37:33,020 contact time. 770 00:37:33,020 --> 00:37:36,150 So you can't affect the change in velocity very much. 771 00:37:36,150 --> 00:37:38,540 That's either v or 2v or something like that. 772 00:37:38,540 --> 00:37:40,340 But the content time you can change. 773 00:37:40,340 --> 00:37:42,020 So with the steel ball, contact time is 774 00:37:42,020 --> 00:37:43,350 actually very short. 775 00:37:43,350 --> 00:37:45,110 Because the speed of sound is so fast. 776 00:37:45,110 --> 00:37:48,470 So the impact force is really high. 777 00:37:48,470 --> 00:37:52,010 So yes, your intuitive model was correct. 778 00:37:52,010 --> 00:37:53,920 And now we're yanking the school model towards it. 779 00:37:53,920 --> 00:37:56,360 So you actually bring the two together. 780 00:37:56,360 --> 00:37:58,540 And you fight the divergence. 781 00:37:58,540 --> 00:38:02,290 So you want to look for examples like that 782 00:38:02,290 --> 00:38:03,970 whenever you can. 783 00:38:03,970 --> 00:38:07,900 And to summarize the lesson-- because it was pointed out 784 00:38:07,900 --> 00:38:10,130 that I should summarize more often-- 785 00:38:10,130 --> 00:38:15,010 the summary of the entire theme about misconceptions is 786 00:38:15,010 --> 00:38:18,380 that you need to understand that so that 787 00:38:18,380 --> 00:38:19,360 you can teach properly. 788 00:38:19,360 --> 00:38:22,930 Because there's no way to deduce this without doing 789 00:38:22,930 --> 00:38:24,620 that, which we're going to talk about later. 790 00:38:24,620 --> 00:38:27,350 But even knowing that, if you don't know this, you can't run 791 00:38:27,350 --> 00:38:29,840 the system backwards and figure out what goes here. 792 00:38:29,840 --> 00:38:34,410 And so then, how do you actually learn about this? 793 00:38:34,410 --> 00:38:35,960 Well you've done one way. 794 00:38:35,960 --> 00:38:37,450 You've done one way already. 795 00:38:37,450 --> 00:38:41,370 Let me write down two or three ways for you. 796 00:38:44,460 --> 00:38:52,810 So the question is, how to learn about s, 797 00:38:52,810 --> 00:38:53,920 the students' system. 798 00:38:53,920 --> 00:38:55,270 Well one way is to read. 799 00:38:58,930 --> 00:39:01,990 So in many fields, there's lots of research about what 800 00:39:01,990 --> 00:39:04,760 particular misconceptions students have. 801 00:39:04,760 --> 00:39:08,430 And I've given you some of those readings on the website. 802 00:39:08,430 --> 00:39:10,450 But there's a whole vast body of literature, and 803 00:39:10,450 --> 00:39:11,650 many fields have that. 804 00:39:11,650 --> 00:39:13,270 So you can look at that. 805 00:39:13,270 --> 00:39:15,920 Another way, and this is-- 806 00:39:21,970 --> 00:39:24,540 so this is shut your mouth. 807 00:39:24,540 --> 00:39:25,540 Now why is that? 808 00:39:25,540 --> 00:39:28,090 Well there's no way to really understand what students are 809 00:39:28,090 --> 00:39:30,030 thinking unless you ask them. 810 00:39:30,030 --> 00:39:33,310 And if you don't let them talk, you're not going to 811 00:39:33,310 --> 00:39:35,690 actually hear what they're thinking. 812 00:39:35,690 --> 00:39:37,330 You'll actually put words into their mouth. 813 00:39:37,330 --> 00:39:40,290 Because you won't imagine that they think f equals mv. 814 00:39:40,290 --> 00:39:41,720 You never would imagine that. 815 00:39:41,720 --> 00:39:43,790 So you have to shut your mouth and listen. 816 00:39:46,430 --> 00:39:50,860 So office hours is a chance for that. 817 00:39:50,860 --> 00:39:53,560 And related to that is the feedback sheet. 818 00:39:58,790 --> 00:40:01,660 If students get in the habit of asking you questions about 819 00:40:01,660 --> 00:40:04,080 what was confusing, you start to build a model of the 820 00:40:04,080 --> 00:40:06,765 students' system, what worked and what didn't work and what 821 00:40:06,765 --> 00:40:07,850 was confusing for them. 822 00:40:07,850 --> 00:40:12,640 So all of these ways are ways to build models of s, which is 823 00:40:12,640 --> 00:40:17,720 absolutely necessary for deducing t. 824 00:40:17,720 --> 00:40:18,970 Any questions? 825 00:40:24,300 --> 00:40:27,070 OK so if you could do two things. 826 00:40:27,070 --> 00:40:29,070 One is to spend one minute filling out 827 00:40:29,070 --> 00:40:31,190 the feedback sheet. 828 00:40:31,190 --> 00:40:37,080 And the other is everyone who has a homework equation 829 00:40:37,080 --> 00:40:40,010 treatment, please raise your hand. 830 00:40:40,010 --> 00:40:40,810 OK that's great. 831 00:40:40,810 --> 00:40:45,070 So now just find one person near you to swap with. 832 00:40:45,070 --> 00:40:46,160 Question? 833 00:40:46,160 --> 00:40:46,610 Yeah, OK. 834 00:40:46,610 --> 00:40:47,100 So that's great. 835 00:40:47,100 --> 00:40:50,620 So just find one person who has one near you to swap with 836 00:40:50,620 --> 00:40:52,300 and give each other comments. 837 00:40:52,300 --> 00:40:57,690 And then just write your name over the next week and bring 838 00:40:57,690 --> 00:41:00,590 it in next time with you. 839 00:41:00,590 --> 00:41:02,690 So basically this is a way-- 840 00:41:02,690 --> 00:41:04,810 and if there's no one near you, or they've already 841 00:41:04,810 --> 00:41:06,370 swapped, just make a group of three. 842 00:41:06,370 --> 00:41:09,360 It's no big deal. 843 00:41:09,360 --> 00:41:11,190 Does anyone not have someone to swap with? 844 00:41:13,700 --> 00:41:14,670 OK great. 845 00:41:14,670 --> 00:41:17,030 And so introduce yourself to the other person. 846 00:41:17,030 --> 00:41:18,980 It doesn't matter if you're in different fields. 847 00:41:18,980 --> 00:41:21,900 It often can be an advantage, because you'll get a fresh set 848 00:41:21,900 --> 00:41:24,360 of eyes on it. 849 00:41:24,360 --> 00:41:26,990 And I'll give you some directions by email about what 850 00:41:26,990 --> 00:41:28,620 to look for and put it on the website. 851 00:41:28,620 --> 00:41:31,150 But generally just give each other suggestions and work 852 00:41:31,150 --> 00:41:33,690 together on what you thought was a good 853 00:41:33,690 --> 00:41:34,350 and what wasn't good. 854 00:41:34,350 --> 00:41:37,300 And just write your name on the person you commented on. 855 00:41:37,300 --> 00:41:38,870 And then you'll bring them both in. 856 00:41:38,870 --> 00:41:42,800 You'll bring the commented sheet from the other person in 857 00:41:42,800 --> 00:41:44,610 the next time. 858 00:41:44,610 --> 00:41:46,470 Or you'll just give it back to them and bring it. 859 00:41:46,470 --> 00:41:48,110 This is not a big police exercise. 860 00:41:48,110 --> 00:41:51,150 I just want you to meet somebody else and have 861 00:41:51,150 --> 00:41:52,630 practice discussing these things and 862 00:41:52,630 --> 00:41:54,770 learn from each other. 863 00:41:54,770 --> 00:41:56,470 So questions about that? 864 00:41:59,910 --> 00:42:04,540 So if you could finish filling out the sheets and bring the 865 00:42:04,540 --> 00:42:07,970 sheets either here or here, I'll pack up. 866 00:42:07,970 --> 00:42:11,120 And meanwhile we'll have any questions that people want to 867 00:42:11,120 --> 00:42:14,830 talk to me about individually outside, because I think 868 00:42:14,830 --> 00:42:17,350 there's another class that 's going to come streaming in 869 00:42:17,350 --> 00:42:20,490 with like 150 people. 870 00:42:20,490 --> 00:42:24,050 So remember, learn about the student system. 871 00:42:24,050 --> 00:42:27,480 It's probably the most important thing to do to make 872 00:42:27,480 --> 00:42:29,220 your teaching stand out and have long 873 00:42:29,220 --> 00:42:30,470 lasting, good effects. 874 00:42:32,980 --> 00:42:36,030 NARRATOR: Answers from lecture four to questions generated in 875 00:42:36,030 --> 00:42:37,280 lecture three. 876 00:42:40,350 --> 00:42:43,660 PROFESSOR: So first, questions from before. 877 00:42:43,660 --> 00:42:44,870 One of the comments-- 878 00:42:44,870 --> 00:42:48,970 well, let's see, two of them were that could we have less 879 00:42:48,970 --> 00:42:52,380 time for questions, but also they're very helpful. 880 00:42:52,380 --> 00:42:53,730 So there's always a tension there. 881 00:42:53,730 --> 00:42:57,740 So what I'm going to try to do is shorten some of the total 882 00:42:57,740 --> 00:43:01,080 answers to the questions and then maybe put some answers to 883 00:43:01,080 --> 00:43:06,050 the questions towards the end of the lecture as well. 884 00:43:06,050 --> 00:43:09,230 I do like to try to answer most of the questions. 885 00:43:09,230 --> 00:43:12,720 And the reason being is that it provides also pacing. 886 00:43:12,720 --> 00:43:14,090 And you'll find that too. 887 00:43:14,090 --> 00:43:17,180 If you find yourself generating four times as many 888 00:43:17,180 --> 00:43:20,150 questions as you have time to answer the next time, it 889 00:43:20,150 --> 00:43:23,160 probably means you did way too much stuff the previous time. 890 00:43:23,160 --> 00:43:26,980 So it gives you feedback signal on how much material to 891 00:43:26,980 --> 00:43:29,080 discuss in the class time. 892 00:43:32,880 --> 00:43:35,690 And so that's one reason I'm reluctant to just put all the 893 00:43:35,690 --> 00:43:36,970 answers on the web. 894 00:43:36,970 --> 00:43:40,620 Because that's sort of like teaching with slides where you 895 00:43:40,620 --> 00:43:43,270 can just flip through all the equations really fast, and it 896 00:43:43,270 --> 00:43:45,070 seems like you got through a lot of equations. 897 00:43:45,070 --> 00:43:47,680 But actually no one really understood the equations. 898 00:43:47,680 --> 00:43:50,240 So it's sort of a symptomatic treatment for a more 899 00:43:50,240 --> 00:43:52,500 fundamental problem which may be that I'm doing too many 900 00:43:52,500 --> 00:43:55,490 things and not allowing enough time for questions in the 901 00:43:55,490 --> 00:43:56,740 class itself. 902 00:43:59,330 --> 00:44:01,120 A general question, which is, how do you 903 00:44:01,120 --> 00:44:02,750 evaluate your own teaching? 904 00:44:02,750 --> 00:44:06,270 When you feel good or bad about how class just went? 905 00:44:06,270 --> 00:44:09,090 And what metrics do I use popularity, covering what I 906 00:44:09,090 --> 00:44:10,650 set out to? 907 00:44:10,650 --> 00:44:11,770 How do I know? 908 00:44:11,770 --> 00:44:14,610 So one major way is I use these. 909 00:44:17,430 --> 00:44:20,690 It's not really that I say, oh did people hate it or like it? 910 00:44:20,690 --> 00:44:22,700 I mean, I want to know if people hated it or liked it. 911 00:44:22,700 --> 00:44:25,680 And I just I don't just add up the likes and the hates and 912 00:44:25,680 --> 00:44:28,690 say OK, 10 likes minus 3 hates equals 7. 913 00:44:28,690 --> 00:44:30,290 That's great. 914 00:44:30,290 --> 00:44:34,200 You're batting 80%. 915 00:44:34,200 --> 00:44:37,940 But I try to get a sense from reading the comments of what 916 00:44:37,940 --> 00:44:40,570 people were confused about, what was interesting to 917 00:44:40,570 --> 00:44:45,330 people, and sort of integrate that up and imagine the whole 918 00:44:45,330 --> 00:44:47,860 class as one giant student mind. 919 00:44:47,860 --> 00:44:51,660 And I want to know how I connect the material to the 920 00:44:51,660 --> 00:44:53,300 collective student mind. 921 00:44:53,300 --> 00:44:57,040 So I evaluate it that way, sort of intuitively from the 922 00:44:57,040 --> 00:44:58,960 comments that people make on the sheet. 923 00:44:58,960 --> 00:45:01,240 Which is why I hand out the sheet every time, and I think 924 00:45:01,240 --> 00:45:02,490 the sheet is so valuable. 925 00:45:05,180 --> 00:45:10,110 Now another way, covering the materials I set out to. 926 00:45:10,110 --> 00:45:13,320 Well I'm probably sort of on one extreme of that, which I 927 00:45:13,320 --> 00:45:16,240 don't really care how much material I covered. 928 00:45:16,240 --> 00:45:19,030 But let me explain why I'm on that extreme. 929 00:45:19,030 --> 00:45:22,370 So some people only care about how much material is covered. 930 00:45:22,370 --> 00:45:25,550 And some people are more on my end. 931 00:45:25,550 --> 00:45:26,925 So I'll give you the reason. 932 00:45:26,925 --> 00:45:28,810 And most people I would say on this edge. 933 00:45:28,810 --> 00:45:30,640 So I don't need to really justify that, because you've 934 00:45:30,640 --> 00:45:32,220 heard lots of reasons for that. 935 00:45:32,220 --> 00:45:37,080 But it's the other side, why do I not care? 936 00:45:37,080 --> 00:45:41,580 Well I don't care so much just because I think class time and 937 00:45:41,580 --> 00:45:46,310 a lecture course is such a small part of a student's life 938 00:45:46,310 --> 00:45:49,860 that if you haven't connected to the students, if you 939 00:45:49,860 --> 00:45:52,450 haven't made it click for them and got them interested, made 940 00:45:52,450 --> 00:45:55,460 them want to continue studying, then you've lost 941 00:45:55,460 --> 00:45:57,210 most of the effect of the teaching. 942 00:45:57,210 --> 00:45:59,760 The effect of the teaching isn't really going to be so 943 00:45:59,760 --> 00:46:01,000 much in the class. 944 00:46:01,000 --> 00:46:04,450 Because what you can do is you can kindle interest, kindle a 945 00:46:04,450 --> 00:46:06,040 way of looking at the world. 946 00:46:06,040 --> 00:46:08,360 But it's really up to the students to carry that on 947 00:46:08,360 --> 00:46:09,060 afterwards. 948 00:46:09,060 --> 00:46:12,600 And if you've actually bored them, or they've decided that 949 00:46:12,600 --> 00:46:14,860 this whole material is useless, they're 950 00:46:14,860 --> 00:46:15,930 not going to do that. 951 00:46:15,930 --> 00:46:18,090 And then it's all just going to go away anyway. 952 00:46:18,090 --> 00:46:20,840 So actually what I'm much more interested in is how much I've 953 00:46:20,840 --> 00:46:21,830 kindled those interests. 954 00:46:21,830 --> 00:46:25,610 And that's what I try to judge using the sheets. 955 00:46:25,610 --> 00:46:28,430 And I think that's actually what produces the long term 956 00:46:28,430 --> 00:46:29,680 change in ways of thinking. 957 00:46:32,310 --> 00:46:37,360 Another comment was that the rock experiment rocked, which 958 00:46:37,360 --> 00:46:39,960 I thought was a good pun. 959 00:46:39,960 --> 00:46:43,190 If you're not willing to write out detailed derivations on 960 00:46:43,190 --> 00:46:46,010 the board, is it more important to have more detail 961 00:46:46,010 --> 00:46:48,070 published lecture notes? 962 00:46:48,070 --> 00:46:49,660 And yeah I think that's right. 963 00:46:49,660 --> 00:46:52,360 In fact, that's generally necessary. 964 00:46:52,360 --> 00:46:54,390 But it doesn't have to be your own published notes. 965 00:46:54,390 --> 00:46:56,850 It could just be a book you refer people to. 966 00:46:56,850 --> 00:47:01,790 So we've sort of gone backwards from Gutenberg. 967 00:47:01,790 --> 00:47:05,630 In the days before Gutenberg, everyone read out their notes. 968 00:47:05,630 --> 00:47:07,250 That was what monasteries did. 969 00:47:07,250 --> 00:47:08,650 You read out the bible. 970 00:47:08,650 --> 00:47:11,420 And at the end of a year or two of doing that, you had 50 971 00:47:11,420 --> 00:47:13,850 bibles, because everyone took notes on what you read. 972 00:47:13,850 --> 00:47:19,410 That was the old photocopier, back in say the year 1300. 973 00:47:19,410 --> 00:47:23,160 And then when Gutenberg came along the universities then 974 00:47:23,160 --> 00:47:24,280 did the same thing. 975 00:47:24,280 --> 00:47:25,440 That's what universities were. 976 00:47:25,440 --> 00:47:26,840 They basically grew out of the monasteries. 977 00:47:26,840 --> 00:47:31,050 Then Gutenberg came along and said, oh actually books are 978 00:47:31,050 --> 00:47:33,340 cheap now, well much cheaper than before. 979 00:47:33,340 --> 00:47:36,570 Books used to cost roughly say $5,000 a 980 00:47:36,570 --> 00:47:38,540 book in today's dollars. 981 00:47:38,540 --> 00:47:41,750 But then with Gutenberg and automatic printing, they could 982 00:47:41,750 --> 00:47:42,670 be much cheaper. 983 00:47:42,670 --> 00:47:45,430 So you can-- and we've developed printing technology 984 00:47:45,430 --> 00:47:46,410 much farther. 985 00:47:46,410 --> 00:47:50,160 So actually for a while until word processing came along and 986 00:47:50,160 --> 00:47:53,090 typesetting on your computer, people would actually just 987 00:47:53,090 --> 00:47:55,190 refer people to books and say, OK you're a student. 988 00:47:55,190 --> 00:47:57,090 You are expected to read these books. 989 00:47:57,090 --> 00:47:59,615 But now that people can make their own notes, people have 990 00:47:59,615 --> 00:48:03,330 sort of forgot how to tell people to use books. 991 00:48:03,330 --> 00:48:07,750 And instead people, the lecturers, just tend to write 992 00:48:07,750 --> 00:48:08,910 their own notes out. 993 00:48:08,910 --> 00:48:12,240 And the students actually seem for some crazy reason to 994 00:48:12,240 --> 00:48:16,760 basically except notes specifically for that topic 995 00:48:16,760 --> 00:48:19,230 and don't want to read books. 996 00:48:19,230 --> 00:48:21,470 So it gets really strange. 997 00:48:21,470 --> 00:48:23,870 You get in these bizarre cycles where you make notes. 998 00:48:23,870 --> 00:48:26,700 And as long as you have notes that seem like they're key to 999 00:48:26,700 --> 00:48:28,310 the lecture, that's great. 1000 00:48:28,310 --> 00:48:29,370 Students will read them. 1001 00:48:29,370 --> 00:48:31,430 And then you assemble all the notes for the very same 1002 00:48:31,430 --> 00:48:33,540 lectures into a book, and now suppose 1003 00:48:33,540 --> 00:48:35,180 the book gets published. 1004 00:48:35,180 --> 00:48:37,870 The students will say, well now we want notes again. 1005 00:48:37,870 --> 00:48:41,560 You can't say, well there's the book which was the notes. 1006 00:48:41,560 --> 00:48:44,170 Because it's not specifically targeted. 1007 00:48:44,170 --> 00:48:46,890 So part of the cure for that is reading memos. 1008 00:48:46,890 --> 00:48:49,890 Or any way of teaching students to do reading is to 1009 00:48:49,890 --> 00:48:52,870 shift back to taking advantage of Gutenberg. 1010 00:48:52,870 --> 00:48:54,650 And then once we take advantage of Gutenberg, we can 1011 00:48:54,650 --> 00:48:57,880 worry about taking advantage of all the web technology and 1012 00:48:57,880 --> 00:48:59,980 word processing technology that we have. 1013 00:48:59,980 --> 00:49:02,710 But yeah, it is important, to answer the question directly, 1014 00:49:02,710 --> 00:49:06,210 to have the details somewhere for the students. 1015 00:49:06,210 --> 00:49:08,620 And the best place generally is in printed material, 1016 00:49:08,620 --> 00:49:12,160 whether in a book or in something you write. 1017 00:49:12,160 --> 00:49:16,160 Because writing it on the board is very noisy. 1018 00:49:16,160 --> 00:49:19,850 By which I mean that it's very easy to mis-copy. 1019 00:49:19,850 --> 00:49:22,030 It's easy for you to make mistakes. 1020 00:49:22,030 --> 00:49:23,850 It's easier for the student to make mistakes when they're 1021 00:49:23,850 --> 00:49:24,940 taking notes. 1022 00:49:24,940 --> 00:49:29,830 It's best to have it all in print, and also it's best to 1023 00:49:29,830 --> 00:49:32,790 put the high level stuff in lecture, and have all the 1024 00:49:32,790 --> 00:49:38,410 details with the students in their own comfortable space 1025 00:49:38,410 --> 00:49:41,780 when they have time to go back and forth. 1026 00:49:41,780 --> 00:49:46,370 OK, so then there were several questions about how do you 1027 00:49:46,370 --> 00:49:50,010 meld the intuitive model and the school model? 1028 00:49:50,010 --> 00:49:51,080 Are their general principles? 1029 00:49:51,080 --> 00:49:54,850 So I talked about how for example with the rock example 1030 00:49:54,850 --> 00:50:00,670 that the school model is that when the rock isn't moving, 1031 00:50:00,670 --> 00:50:03,190 the force is zero because it's in equilibrium. 1032 00:50:03,190 --> 00:50:05,060 So the net force is zero. 1033 00:50:05,060 --> 00:50:08,640 So the reaction force equals the weight. 1034 00:50:08,640 --> 00:50:10,000 So that's the school model. 1035 00:50:10,000 --> 00:50:14,270 The intuitive model, somewhere in there is an intuitive model 1036 00:50:14,270 --> 00:50:16,850 that you don't want the rock to fall on your hand. 1037 00:50:16,850 --> 00:50:19,080 So what's a general ideas behind 1038 00:50:19,080 --> 00:50:20,060 melding those two models? 1039 00:50:20,060 --> 00:50:23,690 So that demonstration I showed of borrowing someone's hand 1040 00:50:23,690 --> 00:50:25,850 and dropping a rock on it, well not quite dropping the 1041 00:50:25,850 --> 00:50:28,210 rock on it, but threatening to, Is one way. 1042 00:50:28,210 --> 00:50:31,020 But what are the general principles behind that? 1043 00:50:31,020 --> 00:50:35,710 So the general principles are that you want to connect to 1044 00:50:35,710 --> 00:50:38,530 where all the mental hardware is. 1045 00:50:38,530 --> 00:50:43,196 So generally symbolic processing it's a sort of 1046 00:50:43,196 --> 00:50:45,460 surface phenomenon in the brain. 1047 00:50:45,460 --> 00:50:47,900 It's linguistic. 1048 00:50:47,900 --> 00:50:52,280 It's only about 100,000 years old, whereas perceptual 1049 00:50:52,280 --> 00:50:56,850 reasoning, visual reasoning, visceral reasoning, those are 1050 00:50:56,850 --> 00:50:58,650 hundreds of millions of years old. 1051 00:50:58,650 --> 00:51:01,790 Organisms have been seeing things for hundreds of 1052 00:51:01,790 --> 00:51:03,010 millions of years. 1053 00:51:03,010 --> 00:51:05,540 So there's a huge difference in the amount of 1054 00:51:05,540 --> 00:51:06,790 hardware that we have. 1055 00:51:14,790 --> 00:51:17,120 So symbolic hardware say has evolved for 1056 00:51:17,120 --> 00:51:19,620 10 to the five years. 1057 00:51:19,620 --> 00:51:29,880 And perceptual, rough estimates 10 to the five 1058 00:51:29,880 --> 00:51:30,710 versus 10 to eight years. 1059 00:51:30,710 --> 00:51:34,640 So there's about 1,000 times more evolutionary help for 1060 00:51:34,640 --> 00:51:35,620 these things. 1061 00:51:35,620 --> 00:51:37,750 So if you want to meld-- 1062 00:51:37,750 --> 00:51:41,040 and this is basically where the intuitive model, the 1063 00:51:41,040 --> 00:51:42,560 internal model lives. 1064 00:51:42,560 --> 00:51:46,170 So if you want to change it, you have to connect to where 1065 00:51:46,170 --> 00:51:46,890 the model lives. 1066 00:51:46,890 --> 00:51:49,000 You have to do things that are perceptual. 1067 00:51:49,000 --> 00:51:53,270 So visual arguments, visceral arguments, auditory 1068 00:51:53,270 --> 00:51:56,930 demonstrations, things they create conflict. 1069 00:51:56,930 --> 00:51:59,440 Somehow some kind of conflict is also another general 1070 00:51:59,440 --> 00:52:03,350 principle, where you expose some kind of obvious 1071 00:52:03,350 --> 00:52:04,650 contradiction. 1072 00:52:04,650 --> 00:52:08,190 And that's so beneficial, because that's self teaching. 1073 00:52:08,190 --> 00:52:10,820 As long as there's a contradiction, the students 1074 00:52:10,820 --> 00:52:13,250 know that the problem isn't done. 1075 00:52:13,250 --> 00:52:16,290 So in the rock example, there's a contradiction 1076 00:52:16,290 --> 00:52:19,740 between, I don't want the rock dropping on my 1077 00:52:19,740 --> 00:52:22,310 hand on the one hand-- 1078 00:52:22,310 --> 00:52:24,100 that's the perceptual view-- 1079 00:52:24,100 --> 00:52:28,760 with the symbolic one, which is net force is zero so the 1080 00:52:28,760 --> 00:52:31,130 reaction force equals mg. 1081 00:52:31,130 --> 00:52:32,630 So there's a contradiction. 1082 00:52:32,630 --> 00:52:34,880 As long as you can't get the same answer by two different 1083 00:52:34,880 --> 00:52:36,830 ways, you know you still have more to learn. 1084 00:52:36,830 --> 00:52:37,800 So it's self-teaching. 1085 00:52:37,800 --> 00:52:38,940 You don't have to wait for the teacher to 1086 00:52:38,940 --> 00:52:40,240 say, oh wait a minute. 1087 00:52:40,240 --> 00:52:41,150 That's not the right answer. 1088 00:52:41,150 --> 00:52:42,720 You know on your own. 1089 00:52:42,720 --> 00:52:43,620 So know on your own. 1090 00:52:43,620 --> 00:52:45,670 You internally feel that there's something wrong. 1091 00:52:45,670 --> 00:52:48,340 So you're tapping in, I would say 1092 00:52:48,340 --> 00:52:50,160 contradictions and puzzles. 1093 00:52:50,160 --> 00:52:52,570 They're somewhat symbolic, but they're also kind of visceral. 1094 00:52:52,570 --> 00:52:54,940 You feel them like oh, my god that can't be right. 1095 00:52:54,940 --> 00:52:57,740 So if you can tap into that as well that's another way of 1096 00:52:57,740 --> 00:52:58,990 trying to meld the models. 1097 00:53:03,130 --> 00:53:03,710 It's interesting. 1098 00:53:03,710 --> 00:53:06,280 I think it's a fundamentally important point about teaching 1099 00:53:06,280 --> 00:53:07,340 these two models. 1100 00:53:07,340 --> 00:53:11,090 And I've never seen it discussed anywhere. 1101 00:53:11,090 --> 00:53:14,040 So you're the sort of first people to hear it officially 1102 00:53:14,040 --> 00:53:15,970 as far as I can tell. 1103 00:53:15,970 --> 00:53:18,980 So any questions actually about that model? 1104 00:53:18,980 --> 00:53:20,330 Because I think it is so important. 1105 00:53:24,220 --> 00:53:24,960 Yes. 1106 00:53:24,960 --> 00:53:27,785 AUDIENCE: I'm just curious, what if you're in a field that 1107 00:53:27,785 --> 00:53:30,731 doesn't lend itself to perceptual-- 1108 00:53:30,731 --> 00:53:33,677 like if you're in a field that tends to be more symbolic. 1109 00:53:33,677 --> 00:53:35,150 And physics is really nice because it has this physical 1110 00:53:35,150 --> 00:53:37,058 aspect to it. 1111 00:53:37,058 --> 00:53:40,050 So I'm just curious if you've thought about other-- 1112 00:53:40,050 --> 00:53:41,960 PROFESSOR: So the question was, what happens if you're in 1113 00:53:41,960 --> 00:53:45,530 a field that doesn't lend itself to perceptual 1114 00:53:45,530 --> 00:53:46,810 reasoning so well? 1115 00:53:46,810 --> 00:53:48,870 AUDIENCE: I guess I wouldn't [INAUDIBLE] 1116 00:53:48,870 --> 00:53:51,450 PROFESSOR: Yeah there's visual in everything. 1117 00:53:51,450 --> 00:53:54,080 And I think it's hard to imagine a field that 1118 00:53:54,080 --> 00:53:55,000 doesn't have it. 1119 00:53:55,000 --> 00:53:55,775 Can you give an example? 1120 00:53:55,775 --> 00:53:58,540 AUDIENCE: Well, some computer science. 1121 00:53:58,540 --> 00:54:01,468 And if you spend a lot of time with computers, then you start 1122 00:54:01,468 --> 00:54:01,956 to develop it. 1123 00:54:01,956 --> 00:54:04,884 But you could think, well I know that it's not possible 1124 00:54:04,884 --> 00:54:06,348 that it could just randomly decide to give a wrong answer. 1125 00:54:06,348 --> 00:54:08,200 There has to be a reason. 1126 00:54:08,200 --> 00:54:11,049 How the program develops, you can develop intuition about 1127 00:54:11,049 --> 00:54:11,580 where the bugs might be. 1128 00:54:11,580 --> 00:54:15,406 But when you're just starting out, that's not something that 1129 00:54:15,406 --> 00:54:19,670 we necessarily have built in to us in an intuitive way. 1130 00:54:19,670 --> 00:54:20,270 PROFESSOR: That's true. 1131 00:54:20,270 --> 00:54:21,520 AUDIENCE: [INAUDIBLE] 1132 00:54:23,630 --> 00:54:27,670 PROFESSOR: Yeah so computers and programming and computer 1133 00:54:27,670 --> 00:54:29,370 science was offered as an example. 1134 00:54:29,370 --> 00:54:30,330 Yeah I agree with you. 1135 00:54:30,330 --> 00:54:31,680 Some fields are harder. 1136 00:54:31,680 --> 00:54:35,230 And what makes computer science harder is that 1137 00:54:35,230 --> 00:54:37,560 computers are symbol processing devices. 1138 00:54:37,560 --> 00:54:40,650 So it pushes them over here. 1139 00:54:40,650 --> 00:54:44,240 But even there I think you can do stuff. 1140 00:54:44,240 --> 00:54:48,550 Like for example, what you could do? 1141 00:54:48,550 --> 00:54:52,800 Well there are some things like for example, loop 1142 00:54:52,800 --> 00:54:55,430 invariants, and you can actually turn those into 1143 00:54:55,430 --> 00:54:56,940 perceptions. 1144 00:54:56,940 --> 00:55:01,810 So once you have get a way of looking at programs. 1145 00:55:01,810 --> 00:55:07,410 You say, oh that can't be right because it has a 1146 00:55:07,410 --> 00:55:11,090 fencepost error in it or it keeps keeps decrementing the 1147 00:55:11,090 --> 00:55:12,810 counter, but there's nothing to stop it 1148 00:55:12,810 --> 00:55:14,540 from having a floor. 1149 00:55:14,540 --> 00:55:16,830 So you start to see problems like that. 1150 00:55:16,830 --> 00:55:20,350 And you have tools like different kinds of magnifying 1151 00:55:20,350 --> 00:55:21,570 glasses to look at programs. 1152 00:55:21,570 --> 00:55:23,720 So you could actually shift it that way. 1153 00:55:23,720 --> 00:55:28,030 And really good programmers actually do look at programs 1154 00:55:28,030 --> 00:55:29,077 perceptually. 1155 00:55:29,077 --> 00:55:31,025 AUDIENCE: See, that's encouragement to build up 1156 00:55:31,025 --> 00:55:32,000 [INAUDIBLE] 1157 00:55:32,000 --> 00:55:33,920 PROFESSOR: Yeah, so you want to encourage that. 1158 00:55:33,920 --> 00:55:35,680 And it's true-- 1159 00:55:35,680 --> 00:55:35,950 pardon? 1160 00:55:35,950 --> 00:55:37,780 AUDIENCE: It's not built in? 1161 00:55:37,780 --> 00:55:41,040 PROFESSOR: I would say that the field itself pushes things 1162 00:55:41,040 --> 00:55:41,970 towards symbolic. 1163 00:55:41,970 --> 00:55:44,570 Just because it is a symbol manipulation device. 1164 00:55:44,570 --> 00:55:47,650 So that's what makes computers so much better at what they do 1165 00:55:47,650 --> 00:55:48,430 compared to humans. 1166 00:55:48,430 --> 00:55:49,890 That's why we use computers. 1167 00:55:49,890 --> 00:55:52,860 Because we're so bad at symbol manipulation, and computers 1168 00:55:52,860 --> 00:55:54,360 are so good at it. 1169 00:55:54,360 --> 00:55:56,430 So our ways of solving problems are going to be very 1170 00:55:56,430 --> 00:55:57,950 different than computers. 1171 00:55:57,950 --> 00:56:01,890 But the problem that we have when we're using computers, we 1172 00:56:01,890 --> 00:56:03,990 have to solve with human hardware. 1173 00:56:03,990 --> 00:56:06,510 So when the computer isn't doing what we want, we have to 1174 00:56:06,510 --> 00:56:10,160 try to find some way of understanding what's going on. 1175 00:56:10,160 --> 00:56:13,170 And it can't be just purely symbolic, because 1176 00:56:13,170 --> 00:56:14,420 we're bad at that. 1177 00:56:14,420 --> 00:56:16,070 So you can't-- 1178 00:56:16,070 --> 00:56:18,390 I mean in desperation sometimes you'll trace through 1179 00:56:18,390 --> 00:56:21,150 every step of a program and see what went wrong. 1180 00:56:21,150 --> 00:56:22,600 But generally that just overflows you 1181 00:56:22,600 --> 00:56:23,330 with tons of data. 1182 00:56:23,330 --> 00:56:26,030 It's much better to actually try to take a step back, break 1183 00:56:26,030 --> 00:56:30,130 the program into parts, see big chunks, and make sure the 1184 00:56:30,130 --> 00:56:31,780 big chunks work. 1185 00:56:31,780 --> 00:56:36,060 And experts do that, not sort of automatically because 1186 00:56:36,060 --> 00:56:37,570 they've been trained as experts. 1187 00:56:37,570 --> 00:56:42,670 So in any field the experts see differently than the 1188 00:56:42,670 --> 00:56:45,760 novices, because their perception is different. 1189 00:56:45,760 --> 00:56:48,750 So yeah, when you're teaching computer science to students, 1190 00:56:48,750 --> 00:56:51,230 you want to teach them ways that are perceptual. 1191 00:56:51,230 --> 00:56:53,840 You want to teach them visceral feelings. 1192 00:56:53,840 --> 00:56:56,420 So one way to do that for example is running time of 1193 00:56:56,420 --> 00:56:59,200 algorithms. 1194 00:56:59,200 --> 00:57:01,710 Different sorting algorithms take different times. 1195 00:57:01,710 --> 00:57:04,855 So suppose you have-- for all the non CS people, a huge area 1196 00:57:04,855 --> 00:57:06,320 of study in CS is sorting. 1197 00:57:06,320 --> 00:57:08,810 Suppose you have a giant list of numbers or list of words 1198 00:57:08,810 --> 00:57:11,370 you want to alphabetize. 1199 00:57:11,370 --> 00:57:12,630 Text indexing uses it. 1200 00:57:12,630 --> 00:57:14,240 All kinds of stuff using sorting. 1201 00:57:14,240 --> 00:57:15,670 So how fast can you sort? 1202 00:57:15,670 --> 00:57:19,070 Well the naive sort of simplest thing to program is 1203 00:57:19,070 --> 00:57:20,320 called bubble sort. 1204 00:57:24,400 --> 00:57:29,090 And if you have n numbers, it takes something times n 1205 00:57:29,090 --> 00:57:31,840 squared, so flip operations. 1206 00:57:31,840 --> 00:57:34,310 So time is of order n squared. 1207 00:57:34,310 --> 00:57:44,620 But then there's merge sort, which is a divide and conquer 1208 00:57:44,620 --> 00:57:46,620 algorithm, which takes n log n. 1209 00:57:46,620 --> 00:57:48,760 Now you could just say that. 1210 00:57:48,760 --> 00:57:51,010 But it's also very helpful for students to have a feel, 1211 00:57:51,010 --> 00:57:52,840 because generally students don't have an understanding of 1212 00:57:52,840 --> 00:57:54,400 functions is what I found. 1213 00:57:54,400 --> 00:57:55,960 They don't have a good feel for what 1214 00:57:55,960 --> 00:57:58,030 functions do and mean. 1215 00:57:58,030 --> 00:58:00,040 So what's the difference between n squared and n log n? 1216 00:58:00,040 --> 00:58:02,510 Well one way is to actually program a sort algorithm and 1217 00:58:02,510 --> 00:58:03,760 then try it. 1218 00:58:06,350 --> 00:58:10,010 Try both algorithms and make your list bigger and bigger. 1219 00:58:10,010 --> 00:58:14,430 When do you get fed up waiting for the damn thing to finish? 1220 00:58:14,430 --> 00:58:15,940 So that's a visceral feeling. 1221 00:58:15,940 --> 00:58:18,730 And actually that will tell you that there's a huge 1222 00:58:18,730 --> 00:58:21,200 difference between n log n and n squared. 1223 00:58:21,200 --> 00:58:24,650 Because this guy, basically when n is around thousand-- 1224 00:58:24,650 --> 00:58:27,430 well, actually now computers are much faster, let's let's 1225 00:58:27,430 --> 00:58:28,800 say 10,000-- 1226 00:58:28,800 --> 00:58:30,300 you're pretty much going to be fed up with that. 1227 00:58:30,300 --> 00:58:32,220 Because that's 100 million sort operations. 1228 00:58:32,220 --> 00:58:37,200 It's going to take five or 10 seconds, maybe even bit longer 1229 00:58:37,200 --> 00:58:39,410 if it's written in a higher level language. 1230 00:58:39,410 --> 00:58:46,980 So maybe you're fed up around n equals 10,000. 1231 00:58:46,980 --> 00:58:49,660 Here 10,000 times log n. 1232 00:58:49,660 --> 00:58:52,850 Well, the log of anything is never bigger than 10, as a 1233 00:58:52,850 --> 00:58:57,520 rough rule of thumb The reason being that if the log of the 1234 00:58:57,520 --> 00:59:01,130 thing were bigger than 10, then n itself will be so big 1235 00:59:01,130 --> 00:59:03,020 your computer couldn't even fit it. 1236 00:59:03,020 --> 00:59:04,970 So you're not even dealing with numbers whose logs are 1237 00:59:04,970 --> 00:59:05,370 bigger than 10. 1238 00:59:05,370 --> 00:59:08,400 It depends on the base, but roughly speaking in base 10 1239 00:59:08,400 --> 00:59:10,260 logarithm, it's never going to be bigger than 10. 1240 00:59:10,260 --> 00:59:12,420 So this is say at most 10. 1241 00:59:12,420 --> 00:59:15,640 And so if you have a 10,000 only times a 10 here. 1242 00:59:15,640 --> 00:59:17,790 Whereas here you have 10,000 times 10,000. 1243 00:59:17,790 --> 00:59:23,340 Here you can go up easily to a million before you get fed up. 1244 00:59:25,880 --> 00:59:29,150 So the fed up is a connection to a visceral reasoning. 1245 00:59:29,150 --> 00:59:31,860 And actually it then makes the difference between these 1246 00:59:31,860 --> 00:59:32,720 really, really apparent. 1247 00:59:32,720 --> 00:59:35,240 So there's ways you can do it even there. 1248 00:59:35,240 --> 00:59:36,572 Question? 1249 00:59:36,572 --> 00:59:43,610 AUDIENCE: I think that in the science fields that, if you 1250 00:59:43,610 --> 00:59:46,875 really understood what you're teaching, then you should be 1251 00:59:46,875 --> 00:59:49,350 able to basically put it into a simplification. 1252 00:59:49,350 --> 00:59:51,740 Because then it's basically summarizing everything, and 1253 00:59:51,740 --> 00:59:55,308 it's just so much better and closer. 1254 00:59:55,308 --> 00:59:58,194 I mean, if you write for example a term paper or a 1255 00:59:58,194 --> 01:00:03,210 proposal or whatever, you can write a one page text 1256 01:00:03,210 --> 01:00:05,489 basically, or you can make a figure or a nice sketch. 1257 01:00:05,489 --> 01:00:08,660 It's the same information. 1258 01:00:08,660 --> 01:00:11,260 PROFESSOR: Right so the comment was that if you really 1259 01:00:11,260 --> 01:00:14,550 understand something, you can make a really compact 1260 01:00:14,550 --> 01:00:15,790 pictorial representation. 1261 01:00:15,790 --> 01:00:16,950 That's generally true. 1262 01:00:16,950 --> 01:00:21,010 And that's why it takes really long time to make those. 1263 01:00:21,010 --> 01:00:24,750 So when you're making talks, for example, when we talk 1264 01:00:24,750 --> 01:00:28,000 about this later in the term, when you're making talks and 1265 01:00:28,000 --> 01:00:31,870 slides, a really good model for making slides is-- 1266 01:00:31,870 --> 01:00:33,120 here's a slide. 1267 01:00:35,210 --> 01:00:37,620 This is the four to three ratio sort of. 1268 01:00:37,620 --> 01:00:39,090 So here is a sentence. 1269 01:00:41,880 --> 01:00:44,060 So this is some assertion, some message 1270 01:00:44,060 --> 01:00:45,310 you want to get across. 1271 01:00:48,690 --> 01:00:49,940 And here is the evidence. 1272 01:00:52,810 --> 01:00:54,560 The body of the slide is the evidence. 1273 01:00:54,560 --> 01:00:57,430 And that's a picture. 1274 01:00:57,430 --> 01:01:00,600 The best is to have some kind of visual evidence. 1275 01:01:00,600 --> 01:01:03,340 But what you find when you do this-- and it is, I think, the 1276 01:01:03,340 --> 01:01:04,600 best way to make slides-- 1277 01:01:04,600 --> 01:01:09,220 it takes a long time to really find the best picture. 1278 01:01:09,220 --> 01:01:11,360 And even to find a good picture because it's just so 1279 01:01:11,360 --> 01:01:13,110 tempting to write a bunch of words out. 1280 01:01:13,110 --> 01:01:15,000 But actually if you look for that picture and you really 1281 01:01:15,000 --> 01:01:18,530 try to capture the main idea in a picture, you will be much 1282 01:01:18,530 --> 01:01:20,500 more successful at conveying the messages 1283 01:01:20,500 --> 01:01:22,280 that you want to convey. 1284 01:01:22,280 --> 01:01:28,200 So in general, try to talk to the perceptual system. 1285 01:01:28,200 --> 01:01:32,730 Now there was a question related to that which is that 1286 01:01:32,730 --> 01:01:37,470 maybe I'm too harsh or too negative about the symbolic 1287 01:01:37,470 --> 01:01:38,950 ways of doing things. 1288 01:01:38,950 --> 01:01:43,270 And the particular example was so if you remember last time, 1289 01:01:43,270 --> 01:01:44,855 I gave this example. 1290 01:01:57,740 --> 01:02:00,050 So this is a pendulum, and this is the extreme of the 1291 01:02:00,050 --> 01:02:01,270 pendulum motion. 1292 01:02:01,270 --> 01:02:04,955 And the question was what is the acceleration vector at 1293 01:02:04,955 --> 01:02:08,950 these various points in motion. 1294 01:02:08,950 --> 01:02:13,090 And I said, well the way to do it is to reason intuitively 1295 01:02:13,090 --> 01:02:15,100 about it, to figure out what it is here and here, 1296 01:02:15,100 --> 01:02:17,590 understand what acceleration is, and do this one by 1297 01:02:17,590 --> 01:02:19,780 interpolation. 1298 01:02:19,780 --> 01:02:23,610 And the comment was well, even if you can't do that, you can 1299 01:02:23,610 --> 01:02:26,950 still use the Lagrangian. 1300 01:02:26,950 --> 01:02:29,130 If I know the Lagrangian for the system, which for the 1301 01:02:29,130 --> 01:02:32,420 non-physicist is sort of the elephant gun. 1302 01:02:32,420 --> 01:02:35,990 You can solve pretty much anything with the Lagrangian. 1303 01:02:35,990 --> 01:02:38,280 So you can solve it with the Lagrangian and work out the 1304 01:02:38,280 --> 01:02:39,440 forces everywhere. 1305 01:02:39,440 --> 01:02:40,810 And that's true. 1306 01:02:40,810 --> 01:02:44,510 But I still maintain that it's better to have-- 1307 01:02:44,510 --> 01:02:46,730 I'm not saying it's bad to have a Lagrangian analysis. 1308 01:02:46,730 --> 01:02:48,590 The more ways of doing things is better. 1309 01:02:48,590 --> 01:02:51,350 But there's a fundamentally important reason why it's 1310 01:02:51,350 --> 01:02:54,460 important to have the intuitive way of doing it. 1311 01:02:54,460 --> 01:02:57,830 And that is a search argument. 1312 01:02:57,830 --> 01:03:00,130 So if you remember from before, we 1313 01:03:00,130 --> 01:03:03,530 talked about chunking. 1314 01:03:03,530 --> 01:03:05,770 The experts have big chunks, and the 1315 01:03:05,770 --> 01:03:07,740 novices have small chunks. 1316 01:03:07,740 --> 01:03:12,220 Well we looked at the effect of that of perception, which 1317 01:03:12,220 --> 01:03:16,400 is that one of the proxies for it was the chess masters could 1318 01:03:16,400 --> 01:03:18,710 remember whole positions, and the novices couldn't. 1319 01:03:18,710 --> 01:03:21,600 Because their chunks were too small, and if you remember 1320 01:03:21,600 --> 01:03:24,130 only seven chunks, which is typical, you can't remember a 1321 01:03:24,130 --> 01:03:25,030 whole position. 1322 01:03:25,030 --> 01:03:27,150 Whereas the chess master's chunks are like three, four, 1323 01:03:27,150 --> 01:03:27,830 five pieces. 1324 01:03:27,830 --> 01:03:28,850 And they can easily fit a whole 1325 01:03:28,850 --> 01:03:30,390 position in seven chunks. 1326 01:03:30,390 --> 01:03:33,000 Well there's another, perhaps even more fundamental 1327 01:03:33,000 --> 01:03:37,360 consequence of the different chunk size, and that is the 1328 01:03:37,360 --> 01:03:41,090 exponential explosion in search space. 1329 01:03:41,090 --> 01:03:46,290 So suppose you're here. 1330 01:03:46,290 --> 01:03:50,160 This is your state now, and you're looking for a solution 1331 01:03:50,160 --> 01:03:51,510 to the problem. 1332 01:03:51,510 --> 01:03:53,420 So suppose the solution is here. 1333 01:03:56,830 --> 01:03:58,155 But you don't know that yet. 1334 01:03:58,155 --> 01:03:59,690 You're looking for the solution. 1335 01:03:59,690 --> 01:04:02,490 Now if your chunk size is really big-- 1336 01:04:02,490 --> 01:04:04,540 well let's say suppose your chunk size is really big-- 1337 01:04:04,540 --> 01:04:07,460 you'll try a few possible chunks. 1338 01:04:07,460 --> 01:04:10,570 OK, do any of those get me closer? 1339 01:04:10,570 --> 01:04:11,890 Well this one looks a bit likely. 1340 01:04:11,890 --> 01:04:15,600 Let me try expanding that one, maybe that one. 1341 01:04:15,600 --> 01:04:17,550 Try this one. 1342 01:04:17,550 --> 01:04:19,760 And then eventually you get your way over here. 1343 01:04:19,760 --> 01:04:25,360 So here suppose there's four possible approaches you could 1344 01:04:25,360 --> 01:04:28,710 try, and because the chunks are big, 1345 01:04:28,710 --> 01:04:30,470 there's only three levels. 1346 01:04:30,470 --> 01:04:36,620 So this gives you four to the three items to search. 1347 01:04:36,620 --> 01:04:39,670 OK, now suppose you make your chunks really, really small. 1348 01:04:39,670 --> 01:04:45,060 Suppose your chunks are only half the size. 1349 01:04:45,060 --> 01:04:49,820 So you now at every half branch you have to search for 1350 01:04:49,820 --> 01:04:51,310 possibilities. 1351 01:04:51,310 --> 01:04:52,820 So this is the large chunk. 1352 01:05:07,610 --> 01:05:11,070 So here we had one, two, three levels. 1353 01:05:11,070 --> 01:05:17,100 Here we're going to have six levels, in other words, four 1354 01:05:17,100 --> 01:05:20,410 to the six possibilities to search. 1355 01:05:20,410 --> 01:05:23,490 So the difference doesn't show up in simple problems. 1356 01:05:23,490 --> 01:05:26,670 Because you may have only one level to go through versus two 1357 01:05:26,670 --> 01:05:28,650 levels to go through, and either method works. 1358 01:05:28,650 --> 01:05:30,210 But when you're trying to solve interesting new 1359 01:05:30,210 --> 01:05:33,890 problems, you want your chunks to be as large as possible so 1360 01:05:33,890 --> 01:05:37,860 your search space collapses to something manageable. 1361 01:05:37,860 --> 01:05:43,870 So you don't want to have to reason with the Lagrangian as 1362 01:05:43,870 --> 01:05:45,500 the default mode. 1363 01:05:45,500 --> 01:05:47,100 It's fine for that particular problem. 1364 01:05:47,100 --> 01:05:49,470 But if that problem is embedded in a bigger problem, 1365 01:05:49,470 --> 01:05:52,020 you want to just know what the answer's going to be roughly. 1366 01:05:52,020 --> 01:05:54,430 So you can continue making progress and decide should I 1367 01:05:54,430 --> 01:05:56,430 go this route or this route. 1368 01:05:56,430 --> 01:06:00,890 So that's why I'm not negative on the standard ways. 1369 01:06:00,890 --> 01:06:04,530 But I think the standard ways are actually genuinely bad for 1370 01:06:04,530 --> 01:06:06,490 problem solving in new domains. 1371 01:06:06,490 --> 01:06:10,610 And you want to augment your perception so that you guide 1372 01:06:10,610 --> 01:06:14,870 your search in the right direction with big chunks. 1373 01:06:14,870 --> 01:06:19,410 So that's fundamentally important. 1374 01:06:19,410 --> 01:06:22,180 So I think that was sort of the main themes in the 1375 01:06:22,180 --> 01:06:26,960 questions, and I'll look over them and see which ones to 1376 01:06:26,960 --> 01:06:29,880 answer probably at the beginning of next time. 1377 01:06:29,880 --> 01:06:33,410 I'll try to leave more time for questions today, so we 1378 01:06:33,410 --> 01:06:35,610 don't generate so many questions next time to 1379 01:06:35,610 --> 01:06:36,930 overflow it. 1380 01:06:36,930 --> 01:06:42,200 OK, so before I continue, any questions about these general 1381 01:06:42,200 --> 01:06:44,140 principles of thinking about your teaching? 1382 01:06:48,470 --> 01:06:49,610 Yes? 1383 01:06:49,610 --> 01:06:52,710 AUDIENCE: So the approach is trying to focus more on the 1384 01:06:52,710 --> 01:06:56,240 perceptual way to understand thing, but on the other hand, 1385 01:06:56,240 --> 01:06:58,550 the symbolic manipulations are useful. 1386 01:06:58,550 --> 01:07:03,060 So how do you make sure that the students will themselves 1387 01:07:03,060 --> 01:07:04,080 engage in the symbolic. 1388 01:07:04,080 --> 01:07:09,240 How do you encourage them to do the symbolic reasoning 1389 01:07:09,240 --> 01:07:12,500 themselves without scaring them? 1390 01:07:12,500 --> 01:07:15,290 PROFESSOR: So how do you encourage the students to do 1391 01:07:15,290 --> 01:07:16,010 the symbolic reasoning? 1392 01:07:16,010 --> 01:07:16,790 Because it is important. 1393 01:07:16,790 --> 01:07:17,830 And I agree it is important. 1394 01:07:17,830 --> 01:07:21,330 Just pure perception isn't quite enough. 1395 01:07:21,330 --> 01:07:24,900 I think most courses I would say, and most teaching, is of 1396 01:07:24,900 --> 01:07:27,970 99% symbolic and 1% perceptual. 1397 01:07:27,970 --> 01:07:33,210 And it depends on the field, but a rough estimate is I 1398 01:07:33,210 --> 01:07:36,980 think it should be say something like 20% symbolic 1399 01:07:36,980 --> 01:07:40,160 and maybe 80% perceptual initially. 1400 01:07:40,160 --> 01:07:44,020 And then as you go to more and more advanced levels, if 1401 01:07:44,020 --> 01:07:47,270 you've done it mostly perceptual earlier on, and 1402 01:07:47,270 --> 01:07:49,380 you've developed the intuition, then you can 1403 01:07:49,380 --> 01:07:50,770 increase the amount of symbolic. 1404 01:07:50,770 --> 01:07:52,700 So I think the way you do it is you build it into the 1405 01:07:52,700 --> 01:07:55,560 structure of the curriculum. 1406 01:07:55,560 --> 01:07:58,300 So one particular course might just have one mix. 1407 01:07:58,300 --> 01:08:01,510 And the more advanced students who got the perceptual 1408 01:08:01,510 --> 01:08:05,070 material earlier are now ready for the symbolic. 1409 01:08:05,070 --> 01:08:10,620 So there's actually a graph that I show for this. 1410 01:08:21,890 --> 01:08:23,864 So that's the fraction of symbolic. 1411 01:08:31,430 --> 01:08:33,942 And that's the fraction of perceptual material as you 1412 01:08:33,942 --> 01:08:35,430 sort of go on in the major. 1413 01:08:35,430 --> 01:08:38,819 So I think this is the ideal structure for a major. 1414 01:08:38,819 --> 01:08:42,220 Because this is going beyond course design to whole design 1415 01:08:42,220 --> 01:08:43,939 of a major in a curriculum. 1416 01:08:43,939 --> 01:08:46,770 So early on, say for the freshmen, it should be mostly 1417 01:08:46,770 --> 01:08:48,220 perceptual. 1418 01:08:48,220 --> 01:08:49,790 And there's many reasons for that. 1419 01:08:49,790 --> 01:08:52,529 First of all their symbolic capacities-- 1420 01:08:52,529 --> 01:08:56,420 we're all bad at symbolic capacities to start with. 1421 01:08:56,420 --> 01:08:58,590 As freshmen, they're worse than the seniors. 1422 01:08:58,590 --> 01:09:00,800 They haven't had time for example get sophisticated 1423 01:09:00,800 --> 01:09:02,740 understanding of differential equations. 1424 01:09:02,740 --> 01:09:04,560 Their algebra's rusty. 1425 01:09:04,560 --> 01:09:06,600 It's not really practiced much. 1426 01:09:06,600 --> 01:09:09,640 So don't rely on those modes of learning. 1427 01:09:09,640 --> 01:09:11,290 Rely on the perceptual modes. 1428 01:09:11,290 --> 01:09:14,069 And the perceptual modes give you the big chunks. 1429 01:09:14,069 --> 01:09:18,180 And so then once you have the big chunks, well then sure 1430 01:09:18,180 --> 01:09:20,220 they're ready for the symbolic more and more. 1431 01:09:20,220 --> 01:09:23,380 Because the symbolic fits into the big chunks and helps guide 1432 01:09:23,380 --> 01:09:24,760 the search. 1433 01:09:24,760 --> 01:09:27,160 It helps actually do the search. 1434 01:09:27,160 --> 01:09:28,910 I have to calculate this. 1435 01:09:28,910 --> 01:09:29,910 Then I calculate this. 1436 01:09:29,910 --> 01:09:32,319 Well each calculation you do you need the symbolic. 1437 01:09:32,319 --> 01:09:34,550 So by the time they're graduate students, hopefully 1438 01:09:34,550 --> 01:09:36,700 the perceptual stuff they've had it up here. 1439 01:09:36,700 --> 01:09:37,729 It's really solid. 1440 01:09:37,729 --> 01:09:40,160 And now they're ready for partial differential equations 1441 01:09:40,160 --> 01:09:41,439 and things like that. 1442 01:09:41,439 --> 01:09:45,550 But earlier on, you don't want to avoid partial 1443 01:09:45,550 --> 01:09:46,310 differential equations. 1444 01:09:46,310 --> 01:09:49,020 You want to solve them in intuitive ways. 1445 01:09:49,020 --> 01:09:54,300 And an example of that is early on, when you're doing 1446 01:09:54,300 --> 01:09:56,510 fluid mechanics, you would solve the cones using 1447 01:09:56,510 --> 01:09:59,130 dimensional analysis, as we did before. 1448 01:09:59,130 --> 01:10:01,390 You wouldn't solve the Navier-Stokes equations. 1449 01:10:01,390 --> 01:10:05,860 And later on, for example, in a graduate class on 1450 01:10:05,860 --> 01:10:08,920 computational fluid mechanics, you'd actually try to simulate 1451 01:10:08,920 --> 01:10:12,480 that and figure out what the drag coefficient is 1452 01:10:12,480 --> 01:10:13,830 numerically. 1453 01:10:13,830 --> 01:10:16,420 So that's the mix between perception and symbolic that I 1454 01:10:16,420 --> 01:10:22,070 think is a very good way to construct a whole curriculum. 1455 01:10:22,070 --> 01:10:25,330 Does that answer your question? 1456 01:10:25,330 --> 01:10:26,580 OK.